Pros and Cons of the CPM Math Textbook Series

Teachers tend to love the CPM math series, currently in use in many SMFCSD and SMUHSD classes, while parents have the opposite opinion. Why??

(Note updated 6/29/2022 – This article, more than any other, gets non-stop daily traffic on my blog from around the country as well as internationally.   As of today, it has been viewed 24,260 times.  Parents, students (and even teachers as you can see in the Comments section following the article below) are looking for information about CPM beyond that provided by the publisher, often when students start struggling with the math program.

I’d appreciate it if readers would leave a comment below about what prompted them to search for and read this article.   You might also want to name your school district in case others in your locale are in the same situation.  There is strength in numbers.

Thank you!  I am also open to suggestions for additions or clarifications to the material contained herein.

Finally I would like to call your attention to probably the most important article that I have written for this blog: Raising our Children – American Society Reflects our Values and Choices.  Other top-ranked articles are listed on the main home page.)


8/28/2017 – This article was prompted by the comments of a parent to my last blog article.  Those comments were made on Nextdoor, not on this site.  (The Nextdoor link will only work for local residents who have Nextdoor accounts.)

Referring to the new math curriculum “pathways” or course sequences from 6th through 8th grades, the parent said:

I would add that these new pathways and CPM curriculum (2014) were unveiled with promises that they would provide a deeper and more comprehensive math program. My observation is that this curriculum is more confusing and less comprehensive. I have two daughters who enthusiastically take Math at RSM and who can explain how little material is covered in CPM textbooks and what a superficial foundation they provide in math.

RSM is “The Russian School of Math,” a private organization with an office in San Mateo and many other locations across the U.S.

The CPM (College Preparatory Math, www.cpm.org) mathematics textbook series is used in many classes at Aragon High School (as well as some other SMUHSD schools such as Hillsdale HS).  According to Superintendent Dr. Joan Rosas, it was also adopted by SMFCSD middle schools to align their curriculum with the high school curriculum.

I first used this math series when I taught 9th and 10th graders at George Washington High School in San Francisco, and have tutored many local students who are using these textbooks in class. I should also note that the unfortunate CPM precalculus textbook trial two years ago at Aragon was the motivation for my starting to blog, first on Nextdoor.com, and then at this site.

Although many teachers will say that CPM is the best mathematics series that they have used, I have very mixed opinions about it and am NOT an unabashed fan of the program as I will detail below.

A Google search on “opinions on CPM math” may interest you and will display the experiences of many other communities.


The CPM Math Program

CPM spans middle school to high school math, previously stopping at precalculus, but lately including Calculus AB and BC textbooks.  The program strongly encourages group work over individual study.

Students are typically placed in groups of four in their classroom and are given defined roles within their group: “Resource Manager,” “Facilitator,” “Recorder/Reporter,” and “Task Manager” (see the beginning of any CPM textbook for details if you are interested in what these roles entail).

CPM literature frequently mentions that in the “real world” people work in teams, and therefore CPM aims to teach and facilitate collaborative learning.  Many teachers have told me that students are more engaged with the CPM math curriculum than with any other series that they have tried.  Students have active discussions about the material and work on group problems in class versus passively listening to lectures, taking notes, and only working actively when they do homework alone after school.

As long as the program works in this manner this is definitely a strong positive in its favor.  Mathematics is definitely learned by working problems actively rather than watching a teacher do them on the board.  It is also a major plus to work problems in class, when others are around to offer a helping hand, instead of finding later, when starting the homework at home alone, that one didn’t understand the material.

A typical CPM lesson works as follows.  Each textbook section begins with a series of guided questions that lead students to discover a new math concept if they answer the questions correctly and in order.  Often these guided questions are quite clever and well-designed.  The books do not simply explain a math idea and and then provide worked examples to imitate, as do traditional math texts.  Students work with their groups to solve the set of problems and learn the lesson that the section intends to teach.

Teachers are supposed to move around the class from group to group, answering questions from each group and making sure that students are on task.  Lecturing is kept to a minimum.  This is in agreement with the “learning by doing” philosophy.   Current teaching practice tends to denigrate lecturing, calling a lecturer a “sage on the stage,” with the implication that lecturing stokes the ego of the teacher instead of really instructing the student.

As I have found with many ideas in education, such theories work great when one has motivated students who actually do the work.  If the student groups are well-structured, the better students help those in their group who struggle with math, and everyone benefits.  The good students benefit because, paradoxically, there is no better way to learn a subject than to teach it.  The less mathematically-inclined students get help from their peers, which is often less intimidating than asking a question from the teacher.

However, if a teacher has a class of mainly lower level students who have not done well in math previously, the CPM method can become problematic.  Putting pre-teens or teens who “hate math” into groups results in a major “classroom management” challenge for a teacher.  The group conversation is often on any popular teenage topic other than mathematics when the teacher is not watching!

Again, if a group is able to finish the guided questions, then they learn the lesson for the day.  In a poorer class, however, the teacher often has to answer the same questions repeatedly for each group and may eventually decide to stop the class and lecture for a while on the topic.

During the one-year CPM precalculus trial at Aragon, two school years back, one of the teachers tended to lecture at the beginning of each class.  That teacher’s students were appreciative of this lecture effort while those in classes where the teachers followed the standard self-discovery prescription were often frustrated.  In fact I had one student, who was totally upset with a teacher’s “hands-off” approach, comment to me, “Don’t they get paid to teach??!!??”

CPM basically is a set of pre-made math lessons which alleviates a lot of lesson planning for teachers.  A motivated teacher can use these lessons as PART of a good curriculum as I will explain further below.  Unfortunately this also means that a burned out teacher can use the CPM program as an excuse to coast.  The lessons are spelled out in the book, the students are supposed to do the work themselves, so “get in your groups, open your book to section X, and do problems Y to Z” is the very minimal teaching effort required.

Typically, the group problems in the first part of each section take up most, if not all, of a class period.  The second part of each textbook section is a set of problems (with no additional explanatory material) entitled “Review and Preview.”  These “review” problems are typically assigned for homework.  Hints for the homework problems are online at the CPM website, and Aragon teachers frequently post answers online.  The “review” problems include some additional practice on the ideas just learned in the guided questions section, but also include review problems from earlier textbook sections.  This practice of frequently returning to older topics in each new section is called “spiraling.”

[Aside: The spiraling concept can also come into play on CPM chapter tests, i.e., the chapter 4 test will include problems not only from chapter 4, but also from chapters 1, 2, and/or 3.  This can turn a chapter test into what is basically a mid-term or final exam.  On the plus side, the constant cycling back can really reinforce the material.  On the minus side, students can feel really stressed as the tests can cover much more material than traditional chapter tests.]

The “Review and Preview” homework section may also include “thought” problems (called “preview” problems) on topics that students have not even encountered yet.  The purpose of such questions is probably to see if a student can discover the solution to a completely new, challenging problem on his/her own.  Unfortunately, “preview” questions tend to confuse all but the very best students.


Critique of CPM

Having now described how the “Review and Preview” section works, I must next note its  most serious drawback.  I have seen many instances where the “Review and Preview” section offers only minimal additional homework practice on the lesson just learned and then “spirals” back to problems from earlier sections picked in a rather random fashion.

Too often I have tutored students who are just beginning to master a new concept when the homework diverts them back to earlier topics without cementing the knowledge just learned.  I then have to use other sources or make up my own problems to help the student.

Traditional texts give a far greater number of practice problems than CPM.  They usually have solutions readily available to odd-numbered problems and have worked examples, both of which allow a motivated student to do extra work if they still don’t understand a concept.

This is much harder to do using the CPM series.  In my experience a teacher who decides to use CPM needs to give students supplemental practice problems.  If one has to find this extra material, then one needs to be convinced that the CPM guided questions are so good that it is worth this extra trouble, instead of simply using a different textbook.

One must also strongly believe in the value of the self-discovery process.  “Self-discovery” as a teaching method is not universally accepted, and I address the issue of self-discovery versus fully guided instruction further below.

In summary, the biggest problems with CPM are the lack of explanations, worked example problems in the textbooks, and insufficient practice problems.  The first two omissions are by design because each group is supposed to discover the concepts through the guided questions.  Worked examples would circumvent this process.

However, if a group does not “get” the topic and fails to complete the guided problems in class, they are left with nothing to explain how they should do the homework!  Essentially the student has a textbook with only questions and little or no explanations.  This is a significant problem in classes with weaker math students and with students who are absent from class.  They have nothing to refer to at home unless the teacher puts additional material on the Web.  However, this means the students have to navigate to other sources instead of just being able to use their textbook.

The CPM books do have small boxed highlight sections called “Math Notes,” usually in sections of the book beyond the section being studied that day, that try to summarize the important points.  These sections are very concise and also do not contain worked examples as do traditional math texts.

Another learning problem can arise because many schools use the cheaper paperback version of the CPM books which are split into two volumes.  The index is in the second of two books, and the student may not have book 2 during the first part of the year.  Only the hardback version is a single volume.  Lack of an index makes it difficult to look up particular concepts when one is “stuck.”


Critique of the Self-discovery Methodology

Finally, and the most important learning issue in my opinion, the self-discovery method tends to work better on easier concepts such as Algebra 1.  As one moves up the math hierarchy and ideas become more complex, self-discovery becomes increasingly time-consuming and inefficient.

I think this was a major reason why the CPM precalculus experiment at Aragon failed, and why at least one of the teachers had to revert to lecturing.

In fact I presented the teachers at Aragon with an article from an American Federation of Teachers journal critiquing “self-discovery” methods when I met with them early in the 2014-2015.  I warned them early in the school year, first by emails and then in a face-to-face meeting with math department staff, the principal, and a vice-principal, that the CPM experiment was in grave danger of running off the rails.  I believe the principal had evidence of this too, which is one reason why the meeting took place.  The precalculus students that I tutored that year were clearly struggling significantly more than in years past when a traditional textbook was used.

The article I gave Aragon staff was in the Spring 2012 issue of American Educator, vol. 36,  no. 1, and it was a through review of numerous educational research studies including academic references.  The article concluded:

Research has provided overwhelming evidence that, for everyone but experts, partial guidance during instruction is significantly less effective than full guidance.

To date, I have no indication that anyone at the school ever took the time to read this article unfortunately.  If the article’s conclusion is true (I think it is, and the research in this article has the great advantage of correlating closely with common sense unlike many teaching fads), this is a damning condemnation of the CPM methodology.

The fact that CPM now has textbooks for Calculus AB and BC makes me shudder.  This would raise self-discovery to the highest level of complexity in high school math.  I would be extremely concerned if any of the SMUHSD schools adopted those books.


Research Supporting CPM

Having discussed the CPM methodology and its pros and cons, one might still wonder what kind of research does CPM tout to promote their program?  If one takes the time to navigate through the cpm.org website, one can find a section detailing research studies behind the CPM program.

Much of the research is older, probably in part because the standardized STAR test base was discontinued with the adoption of Common Core.  However it is interesting to look at one of the later studies from 2013 in 8th grade and high school.

The methodology of this study is very flawed, however, because it appears to only take the results of school districts that CPM knew used their books and compare them to statewide averages.  There is no controlling for differences in, e.g., demographics between districts that adopted CPM and the state as a whole.  Nevertheless, there is no indication that schools using the CPM series did any better (or worse) than the STAR test state averages in Algebra 1 and Geometry and only slightly better in Algebra 2.

This might not seem too bad until one realizes that our local schools have always prided themselves on scoring significantly ABOVE the state average!  Why would they want to adopt a series that only delivers average state test scores, particularly when we know how pathetic state math scores have been??

The illustration below shows California STAR math scores (% scoring proficient or above) from Grade 2 up through Algebra 2 for years up through 2012 before the state terminated this test in favor of  newer Common Core testing.  As one can see, there was a downward slide in math scores from 4th grade through high school in 2012.

Passing rates of 35%, 32%, and 34% are not benchmarks that I would want to use for marketing any product that I developed!!!  Why would one adopt a program if this is the research used to promote it??

In fact, I learned from an administrator that CPM was tried in the SMUHSD years ago, long before Common Core was a glimmer in anyone’s eye, and was abandoned by all high schools except Hillsdale!  The group work encouraged by Common Core appears to be resurrecting it from the dead.

IMG_0226

Postscript

One final aside which I did not mention earlier as it is a lesser, but persistent, irritant for many parents: students in CPM classes also engage in a practice called “group tests.”  All four students in a group work collaboratively on a test.  At the end of the period, the teacher randomly picks one of the group’s four test papers, grades it later, and then assigns that grade to everyone in the group.

The first time this happens during the school year, some of the groups will have a paper selected from the weakest student in the group, and everyone in that group might end up with a bad grade.  During subsequent group tests, the better students in the group will frantically check that everyone’s test papers have the same answers, so that they do not “get screwed over” a second time.  Parents tend to shake their heads incredulously when they learn about this practice, and I can’t blame them.

However, I don’t see that this has to be an essential part of a CPM class and could be eliminated if a teacher so desired.  Of course, this would require grading four times as many tests…  However, I always thought that the purpose of a test was to assess what a student knew, not what his/her group can gin up and copy in a hurry…

NOTE: If you found this article to be of interest, you may also want to read Why Can’t We Teach Mathematics Properly?


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Thanks as always for your participation!

Author: David Kristofferson

Retired Ph.D. scientist, teacher (after retiring from industry, taught in private and public high schools and then worked a decade in my own private tutoring business), bioinformatician (managed both the NIH-funded GenBank National Nucleic Acid Sequence Databank and the BIONET National Computer Resource for Molecular Biology), IT director at Eos and Raven Biotechnologies, software product manager, AAAS Fellow, avid cyclist, and backpacker!

175 thoughts on “Pros and Cons of the CPM Math Textbook Series”

  1. I have used two “discovery” programs. Neither was effective in comnprehensive, deep study , college prep settings because the students do not get enough information, skills,etc to be ready for calculus, nor were they effective with non-college bound who could care less about math and were more interested in mechanics, plumbing, electrical, cowboying, football, or anything else except math. Reading challenged students also struggle if they try at all. Not just a bad choice, but a serious mistake. What blows me away after teaching math for 45 years is that many think that students cannot “discover” discuss, work together, and a pile of other things if they are in a traditional math class. I have seen all of those things in traditional classes and have had great success in sending student into the scientific world.One of the last I taught is now an MIT graduate. His class did all of the things discussed above, BUT A TEACHER HAS TO BE CAREFUL ABOUT WHEN AND HOW MUCH FOR ANY PARTICULAR CLASS.

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    1. Thanks very much for your comments. I have been getting a lot of traffic on this article recently and have been curious how people are finding it. The “referrer” stats on WordPress don’t seem to work very well. Would you mind telling me how you discovered it?

      Liked by 1 person

      1. Hi David,

        Thank you for your article. I found it by doing a google search of CPM + tests + math. Your article was the second listing.

        Sara

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      2. I Googled “Math CPM” to learn more about what a district was using. As a former math teacher (including of IMP), I understand your concerns (and the positives)

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      3. I personally found your article by googling “CPM math reviews” 🙂 Thank you for your article; it is very helpful to me as a parent as both my sons have struggled with this math.

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      4. Hi David,

        I googled ‘College Preparatory Math curriculum for Middle School’ and then followed a series of links to get here, one in particular titled ‘Is CPM Math Good?’. I’m comparing this curriculum option to the OUR for MS Math curriculum adoption to lobby for one or the other. I found your insights very helpful.

        Thank you!

        Jamie Day

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    2. Google searches; “CPM Math Does Not Teach Students” “Frustrated Parents with CPM Math” “High School Kids Hate CPM Math” “Parents must tutor kids with CPM Math”……any of the above.

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      1. I found your article after googling cpm math. My son is high functioning and on the spectrum which means he can be a bit ADD as well.

        He struggles with the team environment and learning in groups (off task a lot).

        I find myself “tutoring him” after school. I lie the comments and the blog.

        After reading more about common core, Japanese systems, etc., I am shocked that teachers get so little training here and do not get the facilitators needed to help with the diverse needs of students.

        Thank you for your work!
        Rhonda

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    3. Hello. I am a high school student who is currently involved in a CPM program in my high school. I consider myself to be a very bright student with a high GPA and enrollment in honors/ AP classes. However, CPM math has been miserable for me. The “spiraling method” completely ruins any form of structure when it comes to learning math, making it way more confusing than it should be. This being paired with an incompetent teacher who refuses to give help and grades on subjects we’ve never formally gone over in class could possibly put me at risk for not being admitted into the universities that I plan to attend. What a terrible program! If your academic level geometry class is harder than your AP classes, then there’s clearly something wrong!

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      1. Starting in 2018, experimentation was begun with the program, this year, they made it mandatory in all math classes (with the exception of AP calculus and precalculus). Last school, a good amount of students transferred out of their CPM run classes because their grades began to plummet, and when they returned to traditional math, they got A’s and B’s. This year, however, there is no where to turn.

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      2. PS – I might be having a technical problem in the blog comment section. Would you please let me know how you saw my replies? Are you using the WordPress app on a phone/tablet to read the blog? Are you using a web browser and seeing the replies there? Or are you getting the replies via email??

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      3. I hope more students speak up and I hope all your dreams come true. Please tell your peers to post and send this and other resource materials to your administrators.

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    4. I have an interview for math teacher at the local high school and found out they use cpm exclusively, so I Googled “cpm math curriculum reviews” and your “Pros and Cons” article was at the top of the search results.

      Like

  2. David, Thank you for this post!

    I’m a 27 year veteran math teacher. Unfortunately the district where I teach is “piloting” CPM in Algebra I and already considering it for geometry and algebra 2. There hasn’t even been enough time to collect, let alone analyze data from the supposed “pilot” but here we go forging ahead with dataless decision making.

    I think you hit the nail on the head in observing that “The group work encouraged by Common Core appears to be resurrecting [CPM} from the dead.” I’m SO glad that I began my career years ago, but I am still very concerned about opportunities for students in the future.

    The tutoring boom will surely continue as long as CPM is keeps creeping in…

    Thanks again
    and best wishes!

    Lisa Jones

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  3. This is a wonderful write up of all the issues I have with CPM.

    I’m a 17 year old junior going through pre-cal right now. I’ve never been extraordinary at math, but I rarely got anything below a B. I always understood all the concepts and by the time finals arrived, I was confident I would pass.

    However my pre-cal class uses the CPM textbook, and I simply cannot deal with the level of confusion I go through every day in that class. I feel like I’m getting no guidance whatsoever, and my only option is to get a math tutor, which is something I’ve never had to do. One of my best friends who’s now attending NCSSM had to retake precal because he simply couldn’t grasp the concepts she was teaching.

    My teacher is also quick to assume I’m simply slower when compared to the rest of the class. She’s constantly singling me out, asking if I’m doing alright and if I need help. Besides being embarrassing, it just feels insulting. I just want one day where I can sit alone and take notes to study later.

    And I can’t turn to my classmates for help either. They’re either way ahead of me and just give me answers so they can move on to the homework for that night, or stutter over themselves trying to explain a concept. Which isn’t their fault, they’re certainly now the ones expected to know how to teach.

    To top it off tests make up 65 or 75 percent of our grade, and they’re some of the hardest tests I’ve ever taken, compounded by CPM. Luckily we’ve done one of the group tests, but I’m sure there will be more.

    I’m not really sure how to express the problems I have with CPM to the school admin. Of course they’re not going to rollback a program they spent money on to train teachers and buy textbooks. So I feel like I’m just doomed at this point.

    Thanks again for writing this.

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    1. Danny,

      I really sympathize with your situation as I have seen students in my location in similar situations.

      Even though I work as a tutor and am essentially arguing against my own business, not only CPM, but much of the AP curriculum almost forces students to get tutors to survive. This gives yet another advantage to students from wealthier families which runs counter to the public school idea of equal opportunity.

      When I went to high school back in the late 60s and early 70s, tutoring was almost unheard of. The curriculum was such that a good teacher could explain it well and students could master it without resorting to expensive after school fixes.

      I hope you can find some help, but you might also consider getting your parents and other parents in your area to read my article. If enough of them get upset and complain to your school district, perhaps the system can be changed?!??

      Liked by 1 person

  4. Mr Kristofferson,

    Thanks for taking the time to so lucidly explain your thoughts and concerns with this teaching approach. I suspect that the reason you are getting so much traffic is because concerned parents, including me, are looking to combat the “kool-aid” tainted view presented by school administrators when they embark on this social experiment.

    In our case, the Birmingham (MI) Public School district has decided to roll this out despite the unequivocally dismal responses from students in the pilot(s) over the last year. Parents are justifiably irate because our schools consistently rank at the top within Michigan and this clearly does not advance learning as purported.

    We are fighting. Parents are rallying to reverse this, but at this point the outcome is unclear.

    Thanks for the well thought out and insightful perspective.

    Steve

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    1. Thanks for taking the time to write your comments above, Steve. Glad that I can be of assistance. You might be interested in another article on my blog in a similar vein entitled “Never Trust Educational Experts (or Me)! It details my local struggles with the education system.

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    2. Hi David! I am part of the Birmingham (MI) parents that are fighting the administration. I, along with 4 other moms, started a FB group titled Concerned Birmingham Parents, please check it out. Have you found, in your research, any other states in recent years that students have been negatively affected by the CPM program. We appreciate your blog and quoted some of your research in our presentation to the board. Thank you for your hard work. Ashley

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      1. Thanks for your comments, Ashley. Unfortunately, I do not use FaceBook for a variety of reasons which I will not elaborate on here.

        In the 8th paragraph of my article there is a hyperlink in the following text:

        “A Google search on “opinions on CPM math” may interest you and will display the experiences of many other communities.”

        That link should lead you to other communities’ experiences, and I would not be surprised if variants on that Google search might uncover more.

        Best of luck with your efforts!

        Liked by 1 person

  5. Hey there I recently read your article and it perfectly describes my feelings about CPM am currently taking geometry again this time using CPM curriculum. I failed geometry last year as a result of being hospitalized for most of the past school year. CPM has confused and has left me feeling so disappointed with myself that I have cried in my geometry class because of my grade in that class. Reading your article has provided some reassurance. Thank you for writing this article I would really appreciate some advice.
    Sincerely a student :^D

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    1. Hi Angel,

      Sorry to hear about your frustration. I have seen similar issues with students that I have tutored in the past.

      If you can’t get a teacher, friend, or a tutor to help you catch up, you might consider looking for a cheap used traditional geometry textbook on Amazon or at a used bookstore that would provide you with instructions and worked examples often lacking in CPM. There is also the paperback Schaum’s Outlines series that have worked examples. You’ll have to try to find the sections in a supplementary book that correlate with your current class topic, however.

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  6. Mr. Kristofferson,

    Thanks for you analysis of CPM. I have taught it for the last 10 years and have been teaching math for over 20+ years, so I have experience with traditional text also. Tustin Unified School District in Southern California is considering adoption for middle and high schools within the next week, with the board voting on it by april 9, 2018. This will effect my 2 young kids who are starting middle school.

    If you could reach out to the district or powers in place, it would be greatly appreciated. Or if the other parents could be better educated, that would be a plus.

    Teacher and DAD

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    1. Dear Mr. Liskey,

      Please feel free to distribute my article to your Board members. Unfortunately I can not get involved in other districts’ battles. I have more than enough to do trying to win policy changes in my own locale.

      It is my philosophy to focus my efforts locally and provide access to my experience in case it might be of benefit elsewhere. Given your level of teaching experience, one would hope that your local Board will listen to you respectfully. However, it is a battle that you along with others in your district will have to undertake. If you search on Google, you will find enough other districts where this battle has also been fought that there should be no shortage of supporting arguments for your side.

      Sincerely,

      Dr. David Kristofferson

      Like

  7. Thank you for this article. You asked how people are finding your article- I googled “CPM is a horrible math book” and your article was the second search result. I’m beyond frustrated for my kid, a freshman in honors Alg 2 in a hyper competitive east bay town. My kid once loved math regardless of the subject or grade, but now feels despondent and left behind. Getting help from the group- laughable, you’re better off asking Jack from Lord of the Flies for help.

    Any suggestions for a decent alternate text book, for Alg 2 and pre-calc?

    Our district is considering doing away with higher math in middle school because they argue kids drop out of higher math at too high a rate in high school- yet have they stopped to consider it’s the text books they adopted a few years ago- nope!

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    1. Thanks for your note, Sarah. Just out of curiosity I googled “CPM is a wonderful math book” and my article also came up 3rd from the top 😉! Nice to see that Google is “fair and balanced!”

      Regarding alternate textbooks, after our local high school tried and abandoned CPM precalculus,they switched to Demana et al. “Precalculus – Graphical, Numerical, Algebraic Common Core” published by Pearson. I have also used Stewart, Redlin and Watson’s “Precalculus – Mathematics for Calculus” which is a bit more rigorous.

      I don’t have specific recommendations for Algebra 2; the Algebra 2 students that I tutor continue to use CPM. As I said, CPM can work **if used by a good teacher** in the lower grades, but I think that the method becomes increasingly problematic as one advances in math.

      In general, I think a good textbook should have the following minimum features:

      1) worked examples – this runs counter to the CPM methodology

      2) a reference after each worked example to a similar odd-numbered problem in the Exercises section so that a student can immediately try the skill and check their answer in the back of the book

      3) answers to odd-numbered problems.

      CPM aficionados will immediately reply that the above means that students will simply be parroting the material instead of thinking about it, but, as I said in the article, guided instruction usually seems to work better than self-discovery for most people.

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      1. Hi David,

        Parents and students can use the Parent Guide–an online resource free to anyone–which has worked examples for the topics in each chapter followed by 20 practice problems with answers included to all of them.

        Sincerely,
        Sara
        P.S. Just google CPM + Parent Guide

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      2. Great article found on Google search of cpm math curriculum. Our son has been complaining about the new cpm program in his 7th grade algebra 1 class in the high performance middle School in the Indianapolis Public schools. His main complaints are that the teacher doesn’t teach anymore and the group centered learning is totally not productive as only the top kids in the group contribute and sometimes they don’t know the material well enough to help the group which puts undue burden on them. He is a straight A student who loved math prior to this year, and his first test resulted in a C, and worse an admission that he doesn’t like math anymore, and this is the same teacher he had last year that he loved. His friends mother told me they are thinking about transferring schools because of this program. We are very concerned about what do do as I doubt they will switch a curriculum that just begun. What do you suggest?

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      3. Jason,

        You are basically stuck with the school’s decision unless parents can show to the principal that overall student performance is negatively impacted in a significant manner. I would begin by talking to his teacher and relating the above to him/her. If that gets you no satisfaction, try the principal and possibly the PTA. Parents have a better chance if they band together rather than engage in solitary protests.

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  8. Thank you for your article. My children’s school district adopted CPM a few years ago. For the second year in a row, the district will not offer AP Calculus because the students aren’t ready. I am part of a parents group organizing to try to convince the school board to use a different curriculum.

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  9. I enjoyed reading your articulate and thorough article on this topic. I found it by searching Google for “CPM Math” and it was the 5th search result. Not bad! 🙂

    I am a parent in southern CA. My oldest child is in the 6th grade. This year, our district piloted CPM vs. Pearson EnVision 2.0 for grades 6-8, and CPM vs. HMH (Houghton Mifflin Harcourt) for High School Integrated Math 1, 2, and 3. This meant that my child’s class used CPM for the first half of the year, and then switched to Pearson for the second half of the year. (Switching mid year posed its own struggles.) So my experience with CPM was limited to seeing the 6th grade book she brought home, comparing it to Pearson, and talking with her about her preferences. She and I both preferred CPM over Pearson because the CPM book is laid out in a more clear and direct way. Pearson is cluttered with illustrations which we find distracting and the instruction seems more convoluted. Also my daughter at this age enjoyed the group work and self-discovery method, however she did say that the teacher also explained things to the whole class. At any rate, the district decided to adopt Pearson.

    Meanwhile the debate over CPM for high school has been raging. Parents were complaining about CPM on a Facebook page so I have been trying to understand why. The most common complaints seem to be about the kids having to do group work and also that the teacher isn’t teaching. This wasn’t our experience and I didn’t understand what was wrong with group work, but now that I have read your article I think I have a clearer picture of what some of the concerns are. However I am hoping for the best because just yesterday the district announced they will adopt CPM for high school for the next 3 years. This was despite a lot of parents being against it. Here is an article about it: http://www.latimes.com/socal/glendale-news-press/news/tn-gnp-me-gusdmath-20180505-story.html

    It occurred to me that my now 6th grader has had 5 or 6 different math curriculums tried out on her during her elementary school career with this district. Her class came of age just as they were starting to implement common core. Sometimes I wonder if this generation has a solid foundation to move forward with. It looks like she will use Pearson for the next 2 years, then CPM for a year, and then they may or may not change the curriculum yet again.

    I’m curious if you have any thoughts on CPM being from a nonprofit, versus these large for-profit corporations like Pearson and HMH.

    Thanks!

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    1. Thanks for your detailed comments, Jessica, and the link to the Glendale News story. Several of the points on both sides in that story correlate with those in my article above, but the newspaper coverage does not go into as much detail (as expected in a newspaper).

      I have been a persistent critic of California’s constant educational experiments using our children as guinea pigs. Please see my article at https://eduissues.com/2018/01/29/never-believe-educational-experts-or-me which focuses on this problem. Personally I think the points that I make in that article are even more important than my review of CPM, though the CPM article is the most frequently read on my blog.

      Math is clearly a cumulative subject, and when one of these experiments backfires, whole groups of children are negatively impacted. Clearly each district has to go through some kind of review process to select a book, but California seems to “refresh” its curriculum way too frequently. This disrupts lesson plans that teachers have developed over years. I suspect that part of the reason for this is the constant lobbying from textbook publishers. We are the largest textbook market in the country and what we adopt tends to influence others.

      Regarding your non-profit versus for profit question, I do not think that CPM’s non-profit status gives it any kind of moral superiority by default. The College Board which runs the SAT and AP tests is non-profit, and they are doing quite well for themselves, hitting up American students for a considerable sum of money in test fees. “Non-profit” does not mean that people at these organizations are not paying themselves princely salaries which then go into the “Expense” category. The important consideration is which textbook and instructional method produces better learning outcomes; other considerations are secondary.

      Because CPM’s textbooks are frequently purchased in paperback format, they may save school districts money compared to more traditional hardcover books, but one would also have to factor in how long a paperback lasts versus a hardcover in the rough and tumble environment that school books encounter. When the Glendale News story said that principals were favoring CPM, I wondered if this cost issue might be one of the reasons. You might want to inquire further on this point.

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      1. Thanks again, Jessica. In some of our local classes paperback copies are used. These are unfortunately split into two volumes. When the kids are working on volume one, they don’t have access to the book index which is in volume two, and they often don’t get volume two until the latter part of the year.

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      2. Hello again, Jessica. I was intrigued by the reference to Edreports.org in one of the slides (#11) in the Board presentation that you provided. I investigated further and saw that slide presented the data for the high school version of CPM. If you go to their site you will see that the ratings for the 6-8th grade material are not as high: https://www.edreports.org/math/reports/compare-k8.html

        One also must be aware of who is publishing the data and what are their financial interests. I have seen a lot of shenanigans in the textbook promotion arena. I would recommend reading Edreports.org About page and the various staff Bios. From an initial read the organization looks legitimate, but it is hard to tell if they are enamored of a particular teaching philosophy that guides their reviews.

        The US Department of Education also has a database called the “What Works Clearinghouse.” The math section is at

        https://ies.ed.gov/ncee/wwc/FWW/Results?filters=,Math

        Unfortunately I do not see CPM in the list of math programs there. That surprises me greatly since the program seems to be under consideration in many districts across the nation.

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    2. P.S. to Jessica – As I also mentioned in my article, I think CPM has a better chance of working at lower levels of math, so the fact that you prefer it for your 6th grader does not surprise me. However, once one gets past the Algebra 1 and possibly geometry level my doubts about the program increase dramatically. It will be interesting to see if CPM is still used in your district by the time your daughter reaches high school…

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  10. Thank you for your article. I came here by googling “CPM Curriculum.” I used this text last year because of its group work aspect. It offered some excellent problems that the students enjoyed and some odd ways of teaching negative numbers (which I am abandoning this year). There are some excellent modeling techniques for introducing algebra. But they overcomplicated modeling for negative numbers. I used it for 1/2 a year and switched to a combo of traditional text and CPM because the pros did not outweigh the con that we were not covering enough material. This year I will be using several resources to patch up where CPM doesn’t deliver. I’m keeping the group work and examples that were creative and interesting. It also requires too much time, and they expect teachers to just move on stating that the students will see the topic later. I ended up using direct-instruction when needed. I am seeking balance this year between direct instruction and group work and between fluency and understanding. I never used the group test approach. CPM has some great goals but it’s not balanced enough for me.

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    1. Thank you for your detailed comments. It sounds like you are trying hard to do a good job for your students and your approach sounds eminently reasonable. Was CPM introduced by your district or did you run across it on your own?

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  11. Thank so so much for this comprehensive and thoughtful analysis. My daughter’s school just switched to CPM this year, and she’s gone from “A” in math in 6th grade to a clueless ball of math-ridden anxiety in 7th. She is not doing well at all with the “no instruction” thing.

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    1. Thanks for you comment, Jon. I hope parents in your area are successful if they protest. In some places parents face a “we know better” attitude from the education establishment. The establishment needs to hear if things are not working out the way “research says” they should. Do you mind disclosing your location and what justification was given for adopting CPM?

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  12. Thank you for explaining the CPM program. My daughter just started her freshman year and math has always been a struggle for her. She kept telling me that her teacher “doesn’t teach” and that they spend their class time in groups trying to figure out the problems. I found this hard to believe until I came across your article and did some further research. Apparently, my daughter is correct. Her teacher doesn’t teach and mainly answers questions should any group have them.

    Since my daughter has a diagnosed learning disorder in Math, she is in a slower paced math group, with others who find math challenging. This program is a disaster for kids who already struggle in math. She is currently being tutored privately a few times a week, costing us $$$. Her tutor actually teaches her the concepts. Luckily for her class group, my daughter shares what her tutor has taught her. I’m at a total loss as to why her teacher is getting paid or why he is needed if he is not teaching.

    Thankfully, my daughter is doing well, only because of her tutor. We attend a small private school so I am hesitant to rock the boat with the administration. However, I will share your post with other frustrated parents.

    Thank you for your thoughtful and thorough explanation of CPM.

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  13. Forgot to mention: I found your article because my daughter’s math class uses an online version of the text book so I googled the CPM Core Connections Algebra to see if I could purchase a physical text book. I scrolled down to see various links expressing dissatisfaction with the program. Your link was about 5 or 6 down the page.

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    1. Thanks very much for your comments. I agree completely that the CPM approach is most problematic if a group of kids has a hard time and/or no interest in math. Grouping them together will not magically solve this problem, and the teacher should obviously not just leave these kids to their own devices.

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  14. Last year was my 8th and last year hired at my charter school as a Saxon math teacher- awesome year as I grew so much learning how to enhance the program with more hands on and some common core ideas kids could “see” the math. Well little did I know my contract along with the other 3 middle school math teachers’ contracts were not renewed….. hm. Only to find out that the principal, in private, all by herself with no info to parents or the board, made the decision to go with CPM (she found 3 teachers who already teach it )
    As the school year is underway more of what the principal’s vision of what a great math program is is coming to light. Despite being treated so poorly and lied to by my principal , I’ve been trying to be open to this new curriculum and just checked your article to find out more about what CPM is all about.
    The history of math fads is interesting. And it’s also interesting that the people in charge won’t have discussions with those of us with degrees and teaching experience in the field – I guess authority gives you insight I’m lacking.

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  15. When I read the title of this blog, I was interested in hearing an unbiased teacher evaluation of CPM. I wonder if you did see some positives to the resource? The developers of CPM have stated that ongoing professional learning is critical for implementation. I wonder if many of the schools/districts were not able to sustain that piece in order to help teachers implement and supplement for learners appropriately? CPM assumes that teachers have a strong workshop model structure at all levels…if they don’t, that is a set-up for failure. It is a viable resource, given that all the cards are in place. I would be interested in hearing your thoughts on Integrated vs Traditional pathways in high school math…(regardless of discovery/constructivist model that is in CPM). Thank you for opening conversations and for your honesty. (20-year, 8-12th math teacher, currently a math content specialist)

    Liked by 1 person

    1. Thanks for your comment/questions, Tracy.

      I listed several of the “pros” right up front in my article, but then went on to the “cons” which predominated the latter part of the article. I don’t think this *necessarily* means that my article is biased, using the mathematical sense of the word. It is very rare to have an equivalent number of pros and cons on any question, and I believe that there are more cons than pros in this particular case.

      Many districts do not have the resources for extensive professional development, and that is a BIG red flag for any program that requires it. In engineering there is an acronym “KISS” which stands for “Keep It Simple, Stupid.”

      Our local K-8 district adopted Everyday Mathematics several years back and abandoned it several years later when it was clearly failing. The publisher called it the best researched elementary math program available! I see daily the ill effects of that program on those kids who are now in high school.

      When discussing this debacle with our local high school district assistant superintendent of curriculum a few weeks back, she tried to defend that program by saying that it was a “complex curriculum” which probably required more teacher support than was given. Sorry, but I can not accept these kind of rationales. Particularly in K-8, anyone who adopts a “complex curriculum” that requires a lot of support should realize in advance that they are gambling with kids’ education. When the program fails, it is cold comfort to the children that the professional support was inadequate. A complex program, particularly for elementary math where many K-5 teachers have math phobias as mentioned recently on EdWeek (see http://www.edweek.org/tm/articles/2018/11/06/how-to-help-students-heal-from-math.html?cmp=eml-contshr-shr and https://scholarcommons.scu.edu/cgi/viewcontent.cgi?article=1058&context=tepas ), should never be chosen.

      I will also note that the “research” base behind CPM strikes me as being very weak. I have intended for some time to write critiques of some of those papers in their bibliography, but have not yet had time to do so. The CPM publisher takes weakly supported claims in some of those papers and grossly over-interprets them to support their practice.

      As I also indicate in my article, CPM’s own comments about the results of their in-house research do not overwhelm. From https://pdfs.cpm.org/research/CA%20CST%208th-11th%20Grade%20Results%202011-13.pdf :

      “Of the 33 comparisons in the tables on the following pages, CPM schools scored equal to or higher than the state average on 27 of them. Since the CST measures knowledge of basic skills and procedures, these results demonstrate that the CPM program is one effective way to teach this part of a complete mathematics program.”

      The state averages on the old standardized tests were pathetic (often between only 30-40% “proficient or above” in algebra and geometry), so saying that CPM’s results are equal to or higher the state averages is very faint praise. You can refer to the actual numbers in the link above and see that the results vary only by about 4% from the state averages in the better cases, and several instances are below the state averages.

      Finally, regarding Integrated versus Traditional pathways, I do not have any significant input here. I have read that Integrated pathways have been used with great success in other countries. During the last six years most of my time has been spent on precalculus, and AP courses such as Calculus AB, BC, and Physics, so I no longer have a personal horse in the Integrated vs. Traditional pathways race.

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  16. David,
    Thank you for your article. I found this blog via google search of Opinions of CPM.

    In the 13 years since our district in Wi adopted CPM our district is as committed as ever. Yet, students and parents have to to find other means to get our kids college ready. The irony is not lost that CPM did not prepare our elder son for college. I agree that for elementary grades up through 6th it can be fine. Anything beyond pre-algebra needs to be taught not experienced. Math is like learning a language it requires instruction and practice. Math is central to chemistry, physics, statistics, lab reports.

    CPM relies on group learning. The premise is that all kids are invested and mature enough to meet the expectation. It also assumes that the teachers are supported and invested too. We experienced a different reality.

    Our son learned math from his physics and chemistry teachers. -For that we are grateful. CPM teachers are taught to not answer the students questions but to ask questions in return. Eventually, our son chose not to ask questions. When students bombed the test, then the teachers would go over the exam question by question and teach. Then the kids were to take the test again. This time most passed.

    In my opinion and experience, CPM does not deliver on higher level math, which prepares kids for STEM careers, And AP calc ,which son took, was not rigorous enough for the AP exam. He retook calculus in college. Many of his peers that went onto study engineering, physics and math also had to redo calculus at the university level. Our son is currently a junior pursuing a degree in Biochemistry. He said he would give the same advice he was given – redo calculus. He also has struggled with Physics as the geometry was not thorough enough and there was no trig at our school.
    His cohort, who did not have CPM , were able to advance quickly to other course work and test out of their first semester or year.

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  17. My 8th grade Algebra students class is switching to CPM at semester. The High School math curriculum is staying traditional. So we are concerned that he will not be prepared for the high schools rigor

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  18. Thank you for sharing all your expertise, informed opinions and research. I appreciate it greatly. Our school district, Klamath Falls City Schools, also uses CPM. Recently, the high school and middle school hosted a Parent Math Night event, and even had a CPM representative on hand. Sadly, there was nearly no opportunity for parents to articulate concerns and it smacked of used-car salesmanship ethos more than anything. I plan to reference your information with the principals and high school math department head at a closed meeting I have been invited to. Note: the closed meeting – not my idea, and I hope to convince them to hear from more concerned parents.

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  19. Hello,
    Thank you so much for explaining CPM and analyzing why it is not effective at times. I am both a frustrated parent and educator. My son who is Gifted was recently placed in an accelerated Math class at his middle school. He is a 6th grade student. Math was always his strength and he excelled and enjoyed being challenged by his teachers. This has changed during this school year. My son is basically frustrated with CPM. He does well with his homework but when he takes the chapter tests he has gotten 56%, 42% and even 23%. This doesn’t make sense to me. On top of this his teacher does not allow the students to take the tests home because she claims she created the test herself and doesn’t want students posting the past test. I don’t agree with this reasoning but after speaking to the admin they can’t change her decision. I have to go and see the test on my own and write down the mistakes my son did. I know my son is capable of achieving higher, and I think something is missing that is not allowing him to succeed. On top of this, she says she will not recommend anyone for accelerated 7th grade Math next year if the student didn’t achieve 87% both semesters. My son has only achieved 82%. I don’t think it’s fair or right that my son is stopped from going to a higher class because of his outcome in CPM. My son has always received a 4 in the CAASPP and was points from achieving perfect score in both ELA and Math, so she can not tell me he is not capable of success in a higher/accelerated class. The problem is not my son, but the logistics of the lesson, and the classroom activities in combination with very few practice problems and chapter tests that are more rigorous than the material she taught . These tests include problems she hardly discussed. I don’t se a parallel in this. Please feel free to respond if you would like to. Do you have any suggestions as to what I should do next? What educational rights does my son have? I don’t want CPM to determine the outcome of where my son will be placed. This can alter his plans in high school to continue taking accelerated courses which he wants to take and is eager to be in.

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    1. Dear olimpaso,

      I am not a legal expert on students rights, and unfortunately I see students negatively impacted by the educational system constantly as you can read in my other blog articles. There isn’t always a ready path to “salvation.”

      I taught math in CA public schools and could easily make up a new test for any math topic that I taught. I think that this desire to secure old tests, particularly at the 6th grade level, is slightly ridiculous to say the least. Kids have to be able to learn from their errors, and making it hard for them to access their old tests hurts them. If this teacher needs to reuse the same exam, I would worry a bit about the teacher’s abilities.

      Obviously by your reference to CAASPP you are somewhere in California. My article https://eduissues.com/2018/09/29/how-students-are-accelerating-in-math-at-aragon shows how parents in the San Mateo area are circumventing school road blocks to accelerate their kids in math, and it might provide you with some ideas. I hasten to add, however, that if the acceleration is pushed by the parent and not the student, then I doubt that the outcome will be a happy one.

      Looking for outside help at a place like Kumon, the Russian School of Math, etc., is another option, although it saddens me when parents have to pay for such help instead of having the school system do right by their children in the first place. You might want to go to a PTA meeting or use a community forum like Nextdoor.com and see if you can ally yourself with other families in a similar situation. It is much harder for a school/district to ignore a group of parents than it is to just brush off a particular family.

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  20. I have been piloting CPM Precalculus and AP Statistics this school year, and fellow teachers in my department have been using many of the Geometry and Algebra 2 lessons. After my and my students’ experiences so far, I strongly disagree with most of your criticisms of CPM. The lessons are very well made and provide students with a deep conceptual understanding of the material. Overall, my students are performing better and retaining more, and their self-confidence and self-efficacy have skyrocketed. And, no, not all of my students are super self-driven and high flyers. The only major changes I’ve made are homework and practice. I throw in extra classwork problem sets in each unit, because I’ve stopped assigning homework (there’s just no good research to support it, and students are doing better without it!).

    There are also plenty of opportunities in each lesson to differentiate up or down, and even more opportunities for teachers to add in their own lessons. I’ve found that teachers who struggle with inquiry-based learning tend to be tied to the textbook and have trouble thinking of their own lessons, and any non-traditional text will be an issue for them.

    I’m not a fan of their assessments. The assessments should mirror the same critical thinking as the lessons, and it’s just rote processes. I often give group quizzes that are challenging and problem-based (like the lessons), so I definitely agree with their approach to that. However, my tests are still unit-like and problem-solving based with an emphasis on critical thinking, instead of their premade tests. However, I don’t think any teachers should ever use a premade test or test bank for an entire test – every classroom will be different, and our assessments should reflect that.

    Your complaint about the text having no worked problems or resources… They’re in nearly every lesson, and there are hints on the homework help. If students are struggling, I see nothing wrong with providing a video recapping the lesson or the main ideas. You want them to stare at preworked problems in a traditional text and figure it out for themselves… A quick YouTube video can work just as well, if not better. Again, this simply sounds like a traditional teacher who doesn’t know how to adapt and be flexible problem.

    Lastly, there are several meta-analyses showing problem-based and guided inquiry learning methods have the highest effect sizes in math and science out of all the common classroom interventions. One article illustrating the scariness of “discovery learning” does not outweigh the rest of the literature.

    I appreciate your thoughts on CPM, but I believe it’s delivery is misguided and is leading a lot of parents, students, and teachers to blame a curriculum when it’s more likely there are many other issues happening in those classrooms.

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    1. Angel,

      Clearly your experience with CPM differs somewhat from mine. You can try to blame it on my “lack of flexibility” but I beg to differ.

      I will leave it to readers to judge if there are worked examples “in nearly every lesson” in the textbooks like you claim to see. The homework help is separate from the textbook and intentionally so. Having worked examples in the lesson would defeat the self-discovery philosophy.

      As I said, the CPM textbook can be supplemented and *needs to be supplemented.* There is nothing inherently wrong with the guided lesson questions themselves (as I also mentioned in my article), but one can’t rely on them solely as you also attest.

      This has *nothing* to do with being flexible or not. You admit that you are supplementing the material which is what I also said was needed to make CPM work. If you prefer to supplement it with Youtube videos, that’s fine.

      You mention the “major changes” that you made: “I throw in extra classwork problem sets in each unit, because I’ve stopped assigning homework (there’s just no good research to support it, and students are doing better without it!).” I would not be surprised if several students end up completing at home the extra problems sets that you provide.

      One of my main concerns with CPM was not that the self-guided problems were necessarily bad, but that there wasn’t enough practice in the lesson. By adding additional problems you are doing precisely what I said needed to be done.

      As to there being “no good research” to support the assigning of homework, I find this a bit astonishing. As a Ph.D. level scientist who has worked in multiple technical fields in addition to teaching (please see the blog link to my bio), I can tell you from direct personal experience, and from that of many others, that many people have learned much more math and science from quiet contemplation doing homework than in the classroom. I really hope your students are in fact “doing better without it” because that comes off as a very glib comment.

      The “one article” that I cite is a serious review of multiple research studies comparing constructivist approaches versus guided instruction. It is most definitely not simply “One article illustrating the scariness of “discovery learning”.” Please read it before you dismiss it out of hand.

      As to your conclusion, “I appreciate your thoughts on CPM, but I believe it’s delivery is misguided and is leading a lot of parents, students, and teachers to blame a curriculum when it’s more likely there are many other issues happening in those classrooms.”, I also beg to differ.

      I am willing to bet that most parents who stumble upon my article do so mainly because their children are already having problems. My article is not “leading” parents to blame the curriculum. They already blame the curriculum before they seek out confirmation on the Internet. They see in my article confirmation of their own experiences which is why close to 8,000 people have now seen it and told others about it.

      You can find several other instances of parental discontent with CPM on the Internet. I agree that there may be several factors responsible for this discontent, but I tried hard to faithfully and accurately relate my concerns about CPM in my article. We can agree to disagree on whether its “delivery is misguided.”

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    2. “The lessons are very well made and provide students with a deep conceptual understanding of the material. “

      Sometimes. Just as frequently, the ‘story’ aspect of the CPM problems obscures the mathematical concepts and students. CPM also seems to devalue procedural fluency in favor of conceptual understanding, as though procedural fluency was but a weak subset of conceptual understanding.

      “You want them to stare at preworked problems in a traditional text and figure it out for themselves…”

      Yes, that is how students gain experience in inductive reasoning, which is a valuable intellectual tool.

      “Your complaint about the text having no worked problems or resources… They’re in nearly every lesson”
      That is not the case in the Integrated 1, 2 and 3 series. There are virtually no worked out examples.

      “A quick YouTube video can work just as well, if not better.”

      While helpful, students cannot find that YouTube video if they don’t know what to search for. CPM often uses non-standard notation and vocabulary, so students may trouble with that search. Also, most of the ‘Review / Preview’ problems have no citations as to where in the textbook the concepts were introduced, so if a student doesn’t remember when the topic was introduced, they cannot check their notes effectively to see what they previously learned. CPM is very poor at indexing concepts.

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  21. Several years ago, I taught (as a student teacher) the same classes that you did at George Washington High School in SF. Contrary to what I read in your article, I found most of the problems in the classroom to be tangential or parallel to CPM. These problems included: insufficient prep and teacher-collaboration time, parenting and community issues, parent-teacher engagement, mental health counseling, administrative necessities and limitations, etc.

    Self-discovery is present in all forms of education. Students that can self-organize and self-manage can more easily experience self-discovery in a lecture-formatted education; I think it is true (but slightly disingenuous) to attribute their success to the lecture-format.

    As a consultant and educator in the field of manufacturing and compliance, I see, everyday, individuals and teams (of all backgrounds and levels of education, from PhD to illiterate) around the world tackling problems, without much explanation from a textbook. I agree with criticism that clear instruction is important, but I have never seen an employee commend a manager (or anyone for that matter) for a lecture. Nor have I seen a manager’s lecturing help solve many employee (or team) challenges.

    A big problem with the lecture format is the need to drive students through a narrow curriculum, rather than allow for slower groups to examine the fundamental aspects of a mathematical concept, question and solution and for more advanced groups to perform a more profound exploration of the same work. I think CPM, at this moment, has the greatest potential to address that problem as the learning model is more active than passive.

    It is worth noting also that, across industry, managers struggle with employee engagement; it may be because: (i) people who are good at tasks are often promoted to (ii) manage people (which involves a very different skill set than completing tasks) to (iii) explore and solve continuous series of diverse problems, sometimes with a team and often with limited resources. (iv) Managers are also expected to be able to frame a problem or solution differently for different people (e.g. regulators, vendors, employees, C-level executives, customers and consumers).

    CPM is designed to prepare students for (ii), (iii) and (iv). It does this by substitute procedural learning through repetition with a more complex and profound learning that emphasizes teamwork, critical thinking and presentation. For example, rather than asking students to individually complete 50 problems that look like: “12 X 16”, a question might be: ”How many ways can one solve “12 X 16” and how many ways can it be presented? Students can work in groups and use props (pennies, colors, a graph or groups of dots on a page) to present their findings to other groups to see which group discovered a unique solution and which group found the most solutions. The class as a whole can decide which one they liked the best.

    Parents who complain about CPM may have good reason, in some cases, but I think they may also overlook the purpose of CPM’s design and use of “struggle”: to prepare kids for complexity, adversity and responsibility, to drive innovation and entreneurship, and to save students from jobs (and tasks) that are increasingly automated or cheaply valued.

    I don’t think anyone expected CPM to roll out smoothly, especially it’s creators; CPM was expected to improve by participation, feedback and creativity. Teachers ought to have latitude to make a lesson simpler or more dynamic, or both at once, and CPM provides the fundamental structure.

    Much frustration around CPM comes from teachers’ lack of time, resources and support. Teachers fill a combination of roles: project manager, mentor, auditor (in some sense of the word) and trainer of technical and managerial skills. However, this problem is tangential to CPM. The role of the teacher is under-appreciated and underestimated in our society and teacher salaries, prep time and administrative-community-parent support have been severely lacking in many school districts.

    I can understand why some people, particularly academics and accomplished industry professionals, might feel frustrated with, or affronted by CPM: “I am successful and so my education is validated by my success, and to question the format of my education is to ignore the validating results.”

    I can also understand why parents are frustrated: “Never in my life have I needed to solve a math problem three different ways. Why should my kid be denied college admission due to a drop in GPA over useless math?”

    I can also understand teacher’s resentment for CPM: “I have been teaching for X years in this format, to question this format is to question the value of my work and the conclusions of my experience.”

    In these logical arguments, perhaps there is something to gain, but also much to lose. While critiquing CPM has value, tossing it away would be unfortunate.

    I am amused though by the outrage around abandoning a traditional lecture-format for a model that prioritizes creativity and engagement. I don’t know many people that yearn for the same level of engagement they had in their sine-cosine-tangent lectures or lament to others, “Kids just don’t get lectured on logarithms anymore, it’s a shame.”

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    1. Michael, thank you for your lengthy reply. You do a good job of explaining the rationale behind CPM.

      Unfortunately I am a bit concerned that both you and Angel before you did not read my article carefully.

      In the section entitled “The CPM Math Program” I mention many of the advantages of CPM. I also mention in my article that CPM is suitable for the lower level high school math classes. I used CPM for Algebra 1 at GW and had no problems with the group lessons though I supplemented it with additional homework practice.

      My biggest concerns are at the pre-calculus level and higher (though I thought a few things like group tests were ridiculous at all levels). Those concerns unfortunately were not addressed anywhere in your lengthy reply.

      Your observations about managers lecturing employees may be valid, but they are somewhat irrelevant to the problems that I tried to address.

      As the level of complexity increases in higher math classes, self-discovery takes more time, and the level of practice required to master the material also increases. This is where the most serious issues with CPM arise. It just takes far too much time to cover higher math classes.

      Defenders of CPM always wax poetic about self-discovery, and this sounds very good in theory until it runs into issues like those above.

      In the end outcomes matter. When our local high school tried the CPM precalculus curriculum, teachers had to supplement it extensively, and the following year the program was dropped in favor of a traditional textbook. I continued to tutor several kids who went through that ill-fated pilot, and it negatively impacted their subsequent high school math progress.

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      1. Hi David,

        Thank you for your reply and I am grateful that you have this dialogue through your article. I found it through a google search.

        I disagree that there is no relationship between CPM, management skills and professional growth opportunities. Despite your negative experiences with CPM in higher level math, from an industry perspective, I still see lecturing as more problematic and a means to rendering math less accessible and less meaningful to students. Thank you for your time and consideration in this comment/response thread!

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      2. Hi Michael,

        I am not sure what industry you consult for, but your experience differs from mine.

        After finishing my Ph.D. and postdoctoral research, I spent about a decade in the scientific software industry and was involved with the Human Genome Project and the early development of the Internet for biology. I then spent another decade doing IT work in the biotech industry. In addition to a Ph.D. in biochemistry, I also have an MBA from the UC Berkeley Haas School of Business and was in management for many years.

        Most of the people that I worked with were highly educated, and I rarely, if ever, recall managers lecturing anyone. There might be an occasional brainstorming meeting, but, in my fairly extensive experience, most knowledge workers tackle difficult problems on their own and consult with others as needed.

        However, the most insightful ideas usually come from quiet solo contemplation. I am not minimizing the need for collaboration, but CPM, in my opinion, greatly exaggerates that aspect and does not give sufficient weight to the quiet contemplation aspect.

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      3. I have taught the CPM curriculum in middle school for about 8 years. I think that many of the concerns expressed here by many of your readers represent the continuum of my feelings over that time. I too have felt frustrated by some of the open-ended lessons and what I have felt are some of the assumptions that the authors make about the proficiencies of the students who may be using the program. I have found myself frustrated by the “lack of practice” and often supplemented. I have been to several conferences from the publishers and been very energized by each of them, to return to the classroom and once again be frustrated by how difficult it can be to make those strategies learned actually happen in the classroom. It is a program with which I truly have a love-hate relationship. It challenges me and my students. It makes us all uncomfortable. That being said, I’m not sure that this is a bad place to be.
        One of the most common frustrations has been the lack of practice in a given lesson that specifically targets the concepts or skills taught in that lesson. I have thought a lot about this over the years and also read a lot about practice and its connection to deep learning. The theory of interleaved or spaced practice is integral to the CPM program and the success of its structure. I know that students can be frustrated that they are doing problems like what they learned that day, but the neuroscience supports giving them time to forget a little before hitting it again. The simple act of recalling something that you have nearly forgotten strengthens synapses and memory. Even if it cannot be fully recalled, the effort expended grows the brain. This is purpose behind the review /preview and the seemingly random chunking of a concept over many chapters rather than just one.
        Like every math program, no curriculum is perfect. You mentioned that it takes the skill of the teacher to make CPM work, and I would argue this for ANY program. Students who struggle with an inquiry based curriculum would most likely struggle with a more traditional program as well. It takes an experienced practitioner to find the emphases and then create engaging opportunities to interact with that content. This is a perfect time to talk about best practices.
        Thank you for your mostly objective review of the program. I would have been a very different math student had I grown up thinking through an idea in context rather than practicing 20 even problems every night. I’m sad that I never had that opportunity as a student but I am very grateful to have it as a teacher.

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      4. Andrew,

        Thanks very much for your detailed comments.

        I only wish to remark on one point that you make about the “neuroscience supports giving them time to forget a little before hitting it again.”

        I think *** very few people have ever taken the time to read the articles that CPM cites to support this contention.***

        I have intended to write a review of this at some time, but it is a lengthy undertaking and I have not had time to do so.

        While it is everyone’s common experience that reviewing subjects at a later date helps cement the concepts in memory (and this is why the CPM claim on its face seems reasonable!), the actual research cited provided very little good quantitative data about how many initial practice math problems are optimal nor does it provide details about optimizing the time period between review “spirals.”

        Instead, during debates about CPM, we hear the usual claims about “Research shows” without anyone checking whether or not the research is valid. Some of the papers cited by CPM to support “spiraling” studied nothing other than the retention of things like the number of city names on a list of such items which is a far different thing than learning math problems!!!!

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  22. Context:
    Schools with high level of student absenteeism/tardiness.
    CPM Problem: CPM textbook does not satisfy the needs of absent/tardy students. Teachers cannot refer absent/tardy students to read in the CPM textbook the explanation of a particular topic, including examples and applications, because those explanation, example, and applications simply do not exist.

    Solution:
    Schools should have a set of “approved” textbooks instead of only one monopolistic textbook. Teachers as professionals –and not clerks– should decide which textbook to use according to the particular reality of their different classrooms.

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  23. Hi David, I am a new to the public schools math teacher in CA, with many years of experience tutoring in the same subjects you cover (Math, Physics, Chemistry) and teaching adults. I have just started using CPM’s Integrated Math 1 curriculum for the first time, and I found your blog via a Google search on “is cpm integrated math good”. My initial experience echoes many of your own sentiments, and I believe that active supplementation is going to be necessary to make it work. I would like to continue this conversation with you and others who have expressed thoughts on this topic.

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  24. Hello, I came here because my son is a sophomore in SMUHSD, he is quite advanced in math, and when he started CPM math last year he could see the weaknesses and did some research on the curriculum. And tonight for some reason he decided to mention what he has found to me, and I did a quick search for “CPM math reviews” and came across this blog. Thank you for posting this, I’m sure it has been of interest to many people!

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  25. I am on year three of a pilot for CPM math in the district I teach in. I am wondering if you or anyone that reads your article have information on how CPM impacts students who have autism. I know the difficulty students with autism have with social skills. I am more specifically wondering about the barriers when compared to traditional direct instruction methods for teaching math. I appreciate any and all feedback.

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    1. Thanks for your comment, Penny. Although I have taught a few autistic children, I have no special expertise in this area. The autistic students I taught were not using CPM for math, but your concerns about how such students would handle group learning are legitimate. Hopefully others can offer their experience. What is your take to date?

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  26. Thank you for the detailed article and posting of comments, both pro and con. Excellent discussion. My daughter is a junior in the Winneconne School District (WI) taking pre-calc. Her (and our) experience finds a level of frustration with “teaching” by the CPM method, leaving her stressed and worried for her future college career. A previous “A student in math is now failing. She no longer thinks she can succeed because she is given tests that include materials that were not taught and “will not be taught” according to her teacher. But have to be learned. The challenges of group teaching by fellow students seems counterproductive to the best use of ones time in the classroom. I have been to so many industry training sessions where the presenter gives attendees 5-10 minutes to discuss power point material just presented. Most of that time is wasted trying to understand the question and wants of the presenter instead of directly asking and allowing questions. Showing examples of their material. But that takes time and effort on part of the instructor. Perhaps if the material was taught 3 of the 5 class days with in-class lectures and questions/discussions and homework, then group discovery or review the other 2 days, CPM could be possibly an effective way to reinforce the formal instruction. But a total lack of formal instruction leaves many students behind with no material or basis to get caught up to their ‘group’ which may also be behind or failing. At some point the learned, educated adult in the room needs to instruct and present the material logically and completely. Math builds upon lessons taught and learned by solved examples and problems, not errantly discovered or falsely copied to appear to succeed. I found your website searching on Bing for CPM math precalculus help. It was a couple of pages in.

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  27. Thank you for your honest evaluation. I am a high school math teacher who has astounding success teaching students as their final stepping stone to college. I use several techniques: lecture, drill and practice, group conversation. I am in constant interaction with ALL students; constant movement around the room. Sometimes the thought-provoking questions come from me, sometimes from them. Great success. ALL students that passed my class passed the AccuPlacer with astounding results (I’ve never seen the AccuPlacer test, so I cannot be accused of teaching to the test). After three years with great results, the college that my students move on to, exonerated students that pass my class from taking the AccuPlacer–they receive automatic acceptance. I switched to middle school this year and my administrator is all about CPM. “We’re going to prove CPM makes great test scores!” I attended the week-long training. Trying to remain positive, I adapted the curriculum as best I could to my teaching style and was feeling good about it. After my first evaluation, my administrator wrote TWICE…”trust the CPM!” In other words, become a robot to the method. I hear kids mumbling about confusion and have to bite my tongue. After two days of CPM group work, I end up spending two days going over the homework so I can TEACH and the kids can LEARN. I don’t know how many times I hear kids say: “Oh! that’s how you do it. I had no idea!” What I can accomplish in one day now takes me four days. But I must “Trust the CPM!” Needless to say, I love the kids, I love teaching, but I am looking for another job. CPM is not teaching. It is a robot, doing exactly as instructed. Thank you for showing that I am not alone and not losing my mind. I teach for the love of it. In three months of this curriculum, my passion is gone. I am drained. The world of math may lose a very effective teacher due to the duress.

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    1. Lara, thank you very much for one of the most interesting comments posted to date! I am constantly amazed at how educational fads overwhelm common sense!! You might also enjoy my related article “Never Trust Educational Experts (Or Me)!”

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      1. Please direct me to your aforementioned article. I have been unable to find it. Thank you for the response. Just this week I was put on a “Performance Improvement Plan” by my administrator because I was not teaching CPM with 100% fidelity. The plan states the I MUST use group tests, I MUST use CPM resources (not my own supplements), I MUST use the CPM-generated tests, etc. I will be reviewed many times throughout the upcoming semester. I’m too old, too successful, and love my calling too much to follow through with this. Job or no job at the semester break, I must walk away. I compare CPM to a cult. Once they get you in their clutches you become the puppet to the puppet master. Common sense is the only thing that steers one clear of becoming a puppet. A good teacher knows how to teach with passion, knows how to tap into her own experiences and knowledge and knows how to love, nurture and teach children. Students respect and respond to a good teacher; they will not respond to a monotone puppet.

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  28. As a special education math teacher and a parent whose child is in middle school and hating this horrible curriculum I am struggling to enforce him to do the work. This quote from your article hits it spot on: ” the biggest problems with CPM are the lack of explanations, worked example problems in the textbooks, and insufficient practice problems”. My son is the child that hates math. So lets make him hate it more, REALLY! When I ask him well how did your teacher show you how to do it? He says what do you mean she just handed me the work sheet and said here do this. So after a full 8 hr day of TEACHING algebra, that is right I said teaching math (where I explain the problems, we work through many problems together and then my students do some problems on their own) (I do, we do, you do, method works wonderful) I go home and teach my son math as well. The math that his math TEACHER is suppose to be teaching him. I am currently so frustrated as a parent and a teacher.

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  29. I read your article with great interest. It’s most interesting, in fact, that you are full of praise for many of the points of CPM’s methods, including group work, the initial guidance provided by the teacher at the beginning of the period, spiralling, and others. You aren’t supportive of Group Tests, though you don’t discuss the amount of a student’s grade they actually account for. (This is a rather small percentage in most CPM classes.) The bulk of your criticism seems to come from your experience as a tutor, and not as a classroom teacher. You speak of student frustration and your concern that students were not understanding deeply enough so creating your perception that more repetitive work was needed, found and provided by you for your tutees to shore up this shortcoming.
    What I would like to point out is that you’ve only incompletely understood the constructivist education agenda. It is not without guidance – it is without dictation. Having an incomplete understanding part way through the book or chapter is common for a constructivist approach and is not a shortcoming. Consider Vince Lombardi’s 1961 instruction to his Green Bay Packers: “This,” he said, “is a football.”

    Do we imagine Vince stopped there? That he was a hands-off coach? Indeed not. The point of constructivism, and telling three dozen professional athletes “This is a football” is to get THEIR brains working on the topic. There is ample data that demonstrates that students at all levels, engaging with, but not as a total substitute to direct instruction, achieve greater in greater numbers and remember the lesson longer. This is especially true for populations that are traditionally underachieving.

    I’ve witnessed first hand many parent, tutor and sometimes even teacher complaints about CPM. Almost universally these are founded on the essential misunderstanding that when a student is struggling there is a problem with the instruction and when they announce they don’t get it that they never will, and that when the materials haven’t outlined the complete lesson -immediately- that it never will. None of these are true. In the string of articles that led me to this one, I saw the article by Clark, Kirschner and Sweller titled “Putting Students on the Path to Learning: The Case for Fully Guided Instruction.” That work, as yours, has conflated several different education styles, to the detriment of your analysis of them. There is no dichotomy in actual classrooms between “fully guided instruction” and “self discovery.” All classrooms, practically without exception, are a mixture of those, with the exception of the teacher who is content with education malpractice who sits at their desk and passes out worksheets all day long.

    The CPM curriculum is no exception. It starts with a challenging question. It may end only with that question – without its full implications – explored. Eventually the bigger, broader picture emerges. Students who have been through the CPM program, and programs like it, are following the Common Core’s eight standards for mathematical practice in addition to the standards, which some have pointed out are not much different than the standards list that came before. The 8 standards for teachers, however, is very new. They create mathematical power. They empower students to own the math and use it as a tool for solving real problems. A student who has been led by the nose, willingly or not, directly through the entire lesson might be a great symbol manipulator – might be – but a student who has thought through the implications of all this and been asked to tackle a variety of problems with it, are truly powerful.

    There is simply no comparison. If you don’t see the outcome and only have a view for the short term struggle, you will of course have questions. As a tutor, rather than a teacher who has followed students through their education perhaps for several years, you will of course miss out on all that.

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    1. Richard, you make many valid points in your detailed comments above, and I thank you for them.

      I think your inference, however, that my criticisms of CPM stem from lack of experience as a classroom teacher are incorrect. I understand the constructivist education agenda. In a world where students are engaged with the material and have sufficient time for the self-discovery process, I agree that the constructivist agenda would probably work. It seems very reasonable under those ideal circumstances.

      In the real world where self-discovery takes longer as the complexity of the material increases, the CPM method tends to fail. Not only are fewer students capable of reaching the endpoint of a more complicated discovery process in the allotted time, the increased frustration with not finding that endpoint also affects student motivation!!

      I mentioned in the article that CPM can work at lower math levels, like Algebra 1, but when it was tried at the precalculus level in our high school district, it was a disaster.

      Also, another problem results in classes where a teacher has a problem keeping groups of children focused on the subject material in the first place. I am not saying that traditional methods would accomplish miracles in such classes either, but CPM methods in such cases can add to the chaos.

      I always find it interesting that CPM defenders focus mainly on the logic behind the constructivist agenda, but don’t seem to be able to acknowledge its flaws. It *does* have some strengths, but it also has serious weaknesses. My views about CPM are based on *both* my classroom *and* subsequent tutoring experience and are also based on watching how it has been implemented by other teachers.

      I also have a Ph.D., have spent years doing scientific research, and have worked in fields other than education, so I believe I have the ability to look at difficult problems with a degree of objectivity instead of looking through a partisan lens.

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    2. I should add after rereading your comment that, unlike teachers, since I retired from teaching and took up tutoring, I have worked with several students through multiple grades, in some cases from 7th grade all the way through 12th!, so I have seen the longer term effects of instructional methods that most teachers do not since students are in their class often for only one year and then move on.

      As long as CPM is part of a curriculum that includes direct instruction when needed, some, but not all, of my objections will be addressed.

      Unfortunately you can read several comments from others above where CPM implementations in their locale have not gone well. I think it behooves educators to listen to these complaints and not quickly dismiss them as being from ignorant parents who do not understand the niceties of constructivist methodology. This sadly seems to be the common response that parents receive…

      I really do not like comparing education to business for many reasons, but in this case the analogy is apt: when many of one’s customers are complaining about their satisfaction with one’s product, a rational person would stop and carefully consider whether or not the criticism is just. I have tried in my article to point out both the strong and the weak points of CPM. However, too often CPM advocates appear to be relying on the “lack of experience / ignorance of constructivism / we know better than you” argument to dismiss any and all criticisms of the program. When one has a monopoly and can not be fired, one can get away with this response. In the business world, customers would quickly abandon that business for a different vendor.

      Richard, you do acknowledge in your comment that “There is no dichotomy in actual classrooms between “fully guided instruction” and “self discovery.” All classrooms, practically without exception, are a mixture of those, with the exception of the teacher who is content with education malpractice who sits at their desk and passes out worksheets all day long.”, so we may be more in agreement than one might think.

      However, dismissing my views with the comment “If you don’t see the outcome and only have a view for the short term struggle, you will of course have questions. As a tutor, rather than a teacher who has followed students through their education perhaps for several years, you will of course miss out on all that.” is completely out of line!

      As I said above, I follow many students for FAR LONGER than the one year most classroom teachers are with them. Several of my students who were with me during the ill-fated Aragon HS CPM precalculus experiment continued with me for their remaining two years in high school. That group had the most problems in subsequent math classes out of all of the 8 years that I tutored Aragon precalculus students. Please don’t preach to me that CPM works out better in the long run… It may work in the lower grades as I stated in my article, but I remain completely unconvinced about its efficacy at higher levels of math instruction for reasons that I also detailed in my article.

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      1. David:

        If I was out of line I apologize. I don’t see it, but as they say, if the offended feels it, we shouldn’t argue whether they should or not.

        You mention your degree and your experience – which was not part of the article so I did not previously know. As a scientist, you understand the need for real data; data, that is, that goes beyond personal experience. I mentioned that I also had met with many people who had complaints. I have also read longitudinal data and large group data on CPM. The data is clear: it very effective, and effective at all levels. Your complaint about its effect at higher grades just isn’t true for most students.

        I don’t mean to blithely ignore the complaints of my clients. I listen and adjust my instruction daily. I AM aware, however, that there are many students, good at “traditional” instruction, who try to summarize one-on-one tutoring with “Oh! You JUST (fill in the blank with some mechanical method)!” To me this is very troubling and disappointing. I have taken to stopping this exclamation of oversimplification with the observation that “Yes, that is all if all you wish is to finish this problem – or problems that look just like it. The real question is do you understand the ‘why?'”

        The problems with the Pre-Calculus course in your local high school are regrettable. I have seen CPM fail in other high schools with a well-educated group of parents who attack the problem as Ph.Ds instead of as educators. I have personally debated the pros and cons of a constructivist approach in the university setting (though I am not a Ph.D.) with folks with Ph.D.s in Mathematics. The one thing I like to remind them is that they are experts in their field; I would not debate them there. But when you expound upon education methods, you are in MY wheelhouse. There are very few Ph.D.s outside of the Education Department who have done even the essential readings of a first year Teacher Education student. I don’t know if you have or not, of course, but I would remind you and your audience that your Ph.D. does not make you an expert in THIS field. No offense. These are just the facts.

        I would also remind you of the reality of the Expectation Effect. Without real data, your interpretation of events around you are of course filtered through your prejudices. Also not meant to be an offense, this is basic science – as you are more aware than I, to be sure. You have reported the students who complain, but have given very short attention to the fans. Have you never met any? I’m pretty sure that is not the case.

        Regarding your tracking of some students over several years, I would argue again that this is not real data. Have you never tracked a student who was succeeding under CPM or similar programs? Again, I’m pretty sure that is not the case – whether you are aware of it or not.

        Since I have mentioned data several times, I offer some for your perusal. In the linked article, the researchers found that students perceptions of their successes under a constructivist regimen does not match the reality. They offer an explanation of why that might be so. I recognize the inherent danger of theorizing why we might ignore the complaints of our clients, as this article might empower us to do – but on the other hand, we DO have to explain to ourselves why a program that performs so well for so many has so many detractors. Success should, by itself, be apparent, correct? Why do so many not recognize it? Personally, I think the parental tutorial moment explains much of it, and doubly so for well-educated parents. If they are attempting to help students through public school mathematics and find it difficult, they will surely blame the authors, the method of instruction, the theory of learning… In my experience, Ph.D.s included, defensiveness and attack are part of that package.

        Best regards. I’ve enjoyed your blog.

        Here is the link to the article I mentioned: https://news.harvard.edu/gazette/story/2019/09/study-shows-that-students-learn-more-when-taking-part-in-classrooms-that-employ-active-learning-strategies/?fbclid=IwAR3NUPK7EsxgzfO0RTmgY0rICOtwnjZ_Cu65vTighvllIZlwptYfBeaYtvg

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      2. Thanks for your response, Richard. Again, you raise several valid points above. I will read the article you list above and comment in detail later. In fact I would like to speak with you by phone in the near future and will contact you privately.

        If CPM has all of this amazing research, why is it not quoted in their online “research base?” I looked through many of the articles that they quote supporting their methodology, and the quoted research support struck me as being EXTREMELY thin.

        I have had the goal of writing a detailed review of that research for quite some time, but as this would be a very time-consuming undertaking, I have not been able to undertake it yet (perhaps next summer break).

        I mention the CPM analysis of school STAR test results in my article and the wishy-washy language that CPM uses to describe them. The last time I looked at their research list (about a year ago) they were soliciting research proposals, but most of their research references still dated back to the STAR testing era…

        Their evidence for spiraling, for example, came from a psychological study that focused on recalling things like obscure city names from a list, a skill which is extremely different from learning mathematical concepts. Extrapolating from this list memorization research foundation to the CPM math instruction methodology requires a GIGANTIC leap of faith!!

        As I said above, it would take some time to type all of this up, so I would like to talk to you personally and discover any other research that you are aware of that supports their methodology.

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      3. I read your article cited above, and I am in complete agreement with its results!

        HOWEVER, it does NOT validate the specific CPM methodology! Any teacher who has ever taught should realize without needing to conduct a study like the one cited above that students do not learn topics like math and physics only from sitting in a lecture. Mastery does not come until they not only attempt the homework problems themselves, but, in almost all cases, try to explain/teach the concepts to someone else. In this case, if this is your point, you and I are in complete agreement!!

        My concern is that CPM in the higher math levels does a crappy job of cementing the initial concepts before they move on to the next topic. The subsequent spiraling is poorly implemented and in practice does not work! They merely compound confusion over time and THIS is the reason for student and parent dissatisfaction!!

        Again, I think it would be much more productive if we talk in person. You sound like an intelligent person who has seriously considered this topic, unlike some CPM teachers I have met who just mindlessly spout the CPM party line after attending a company-sponsored training. I have several questions about mathematics practice and spiraling techniques that I would love to discuss with you, and I will contact you by email later today.

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    3. Hi Richard,
      Pardon my being late to the discussion. I got to it after scrolling through a hundred other comments. I see you are trying to convince Mr. Kristofferson (and perhaps were successful via email or phone) of the benefits of CPM. I’m sure you’ve read many other comments on this blog, so you’ve seen that it’s not just Mr. Kristofferson whose mind you’d like to change. It’s parents, teachers, and students all over the country facing problems with CPM. I tried CPM for two years, and it became so monotonous and stressful circulating the room day after day, trying to motivate students who didn’t want to do any work at all, that it took a physical toll on my health. I spend many a night researching CPM online, and found so many people who had similar issues. I vowed after my 2nd year never to touch CPM and just teach it my way, and I have enjoyed teaching ever since. I don’t think anybody would contradict that constructivist learning, and actively engaging with the material, is the best way to teach. I read the Harvard article you sent as well, and have noticed that it is geared toward college learning. High school learning is not lecture style, so I’m not sure what the high school takeaway would be, other than “keep teaching the way you’re teaching, and maybe college professors will start to make changes to be more like a high school classroom.” The issue with CPM is that it takes the fun and the creativity out of teaching. It tries to control all aspects of the classroom except one of the most important pillars – classroom management! CPM tries to convince you that it knows best about everything in your classroom, but when it comes to motivating kids to stay on task, CPM has nothing for you. I even attended a CPM professional development that showed us how we shouldn’t grade kids with number, but symbols that direct them to a rubric with feedback. Now CPM thinks it knows best how to grade my students’ tests! It thinks its method will fit all teachers and students, when in fact, only teachers themselves know what fits best. Lesson planning is, after all, part of what our degrees train us to do.

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  30. Hi David!

    I’m a future Math educator and plan on entering the credential program in Fall 2021. I found this post when I Googled “i don’t like cpm curriculum.” My university and the surrounding school districts focus primarily on CPM. I’ve been feeling really disappointed with it. At first I liked the emphasis on group work, but as I’ve become more familiar with it I’m finding it’s just a series of equations bookended by “a student thinks THIS. what do you think of THIS?” It still paints Math as being boring and academic instead of having students find real life examples of it. I also don’t like how little flavor it has and how prescribed it is. The teachers I have shadowed do exactly as you’ve mentioned.. “open the book to page blah and do 106-110 or whatever.” It makes me feel like I’m going to school to be a glorified babysitter.

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  31. Hi David,

    I found this post after searching for “CPM Educational Program” on Google. After reading through this post and the comments, I have a some remarks.

    It’s clear that there’s a lot of discontent among students, parents, teachers, and other stakeholders over the fact that CPM causes many students to struggle in their math classes. Previous high-achievers struggle, previous strugglers struggle even more. In my view, this is not a failure of the curriculum.

    Math education researcher Manu Kapur (2016) developed a theory of productive failure, reviewing research commonly touted by proponents of both direct instruction (who he calls instructivists) and constructivists. He examined a number of studies that found that students who generate solutions to novel problems prior to being instructed learn better from the instruction than students who receive instruction and practice procedures. “Better” in this sense means that students who struggled with the problems prior to instruction performed better on measures of conceptual understanding and transfer AND that there was no difference in procedural knowledge between the two groups. Kapur calls this kind of struggle “productive failure.” He also provides compelling theoretic arguments for why this should be the case. “Productive failure” activates prior knowledge and leads students to be conscious of its limits, deficiencies, and any misconceptions, priming students for later instruction. Students can compare their naive solutions to correct solutions, making the critical aspects of the new knowledge more apparent and more easily retained. And it also affords greater agency to students, which has positive effects on motivation and engagement (given that students do not have failure-averse mindsets).

    In his own research (which has been replicated), he found that students who received no guidance during this initial period of problem solving outperformed both students who received guidance and students who received only direct instruction on post-instruction tests, even though none of the students were initially able to solve the problems, unlike students in the guided problem solving group. This is consistent with the theoretic framework that he proposes.

    He also gives strong arguments as for why direct instruction should not be considered “productive success” (he gives PBL as an example of productive success). I think it’s safe to say that there are many acknowledged deficits of direct instruction in math, including low retention over time, low transfer, lack of authentic problem solving, and, most worryingly, negative impacts on students’ creativity. There is a large body of research that supports these criticisms, but even without this research, these effects are logical consequences of a pedagogical method that promotes rote memorization of and blind adherence to standard algorithms.

    Here’s the link to Kapur’s article: https://www.tandfonline.com/doi/abs/10.1080/00461520.2016.1155457.
    It’s unfortunately not open-access, but if you don’t have access and would like to read it, feel free to email me.

    I’m very familiar with the work on cognitive load theory that you cited in the AFT article by Clark, Kirschner, and Sweller. Cognitive load theory has a lot of utility in learning design, but it is not without its faults. Ignoring any empirical challenges to CLT’s research base, one of the major issues is that Clark, Kirschner, and Sweller very purposefully use their research as a weapon against constructivist methods, setting up straw-man arguments and false dichotomies that suggest that, since discovery learning is a failure, some version of direct instruction is the only viable pedagogy. They make fundamental, non-empirical assumptions to justify their crusade against constructivism, including that “learning” can and should be measured solely by tests of procedural knowledge. This is an assumption that pervades American education and is propped up by our worship of standardized testing, but I hope we can agree that it is nevertheless an incorrect and counterproductive assumption.

    I think we both agree that math teachers should make use of a variety of traditional and constructivist teaching methods. The method that you describe in this post at “self-discovery” is only one of many constructivist/reform methods, and I agree with your assessment that completely unguided discovery is a highly inefficient way to teach math, especially at higher levels. However, efficiency of information transfer is not the sole metric by which we should judge pedagogical methods. As research has shown, contrary to what Clark, Kirschner, and Sweller argue, complete guidance, as efficient as it is, does not produce deep learning. For many students, it is joyless and intellectually stultifying. Without a variety of behavioral tricks that have nothing to do with math itself, direct instruction, at best, leads to apathy towards math, at worst, results in a hatred of math that is retransmitted to generations of children and students. Students who enjoy classes conducted entirely through direct instruction often do so because they enjoy algorithmic thinking, the feeling of proficiency, and/or have exceptional teachers, not because they actually like math. If we can agree that the goals of math class should include giving the opportunity to all students to participate in the wonder and beauty of math, which entails getting them to ask questions, pose conjectures, construct rigorous arguments, and participate in mathematical discussions, then the sole use of direct instruction fails completely in this regard. Unfortunately, based on my personal experience and based on research, direct instruction is seen as the be-all and end-all by many stakeholders in education. This is reflected in the rigidly traditional practices and negative attitudes towards math seen in so many math classrooms. Reform is necessary.

    My issue with your article is that, for many who read it, it is yet another article that validates their beliefs that “Common Core math,” “New Math,” “reform math,” “CPM math,” or any kind of attempt to reform the mathematics curriculum are abject failures and that we need to return to the way things were post-haste. From a sociopolitical lens, traditional curricula and reform curricula are not on equal footing. If there is “balanced criticism” of each, the default will always be to revert to traditional curricula, since so many stakeholders fear change, and so many stakeholders fear struggle. Of course, curricula are not “traditional” or “reform” but lie on a spectrum, and the pendulum of education policy has swung back and forth between the extremes several times over the past few decades. There’s been some progress, but it is agonizingly slow, to the detriment of students. That’s why I believe that any criticism of new methods should also make it abundantly clear that reverting back to traditional methods, appealing as they are in their familiarity and safety, is not a viable alternative in the long-term. I am not contesting that, in many schools, the implementation of CPM has been/would be disastrous for students’ learning. These are schools that are facing a host of other problems that need to be resolved in order for any kind of reform curriculum to succeed.

    Having read some of your other posts, I think we agree on what a lot of these problems are. There’s too much acceleration and too much emphasis on standardized testing. Math teachers are under-paid, under-appreciated, and often under-prepared (especially at the elementary level). Many predominant (and traditional) beliefs about education in American society, especially about math education, are deeply harmful to children. All of these are much bigger issues than what textbook schools choose to use. Unfortunately, curriculum reform is the most visible aspect of education reform, especially to parents, which is why I think it occurs so frequently, attracts so much attention, and stirs up so much controversy. Additionally, reform curricula often require teachers to have deeper and more flexible knowledge about math than traditional curricula do, exacerbating existing problems of teacher quality.

    I wish this post had taken the opportunity to explain a more nuanced view of the CPM program’s pros and cons in relation to the broader landscape of math education reform. All too often, these kinds of posts are read outside of this context. Parents seek to validate their view that the new curriculum is at fault for their children’s struggles, and they don’t see the problems with the status quo. Traditionalists revel in another victory over the reformers. I’m not advocating for everyone to forego criticism of reforms, be more explicitly optimistic about their potential, and to promote a vision of math education that is better than what we have now.

    I can empathize with your concern about reforms using students as “guinea pigs.” Top-down reforms imposed by ignorant, overzealous policymakers being directed by greedy, shameless education companies have wrought havoc on generations of students in the United States. However, CPM is an organization that was founded by teachers. Their materials are reasonably priced, they provide free professional development, and I believe that they have genuinely good intentions. They produce materials that, in their experiences and in the experiences of many students, parents and teachers, are more effective than traditional textbooks. Their curriculum is based on beliefs about math education that I respect and believe. I don’t hold them at fault for the fact that their curriculum is used in hasty, half-baked attempts at reform that disrupt students’ math learning, restrict and confound teachers, and generate unmerited animosity towards reform efforts. But by promoting awareness of the broader context of math education reform, emphasizing the failures of the current system, and getting more teachers and parents invested in the work of reform, communities can create reforms that work for them, just like CPM did. That’s why I respect the work that your blog does. But, I worry that many people find this post (and other criticisms of reforms) through Google, don’t get any of the nuance, feel validated in their belief that “reform=bad,” and never read any of your other posts. I don’t fault you for their actions, but I think that this can be avoided by emphasizing the harm of solely using direct instruction.

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    1. Justin, I read your eloquent, detailed response carefully. Thank you for taking the time to write such a great critique! I think you and I are probably in pretty close agreement on many points, but the “devil is always in the details.” I will make only a quick comment here, but then would like to talk with you by phone to discuss this at greater length.

      My article title contains the words “Pros and Cons.” I tried to make it clear that there are aspects of CPM that I like, but I agree that people who are pissed at CPM might gloss over that part.

      The way that CPM is often implemented in practice is a part of the problem, but I think that their implementation of spiraling is another part. I have been meaning for some time to go through their “Research Base” and write a detailed critique, but this is an extremely time-consuming task that I just have not been able to undertake given my numerous other responsibilities (particularly right now in the middle of the COVID-19 pandemic – I recently dropped ten pounds in four days due to the stress of getting my wife’s business set up for remote access. I also do computer systems administration work in my “retirement”).

      I know that CPM is founded by teachers, is reasonably priced, as you mention, and that is part of the reason it is popular among teachers.

      I also agree, to use simpler language, that just talking at students all day long does not lead to quality learning. However, neither do the great majority of students learn if they are left completely to themselves with only a book of questions. Both of these positions are extremes.

      The problem with educational “factions,” just like political parties, is that they tend to take “evidence” (that few members of the faction ever actually read!! “Research says … “ once again) and use it to justify why their side is completely right and why the other side is foolish.

      There are also fundamental philosophical issues about the lack of good controls in educational research, so, as I have written elsewhere, I am immediately suspicious of results that do not correlate with “common” sense, but also realize the dangers inherent in that attitude.

      Hopefully mature people understand that the truth is never a monopoly.

      I look forward to discussing this with you some time soon!!!

      Like

  32. I read your article after googling CPM reviews. I have found myself teaching at home in recent weeks due to the pandemic and I am appalled at the CPM curriculum model and the lack of instruction my child has been given thus far in his education. Furthermore, it has proven to be an impossible program to use at home without the everyday instruction from the teacher that we need. I have no credentials that would qualify me to teach math at home but it is already clear to me that there are distinct holes in my child’s math education so I am switching him to a program that I feel will work at home and fill in the areas that have been lost through CPM. He is currently in pre-algebra so, for now, we will be using the Saxon Pre-Algebra program as it is easy to teach from home and I have access to online lessons that I can sit through with him but Saxon also implements a spiraling method for homework that could potentially be frustrating. I’d love to know of any recommendations you have as so many of us have suddenly been thrust into educating our kids at home when we had previously assumed our children were receiving a stellar education in mathematics at their schools.

    Like

    1. Hi Emily, sorry for the delay getting back to you.

      As I do not know your personal math background, I am going to make a pre-algebra recommendation that I think would be suitable for most interested parents who now have to teach their children alone.

      This recommendation is based on using the California Math Standards that pre-dated Common Core and will undoubtedly draw the ire of the numerous “drill and kill” opponents. This is because this textbook series is almost mechanically tied to the CA standards and *provides numerous worked examples* which I am sure that parents will find extremely helpful if their math skills are rusty. However, many teachers might claim that this series makes math just a stream of uninspiring drills.

      My background is in science, and I always found my interest in math derived from its applications to science.

      Professional mathematicians generally go into the subject because they are enthralled by the logical beauty and unexpected interconnections between different parts of the field. They sometimes tend to look down their noses at applications, but I believe that most students generally ask the question “why am I studying this and what is this good for?”

      The answer is that a tremendous amount of math was developed to solve problems in commerce/finance, astronomy, and physics, but the mathematicians who write textbooks (and even more so, the K12 teachers who teach the subject) aren’t always conversant with the interesting human stories behind these developments, and thus far too many students never get satisfactory motivational answers.

      The series that I will recommend is from McDougell Littell which is a division of Houghton Mifflin, and the lead author on these books is Ron Larson. This series includes books in Algebra 1/2 and Geometry as well.

      I have not used the Pre-Algebra book as I generally teach AP classes, precalculus, and less frequently, algebra and geometry, but here is a link to the 2005 edition on Amazon:

      https://www.amazon.com/McDougal-Littell-Pre-Algebra-Student-2005/dp/0618250034/ref=sr_1_1?dchild=1&keywords=mcdougal+littell+pre-algebra&qid=1585758202&sr=8-1

      There is also a 2008 edition which is probably very similar, but there is only one left in stock and it is slightly more expensive. Odds are that either book will work fine.

      Many home schoolers like the Dolciani series (lead author Mary P. Dolciani) which was used in many states across the U.S. when I was in high school in the late 60s. Personally, I think that your student would likely be a bit put off by the “ancient” format of these books though. In addition, Dolciani made use of set theory and other mathematical formalisms that I think math purists might love, but, IMHO, might make life slightly more difficult for parents struggling to keep kids focussed under our current extraordinary conditions. If you are interested, you can also find Dolciani books from used booksellers on Amazon.

      I think the McDougal Littell series would be good to try first though.

      Like

  33. Now that I’ve had more experience teaching the CPM curriculum, there’s one other difficult aspect to the program I wanted to mention, one that I didn’t see raised in the article. The problem-based exposition of material in CPM is very text heavy, and the close reading skills required simply to get through all the descriptive stories accompanying the problems are not as prevalent among high school students as they once were.

    Some would cite this as a strength, pushing students to read in a manner that they may be unaccustomed to, but it is most certainly too much of a good thing, if we can even call it that. The passages are given such elaborate context that it detracts from the underlying mathematical point, and for all their wordiness, they often include mathematical ambiguities that allow divergent lines of thinking to extract very different takeaways from each problem or activity. Again, this is praised by CPM’s proponents as a feature rather than a bug, but in my opinion, too much of this means there are not enough precise and common skills being practiced and honed.

    More to the point, I see many students glaze over when reading problems that are worded in long paragraph passages rather than two or three simple sentences. When I probe them to assess how they’re approaching a given discovery exercise, more often than not I’d find they haven’t had the patience to read the entire passage. At best, they have completely missed some important details, and many times I’d find groups that were completely lost in a story problem. In these cases, there is so much story to process that even the act of reading it to them verbally doesn’t help.

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    1. Interesting point, Mike. Thanks for your comment! I believe, and am sure you agree, that solving “word problems” is an extremely important skill for students to develop as that is obviously how all real mathematical applications initially present themselves to people. However, your observation about students getting lost in the details is one that I saw too.

      My response was to spend extra time on word problems with the consequence of falling behind where my colleagues teaching the same class were in the curriculum. But when the CA standardized test results came out, my students scored higher on word problem solving. It came at that cost of covering less material though, but I continue to think that each teacher should be free to make that decision for their classes. Being trained as a scientist originally, I thought the extra time on word problems was worth the effort. However, if the written presentation in the text obscures learning required math skills, that is clearly a defect.

      Like

  34. Hi David,

    I am a newly admitted math Ph.D. student at a university in California, who also has a strong interest in math education. First, let me tell you: thank you so much for putting a detailed, strong criticism of this messed up CPM curricula. For reasons, I have had a huge amount of free time, during which I discovered that there’s an underbelly of reform movements that have taken over school districts. Here are some things I discovered that would serve as a useful background for you.

    CPM and IMP (Interactive Math Programs) started out in the early 1990s I believe, and were designated as “exemplary” by the California state government at that time. These “exemplary” curricula were criticized by mathematicians as “watered down” and even “defective,” to the point that 5 mathematicians wrote an open letter to secretary Riley to not use the word “exemplary” or “promising” to describe these curricula – an open letter that garnered 200 signatures from mathematicians and educators.

    For more information, see
    Criticisms of Reform Math Curricula
    http://www.csun.edu/~vcmth00m/usnoadd.html

    Open Letter to Secretary Riley
    http://www.csun.edu/~vcmth00m/lausd.html

    This watered-down CPM curriculum was – to my knowledge – were dead until various “reformers” like Jo Boaler and Alan Schoenfeld revived them in San Francisco (in fact, on Quora posts, a self-professed CPM director endorsed Jo Boaler, saying “viva la revolution,” a phrase Boaler uses frequently). They rile on about how their approach to teaching emphasizes “conceptual understanding,” “creativity,” and have problems that are of “low-floor, high-ceiling”; however, when I took a closer look at Boaler’s website youcubed.org, I could tell that they fail on their own terms. I didn’t read CPM or IMP, but similar criticisms were made by a mathematician Hung-Hsi Wu in 1997 of IMP https://www.researchgate.net/publication/2466737_Review_of_the_Interactive_Mathematics_Program_IMP.

    This brings me to my next point – I think you’re giving CPM more credit than it deserves. There are a couple things in your post I disagree with. First, you briefly mention that many teachers find that students are more engaged in CPM than others, and cite that as a good thing, but engagement in an activity is good only when it is the thing you want students to engage in. As an example, (see first link) CPM would waste time asking students to see a candle burn, plot the “time vs length graph” and only THEN, they use linear equations to model the burning. This is a waste of time that could be better used to learn more in depth about graphing linear equations (and understanding their properties). So, engagement is only useful when it forces students to think about deep concepts.

    Secondly, there’s a brief mention of how “in the real world” people work in groups, but that’s also misguided. Math classes, by virtue of being math classes, should prepare students for higher level math and not “real world.” I think at least in math, collaboration is the most fruitful when each party has had a significant time to think about the problems at hand before sharing ideas (and when each party is given enough autonomy to follow through with shared ideas). So even by this standard, CPM seems to be a massive failure, which brings me to my third point. I don’t quite agree with you that CPM’s emphasis on “group” work is only problematic for lower-achieving students. From what I gather (and again, I’m taking Dr. David Klein’s word for it), CPM content is kind of watered down, and high-achieving or talented individuals won’t benefit from a watered-down content. What they need, is a hard and abstract curriculum that emphasizes problem-solving and systematic development.

    Anyways, I think your criticisms of CPM is spot on, especially your point about lack of practice in CPM books and shoddy research (often with financial conflict of interest) backing CPM. Also, I thank you for writing out this rather long, but strong criticism – this helped me in gathering more information about these watered-down “reform” curricula. Since you’re curious, I found this post by searching “CPM criticism” or “College Preparatory Math criticism” or something along the lines (and interestingly enough, you can find this post by searching “criticisms of math textbooks”).

    John S

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    1. John, thanks for your response here and also to my other article about 10^0 = 1. You caught me right in the middle of a big controversy about school reopening as indicated in the two most recent articles on my blog, so I only had time last night to approve your comments, but not respond. I still am swamped this AM, but will respond at more length this weekend.

      For now, please let me mention that I am aware of the Boaler controversy and think it was discussed briefly somewhere in the now extensive list of Comments above. I will respond to your other points this weekend. Thanks again!!

      Like

      1. Hi David,

        Yeah, no worries about responding. Take your time -really! Technically, you don’t owe me any response what so ever, and certainly, not by a “specified time.” That said, I’ll be happy to see what you’ll have to say.

        As for Jo Boaler, I wasn’t sure how much you knew about her (I did see a glimpse of her mention as I scrolled through the looooong comment section haha. I guess I wasn’t sure if you knew about these “reform” math controversies, though it seems like you know about it a fair bit from your other posts.

        John

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      2. John, I usually try to reply to all posts, time permitting. That is one of the perks of being retired. Most readers do not comment or comment on Nextdoor. Even though the Comment section on this article seems long, the article has been viewed almost 15,000 times since it went online.

        Like

    2. Hi John,

      As I said briefly earlier there was a brief mention of Boaler in the admittedly long comments section above and I also wrote the following article https://eduissues.com/2017/10/09/here-we-go-again in response to media coverage of her. I first learned about the controversy over her CPM results either from visiting her website where she had a prominent section defending herself from attacks, or possibly from discussions on Nextdoor. I announce my articles locally there and a lot of my local readers also post comments there where they have ready accounts instead of coming here to post unfortunately. Occasionally I will excerpt parts of that discussion ad also put it here. Nextdoor unfortunately doesn’t even cover our entire local school district which is why I started this blog, but there are many more Nextdoor users in our area than people with WordPress accounts unfortunately.

      I agree with your candle/engagement remark above However, the teachers that made the engagement comment were talking about how their students were actively discussing and working together on the CPM problems in the textbook versus just sitting passively listening to a lecture in an “old-school” classroom. I do agree with them that active problem solving rather than passive listening is better.

      I agree that CPM’s view of how collaboration works in the “real world” is misinformed and your description is better. As for Klein’s comment, it is too general and vague for me to respond. My experience was that the CPM curriculum seems suitable for algebra I and perhaps geometry, but by the time that one reached the precalculus level, the program broke down and could not successfully handle the required level of complexity using self-discovery in the time allotted.

      Like

      1. Hi David,

        I should probably respond by now. Looking back at my comment now (and also your original post), I realize I haven’t really articulated my larger point properly. Admittedly, I don’t even own a copy of any of CPM books, so really I’m in no position to voice my opinions. That said, I do stand by much of what I said (especially about Boaler, and I think you agree too).

        So if I understand you correctly, you seem to think that content-wise, the CPM curriculum is good up to certain level (i.e. before precalculus) and that the guiding questions and activities really do get at mathematical concepts, but the pedagogy is unsound (mainly because it lacks explanations, worked examples and practice, and focuses too much on group work, which might not work on “math-phobic” people). I’m more than willing to take your word for it that CPM fails in pedagogy, but I’m extremely skeptical that CPM is solid on contents either. I’ve ended up spending several weeks looking at Boaler’s website youcubed.org, and her “rich, creative, low-floor high-ceiling” activities, and I can tell you that what Boaler touts as problem-solving, “low-floor high-ceiling” activities are really disconnected activities with little mathematics (and I’m more than willing to tell you all the ways in which Boaler fails in her own words, but it’ll be too long). And I’m more inclined to believe mathematicians such as Dr. Klein, Dr. Wayne Bishop, etc. who claim that much of reform math (including CPM), like Boaler’s work, is watered-down, or even defective curricula (see http://mathwise.net/?p=1262 for Dr. Wayne Bishop’s critique of CPM). In fact, one of your guest post writer agrees.

        Oh, btw, the open letter was from a wrong link. Here’s the correct link: http://mathwise.net/?p=1248

        My comments about candle problems and engagement is an extension of this. If you have students really engaged in cool activities, but have little mathematical content, I think that does more harm than good, for they are *distracted* by the cool, but non-mathematical activity (in fact, Bishop calls it “math avoidance activities”). This was lightly touched upon by another commenter Mike, but in pointing out how the long text obscures mathematical relationships. Similarly, I believe CPM curriculum just doesn’t serve mathematically talented students very well (and much of it could also be said of the “traditional” curricula).

        As you alluded, I think it’s not easy to convince one way or another without specific set of examples (and that is, if I’m even qualified to disagree with you :)), but for now, I’ll have to take mathematicians’ words. We can agree to disagree. If I ever get my hands on these books (and heck no, I ain’t paying CPM scammers hundreds of bucks; maybe used books?), I’ll be happy to go through the specifics in a suitable medium and share it with you.

        Like

      2. To clarify one point out of many above. CPM purposely lacks explanations and omits worked examples from the main textbook because of their self-discovery philosophy. When I say it works for Algebra I, and possibly geometry, it is because, particularly in the case of Algebra I, it is not so hard to figure out the concepts presented in the available time. At higher math levels, self-discovery is more problematic and that is where I believe worked examples and explanations are more beneficial than CPM’s approach.

        Also remember that when teachers praise the program it is based on its use with the bulk of their students. You mention the mathematically talented students in your comments above. In the CPM program they usually end up teaching (in the best cases) or doing the work for (in the worst cases) the other students in their groups.

        Finally CPM is a non-profit organization and their paperback books are reasonably priced compared to many other publishers. This is another reason that teachers like CPM.

        As to the Boaler controversy I have too many other things that I am working on currently to wade into that one as well. I expressed my brief opinion in the article “Here We Go Again…,” but will admit that I am a Ph.D. scientist, not a mathematician.

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      3. Without CPM materials at hand, I will have to concede.

        As for mathematically talented students, you and I probably disagree, but I think talented students should be given stronger, more challenging material that goes beyond school math. Admittedly, being a gifted student myself, and someone who has TAed for multiple courses (both in person and online) designed for gifted people, I’m a bit biased (Btw, I TA at Art of Problem Solving courses; you should check them out – they’re great!). Many of them thrive at a challenging environment filled with like-minded peers, and they really should not be in a position to constantly “teach others” (even if teaching reinforces material). Understandably, gifted programs do pose an “equity issue,” but I think that should be solved by self-selection and potentially a diagnostic testing/counselling. Lack of support for gifted students is an issue both with the traditional curriculum and CPM.

        I’m still not willing to pay CPM organizations; CPM director has said some really unprofessional things to others, and they’ve lost all of my sympathies. I think you’re at least aware of some unprofessional behaviors, so I’ll just end it at that.

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      4. Sadly, in many cases, high school
        is something that talented students need to survive until they get to an environment where they can hopefully thrive. Be thankful for every good teacher that you encounter on the journey!

        Like

    3. Hi John,

      I enjoyed reading your comment. Like you, I’m quite aghast at much of the reform curriculum agenda, though I try to look at both sides of it as much as possible. The “math wars” of the 1990s that you’ve alluded to were indeed detrimental, and they have taken new life in the Common Core Standards, which now have an incarnation of the NCTM Standards baked right into them. I have great respect for the work of David Klein, Barry Garelick, and others, and I do generally feel the reformers have misunderstood the strengths of the traditional approach, replacing them with unproven alternatives and wishful thinking on a grand scale.

      Perhaps you’ve seen my guest comment on this blog, criticizing the Common Core Standards. I do recognize the benefits and reasoning behind CPM and other such “exemplary” curricula, but as a whole, I find them to be largely overstated. More to the point, for people like you, me, and the author of this blog, who pursued STEM careers and learned math accordingly, the reform coverage of math is entirely inadequate. The fact of the matter is, without additional tutoring or other supplementation, this current crop of students will not be prepared for top-level work in these fields.

      Like

      1. A lot of CPM-oriented school teachers might respond that they are more concerned with teaching the bulk of students who become turned off by math taught in the “traditional” manner and are trying to increase engagement using the program. My bet is that your final sentence regarding doing “top-level work in these fields” is not a frequent concern at the K-8 and possibly even the K-12 levels.

        Like

      2. Hi Pareto Educator,

        I appreciate that you’re trying to look at “both sides” but actually, I think there shouldn’t even be “sides” at all. And make no mistake – I’m not really happy with the “traditional curriculum” as I understand it (and I don’t know if you are… I just thought I should point that out). I went to a high school in NJ, where arguably, they used “traditional methods” and much of the material was not challenging at all, and often repetitive, with little regards to mathematical problem-solving and proofs. And that was in a supposedly “gifted” classes (actually, my district didn’t call it that – they called it “honors and accelerated”). In some regular or even “honors” classes, it was worse: in a “proof” based geometry class, they would force students to memorize definitions in exact way, and in an AP Stat class, we would be asked to assume provable facts about various distributions without proofs or any explanations. Moreover, it was clear that teachers weren’t knowledgeable beyond what was required to teach, which impacted their teaching. Remember – this was in a relatively wealthy neighborhood, so I can only imagine what people in a poorer neighborhoods go through.

        I don’t think I’m alone in this experience, too. At my alma-mater CMU, there is a course called “Concepts of Math” (or honors level “Mathematial Concepts and Proofs”) that effectively introduces proofs and problem-solving to freshmen. As someone who tutored for that course for 3 years, I can tell that many of them have not had much problem-solving and proofs either, even though, arguably they should have (it’s almost like having no lab experience in science). So the course is a huge jump for them, and many of the struggle (and I presume most would have had “traditional” math classes before then).

        So the issue seems to be more in the lack of deeper math problem-solving based knowledge of teachers than say, reform vs traditional curricula (although, I agree that reform curricular do far more damage). Regarding your point on Common Core, some, including mathematically literate people, have said that Common Core itself is a positive move in a right direction, but it’s just that teachers, textbook companies, and educators aren’t ready to actually teach to the standards. You seem to think that the standards itself is at least partly responsible for the problems – I’m not educated enough to comment on that.

        Any reform must address the issue of teacher education and professional development. I do have a great respect for Dr. Klein and others, but not so much for advocacy for traditional curricula (though I don’t know if he would favor what my classes were like), but for rebuttal of many dangerous claims made by “reformers.”

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  35. Hi John,

    Just responding to your reply. I hear you about teacher education and professional development, but that’s a really tough ask. Here’s an essay written by the late Ralph Raimi of the University of Rochester on the topic, based on a teacher PD effort he conducted back in 1960. It’s pretty grim, and the situation has only worsened since then. https://web.math.rochester.edu/people/faculty/rarm/institute_60.html

    I mention the math wars because David Klein was a major player in them, and all of his references that you’ve cited date to that particular era. I’m also thinking about the shortcomings of “traditional curriculum” as you’ve described it, and I suspect that we’re not using the same definition for that. After the New Math became widely viewed as a failure, there was a back to basics movement in the early ’80s that went overboard on traditionalism. The frustrations you’ve described in your own experience (so sorry to hear about that) strike me as falling under an overemphasis on “back to basics” rather than the best practices of a balanced, traditional approach. (Which everything you wrote in your first comment struck me as being in strong support.) It’s a pretty heated topic these days, with a lot of debate over definitions.

    As for Common Core, in many ways it is a wolf disguised in sheep’s clothing, and that sheep’s clothing is indeed very nice. In a large number of states, much of what’s included in those standards is better than the previous standards that existed before. However, the State of California is one grand exception to that general rule, thanks the great work in the late ’90s that you cited in your references, and the standards that emerged from them.

    This is a long and involved topic of conversation that I’d like to continue, for sure.

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  36. Thanks David, I have no doubt that CPM-oriented teachers feel that way. I have heard it from them directly, and it is a valid argument. But with the tracking of students according to ability effectively ended, where does that leave the STEM-talented student who can’t afford private tutoring or expensive extracurriculars?

    I also wanted to point out that the entire CPM curriculum is now available for free and open use until the end of June, to help aid the COVID-19 shelter in place, so John and others who haven’t actually seen it may have a free peek. Here is the link: https://open-ebooks.cpm.org. In recent months, I’ve been doing more work with middle schoolers, and I have to say, similar to what David indicated in the article, I’ve found more appropriateness in the CPM materials at these middle school levels than I saw at the high school levels. A good portion of the discovery explorations are very well-designed for that purpose.

    So CPM does serve a certain purpose, and it serves that purpose quite well. That said, I agree with John that, for the most talented of students, even if they follow, enjoy, and engage in CPM throughout several years of instruction, they will still find themselves inadequately prepared as a result of formative time and effort that was wasted for them. All so that they can pay thousands of extra dollars in college tuition to build skill that used to be imparted upon them for free in K-12.

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    1. Actually I agree with both you and John about the fate of the most talented students under CPM. In fact, my parental quote at the very beginning of the original article was an example of exactly the problem of which you write. Public school teachers have to minister to the needs of many different groups (which is the reason for the constant call for class size reductions), and consequently tend to take measures that will benefit the greatest number. This is reasonable and is also the main tenet of “utilitarian” philosophy.

      There are teachers, thankfully, who will supplement their course material to keep their best students engaged, but this is not always the case. I made frequent mention in the article about the need to supplement the CPM textbook and, sadly, this does not always happen even for the larger groups of students in the middle.

      Thanks for calling the free CPM offer to our attention. I hope John finds it useful, but he will also need to experience the curriculum in an actual classroom to get the “full flavor.”

      Finally, I just reread the entire original article this morning. Since it was published going on three years ago, I have read every one of the comments in this long section and have responded to almost all. I continue to think that the article covered all of the important points about CPM, but, as a result, it was rather long. I found rereading the entire article valuable, just like when one views a movie a second or third time, one usually notices items that one missed the very first time through.

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      1. Wow, I didn’t know there was this discussion going on :). Thanks to Pareto Educator for the link – that helped me tremendously.

        Immediately, from my peek at 1.1.1 of Algebra lessons, I actually agree with pretty much everything you (David) said, including how the guiding questions are actually somewhat cleverly designed. They’re essentially introducing function composition without the use of the term. Whether they’ll actually follow through with it, I’ll have to check.

        Actually, reading through 1.1.2-1.2.x, I’m starting to have some doubts about the content, but for now, I can’t tell if it is the content or the pedagogy. I’ll have to get back to you once I have read a bit more thoroughly.

        I also notice the lack of practice, and “spiralling” that’s going on, which is really annoying. And given practice and conceptual understanding can often go in hand-in hand (and I firmly believe that’s how much of, but not all of, practice problems should be designed – to illustrate larger points, and to allow conceptual understanding to be applied in clever ways), that’s really troubling. This is spaced repetition theory gone wrong.

        Honestly, thank you again for these insights. I should mention, the reason why I ended up in this blog post in the first place is that I was collecting every information I can about Boaler’s work (because of concerns of harms she could/have done), and I found out that Boaler’s ideas have pretty much entirely been adopted there, which is really troubling. And there, I found that the curricula there borrows heavily from CPM! (with some from Boaler’s own activities and IMP as well) So the discussion isn’t complete without understanding CPM and its impacts on the classroom.

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      2. John, “Pareto Educator” suggested that a three way email exchange might be easier at this point if you want to continue the discussion, since the comments section on this article is overly long.

        If interested, please use the Contact form on my blog to send me your email address and I will set this up.

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  37. Thank you for writing about this!
    I had a lot of support and small class sizes when I first started teaching CPM, so I was able to handle low-skills kids and learned when to step in, draw the whole class together and do direct instruction. In trainings, they now teach you to do that. I agree with your critiques – they are all valid (esp. not enough problems to gain skill fluency). A critique I would add is that when you have a large new-to-the-country ELL population, the high level of text makes the math totally inaccessible, whereas a non-wordy textbook is far more doable. I think the habits of mind, critical thinking skills, ability to work in a group are invaluable skills, and I continue to stand by CPM. It does need to be supplemented by extra worksheets, review lessons, etc., and is totally inappropriate for students with low English literacy. I did have students with poor reading skills but taught them all kinds of reading strategies and even grammar, so their skills improved.

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    1. Thanks for your comment, Harmony! What CPM math classes did you teach? I am glad to hear that they are encouraging direct instruction in the cases that you mention above. I’ve taught ELL students before, but not in science, not math, which is why I didn’t think about this issue. You make another excellent point.

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  38. I read your article. Accidentally came a crossed it. Read though some of the comments as well. We adopted CPM in my district last year. We are on year two. We did this because traditional books where not working in the district. (We are minority majority school with 95% students on free and reduced lunch.) I made a huge effort before CPM to merge our text book and Carnegie. I was having a lot of success in my room. Carnegie is also like CPM. Our district used Springboard for a few years also. So I have looked at a lot of problem based curriculum and have tried to teach it. I teach Secondary Math 3 (Algebra 2 with pre-calculus concepts). The grade level below me makes an effort to supplement CPM with drill and kill worksheets. I think what CPM has posted in the Parent Guide and Checkpoints provide plenty of supplemental problems for skill based development. I believe that these problems are well picked. Not randomly being generated with Kuta software to give a worksheet and so many problems.

    We can argue about gifted students and correct curriculum for them all day.

    I do feel that CPM develops and strengths core math skills. If these core skills are developed students should be able to have success in any math class.

    At the grade level that I teach, some things have to be direct instructed. It is a learning process of where that needs to happen in the CPM curriculum. I direct instruct all math notes so my students have worked examples.

    I find the issue that I have with CPM is time with the instructor. It is definitely designed and works best as a daily curriculum. The review and preview problems need to be given with time in the room so the instructor can help students who need help. I do think that Review and Preview helps student develop the ability to be autonomous self learners outside of the classroom. Students need that skills set. The corona-virus shut down last spring painfully showed that students don’t have the skill in spades.

    The next thing is that CPM is not for lazy teachers. You have to move around your room. You have to be prepared to teach. You have to understand what the goal of the problems are. You have to interact with your students and guide them to the learning goal for the day. You can’t work problems and then have students copy what you did on the board.

    So in reading your article, some of the criticism is well placed. In upper level math at a high school, it is hard to have discovery lessons. The professional development for teacher in these concepts is very week and light.

    If everyone could teach math, then we would not have a math teacher shortage. You need a curriculum to support teachers. They can’t do it with out it. All of your arguments could be said about any discovery based curriculum (CPM, Carnegie, Springboard, Mathematics Vision Project, Engage NY).

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  39. This is my 33rd year of teach math. I’ve decided that it doesn’t really matter what curriculum you use, it’s the individual teacher that has more of an impact on learning outcomes than what book you use. In my longest job (17 years) we changed curriculum at least 4 times with the same results. What did matter was the learning habits of my students. You can watch a class work for a month and tell who is going to pass and who isn’t. If you can convince your students to adopt more effective ways to approach their education, you can change outcomes.

    As for CPM, I mostly agree with you. It delivers challenging and thoughtful problems, but unless you can train your students to take organized and thorough notes, it’s easy to lose them. Parents almost always hate CPM for the same reason some student do, their is no effective way to review from the books. It’s also torture for people who don’t like to read or don’t speak English as a first language.

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  40. Thank you so much for articulating the concerns I have had with this curriculum. I am a 20+ year teaching veteran with a history of excellent results, in both remedial and advanced classes. My small private school merged with another larger school, and we were “bullied,” if I may, into using their math curriculum – CPM. I am a more experienced teacher than any of the teachers at this new school, but my views on the curriculum were disregarded. (Clearly, I’m a little salty about this.) We are now five years into this merger, and I’ve successfully argued myself out of using these books personally. However, the fact that other classes continue to use these ineffective books has many ramifications for my classes. 1. Less than 10% of the student body attempts Calculus. One of my top kids in 9th grade geometry did NOT sign up for Calc this year; I asked him why, and his answer was “I haven’t learned anything since 9th grade.” This is massively sad. 2. The students I do get in AP Calculus are _woefully_ unprepared. Every. Single. Concept. we attempt in class needs to begin with me giving some direct teaching on what the topic is to begin with – from rational functions to logarithms to nonlinear inequalities to even graphing basic polynomials. We move SO slowly because I’m basically reteaching all of Algebra II and Precalc while trying to get to Calculus. 3. Students flock to my “remedial” geometry class instead of the regular class because we don’t use CPM. My roster is overflowing, which is actually a disservice to the students who need the remedial class. It’s become an escape from CPM.

    I do appreciate this article, and the data you link, to support what I have noticed colloquially. I hope that one day, we will move away from this curriculum. If not, it may be the nail in the coffin for my career at this school. It is simply too hard to teach hindered by CPM, even if it’s not in my own classroom.

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    1. Rachel, thank you very much for posting your observations. I hope teachers everywhere who are using this curriculum are objectively evaluating their results and not just “going along” with the inertia of the “party line” in their school/district.
      I thought that the CPM curriculum wasn’t bad for Algebra 1 but became increasingly problematic as one increased in complexity beyond that class. Sadly these issues are never explored in an unbiased systematic fashion, so we all have to make our decisions based on our personal observations.
      As I mentioned in the article, the CPM organization’s “research base” is woefully inadequate, and, in any event, these issues should be studied by an independent education organization, not the publisher (nor by those predisposed to push a particular teaching philosophy).

      In the meantime kids are caught in the middle of these teaching battles.

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  41. Love the article…but be sure to read this whole reply! Thank you for the words and research to say what my gut feels. Actually, even more powerful than my gut is what my students say (Algebra 2 and PreCalc)…they beg for worksheets on the topics that we cover in our CPM book. They recognize they don’t know it well enough yet after the few problems they do in their homework. And for this reason, I actually really like CPM. 🙂 I do. I don’t buy the party line that we should just keep moving forward and the kids will understand it better as it spirals back in their homework. (But I do love the spiral though.) We do some lessons from the book and some homework from the book, and we also do some direct instruction (for both efficiency and understanding) with repeated practice kind of work on important topics. I use the CPM book and some of the discovery based problems as a primary resource…what they get the kids to do and discuss and struggle with is really good stuff, but they’re not the only resource. Isn’t that true of all textbooks…they are resources, not the curriculum. I appreciate that you highlight the pros and cons of the curriculum. If a teacher is aware of those things and supplements as needed, I do believe the curriculum can still be used to teach quality mathematics. (But don’t let the CPM people see that I just said to supplement…that’s kind of an s-word. 🙂 )

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    1. Brad, thanks for your comments. I agree that CPM is not bad for the lower levels, possibly up through Algebra 2, but needs additional supplemental practice problems. I also think that teachers using more traditional textbooks which contain lots of practice problems can design similar thought provoking in-class lessons over the course of their career; for example, they can use the CPM book as a lesson planning guide instead of making it the main text that is given to the students.

      There is no doubt that CPM does this lesson plan work for new teachers, but it also can become a crutch and make teaching a less creative craft.

      As I mentioned in the article, though, my experience with the CPM precalculus program was that it not only required a lot of supplementation but that the self-discovery method loses it effectiveness due to the excessive time it takes as one advances to more complicated material. When our local high school experimented with the CPM precalculus book that entire year of students took a substantial learning hit before the book was dropped at year end. Your experience may be different, of course.

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  42. I have taught math for 30 years now. I used CPM one year for my 7th grade class and at the end of the year they stated that they had learned virtually nothing. My district two years later adopted it for all 6,7,and 8 graders. Because I teach in an alternative program, I have not had to use it. It is poorly written and has so little in accountability that it is extremely difficult to assess which students are understanding and which are not. Even teachers I have talked to who love it, admit they have to supplement tons in order to make it work.

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  43. As a parent of Bay Area high school triplets (one on an IEP for OHI, ADHD, slow processing speed) I am struggling to find explicit instruction (per my one son’s IEP accommodations) to help my son learn this curriculum. All three are struggling in math. The group work strategy is detrimental and I can’t imagine how any District/school could justify teaching students with learning disabilities using this teaching strategy. My son is in a class with two teachers and advises that the main teacher is inconsistent in circulating and asking questions. I’ve also learned that the support teacher spends most of his time with one special ed student who is lower functioning. I am trying to find and connect with resources (special educators who teach high functioning students) who can point me to resources that will help my son. I’m also wondering if anyone has successfully challenged this teaching strategy (group work) as a denial of FAPE? I truly appreciate you sharing your perspective.

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    1. Thanks for your comment, Gloria. Although this article which was published a few years ago still gets a lot of traffic, discussion has trailed off substantially. I hope someone reads to the end of the long comment section and can answer your question. It saddens me constantly that obtaining adequate education for one’s children can be such a struggle.

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  44. Thanks for this forum! I found it by searching Google for Integrated Math Pros and Cons.

    Four questions are on my mind:

    1) Do schools of education prepare future teachers to teach Integrated Math?

    2) By bringing more subject areas into the mix of what is taught in math class, doesn’t this necessarily mean each is covered in less depth? In other words, isn’t taking Integrated Math more like taking a survey course?

    3) If Integrated Math is the best approach to teaching mathematics, I wonder if colleges and universities have transitioned all their math courses to that model?

    4)Before we can say if a process is successful, we have to define what success will look like. So, before we can say whether Integrated Math is successful, it seems like we would have to state its goals, which leads me to the broader question – what are the goals of math education?

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    1. Thanks for your comment, David.

      In response to 1), I highly doubt it. Education classes usually don’t go into that level of subject matter detail.

      2) Not necessarily. For example, instead of taking Algebra 1 and Geometry as separate courses, there is no reason why some of the topics could be intermingled and simply taught in a different order over the same time period. I’ve been told that this is common in European schools. It only becomes a “survey” if topics are omitted or covered more quickly than in the traditional separated classes.

      3) I think this is a pedagogical approach restricted to high school.

      4) That is too big/philosophical of a question for me to tackle this evening 😉!!

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      1. Dr. Kristofferson, thank you so much for taking the time to talk with me about this. I’d like to address 2 of the points you make.

        First, to restrict this pedagogical approach to high school students comes across as hypocritical. We’ll put students through it in high school, but then conveniently forget about it when they go off to college or tech school, where we REALLY want them to learn the math. For this real learning, we return to the traditional, rigorous instruction in semester-long courses in Algebra, in Geometry, in Statistics and Probability, in Calculus. But why? If Integrated Math is the ideal instructional model for a 17-year-old high school senior, why is it not equally ideal for an 18-year-old college freshman? And then by extension, why would it not be ideal right on up the line to college seniors and graduate students? Is it ideal, or isn’t it?

        Secondly, regarding teacher education programs not preparing future teachers to teach integrated math: if the ed. schools don’t address it, they must be assuming the future teachers in their programs know, in addition to the mathematics, all the peripheral subject matter being drawn in – biology, sociology, auto mechanics, whatever – in order for the future teachers and their students to have a chance at making sense out of the connections being drawn. But we should not base our kids’ futures on an assumption. We should be sure. We should instruct future teachers in Integrated Math AND in the peripheral disciplines. This will be difficult and demanding, but if we insist on teaching math “across the curriculum,” then aren’t our kids worth the extra effort?

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      2. David, my understanding is that Integrated Math does not include calculus and other higher subjects. I am not an expert on the history/development of Integrated Math, but my understanding is that this approach has been used successfully in high schools in countries that have scored higher on international tests than the U.S. and this led to interest in adopting it here. I might be mistaken in this belief though as it has been sometime since I read about this subject.

        However, there is absolutely no doubt in my mind that the “audience” in college classes has been significantly narrowed down from the high school student population, so there is absolutely no necessity that what might work well in high school classes will necessarily be equally effective in college classes, especially when one tries to extrapolate from high school teaching methods to upper division (or graduate) math classes.

        As to your second point, high school teachers learn their subject specialty in their major classes in college. Education courses for a teaching credential are usually focused on topics like cognitive psychology / “how children learn,” etc. If teachers are given a new math curriculum to teach, they are usually given some kind of professional development training by their district if they are lucky, but quite often they may simply have to read the books and train themselves. I doubt that most education schools will ever go beyond teaching elementary school math instruction techniques (but I have not done an extensive survey of course offerings in U.S. schools of education.

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  45. Hi there! As a parent of a middle schooler, I have struggled to see the value of this text book series. I found your article insightful and interesting. I didn’t realize the book depends on group work and lessons in class for content explanation. I confess that I rely heavily on the CPM parent links to help my kid out.

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  46. My daughter loved math and was really good at it. After a few weeks into CPM she hates it and isn’t finding the success she used to have. It’s confusing and there isn’t any instruction from the teacher.

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  47. This thread is overwhelmed with folks who were ready to find CPM lacking. Despite your efforts in the article to present a balanced presentation, it was the negatives that resonate with these respondents. Such is always the case – the naysayers are always more motivated to find fellowship in their argument then those who are quite happy with a program/policy. There are very few fan sites for Math textbooks in particular. I mention this because a blog/website that attracts like-minded people like that will give the impression that there is virtually no support for the program/policy. I’m here to tell you that CPM is adored by many – they just aren’t here to tell you about it.

    Also, however, there were a lot of points brought up that beg for a response. I can’t respond to all of them from the last four years but I will try to hit some of the ones that stand out.

    1) Integrated Math is the preferred curriculum by ALL the countries that are beating us in international test scores.
    2) That being said, CPM is NOT an integrated program.
    3) Group work in CPM is not the default arrangement. It is used when appropriate, which is not even daily.
    4) Group Tests are meaningful practice tests and group projects. They are not weighted in a student’s grade equal to an Individual Test.
    5) There is definitely a connection between recent increases in interest in CPM and Common Core. This is because Common Core is encouraging us to move instruction onto the levels achieved by other countries, using methods that international data demonstrates to be superior. The fact that Common Core has been met with some negative reactions is very similar to the kind of reasoning presented in this thread – which I will discuss shortly.
    6) “Individual Discovery” is not a thing. There is a such thing as “Discovery”, which ANY good teacher will do, whether they are teaching CPM or not. If a teacher relies 100% on lecture, they are doing a poor job. If you look back at teachers you’ve had who performed well, you will recall activities that broke up the week. There has to be SOMETHING to give you a break from the teacher’s voice. But also, we’ve known for decades that if a student is led to a conclusion rather than told about it, they retain it longer. This is not CPM data, so you won’t see it on their website. This is a very large body of work going back to Socrates.
    6) Complaints from the successful: there are, in this thread, declarations from students who are self-reported successes in Math who don’t like CPM. There are also numerous declarations from parents who don’t like it for similar reasons. In both cases these are people who’ve been successful in a traditional mode and so want to continue in that mode. This is completely understandable. But the result, sadly, is a Calculus student who has no idea about what Slope is. No sense of it as rate of change. They know how to graph the equation y=mx+b and they can tell you m=slope; they are capable of doing various manipulations of the equation. But if you ask them a “Depth of Knowledge” question of Level 2 or higher, they draw a blank. The simple truth of traditional teaching (i.e., lecture based) is that you learn to obey a teacher and if you do that well you score well. In the meantime it is still open to question how much you actually understand what you have learned. An easy way to recognize such a student is that when faced with a novel approach to a problem, they turn away with the observation “That’s not how I learned it.” As the famous educator George Polya put it, ““It is better to solve one problem five different ways, than to solve five problems one way.” It is very much like the person who has a hammer and sees the world as full of nails. Their method will inevitably fail, whether they know it or not.
    7) Teacher Education programs that prepare teachers for teaching Math are actually two or three classes that a prospective teacher takes while working on their credentialing process. These will absolutely talk about Integrated Math, but they also have many other things to discuss so it is probably a pretty thin preparation. This is why continued Teacher Training is necessary and exactly why CPM offers that, in fact requires it, for schools who are adopting their textbooks. No other textbook can make that claim that I’m aware of.
    8) A teacher in the thread above points out that they augment the CPM text with worksheets. This is pretty normal. If you hear from a teacher that they have been forbidden from doing that, or forbidden to help students, or required to sit them in groups at all times, they are simply prevaricating. When my district decided to adopt CPM we saw some teachers who were in the camp of “I’ve succeeded in a traditional mode so that’s all I want to continue with.” Those teachers, when saddled with a CPM text, were the ones who made these outrageous claims. I don’t know if the teachers in this thread are those or not, obviously, but I can report from my experience that such were the majority of the naysayers. Bottom line: any teacher who claims their hands are tied is being defeatist or is purposefully trying to scuttle a program/policy. Teachers hands are RARELY tied in any fasion. Certainly never in the decision to assign an occasional worksheet. What CPM offers, is lots of alternatives to worksheets. Which is a very good thing.
    9) There is a reality that some students will succeed in Math no matter the curriculum or teacher. CPM can’t use them as proof of concept any more than any program/policy. However, what CPM DOES bring to the table is the broadening of the Mathematical practice such that a wider variety of students can access it or are interested in trying. A parent in the thread above wondered how CPM could be helpful for students with IEPs or otherwise have been shown to be Math low achievers. These students will actually benefit MOST from the program, and I can’t imagine why anyone would question that. Remember the definition of crazy (endless repetition with the expectation of a novel outcome). Of course, these students will not benefit from CPM if they have a less than adequate teacher (for whatever reason) or have been warehoused in a class full of students with similar problems. These two are problems are, let’s admit, as common as sunrise. But to be clear: neither is the fault of the curriculum.

    What CPM proposes to do is to open the door to mathematics for as many as possible. It does this very well.

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    1. Richard – you make many good points in your reply. I agree with several, but not all of them, and will find time in the next few days to compose a detailed response.

      Please note that CPM boosters were never barred from this blog, and i disagree that this blog is a site for one-sided thinking. Instead I think it shows that CPM repeatedly has “image problems” with the public in many places where it is implemented, and a concerned teacher might want to sincerely determine why this happens. I am concerned that too many educators don’t like to listen to feedback that doesn’t correlate with what they were taught in education school and simply adopt the attitude that the other side is ignorant/misinformed. That (as well as only seeing the flaws in CPM) is a mistake…

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      1. I have no doubt that your intent was to be open to all opinions and you made an effort at being even-handed in your reporting. What I mean to suggest is that the fans didn’t seek you out – only the opponents. That is the nature of the beast and is beyond your control.

        I think I’m pretty open to feedback, both positive and negative. However, I am also acutely aware that a lot of the feedback is pretty uninformed or are prejudicial. Most respondents have not been trained in nor are even trying to follow the scientific method or even the rules of evidence.

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      2. Hi Richard. It will be at least this weekend before I have time to compose a detailed response, but I have one immediate point to make for your consideration.

        If you want to defend CPM, why do it by trying to paint the people who responded to my article as misinformed? That strikes me as use of the “straw man” technique and is the common reaction that I usually see from CPM advocates when they are questioned by parents.

        *** Why not respond directly to the criticisms of CPM in my main article??? *** I’ve taught CPM and seen it in action at several different schools and districts in “the City” and on the San Francisco peninsula. I am also very familiar with scientific method having a Ph.D. in biophysical chemistry and numerous research publications.

        I should also note that several experienced teachers are among the people who commented on the article. While dismissing some CPM critics as “old school” may at times be valid if a person is truly “set in their ways” and doesn’t want to try new things, that is not uniformly the case. Sometimes experience counts for something.

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      3. I use the word “misinformed” because some of the statements of fact were false. I am not tossing epithets. It is strictly a clinical analysis. Such is not a straw man. I actually DID respond to criticisms – and if I had taken even more time I could have responded to them all! I’m sure you, and many of the respondents here, are familiar with the scientific method. A reminder to adhere to it should truly resonate, then, no? I don’t recall saying either ‘old school’ or ‘set in their ways’. Did I do that? I’ve students walking in the door right now so I can’t scroll up right now to check…
        Regards.

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      4. No, you didn’t use the term “old school” but point 8 in your original reply about teachers who resisted the CPM curriculum in your area strongly implies this. I am well aware that there are “old-timers” who resist all kinds of change as they approach retirement, particularly if they have been teaching something the same way for years. However, that is definitely not a factor in my essay on CPM. I also have to run now. More on Sunday or Monday.

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      5. Hi Richard. This has also been a very busy week. I have not forgotten about your comments but have not had time to finish a detailed response yet. As I said above, I wrote **my** objections against CPM in my article, and I do not want to take time to rewrite the same material. You were objecting against items brought up in the discussion, some of which like Integrated Math were not even discussed in my article.

        While I mention several strengths of CPM, I should also note that several of my most important objections against CPM were not even addressed in your comment. I will try to get a response out by this weekend, but I remain concerned that parents attending school meetings are frequently told that they do not understand CPM and that the program is actually really good. Any unbiased person can appreciate why this kind of reply leads to massive frustration on the part of parents. This frustration is vividly illustrated by the fact that people are searching and finding this article in Google on a **daily** basis. It has been viewed over 25,000 times now by readers.

        If you ever have taken the time to look into CPM’s “research base” it was very thin and flawed, and I find their approach to “spiraling” very poorly executed. I do agree that for subjects like Algebra 1 it is a viable option, but it rapidly becomes less so as students advance to more complicated material for reasons stated in my article.

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      6. Richard, in the interest of avoided “uninformed” or “prejudicial” remarks, I will point out a few things.
        1) Integrated math may certainly be the preferred curriculum in other countries, but they don’t implement anything resembling the CPM approach, so that point is irrelevant to the question of evaluating CPM. However, your referencing it seems to belie a common trend, that proponents of CPM are also proponents of the integrated pathway.
        2) CPM offers both the traditional program and an integrated program. Districts decide which they’ll use, and many in my state have switched to the integrated pathway. To a student, parent, or teacher who’s experienced a switch both to CPM and to the integrated pathway simultaneously, these two changes will be conflated. The publishers of CPM have played a role in this shift, and it is misleading to claim otherwise.
        3) The materials were originally written to have a group discovery focus as the default. That may have changed when the CPM publishers had to adjust to distance learning under COVID-19, but it is baseless to judge any comment made before 2020 as false or misleading on these grounds.
        4) Nobody said that group test were being weighted as much as individual tests, but these practices vary from district to district and classroom to classroom. A claim as strong as yours cannot be justified across the board.
        5) International data never demonstrated the Common Core methods to be superior. It’s been 10 years since Common Core Math came onto the scene, and virtually no gains have occurred with American students vis a vis their international counterparts.
        6) The calculus student who has no idea what slope is is not necessarily the failing of the traditional curriculum. There are many other alternate explanation (bad teaching, perhaps), and studies that have been done to try to isolate the cause to lecture vs. discovery have not been conclusive enough to support your claims about this.
        7) So noble that CPM offers teacher training, and if they are the only textbook that does so, is that not telling in itself? Shouldn’t better math teaching be a good that goes beyond any one particular textbook or curriculum. This is also irrelevant to the question of whether or not CPM’s claims of effectiveness can be supported. If extensive training is required in order for a product to be employed effectively, that’s usually not a selling point.
        8) Stances on providing supplementary materials vary from district to district. Can you back up your claim that any teacher who claims they are not allowed to supplement is “simply prevaricating”?
        9) This is your best point, and the one with which I most closely agree. That said, there are studies that show the discovery method does not work as well for many students who find themselves needing an IEP. They are not one-size-fits-all, and so their mileage with CPM will vary.

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    2. Hi Richard. I finally have time to reply to your detailed comments in your response on 9/13/22.

      First, please let me thank you for taking the time and having the concern to respond in detail. Here are my responses to your numbered comments.

      1&2. Michael Malione addressed points 1 & 2 about Integrated Math. I merely want to point out that I made no mention of integrated math in my original article as this was not a CPM issue at that time. I also think that international test scores can be difficult compare. The PRC, for example, rocketed to the top of the list at one point, but only included students in some schools in Shanghai if I remember correctly. This incident strikes me more as a political propaganda move than a scientific study. However, I am now retired and am not following these developments as closely as I used to.

      3. I find the assertion in point 3 extremely puzzling. At the time of writing the article, CPM most definitely was founded on group work. Every class I was involved with or observed had students grouped in fours in accordance with the group roles *as defined in the CPM textbooks.* Michael implied that this methodology might have been updated during remote learning necessitated by the pandemic, but it is clear from reading the text books at the time that I wrote this article that group work was integral to the CPM program. This is what I stated in the original text:

      “Students are typically placed in groups of four in their classroom and are given defined roles within their group: “Resource Manager,” “Facilitator,” “Recorder/Reporter,” and “Task Manager” (see the beginning of any CPM textbook for details if you are interested in what these roles entail).”

      Perhaps the implementation was different in your district or changed as the result of the pandemic, but I still believe that most locations follow the methodology just described and set up classroom desks in groups of four.

      4. This point on the value of group tests is where there is significant disagreement between many students and CPM teachers. I agree that there can be value in group *projects,* but too often I have seen group *tests* devolve into the better students in a group quickly doing the test and the weaker ones hurriedly copying the work before time is called, followed by the better students making sure the copies are faithful because they don’t want to get burned again by the teacher randomly picking a weaker student’s test paper and then assigning that grade to the entire group. I believe that this teaches students practices that can only be described as marginally ethical at best. The cynic in me wonders if this practice continues mainly because it lowers the number of test papers to grade by a factor of four…

      5. Michael Malione and you are probably more current on the Common Core debate that I am as I have been teaching/tutoring AP subjects (physics and calculus) almost exclusively for about the last 5 years (except for some non-CPM precalculus students after Aragon HS dropped their one year CPM precalculus trial). Common Core was not my central topic in the original article other than to mention that it revived interest in CPM.

      6A (there are two comments marked with “6” in the comments to which I am replying so I am relabeling them A and B). I agree with this remark and have been a fan of the Socratic method for years. However, I also quoted a review paper in my original article that surveyed the literature on discovery-based learning and noted at the time that, despite providing this article from the American Federation of Teachers journal, none of the teachers that I gave this too ever gave any indication that they read the article and obviously made no attempts to refute the important points made there.

      6B. I agree that there are students who go through the mechanics in math without understanding deeply what they are studying. I also agree that in lower level classes like Algebra 1, **IF** an experienced teacher has good classroom management skills **AND** has not been handed a class full of students who have already failed/hate mathematics, that CPM can engage students to a higher degree than the traditional lecture mode of instruction. You use the analogy “It is very much like the person who has a hammer and sees the world as full of nails. Their method will inevitably fail, whether they know it or not.” in this section.

      I could basically say the same thing about those who try to employ CPM methods for higher levels of math. While it is a noble goal to try to make everyone as proficient in mathematics as possible, I believe that this goal is increasingly hard to accomplish the higher up in the subject one goes.

      In my tutoring work, I worked with several students for multiple years, in a few cases from 7th grade all the way through 12th, so I had the unique opportunity to see the long term results first hand of various curricula. The students who went through the CPM Precalculus experiment at Aragon HS definitely appeared to me to struggle in subsequent math classes more than any other cohort that I worked with over the course of a decade. None of my students in that class went on to Calculus BC. After taking Calculus AB in junior year and not having a strong precalculus background due to the experiment, those who continued on with math senior year opted for AP Statistics (which has its own set of issues which I have described in other articles in this blog). This is not due to a single poor teacher either as there were three CPM precalculus classes with three different teachers in this curriculum trial and the majority of these teachers were fine. One of them had to revert to lecturing and staying after school to provide additional help to make up for deficiencies in the curriculum.

      There is a natural winnowing out of math students for a variety of reasons, some due to innate ability but also sometimes due to unfortunate encounters with poor teachers. etc. At some point students have to sink or swim in math as the subject becomes more difficult and a greater amount of material must be covered. Collectivist attempts to hold the best (and sometimes only luckiest) students back to help weaker ones, though seemingly “equitable,” will only hurt our country’s competitiveness in the long run, ultimately damage the economy, and result in fewer resources/services for *all citizens* as a result. Trying to deny that a division of labor is needed and assert that everyone can be treated the same is a utopian dream that is never successful.

      Having said that, I also support efforts to address inequities in our society as I wrote about in my latest blog post several months ago ( https://eduissues.com/2021/11/26/reducing-the-achievement-gap/ ). Too often though these interventions come too late in school, and steps need to be taken *before* “achievement gaps” appear.

      7) Teacher training is one of the points that I often hear CPM teachers laud, and I support this. Too often in-house professional development is a joke. That said, I refer back to my response in 6B above. Don’t view CPM as the only hammer and all math topics as similar nails…

      8) This unfortunately boils down to your experience in your district versus the reports from other teachers elsewhere. Not being in a position to know any of you closely, it is very hard to know who is correct here.

      I have always had a problem with CPM’s implementation of spiraling, and in my tutoring work where I am working one-on-one with a student I have seen all too often how the CPM practice problems following the guided questions in the main lesson section are barely enough to make a student comfortable with the latest topic before they start jumping back to earlier questions or, worse, jump to preview questions with zero instructions and simply bamboozle and frustrate weaker students. I think that this is one of the main reasons why teachers supplement the course with worksheets.

      I also commented in one of my brief replies to you earlier in the week that the CPM research base, particularly on spiraling was very weak when I wrote this article. If I took the rather large amount of time required to write an article exposing that, people would see even more clearly the paltry evidence provided for the structuring of the Review and Preview problems. Some of the research they cited involved psychology tests which studied the most effective means for a person to memorize lists of items like foreign city names which had only marginal, at best, applications to mathematics instruction.

      Of course, we all know from common sense and personal experience that reviewing past material helps cement it in one’s mind, but making this statement and using it to justify spiral methods where sometimes only a couple of problems on the new topic are practiced before spiraling back to earlier topics is a bridge too far IMHO. It is also obvious from personal experience that the more one reviews, the less time one has to learn new topics, so a trade-off between review and new topics is inevitable. The fact that so many teachers feel the need to supplement CPM with worksheets says pretty clearly to me that CPM has NOT struck the proper balance here. Finally I also noted the following in my original article:

      “Much of the research is older, probably in part because the standardized STAR test base was discontinued with the adoption of Common Core. However it is interesting to look at one of the later studies from 2013 in 8th grade and high school ( https://pdfs.cpm.org/research/CA%20CST%208th-11th%20Grade%20Results%202011-13.pdf )

      The methodology of this study is very flawed, however, because it appears to only take the results of school districts that CPM knew used their books and compare them to statewide averages. There is no controlling for differences in, e.g., demographics between districts that adopted CPM and the state as a whole. Nevertheless, there is no indication that schools using the CPM series did any better (or worse) than the STAR test state averages in Algebra 1 and Geometry and only slightly better in Algebra 2.

      This might not seem too bad until one realizes that our local schools have always prided themselves on scoring significantly ABOVE the state average! Why would they want to adopt a series that only delivers average state test scores, particularly when we know how pathetic state math scores have been??

      The illustration below shows California STAR math scores (% scoring proficient or above) from Grade 2 up through Algebra 2 for years up through 2012 before the state terminated this test in favor of newer Common Core testing. As one can see, there was a downward slide in math scores from 4th grade through high school in 2012.

      Passing rates of 35%, 32%, and 34% are not benchmarks that I would want to use for marketing any product that I developed!!! Why would one adopt a program if this is the research used to promote it??

      In fact, I learned from an administrator that CPM was tried in the SMUHSD years ago, long before Common Core was a glimmer in anyone’s eye, and was abandoned by all high schools except Hillsdale! The group work encouraged by Common Core appears to be resurrecting it from the dead.”

      This was CPM reference noted earlier above in my response to comment 5.

      9) I have no issue with your comments in 9 and as I said at the beginning of my response to 6B, I agree that CPM has benefits in some math subjects like Algebra 1 for reaching a wider group of math students than traditional textbooks, with certain caveats about the quality of the instructor’s classroom management skills. The fact that you are willing to write so much to defend it indicates to me that you belong to the group of dedicated instructors. I don’t know what math topics you teach, but I have always felt that the quality and caring of a teacher is the most important factor. That said, education can not be imparted solely by the instructor. A student must be a willing participant and any curriculum can only go so far if this care on the part of a student is lacking.

      My purpose again in writing this article was to show that, despite some of the strengths of CPM, there ARE legitimate reasons for parents to be concerned about CPM. These reasons should not be brushed aside by educators without careful consideration, but sadly, as is so common in our public discourse now, both sides seem merely to dig in their heels. The fact that this article is still read almost six years after it was written on average between 10-15 times a day (primarily as a result of Google searches) illustrates the extent of this frustration.

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      1. 1-2) International scores might be difficult to compare in some instances and not so much in others. The National Center for Educational Statistics has produced a number of guides on productive comparisions here: https://nces.ed.gov/surveys/international/cross-study-comparisons.asp. As with all statistical studies, there is room for doubt but repeated stats that point in the same direction seem to quell those, as international ed data has done over the years.

        3) “Founded on”, “default” etc., I don’t know how to respond to these questions (I tried when responding to Michael Malione elsewhere in this thread). I don’t know if we are having a genuine disagreement or if this is purely semantics or just a matter of emphasis. CPM, even if it asks for daily team work, does not necessarily earn “founded on” or “default” if it has significant work done by the individual. Note, just as food for thought, that there is daily homework in CPM courses and that is pretty much an isolated experience. But even in the classroom there is ample individual work/experiences/writing/reflection… I think your confusion (and Michael’s) is based on the assumption that I’m denying team work’s existence, or even its prominence. What I meant to point out is that it is well balanced by individual work, which takes up the majority of the clock time in class – this is not to counter that team work may even happen daily.

        4) I’m sure I don’t have to point out to you that your experience doesn’t replace data nor the stated policies/procedures espoused or supported by CPM. Forgive the lengthy reference, but here is the CPM word on Team Assessments (taken from the CC1 Teacher Notes, but is pretty much the same in all the texts):

        “Team Assessments
        Quality teamwork is essential to an effective CPM classroom. The use of team roles promotes better teamwork, as do the Study Team and Teaching Strategies (STTS). Sometimes, though, it is necessary to emphasize the teamwork by assessing it in some manner. Therefore to build a class culture in which teamwork is truly essential for learning, it is important to assess teamwork. A Team Assessment is an assessment in which collaboration is essential for mathematical success.

        This activity is referred to as a “Team Assessment” but this does not mean that a score or a grade must be assigned to it. In fact, CPM believes that a Team Assessment should not be graded, certainly not in the traditional sense; a Team Assessment should not be part of the students’ grades. Despite the fact that this assessment is not graded, a Team Assessment is still an excellent activity because it provides opportunities for certain kinds of learning and collaboration.

        A Team Assessment does not necessarily need to have a score that counts toward an individual student’s grade. If you must include Team Assessment scores in individual student grades, be sure it has minimal influence on the student’s overall grade and is not detrimental to any individual student’s grade.”

        Re: your ‘cynical’ view (don’t be hard on yourself; folks try to limit their workload in various ways across all professions daily) remember that there are still quizzes, tests, homework and in-class activities. No one would claim that CPM has lighter grading requirements who has taught the course.

        6) Re: hammers and nails- I think you’ve missed the point (unless you were using the experience of students as a simile to the experiences of teachers). When I said hammers and nails I was talking about the mathematical power of students. Traditionally taught students (or any student who is dependent on their favorite trick or memorized algorithm) can be very good with those methods (their hammer) but then are led to think that mathematics is ONLY those things that yield to it (their nails). Such, of course, is not what mathematics IS nor should we present it that way.

        About passing CPM students on to Calculus (or whatever form of advancement one might be interested in). I can’t detract from your experience. I can only suggest it is, as our experiences always are, multi-causal. I could quote you my experiences with CPM creating a Calculus class where there was none, and the following year we had two sections and the following year three and the next four. Our AP Stats classes similarly grew. But I don’t think my experience proves the point, either.

        About Collectivist attempts to create equity: not sure how this featured on our discussion, (do you see CPM as a collectivist attempt?) but I would agree, in the abstract, that any policy comes at a cost. If you try to educate more people through the high school level, you are spending money and time on that goal and that might come at the expense of whatever students you decided not to spend that time and money on. The goal of CPM (and equity in education) is to raise the total number of educated people. I think this is an admirable goal for public education. The talented few who hoped to get more from the public may have to dig deeper and find a private alternative. I’m not happy with that outcome, but I acknowledge that policies have costs. We are working for the greater good, no?

        7 and passim) In regards to the proof we hope to see (hard data) that CPM works, that spiralling works, that Team-based education works: there have been many research articles supporting all of the methods CPM uses. Folks who are displeased with CPM (and have the time and energy) have referred to the list of research articles that CPM posted (years ago). My understanding of CPM’s effort in that direction is that they decided to not get bogged down in that debate – as the research is available in professional journals and has been for years. For ease of access I will list a few here, but my point is this: when I (or any CPM enthusiast) uses words like “uninformed” or the like, we are referring to the fact that our interlocutors haven’t dug through this body of the literature.

        I have often taken up the defense of CPM with professors of Mathematics, who are sometimes aghast at my presumed arrogance (I am not a Ph.D). I ask them if they have a degree in education. (They rarely do). I ask them if they’ve read the pertinent literature. (They rarely have. Actually, I’ve not met one yet who is both critical and has read the material, but that’s MY experience.) It is easier to have these discussions with such professors, though, because when you refer the literature they know your reasons for doing so. When speaking to the public, it is trickier. Of course they haven’t read the literature, or if they’ve read anything it is surely in the form of internet “research” which we have all exercised – we sought out the article online that agreed with us most.

        You made reference to some of this material and that you had read it. You also complained that it was sparse and perhaps not relevant to math and perhaps also methodologically unsound. I think these observations might lead you to deepen your search. The literature is there. Here are a few examples (just on the subject of spiralling, as that had come up earlier).

        Bruner (the original piece on spiralling)
        On learning mathematics
        JEROME S. BRUNER
        The Mathematics Teacher
        Vol. 53, No. 8 (DECEMBER 1960), pp. 610-619 (10 pages)
        Published by: National Council of Teachers of Mathematics

        A piece which compares spiralling to networking (thus exemplifying that some people think traditional practices are “spiralled” because topics are repeated year after year, which is not what CPM means when it says spiralling, which is iterations within the same course of similar material)

        Click to access 598388-perspectives-on-curriculum-design-comparing-the-spiral-and-the-network-models.pdf

        Another that compares spiral to strand curriculum. The authors seem to have a slight preference for strand, but CPM does both. Otherwise, the article lists several advantages of both.
        https://www.nifdi.org/research/journal-of-di/volume-4-no-1-winter-2004/459-a-comparison-of-spiral-versus-strand-curriculum/file#:~:text=Many%20topics%20are%20cov%2D%20ered%20in%20each%20lesson%20in%20strand,math%20period%20for%20easy%20concepts.

        An explanation of spiralling and its defense:
        View at Medium.com

        A study done in Mauritius extolling spiralling:
        https://eric.ed.gov/?id=ED604303

        There are many more but I guess this short list is enough to demonstrate the existence of a body of literature.

        Regards.

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      2. Thanks for taking the time to compose your extensive response, Richard. I’ll consider your points carefully, but, as before, it may take several days before I can respond in depth. Thanks again!

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      3. Hi Richard,

        I am getting a bit concerned that we are starting to talk past each other to some extent instead of addressing the points that the other party makes. This is a frequent occurrence in debates, so it is not too surprising.

        Despite the fact that I (age 69) am retired now, the additional time it takes me to exercise to maintain a desired level of health, honor my commitments to my wife, two daughters and their families, including three grandchildren whom I am starting to help educate, keep up with routine household chores, etc., actually leaves me with less time than I used to have for these debates. This may very well be my final response in this thread, and I will have to leave it to the readership to tally their own score of “debating points.”

        To illustrate my concern about speaking past each other, I am going to begin by addressing your following comment near the end of your remarks:

        “7 and passim) In regards to the proof we hope to see (hard data) that CPM works, that spiralling works, that Team-based education works: there have been many research articles supporting all of the methods CPM uses. Folks who are displeased with CPM (and have the time and energy) have referred to the list of research articles that CPM posted (years ago). My understanding of CPM’s effort in that direction is that they decided to not get bogged down in that debate – as the research is available in professional journals and has been for years.”

        My points 7 through 9 contained some of my most specific serious concerns about CPM and several of these were not addressed by lumping them as “7 and passim.”

        Your main response to my issues was to provide more references. I thank you for providing these references, but they suffer from problems similar to those that I looked at earlier. With the possible exception of the 1960 Bruner article ( I could not read it without paying $15 as I do not have a JSTOR subscription), all of these articles **are simply descriptive instead of providing quantitative studies** or else they make claims that are supposedly backed up simply by citing by other references. ** There is little or no data.**

        I suspect that the Bruner article is probably of a similar nature, talking more about the common sense advantages of coming back repeatedly to material at a higher level, but not providing actual implementation details. The devil is always in the details (is a free copy of that article available anywhere? If not I will purchase a copy at some point.).

        The Mauritius study was unfortunately a great example of worthless “research” as it consisted of a sample size of **8** students. As I have said elsewhere ( https://eduissues.com/2018/01/29/never-believe-educational-experts-or-me/ ), too much of educational research not only has serious methodological flaws but often also has financial conflicts of interest (e.g., publisher sponsorship).

        One of my major concerns in response to point 8 was:

        “I have always had a problem with CPM’s implementation of spiraling, and in my tutoring work where I am working one-on-one with a student I have seen all too often how the CPM practice problems following the guided questions in the main lesson section are barely enough to make a student comfortable with the latest topic before they start jumping back to earlier questions or, worse, jump to preview questions with zero instructions and simply bamboozle and frustrate weaker students. I think that this is one of the main reasons why teachers supplement the course with worksheets.”

        Unfortunately in many instances like the above with my tutoring students, the teacher did not provide additional exercise material and none was available in the CPM textbook, so I would have to make problems up on the fly.

        Instead of more philosophical references like those that you cite, I would like to see extensive *quantitative* educational psychology studies which address **on average** in a **large** group of students, **how many** math exercises should be assigned in the initial introduction to a topic to produce reasonable retention and also some attempt to **determine how soon there after** a teacher should return to a topic. I realize that designing such a study would require a lot of funds and would be difficult to implement, but without such data, we are only bandying about buzzwords like “spiraling” and “stranded” and leaving it to individual teachers to sort out this difficult problem on their own.

        Teachers have been grappling with this problem in their own math classes for ages, but at times the solution is nothing more than following the assignments of a mentor teacher who taught the subject before them. When a new curriculum gets foisted upon them, then even this personal experience is lost.

        Every night math teachers have to determine how much homework to assign. Obviously this will depend on the complexity of the subject as well as the skill level of the class involved. Note however, that a “traditional textbook” usually has a large number of practice exercises in each section which gives a teacher flexibility in their homework assignments. *** CPM sections do not have these extra problems unless the student looks up other online resources. *** That additional barrier is usually enough to deter unmotivated students.

        Another major wrench that often gets thrown in the mathematical works is the perceived need to cover state standards which results in classes “moving on” whether the bulk of students have mastered a concept or not. It strikes me that a really good teacher would be monitoring student work on an ongoing basis, and, if a concept is not sinking in for the bulk of a class, then the right thing to do would be to slow down and provide additional practice. However, how often does one see classes where the homework is collected at the end of a section of lessons and the homework “packet” is then simply “checked for completeness.” In my tutoring work I have seen cases where the class average on a test for a particular topic was dismal, and yet “on we go.” Had homework actually been examined before the test was administered instead of turned in at the end of the section, the dismal test scores would have been anticipated in advance of the test. The problem mentioned in this paragraph is not uniquely specific to CPM; it can clearly happen with another textbook series if the teacher does not monitor progress. I think that my main point here is that ultimately in all of these debates, *** it is the quality and dedication of the teacher that is really the external determinant in student success ***; the internal determinant is the student’s personal motivation over which the teacher has a smaller degree of control.

        All of the references you provided seem to indicate that repeated review is a “good thing,” but then acknowledge the problems it imposes on introducing new material, again leaving it to practitioners on the ground to sort this out for themselves.

        You mentioned in point 8 of your original comment that teachers often supplement CPM with worksheets. This basically means that the ***success or failure of the CPM implementation in a school may depend more on the diligence and caring of a teacher than the quality of the textbook itself.***

        You strike me as being a concerned and caring educator, and I personally would probably attribute your success with CPM more to this factor than to the quality of the textbooks. Unfortunately, not every school district is lucky enough to have a math instructor like you.

        When I made my response quoted above starting with “I have always had a problem with CPM’s implementation of spiraling,” I was referring to my reaction to the material I saw in the textbooks, not to a class that is getting extra supplementation. This was particularly my impression during our local CPM precalculus trial that I have mentioned several times previously. The number of Review and Preview questions was often insufficient and not all teachers were equally diligent in supplementing material.

        Regarding research supporting CPM which I also addressed in my reply to your point 8, when you stated that “CPM’s effort in that direction is that they decided to not get bogged down in that debate.” I should note that CPM’s website for several years **was** soliciting research proposals. I do not know what the results were, if any, of this solicitation, as I finally stopped looking when I kept seeing only the same references dating back to 2103 or earlier.

        I quoted in my response to point 8 that CPM’s attempt to generate data by selecting STAR scores from districts which they knew were using their textbooks, and noted that, even in this methodologically flawed approach, they were not able to give convincing numbers that CPM made a significant difference in STAR scores. For any reader who missed this, I refer back to my earlier reply and do not want to repeat this again. If this was the best that CPM itself could come up with to promote their program, it was pretty thin… no wonder they bowed out of the debate.

        Having spent quite some time on the above, I need to wrap up the earlier points before 7-9 quickly.

        1-2) I agree that the international trend looks unfavorable to there U.S., but I have not bothered to look into the gory details here. As I have mentioned several times earlier, digging deeply into educational research usually finds problems, so I listen to, but tend to suspend judgment on, items that I do not have time to delve into. Instead I focus on my locale where I personally can make a difference. In the case of international studies, not only are comparisons complicated, but national pride is yet an additional factor which can lead to complications as I mentioned in the specific case of the PRC.

        3) I think that our three experiences with CPM group versus individual work are clearly different here. Nothing further to add.

        4) Thanks for the information from CPM, but I stand by what I said regarding group tests in my earlier response to 4.

        6) Regarding the hammer and nails analogy, I understood completely how you were applying it to students as you described again above. My point (I was trying to be slightly humorous here) is that the same analogy can be applied to CPM adherents who think that because the CPM methodology works in, say Algebra1, it will continue to work up through pre-Calculus and Calculus (not to mention possibly in higher math subjects as well). Your experience with Calculus development at your school is commendable, and I applaud your efforts. Whether other districts are as successful as you have been is a question that I can not personally answer. CPM was tried and abandoned for Pre-Calculus at our local high school. I remain confident of my statement that as the complexity of a math subject increases and the amount of material attempted also increases, self-discovery takes too long.

        I agree that our goal in educating as many people as possible in high school is noble and appropriate. The “great sorting” usually occurs in college where students have to choose a major. Because however, the AP curriculum moves more college material down into high schools, conflicts occur between these two goals.

        I do not think that CPM is “collectivist” per se, but there are collectivist influences who are rallying around CPM as evidenced by insertion of agendas into the latest CA math framework drafts of topics that previously would have been considered partisan and outside of the school curriculum. Sadly, it appears that people on the left and the right are trying to inject their political philosophies into mathematics education. We do not need either Proud Boys or union organizers on school boards…

        When you say “The talented few who hoped to get more from the public may have to dig deeper and find a private alternative. I’m not happy with that outcome, but I acknowledge that policies have costs,” I think there is another way off looking at that problem.

        Lyndon Johnson said that education was the “only valid passport from poverty.” This means that students who were smart but came from a poor background still had a chance through the public school system of improving their lot compared to their parents. If public schools can spend a lot of resources on special education and equity issues, it seems like they should also not shut the door on such students. If providing GATE education also benefits “privileged” students, that is not a reason to shut it down. But this could be the subject of yet more hours of discussion and I have to draw this to a close.

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      4. I don’t see the talking past each other – I think we are getting a bit repetitive perhaps. Just a few quick points, as I understand you don’t have the time to continue this conversation.
        “Passim” means “throughout the text”, not “from here and onwards”. I didn’t respond to 8 and 9 because I thought doing so would be redundant; I have already done so elsewhere.
        You’re correct that my main response was to provide references. I confess that I, too, don’t have the time the subject deserves so my search for same was fairly cursory. I only meant to point out that there is a lot out there beyond what CPM has posted.
        I appreciate your criticism of several studies based on sample size. That might be a problem – but isn’t by itself a solid criticism. There is an accepted protocol for small sample studies and these are accepted throughout the sciences daily. A small sample size cannot by itself be the support for a declaration of “worthless” for a study.
        About data: we know that the US is faring poorly in comparison to other nations. (Your exception to this data is noted, but since it is accepted around the world as valid, you and I may be forced to deal with it). We had been losing ground to other countries for decades – that data is VERY clear. Something had to be done to improve our mathematics instruction. I point this out because you don’t believe CPM-supporting data (I have the feeling that would be the case no matter how much was presented to you) and you don’t believe the data that says a change was necessary. Are you sure your problem is with the data and not with your desire to find data that agrees with your position?
        You have gone back to your own experience repeatedly. I understand that many folks value giving witness over analyzing the data, so I offered my own experience as a counterbalance. But you poo-poo my experience and return yours to center stage. I can only suggest to you that, as I’m sure you actually already know, your experience cannot provide greater insight than any data – even when it is a small data set (as is your experience, after all).
        You point out the troubles of every day teachers. There are troubles with teachers regardless of curriculum. These cannot be held up as a problem with CPM or any curriculum – especially when CPM provides extra training!
        Throughout your argument, however, we see a strand of thought that deserves correction: you clearly think that teaching to mastery is the only valid teaching method. You apparently have never been forced to look at the flaws of that method, namely (among others), the mind-numbing boredom that it inculcates among the majority of recipients; the fact it is a myth (you may THINK you have taught to mastery and then, at Final Exam time are shocked to see how reteaching is, in fact, necessary in the form of a Final review; the fact that it supports the “mile wide and an inch deep” curriculum that has been the death knell of math achievement in this country (again, in international data comparisons, of which there are many and not subject to national pride or ‘official’ meddling); the fact that it injures the most talented students the most; and is almost always redundant, as individual teachers who are attempting it often don’t have multiple representations in mind when they attempt this strictly focused methodology.
        So, a few quick points to close: the info I provided from CPM about grading Group Tests directly contradicted you stand. Don’t know how in the face of that you could stand by your earlier statement.
        If you meant to say CPM was using a hammer and nails approach, I think you’ve lost the track of your own argument. Elsewhere you have said they move from topic to topic too much. If the methodology is to teach the use of various tools, and your attempt to call THAT a hammer and nails approach I think you’ve stretched credulity to the breaking point. It isn’t a narrow approach if it is defined as trying a variety of things.
        Proud Boys vs Union Leaders? Is that really a microcosm of our current partisan politics? Indeed, is inclusive education practices a partisan philosophy At All? We are involved, here, in Public Education. From its very inception, that was meant to be a system for every child it meets. Schools are defending themselves from partisans because of this, to be sure, but the resistance against those attacks isn’t partisan. It is the very essence of our original, century+ project.

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      5. Richard, when I read your latest response I don’t think that I am making myself clear to you with my writing. Perhaps if we were to sit down and have a face-to-face chat, this might change. Despite spending several hours writing my response yesterday, I still come away with the feeling that you are not understanding me, so I am going to call it a day here. Perhaps my writing is unclear or perhaps you have been through this argument so many times that you assume in advance what the other party is communicating. In any event, further written communications appears futile, and I will leave it to any readers who are diligent enough to read all the way through to the end of these lengthy comments to draw their own conclusions. All the best and keep up your good work at your school.

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      6. After mulling it over, I am going to try to make one last attempt at clarification and try to keep it briefer than before.

        I wrote extensively in my last comment about point 8 which you simply summarily dismissed by saying “I didn’t respond to 8 and 9 because I thought doing so would be redundant; I have already done so elsewhere.”

        Where???? I clearly stated that 7-9 contained several of my biggest problems with CPM. To quote: “My points 7 through 9 contained some of my most specific serious concerns about CPM and several of these were not addressed by lumping them as “7 and passim.””

        Do you understand why this conversation is getting frustrating for me???

        Regarding the international data, since this is an issue that you keep mentioning, I have taught in Malaysia, been to Singapore, and used CPM in San Francisco where there were many recent Chinese immigrants. These immigrant students would repeatedly say that math in the U.S. was very easy compared to the PRC.

        There are ***vast*** cultural differences in the attitude towards teachers and education in several of these Asian countries compared to the U.S. In fact it was very apparent that after migrating to America, Chinese students in S.F. were already being corrupted by the local “culture.”

        Having American students switch to a curriculum like the one used in Singapore may make some difference in test scores, but I am concerned that there are many other societal factors at play here other than just the teaching methods. I am not going to spend the time to elaborate this point further here, but I doubt that CPM or spiraling by itself is the solution to this test score issue.

        As to your quote “I point this out because you don’t believe CPM-supporting data (I have the feeling that would be the case no matter how much was presented to you) and you don’t believe the data that says a change was necessary. Are you sure your problem is with the data and not with your desire to find data that agrees with your position?”

        This is a total misrepresentation of my position. I HAVE NOT SEEN ANY GOOD *QUANTITATIVE* CPM-SUPPORTING DATA, just verbally descriptive articles extolling the benefits of spiraling. None of the references that you provided were of this quantitative nature. Possibly there may have been such studies buried in some of their references, but I only had time to go through the references provided.

        I mentioned more than once that CPM’s quantitative attempt to support its own curriculum via STAR scores did not give a compelling reason to adopt the curriculum. If the publisher who has a clear financial interest at stake cannot provide easy access to quantitative data support, why should I believe your claim that such quantitative data exists??? Please provide a reference to such a study if you can.

        I do not doubt that there are benefits to spiraling, but as I said in my last comment I would like to see **quantitative studies** indicating how to implement it properly. I do not believe CPM gives enough initial practice when a subject is introduced and is too quick to spiral back to other topics or advance to preview problems. The fact that so many teachers supplement the textbook is a good indication that this problem is NOT solely my personal experience.

        You state “Throughout your argument, however, we see a strand of thought that deserves correction: you clearly think that teaching to mastery is the only valid teaching method. You apparently have never been forced to look at the flaws of that method, namely (among others), the mind-numbing boredom that it inculcates among the majority of recipients; the fact it is a myth (you may THINK you have taught to mastery and then, at Final Exam time are shocked to see how reteaching is, in fact, necessary in the form of a Final review; the fact that it supports the “mile wide and an inch deep” curriculum that has been the death knell of math achievement in this country ”

        I “clearly think that teaching to mastery is the only valid teaching method.” ???????????? Where the heck did you come away with that conclusion???

        I clearly stated multiple times that the CPM method is applicable for the lower level math topics like Algebra 1. My doubts arise when the subject material becomes more complicated as I am stating again for the nth time.

        You are very quick to assign to me the motivations of your usual sparring partners.

        Finally your response to the “hammer and nails” analogy again completely misses the point that I was trying to make. I will simply repeat the relevant text here: “Regarding the hammer and nails analogy, I understood completely how you were applying it to students as you described again above. My point (I was trying to be slightly humorous here) is that the same analogy can be applied to CPM adherents who think that because the CPM methodology works in, say Algebra1, it will continue to work up through pre-Calculus and Calculus (not to mention possibly in higher math subjects as well). ”

        The point is that just because self-discovery may work in Algebra 1 and possibly geometry or algebra 2, does not mean that it will work on higher math topics. “Self-discovery” is the “hammer,” and there are adherents who think that it is the solution to all the nails, i.e. to all mathematics teaching issues.

        Enough!

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  48. Michael: Quick responses to your points here.
    1) I don’t think there is a type of teacher. There is undoubtedly much overlap in the groups “teachers who like CPM” and “teachers who approve of Integrated Math”, but as I mentioned, CPM is not integrated so surely they are not exactly the same group.
    2) I wouldn’t call any of the texts CPM has produced “traditional”. You are correct that CPM is working on some new “integrated” books, but these are not as yet in wide circulation, and definitely not the ones being referred to in this thread. If someone were one of the relatively small number who have seen the new integrated texts then, yes, I can see how the two would be conflated. For the vast majority this is not the case. At any rate, I stand by my statement that CPM does not represent integrated math – that they are producing some books that fit that mold doesn’t change their nearly 30 year history of doing otherwise.
    3) Group discovery is not the default. It is common, but default means automatic unless otherwise instructed, right? Such is not the case.
    4) I am an ex-trainer of CPM courses. I can tell you what the teachers are told are best practices in a CPM classroom. If there are some who don’t follow those, I don’t think that is the problem of CPM. There are individual teachers who can make a mess of any curriculum, right?
    5) The NAEP data went down for the first time EVER this year. We have made gains in this country – if your claim is that internationally they have made gains that outstrip ours, then I can only repeat that they are doing so with Common Core – like standards. They are not beating us, if they are, with traditional methods.
    6) The calculus student I mentioned was fictional. I was just trying to point out a phenomenon in terms all the readers of this blog might understand. The point was that there are a lot of students who are successful in a traditional method who are expert at mathematical ‘tricks’ and algorithms but have only the faintest idea what it’s all good for. You hear them ask that very question often. Students who are asked to follow a discovery, or problem-solving method, will know immediately that there is a problem-solving use for these mathematical tools. One might complain – as some have done – that teaching problem solving uses classroom time that could have been used for teaching more algorithms. This is more or less true; there is a cost for every policy. The question is, for teachers and parents, is: “Which is a better Math student? One who is ready and willing to pursue problem solving, especially novel problems, or one who is less able to do so but knows some arcane algorithms (some of which are truly not practical in practice).
    7) Thank you for declaring CPM noble, but I think that’s bit of an overshot. It is responsible and shows dedication to a good outcome (over just profiting off it, for example). Your desire to have good teaching win the day regardless of textbook is idealistic – maybe even naive. Even good teachers will benefit from a rehearsal of the curriculum in question. Requiring it is not to improve their teaching as much as it is a check that administrations supply the time they ought to – always.
    8) But that was not my claim. I said it was often the case. What I DID say was that any teacher who says their hands are completely tied was prevaricating. Even very strict districts expect creativity from teachers. They expect them to meet students’ needs – which don’t always match across the grade or classroom. No teacher, anywhere, (to my knowledge – I will include the disclaimer here in case there is, unknown to me, some hellish administrator somewhere who is actively working against students) is expected to stay lock-step within the boundaries of some imagined corridor of learning/methodology.
    9) I’d love to see the reference for these studies. Getting a chance to do Math differently has been the one best answer for the students I’ve taught who are not already model Math students.

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    1. Richard, I am confused by your reply. Are you sure you’re talking about CPM and not the similarly-acronymed CMP curriculum?

      If not, there are so many factual errors in that response, all I can do is point out the most basic. I taught CPM Integrated Math 1 (https://shop.cpm.org/pc_product_detail.asp?key=752228E69CCA47BBBF28742853E884B0) to 9th graders in my district.

      That district adopted CPM INTEGRATED MATH curriculum over 6 years ago, so these materials have existed for at least that long. The first section of Chapter One details how group work is to be the default configuration for the discovery exercises in the book. Students are given roles in their groups of four (Resource Manager, Facilitator, Recorder/Reporter, Task Manager). This is right out of the CPM textbook.

      When I speak of the “traditional” pathway. I am referring to the Algebra 1, Geometry, Algebra 2 sequence. If the pathway is not “integrated,” as you’ve indicated about CPM, then it must be traditional. This verbiage comes right out of the California Common Core State Standards for Mathematics.

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      1. I don’t have access to that textbook so I can’t reply. If CPM has started using the term “default” that would be a change. I would argue that even if CPM suggests “daily” team work, that is still not the “default” arrangement. Do you think I’m splitting hairs by claiming that? I don’t.
        What you mean by traditional is what I mean as well, for the most part. However, while CPM has produced a series of books in the ‘traditional’ sequencing, these are not ‘traditional’ books in presentation or pedagogy. Perhaps it is the multiple uses of the word that cause the confusion.

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