## An update on the campaign to defend serious math education in California

Update (April 27): Boaz Barak—Harvard CS professor, longtime friend-of-the-blog, and coauthor of my previous guest post on this topic—has just written an awesome FAQ, providing his personal answers to the most common questions about what I called our “campaign to defend serious math education.” It directly addresses several issues that have already come up in the comments. Check it out!

As you might remember, last December I hosted a guest post about the “California Mathematics Framework” (CMF), which was set to cause radical changes to precollege math in California—e.g., eliminating 8th-grade algebra and making it nearly impossible to take AP Calculus. I linked to an open letter setting out my and my colleagues’ concerns about the CMF. That letter went on to receive more than 1700 signatures from STEM experts in industry and academia from around the US, including recipients of the Nobel Prize, Fields Medal, and Turing Award, as well as a lot of support from college-level instructors in California.

Following widespread pushback, a new version of the CMF appeared in mid-March. I and others are gratified that the new version significantly softens the opposition to acceleration in high school math and to calculus as a central part of mathematics.  Nonetheless, we’re still concerned that the new version promotes a narrative about data science that’s a recipe for cutting kids off from any chance at earning a 4-year college degree in STEM fields (including, ironically, in data science itself).

To that end, some of my Californian colleagues have issued a new statement today on behalf of academic staff at 4-year colleges in California, aimed at clearing away the fog on how mathematics is related to data science. I strongly encourage my readers on the academic staff at 4-year colleges in California to sign this commonsense statement, which has already been signed by over 250 people (including, notably, at least 50 from Stanford, home of two CMF authors).

As a public service announcement, I’d also like to bring to wider awareness Section 18533 of the California Education Code, for submitting written statements to the California State Board of Education (SBE) about errors, objections, and concerns in curricular frameworks such as the CMF.

The SBE is scheduled to vote on the CMF in mid-July, and their remaining meeting before then is on May 18-19 according to this site, so it is really at the May meeting that concerns need to be aired.  Section 18533 requires submissions to be written (yes, snail mail) and postmarked at least 10 days before the SBE meeting. So to make your voice heard by the SBE, please send your written concern by certified mail (for tracking, but not requiring signature for delivery), no later than Friday May 6, to State Board of Education, c/o Executive Secretary of the State Board of Education, 1430 N Street, Room 5111, Sacramento, CA 95814, complemented by an email submission to sbe@cde.ca.gov and mathframework@cde.ca.gov.

### 68 Responses to “An update on the campaign to defend serious math education in California”

1. Cerastes Says:

Honestly, even this undersells how many STEM fields kids will be shut out of. I’m in a biology department and these fundamental mathematical skills are used by my colleagues in cell biology, ecology, physiology, evolutionary biology, and more! I was literally just talking to someone in forensics about using the tangent, normal, and binormal to characterize the shape of ribs. I’m pretty sure you’re legally required to include a photo of an ammonite or nautilus shell if you talk about logarithmic spirals.

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3. David Karger Says:

I fully agree with the main argument of this statement, that Algebra II is absolutely essential for high school students, and that coming to college without it is a disaster for just about any student.

I will quibble though with a subtext implying that calculus is more important than stats. For students *in* college, who are *not* going into STEM, I would advocate for stats as a more valuable topic than calculus to prepare for life in general. For that reason I’d advocate for making stats that default math requirement in college, rather than calculus.

4. Scott Says:

David Karger #3: You might be right … that’s of course a completely different question, whose answer might depend on the specific major or program. Our goal has simply been to ensure that a rigorous math education remains available to California K-12 students whose parents can’t afford fancy private schools or tutors. I’m surprised and delighted that we’re having some actual success with that goal.

5. Peter Gerdes Says:

I’m not clear why statistics and data science can’t introduce and use tools like logarithms and algabraic reasoning.

I agree that the concepts uses in algebra II and the like are important but, in my experience, most of what happens in those courses is pointless memorization of mindless computational steps. A student who needs to look up the quadratic formula isn’t any more at a disadvantage in learning higher mathematics than one who can apply it by heart. Hell, every time I have to teach calculus I have to go relearn/rederive a bunch of the basic formulas and if mathematicians don’t need to memorize this stuff why should students?

Look, I agree with the importance of learning these concepts but the truth of most HS education is that only the barest amount of time is spent on concepts and the vast majority on pure calculation. There is some role for computation. But if that’s the reason for all the rote/algorithmic work then why can’t that happen equally well as part of a data science class as a algebra II class?

In fact, I sometimes think that HS math classes actually are as much of a problem that we have to overcome to teach advanced mathematics as a benefit. Indeed, in every calc course I’ve ever taught the biggest stumbling block students seem to have about the conceptual material comes up with their difficulty grasping the epislon-delta concept: largely bc they’ve had it drumned into them that math is the algorithmic solution of equations which prevents them from even being willing to just try a potential rule for delta in terms of epsilon. I find that the biggest obstacle is often fighting the expectation that they must be doing something wrong if they aren’t proceeding by following an explicit algorithm for finding solutions.

6. Peter Gerdes Says:

Let me add that there is a huge cost to trying to have all students take algebra II etc in preparation for college. Once that course is something we expect all college students to do it must be taught in a way that doesn’t fail out many diligent, promising students.

The problem is that means the course must be taught in a way that even someone who hates math, thinks they are awful at it and doesn’t want to think about it can do well if they force themselves to sit down and put in the effort. But, when you hate and dislike a subject it becomes virtually impossible to think creatively about it. You can force yourself to do rote manipulation you detest but not to creatively and intelligently engage (eg intelligently make and evaluate hypthesises etc).

I think we might serve everyone better if we taught classes like algebra II in a way that wasn’t designed to ensure that even students who despised thinking about it could succeed since that forces it to be boring rote repetition and more students then despise it.

7. Jackson Says:

You elected these people, and you will vote for them in the next election too. Not just that, you will enthusiastically support them and encourage others to do the same. Maybe you should consider other options, but you think they’re literally worse than hitler.

8. Bob Says:

I suppose it’s politically savvy to praise the regulations as well-intended but that just kind of delays the conflict. What were the intentions? If you can articulate something good about them, let us know. My guess it was it was pure jealousy because some kids are able to handle the math and some aren’t. The folks behind the regulation see themselves on the side of the kids that can’t and they think that stopping the high flyers will help their side “catch up.” It’s a very sad ploy.

9. Jess Riedel Says:

This is certainly a laudable effort to hold the line. It’s upsetting though that it is hard to do anything about the root problem, which is that the educators in charge of managing the schools and creating these proposals believe they are pursuing reasonable policy. Until that changes (almost certainly requiring a change to the educator training pipeline, rather than through persuasion), this will be never-ending whack-a-mole against a tireless adversary.

10. Scott Says:

Jackson #7: Please check your assumptions! I haven’t lived in California for nearly 20 years. If I did live there, I would vote whenever possible for local officials who are Democrats but also support rigorous math education. You know, as crazy and paradoxical as this will sound, ones who want neither a fascist theocracy nor a Marxist-Leninist utopia. In other words, I’d strongly support measures like San Francisco’s successful recent school board recall (to take one example).

11. Aaron Denney Says:

David Karger #3: I am convinced that statistics cannot be adequately taught without calculus. A full Kolmogorov measure-theoretic treatment is clearly not required, but doing anything with continuous distributions really needs an understanding of calculus.

Peter #5: Your first sentence is completely correct. But note that there is no claim being made that the core content of Algebra II can’t be learned through well-developed data science contexts (the statement was carefully written not to make such a claim). By all means, modernizing the motivation for fundamental content (and removing tedium) is very apt. Some day there can and should be courses which teach kids these essential math tools using contemporary contexts such as data-oriented problems or statistical tasks. Such overhauls need to be carefully developed by coordinated efforts of K-12 teachers and experts in colleges and industry.

But the reality is that currently available course material for high school teachers don’t remotely do this (more lucrative for textbook publishers to have two separate streams of books?). And high school teachers are generally too swamped to create their own such synthesized course material from scratch (whereas university faculty working together are able to do such things at the college level). Unfortunately, what are currently being offered as “data science” courses do not come anywhere at all near to the approach what you are suggesting should be done (e.g., they don’t develop fluency with function concepts and further algebra skills).

Those courses are being pitched to parents and school districts in California as fulfilling an “Algebra II” requirement even though the content is very far from achieving that, and it leads parents and school districts to think these courses are paving the road for preparedness to earn a 4-year data science degree in college. But it is false. (Those courses have been officially approved as such for the UC system, but the story behind that paradox is too long to fit in this comment box.) There has been a lack of transparency about the actual math in such courses currently most widely offered and the math relevant to preparedness for 4-year college degrees in data science and statistics (for example). This is one thing the statement is aiming to clear up for a wider audience.

13. Yisong Yue Says:

I’m having trouble navigating the extremely (and unnecessarily) dense text in the CMF. Is there a cliff-notes version that points out the most problematic aspects? I read Lines 1225-1239 in Chapter 5, and I do agree that is problematic. Is there a summary or itemized list of the other sections?

14. Boaz Barak Says:

David Karger #3: I don’t there is any subtext saying that calculus is more important than stats.

My position (and I believe the position of most letter signatories) is that stats and data science are extremely important, and some literacy with data is of great benefit to all citizens. I do think that such content should be taught to all students at both the high-school and college level.

The problem is when high-school data science courses are promoted as (1) pathways to a career in data science or tech and (2) equal or better alternatives to Algebra II/pre-calculus for students that are interested in a career in this field. Both are bad, but (2) is especially pernicious, since the students that will end up falling for this misinformation are likely to be those with fewer resources.

For students who will NOT go for a STEM major, I agree that statistics is more important than calculus. Indeed, MIT is an exception, and I think most colleges do NOT require calculus as a default requirement. (And in contrast many fields such as social sciences etc have some required intro statistics course.)

But for any college statistics course, students will still need to know Algebra II topics such as polynomials, exponentials, logarithms, etc..

15. Leo Reyzin Says:

Agree with Peter Gedes #5 and #6 and Brian Conrad #12.

Having seen my own child work through the US high school curriculum, I firmly believe that Algebra II has with too many topics and way too little in the way of actual understanding. Many kids taking AP Calculus, and even doing well in it, don’t really understand Algebra because understanding is not the goal of the curriculum. Cramming their brains with content is. My own kid, despite doing very well in accelerated classes, was turned off by the terrible curriculum.

I am not saying that what California was planning to do instead was any good. But the status quo is pretty terrible, and advocating for retaining the status quo — politically, the easiest position to coalesce around — is a big mistake. The status quo is not “serious math education.” It’s rote math. We should try to advocate meaningful reform.

16. Sigmund Says:

Of COURSE teach calculus in high school.

Calculus has two parts, differentiation
and integration, and each undoes
(reverses) what the other does.

derivative of the odometer. The odometer
is the integral of the speedometer.

When accelerate quickly, stop quickly, or
go around a corner quickly, each of which
is a case of acceleration, a passenger
gets pushed back in their seat, thrown
forward against their seat belts, or are
pushed to one side. Here they are feeling
the force in Newton’s second law, the
force that results from the acceleration.

That acceleration is the derivative of the

Right: Don’t teach AP calculus — it’s
too elementary. There is no reason for
it. Instead, just teach calculus. Get
one or a few good texts intended for use
with college freshman and have the
students dig in — study the text, work
the exercises, get the answers in the back
of the book. Can get some really good
calculus books, used, cheap.

Then move on to linear algebra,
multi-dimensional calculus, Laplace’s
equation, Stokes theorem, etc.

Do this studying essentially
independently, that is, at home, on own
time, with occasional contact with a good
mathematician.

For the public schools? Work hard to
minimize the damage they do.

For California, it is a hopeless political
cesspool. On nearly every subject their
pattern is to try every stupid,
destructive policy they can think of
before they finally, reluctantly accept
the obvious, common sense policy. For
citizens, don’t hope to clean up the
cesspool and, instead, just work to avoid
it.

For what to learn in math? No one knows
what role math will have in civilization
in 20, 30, 40 years, but generally for
progress it is “Queen of the sciences” and
one of most promising subjects.

17. Caleb Kemere Says:

I’m trying to understand the precise disagreement here. It appears the conclusion of the CMF is that many students will be better served getting a better foundation of algebra rather than being pushed into Calculus. I’ve definitely encountered talented electrical engineering undergrads who started Calculus as freshmen and struggling ones who took it in high school without apparently getting it. (There are apparently public high schools here in TX that don’t offer calculus.)

It seems like the dispute is about what to do if you have a mixture of student-understanding. I think that Scott – you’re saying they should track students into the calculus vs not tracks, while the CMF is saying that the cost of that tracking is that there are students in the lower track that with extra time would develop into STEM careers, and people in the higher track that will struggle because they scrape through without an in depth understanding. I think no one disputes that some students, if offered the chance, will take multivariate calculus in high school and go on to highly successful careers?

18. Scott Says:

Caleb Kemere #17: Let me put this in the simplest terms then. My experience, and the experience of almost everyone I know who works in STEM, was that precollege math was forehead-bangingly, eye-wateringly slow, taking years to review and review and review what could’ve been done in weeks. The motivating impulse behind the CMF seems to be to make it even slower—witness their constant denigration of “the rush to calculus”—and, crucially, to foreclose any option even for individual advanced students to move faster, because that would harm “equity,” and because the CMF authors “reject the idea of natural gifts and talents” (their words). At that point, you might as well tell me that the CMF plan is to shoot puppies and orphans, for all I need to know further details. It’s the diametric opposite of everything my life experience tells me is good.

The fact that the CMF’s “data science pathway” seems to be comically ill-thought-out, that CMF author Jo Boaler has giant financial conflicts of interest and has been charging poor school districts \$5k/hour for this stuff, etc, are then just additional icing on the turd cake.

[disclaimer: opinions are my own and are not necessarily shared by the organizers of the linked statement 🙂 ]

19. Boaz Barak Says:

Leo #15: The idea that because the status quo is not good, then any experimentation would be for the better is flawed. The CMF recommendations, if adopted, will make things worse: not for the kids of my CA friends and colleagues (who will be able to either take kids out of the public system or complement any shortcomings) but for the kids who have fewer resources.

Caleb #16: This is not at all about having all kids take calculus in high school. This is about using the hype of “data science” to create what would be effectively a lower tier track by another name. I am against tracking (pro acceleration, which is different) but at least explicit tracking is honest: if you are in the lower track, you know you are in the lower track. No one tells you that you are in the special track that teaches you 21st century math that will land you a tech job without having to learn algebra.

20. bagel Says:

In my (private, Jewish) middle school I was subjected to a curriculum without an advanced math track. Its pace was above average, but had no accelerated option. I don’t think it was ideological, just what they had resources for, but it was not a good fit for kids whose parents averaged two degrees or more.

The people who succeeded did so by turning to private math tutoring, on top of private school. While many of our classmates turned out to be interested in STEM, and some went to very good schools, there was a correlation between the kids who got tutored and the kids who got into the best schools. Causal because they did better on math? Correlation because they had more organized and informed parents? Something else? Who knows!

But I suspect that as long as such private supplements exists, slowing everyone down won’t even increase equity, and may well decrease it.

And private options are getting better, cheaper, and more available! Kahn Academy! FIRST Robotics! MIT OpenCourseWare! Tutors still exist, of course, but the skill floor is climbing rapidly.

21. Patrick Rogers Says:

This is so disappointing. The discussion in the CMF document about how mathematics tracking has directly led to socioeconomic inequities in education is heavily cited, but complaints are grounded in emotion, anecdote and peoples personal impressions of the role high school played for them–mostly coming from people years or decades removed from high school and subject to all the cognitive biases that involves. If you want to challenge the policy recommendations properly, let’s have a discussion of the evidence rather than subjective memories. Even the “detailed analysis and critique” cited in the earlier post only cites two additional studies, and provides superficial one-sentence critiques of five studies cited in the whole CMF document. Chapter 8, the section specifically on high school pathways has 57 items in its works cited list. I’ll be the last one to argue that science or policy should be evaluated based on citation counts, but it is concerning how little evidence-based push-back there is.

Worryingly, Scott (#18) seems to have the exact backwards understanding of the document. Referring to Ch 8, Lines 806-810: “…it would be desirable to consider how students who do not accelerate in eighth grade can reach higher level courses, potentially including Calculus, by twelfth grade. One possibility could involve reducing the repetition of content in high school, so that students do not need four courses before Calculus.” It’s not about making it “eye-wateringly slow” it’s about removing 8th Grade as the sole opportunity to graduate High School with Calculus credits. Lines 817-818 adds “…it should continue to be possible for students who are interested and ready to take Algebra I in eighth grade to do so…”

Realistically, the CMF is not anti-Algebra I in Junior High, it’s against taking Algebra, then Geometry, then Algebra II, then Pre-Calc as separate classes in a fixed sequence. The document argues instead for an integrated curriculum where students are introduced to elements from all those courses together, “Integrated Math I, II and III.” The document also points out that is how Japan, Korea, Estonia and Finland (all countries that outperform the US in PISA exams) structure their mathematics curriculum.

The document also suggests a separate pathway that focuses less on abstract mathematics and more on applying mathematical reasoning to the concrete world: “Mathematics: Investigating and Connecting (MIC).” This comes in for particularly disingenuous criticism in the earlier critique. The CMF document is explicit that the MIC/Data Science pathway is meant to drive equity because it provides a way for students who struggle in the traditional pathway to participate in mathematics instruction that less abstract and more grounded in real world examples. The previous critique ominously links to the Amazon page for “Weapons of Math Destruction,” I assume trying to imply that data science is inherently inequitable? I hope the authors simply misunderstood the point being made in the CMF, because no one is arguing that data science can’t be misapplied, the point being proposed in the CMF is that teaching mathematics alongside data science can provide another opportunity to link mathematical concepts to the real world and to things students care about. It’s about maximizing the opportunity for every student to have a positive experience learning mathematics in whatever way works best for them. This is where the concern about the “rush to calculus” comes in. It’s not saying no one should learn calculus in high school, it’s just saying that designing high school mathematics instruction as a single-minded march to calculus doesn’t work for a large enough number of students, that maybe we should have some alternative options for them. Ones that don’t exile them to remedial math or innumeracy.

It’s hard not to conclude that criticism is reactionary, driven by the impulse that “people should learn math the way *I* did.” Which ignores the reality that many people couldn’t learn math that way, were labeled as bad at math, and taught to avoid it. This is troubling when we see other educational systems that do not have the same inequities, particularly as those same systems seem to outperform the US fairly consistently in mathematics instruction more generally. In other words, we’re failing a large number of students, and not getting much benefit for the lucky ones either.

As near as I can tell, the only real benefit of the current system is the inequity itself. If you’re lucky enough to perform well in the traditional tracking system, then you have a very valuable academic advantage that translates to real-world benefits in hiring and salary. Which raises the concern–to me at least–whether this opposition is simply people defending the current system because it’s one that allowed them to succeed, despite that fact that it also led too too many to fail.

Addenda: Apologies if this is overly confrontational. I’m not in any way involved with the creation of the CMF, but I am very invested in evidence-based policy-making more generally, and I tend to get somewhat animated when discussing and defending it.

22. Boaz Barak Says:

Patrick #21: Thank you for reading the analysis document and the CMF but I do think you have it wrong (which is not your fault! The CMF is a very long document and written in an unclear way, also many citations look impressive until you chase down the original papers).

I will write more below, but here are some links about the YouCubed research which is most related to the data science issue:

Some analysis of the CMF as a whole: (a much more comprehensive one from
Brian Conrad is in the works)

https://blog.mathed.page/2021/12/14/yet-more-on-the-california-framework-part-1/

23. Wujing Harrison Says:

Thank you for the information regarding sending in concerns.
Also thanks to Patrick Rogers’s Comment #21, it seems that the CMF authors intend to improve student math learning experience. I would like to suggest that calculus and some essential modern mathematical tools be introduced in preliminary forms with appropriate approaches at elementary levels.
For Isaac Newton, calculus is a logical thinking tool he discovered to address philosophical matters. I am yet to meet a person who doesn’t understand the calculus logic in a preliminary form. I like the examples in Sigmund’s Comment #16,but wonder if it unfortunately may be too challenging for some to comprehend. I try to use elementary examples, like the number of stickers per day and the total number in a week or month.
I think we need to develop alternative curriculums that shows how calculus can be taught as an organizational and logical thinking tool starting from elementary levels, then gradually progressing into advanced forms in high school. Then we will have new generations who embrace calculus and many other modern mathematical thinking tools from young, to thrive in ever more complex systems.

Patrick #21:

I read the entire document during the week it appeared. I will have many details to say about it elsewhere (it takes time to type up comments on a 900-page document). But unfortunately, what is most disappointing is that after so many months of waiting, what has emerged is plagued with serious problems (some concerns with the previous version have been addressed, though not generally in full).

Since you brought up MIC, I will first say that it is not as it seems. What details are provided (really in Appendix A, oddly not in Chapter 8 on high school pathways) serve to show that there is no real plan for it. MIC 1 and MIC 2 are now Integrated 1 and Integrated 2 by new names; e.g., their tables of content standards in Appendix A are *identical* to the Integrated ones there, even down to the cut-and-paster forgetting to change the column headers for the course names (check for yourself). MIC 3 is only very loosely described, in a rather shape-shifting way. In one place it is said this will cover all learning standards of Integrated 3 and is specifically not a data science course.

But if MIC can cover everything in the Integrated sequence then what is the point? If it can all be done with better motivation, then there is no purpose to branding a new sequence: just give guidance on how to incorporate better motivation into the existing sequence (and so the same with the traditional one: the content in the two are just rearrangements anyway). Yet that has not been done. I see no evidence of any real detailed plan; it is wishful thinking. (This isn’t to say such an improvement can’t or shouldn’t be done — it can and should. But the CMF provides no real contribution towards such a goal. And there is zero evidence in the CMF for why a new 3rd sequence is needed for this; just incorporate improved motivation into what exists.) I will have much more to say about this elsewhere.

Coming to the citations, I read many of the papers cited. In nearly every case, the citation did not at all support what was being claimed. I offer two examples here (which have the merit is being more succinct to explain), and will have much more to say about this elsewhere soon.

1. For the 2013 paper of Park and Brannon cited in Chapter 1, the CMF writes that it “found that when students worked with numbers and also saw the numbers as visual objects, brain communication was enhanced and student achievement increased”.

This is false in multiple ways. The paper is not about students at all — the subjects are adults (see Experiment 1, 2nd paragraph) — and doesn’t mention brain imaging anywhere. The word “student” never occurs in the entire paper (outside one bibliography entry).

So there is nothing in the paper about “student achievement” or “brain connections”.

2. Chapter 9 cites a 2009 paper of Woessmann for evidence about damaging effects of tracking in schools. But the definition of “tracking” stated on the very first page of that paper (putting kids into different types of schools, depending on ability) is nothing at all like “tracking” as is meant in the context of the US education system.

The paper even says in a Table on the first page that “tracking” in the US generally doesn’t begin until age 16, already a tip-off that it can’t be discussing the same thing. Figure 3 shows US data in red, where red is for countries which are “not tracked”, and indicates educational inequality *decreases* from primary to secondary levels in K-12. Needless to say, the CMF didn’t cite that Figure 3 information (wouldn’t fit the narrative).

The two preceding examples of mischaracterized citations are unfortunately not at all atypical in the CMF. The document is full of many other things like this, as I will make public soon. Moreover, in cases where there is reasonable debate in the literature, the CMF passes over that in silence and only mentions papers it claims support its preferred view (though ironically some such cited papers demonstrate the opposite of what is claimed in the CMF). So the CMF is not “evidence-based” or “well-researched” in the usual meanings of those phrases. There is only value in citations if one reads them to affirm the relevance and quality, but the 20-person oversight team appears not to have done this, given all I found working by myself in a single week compared to what they missed despite working on it for very many months. More to come.

25. Akhil Jalan Says:

You probably are aware of the recent Florida decision to reject certain math textbooks as part of a campaign to ban the teaching of critical race theory. What the relation between those two topics is, I have no idea. But anyways, your readers might appreciate this satirical article about it.

https://www.mcsweeneys.net/articles/math-concepts-the-state-of-florida-finds-objectionable

26. Scott Says:

Akhil Jalan #25: Indeed, between a leftist faction that wants to dismantle advanced math classes so that everyone can be equal, and a rightist faction that wants to ban math textbooks if they contain word problems involving racial demographics, I know it’s crazy to imagine that there could be a third way, but some of us can’t help but persist in that craziness! 😀

27. Elizabeth Statmore Says:

Patrick #21 – I truly wish the CMF document and authors were doing what you are claiming they are doing.

But they’re not.

As Brian and others have pointed out above, an embarrassing number of citations made by the authors don’t prove what they claim the articles show. In fact, there are far too many citations in the CMF where the divergence between what is claimed and what the article says is so wide it’s disturbing. I don’t know whether the misinterpretations and misrepresentations were deliberate or inadvertent, but they are disqualifying. In addition, the authors cite themselves as experts on fields as wide-ranging as neuroscience and machine learning.

This is bonkers. And California’s students and families deserve better than this.

I am all for having a robust, evidence-based debate about what works to deepen student understanding. But the CMF is not providing that debate. It’s being used as an occasion for the authors to advance their own agendas, and I find this unacceptable.

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29. Steven Gubkin Says:

I do not know the details of the proposal in California.

As a high school student, I had a hunger for mathematics. I took Calculus BC in 10th grade from an instructor who ran the course more like a real analysis class. He expected us to offer a rigorous proof the the Mean Value Theorem for integrals and the Fundamental Theorem of Calculus on the exams, for instance. This experience was utterly transformative for me, and set me on the path to becoming a mathematician. However, other students at the same school were forced to experience the same boring memorized drivel year after year, gaining nothing of value.

I do believe that we can design mathematics courses at “lower levels” which are meaningful for students. There is little point learning to solve an equation if you cannot set one up based on a real world context, or interpret the results.

I am open to the possibility that other systems of organizing mathematics content delivery could be more equitable while still ensuring that passionate students can engage at a high level. I even can imagine a system without “tracking” where people with different strengths/passions could interact and collaborate in the same space. This is especially possible when the instruction is centered around solving open ended tasks which are “low floor, high ceiling”. If the task is to model disease spread, one student could literally just double numbers with their calculator while another could be solving the differential equation for logistic growth. I think that the kinds of contributions both of these students are making could be honored and valued, and the exchange of ideas could be valuable for all parties.

Such a system would require an extreme level of planning to design appropriate tasks, and also herculean mathematical/communication/empathy/moderation skills on the part of the instructors. I think with a few years of dedicated work I could maybe create such an experience and run it myself. I very much doubt that after doing this work I could “pass it off” to anyone else. So I am pessimistic that such an approach could work en masse.

30. Scott Says:

Steven Gubkin #29:

I very much doubt that after doing this work I could “pass it off” to anyone else. So I am pessimistic that such an approach could work en masse.

Right, that’s exactly the issue. Imagine trying to design a varsity football program that would benefit both a young Tom Brady and me, or a music program that would benefit both Yo-Yo Ma and me. Even if such a thing were possible in principle, no one would think of inflicting that ludicrous-sounding task on overworked high-school teachers … so why does it suddenly become obligatory when the subject is math?

31. Edith Cohen Says:

Patrick #21:
The CMF second revision is a convoluted long document with some forced edits done on a prior version without removing much of “evidence” (in double quotes for a reason) to the contrary. Beyond what the dry text states (which is problematic) there is also media statements by its most prominent author. That prominent author is also a curriculum developer for a proposed “new pathway” course and is marketing it in a certain way.

Because the CMF is essentially just a “recommendation” it is important *how* it is stated. If it says that algebra 1 in middle school can benefit some students (this was forced addition in the revision) but then provides a long discussion that claims it is problematic, then it is not that much of a recommendation. It re-enforces at best the situation that affluent school district with involved parents will continue to offer Algebra 1 in middle school and this document will provide a stamp of approval to poor districts (and the state) to continue or even eliminate providing such an option at their students. Overall, about 25% of US students take Algebra 1 in middle school and most have access to it. It is important to expand access to that option but the CMF only goes as far as saying that it is allowed but also continues with “evidence” not to do so. The “evidence” is not only extremely biased but is also flawed (see Brian #22). The expectation of a document of this importance is that it *should* make unbiased recommendations based on a careful scientific synthesis of research and evidence. The draft CMF is the opposite of that.

You make multiple statements. First, regarding taking Algebra 1 in middle school. You point into Ch8 806-810 on options to reach calculus to students that did not take Algebra 1 in middle school. I (and I am sure authors of both open letters) agree that such options are important — students that are ready for more and can learn at a faster pace or put more time should have options to do so. But what the document fails to do is (1) clearly emphasize that this is the LESS DESIRABLE option. That all effort should be made to provide access and support to students to take Algebra 1 in middle school. Then point on evidence (the SFUSD data is actually strong evidence for that) that this is the best path to reach calculus. Instead, by just casually mentioning (without real evidence and data) that such recovery is “possible” it still signals that offering Algebra 1 in middle school is not important since there is still a way to recover. (2) After eliminating Algebra 1 in 8th grade (under the consultancy of one CMF author) The qualified teachers at SFUSD tried hard to make “reduce repetition” work and in fact tried hard to trim content so that just the essentials necessary to “AP Calculus AB” (not even BC) are covered. This FAILED. The district eventually resorted to doubling math courses in high school (also problematic from wellness and other reasons like giving up electives). Many families used external math. Elizabeth #27 can tell you more on that. This suggestion of “removing repetition” is also problematic because what is important in the Algebra 2 content goes beyond just being a preparation for calculus. For example, probabilities and counting and complex numbers are not needed for AP calculus but are important foundations nonetheless. Also, “repetition” is known to be helpful in pedagogy where the same topic is introduced in one iteration and learned in greater depth at the next iteration (I think the technical term is “spiraling”). But the draft completely fails irresponsibly to mention that when suggesting “reducing repetition”. Please also look at the long paragraphs around the hesitant phrasing in lines 817-818 “it should continue to be possible…”.

Second, you state that the CMF only seems to support an “integrated pathway” rather than the (uniquely American?) “traditional pathway” That is, the sequence Algebra1,Geometry,Algebra2. This is a valid point that is not new and many places use integrated pathways. BUT the directive to the CMF authors is that they are actually not allowed to express preference on one pathway over another… But you seemed to have picked that bias up towards “integrated” regardless (!) And somehow you did not pick up the not-so-nuanced bias against middle school Algebra 1? My understanding of that sequencing is that there is a huge amount of inertia (textbooks, teacher training) in that “traditional” sequence. Changing that is immensely costly and it is also not the heart of the problems in math education here, so better perhaps to focus resources elsewhere. Additionally, the traditional sequence also allows for doubling for acceleration (Geometry can be taken concurrently to either Algebra 1 or Algebra 2 without changing the course offering of the school. But this is not possible with the “integrated pathway”.

Third, the “MIC” pathway. Brian #24 addressed that very well and there is more to say on that (in another post).

32. Scott Says:

Patrick Rogers #21: I’m grateful that Boaz Barak #22, Brian Conrad #24, Elizabeth Statmore #27, and Edith Cohen #31 have already answered you at length about the CMF document being “heavily cited.” Long story short, when you actually track down the citations, again and again they don’t even come close to supporting the uses that the CMF wants to make of them. And after you’ve seen enough examples of that sleight-of-hand, you stop being intimidated by the next one. So, that’s the answer to “how dare I put my mere lifetime spent in actual STEM education up against the ed-school folks’ piles of citations and studies?”

Having said that, I also wanted to reply to the following paragraph, whose “overly confrontational” nature you’ve already pre-apologized for:

As near as I can tell, the only real benefit of the current system is the inequity itself. If you’re lucky enough to perform well in the traditional tracking system, then you have a very valuable academic advantage that translates to real-world benefits in hiring and salary. Which raises the concern–to me at least–whether this opposition is simply people defending the current system because it’s one that allowed them to succeed, despite that fact that it also led too too many to fail.

Here’s a different way to describe the same situation. Some people, of all backgrounds, have put in years or decades of hard work to gain honest understandings of math and science, and now have careers reflecting that. Other people, resenting the “inequity” of this, want to redistribute those careers to people who haven’t put in the hard work. If they ever actually succeeded in their goal, it would be like the dog finally catching the car: bridges would collapse, planes would fall out of the sky, surgery patients would die. The much likelier scenario is that the students they’ve coddled will simply be in for a world of disappointment and hurt, when they discover that the “data science pathway” was not an equally-rigorous substitute for traditional math, and the colleges know that, and if they don’t then the graduate programs do, and if they don’t then employers do, and so STEM careers are closed off to them after all. Both of these scenarios suck and both should be prevented.

33. M DeYoung Says:

1+1 equals 2. Your PC, WOKE garbage will never change that (-:

34. Matt Says:

Scott #32

In all honesty, I think that’s far too charitable a reading of what Boaler et al are trying to do. If this were a good-faith attempt to address inequality, we could engage them in a conversation about what math education might actually look like in a fairer society.

Rather, it’s a blatant attempt to capitalize on genuine grassroots anger over inequality. The idea that achievement gaps stemming from poverty and historical inequalities can be fixed with pedagogical changes is a very attractive one to elected officials. You can tell voters you’re doing something to narrow the gap, and you don’t have to talk about addressing any of the economic factors that cause it. And when (predictably) nothing changes because childhood poverty still exists and school funding is still coupled to property taxes, you can just blame teachers for not implementing it correctly! If a consultant can slap enough buzzwords and citations together to sell this idea, they can make a killing.

It’s textbook rent-seeking.

35. AZ Says:

I very much admire your desire to fight against misguided attempts to replace important mathematical concepts with vague notions of data science or other more “marketable” mathematics. As a scientist myself, I’m all on board with your pushback against this ill-conceived restructuring of math: like you, I trust the judgement of scientists and mathematicians completely in what to teach, not the educators. But I think I’m hung up on what is perhaps a misunderstanding of what you’re pushing for, and I’d like to spread the message of your campaign if you can ease my fears. In fact, what I nitpick might be me looking too deeply at some comments you’ve made that are tangential to your main points.

In previous posts on this topic, you seem to imply that one your fears is the elimination or reduction in advanced, special-placement courses/tracks in K-12. But I think this conversation is lacking perspective from an academic who was -not- lucky enough to enjoy private education or special placement. In my youth in the US public school system, these “talented and gifted” advanced placement tracks were evil, discriminatory programs that gave teachers freedom to completely ignore the 95% of of us regulars; indeed, every day, they would make a massive show of the “smart” kids being taken out of class to get real education, while us normies were told to read out of the book or something. They selected 6-year olds (!), sealing everyone’s fate at such a young age and arbitrarily identifying some as smarter. This caused such incredible resentment in me that I was determined to become a scientist just to prove wrong all of those who shunned me over the course of 12 straight years, implicitly telling me how worthless I was because I wasn’t in the advanced track courses.

To be perfectly honest, I don’t think many people here have experienced the public education system, and suffered from the complete neglect and shunning from all educators that results from these fast-track, special placement programs. Private school experience does not give one a sense of the dark reality of what most students face. To support these programs almost always results in giving up on all of the regular students, at least in practice. To use the p-word: I think there’s too much privilege in the way to understand the terrible damage these programs do.

Any reform needs to be composed primarily of comprehensive protections for regular students, ensuring that the system doesn’t continue to focus on the top 5% exclusively. Then, only then, can reforms be considered for the top 5%: once the majority’s needs are met. Ignoring the bottom 95% is not a viable strategy. The needs of the 95% will always be more important than those of the supposedly elite 5%, and 95% are definitely suffering the most in public education. How many millions of physicists and mathematicians have been born but ignored, because they weren’t identified as adolescent geniuses at the absolutely insane age of 5 or 6?

36. Scott Says:

AZ #35: I confess that I’ve never, in all my years, heard of any public-school program anywhere in the US remotely like what you describe. “Gifted and talented” programs tend to be wildly under-funded relative to normal and remedial programs in the rare cases when they exist at all.

So I hope you don’t mind if I ask for more details: how were the 5% selected for the gifted program at your school? Was it on the basis of standardized tests, or just the subjective evaluations of teachers? If the latter, we may have located the problem right there.

Incidentally, a rigorous math track that includes algebra by 7th or 8th grade, and AP Calculus by 12th grade, isn’t the right thing for all students (or even a majority), but I think it’s definitely right for much more than 5%. It’s not merely the super-ultra-gifted who are at issue in this discussion: those I’d hope would’ve finished calculus well before 12th grade!

37. AZ Says:

Scott #36: While I can’t remember the exact details of my formative years in a midwest public school system circa 1990, I remember that in K-5, every day a staff member would drop by the class in the morning, loudly announcing “I’m here to pick up X, Y, and Z for talented and gifted lessons today,” after which they would disappear for the day. Maybe this stuff is more prevalent in, shall we say, rural areas? But maybe that isn’t the most important point.

But I think the relevant point is this: there is -no- criteria by which you accurately judge most people’s intelligence prior to middle school school, and any such judgement call is going to severely damage the lives of the highly intelligent people that are late bloomers and get passed over—because they do exist: every single person who was not selected as gifted, but becomes successful, has been failed by this system. Who knows, maybe half the population could be high achievers if they got the attention they needed? And while I have no recollection of such gifted programs beyond K-5, it is also true that once you reach 6th grade and your math class has been selected for you by some standardized process, it is exceedingly difficult to move ahead, as math classes (in my school) are year-long and considered sequential, such that skipping to reach higher classes is exceedingly difficult. The key is that your fate is likely sealed at an early age, and any such selection is wrong. And even if something like 20% of students have the pleasure of partaking in calculus by 12th grade, it’s society’s duty to be primarily concerned with the lower 80%. Not only is it necessary to fight back against misinformation and the demonization of education in America, that 80% surely contains geniuses waiting to emerge from their shell.

Suffice to say, I don’t think the education of the best students is America’s biggest problem right now: the challenges we face are entirely because an uneducated and antiscientific underclass has been allowed to rise because they effectively weren’t taught anything for 13 years, while the smartest students were focused on. Everyone deserves quality education. Focusing on the minority isn’t what the education system needs the most, that seems totally obscurantist to me.

38. David Karger Says:

Scott #4. I’m pleased but not surprised. Activism does work, at least some of the time.

39. Scott Says:

AZ #37: The trouble with that argument, it seems to me, is that it can be extended arbitrarily forward in time. At whatever age you pick out some people as being better at some activity than others, there are going to be people who would only “bloom” at that activity at a later age, and who you’ve in some sense unfairly passed over. So other than “Harrison Bergeron,” what’s the alternative, other than simply to recognize the late bloomers whenever they do bloom?

40. Zen Cheruveettil Says:

@Scott

Once, I happened to read a blog which was published by Keith Devlin – a colleague of Jo Boaler. In that article, he was mentioning that he is OK with the concept of “giftedness”. However, he was suggesting not to use calculus in K-12 because of the difficulties involved in teaching (and learning) it at a suitable and required depth.

That is an argument to which I can personally relate with. My personal experience was that it was possible to achieve certain level of procedural mastery of basic calculus in high school, but a conceptual mastery was possible only after achieving a higher level of mathematical maturity (even then delta-epsilon definition was sort of a brainf**ck). So I suppose those who skip Calculus in college (which as far as I understand, is a common practice in US) may end up having inadequate foundation.

In any case, I*ve often wondered when I read stories about Terence Tao or Von Neumann — those who had taught themselves Calculus by 8 or 10 — genius they are, however to what level and how deep they could learn by that age is not very clear from various anecdotes.

41. Aaron Denney Says:

> there is -no- criteria by which you accurately judge most people’s intelligence

This is generally true. It’s a lot less true about specific subsets such as “math” or even more specific “calculus”.

The general tying of all curricula to a specific “grade level” is a great tragedy. People are different and handling this diversity is vital for treating the students well.

42. Mark Hillery Says:

I was not aware of Section 18533 and was going by what was stated on the California Department of Education website: “Commenters may submit comments using the following methods:

Via email to mathframework@cde.ca.gov. This is the preferred method of receiving comment.”

It then goes on to give a physical address. I wonder what happens to the comments that are only submitted by email.

I would like to thank everyone who organized the petitions and produced the statements pointing out the flaws in the CMF and for actually checking the validity of the citations. Here, for what it is worth, is part of what I sent in via email to the above address:
One of the main reasons the framework is fatally flawed, is that it did not seek any input from people who teach mathematics in college. The designers of the framework claim that they want to emphasize 21st century mathematics, in their view data science, over 20th century mathematics, the current curriculum. Is there anyone on the committee that wrote the framework who knows what data science actually is? Something like this clearly calls for the involvement of university-level people who know the math, which is quite sophisticated, and should not just rely on people from mathematics education who do not have the necessary knowledge. Computer scientists who know data science say that the current curriculum is far better preparation for studying data science at the university level than the curriculum proposed in the new framework. And another point, are people in mathematics education the best ones to decide what is 21st century mathematics? This seems rather presumptuous, since they do not do actual mathematical research, and, furthermore, it is doubtful that this phrase even has any meaning. The point of high school mathematics should be to provide a solid foundation so that when students get to college, they can go off in any direction they choose. So, in view of these comments, why were university-level people excluded from the design of the framework? This was not an oversight. In one article I read, Prof. Jo Boaler from Stanford, the animating spirit behind the framework, said it was deliberate: “We understand education, and they have no experience studying education. Mathematicians sit on high and say this is what is happening in the schools.” While Prof. Boaler, who does not have any degree in mathematics, may be a “rock star” in math education circles, in my experience, she is viewed very negatively in university STEM departments to the extent she is known there. Her work has been justifiably criticized by mathematicians, criticisms she never attempted to answer. A study by Bishop, Clopton, and Milgram showed that her “reforms” at “Railside High School” (a pseudonym) had the effect of significantly increasing the percentage of students from that school who needed to take remedial mathematics when they got to college, and I am afraid the proposed framework would have the same effect. Her papers and books look impressive at first glance, but contain unjustified claims, and the papers she cites to support those claims often do not say what she claims they do. This is a pattern of carelessness at best. She didn’t want college professors from mathematics or computer science departments on the framework committee, because, if her past tangles with mathematicians and two recent petitions are a guide, they would have strongly disagreed with what she was trying to do. Why California chose someone with her record to revise their mathematics framework is beyond me.

Mark #42:

If you’d like to see how public comments seem to be treated, you can see what happened to the public comments for the first draft in a .doc file in the agenda for the May 19-20, 2021 meeting of the Instructional Quality Commission at

https://www.cde.ca.gov/be/cc/cd/may2021iqcagenda.asp

Within item 3.A.1 in the Agenda near the bottom, click on the “Attachment 1” link. Scroll down to see in the right column what decisions were made.

44. Claire Ralph Says:

Does anyone know of a side-by-side comparison between the proposed standards buried within this rambling document and the current standards? I’m trying to better understand what concepts from algebra one and two, specifically, would be discarded.

Every critique I’ve seen that cites a specific line in the document is critiquing the philosophic prose, which I agree is often problematic, and a good editor would have cut it. That said, if this document had, had a good editor and been cut down to the 150ish pages it should have been, would there be so much objection to it?

Several of the critiques I’ve seen addressing the curriculum itself, rather than the justification/ rationalization for this curriculum, equivocate ‘data science’ and ‘data literacy,’ defining the latter and then arguing against the former. It’s not hard to agree that ‘data literacy’ should not replace a traditional algebra one or two course, but one could easily teach the same mathematical concepts taught in algebra one and two through the lens of ‘data science.’ When I read the actual standards being proposed for these new classes, that seems to be what they are doing. Am I missing something?

45. Mark Santolucito Says:

Just want to say I appreciate the discourse here. It is great to see STEM experts (and really, absolute pinnacle of human understanding, experts) dedicating time to k-12 education policy. I’m heartened to see it having some impact – hopefully more to come.

Claire #44: The tables of learning standards for MIC 1 and MIC 2 (see Appendix A) are literally identical to those for Integrated 1 and Integrated 2, and in Chapter 8 (lines 914-915) it is says that MIC 3 has the same learning standards as Integrated 3 (though MIC 3 is never described in any actual detail, neither in Chapter 8 (despite line 912 having the phrase “MIC 3 course described below”, there is no such “below” in Chapter 8) or Appendix A, and it is left somewhat up in the air what it is, though lines 912-913 of Chapter 8 says “MIC 3 is *not* a data science course.

In a nutshell, MIC is just claimed to be exactly the Integrated sequence with everything motivated by problems in statistics (CMF always says “data science”, but that’s just because CMF is enamored of that buzzword; everywhere it says “data science” is just topics which have always been called part of “statistics” at the high school level). This is not a serious proposal in several ways:

1. The CMF doesn’t give any details on teaching the Integrated courses through the lens of statistics. Nowhere does it go through a lot of the standards and provide suggestions like “here is a statistics topic motivated that”. So they haven’t done any of the hard work to illustrate how this would be done. One may as well say “We encouraged to teaching the integrated pathway through problems coming from [pick your favorite field]”.

2. There is no reason for this to be specific to the integrated pathway (except that one of the CMF writers is on the record as wanting to get rid of the traditional pathway because…well, something about it having had problematic outcomes somehow being the fault of the content rather than of archaic teaching methods).

3. Given that this is just the usual content with improved motivation, there is no reason *all* the motivation should have to come from statistics; could be some from chemistry, or probability, or geology, etc. So in the end the entire meaning is “teaching with improved motivation” — which is very apt advice, yet when it isn’t accompanied by detailed guidance (setting aside a smattering of vignettes which hardly constitute guidance on how to do this coherently for an entire course) there is no substance behind the proposal. One may as well also say “All teachers will be good” and give no details either.

There is no serious reason for the CMF to speak of a new third pathway; it should just give guidance on how to enrich the existing pathways with improved motivation in a coherent way (the CMF writers were warned by the State Board of Education and a staffer back in August 2020 that it is forbidden for them to show a bias towards a particular pathway.

So MIC is a backdoor to create the (false) impression that somehow the integrated pathway is better-suited to modern motivation than the tradition one (should help in trying to kill off the traditional pathway, which one CMF writer wants to do). And by promoting specifically the lens of statistics — oops, data science — to supply all motivation, it helps to support the narrative of the magic of data science which in turn will help in convincing ever more schools to offer data science courses in future years (ideally in place of that pesky Algebra II, a replacement the data science advocates love to promote). That in turn leads to more professional development fees and high profile for the data science advocates. Follow the money.
.

Claire #44: For the omission of Algebra I and Algebra II topics, that is for the “data science” courses. For example, the most widely-taught one (IDS from UCLA) has its entire contact with the Common Core contained within statistics standards. This is shown at

So there you see it has no logarithms, no exponentials, no unit circle, no anything in Algebra II that isn’t statistics standards. The other big data science course is similar. In Chapter 5 there is a description of an “advanced data science” course from page 66 onwards; the first 10 pages there are rambling without content, and then it gets into the proposed standards, which are again all statistical (or CS), except for some brief mention of “vectors and matrices” (a pre-calculus standards). It’s not clear how this is coherent, but anyway that too omits logs, exponentials, unit circle trig, and everything else in Algebra II that isn’t statistical standards.

Coupled with the advocacy to take data science in place of Algebra II, it leads to the concerns which are addressed by the recent statement on math & data science that is the focus of this blog post.

48. Matt Says:

Zen #40:

That doesn’t match my experience at all.

I teach intro physics at a highly selective US research university. My students have a mix of math backgrounds—most took AP calc at some point in high school, others learned it their first year of college. Almost universally, the students who struggle the most are the ones who waited the longest to take calculus. If our goal is to produce students with a deep understanding of calculus, we need to ensure that they see it as early as possible.

“Mathematical maturity” is a big part of this. But crucially, maturity comes from seeing lots of new math and trying to build things with it. Delaying advanced math doesn’t mean students will be “mature” when they take it. It just means they’ll take longer to achieve that maturity in the first place. Not to mention, a lot of them will be bored to tears in high school (ask me how I know this!).

49. Andrew Says:

I took algebra in 8th grade at an international school and to this day it was one of my hardest classes (I have CS degree). Kinda stressful but I see no reason to change anything. Calculous is beautiful in a way, it’s not just about rules and solving problems but learning to see the world differently. Why would we take that away?

Andrew #49: The answer to your question is that those who are loudest in denigrating calculus (and algebra) don’t really know the subject, nor how it is relevant to anything outside physics. That makes it easy for them to advocate for all sorts of things without genuinely addressing downstream effects in college courses that are evident to those who do know such things.

51. lin Says:

Matt #48: this is a great point. The way to achieve mathematical maturity is to repeatedly attempt to learn math that is too advanced, fail (e.g. by only picking up some rote formulas without real understanding), and try again until it clicks the third or fourth time. People are always looking for ways to get around this process because it sounds and is very annoying, but as far as I can tell it is the only way to do it (even into grad school and beyond…)

52. Ralph Kelsey Says:

Aaron Denney #11

Agree.

But some very simple calculus goes a long way.

1. I am evidently in the minority (of those with PhD’s in math) in thinking that there is way too much theory in beginning calculus. The key idea in calculus is linear approximation, which is EASY, EASY, EASY. Proving stuff with deltas and epsilons is 100X harder. It is certainly essential for those who will go on and study math, but for those who are doing statistics???

2. A huge problem in teaching stats and data science is that any nontrivial problem requires a lot of arithmetic, so anything non trivial tends to get lost in the woods. This is different than physics, where you can go a long way and still get simple numbers.

3. A reasonable way of dealing with #2 is doing most things with MS Excel. Today’s college students all seem to have grown up with Excel, so they are not intimidated, they all have it (maybe on their phones?), and it is a really good skill in general. A lot of us CS types are not fond of MS products, but if you have had to use Blackboard, you learn to like MS.

53. mls Says:

Thank you for your summary in #47

@ all

There is a deeper problem at work here. A paradigmatic example of it might be found in Comment #8 by Jon Awbrey in the Form and Meaning thread. Or it can be found in a Lubos Motl post which had been motivated by comments made by Timothy Chow in one of Dr. Aaronson’s posting on artificial intelligence. By identifying the exclusion of transcendental functions from the proposed curriculum, Mr. Conrad’s summary places the proposal squarely into a historical debate between empiricism and idealism.

When Matt #48 speaks of “mathematical maturity” and Andrew #49 speaks of “seeing the world differently” they are making comments eloquently ridiculed with Jon Awbrey’s “Elf on the Shelf” characterization of any philosophy trying to portray semantics as meaningful. (Personally, I have learned to differentiate between syntax, semantics, and pragmatics because semantics ought to refer only to the stipulated interpretations justifying steps in formal derivations. Meaning for language users — that is, pragmatics — is only tangential to the role of semantics in a proof theory.)

So, why are transcendental functions important for a broad view of mathematics?

Vaughan Pratt is a proponent of an algebraic foundation for mathematics. He has written an extraordinary entry on Algebra for the Stanford Encyclopedia of Philosophy. Ultimately, however, the relationship of algebra to the debate between empiricism and idealism rests with the evolution of universal algebra from de Morgan’s work on symbolic algebra. Read George Graetzer’s book on universal algebra and you will discover that everything is stipulated to be well-formed prior to any mathematical analysis whatsoever.

This is contrary to ordinary mathematical practice where definitions are not assumed to be automatically satisfied. Ordinary mathematicians substantiate definitions with examples demonstrating that the proposed definition is not vacuous. This is precisely why I attempt to direct people’s attention to what is written in Section 74 of Kleene’s “Introduction to Metamathematics.” The first sentence or two is the only place in the entire corpus of foundational literature (with which I am familiar) that explains how a correct formalization of ordinary mathematics ought to appear.

Where transcendental functions meet algebra as it relates to the foundations of mathematics is undecidability. Alfred Tarski showed that the theory of closed real fields is decidable. As described in the Wikipedia link on o-minimal structures, this decidability result can be partially extended with restricted analytic functions. However, the unrestricted use of trigonometric functions introduces undecidability. I believe this is stated explicitly in the Wikipedia link on Richardson’s theorem.

There is another subtle effect of algebra from a different direction. Category theory has basis closely related to part-whole relations. This is contentious in the foundations of mathematics because advocates of the first-order paradigm will criticize this as a second-order construct for which no definite semantics can be given. Following remarks found in Russell, significant import is given to Peano’s introduction of the symbol for membership. Lawvere and Rosebrugh will say that philosophy departed from mathematics when veering away from Dedekind. Michael Potter accuses Dedekind of a mereological conflation made evident by Peano’s membership relation.

The problem exposes itself when one asks whether the subset relation from set theory can be a mereological part relation. Joel David Hamkins studied this. The resulting theory is decidable.

To the best of my knowledge, the fundamental theorem of algebra has not yet been proven with a purely algebraic proof.

Having been ridiculed by “the finest minds in the world” for my understanding of mathematics and my attempts to have competent discussion, I could give a flying F___ about what gets taught. Historically, this has never been decided.

54. Peter Gerdes Says:

Fair enough, it may well be that those materials for an appropriate data science based course don’t currently exist.

55. Sniffnoy Says:

To the best of my knowledge, the fundamental theorem of algebra has not yet been proven with a purely algebraic proof.

This is false. (After all, it works not only for C, but for K(i) for any real-closed field K, so there had better be an argument not depending on particularities of C like its topology!) Wikipedia discusses algebraic proofs here.

Of course, it leads with the following disclaimer:

These proofs of the Fundamental Theorem of Algebra must make use of the following two facts about real numbers that are not algebraic but require only a small amount of analysis (more precisely, the intermediate value theorem in both cases):

• every polynomial with an odd degree and real coefficients has some real root;
• every non-negative real number has a square root.

…i.e., you have to make use of the fact that R is a real-closed field. Which, I mean, of course you do. But note that that is an algebraic property of R; the proof assumes an algebraic property of R (that it is real-closed), and, using solely algebraic reasoning, concludes an algebraic property of C (that it is algebraically closed). It is, undeniably, an algebraic proof of the algebraic fact that if K is real-closed then K(i) is algebraically closed.

I do not know if you will find that sufficient, or if you will object that analysis is still be used to prove that R is real-closed. But I will note in advance that I do not consider this a reasonable objection. No proof that is meant to prove things purely about R, as opposed to other fields or ordered fields, can be purely algebraic, because the fact that makes R distinct from other ordered fields is its Dedekind-completeness, and if you use that, you’re no longer writing a purely algebraic proof. So things like “if K is real-closed, then K(i) is algebraically closed” can be proven in a purely algebraic manner, but to apply this to R in particular will always involve an analytic step, because you have to use R‘s distinguishing property somewhere, otherwise you’re just proving a result about ordered fields in general.

So, by any standard that I would consider reasonable, yes, the fundamental theorem of algebra absolutely has been proven in a purely algebraic manner.

(Hell, and then you’ve got the Artin-Schreier theorem, which goes further, and says that real closed fields are the only fields other than algebraically closed fields such that $$[\overline{K}:K]$$ is finite, and of course that’s proven purely algebraically…)

56. Tech roundup 139: a journal published by a bot - Javi López G. Says:

[…] An update on the campaign to defend serious math education in CaliforniaUpdate (April 27): Boaz Barak—Harvard CS professor, longtime friend-of-the-blog, and coauthor of my previous guest post on this topic—has just written an awesome FAQ, providing his pers… […]

57. Bystander Says:

Upstairs, in a combined history and math class, students use statistics to find patterns in the rise and fall of nations.

Some schools in China apparently teach statistics, yet I guess that they have math background for that. BTW With the decay of US education, you will be an example in that class.

58. SFUSD Teacher Says:

I’m a San Francisco math teacher in the public school district. I’m 100% liberal, anti-racist, pro-union, and a life-long Democrat. For what it’s worth, I’ve been a math teacher for 26 years, and I’ve taught every course from middle school to high school. I have seen a marked decrease in the core math skills of my students over the past 10 years, and this exactly correlates with the adoption of Common Core, and it’s gotten worse ever since Stanford University (i.e. Dr. Boaler, et. al.) started a strategic partnership with the district.

From my own admittedly cursory research, it seems like the problems begin in elementary school, where teachers are no longer encouraged to have students memorize their multiplication facts. In middle school, teachers are being encouraged to de-emphasize algorithms in favor of “meaning making,” which appears to mean talking to one another about math ideas in small groups. Any kind of skill concentration or practice is derided as “drill and kill”. Testing is also disappearing, in favor of “formative assessments” which don’t count toward students’ GPAs. Students are entering 9th grade not having multiplication facts memorized, not knowing how to do operations with positive and negative integers, or knowing how to solve one- and two-step equations. It’s never been this bad, not in my entire career.

Finally, imagine this situation, and maybe it’ll shed some light on such things as the achievement gap:

A student does nothing in 6th grade. No work in class. They roam the halls every chance they can get. They do no homework. They’ve learned no math in 6th grade. Since there is no remediation of any kind, or ability grouping, they are passed to 7th grade. The child repeats the maladaptive behavior in 7th grade. Again for 8th grade. Now the student is in 9th grade, having learning nothing in the last three years, and they are woefully behind their peers. Again, no remediation of any kind, no ability grouping.

Imagine a different student whose parents pay for them to attend online math tutoring courses, or tutoring programs like Kumon, every year, from 6th grade to 9th grade, because the parents feel that the district is not teaching enough math. Sometimes these students work straight through their summers.

Now, imagine that these two students are in the same 9th grade class. This is the real achievement gap. It has nothing to do with race, ethnicity, gender, sexual orientation, or socioeconomics, but rather the district’s response to avoidance, dissociation, and other maladaptive behaviors, including disruptive ones, that enable students to fail. Intervention should have happened in 6th grade. It didn’t.

I just attended a PD at my school where the speaker – a non-math teacher – claimed that “math is racist” and that we need to start “equitable” grading, which apparently translates to passing students who do nothing.

This is the state of SFUSD math.

59. Scott Says:

SFUSD Teacher #58: I’m sorry to hear that things have gotten so bad. Nevertheless, thank you for sharing your on-the-ground experience with this.

It shouldn’t surprise anyone that, when you remove the incentives for teachers to teach and kids to learn, things eventually regress to a state where the teaching and learning don’t happen. Even then, though, some people might be surprised by how quickly they regress.

60. Wujing Harrison Says:

Thank you, SFUSD Teach #58, for sharing the reality some just don’t see or judge with their reasonings we don’t buy.
Thank you, Scott, Boaz Bara #22 (I read more in your blog), Brian Conrad #24, #43 (especially the link to the document on how public comments were treated), #50, Elizabeth Statmore #27, Steven Gubkin #29, Edith Cohen #31, Ralph Kelsey #52, and more, for the informative and insightful sharing.
At this stage, I will send in my comments for the sake of records. However, we need to lead an effort to offer a concise, clear and coherent standards framework (less than 100 pages if possible, appendices can be used for supplemental information). @Stevn Gubkin, if you could dedicate and develop a system with appropriate tasks, I think it will be super valuable and I will use it. I will like to be part of the effort to create better alternatives. Then, we are in a more effective position to compare notes with stakeholders in a professional manner.

61. Elizabeth Statmore Says:

SFUSD Teacher #58: Thanks for sharing your experience. I think we’re in the early days of a sea change in SFUSD. It pains me that our least-reached students are being harmed in these ways. This is why we’re fighting to improve programs for our students, especially in math and reading. I’m cautiously optimistic, at least more so than I was before the BOE recall, but we need to keep fighting. Keep fighting for our students.

62. Brian Conrad Rips into the California Mathematics Framework – BAD MATHEMATICS Says:

[…] well as providing links to commentary critical of the CMF (here, here and here), Conrad has posted two articles of his own.  Conrad’s first article, written with […]

63. Veronica Says:

Sfusd and Scott:

You may want to check out Philip Bobbitt’s Shield of Achilles. He and a bunch of other ir professors argue that, since the 1980s/1990s, the Federal government has had a very specific vision of what it wants American people and government to be like in 2030 — friendly consumers that work pointless retail jobs like on Raising Hope. Dumbing down the curriculum, small group “meaning making”, and so on is really what that’s all about. It is how you get a new generation of good retail employees while opening a whole new market (i.e., schools).

Wujing Harrison #60: I have posted my initial document of comments about the current CMF at a new website https://sites.google.com/view/publiccommentsonthecmf/ . More coming soon.

65. Claire Ralph Says:

Brian #46-47: Thank you for responding so thoughtfully. Your point about the CMF not having done the hard work of developing good examples of teaching algebra I & II through the lens of data science/ statistics resonates.

Personally I would very much like to see that type of curriculum developed. Many concepts in ml do not require calculus and could be taught to a much wider audience as part of the traditional or integrated pathway. The best work I’ve seen in this area is that done by AI4All. I was hopeful the CMF would be another step towards that, but unfortunately not…

66. Yisong Yue Says:

The CMF website seems to have a slightly different process for submitting comments compared to Section 18533.

Specifically:

• Although the street address is the same, the CMF website suggests to send it to the “Instructional Quality Commission” whereas Section 18533 is to the “State Board of Education”
• The CMF website states that “Please note that for security reasons we cannot forward hyperlinks to personal mailboxes or online storage sites; any such links will be redacted. Please include the full content of your comment either in your email or as an attachment.”. I interpret this to mean that one should not link to Brian Conrad’s comments, but attach them in full as an appendix to a letter one writes.

Thoughts?

67. Wujing Harrison Says:

Thank you, Brian Conrad, #64, for the link. Thank you for standing out! Salute!