Wednesday, January 12, 2022

On the California Math Framework

Guest post by Boaz Barak and Jelani Nelson

In a recent post, Lance Fortnow critiqued our open letter on the proposed revisions for the California Mathematics Framework (CMF). We disagree with Lance’s critique, and he has kindly allowed us to post our rebuttal here (thank you Lance!).

First, let us point out the aspects where we agree with both Lance and the authors of the CMF. Inequality in mathematical education, and in particular the obstacles faced by low-income students and students of color, is a huge problem in the US at large and California in particular. As a Black mathematician, this portion of the CMF’s introduction particularly resonated with me (Jelani):

Girls and Black and Brown children, notably, represent groups that more often receive messages that they are not capable of high-level mathematics, compared to their White and male counterparts (Shah & Leonardo, 2017). As early as preschool and kindergarten, research and policy documents use deficit-oriented labels to describe Black and Latinx and low-income children’s mathematical learning and position them as already behind their white and middle-class peers (NCSM & TODOS, 2016).

We agree with the observation that bias in the public education system can have a negative impact on students from underrepresented groups. Where we strongly oppose the CMF though is regarding their conclusions on how to address this concern. 

The CMF may state that they are motivated by increasing equity in mathematics. However, if we read past the introduction to the actual details of the CMF revisions, then we see they suffer from fundamental flaws, which we believe if implemented, would exacerbate educational gaps, and in particular make it harder for low-income students and students of color to reach and be successful in college STEM.

You can read our detailed critique of the CMF, but the revisions we take issue with are:

  1. Recommendation to drop the option of Algebra I in the 8th grade
  2. Recommendation to offer (and in fact push and elevate above others) a “data science” pathway for high school education as an alternative to the traditional Algebra and Geometry curriculum. While data science can be a deep and important field, teaching it for students without a math background will be necessarily shallow. Indeed, the proposed data science courses focus on tools such as using spreadsheets etc., and do not provide mathematical foundations.

1 and 2 make it all but impossible for students that follow the recommended path to reach calculus (perhaps even pre-calculus)  in the 12th grade.  This means that such students will be at a disadvantage if they want to pursue STEM majors in college. And who will be these students? Since the CMF is only recommended, wealthier school districts are free to reject it, and some already signalled that they will do so. Within districts that do adopt the recommendations, students with means are likely to take private Algebra I courses outside the curriculum (as already happened in San Francisco), and reject the calculus-free “data science” pathway. Hence this pathway will amount to a lower-tier track by another name, and worse than now, students will be tracked based on whether their family has the financial means to supplement the child’s public education with private coursework.

Notably, though the CMF aims to elevate data science, we’ve had several data science faculty at the university level express disapproval of the proposal by signing our opposition letter, including a founding faculty member of the Data Science Institute at UCSD, and several others who are directors of various undergraduate programs at their respective universities, including four who direct their universities' undergraduate data science programs (at Indiana University, Loyola University in Chicago, MIT, and the University of Wisconsin)! 

One could say that while the framework may hurt low-income or students of color who want to pursue STEM in college, it might help other students who are not interested in STEM. However, interest in STEM majors is rapidly rising, and with good reasons: employment in math occupations is projected to grow much faster than other occupations. With the increasing centrality of technology and STEM to our society, we urgently need reforms that will diversify these professions rather than the other way around.

As a final note, Lance claimed that by rejecting the CMF, we are “defending the status quo”. This is not true. The CMF revisions are far from the “only game in town” for improving the status quo in mathematics education. In fact, unlike these largely untested proposals, there is a history of approaches that do work for teaching mathematics for under-served populations. We do not need to change the math itself, just invest in more support (including extracurricular support) for students from under-resourced communities. For example, Bob Moses’ Algebra Project has time and again taken the least successful students according to standardized exams, and turned them into a cohort that outperformed state averages in math. One of our letter’s contact people is Adrian Mims, an educator with 27 years of experience, whose dissertation was on  "Improving African American Achievement in Geometry Honors" and who went on to found The Calculus Project, a non-profit organization creating a pathway for low-income students and students of color to succeed in advanced mathematics. 

To close, a critique of the proposed CMF revision is not a defense of the status quo. Even if change is needed, not all change is good change, and our letter does make some recommendations on the front, one of which is a matter of process: if a goal is to best prepare Californian youth for majors in data science and STEM more broadly, and ultimately careers in these spaces, then involve college-level STEM educators and STEM professionals in the Curriculum Framework and Evaluation Criteria Committee.


  1. The major problem with K-12 education is the content. Until principles-based school mathematics is being taught, fiddling with the course structure won't make much difference.

    The main author of the latest version of the California Mathematics Framework is Jo Boaler. Regarding her past work, see

    A Close Examination of Jo Boaler's Railside Report

    1. The current CMF Framework is written in an utterly disorganized and unprofessional way, and it is THE FIRST framework in history that avoids accountability and degrades the standards of an entire academic field of study and the process of establishing learning outcomes by DELETING ALL bullet points identifying learning outcomes that ALL prior frameworks published!!!!! The CDE dismissed the request by stating for parents that the document that accounts for learning outcomes in 2021 DOES NOT exist. Great answer, right? Talk about fraudulent attempts to shortchange the public!!! please read the FACTS and compare 2013 to the loosey goosey 2021 CMF documents Who is the one who does NOT collaborate here, Lance? There are a number of highly qualified mathematicians, professors and parents/professionals who are ready to help. WE ARE THE ONES who are excluded while they are writing money making opportunities into the curriculum, selling data science classes by youcubed and UCLA while simultaneously locking kids out of economics and data science college courses by replacing Algebra II with "data science". Jo Boaler is personally profiting and selling her books full of error (Mathematical Mindsets, 2016 edition- just read it -page 19, and the page discussing Maryam Mirzakhani). She is brainwashing girls into her fraudulent ideas; Watch this starting 2:20 She is a disgrace to humanity, and especially women.

      Lance instead of accusing parents/professionals and professors of mathematics make it happen for us! Create a public meeting for us all, high school students, teachers, professors and professionals to attend. YOU should be part of the solution TOO!

  2. To me, the entire discussion and the multiple posts and public signatories to statements, etc., come down to this sentence in this post:

    "One could say that while the framework may hurt low-income or students of color who want to pursue STEM in college, it might help other students who are not interested in STEM."

    It is important to accept that both parts of this sentence are important.

    (1) Any attempt to eliminate the option of calculus in any HS in any CA school will definitely hurt lower-income or students of color who wish to pursue STEM in college and beyond. Thus, under no conditions should those options be removed.

    (2) There are a lot of low-income and students of color and other students who -- for any number of reasons -- don't have an interest in STEM and calculus often gets in the way of their dreams / plans to become nurses, kindergarten teachers, or a whole bunch of valuable respectable jobs and roles in society. They too, need a path, where calculus is not FORCED on them. (I have anecdotal evidence: I tutored a bunch of HS students from a low-income area in Silicon valley last year, and their preparedness for calculus was abysmal. It is true that we need to look further back and fix the preparatory years education, but that's a different discussion.)

    I read HS-level data science as "basic math/stats numeracy as needed in society" -- something very concrete and something that's almost as important as the power of abstraction one begins to learn in algebra.

    TO a mild degree, the response to the CMF revision is as much "throw the baby with the bath water" as the authors suggest the CMF revision itself is. I sincerely hope that elite STEM academics speak not only for the STEM-aspiring students but also for students for whom HS math is a hurdle, not an enabler.

    1. There already is such a path, calculus and even pre-calculus are not forced on any high-school student, and many students don't take it. The CA requirements to graduate are 3 years of math including algebra I (For example, Algebra I + Geometry + Algebra II.) Plenty of students also get into college without having taken calculus. Even at Harvard there are plenty of students that are placed into pre-calculus.

      It is true that some colleges look at calculus as one of the many factors in their admission criteria. You can argue whether or not that's a good idea (which likely depends on the college in question) but this letter is not about college admissions policy but about high-school math curriculum.

      I agree that offering some high-school data science course for students that will not go on to STEM degrees can be good, but this course should be decoupled from the math curriculum.

    2. Is there any first-principles reasoning why a basic data science course can't be a math option instead of geometry or pre-calculus or calculus?

      If data science can be taken instead of Geometry or (pre-)Calculus to satisfy UC/CSU's 3 years of math requirement, what's wrong about it?

      The main point of the letter -- that no student, especially students with less privileged backgrounds, should be denied the opportunity to learn calculus (maybe even AP calculus) at their school -- is very reasonable, but I don't get why it's a one-or-the-other argument (between calculus/geometry and data science).

      I think the open letter mixes two separate issues: the issue of ensuring access to advanced math classes for every HS student (which is strongly related to equality, social mobility); and the issue of creating a lighter pool of math classes for students with insufficient preparation who're not going the STEM path. And taking a condescending tone in the letter about a proposed math class in data science ("trendy", "shallow", etc.) doesn't help. (Equally, I'll maintain that if the CMF revision even hints at eliminating access to calculus for any CA student, it should be DOA.)

    3. The letter is not about first principles but about a very concrete proposal, that could effect the children of our largest state and influence other states as well. For a fuller analysis of that proposal, see

      In particular, we say that the proposed data science class is "shallow", we don't mean data science in general, but courses such as by the CMF's main author.

      However, there is also a first-principle reason why data science should not be considered as an alternative to the basic math foundations courses, but rather as part of science and social studies. This is explained very well by Henri Piccioto here: "A key difference [between data science and math courses] is that in math we should not teach black-box formulas and software packages that students cannot possibly understand thoroughly. We have been moving towards teaching math for understanding at all levels. There is no reason to use data analysis as an excuse to backtrack.""

      See also Piccioto's analysis of the CMF here

    4. The main concern with the so called "data science" classes identified in the curriculum relate to the fact that data science is a replacement for Algebra II. I suppose if we exclude a large number of majors in undergraduate school, ranging from economics to biology calculus at UCLA, to traditional STEM majors, data science is a choice to get you through high school. It is widely circulated in the media how they are enticing especially minorities into enrolling into these specific classes without informing them of the truth that Algebra II is needed in many undergraduate courses to succeed, and most definitely in real data science. Read the IDS website, full of student testimonials, how they are led to believe that they have a future in data science, coding etc.

      How do you succeed even in an economics course without understanding functions never mind in majors that require heavy quantitative courses? All these topics are taught in Algebra II. Now, this UCLA designed IDS course and youcubed data science is replacing Algebra II. The irony of this all is that while UCLA's Statistics Department created the IDS (Introduction to Data Science class), UCLA is the first to reject a high school background that does not have Algebra II for any undergraduate major requiring quantitative analysis. I am deeply ashamed of my own alma mater, and will be writing complaints to them.

      There is no precedence in history when profit motivations are written directly into a public education curriculum. Historically, independent, financial non-conflicted members of the academia wrote the curriculum framework. Today, Jo Boaler, who is selling youcubed data science classes writes articles with the UCLA Professor of Statistic from UCLA, also selling the Introduction to Data Science class, both a replacement for ALgebra II, are writing and/or influencing the curriculum. The first version of the curriculum does not even disclose that this so called data science class will not allow children to pursue a major in data science or anything but an entry level job that ultimately has no upside potential in data science. This is deeply unethical and misleading.

  3. The paragraph for "2 Recommendation" may need to end before the key word "While data science", as the rest looks like the author's opinion.

    I appreciate very much the last sentence by D. Sivakumar, to quote: " I sincerely hope that elite STEM academics speak not only for the STEM-aspiring students but also for students for whom HS math is a hurdle, not an enabler."
    This point holds for all races and all sections of students alike.
    Practically, and perhaps theoretically as well, it is most efficient and fairest for all students to set up different standards and different pathways for different levels of STEM-aspiring (and talents)

  4. Lance in his original post writes: "Many of our students didn’t have the opportunity for advanced math courses in high school. Some struggle but most do well and graduate into a solid job. I do not believe high school calculus is needed to succeed in college."

    Unless there is evidence otherwise, perhaps this actually argues for the open letter: Many of them didn't have to take advanced math courses exactly because the regular courses perhaps had some meat in it!

    Saying "I do not believe high school calculus is needed to succeed in college" is a tautology - of course it is not semantically "necessary". There are people who do amazing things in math and/or programming after studying music or without even going to high school. (Just as one could argue "college education is not needed to land a solid job".)

    The point is whether or not it is noticeably helpful (there is perhaps some debate to be had here although in my mind it does a lot) not whether it is "necessary".

  5. Larry Summers recently tweeted that stronger math instruction and expectations for American youth is an economic and a national security imperative. He hints that America's math standards have been subject to continued erosion by social justice warriors who can’t themselves define exponential growth or solve quadratic equations.

    Jo Boaler is a leading figure of such social justice warriors. What a tragedy that American's K-12 math is guided by such a person who has a very limited knowledge of real math and holds a very distorted view about real math!

    People would naively believe in Jo Boaler's snake oil because of her status as a Stanford math education professor and her superior tactics in peddling woke math/reform math/fuzzy math.

    The following documents are sort of sagas about how American K-12 math has been dumbed down since the early 20th century:

    Jo Boaler's Fame, Stanford's Shame; Students' Gloom, America's Doom
    Jo Boaler's Reform Math Fallacy
    A short version:

  6. Siva --

    Just some thoughts on your points.

    "If data science can be taken instead of Geometry or (pre-)Calculus to satisfy UC/CSU's 3 years of math requirement, what's wrong about it?"

    First, as Boaz said, the minimum mathematics requirements do not include precalculus. They only include Algebra2. Here are the standards for Algebra2 so you can judge for yourself:

    The standards include basic statistics and probability, exponential growth models (e.g. how pandemics behave), concept of functions, concept of a complex number and logarithm, very basic trig (e.g. periodic phenomena, like day light or biological processes). Geometry is important on its own as an introduction to logical reasoning and also because it often reaches and empowers students that do not find algebra appealing.

    These standards of geometry and algebra are in line with minimum standards of math in the rest of the western world. These standards were augmented in 2013 to include more "data science." They include statistics and probability and exponential (and linear and quadratic) models. Are there topics here you find excessive?

    You are making a good point that some kids are so under prepared when starting high school that they need an option. The CMF proposal provides such lower pathway option (but does not market it for what it actually is). But please consider the circular effect of institutionalizing this option as being adequate (and misleading students and teachers and politicians that it is). Then the *incentives* to actually better prepare kids at the level of minimum standards of the rest of the western world will be eliminated and you are giving up on another generation. Traditionally, the demographics of students that had been pushed to the lowest possible option will be pushed even lower. More of the students that could be prepared by a better system will continue not to be.

    You say "I don't get why it's a one-or-the-other argument (between calculus/geometry and data science)."
    Well, we agree it seems that it should not be. The issue is that the proposed CMF made it just that.

    The typical minimum requirement for HS include only of 3 years of math (including Algebra 1 that many take in middle school) and sometimes only 2 years of science versus 4 years of English and Social studies. The standards includes data science topics introduced in an integrated way with the math foundations.

    Indeed it is important for students to learn basic coding and spreadsheets skills. But they should not be replacing math foundations. They can be included in data analysis done in lab science or social studies/economic classes. Because there is so little required STEM, there is room for a dedicated STEM course alongside math and science for students that otherwise do not take higher level coding or statistics.

  7. I would rather not read the proposal but I was surprised by your comment: "Recommendation to drop the option of Algebra I in the 8th grade"

    The FAQ says:
    "Does the draft Mathematics Framework eliminate middle school mathematics acceleration programs?
    No. The draft Mathematics Framework does not eliminate middle school mathematics acceleration programs (including programs that offer Integrated Math 1 or Algebra 1 courses to grade eight students)."

    Am I missing something?


    1. See, the link at “By discouraging the 8th grade Algebra I option”, which redirects into the relevant part of the proposal.

      Generally school districts have a lot of leeway to do whatever they want, hence what you see in the FAQ. Basically what they’re saying is “they don’t have to follow our recommendations if they don’t want to”. In reality, more affluent school districts would probably have such programs, and those strapped for cash may elect to drop activities that aren’t part of the new state guidelines, such as these types of acceleration paths.

  8. Interesting dialogue and some strong words voiced by commentators.

    I think the watering down of mathematics
    content is a global phenomenon seen across different
    syllabi (ranging from I/GCEs to Pre-I.B., etc...).

    Obvious or not, the situation
    is driven by various political motivations;
    of particular interest are exogenous forces
    that are trying to seize opportunities to dump
    their software products onto the education market.

    Whether this is a good or bad thing, one can make the
    cliche either way -- the truth of the matter though
    is, various concepts in algebra 1 shouldn't
    be outsourced to software packages
    (whether proprietary or open source).
    It's crippling or, voiced more emphatically,
    corrupting the human mind at that early stage.

    That aside, @{Boaz, Jelani}, we are no longer living
    in content restrictive times; i.e., 80s and 90s.
    Content for various algebra levels can be found free online,
    in the worst case scenario.

    But as the brilliant Jonny von Neumann slices through
    the topic with surgical precision on ""America's Youth Wants To Know" -- he goes all the way down to ... high school levels.


    More training in science ... we better do something about it quickly and then continuously ...
    "Otherwise, you might never discover that you may have an 'ability'"

    I guess that would imply, don't skip Algebra 1!