## An alarming trend in K-12 math education: a guest post and an open letter

Updates: Our open letter made the WSJ, and has been tweeted by Matt Yglesias and others. See also Boaz Barak’s thread, and a supportive tweet from Rep. Ro Khanna (D-Silicon Valley). If you’re just arriving here, try TodayMag for an overview of some of the issues at stake. Added: Newsweek. See also this post in Spanish. And see a post by Greg Ashman for more details on what’s been happening in California.

Today, I’m turning over Shtetl-Optimized to an extremely important guest post by theoretical computer scientists Boaz Barak of Harvard and Edith Cohen of Google (cross-posted on the windows on theory blog). In addition to the post below, please read—and if relevant, consider signing—our open letter about math education in the US, which now has over 150 now 535 746 952 1225 1415 signatories, including Fields Medalists, Turing Award winners, MacArthur Fellows, and Nobel laureates. Finally, check out our fuller analysis of what the California Mathematics Framework is poised to do and why it’s such an urgent crisis for math education. I’m particularly grateful to my colleagues for their writing efforts, since I would never have been able to discuss what’s happening in such relatively measured words. –Scott Aaronson

Mathematical education at the K-12 level is critical for preparation for STEM careers. An ongoing challenge to the US K-12 system is to improve the preparation of students for advanced mathematics courses and expand access and enrollment in these courses. As stated by a Department of Education report “taking Algebra I before high school … can set students up for a strong foundation of STEM education and open the door for various college and career options.” The report states that while 80% of all students have access to Algebra I in middle school, only 24% enroll. This is also why the goal of Bob Moses’ Algebra Project is to ensure that “every child must master algebra, preferably by eighth grade, for algebra is the gateway to the college-prep curriculum, which in turn is the path to higher education.”

The most significant potential for growth is among African American or Latino students, among whom only 12% enroll in Algebra before high school. This untapped potential has longer-term implications for both society and individuals. For example, although African Americans and Latinos comprise 13% and 18% (respectively) of the overall US population, they only account for 4% and 11% of engineering degrees. There is also a gap in access by income: Calculus is offered in 92% of schools serving the top income quartile but only in 77% of schools serving the bottom quartile (as measured by the share of students eligible for free or reduced-price lunch). Thus minority and low income students have less access to STEM jobs, which yield more than double the median salary of non-STEM jobs, and are projected to grow at a 50% higher rate over the next decade.

Given these disparities, we applaud efforts such as the Algebra Project, the Calculus Project, and Bridge to Enter Advanced Mathematics that increase access to advanced mathematical education to underserved students. However, we are concerned by recent approaches, such as the proposed California Math Framework (CMF) revisions,  that take the opposite direction.

See this document for a detailed analysis and critique of the CMF, but the bottom line is that rather than trying to elevate under-served students, such “reforms” reduce access and options for all students. In particular, the CMF encourages schools to stop offering Algebra I in middle school, while placing obstacles (such as doubling-up, compressed courses, or outside-of-school private courses) in the way of those who want to take advanced math in higher grades. When similar reforms were implemented in San Francisco, they resulted in an “inequitable patchwork scheme” of workarounds that affluent students could access but that their less privileged counterparts could not. The CMF also promotes trendy and shallow courses (such as a nearly math-free version of  “data science”) as recommended alternatives  to foundational mathematical courses such as Algebra and Calculus. These courses do not prepare students even for careers in data science itself!

As educators and practitioners, we have seen first-hand the value of solid foundations in mathematics for pursuing college-level STEM education and a productive STEM career.

While well-intentioned, we believe that many of the changes proposed by the CMF are deeply misguided and will disproportionately harm under-resourced students. Adopting them would result in a student population that is less prepared to succeed in STEM and other 4-year quantitative degrees in college.  The CMF states that “many students, parents, and teachers encourage acceleration beginning in grade eight (or sooner) because of mistaken beliefs that Calculus is an important high school goal.” Students, parents, and teachers are not mistaken. Neither is the National Society of Black Engineers (NSBE), which set in 2015 a goal to double the number of African American students taking calculus by 2025. Calculus is not the only goal of K-12 math education, but it is important for students who wish to prepare for STEM in college and beyond.

We agree that calculus is not the “be-all and end-all” of high-school mathematics education. In particular, we encourage introducing options such as logic, probability, discrete mathematics, and algorithms design in the K-12 curriculum, as they can be valuable foundations for STEM education in college. However, when taught beyond a superficial level (which unfortunately is not the case in the CMF “data science” proposals), these topics are not any easier  than calculus. They require the same foundations of logic, algebra, and functions, and fluency with numbers and calculations. Indeed, the career paths with the highest potential for growth require more and deeper mathematical preparation than ever before. Calculus and other mathematical foundations are not important because they are admission requirements for colleges, or because they are relics of the “Sputnik era”. They are important because they provide fundamental knowledge and ways of thinking that are necessary for success in these fast growing and in-demand fields.

We also fully support incorporating introductory data analysis and coding skills in the K-12 curriculum (and there are some good approaches for doing so).  But we strongly disagree with marketing such skills as replacing foundational skills in algebra and calculus when preparing for 4-year college degrees in STEM and other quantitative fields. These topics are important and build on basic math foundations but are not a replacement for such foundations any more than social media skills can replace reading and writing foundations.

Given the above, we, together with more than 150 scientists, educators, and practitioners in STEM, have signed an open letter expressing our concerns with such trends. The signatories include STEM faculty in public and private universities and practitioners from industry. They include educators with decades of experience teaching students at all levels, as well as researchers that won the highest awards in their fields, including the Fields Medal and the Turing Award. Signers also include people vested in mathematical high-school education, such as Adrian Mims (founder of The Calculus Project) and Jelani Nelson (UC Berkeley EECS professor and founder of AddisCoder) who have spearheaded projects to increase access to underserved populations.

We encourage you to read the letter, and if you are a US-based STEM professional or educator, consider signing it as well: https://bit.ly/mathedletter

Unfortunately, in recent years, debates on US education have become politicized. The letter is not affiliated with any political organization, and we believe that the issues we highlight transcend current political debates. After all, expanding access to mathematical education is both socially just and economically wise.

Note: The above guest post reflects the views of its authors, Boaz Barak and Edith Cohen. Any comments below by them, me, or other signatories reflect their own views, not necessarily those of the entire group. –SA

### 183 Responses to “An alarming trend in K-12 math education: a guest post and an open letter”

1. Boaz Barak Says:

I hope this can make a positive change towards increasing access to advanced math education to more students.

2. Boaz Barak Says:

Also, the “Bridge to Enter Advanced Mathematics” (BEAM) project is currently having a 10x match program for new donors (and πx match for returning donors). If you want to support projects that make a real difference in expanding access to math, take a look at https://www.beammath.org/donate

Another project worth supporting is (co signer) Adrian Mims’ calculus project https://secure.actblue.com/donate/thecalculusproject

3. Wolfgang Richter Says:

Scott thanks so much for posting this!

I consider Computer Science to be on the tree of mathematics, and wondered if you do too?

Mathematics just encompasses so much of all that everyone does today! I definitely do not want to see any steps back in math education, and would much rather see steps forward.

4. DavidM Says:

They’re mad, these Americans. Can it really be true that in the world capital of progressivism people who would presumably talk your ears off about equity and social justice have created a system that allows kids to test out of easy classes, but only if they went to a private school or could afford to pay for an accredited course?

5. Scott Says:

Wolfgang Richter #3: CS is famously a hybrid field, with some aspects that are fundamentally math, some aspects that are actually empirical science, and some aspects that are engineering or even art—united only by the shared interest in computation. You even see that in university organization: CS departments are sometimes part of arts and sciences colleges, sometimes part of engineering colleges, and sometimes even adjoined to math departments. I’ve personally worked or studied at universities with all three arrangements.

6. DR Says:

I keep hearing about “Why Calculus?” in connection with high school math. My response is always, why a one-size-fit-all? Different people might want different things in their child’s education. They are all right, for THEIR respective kids. I think the bigger question everyone is hesitant to ask, is why there is a one-size-fit-all solution. I think that is the right question.

7. Stefan Krastanov Says:

I do not believe this set of signatories is what would give credence to this letter. University professors are good researchers and in some cases they are even be good college instructors, but they are very rarely good K12 instructors. At least in my experience of organizing K12 outreach events on university campuses. It would be much easier to convince me this is indeed a worrying trend if I see K12 educators and researchers studying K12 education and child development sign this letter. Otherwise some kind of logical fallacy is creeping in. My (admittedly flawed and biased) reading of the new CA plan says “gifted programs are fine, but we need to focus on the basics in a way that works for all K12 students”. Lifting all boats, etc etc.

8. Jr Says:

I am skeptical that there is some great benefit to be gained by trying to extend how many minority students take advanced math courses. In particular, the framing contributes to a major problem in western society today, where underperformance of minority groups is seen as a sign of societal oppression, instead of a reason for the minority group to assimilate better.

9. Scott Says:

Stefan Krastanov #7:

My (admittedly flawed and biased) reading of the new CA plan says “gifted programs are fine, but we need to focus on the basics in a way that works for all K12 students”. Lifting all boats, etc etc.

Alas, that’s indeed a flawed and biased reading! Joanne Boaler, the architect of the new CA plan, has gone on record wanting to do away with gifted programs and academic acceleration entirely; she does not regard them as “fine.” And if you want to focus on basics in a way that works for all students—well, what could be more basic than algebra?

University professors are good researchers and in some cases they are even be good college instructors, but they are very rarely good K12 instructors … It would be much easier to convince me this is indeed a worrying trend if I see K12 educators and researchers studying K12 education and child development sign this letter.

Among our drafters are people like Boaz Barak, Jelani Nelson, Adrian Mims, and yours truly, all of whom have been directly involved in various kinds of K-12 outreach.

By contrast, the new California math framework seems to have been developed entirely by ed-school types, without meaningful input of any kind from STEM professionals. And that seems directly related to why the framework document is so wilfully oblivious to what students pursuing STEM degrees in college will actually need as preparation.

10. arcana Says:

The California plan is bad IMO but it the problem is very complicated to solve and I’m not sure the signatories deal with that at all. Every child can’t and shouldn’t master algebra by 8th grade. Not that it matters since no existing form of modern education could achieve that anyway.

First we shouldn’t be teaching a ton of standardized math until ~11 anyway. This strongly results in a perhaps majority of people who think they are “bad at math”. They strongly believe they aren’t and can’t be good at it. Even Scott Siskind famously can’t believe he could be great at math.

It is possible we could identify the causes of some kids having an aptitude for and more importantly an interest in accelerated math development but we haven’t and force feeding math to people who aren’t yet ready makes them hate it. Something similar operates on the infamous high school “english lit and text interpretation classes” although that problem is not nearly as critical.

Having tools for emotional self-regulation, stronger executive function, and better abstract thinking would allow even kids who are not ideal candidates for mathematical acceleration to at least avoid deeply ingrained anti-math thoughts. Additionally there is strong evidence that students wouldn’t particularly suffer by removing standardized mandatory math classes during elementary school. And you could still allow kids who are talented at or enjoy math to take early advanced courses. One size fits all math education is terrible inherently and our current implementations are extra bonus bad.

11. Boaz Barak Says:

This is now on Hacker News as well https://news.ycombinator.com/item?id=29432816

12. Boaz Barak Says:

Stefan Krastanov #7: STEM University educators should not be the only or even main people making choices about K-12 math education, but our input is relevant. After all, one of the goal of K-12 math education is to prepare students for college, and we know what will be needed there.

Also, I don’t think the CMF approach can be called “lifting all boats”. There are approaches (such as the Algebra Project and the Calculus Project mentioned above) that are aimed at “raising the floor”.

For example, in the context of Algebra I, raising the floor is to find ways to increase the fraction of students taking Algebra I before 9th grade by allocating more resources to under-served schools. This however is not the approach taken by the CMF. As we’ve seen in SF, for districts that adopt the recommendations, only students that go to private middle school or take a private out-of-school course will be able to take Algebra I before 9th grade. (Realistically, since these are only recommendations, what will happen is that wealthy public school districts will likely keep offering Algebra I in middle school, while poorer ones won’t.)

Instead the claim is that it’s not important to take Algebra I before 9th grade. What we need is to improve math education, not shift the goalposts.

13. John Lawrence Aspden Says:

> any more than social media skills can replace reading and writing foundations.

This is not a very good analogy, almost all modern reading and writing is done on social media, and if I met someone who was brilliant at social media I’d expect them to be a brilliant reader and writer too.

Someone who can write a good blog probably has all the skills necessary to write a good book. Certainly more so than someone whose only qualification is that they suffered through school or college English.

But if you can’t do calculus or algebra, you can’t understand very much maths or very much of any kind of science, or indeed very much about the world around you at all.

14. Scott Says:

DR #6 and arcana #10: Speaking for myself, not necessarily for all signatories, I’ve always been a huge proponent of choice in education and the impossibility of a one-size-fits-all solution. Mathematical ability is not the yardstick of human worth. Some students are more into carpentry, or literature, or music, or sports, and want to learn only the limited math they need for their chosen direction in life, and that’s completely fine with me, and my ideal education system would make it easy for them. But certainly, certainly, my ideal system also wouldn’t deliberately hold back the minority of students who are desperate to accelerate in math, to learn trig and calculus and statistics and algorithms and combinatorics and set theory and all the other treasures of the Platonic realm that there are, and most of all, to escape the years of endless drudgery, the busywork, that stands between that and them. This minority happens to have included me and most of my colleagues.

To say that the “California Math Framework” would fail students like us is an understatement. I read the document, and its central animating principle seems to be contempt for the possibility that anyone like us could even exist. At the core of the desired change is to force everyone into the same even slower than before curriculum at least through 10th grade, to eliminate tracking based on ability, and to eliminate acceleration as well, just in case anyone got the idea to use that as an escape hatch. There are pages and pages of vague educratese to obfuscate these goals, but the writers didn’t succeed quite well enough at being unclear!

As a grudging concession, some students at some schools might still be suffered to do calculus by 12th grade, but only if they double up on math in 11th grade, hire a private tutor, or take other crazy measures to compensate for how the CMF failed them earlier, sacrificed them on the altar of equity.

The part that gets me about this, every single time, is the double standard. Where is the nationwide movement to keep the kids who are better at football off the field, keep the talented musicians off the stage, for the sake of equity with those of us who lack their gifts? But no, that would be crazy. It’s always and only with academic pursuits—especially with math—that we see these sorts of movements gain traction, that they get to present themselves as part of the moral progress of humanity rather than something out of creepy science fiction.

15. Stefan Krastanov Says:

Scott #9: The vast majority of the signatories are university professors and younger academics. Our input is of course important, but it is hubris to believe that being involved in a couple of K12 outreach events with gifted kids per year gives us insight that supersedes that of people who have dedicated their professional lives to studying how the K12 system works. I would certainly consider “do away with gifted programs and academic acceleration entirely” a ridiculous idea, but I hardly see that idea having much of a hold in this proposal.

16. anon85 Says:

Many K12 teachers are also against the proposed changes. See this tweet, for example:

Or try this blog post by a different K12 math teacher, complaining about Boaler’s math ed research:

17. arcana Says:

@Scott #14:
The current system is quite bad and some other system has to be tried. I happen to agree that the CMF is even worse than the current system. But with an existing bad system and a desire for change you often get a change regardless so the best bet is to provide a better change.

Like, not to get too political, Clinton/Bush/Obama was terrible, second Clinton seems more of the same, let’s give Trump a shot. That was, empirically, a message that really resonated.

Writing this letter won’t achieve much. These kinds of letters mostly don’t. How can you reform public schools to improve overall results, move away from the current one size system, and satisfy by math[replace with x subject] lovers and math[x] sufferers?

Getting rid of public school isn’t an option since you’d have to fight entrenched interests.

How do you keep the unions happy, at least sidestep, if you can’t placate, the “woke reformers”, soothe the worries of parents who are less ideological and more just want the best for their kids, and also improve our education as far as math/engineering/etc.?

That is what you have to answer to stop this from happening and likely spreading. Once California adopts the CMF that is a lot of money, prestige, and precedent to spread this all over the country.

18. DR Says:

Scott #14:

Oh, agreed, that keeping gifted kids back is very punishing to them. They are trapped and actually underachieve being forced to do busy work, and don’t focus anymore, if not challenged. It serves no purpose for society either.

In India too, school math and science are not sufficient to prepare for hard exams like the IIT JEE. Top schools (not that there are many of those) compete with one another on how many 12th graders get into IITs, etc. Firstly, they admit students selectively based on 10th grade scores. In 11th and 12th grade, the way they encourage students to study for IIT, is have them finish the entire academic year’s schooling in the summer before it, with special classes during summer holidays. During the academic year, you are marked present even if you are staying home studying for IIT. There are better places than regular public (and even private) schools that provide coaching for IIT JEE exams, and they encourage you to attend those instead!!!! You only come in for science labs. This is how my husband’s 11th and 12th grade were. I went to a real school that insisted I show up, and burdened me with busy work. It was unproductive.

This was all in my time, 30 years ago. Not much has changed except students start studying for IIT and other competitive exams now by 9th grade. 11th grade is too late.

This is how education works in most schools in Bombay! The state has these silly rules about attendance, which the schools find their way around.

For gifted kids, the main challenge is finding a way to make the time spent in formal education productive. I rather like Bombay’s solution, since the state won’t budge on its rules.

19. Scott Says:

Stefan Krastanov #15: I dunno man. Imagine that I was looking for a challenging curriculum for my kids in some subject where I wasn’t myself an expert—piano, let’s say, or gymnastics. Obviously I wouldn’t screen potential teachers purely on musical or gymnastic talent: I’d care at least as much about whether they enjoy working with kids, whether they’re good at working with kids, etc. All the same, would I pick a program that was vociferously opposed by the world’s professional pianists or gymnasts, that almost all of them regarded as an insult to their field, as something that would’ve crushed their ambitions as children and prevented them from getting where they were? Obviously not: that alone would be 100% disqualifying. I’d listen to what the people who actually succeeded in the field told me was good, what they’d want for their own kids.

20. Matt297 Says:

Jo Boaler is all over these reforms. I think there’s value in printing what Boaler had to say about mathematical ability, giftedness, and Einstein in her book “Mathematical Mindsets,” in a section called “The Myth of the Mathematically Gifted Child.”

Her statements about Einstein, in particular, are hilarious. (For the reality, see: https://www.washingtonpost.com/news/answer-sheet/wp/2016/02/11/was-albert-einstein-really-a-bad-student-who-failed-math/) And also her statements about gifted people “going on to average lives and jobs,” as if the Study of Mathematically Precocious Youth never happened.

Here’s Boaler:

21. TL Says:

As Jo Boaler is an architect of the California Math Framework, I thought some people might be interested in this list of some critiques of her research.

Here’s a quote:

Three “traditionalists” were highly skeptical of Boaler’s findings and decided to go digging into the details: James Milgram, math professor at Stanford University, Wayne Bishop of CSU LA, and Paul Clopton, a statistician. They evaluated Boaler’s tests, the primary means by which Boaler demonstrated Railside’s apparently superior performance, and found them seriously wanting. They identified the schools and compared the various metrics (SAT scores, remediation rates) and demonstrated how Railside’s weak performance called Boaler’s conclusions into question.

22. Scott Says:

arcana #17: In that case, I look forward to learning from you as to what we ought to do or advocate instead … don’t hold back! 😀

FWIW, among the various drafters of this statement, I was one of the most pessimistic: I said we’re inevitably going to lose the battle for serious math education, but we have to fight it anyway because it’s a moral obligation. But people closer to the situation urged me not to be so fatalist—pointing out, for example, just how much pushback there’s been against the CMF from parents in California, and how ultimately these decisions rest with Governor Newsom and his staff, and they’re not anti-accelerated-math ideologues, just people who aren’t particularly well-informed about this issue, and they’ll be willing to listen to a nonpartisan group of parents and STEM professionals with a better alternative. I hope so!

Incidentally, I thought Obama was an excellent President—at any rate, the best since I’ve been born. That’s what made the idea, back in 2016, that “anything, even an authoritarian conman, would be better than our current nightmare!!” particularly risible.

23. Jake Andrews Says:

Dunno Scott, seems to me that you’re be wanting the creators of the future to be answering, not the creators (aka people who have been successful before in a previous paradigm, if that is what is actually necessitated) to be answering?

24. STEM Caveman Says:

It’s sadistic to teach calculus as a single year class (assuming you want differential, integral and series by the end). That makes it an IQ test suited only for the smartest students, when the actual ideas are simple, and it forces a focus on empty technique and exercises as an end in itself. Remove the junk and preliminaries and replace with 2 years of calculus done in good faith, which means a slower pace focused on meaningful use cases (not invented “applications”) with ancillary content added as needed. Ban “pre calculus”.

This is how proficient users of calculus actually learned it, only in the forward direction, by additional learning of multivariable calculus or real/complex analysis, differential geometry, etc until the calculus itself was really understood. My suggestion here is to do it in the backward direction, by replacing bogus content prior to calculus with another year of genuine calculus.

Note that this analysis implies the California educationists are right on their own terms. They noticed the IQ test aspect of calculus is not doing much except filtering out people who could master it in a different setting. Of course they want to replace it with equity, making things horrible for 100 percent rather than 90 percent of the students. But that’s not to say the criticism of one year of calculus as a sort of valorized appendix to the curriculum is in any way wrong.

25. Scott Says:

Jake Andrews #23: So when the creators of the future arrive here in their time machine, I’ll certainly listen with keen interest to whatever they have to say about what kinds of education helped them to establish new paradigms. Do you expect them to say “oh it was totally, 100% the California Math Framework”??

From everything I know of the history of science, the people (including Einstein) who established new paradigms, almost invariably had a thorough understanding of the old paradigms. That’s how they were able to realize that something new was needed. And certainly Einstein had to master calculus! 🙂

In the meantime, it might be relevant that I’m actually starting an NFL training camp in my backyard. If any NFL coaches or players complain that I’m an out-of-shape theoretical computer scientist who’s never played American football and barely even knows its rules, I’ll reply that those players and coaches were merely successful in the old paradigm of how to play football, whereas my new paradigm is the only one that matters.

26. Scott Says:

STEM Caveman #24: I could certainly get behind two years of calculus in high school for students who wanted that, perhaps with some multivariable and differential equations in the second year. God knows there are enough years of repetitive drudgery before then that you could make room by simply eliminating one of them, and (say) starting Algebra I in 7th grade or earlier.

Two years of calculus is, in fact, how things are already done at the country’s top magnet programs like Thomas Jefferson High School—the programs that, not coincidentally, are now being dismantled by the same ideologues who are also taking a hatchet to math education for everybody else.

More broadly, yes, there are countless positive changes that one could make to K-12 math teaching in the US. It’s just that those changes are all in a diametrically opposed direction to the one that California and the rest of the US are now moving in! If we were serious about positive change, we’d look at how they used to teach math in the USSR or Hungary, or at how they teach it now in Singapore or at Canada/USA Mathcamp.

27. STEM Caveman Says:

@Scott

we’re both in the narrow slice of population that would have had fun being cognitive alpha dog at a STEM high school for the top couple of percentiles (with the other alpha dogs there likely being alpha for the same reasons as at every other high school: looks, a nice car, athletic ability, charisma, political machination ability, etc … on top of having made the cutoff). Does wonders for one’s precollege sex life! But I can’t for the life of me justify the existence of those schools as public policy, especially with all the inexpensive and spatially distributed resources that now exist and didn’t in our youth. For the non alphas these IQ shredders can reduce their long term performance, and certainly their confidence, the end result being they get business and law degrees rather the PhD and engineering degrees they could have done otherwise.

The writing has been on the wall for public exam schools for a long time now, and rightly so. Why can’t they just be slightly selective STEM interest magnets, rather than severe IQ filters to enable self congratulation by the winners? It’s anachronistic, and pointless with an internet and all manner of interwar Budapest style STEM enrichment activities and material now available. More than the hungriest bright kid could consume by age 17.

28. JimV Says:

A big, friendly guy I knew briefly in high school (I think we were in the same Geometry class) once asked me if I believed in–I forget the term but it meant segregating classes according to ability, gifted vs. non-gifted. (We didn’t have that at my school but some schools did.) I thought it made sense, and said so, but his reaction made me reconsider a little.

He said something like, “It is so discouraging when the teacher asks a question and nobody in the class can answer it. I feel better when there are some smart kids in with us.”

(I don’t know what the best policy is.)

29. Scott Says:

STEM Caveman #27: Err … if your goal were to be “alpha dog in STEM,” wouldn’t it be incomparably easier to achieve that goal at a non-magnet school, where the pond would be so much smaller? And yet, when I was an adolescent, that wasn’t how I thought about it at all. I remember being desperate to be surrounded by other people who were like me, or who were even better than me (and I was thrilled once I finally was). I remember wanting to improve by testing my skills against theirs. Most of all, I remember being desperate to start learning the math and science and programming and writing that I’d need to do something important with my life, rather than spending day after day in pointless drudgery. Having read dozens of biographies of scientists, I was acutely conscious that if I would achieve anything worth remembering, then I might already do it in my 20s, so I was terrified of wasting precious years.

Of course I also desperately wanted a girlfriend, and there was a direct conflict between having a normal social life and skipping three grades as I ended up doing. But if I’m being honest about it? As badly as I wanted a girlfriend, I probably wanted to learn math and CS even more. And I was terrified that if I stuck around in junior high and high school, I’d end up with neither intellectual stimulation nor a girlfriend … so really, very little reason to wake up or to continue living. So I decided that I might as well go for the intellectual stimulation, and I’m still grateful that I had the opportunity!

30. Scott Says:

JimV #28: That’s interesting, because I’d thought that the much more common problem was students getting discouraged when one of their classmates always knows the answers and they don’t. For the latter problem, tracking is a huge help for the slower students and the faster ones alike.

If I were going to study, let’s say, weightlifting, or a foreign language, I’d definitely want to be tracked into one of the slowest classes.

31. Mike Maxwell Says:

“Calculus and other mathematical foundations are not important because… they are relics of the “Sputnik era”.” I grew up precisely in the Sputnik era, and what came out of that was the New Math. The New Math in Junior High set me back a year in math, as compared with the students in other feeder schools when I got to High School–they had taken a traditional algebra class in 8th grade instead. Fortunately, one of my math teachers had me double up on math in my Junior year, and I was able to join students from the other feeder schools in calculus my Senior year. (And of course we all know what Feynman said about the New Math…)

All that said, I never used calculus after that. (I majored in zoology, and later got into linguistics, finally computational linguistics.) What I think would have been more useful would have been a full year course in statistics–that is indeed something that few people understand, and which can be counter-intuitive, but which is essential to making sense of the world. Of course for some students statistics and calculus need not be an either-or choice, but for most it probably would.

32. STEM Caveman Says:

@Scott, yes, I’m precisely saying that being bigger in a small pond is generally better, except for the people who can still be alpha in a larger pond. So the Thomas Jefferson type schools end up as “IQ shredders” by lowering the status of the students who could have been at the very top of their neighborhood schools but are mediocre or inferior performers at TJHS. The road to bullying the insecure is paved with antibullying intentions.

You advocate for these schools as providing Nerd Youth with a feeling of belonging, lack of ambient thuggery, and possibly enhanced dating options. But why does that require a very high IQ filter rather than mere STEM interest? Why a Stuyvesant rather than Staten Island Tech?

33. arcana Says:

@scott #19:
I do have a personal strategy I would employ were I in a decision making position. It isn’t necessarily the only way to go, though.

I’d look to connect a few key groups for both instrumental and imaging purposes.

I’d like to get organized support from homeschoolers, relevant unions, and leftist/non-partisan school reformers.

I’d also like to find a way to involve homeschoolers both independent and organized. And of course successful professors and professionals would be helpful.

I’d propose, at the high level, something like the following:
Fed/state cooperation to building new 100-200 student schools|
++
-use a smaller number of closed bathroom cubicles to sidestep gender issues
–with smaller schools allow more open bath room policy to reduce “prison/institutional” feel
—achieving multiple goals with other construction choices in a similar way if possible
++
-smaller schools allows for more schools allowing for more varied curriculums at every level
–something like age 7-8-9-10 for elementary, 11-12-13-14 for middle, 15-16-17-18 for high
—separate “daycare” for kids under 6 from “school” since lots of data especially from japan shows starting kids too young is bad
—-split the 3 4-year levels into distinct types of schooling, with some small variance for “advanced” class
_1-4(7-10)
=handles basic stuff like reading, especially focused on phonics, and maybe a little writing, just basic stuff
==broad light exposure to different hobbies and career paths, social skills like boundaries and consent(not sexual consent this early), sharing, etc
===you probably want to start letting kids explore foreign language options in this age as well in a more casual way compared to the current system
====probably aside from mostly reading training you’d do directed play, a bit of light civics, some anodyne cultural stuff
=====you might introduce a semi-GATE-like accelerated set of classes here for kids who find a strong interest or have an obvious talent. sort of a prototype of the “specialized” classes that start in the second phase. maybe we call this free time or w/e and the kids without an interest get recess or some other fun activity so they aren’t feeling “judged wanting” by the system. this is also where the inter-school cooperation really comes into play. you can have “activity sections” in the computer lab where a small number of staff can supervise digital stuff. online/distance learning is far less of a problem when it is focused on special interests as anyone who found their people on the internet can attest, vs generalized usage in classes you are forced into.
_5-8(11-14)
=start adding more writing skills in the first 2 years
==begin “exposure seminars” 1 hour a day 3 hours a week same career/field/hobby for the whole week, so kids really get a broad idea of their options. this would continue all the way to the end of the final year of school. you’d get 864 hours in 288 fields of interest just using a 36 weeks of school a year model. the specific year length is pretty flexible
===you’d start doing more formal math work in this stage now that brain development is more on your side. no hard opinions on which year except not the last one. plenty of data shows that kids can advance through truly developmentally appropriate math fairly quickly and ideally we have really reduced “i suck at math” effects by not pushing everyone too early
====this is the phase where the aforementioned “specialized classes” come in. you’d shift a lot of “gifted education” here probably. again the value of several small schools with good digital cooperation comes in. at this age you might also be able to physically transport kids around.
=====teachers with special skills or interests would be teaching these classes. classroom teachers could handle “average age level” math, with some enrichment from physical or digital special guests sometimes. if one teacher had a major or minor in math education or w/e they could handle all the kids from a group of schools who do accelerated math. again we could combine physical movement with distance learning here. maybe friday you have gifted kids bussed somewhere while for monday and wednesday they do distance accelerated in the activity sections of the computer lab. bussing gifted kids on particular days already existed when i was in traditional broad based gifted programs in the 90s. scaling it up will require some investment but it can be done.
_9-12(15-18)
=we keep up the exposure to areas of interest/job fields/hobbies here. i really think a truly serious focus on this brings huge gains. kids should see lots of different people achieving different stuff. parent job day is in no way sufficient. kids can only do so much rigorous mental work in a day. i’d argue 3-4 hours is best. then we are able to focus on other important stuff that isn’t so mentally taxing. especially for this age group we can do stuff like this in the morning and the brain drain stuff when kids are actually awake.
==i actually think we should have a reversed time to start in schools. 1-4 start at 8, 5-8 start at 9, 9-12 start at 10. getting up at 6:30-7:00 or even earlier for bus kids, was extremely deleterious to effective learning. we’d provide some sort of before school program if necessary and ideally a superior version of the bus system. actually having more smaller schools would allow us to really make our school commute system more efficient. not sure how easily we could sell some parents on trusting their kids to walk a few blocks to school or a block to the bus at 9:30. but we should try to do it because the potential gains are big.
===we would increase the usage of specialized classes. this would be super easy for urban and inner ring schools. because you’d have a much greater concentration of schools to choose from. our ring suburbs would have some trade offs cause they are built for cars. rural areas would need serious and special consideration to provide similar benefits.
====both at this phase and the second one we derive a lot of incidental benefits for integrating homeschooling into the broader system since we have to deal with more smaller schools, potentially frequent student transfers plus special classes transport as well.
=====in this phase of school we’d also start apprenticeship programs. we want german/swiss levels of access to and respect for so called “blue collar” jobs and “trades”. college as an option and not a universal goal and the attendant changes in attitude would benefit our society enormously just for social and political reasons. public school at the high school and even middle school level should be just as supportive of kids who love automative activities or plumbing or sub/pre-university electrical work as kids who love literature or art or w/e. we’d work with community colleges, appropriate local businesses ,and such to allow the option for kids to get placed into apprenticeships. just as much as we do for university stuff. maybe university is a bad fit or maybe a kid just wants to do something else even if they could succeed there. community college should be as well integrated into the system as the previous three phases. plus we could do central european style trade schools as the other half of community college. and community and trade colleges are much easier to normalize than traditional 4 year university.

I want to actually split social studies into a separate section of the comment. Having a more “modular”, for lack of a better word, school system would allow for us to provide for a broader spectrum of people. With say 200 kids in 4 grades with 3-4 classroom teachers per grade per school we could have a variety in all subjects. I’d be willing to support, even as a left, of a somewhat idiosyncratic style, people who wanted a history course that is more “southern” or “conservative” as long as there were other options, even the infamous crt-inspired reading of history, and specifically for me a union/organizing focused version of American history, and other types as interest warrants. The same goes for English classes. You might have one or two teachers focused on Langston and Butler and similar, one who was super into the standard western canon, another who looked at modern writers, even a teacher who taught English using YA and YA adjacent books.

In a midwest major city for example you might have a few hundred thousand people in a metro area plus a million plus in surrounding suburbs. That means you could have double digit numbers of 200 person schools for all 3 phases. Schools would be controlled by a board by school parents or something. They wouldn’t be explicitly linked to residence. Of course many parents might choose a closer school over other trade offs but with schools with relatively small sizes you’d still have options. Just my municipality alone could support 4 200 student schools at each phase. Well you’d likely have 7 elementaries, 4 middles, and 3 highs, becauseof evaporation to private schools of various kinds. But parents could easily choose a school in a neighboring municipality. Obviously there would be a pretty big adjustment period to such a different model vs the current one. You might need to have a system to handle varied demand or something.

Having homeschoolers connect with a school for a few particular subjects would be much easier under such a flexible system. You may or may not want to have some sort of private/religious school exchange as well. Homeschoolers are a much smaller population with more varied desires so that would be easier to accomodate.

You’d have to establish some norms of not demonizing parents or boards who made certain choices on sensitive topics, for the integration of schools to work well.

+=–=+
Whether you agree with my particular compromise reform of public education or not, you need something to advocate for instead of the CMF strategy. They have a movement, at some scale or other, a somewhat comprehensive ideology, and a set of goals. They are also attempting to address a problem that really does exist with the current school system, even if their solution is dubious. Arguing for the status quo won’t work. You need a comparable level of public and activist support to counter them. Which probably requires an alternative model of reform.

Gavin Newsom, like most politicians, has to please his constituents. It isn’t enough to do an open letter. He’d probably support an idea he didn’t personally like if he felt there was no equally viable alternative. He’d rather support an idea he considered good but he’ll settle for a popular but mediocre one if he has to.

My proposed reform is based, from my perspective, around building a coalition of groups sufficient to repel the oomph behind CMF, or other reforms I don’t like in other states. I didn’t write it up specifically targeting California’s current issues. The CMF appeals to unions, or a sufficient subsection of the unions that has a significant majorty of power in the unions, so you need something that can counteract that. The CMF appeals to “woke” liberals. You need a constituency that are potential Newsom voters of equal and ideally greater scope.

The optics of getting homeschooler support for an “social investment” state(i hate the welfare framing) is a strong signal because you’d normally expect their opposition. Academic and industry experts can send a signal and impact media coverage, but they aren’t ballots in boxes, in relative terms. You’d like them but you don’t need them, some other academic discipline can stand in for them, social sciences educators vs great mathematicians and users of math. I tailored my proposal for the latter, the CMF looks at the former.

The CMF has a, fragile, appeal to parents who are failed by the current system. Aside from trying to get support of unions and academics and industry people I tried to configure my reform such that it provides generally popular abstract concepts to out appeal to concerned parents. Parental control, parental choice, flexibility. Etc.

The CMF wins somewhat on degree of change and ease of implementation. That is why they can employ a smaller coalition and mostly provide shallow words since they aren’t trying to spend money or build infrastructure.

Presumably you could build a smaller coalition around a reform that is less materially expensive. I don’t think that could improve education, math or otherwise, but if your goal is purely to derail the CMF you have some options. More power to you if you can devise a simpler/cheaper strategy, while being more robust than a letter of concern. “Fields medals” may or may not impress a wealthy technocrat like Newsom, they don’t mean crap to working class parents without a counter-proposal for reform. Turing means even less to most people. Nobel has a bit of cache in the popular consciousness I guess. Whatever ends up happening I hope the CMF is not implemented but I don’t think opponents are serious enough about stopping it to succeed.

34. Scott Says:

STEM Caveman #32:

You advocate for these schools as providing Nerd Youth with a feeling of belonging, lack of ambient thuggery, and possibly enhanced dating options. But why does that require a very high IQ filter rather than mere STEM interest? Why a Stuyvesant rather than Staten Island Tech?

Oh, I’d love to have 10 Staten Island Techs for every Stuyvesant. I’d love to have K-12 schools catering to every niche interest and every level of seriousness and ability, just like we have colleges—or should have—colleges and universities for every type of student.

35. Mg Says:

@Scott 22

If you’re really desperate for ideas, here is one that came from *my* mind 🙂

As STEM Caveman pointed out, the knowledge is out there in books (available for free on Library Genesis). So, what do the damn kids have a problem with :)? I can come up with:

– Motivation. Can come from oneself and/or from a parent and/or from a teacher and/or from competition with peers and/or ???

– Answers to questions. As we know books ~= NP < PSPACE = IP ~= a conversation with a good teacher :). Sometimes you go on the wrong track with understanding the book, or have some (usually stupid, otherwise usually obvious, otherwise usually well known to Gauss 🙂 ) idea to extend the material. To some extent solved by the internet (for example stack exchange).

– A superset of Answers: Direction. Say you finished the exercises from the Linear Algebra textbook. What now? There are so many directions and all interesting. I guess a natural choice would be to do a whole undergraduate curriculum, but on a 1.2th thought it's a terrible choice if you'll later go to undergrad college. It would be nice to have someone tell you what best to do, and also how well you're doing, if you're approaching it all in right way etc.

So, why are you not offering free tutoring for (ultra-)talented high schoolers :)? (if you aren't; at least you didn't advertise it to me, a non-member of the target audience). Problems include, you probably aren't Pareto optimal, and it would be hard for you to reach and select those most in need. However

– According to my uneducated guess, you could still create a large positive delta for the cost of only, say, 3 hours per week closer to insanity and death from overworking

– You could be working on the ground of something you care about. Which is a great poetic advantage over just signing letters!

36. Scott Says:

Mg #35: While I could certainly be doing more, I do spend a pretty large fraction of my life trying to educate young people (meaning, even younger than my undergrads at UT) about math and CS … from the readers of this blog, to the high school students who email me, to the ones who attend my public lectures or read my book or my lecture notes, to hopefully the ones who will read the graphic novel that Zach Weinersmith and I are planning to write (stay tuned)!

37. STEM Caveman Says:

@Scott 34, college has the same problem, only much more socially destructive. Just because it was fun to have math and babes in one convenient location doesn’t mean IQ clusters can be justified as a public expense. Which part of the Staten Island Tech solution (plus books by mail, internet, local college, math camps/contests etc) would not have worked for a 30 years younger version of ourselves? Why does the Social Darwinist masturbation provide so much marginal benefit (to society, not the Gini winners thus artificially created) that it must be subsidized by operating a Stuyvesant or for that matter a Harvard? High IQ clubs can be organized at even higher levels of selectivity over the Internet for those who want them. Why in this day and age should taxes pay for it though, especially at ages where “research also gets done” doesn’t apply?

38. Scott Says:

STEM Caveman #37: It seems to me that public magnet high schools have paid for themselves a hundred times over in terms of the geniuses they’ve helped produce (did you know that Steven Weinberg and Sheldon Glashow were classmates at Bronx Science)? Whereas, even if you regarded the provision of sexual opportunities to nerdy teenagers as a social good (as presumably most people wouldn’t), it’s far from obvious whether magnet schools have actually delivered on that metric. 😀

39. Boaz Barak Says:

Should mention that the letter does not take a position on magnet schools or gifted and talented education. Our focus is far more prosaic but affects far more students – the opportunity to take Algbera I before 9th grade and to have a path that supplies you with basic mathematical foundations in high school. Nothing fancy and nothing that is out of reach (though perhaps out of interest) for the vast majority of students.

Also, while the current system has many failings, unless one is creating a small pilot school, changes at the state level should be incremental: adopt a course that already exists and has strong record etc.

One lesson from history is that no matter how bad things are, they could always be worse.

40. V Silver Says:

Scott,
Have a look at the 2021 white paper “Rethinking Middle School Math Acceleration.”

Curriculum Associates, the providers of the iReady math diagnostic and learning resource, has found that the idea that all students should master Algebra 1 by 8th grade is having the OPPOSITE effect on STEM preparation in the US. Students are being rushed through a curriculum which is a mile wide and an inch deep, at rocket speed, and NOT mastering the foundational skills that would prepare them for higher math later in high school and college.

Algebra 1 has become a stumbling block for a huge percentage of students because of how we teach that and prior math as a set of disconnected procedures, and most students will never use anything beyond pre-Algebra in their real lives. (We ought to define “Algebra 1” since it now contains standards that used to be part of Algebra 2.) The race to calculus, partly or largely caused by the college admissions process, is not resulting in more students being prepared for or choosing STEM studies or careers.

Data science is in fact the math that most people actually encounter in life and it would be much more beneficial to emphasize that over calculus. The Algebra 1 course in Florida actually includes a unit on statistics, but my students haven’t had time to master it due to the double-time pacing.

Asking employers of STEM grads what skills they are looking for usually brings up the ability to APPLY the math and science they learned to solving problems, as opposed to knowing the procedures used to pass the standardized tests.

V Silver
Middle school math teacher (Algebra 1, Geometry)

41. STEM Caveman Says:

re: Bronx Science, the word from education studies controlling for IQ (or a proxy such as SAT or in this case SHSAT) is that outcomes are driven by individual ability with little or no effect of the school. The marginal effect of going to higher scoring Stuyvesant versus Bronx Science was actually studied a few years ago, and is zero, at least for students near the cutoff so that they could be compared. Another consistent finding is that it’s much better to be the big fish in a small pond so for students well above the cut one can surmise they are better off with weaker competition at Bronx Sci than having the smarter peers at Stuyvesant.

To see what creates a Nobel or Fields medal you need to look not at the biggest stars like Weinberg or Witten, who would plausibly have dominated no matter what, but the “average” guys for whom it was less obvious that they would be on a path to greatness. Laughlin of the fractional quantum hall effect was on the swim team in a farm town. Most Nobelists did not have a science high school where they lived and for those who did, the effect is from early specialization, not the higher IQ of peers (which has the negative IQ shredder effect; the ideal is to have the specialization without the heightened competition and workload, which incidentally is what self study or homeschooling provide. Or a Staten Island Tech.)

42. Jungshik Shin Says:

I have a rather different issue with math education in the US. The US is virtually only country in the world
that does NOT offer integrated math curriculum at secondary schools with a small set of exceptions – public school districts that adopted integrated math curriculum recently. (I wish my school district had adopted it, but it didn’t) and IB programs. Learning algebra all year long, followed by a year of geometry and another year of more advanced algebra(algebra 2, precalculus, etc) doesn’t make much sense to me. All other countries teach algebra, geometry, topology, probability, statistics, set&logic and so forth integrated in any given year of secondary school math.

43. arcana Says:

On a separate topic from large scale school reform, I’d like to comment on my experience in the existing, well it was like 10 years ago but not much has changed, “accelerated” math track in standard public school. This involves finishing AP Calc BC during senior year. Although we had one or 2 kids a year ahead of that schedule. I believe they did dual enrollment to take more advanced classes senior year.

The vast majority of students even if they finish with As or Bs and get 3-4 on the AP test don’t really understand calc at all. They are “academically involved” rather than “nerds”. That was how we used to talk about it. They aren’t particularly interested in math outside of school. Their hobbies don’t involve math. They don’t find it fun or interesting. There was some gender imbalance. Say nerds are 70/30 boys and academically involved kids are 70/30 girls. The two don’t quite match up that evenly but the general idea is important.

AcIn kids could pass get 80+ test scores and they could pass but not excel on the AP test but they weren’t generative math thinkers. They were the kids who learned the “standard algorithm” for multiplication and left it at that.

When the teacher would do boardwork and be asking questions she’d often say, “what’s wrong with this” part way through a problem and those kids really didn’t know. They’d also never correct the teacher if they made an error, though I suspect our teach made many of her errors on purpose.

They were not good at mental math and they needed a lot of “review” during class. They’d email the teacher for help a lot. But they never got better at calc. This was the top tier of motivated students. Not motivated by loving math of course, they had the motivations of an “academically involved” student. I knew most of these kids moderately well. I know who got into Harvard or Stanford or MIT, although my school didn’t have a ton of those since it was a pretty normal public school in the Midwest. A heavily disproportionate number of acceptances to those schools were nerds not academically involved kids. And generally that second group doesn’t *need* to go to those schools. They become vets or teachers or something in that range.

A lot of the AcIn kids were not quite the people who thought they couldn’t be good at math but they didn’t enjoy math. If our teacher wasn’t uncommonly good they would have done much worse.

This is relevant because Boaz wants to be incremental. I don’t think that would work. Just opening up these courses as options for more kids is useless. You are already getting most of the marginal kids into the honors track. Especially looking at Algebra 1 or 2 vs Calculus. Many, many kids took the off ramp to pre-calc junior year and never took even AP Calc AB. A small number did go from PC to AB.

Keeping the current system for lower grade students will continue to push kids away from math by pushing them too early developmentally as I mentioned in another comment. There’s also no attempt to provide interest or motivation which is separate from capability.

“Providing access” is and always has been insufficient to achieve the stated goals and that position isn’t going to get you the oomph you need to fight the momentum of the CMF push. I suspect that the people behind it will be getting similar changes into more states in the future rather than things moving in a good direction regarding math education.

Again you don’t have to go as far as total reform of education but “access programs” and brow furrowing letters won’t cut it.

44. Scott Says:

STEM Caveman #41: You’ve actually put your finger on one of the central ironies of this discussion. People who support tracking in math, or academic acceleration, or magnet schools, or anything of the kind, are constantly accused of something called “genetic determinism.” And yet if you think about it, the reason I support all those things is precisely that I’m NOT a genetic determinist. It’s that I’m acutely conscious of all the environmental factors that matter enormously if we’re serious about producing the next generation of scientific and technological revolutionaries.

It’s like, presumably the people in ancient Athens didn’t differ too much genetically from the people in all the other ancient Greek city-states. Nor did the Jews of early-20th-century Budapest differ much genetically from the Jews of all the forgotten shtetls that the Nazis also wiped out. So then why was it Athens that produced Socrates, Plato, and Aristotle, and why was it Budapest that produced von Neumann, Erdös, Teller, Wigner, and Szilard? It must have been something about the social environment. Something, perhaps, about the pressure-cooker effect of taking so many people who are talented and passionate about the same topics and putting them together in the same place, where they all have to outdo each other to earn each other’s respect. Why don’t the people who decide K-12 education policy wake up every morning asking themselves how to reproduce that today?

45. Boaz Barak Says:

V Silver #40: Thank you for writing. I do in fact agree that a “race to calculus” due to colleg admissions is not great. In Harvard you sometimes see students that took calculus in their sophomore year of high school, skipping one or more of algebra and precalc, and their algebra skills are really not up to par. However here we are taking about calculus in the 12th grade and am also not advocating that every student takes it. I just object to the state of California making it all but impossible for students to reach this without going to a private middle school or paying for an out of pocket private course.

I agree that high school should provide a basic floor of math skills for students who will not go to college or will not major in quantitative fields. But preparing students for majoring in such fields (which are the most popular and also the ones with best job opportunities) is an important part of high school. For these, students actually will need the material that is taught in Algebra II and (most often) AP calc. Regardless of admission criteria, students that don’t have this background will have a harder time doing STEM, which are hard majors anyway, and so are at higher risk of dropping out or simply having fewer opportunities. (E.g., can’t get that machine learning internship since they are still working on basic math while their peers already took algorithms and ML.)

You are right that a lot needs to be done in grades K-7 to reach the goal of universal Algebra I by 8th grade. There is probably not a one size fits all solution, but artificially holding back kids who are ready to take it, and hence limiting their options (unless they have the means to do it privately) does not make sense to me.

Jungshik #42: there is actually a well established integrated pathway, which is a reorganization of the same material. See https://bit.ly/cmfanalysis

46. JimV Says:

Scott @30–yes, he was an exception. To him a class was a small community, working together to learn. He was one of the few people in high school that seemed to like me despite my lack of self-confidence and being a principal’s kid and being a know-it-all, and to consider me a resource instead of a target. So I wanted to drop his position on the other side of the scales, for all its lightness. If everyone were like him, the world would be a great place, but of course we aren’t.

As you know, the best way to learn something is to teach it to someone else, and he provided that opportunity.

47. Boaz Barak Says:

Should also say that I am pro data science being taught in K-12 and there are good courses such as https://coursekata.org/

It’s just that the CMF tried to use such courses in a way that their designers didn’t intend to, as replacement for basic math skills. They somehow think they can “kill two birds with one stone” and use data science as a solution for equity and educational gaps. I think these are two completely separate issues.

48. Tu Says:

Scott,

Thank you for signing this letter, and for making the readers of this blog aware of this very bad proposal. As far as your commitment to improving mathematics education in the US is concerned, I think that your actions (including running this blog) speak for themselves. Your choice in this instance was between signing this open letter or not signing it– not between signing the letter or giving up your career in research to start a charter school in California to show people how to really teach math .

I am going to take a moment to advocate for the following (completely insane, extremist, crack-pot) view: the current mathematics track in the US (algebra I, plane geometry, algebra II, trig/pre calc, calculus) is completely fine.

I am not saying there are no problems with mathematics education in America, but the basic sequence outlined above is not the source of any of them. I was not a mathematical genius as a highschooler. I liked my math classes, but I certainly found them challenging. I was not bored by the easiness or low-level of the material. I would say more or less the same attitude/experience applied to all of the courses that I took in high school (except chemistry, which I hated and almost failed).

I liked my math classes and my math teachers (more or less), but as a highschooler studying mathematics was a lower priority than:
– playing sports
– playing video games
– losing my virginity
– spending time with my friends
– watching TV

Nevertheless, I completed the above-mentioned track on time (finishing calculus my senior year of highschool). When I arrived at college I was still unsure of what I would study. I didn’t really know what it meant to study mathematics in college, but I figured I would keep taking classes until I didn’t want to anymore.

Long story short, I was able to keep going, major in mathematics, work in a technical field for 5 years, and am now enrolled in a PhD program in engineering despite not really expressing particular interest in math before the age of 20. I can attribute this largely to there being a well-trod path in place for me to follow– a path that ensured that I had the mathematical literacy and foundation necessary to keep going.

Much of the discussion in the comments section so far is either explicitly or implicitly saying something like : “the current math curriculum is bad because the really smart kids get screwed over.” That may be. But if we are talking about a curriculum aimed at every student (leaving the debate over whether we should have on at all or not aside), then I think the experience of a non-outstanding math student should be taken into consideration. The boring, lame, very bad, old-school, nonintegrated ended up being a legitimately great thing for me.

Of course, I glossed over the most important part of the story here, which I also think is the most important part of any story about this, which was that I had great teachers. A bad curriculum with great teachers > good curriculum and bad teachers. I know that not everyone else is so lucky.

49. Scott Says:

arcana #43: Your comments are interesting, but I confess I’m still confused about what you would have us do differently from what we’re doing. My feelings about how to improve the most stultifying math curricula are a little like my feelings about how to improve the Communist regime of East Germany: it’s hard to know where to begin, but letting people opt out would definitely be a huge step in the right direction. When people are allowed to opt out of your system—say, by moving to West Germany, or by signing up for a still-viable traditional math track that includes calculus—it sets a floor on how bad your system can possibly become, lest everyone start voting against it with their feet.

50. Marco Says:

To raise a different perspective, proposed CMF would be a dramatic improvement over the middle school – high school math education I experienced in the USA circa 2000 – 2006. We learned math as rote symbol manipulation. My questions about how the methods worked and why we were using them met with deflection, hostility, and, eventually, punishment. By contrast, Chapter 4 of proposed CMF (especially pages 38 – 53) describes a classroom that integrates why-and-how questions into the math curriculum in a way I never experienced until college. It sounds like solid preparation for open-ended mathematical work.

I have a PhD in theoretical computer science, but I barely graduated from high school because of poor grades in math. A fantastic Discrete Math course totally changed my perspective on math and (by extension) my entire career. I was very lucky. But how many people are we missing from the professional community today because secondary education taught them not to ask questions about math? As far as I could tell, proposed CMF focuses on fixing this. It seemed like a net-positive change.

I am surprised that so many professionals think the current system is good enough to reject proposed CMF. Now I don’t know what to think. This discussion has dramatically increased my uncertainty about the baseline question: “how good is secondary math education in the USA”?

51. arcana Says:

@marco #50

I actually agree. When I went to community college later on and took math kids were shocked that I would ever question the teacher. Even the academically involved kids in AP calc lacked a certain kind of confidence in their math, and were unhealthily deferential. Our calc teacher was great and she would have been happy to get more of that stuff but many teachers early on were quite hostile to it. Probably because a lot of them didn’t understand all that deeper stuff themselves. My district was blessed that we alternated between pretty cool math teachers and the more standard mediocre kind who were just punching a clock. I liked several of the CMF recommendations actually. However I’d have liked many of the integrated things to be exothermal. Bring math out into other classes, like the example of calculating the local community living wage. Do that in social studies. I think the endothermal bring political topics into math strategy is bad. I’d bring in stuff relevant to trades or something to vary word problems from the infamous trains from city a and city b.

In talking to at least Scott I got more of an impression that the current system is far from ideal for them in fact but that certain parts of the CMF are in their minds even worse and the tradeoff for the improved areas isn’t worh it. Also for obvious reasons highly educated stem and math professionals have a bit of a typical minding limitation. Not all of them and it isn’t super extreme but it is there. I’m a bit in the middle personally. I liked math but hated math education whereas many top professionals excelled in advanced hard science courses on top of also liking math. That puts them two degrees of deviation out from the majority of kids who had a lot of dislike for the system as a whole on top of hating math.

The CMF has some of the right higher level ideas that I personally like but their execution is awful and there are some bad parts as well.

52. Sniffnoy Says:

Marco #50:

I’m not sure you’re correctly attributing the source of the difference. You say,

By contrast, Chapter 4 of proposed CMF (especially pages 38 – 53) describes a classroom that integrates why-and-how questions into the math curriculum in a way I never experienced until college.

The problem is, you are comparing the CMF’s description, to the current system’s common practice.

The feature of the CMF’s description that you highlight is not distinctive. Basically, every math curriculum talks about this in its description, and, in the hands of competent practitioners, actually includes it. The reason this gets so often omitted in practice is not due to differences in curriculum but due to differences in who’s teaching the curriculum. I expect the teachers you had that you complain about would not have done any better with CMF than with whatever curriculum they had at the time.

53. Mark Holum Says:

One thing that’s interesting to note is that american, and canadian, math curricula (IN HIGH SCHOOL!!) are actually extremly difficult compared to others in the anglophone world. Its instructive to compare the GSCE exams with say the New York algebra regents. The algebra regent should be taken in 8th or 9th grade. The GSCE exam is taken after the equivalent of 10th grade. And the algebra regent is shockingly more rigorous.

https://filestore.aqa.org.uk/sample-papers-and-mark-schemes/2019/june/AQA-83003H-QP-JUN19.PDF

https://www.nysedregents.org/algebraone/621/algone-v202-exam.pdf

American calculus standards are also much higher than in england. This is the A level higher math exam:

https://filestore.aqa.org.uk/sample-papers-and-mark-schemes/2019/june/AQA-73571-QP-JUN19.PDF

Which is taken by students who are bound for math programs! And is taken by students a year older than american high shool studnets! And is significantly easier than a calc ap exam

It’s also interesting to note how much wider in covered material the british exams are. American high school math has shed topics that often are extremely important: complex numbers (which is crucial for engineering), much of geometry, a lot of the analytic geometry that people used to study. A huge amount of work is done preparing students for the requirements of calculus, but in so doing, a lot of very important mathematics is undercovered.

54. Boaz Barak Says:

Marco #50: The letter does not claim the current system is perfect or even close to it. Not do we claim that all of the ideas in the CMF are bad. There is certainly lots of room to improve teaching of mathematics.

The particular issues we have with the CMF are spelled out in https://bit.ly/cmfanalysis
These are focused on the 8-12 aspects of the proposal and particularly the “data science” pathway promoted there. It’s not a good pathway for students that want to major in STEM, even for students that want to become data scientists!

55. Mark Holum Says:

To give an example, here is the calc bc exam from last year!

https://apcentral.collegeboard.org/pdf/ap19-frq-calculus-bc.pdf

I don’t think that the relative ease of especially the GSCE is a problem… the point of the GSCE is to demonstrate the amount of mathematics required for the trades, which it ably does…

That being said, the relative difficulty of american high school mathematics combined with the relative ease of pre secondary mathematics might actually have some very weird effects on how math is perceived in america, and how well americans learn the content that they are supposed to know!

56. arcana Says:

@ Mark #53:
My understanding is that while A-level math is much easier than AP calc, AP chem is much easier than A-level chem. Additionally A-level math is something of a joke according to Brits I knew.

Also we learned complex numbers in school. Granted this was the latter half of the first decade of 2000 so it had been a few years. Geometry was a whole year of high school advanced math. It is important to remember that the US is of a size and scale comparable to the totality of Europe, with about 65% of the population. It is hard to generalize the US vs tiny countries. Even the largest Euro countries are less than 1/4th the US pop.

57. Mark Holum Says:

Finally, this is the PISA exam, that american students famously don’t do well on.

http://www.gov.pe.ca/photos/original/ed_PISA_math1.pdf

Note this is supposed to be taken by 15 year olds, who thus are at the end if 9th grade. Again compare this with the california algebra standards exam, which until recently was supposed to be taken at the end of 8th grade!

http://www.tutormemath.net/assets/cstrtqalgebra.pdf

The comparison is instructive. For one thing… the american standardized tests require students to solve a very large number of problems in a relatively small amount of time. The tacit assumption is that students will have very low scores on the exam (and a low score is actually fine). The foreign exams have smaller number of simpler problems, with the assumption that students will generally score very highly on them. 50% of british students who take the maths a levels score at the highest level. The american exams have many more puzzle questions, which are actually much more difficult than the equivalent example of the problem that might solve in real life. The british exams use very simple examples (i.e. find the taylor expansion of 1/sqrt(4-x) vs find the taylor expansion of xln(1 + x/3) on the ap bc exam)

I don’t want to suggest that its bad that the american standards are more rigorous! Just that when we examine the issue of americans learning math, its important to understand that mathematics education in america is unusual in comparison with the rest of the word!

58. DavidM Says:

Mark #53

A closer comparison to AP calc BC (which the internet tells me is the harder of the two AP calcs) is probabiy A-level _Further_ Maths, example paper https://filestore.aqa.org.uk/sample-papers-and-mark-schemes/2019/june/AQA-73672-QP-JUN19.PDF . Also I’m not sure what you mean by `And is taken by students a year older than american high shool studnets!’ – are US students not (typically) 18 at the end of their final year of high school?

59. Tatterdemalion Says:

Mark Holum #53:

In England the maths curriculum is split across two A-levels: “Maths” and “Further Maths”. The former is a prerequisite for the latter; anyone hoping to go on to study maths will need to have an A-level in Further Maths. The thing you’ve linked as

“American calculus standards are also much higher than in england. This is the A level higher math exam, which is taken by students who are bound for math programs! And is taken by students a year older than american high shool studnets! And is significantly easier than a calc ap exam”

is a large-M Maths exam, not Further Maths (so far as I’m aware there is no such thing as “high math” here?), so it’s not surprising it’s easy, and while it’s technically true that it’s taken by students who are bound for maths programmes, they’ll also take another, more advanced exam as well (and they’ll take the one you linked a year earlier than the students who are just taking one maths A level, I think, or at least that was what happened 20 years ago when I did it). Some past papers for that are at https://revisionmaths.com/level-maths/level-maths-past-papers, if you follow the links for “Further Maths” rather than just “Maths”.

Another pitfall is that you need to be slightly careful judging its difficulty even from /those/ papers, though, because the different modules are sequential, so “Further Pure Mathematics 1” is less advanced than “Further Pure Mathematics 2”.

I /think/ that https://revisionmaths.com/sites/mathsrevision.net/files/imce/9FM0_4A_que_20190626.pdf is probably a good sample of how far the English maths curriculum goes at A-level, but I wouldn’t swear to it. As you’ll see, it’s an awful lot harder than the paper you were misled by!

60. Scott Says:

Marco #50: My own view—and here I have to stress, once more, that I’m speaking only for myself—is that current K-12 math education in the US is indeed mostly a disaster, mitigated by some individual amazing teachers, who succeed despite rather than because of “guidance” from the bureaucrats who design curricula. However, I also believe that the CMF would manage the nontrivial feat of making it an even bigger disaster, by eviscerating the few rare pockets (like AP Calculus and other accelerated courses) that actually work.

You say you want math education that addresses the “why” questions, rather than just shoving rote procedures down students’ throats. So do I. So does anyone who cares about math. Since the desirability of this is so apparent, it doesn’t surprise me that the CMF document makes some noises in that direction too. The problem is that they have less than zero credibility … because they’ve been trying to take a hatchet to the rare math classes that actually do engage nontrivial ideas. If you read them, they make it crystal clear over and over that what they really want, above every other goal, is “equity”—and they want to achieve it the Harrison Bergeron way, by pushing the highest-achieving students down, if they can’t figure out how to get it by lifting the slower students up. That deep-rooted impulse on their part is why I’d never trust them to design or implement a math curriculum that I agreed with, let alone one with which I have so many disagreements!

61. Wes Hansen Says:

In 2016 Anthony Yom, a math teacher at Lincoln High in LAUSD, had a student, Cedrick Argueta, ace the AP Calculus exam while all 21 of his AP Calculus students passed; it was the third year in a row that all of his students passed. From LAUSD teacher leads student to ace AP calculus exam with perfect score:

“Yom says he keeps getting asked if there’s some secret recipe for getting students to perform at their highest potential.

“This may sound corny, but you really have to love them,” Yom says. “You build this trust, and at that point, whatever you ask them to do, they’ll go the extra mile. The recipe is love.””

I live in Los Angeles, East Los Angeles, and the big problem here is legal marijuana . . .

62. arcana Says:

So in my email, which went to scott@scottaaronson.com which is hopefully the correct one, I described what you could do if you wanted to really push for something. You personally are in Texas yeah but you can talk to people and stuff.

If you think the stuff I mentioned is too much then I suppose you could pray. With real political pressure from a lot of groups in California there’s just a pretty big limit to what you can do that is less effortful than what I suggested.

63. Lautaro Vergara Says:

Hi Scott,
Do you allow me to replicate your article in Spanish, in my website?
Cheers,
Lautaro Vergara
Associate Professor
Physics Department
Usach, Chile.

64. Richard Lowery Says:

I can’t agree with this part:

“While well-intentioned…”

This is not an attempt at humor. It is essential to get away from this default approach to understanding these efforts because it undermines the search for solutions. Reading through the literature underlying such movements quickly disabuses one of such ideas.

65. Scott Says:

Richard Lowery #63: Once again speaking for myself, I specifically questioned whether the word “well-intentioned” might be conceding too much, but was overruled. 🙂

But really it just comes down to definitions. As one example, I think that the authors of the CMF want to hold the most gifted students down for the sake of equity … and I also think that, in their minds, in their worldview, that is well-intentioned.

66. Edward M Measure Says:

There is a widespread academic critique of all accelerated education programs in the US, as we have seen in the abandoning of strict standards in Virginia and New York City. The animating principle of this critique is that too few disadvantaged minority students enter or succeed in these programs, or, alternately, that too many Asian and white students do enter.

The goal is that no students succeed beyond the success of the slowest.

I think that this is a ridiculous and suicidal way to run a society.

Students vary in their interest and aptitude for math and other things. Rational education should take advantage of this, not punish it.

67. arcana Says:

@ Edward #65:
That is one of the organizing principles of my large post earlier on pre-college school reform. Of course getting that done would be very difficult but the proposal satisfies majority of people who want “equity”. It is important to note that most equity supporters aren’t ideological, it is primarily the leaders/drivers of these programs and a different program that makes significant changes designed to promote equity would satisfy most CMF proponents. All the effort to say that the CMF is voluntary and still allows for some variation and all that is because only a small “elite” group wants to go full Bergeron and they need plausible deniability. That is why eventually you have to provide a true alternative that is comprehensive and serious, although you could argue for something different than what I’ve proposed.

Note that people on the left feel the same about something like charters as you feel about the CMF. There are the public claims and then the private desires. Those private desires, like moving education out of the government and crippling teachers’ unions are what eventually sank the charter movement, when people became aware of them. Charters are a liberatrian ideological project that is part of a larger anti-government plan and parents/teachers/reformers who are not libertarian became hostile to them for that reason.

But both the majority of parents who support charters and the non-top level people who support the CMF are not rigid in their goals and would support an alternative system if one was available. You’d just have to provide a plausible model and a potential coalition for reform of the current outdated system for them to connect to.

You can do a smaller targeted effort to defang the CMF but you’ll still have to deal with parents and teachers and students who are very ready to drop the existing system. Think of it like a sphere of space teleported into the atmosphere. Something has to fill the space left by the failures of the old system so a purely anti-CMF campaign can only stall the rise of “equity education” temporarily.

68. Eitan Bachmat Says:

Hi Boaz and Scott
If you are serious about this and are willing to actually work on this, we can set up a zoom to discuss and if you can persuade me about this issue (requires more detail) I am willing to try and help you, don’t know if I can, but I have some experience as a cleaning person and there seems to be a mess.
Thanks eitan

69. STEM Caveman Says:

@Eitan, can you elaborate on what you consider as “mess”? I think the authors shoot themselves in the foot with the pandering to Minority Education Issues Of Great Social Weight, which is likely to backfire in real terms (higher math requirements can deny high school diplomas to more minority students than would be added to calculus), and is the wrong framing and targeting of the message.

70. Kathy Says:

Using the word ‘access’ and then saying that certain racial groups have ‘less access’ to math classes is misleading.

Calculus is offered in 77% of (high) schools serving the bottom quartile? Fantastic! That is a very high %. What % of those school’s students in the bottom quartile take calculus? How are the students in those schools performing in math — say on the NAEP? Are they proficient in math that is lower level than calculus? Might this lack of proficiency in lower level math have something to do with whether calculus is offered in those HSs? What about the influence of non merit based pay for teachers? Does that make it harder to attract well qualified math teachers, whose skills are in demand in private industry?

Does whether calculus is offered in one’s HS translate to whether students have less ‘access’ to STEM jobs? Evidence for that assertion? Might there be other reasons for this under representation in STEM jobs by certain demographics?

Are there other ways to ‘access’ calculus aside from in one’s own HS? Are there textbooks on calculus in the public library, for example? What about Khan Academy – does it offer a course on calculus? Or what of other online vendors and courses – do they offer course in calculus? Are there community colleges nearby that offer calculus courses?

71. Eitan Bachmat Says:

Dear STEM caveman
By mess I mean the California board proposal, some things seem reasonable while others seem not reasonable, probably worth cleaning up, but can’t say I understand well enough.

72. Boaz Barak Says:

Eitan: thank you but I think that since this is a US issue, it should be addressed by US based people.

Everyone: if you are a US based STEM educator, researcher or practitioner please consider signing the letter as well as forwarding it to other people you know.

73. TiredOfYourSanctimoniousHorseshit Says:

It’s almost comedic to watch you and your colleagues push back against the result of woke politics. Yes, I know, you think you’ve pushed back against that movement, but it’s inexcusably naive to think that voting for Democrat leaders wouldn’t lead to this situation.

For example, you praise Obama, but his focus on disparate impact of discipline in schools was a huge red flag. You get what you vote for, although surely you won’t see it that way.

I hope the CMF is only the first in a series of catastrophically stupid revisions to education in California, if that is what it takes to wake up voters in that state. Combined with the ideological filtering ongoing at UC schools (thankfully, Dr. Thompson had the spine that too many academics lack), I suspect the education of future CA students is looking fairly grim.

74. Eli Says:

Can’t bring myself to sign this kind of thing without someone putting forth an actual platform of reforms that ensure fundamental math is actually taught to everyone, not used as a weed-out course or a class sorting mechanism.

75. Scott Says:

Eli #73: What if every math class, from the easiest to the most advanced, were to be completely open to any student who wanted to take it—with the one proviso that no class would change its level based on the students who were in it; taking a faster-paced course would be “100% at your own risk”?

76. Scott Says:

TOYSH #72: I’m reluctant to engage someone whose handle is such a crude insult, but I’ll remark that even in ultraliberal California, and even in now-solidly-blue Virginia, what I’ve come to think of as the “Harrison Bergeron theory of education” seems to fare rather poorly when it’s put directly to voters, especially parents. So I haven’t abandoned hope; I see plenty of room for politicians of both parties to do well while talking sense about this issue.

And if, as it seems from your comment, you’d rather see blue states like California burn to the ground as deserved punishment for their blueness, than succeed by adopting policies that you and I agree are correct … well, isn’t that sort of the right-wing mirror image of the Communists who used to cheer when conservatives won elections and mourn when progressives won them, since all that mattered was “heightening the contradictions” until capitalist society collapsed? I confess to having zero time for any ideology, left or right, that seeks to increase human misery as a way to “wake the masses up.”

77. Edward M Measure Says:

Scott #74. Yes, but, the carnage in such classes can be terrible, and advocates of equity will be quick to assert that those who fail are victims of discrimination. Not everyone can dunk a basketball and not everyone can pass advanced calculus. Advanced classes select the most dedicated and talented students and discriminate (or select against) the less talented and dedicated.
One of the most egregious sins of the California document is the outright rejection of the notion that their could be such a thing as mathematical talent. It is false, outrageously false, and profoundly destructive.

78. Craken Says:

“Everyone is conservative about what he knows best.”

Scott Aaronson on conserving mathematical education reminds me of Harold Bloom on conserving the literary canon. Both are/were Leftists happy to set fire to everyone else’s values, but enraged when their very own political philosophy comes ineluctably to fire the centers of their worlds. I suppose they had faith that their cloistered delicate dependent worlds would remain intact, as happy unprincipled exceptions, amidst the general decivilising of the Western world. At least Bloom, in his final years, recognized the futility of reason in face of the Red monster; yet, he continued to feed it until his end, grotesquely sacrificing his Precious to his viscera. But, Don Quixote of Austin, perhaps too close to the monster to perceive its vast deceptive dimensions, mysteriously conducts a public struggle to feed and fight the self-same beast.

Most of the American elite knows nothing best, has nothing to conserve, finds conservation a notion with which they cannot identify for lack of relevant experience and the comprehensible analogies it ought to generate. Thus, they have an excuse for their failure to comprehend Chesterton’s fence or Gell-Mann amnesia.

I wonder if the “autonomy first” wing of the Left is doomed to be destroyed the “equality first” wing, just as the anarchists are always destroyed by the communists at first opportunity, whereas, by definition, the destruction never goes the other way.

79. Scott Says:

Craken #77: Speaking, once again, only for myself and not for the other signatories—I find myself to be temperamentally conservative on many, many issues, not just this one. It’s the same temperamental conservatism that causes me to be horrified and depressed by (to take a few examples) the destruction of rigorous math education, the destruction of the earth’s ocean life and its glaciers and the Amazon rainforest, and the ongoing destruction of American democracy aided by the January 6 insurrectionists.

80. Michael Weissman Says:

Perhaps it will help to make a constructive suggestion. I agree with the trendy view that for most students a good grounding in stats is more useful than some other topics, e.g. trig, parts of algebra, formal geometry, and even calculus. Stats are useful both for many technical areas and for general citizenship. This does not, however, justify replacing any math with schlocky “data science” courses.

One problem with teaching stats in middle school or high school or even college is that when it’s taught poorly students go away with serious misconceptions, unlike say trig where most students go away with nothing. So how to get decent stats to students who can’t get it from their teachers?

Ellen Fireman (my wife) has taught several 10s of thousands of UIUC students in a fairly non-technical course, descended from the famous Freedman text from Berkeley. The general approach is described here: (https://stat.illinois.edu/news/2021-11-18/netmath-expands-statistics-course-offerings-include-stat-100)
I think a version of a colleague’s lectures are available online, perhaps along with access to automated homework problems. I’ll scrounge the links soon.

To take the course students really should have facility with basic algebra. No calculus techniques are needed though it helps to understand the concept of area under a curve.

Ellen’s lectures in a more technical and advanced intro course with the same philosophy are available here (http://courses.atlas.illinois.edu/fall2020/STAT200/), and I think that soon links to the homework will also be available. One semester the highest grade was obtained by an online high school student.

UIUC also will let anybody take the courses for credit (https://netmath.illinois.edu/stat-200-statistical-analysis), but they charge tuition.

81. Allan Dobbins Says:

Kurt Vonnegut once wrote a story, or had a Kilgore Trout story within a story, about a society in which those with the clearest thought processes were subject to loud continuous noise in their ears and the most physically gifted were attached to heavy weights which they were compelled to drag around, all in the interest of making everything equal for everyone. It sounds like the California math education proposal is an instantiation of that idea. I wish I had had access to more mathematics when I was in school — I doubt that there was even one teacher who had the vaguest sense of what calculus is. If our goal is to maximize human flourishing then we need to design a system in which there are opportunities for learning geometry (yes, often left out of the discussion), algebra, probability, calculus … My view is that calculus is not necessarily AP calculus but, at minimum, learning what the derivative and integral mean in one’s bones, even if the student doesn’t learn about limits, convergence of series, etc. This implies that there should be multiple modules of calculus and students can do a variable number of them in high school.

82. Scott Says:

Allan Dobbins #80: Yeah, people have commented to me over and over that they thought Harrison Bergeron was kind of over-the-top when they read it, and didn’t expect Vonnegut’s story to become our actual reality.

83. TL Says:

Perhaps it may help to reframe some of the issues involved? Should a goal of the public education system be to help all students reach their potential? From that perspective, inequity is not in itself a problem. Rather, a high level of inequity is a problem because it indicates a great amount of wasted potential. Reducing inequity by holding some students back is thus counterproductive because it simply compounds the problem of wasted potential.

84. Topologist Guy Says:

Scott,

You’re a rationally minded person. So I continue to be astounded that you repeat the dangerous authoritarian and, may I say, anti-democratic perspective on the January 6 “insurrection.”

There are things about the sixth that are absolutely clear:
It was a demonstration, NOT an “insurrection” or even an “attack”.
The protesters were unarmed, had no plans of attack, and were not arranged in a combat formation.
The protesters never attempted to seriously harm or detain the people in the building or outside it, and what violence occurred was virtually all scuffling at barriers and thoughtlessly throwing small found objects, activities that are normal occurrences at any large crowd protest. No deaths or serious injury resulted from this behavior, and it was not widespread or pervasive.
Even the property damage was limited to that associated with any large uncontrolled crowd. Some broken windows, some petty theft, and some vandalism — and much of that was at the hands of infiltrating agitators whom members of the crowd identified to police and in some cases even intervened to stop.
Threats and rhetoric were of a taunting and brash, rebellious, and disrespectful nature, not calculated messages of incitement, calls to commit immediate bodily harm, or intimidation beyond the type common to any protest.
Protesters had good reason to believe that they were granted access to the building and grounds, both by precedent of previous ‘occupying’ protests, and by observing the actions and words of Capitol Police ‘waving them in’. Instances where trespassing might be more obvious consisted of staged symbolic acts, petty souvenir-taking, and in a few instances, seemingly genuine if misguided attempts to locate evidence of a national security crime to deliver to authorities.
These facts are known and provable with video footage.

January 6 was an unruly but mostly peaceful protest, in which a small number of protesters broke into a government building. The media narrative that this disruptive protest was a planned “insurrection” is not only manifestly and egregiously untrue, but is itself a far greater threat to our democracy and our constitutional freedoms of speech and assembly than these unfortunate protesters.

Imagine if there were large disruptive protests after Trump’s inaugaration in DC, and a few dozen unarmed left-wing protesters broke into the Capitol building, chanted some slogans like “black lives matter” and left peacefully. Imagine if Trump and Fox News called them terrorists and Trump detained them in brutal conditions in a DC prison camp for months and sentenced them to years in prison. Imagine if Republicans used the “attack” as a pretext for banning left-wing speech from social media and the internet and putting left wing protesters on no-fly lists.

It’s so clearly obvious which political party is the anti-democratic one. January 6 was the Democrats’ Reichstag Fire, and they’re using it cynically as a pretext for establishing a biomedical surveillance police state, controlling all anti-authoritarian speech, branding their political enemies as terrorists and forcing experimental medical treatments on the wntire population.

85. Scott Says:

Topologist Guy #83: Sorry, a colleague who saw this thread made me promise that I won’t get drawn into a debate over January 6. 😀

For 16 years, my tendency in this comment section has been that, when people tug at various threads of my worldview, I simply throw as much of the rest of my worldview at them as they care to know … refusing to compartmentalize, no matter how much it escalates or derails the situation. But in deference to my colleagues and fellow signatories, this time I’m going to resist!

86. Topologist Guy Says:

Scott,

I do respect your desire to avoid getting involved in a debate about Jan 6. Something I’ve always admired about you, which seems to be in quite rare supply these days, is your willingness to interrogate all different points of view rationally, to avoid partisanship, to stand up for what’s right, even if it’s unpopular, and to defend your principles. You’ve certainly found yourself at odds with the progressive-left consensus in academia on a number of different issues. Recently you’ve apologized for dismissing the possibility of a lab-leak origin for COVID-19, which I do deeply respect. Nonetheless I wish you would extend your openness to question, for instance, the “woke” agenda in academia, intersectionality, race politics etc. to such “untouchable” and arguably more important controversial topics as the dangers of mass vaccination, ivermectin, censorship of “disinformation,” the collapse of the Russiagate narrative, election fraud and more!

The more I’ve kept my eyes open, the more I’ve come to the conclusion that the QAnons and the Alex Joneses of this world grasp some deep truths, despite their many deeply flawed ideas. I invite you to read this article: https://paulkingsnorth.substack.com/p/the-vaccine-moment-part-one and reflect deeply about who the greatest threat to our democracy really is. When I see a minority singled out, skapegoated for disease and confined to their homes, when I see calls for universal passports to identify these minorities and surveil their every move, to isolate them from society, even to send them to camps—it gives me shivers down my spine. And maybe, just maybe, those Jan 6 “insurrectionists” were on the right side of history.

This mass vaccination campaign, it’s the largest human experiment in history. I’d be happy to send you reputable peer-reviewed articles by reputable immonologists and virologists in reputable journals, outlining some of the (potentially catastrophic) theoretical risks of this vaccination campaign. We have very little idea how the vaccines will interact with a rapidly evolving and highly complex viral pandemic. Theoretical risks like original antigenic sin and antibody-dependent enhancement were not evaluated during clinical trials, nor could they be in such a short period of time. We are the lab rats, and the vaccine is a livestock vaccine—dangerous, non-sterilizing and poorly efficacious. When powerful elites and instiutions want to turn you into lab rats and livestock, well…breaking into their buildings and stealing a desk or two seems like quite the underreaction 🙂

87. Data Science Collides with Traditional Math in the Golden State - Today Mag Says:

[…] an accompanying post on the blog run by University of Texas at Austin Computer Science Professors Scott Aaronson, Boaz Barak of […]

88. Data Science Collides with Traditional Math in the Golden State - Datanami - Free Courses Guru Says:

[…] algebra or calculus, and many documented uses of  ‘data science’ amplify inequity.”In an accompanying post on the blog run by University of Texas at Austin Computer Science Professors Scott Aaronson, Boaz Barak of […]

89. Boaz Barak Says:

Michael Weissman #79: Yes, we are definitely not against statistics or data science, but it is important to (1) teach it well, and (2) keep students’ options options for majoring in STEM in college, especially since these are fast becoming the most popular majors (and the ones with best job prospects)

See https://gdoc.pub/doc/e/2PACX-1vQvuzlJ8MWthsqOhRLxQc5akGS0JkgThz3umqO3K-WQiXFhWiq9qw-9iYdTyC652Ftjvv5nHvgGYTEv for the issues we take with the particular approach for data science in the CMF revisions.

90. Some Math and Physics Items | Not Even Wrong Says:

[…] as the California Mathematics Framework. For more about this, see the blog entry posted here and on Scott Aaronson’s blog, and more detail […]

The way the mathematics is taught in the United States is largely a disaster for both the students and the teachers, with topics taught out of order, and neither the current curriculum nor the proposed curriculum in California really address the issue here.

Functions and graphs of functions should be taught around the same time as equations in an algebra class (and not in precalculus), that way one has a visual representation of an equation (i.e. the intersection of two functions on a graph), and an understanding of the relationship between inverse functions of linear functions and solving equations of linear functions, and both linear functions and equations could be introduced when rational numbers get introduced into the curriculum (i.e. in pre-algebra rather than in algebra/precalculus), as they provide yet another representation of the rational numbers through the slope and solution set of the function.

Basic naive set theory also needs to be in the curriculum in an algebra class, so that students could understand what the symbols $\mathbb{N}$, $\mathbb{Z}$, and $\mathbb{Q}$ are in mathematics and what a solution set of an equation is, and so that in the future students could understand that the factorial function and the binomial coefficient are functions defined over only the natural numbers (to avoid thorny questions about the Gamma function), and when defining polynomials and polynomial functions, one could sidestep the issue of defining fractional exponents in powers if we restrict exponents to natural numbers or integers for the time being.

Square roots, fractional powers, infinite decimals, e, pi/tau, and similar things usually introduced in an American pre-algebra class should be delayed until real numbers get introduced in a real analysis class, which should be an actual class instead of largely split between pre-algebra, algebra, and pre-calculus. (I like the name real analysis better than pre-calculus, as it is a topic in and of itself, rather than just being preparation for calculus) And one first introduces sequences and limits of sequences, and define real numbers as infinite decimals, which are actually limits of a sequence of finite decimals. One could then extend the domain and range of functions to cover the entire set of real numbers. Since students should already understand the pointwise definition of functions from their algebra w/functions class, one could likewise introduce sequences of polynomial functions and then define various analytic functions (exponential function, trigonometric functions, inverse power functions and rational functions, logarithms, fractional powers) as a limit of infinite sequences of polynomial functions. The relationships between all the various functions could be established later. Various other functions such as the sign function, the absolute value function, and other piecewise (linear) functions would help establish what it means for a function to be continuous or discontinuous, and continuity and limits are the prerequisites for defining the derivative in a calculus class.

I suppose one could also introduce in the real analysis class above the quadratic formula to solve equations involving quadratic functions, after introducing square roots. However, in practice, most people would end up using calculators or computer code/scripting to solve equations over the real numbers, and the answers from the real world tend to be represented approximately in decimal format rather than in exact algebraic form, so more illuminating for the student than the quadratic formula would be the algorithms used in numerical real analysis (i.e. possibly in the calculator), such as the bisection method or the secant method, which apply to more than just quadratic equations. The fundamental theorem of algebra should be eliminated completely from the curriculum, it is a theorem of complex analysis rather than being anything fundamental to algebra.

Geometry should be delayed until after the real analysis class above, as many of the functions and constants used in geometry (trigonometric function, pi/tau) are defined in that class first, and should be taught not as formal geometry, but as analytic geometry, for the benefit of those who enter some science or engineering field rather than pure mathematics, which is the vast majority of students who go through the school system. For the same reason, geometry should be an elective class, rather than required for everybody to take, as not everybody will become a scientist or engineer. Also to include in geometry are trigonometry, coordinate systems, 2D and 3D vectors and their relation to translations, the various vector operations (dot product, outer product, cross product, geometric product) and their relation to trigonometric functions, bivectors and their relation to rotations and complex numbers in 2D space and quaternions in 3D space. All of these are far more important in physics and engineering, where the translations and rotations in geometry become continuous trajectories in a phase space rather than discrete operations, than any discussion about conic sections and formal proofs of Ceva’s theorem which seem to prevail in the classes today.

For those who do not wish to take geometry, than introductory classes in probability and statics, data science, or computer science should be offered as an alternative after taking the real analysis class, as being able to understand and manipulate data and write code and scripts are fairly important skills to have in today’s digital world. Since the real analysis class already provides the necessary requirements for differential calculus, calculus could also be offered as an elective as well, for those students who need it.

92. Doug Says:

Geometry, an elective? Yikes! I say four years of geometry.

You know what, why not? Algebra w/applications to geometry, real analysis w/applications to geometry, differential calculus w/applications to geometry, integral calculus w/applications to geometry, and for advanced enough students, differential geometry.

94. Non-Woke TCS Says:

@ Topologist Guy: “to such “untouchable” and arguably more important controversial topics as the dangers of mass vaccination,”

I find this part extremely off putting. While I sympathize with all other points you’ve raised, I simply reject the idea that vaccinations are part of any political debate. While, the 6th Jan “insurrection” narrative is driven by the false political agenda to de-legitimize the right, in order for one socio-political group to gain political control for self-serving causes, the mass vaccination campaign on the other hand is an international effort to save lives, and has almost nothing to do with political power gaining. Is this effort going to succeed, or does it hold some risks? Maybe. But it’s simply not a political debate (or should not be one).

95. Scott Says:

Non-Woke TCS #93: See, but I wouldn’t have imagined that the question “who won the 2020 US presidential election?” (as opposed to, “who should win it?”) would become a political question either, if neither the Electoral College nor the popular votes were especially close. I also wouldn’t have imagined that the question “should high school students be suffered to learn calculus, even though learning it might put them ahead of other students who don’t learn it?” would become a political question. But it’s a defining feature of our time that the ideologues of left and right have successfully politicized question after question that one wouldn’t have imagined could be politicized.

96. Thomas E Hastin Says:

Thank you for this piece and all the signatories to it. California’s mathematics is on the verge of taking an entire generation off the map. The teaching of algebra early on is critically important. The Ca dept of edu. is running off the rails trying for “equity”.

I’m just a grandfather who has had his eyes opened during covid to how and what is taught in public schools and how poorly those districts and schools in poorer neighborhoods teach math in K-8 grades. When only 25% of the students in a district are comprehending math AT GRADE LEVEL, 3% above and the rest left behind it is worrisome to say the least.

While I applaud the effort to keep algebra in grades 7-8 I would ask those who signed and are reading this to take it one step further and advocate for more resources brought to bear in grade 1-6. Algebra isn’t even on the table when the very basics are neglected and only 35% of 3rd graders can do the simplest addition, subtraction and multiplication. Please, advocacy to bring more RESOURSES early on would help achieve equity but also better prepare students for algebra and beyond that STEM.
The whole of the new 10 year plan for California is wrong. Learning algebra in the 8th grade is impossible to achieve if the students can’t do Multiplication in the 3rd.
Bravo on this letter and I hope you can help make math more of a center point in the younger grades.

97. Nilima Nigam Says:

It may be useful to see what the curriculum is like in other places.

Here’s the Grade 11 and 12 mathematics curriculum in India’s Central Board of Secondary Education. It’s one of the many ‘Boards’ of education, and a major federal-level one. There are a lot of kids that learn from this curriculum, and they are from one of the most heterogenous societies you could think of.

I’d reproduce the content, but would hate to clutter your comment section (this is all tangential to the main discussion!) I will note that some complex analysis, probability, vector algebra, geometry and calculus is considered entirely reasonable.

https://ncert.nic.in/pdf/syllabus/desm_s_Mathematics.pdf

Not everyone in the CBSE system needs to take math beyond grade 10 (it’s an ‘elective’ in the system).

98. Zen Cheruveettil Says:

@scott @boaz

During the last few decades, those who shone in Mathematics did quite well in life by working in finance and computer industries. But, during the same period, we also saw some worrying levels of growth in social and income inequalities, which were further exacerbated by financial crisis and globalization.

California math reformers seem to be keen on addressing those concerns by prioritising solving those problems. Your concerns are quire legitimate as well, but underrepresentation is not something which you can address by charity projects and non profit groups alone. The whole system has to change to increase the participation among women and minorities.

Personally, I grew up in India in a community known for its non participation in Maths. My mother was really good in mathematics, but as a Muslim woman growing up in 70s, her wings were clipped early on. In India, lower castes and Muslims take the place of Blacks and Latinas in USA. Positive discrimination had only limited success in the current system which is with preoccupied with the question of “who has the math genes”.

One of the critiques of California program which I read stated that “If you want a job in data science that isn’t replaceable by a computer in the next couple of years, you need to take calculus,”. While this might be true, the point which they are missing is that if you are a data scientist who lost job thanks to automation, you are already in a stronger position to figure out what next to do. If you never allow her to become a data scientist, (s)he might be doing some blue collar job where she has even less control over her life.

99. Zen Cheruveettil Says:

@DR

“In India too, school math and science are not sufficient to prepare for hard exams like the IIT JEE. Top schools (not that there are many of those) compete with one another on how many 12th graders get into IITs, etc. Firstly, they admit students selectively based on 10th grade scores. In 11th and 12th grade, the way they encourage students to study for IIT, is have them finish the entire academic year’s schooling in the summer before it, with special classes during summer holidays. During the academic year, you are marked present even if you are staying home studying for IIT. There are better places than regular public (and even private) schools that provide coaching for IIT JEE exams, and they encourage you to attend those instead!!!! You only come in for science labs. This is how my husband’s 11th and 12th grade were. I went to a real school that insisted I show up, and burdened me with busy work. It was unproductive.

This was all in my time, 30 years ago. Not much has changed except students start studying for IIT and other competitive exams now by 9th grade. 11th grade is too late.

This is how education works in most schools in Bombay! The state has these silly rules about attendance, which the schools find their way around.

For gifted kids, the main challenge is finding a way to make the time spent in formal education productive. I rather like Bombay’s solution, since the state won’t budge on its rules.”
——

so, what is your conclusion here? Is that a good or bad system? You mentioned a system which invests most of its educational resources in such a way that a few kids whose parents are privileged (even if those kids themselves are fast in doing mental arithmetics) are channelled to subsidised, elite engineering colleges whereas significant portion of the rest of the population have problems in basic reading and writing. In India it is also a common thing that a few kids who don’t make it to IIT after repeated attempts kill themselves by hanging in their dorms where they had spent past 3 or 4 years of their lives.

These dark aspects of IITs are barely mentioned in US.

100. Scott Says:

Zen Cheruveettil #97:

During the last few decades, those who shone in Mathematics did quite well in life by working in finance and computer industries. But, during the same period, we also saw some worrying levels of growth in social and income inequalities, which were further exacerbated by financial crisis and globalization.
California math reformers seem to be keen on addressing those concerns by prioritising solving those problems…

Then this is the crux of it: I don’t see it as the role of math class to try to solve those problems. It’s a high enough aspiration for a math class to “power up” each and every student to the best mathematical understanding that they’re capable of. Right now, even that limited goal is incredibly far from being achieved, and progress on it has all sorts of social benefits, so why not work together toward achieving it? And what’s the alternative: that some students’ mathematical understanding should be deliberately crippled for the sake of fighting income inequality? “No, of course that’s not what I mean” … well then, what do you mean?

If we’re concerned about income inequality, then we should debate things like tax policy, and raising the minimum wage, and universal basic income, and (an elephant in the room) the credential and licensing requirements that keep so many otherwise qualified people out of high-paid professions. I don’t see the problem as one of some kids learning too much math.

101. Zen Cheruveettil Says:

I don’t see the problem as one of some kids learning too much math.
—–

Me neither. Jo Boaler and her team states several motivations and the reasons I mentioned are not there, so income inequality is anyone’s guess by reading between the lines.

I have mixed feelings about the whole Calculus debate – I certainly agree with your [assessments](https://gdoc.pub/doc/e/2PACX-1vQvuzlJ8MWthsqOhRLxQc5akGS0JkgThz3umqO3K-WQiXFhWiq9qw-9iYdTyC652Ftjvv5nHvgGYTEv) about the importance of LA and MVC for becoming a good data scientist. But in the future, data science is probably going to be the equivalent of clerical job of the 70s and 80s. Everyone should understand data even if they do not understand behind the scenes algorithms and mathematics. And by removing calculus as a gatekeeper to higher education, many people who might have otherwise been left out are likely to profit.

102. fred Says:

Scott #99
in other words, it’s equality by the lowest denominator.
If raising opportunities for all is too hard, or if opportunities are the same but outcomes are different, then closing up opportunities is acceptable, apparently. I.e. reducing the variability of the human experience in the name of equality.

It’s about the long term goals and priorities of humanity. Maximizing average quality of life, maximizing the best possible experiences (even if rare), minimizing the worst possible experiences, minimizing the variability in quality of life, maximizing the total happiness (the bigger humanity is, the better), etc? All while realizing that life will always be fundamentally unfair, a lottery, no matter how much we try to control it… unless we’re all identical clones (thanks to genetic programming), living in identical physical pods in the same identical virtual reality, told exactly what to do and think.

103. fred Says:

Progress has always relied on the fact that life is luck.

Species do not evolve until a few individuals are the recipients of random mutations that give them an edge thanks to better adaptations to their current environment. Eventually the whole species inherits that adaptation (assuming that better adaptation means higher chance to produce offspring).

The same dynamics happen for civilizations.
It takes a certain population mass to increase the chance of producing exceptional individuals and allow them to dedicate their life to their gift.
The hope is that those exceptional individuals will be able to live their life in a way that produces things and ideas that will then benefit the whole society. It’s the case with scientific discoveries (producing technological marvels that everyone can enjoy, like cell phones and advanced healthcare) and sport (the exploits of a few producing national pride) or politics (the exceptional courage and dedication of a few leading the way for big society changes).
We want to maximize opportunities to everyone not as a way to insure equality of outcome, but simply to maximize probability of quality of outcome. The more exceptional individuals can realize themselves, the better for everyone. Recognizing also that “non-exceptional” (whatever that means) individuals are needed just as much as “exceptional” individuals. A society that only has top professional athletes or top mathematicians would simply not function. And a society that just suppresses all top performers would have a much tougher time to progress and adapt.
I also think that it doesn’t take that much to snuff out the capability for a society to make true breakthroughs… currently we produce a lot of physicists, but it’s not clear whether the academic system and private job market can really scale up to let the truly exceptional minds thrive. There’s a tension between quantity of outcome and quality of outcome.

104. Scott Says:

Zen Cheruveettil #100: I agree that calculus shouldn’t be a gatekeeper to higher education. But it is, and will be, a gatekeeper to STEM. Not because it was arbitrarily decreed as such, but because without calculus, you’re still stuck on Zeno’s Paradox and are unable to progress beyond the level of the ancient Greeks, in modeling any of the apparently continuous phenomena of our universe (apart from the very simplest ones, like straight-line and circular motions).

It’s true that the specific rules of differentiation and integration, like other mechanical rules, can generally be forgotten once learned, to be looked up again, or given to a computer, or best yet rederived whenever needed. But the conceptual understanding? We need that even in CS, the most “discrete” of all STEM subjects, so imagine how much it’s needed in engineering and economics and atmospheric science.

105. Anti-Woke TCS Says:

@Scott #99:
“Then this is the crux of it: I don’t see it as the role of math class to try to solve those problems.

If we’re concerned about income inequality, then we should debate things like tax policy, and raising the minimum wage,

I completely agree with this! The goal of math-education is not to achieve “social justice” (which is a politically dubious goal by itself in my opinion). It is to achieve — surrrpr-a-ah–ise — math education!

That’s indeed the crux of the whole “Social Justice” (i.e., Woke) movement: hijacking each and every pure and noble human pursuit to their own subjective political agenda.
Hollywood is now not about the art or business of cinema, but about “diversity of cast”. Math education is not about math, it’s about forcing oppressive equity by slowing down ambitious students. University is not about truth seeking or advancement of science and education but about promoting “equity, inclusion & diversity”. By now, it seems that the far-left has succeeded to politicize almost every human endeavor.

106. Zen Cheruveettil Says:

@scott

I do not question the importance of calculus in STEM. Also I believe that calculus brings a unique perspective to maths, which cannot be gained by studying about sets and number theory alone. I am a person who is taking calculus lessons for a second time in life because I was taught these topics in a rush and sloppy way and conceptual understanding which you mentioned was precisely missing.

I once read a commentary from Keith Devlin (of MAA) that calculus can never be “taught” properly in K-12, so I was relieved that its not just me. Maybe Terence Taos of the world don’t have that problem, but some of us need time to digest and appreciate its beauty.

One of the ideas which I heard recently is to rethink the idea of education completely, given the trend in longevity and pace of change in science. So instead of cramming these complex topics between 11th grade and final years of undergraduation, a motivated student could study it over longer phases of his life.

107. diet pepsi Says:

Thank you for spreading the word about this issue. Both this letter and the other letter that is restricted to CA-based signers have much more support among faculty at CA public universities than the number of signatories may indicate.

The trouble for untenured STEM like myself in the public U system here in CA is that the admin blob supports the lowering of standards. Worse, tenured senior faculty with aspirations of being absorbed into the admin blob have also sent a strong negative signal against speaking out against erosion of standards in CA.

The admin blob’s logic isnt hard to follow. Lowering of educational standards is a leftwing cause, and students — I mean customers — skew left, therefore the admin blob takes lefty positions on all issues because the customer is always right.

For faculty who seek to become one with admin blob, there is no choice but to imitate it. There are only so many openings for Assistant Vice Dean of No Importance, and even with new admin blob appendages sprouting regularly, there are still far more vying to get away from research and teaching. Who can blame them for not wanting to teach classes where 25% of students don’t know its the same as 1/4?

Unfortunately our customers dont see that the prestige of their degrees erodes as the standards do. A degree from say UC Santa Barbara is prestigious today in-state even if not so much beyond the borders. A few more years of no standardized tests, no high school calc, no grades, no English proficiency,… a UCSB degree will be what a Cal State San Bernardino degree is today. Already, smart California kids dont see UC Berkeley as desireable for undergrad as they once did. The out-of-state flagship universities look better and better by comparison every year. Who’da thunk our smart kids would take more pride in attending schools that have admissions standards?

108. TiredOfYourSanctimoniousHoreshit Says:

@Scott #75

“And if, as it seems from your comment, you’d rather see blue states like California burn to the ground as deserved punishment for their blueness,…”

You’re the only one using colorful language about letting blue states “burn to the ground”. At the same time dodging the issue of disparate impact with the platitude “I haven’t abandoned hope”.

No, blue has little to do with it. Rather, I’d like to see punished the specific subset of woke-ists who are destroying other kids’ educations, along with the stooges like yourself who give them power/cover by voting for politicians (who facilitate things like CMF) and then pat themselves on the back for signing an open letter.

Preferably, the punishment would be their/your kids facing the costs for being stuck in a classroom of unruly kids who make learning impossible, with no recourse for the teachers. It would be their/your kids discriminated against in university admissions because their grandparents were Asian or of some other disfavored race. That might wake you and others up.

But it won’t be your kids, right? Because you’re relatively affluent with lots of options, and so you can afford to put on airs about having Enlightenment principles or scrutinizing your worldview, all the while offloading the costs onto other peoples’ kids by voting for the people who come up with these harmful policies.

“I’m reluctant to engage someone whose handle is such a crude insult…isn’t that sort of the right-wing mirror image of the Communists”

I’d take your reluctance more seriously if you didn’t just turn around and compare me to a murderous regime. Pot, meet kettle.

109. Zen Cheruveettil Says:

@Anti-Woke TCS Says:

“Math education is not about math, it’s about forcing oppressive equity by slowing down ambitious students. University is not about truth seeking or advancement of science and education but about promoting “equity, inclusion & diversity”
———-

The main goals of funded education should be targeted towards building a healthy society which protects the interest of all. So there is some public interest involved there. Also, there are lots of mischaracterisation involved in this debate about math reform. Outliers and prodigies will still be able to pursue their talents and interests, just like in any other fields such as music or sports.

110. Scott Says:

Zen Cheruveettil #105: On the one hand, you’re absolutely right that learning is a lifelong process. I’m now 40 years old, but just during this pandemic, I tried once again to learn forcing in set theory, quantum field theory, and holographic error correction, getting a bit further with each than all the previous times I tried!

On the other hand, in the modern world, formal schooling is stretched out for a ridiculously, eye-wateringly long time … to the point where the notion of stretching it out for even longer boggles the imagination.

I first learned calculus when I was 11, although I was 14 before I was allowed to take BC Calc in school. I’m still at least as proud of, when I was 13, figuring out the integral for the length of a curve on my own, as I am of any research paper I’ve written. Looking back, I have trouble reconstructing what the difficulty even was—obviously, you put √((dx)2+(dy)2) in front of the integral sign, then you rewrite it to √(1+(dy/dx)2) dx, what else even is there to say?—but at the time it rocked my world.

And yet, I was not obviously better than math than all the other kids at my school. If I was an extreme outlier, it was mostly just in the burning desire to know—a desire that many of the school bureaucrats, with their beloved frameworks and curriculum plans, tried to frustrate at every available opportunity. This makes me believe that, if we actually tried, quite a few other kids could learn calculus at age 10 or 11. Or OK, if that’s too radical, then let’s compromise at age 14 or 15. To facilitate this, I propose that the mind-numbing repetition of basic arithmetic and memorizing the names of shapes over and over and over, in grades K-6, be compressed to a few years, and that simple algebra be introduced starting in grades 3 or 4. (Crazy? They’ve done it in other countries!)

Then, if students are really going to stay in high school until age 18, multiple years could be spent on calculus, to let students gain the deep understanding that you talk about. And they’d also have time for courses on statistics and data science, and mathematical logic, and algorithms and combinatorics.

Anyway, that would be the STEM track. Not for everyone, but other tracks would also be available, including a humanities track, an arts track, and various professional/vocational tracks. What do you say?

111. Scott Says:

TOYSH #107: I see, so you don’t want to burn California to the ground, you just want my kids to suffer in order to punish me for voting for Democrats, when the alternative is … well, the whole world has seen what the alternative is.

There’s a saying that the reason why Democrats lose so many elections, despite popular support for most of their policies, is that “the Right looks for converts, while the Left looks only for heretics.” You’re a counterexample: someone from the Right who’s hunting for heretics, even to the point of wanting to punish my kids because I dared champion a cause that you agree with (!) while not agreeing with you about everything else.

You are permanently banned from this blog.

112. Scott Says:

Zen Cheruveettil #108:

Outliers and prodigies will still be able to pursue their talents and interests, just like in any other fields such as music or sports.

HAHAHAHAHAHAHA

How familiar are you with the American public school system? This is a system that, by and large, does everything it can to grind “outliers and prodigies” to a pulp. I know this more thoroughly than I know almost any other topic, because it was my reality. No one can gaslight me about it.

Having said that, here and there some “outliers and prodigies” do manage to escape—for example, by going to magnet schools, or by skipping grades, or at least by placing into faster-paced math classes. By a crazy coincidence, the “reform” people who you idolize are hard at work trying to seal off every last one of those escape hatches. Given this, and given my experience, do you see why I questioned whether the word “well-intentioned” really belonged in our open letter?

113. Anti-Woke TCS Says:

Zen Cheruveettil #108:

The main goals of funded education should be targeted towards building a healthy society which protects the interest of all.

Yes, that’s what politicians and administrators use to say. It’s natural: they want to control universities, to call the shots. But the absurdity of this idea becomes evident when you realize that what they call a “healthy society” and “the interest of all” is nothing more than their own party line. And when the party in charge changes, suddenly “health” changes its meaning as well. Once it’s ‘equity’, and next it’s ‘freedom to choose’, or ‘equal opportunity’, and then it’s the opposite ‘excellence’! So eventually you are left with two options: a politicized education system working to satisfy temporary political goals, the contributions of which would largely be ridiculed by future generations as a form of grotesque Soviet-style establishment propaganda (“equity!”), and on the other hand a neutral as possible apolitical science that is rooted in eternal principles, revered and admired for centuries to come. Tough choice.

114. Suomynona Says:

Some of this discussion reminds me of a guy I knew in college. He was studying to become a K-12 teacher and was reasonably smart; he had a 4.0 GPA and had taken calculus and statistics. However, something that he insisted was true was that it’s a biological impossibility(!) for people to learn calculus or other “advanced math” before they are 17 or 18 since the concepts are too abstract for a young human brain to process them. A biological impossibility! When I told him that there were 14 year olds in my high school calculus class, he dismissed them as being math prodigies who are genetically different than normal humans and thus don’t count against this “biological fact”.

So according to him, he might think there’s no point in having a high school curriculum which has an allowance for “advanced math” since the typical student is biologically incapable of learning those concepts anyway.

In my experience, many K-12 educators do not have STEM backgrounds and/or are humanities-types that even have an aversion to STEM subjects. It’s unfortunately not hard for me to see how the CMF could come into being and be considered a good idea by K-12 educators.

115. Boaz Barak Says:

Zen Cheruveettil: I agree that inequality, in terms of income, food and housing, educational opportunities, and more, is a huge problem in the U.S. I also don’t have an issue with math lessons using data from these topics. However, when you are designing math curriculum, the first and foremost responsibility you have to this children is to teach them math. This is doubly so in a public system, where many of the kids will not have the resources to make up for what you fail to teach them.

Now, not everyone goes to a 4-year college, and not everyone who goes to a 4-year college majors in STEM. However, preparing for college is one of the goals of the K-12 system (not the only goal), and STEM majors have been greatly growing in popularity. Funneling kids at 9th grade into a track that restricts their options is doing them a great disservice, particularly because the kids that will be funneled are likely to be those that have less resources.

116. JimV Says:

Without calculus “you’re still stuck on Zeno’s Paradox”–Scott

I have never understood why many brilliant people think things like that. Zeno’s first argument “The Arrow” proves that an infinite sum can have a finite limit (L/2+L/4+L/8+…) <= L). Democritus understood Zeno's argument and used it to predict the existence of atoms (continuous matter is an illusion). Minus some speculative mysticism, Zeno was simply asking how a natural process gets to the end of an endless series of steps. One obvious (in retrospect) solution is quantization.

Calculus is great, but it is simply the limit of discrete systems as the minimum increment goes to zero. Most systems in nature, such as fluid flow, are actually discrete, and work fine that way.

117. Scott Says:

JimV #116: In some sense, there’s a very simple proof that before the discovery of calculus, people didn’t really understand limits. Namely that, if they did really understand them, they would’ve discovered calculus.

This is directly analogous to the proof that before the Bell inequality, people didn’t really understand entanglement.

118. Zen Cheruveettil Says:

my last comment somehow ended up in /dev/null

@scott

On CMR:

Can we think of nurturing precocious children in a way that doesn’t harm the majority, in a way that doesn’t push the lower end of the mean further to the bottom. CMR or otherwise, this is worth addressing.

On Calculus:

This is the article I was mentioning (https://www.mathvalues.org/masterblog/2019/6/27/mathematics-is-a-way-of-thinking-how-can-we-best-teach-it). Relevant paragraph is quoted below:

For sure, no calculus, which has no place in K-12 education. Not least because there is no way it can be done well at that stage. (In large part, because it operates at the third level of abstraction, with its fundamental objects being operations on functions, which are themselves operations on numbers. That’s a huge cognitive leap that takes most of us several years to achieve.) The student who typically has the most trouble with university calculus is the one who has learned it poorly at high school, and comes to Calc 101 at university with a false belief they understand it and can do it, only to crash and burn in Calc 102. (Dealing with that crash scene was a large part of my life for over 25 years!)

My personal experience has been that I could only master the mechanics behind differentiation and integration in high school. Beyond that, it was not possible. I cannot really judge whether this was an intellectual limitation at that time or lack of time though.

Given that Calculus has played a major role in science, its history and high level concepts should be taught to everyone, probably as a part of general history of science, and by ideally including references to contributions from Hindu and Islamic cultures as well to the development of mathematics.

119. arcana Says:

Didn’t the Archimedes Palimpset show that the Greeks were pretty close to calculus? Or at least a few at the top were?

In any case a mathematics curriculum *should* be integrated into the broader school curriculum, but exothermically, rather than endothermally as the CMF does.

Bascially every subject is also a reading class and often a writing class. By exothermic I mean that math pushes out into “cold” math deficient classes rather than having math suck in other subjects. That would put it truly on par with reading and writing, per the “three Rs” concept as well as improve math usage.

Math, much like skateboarding or any other activity, requires practice. Ideally without homework because homework is a bad educational concept. Plus by bringing math out into other subjects as we do with reading and writing we get people used to casual math and they’ll be more likely to pick up hobbies that use math or use math outside of school generally.

Games often push an interest in history for instance, or military knowledge. They have math but are much less likely to lead to the use of math since people think of math as a school thing.

120. JimV Says:

Scott, thanks for the reply. It seems to me Archimedes was well on his way to discovering calculus when he was killed by a Roman soldier. Anyway, I guess empirically I have to agree that many people got hung up on their interpretation of Zeno’s argument and that may have hindered calculus development. Personally I never saw it as a refutation of infinite sums having finite limits, but a question of how nature could process an infinite series of steps–sort of similar to the action-at-a-distance issue that bothered Leibniz and Newton about gravity. (How does nature do it?)

For the record, I know of no mathematician or computer program that has ever summed an infinite series term by term, all the way to the end. We always stop somewhere (or use other tricks, like the Geometric Series formula–is that what nature does?), so it makes sense to me that nature would also. And anyone who has seen a movie or a TV program knows that continuous motion can be an illusion.

121. Scott Says:

Zen Cheruveettil #118:

Can we think of nurturing precocious children in a way that doesn’t harm the majority, in a way that doesn’t push the lower end of the mean further to the bottom. CMR or otherwise, this is worth addressing.

Why would someone even imagine that “nurturing precocious children” would “harm the majority”? Once again, does nurturing basketball or piano or ice-skating prodigies harm those who aren’t similarly gifted? Absent strong evidence otherwise, our presumption should be “of course it doesn’t! on the contrary, crushing kids with unusual gifts harms the majority, by depriving the majority of the eventual benefits of those gifts.”

122. Will Orrick Says:

The clarity and rigor with which Eudoxus handled limits were not equalled until the 19th century. The Greeks could not have invented calculus because they were lacking symbolic algebra and the use of symbolic algebra to describe geometry. After those ingredients were supplied by Viète, Descartes, and Fermat in the late 16th and early 17th centuries, calculus followed fairly quickly.

123. Scott Says:

JimV #120: Archimedes was basically a modern who happened to live in the 200s BC. He didn’t possess the Fundamental Theorem of Calculus, but yes, he certainly knew how to evaluate many nontrivial integrals (via sheer cleverness and what today we’d call Riemann sums), and in that sense, could be argued to have “known calculus.” But alas, as often happens, his knowledge about calculus didn’t stay discovered! It would only be matched and then (greatly) surpassed with the likes of Galileo, Newton, and Leibniz.

124. anon85 Says:

Scott #117: if by discovering calculus you mean “understanding limits”, then calculus wasn’t discovered until the work of Cauchy in 1821, when he first defined the limit. Before this, people were unsure about how to rigorously define the infinitesimal, something that many mathematicians tried to do and something they were (in hindsight) quite confused about.

Were Newton and Leibniz still “stuck at Zeno’s paradox”? I mean, one can certainly claim that they didn’t fully understand calculus, but I don’t think “stuck at Zeno’s” is a fair description.

125. Boaz Barak Says:

Re calculus – the discussion seems to oscillate between talking about 100% of students and few “geniuses”. Neither makes much sense:

1) It doesn’t make sense to me (and it is not proposed by people opposing the CMF) that 100% of the students take calculus, certainly not at the college level (AP calculus AB or BC)

2) That said, it’s not a few geniuses that can take it in high school. There are tens of thousands of students that take calculus in high school, score well enough on the AP exam to skip it in college, and do just fine.

The issue with the CMF is not just calculus. Calculus is useful because it actually requires the knowledge from prior courses, so it keeps the system honest. If you are designing a curriculum, especially in a new and undefined area such as “data science”, it’s easy to convince yourself that you are giving students basic mathematical skills in reasoning and algebra when you aren’t. Testing whether this curriculum prepares students for taking calculus is a good “sanity check”.

Some actual data on calculus education is here. Generally for those students who do well on the AP exam, it does allow them to skip a course in a university. It is indeed the case that because of its role as a gatekeeper, some students take calculus in high school despite not being ready for it. But those students would be better served by building basic mathematical foundations in algebra and mathematical reasoning. I find it hard to believe that they are not ready for calculus but could understand the subtle issues of probability.

https://www.maa.org/external_archive/columns/launchings/launchings_06_09.html

126. Scott Says:

anon85 #124: You could say that Newton and Leibniz found an “effective” workaround to the issues of Zeno’s Paradox—a “good enough for physics” workaround—which was only turned into rigorous mathematics with εs and δs over a century later. That’s not the only such case! With quantum field theory and its infinite-dimensional path integrals, arguably there still isn’t the rigorous foundation.

127. Scott Says:

Boaz Barak #125:

Calculus is useful because it actually requires the knowledge from prior courses, so it keeps the system honest.

That’s … an extremely important point that I’d never before seen spelled out explicitly.

128. STEM Caveman Says:

Calculus, and calculus based physics even more so, require better qualified teachers than the easier math classes, and that has “disparate impact” on hiring. This is a core, bread and butter issue for teacher unions and the diversity industrial complex. The preoccupation with Equity Or Else in hiring is already institutionalized in blue city public school systems.

129. DR Says:

I understand homeschooling kids is not a realistic option for most people in America. Maybe we should work on understanding why and changing that, instead of trying to improve public education. The former seems a lot easier than the latter.

I think education is going to improve only by looking a lot more like homeschooling. Now remember that that doesn’t mean schooling at home literally. You can hire teachers, help your kids join groups of other homeschooled kids in classes at their level,travel to field trips, and customize it all however you like. The internet makes it so easy to find social groups to do this with, even in your own city.

130. Timothy Chow Says:

In the spirit of keeping ourselves honest, I’d like to push back a bit against the truism that to do research in (to pick a somewhat arbitrary example) deep learning, you “need calculus.”

I can’t deny that calculus is needed in the “trivial” sense that just about any advanced textbook on the subject is going to use standard notation for integrals and partial derivatives at some point, and you can’t fully understand those books without understanding that notation. But is that notation strictly necessary? In most if not all applications to physics, we really do want to “think continuously,” so I accept that you do need calculus for classical mechanics, electromagnetism, etc. But in computer science, it’s not as clear to me.

Let me put it another way. Suppose you wanted to develop deep learning discretely from the ground up, without calculus. Could you do it? You would of course need the concept of gradient descent, but this can be described discretely. Backpropagation involves the chain rule, but I think that can also be done discretely. I suppose that at some point the normal distribution is going to show up, but I suspect there are ways to work around that too.

Now you might say, if we’re going to introduce discrete versions of concepts such as small changes, minima and maxima, the chain rule, areas and volumes, and so forth, then we are in effect teaching calculus; to insist that it’s not calculus is just a linguistic quibble. But Scott mentioned Zeno’s paradox, limits, and continuity. I’m not convinced that one needs to grapple with all those subtle concepts associated with infinity in order to do creative work at the highest levels in fundamentally discrete areas of STEM.

Admittedly, this is something of a idle thought experiment, since even if it were possible, I don’t see a lot of practical value in rewriting all the textbooks in a strictly discrete, calculus-free manner. But again, in the interests of intellectual honesty, it’s worth thinking a bit about whether claims that calculus is necessary are strictly true.

131. arcana Says:

@DR #129: I actually proposed a system in these comments to do that. However that is perhaps beyond the scope of effort regarding the CMF even if you account for a downscaled accelerated math only version. Although it still starts with the public education system for simple reasons. Many political. The teacher lobby is strong for dems and it is relatively easy to appease them in education reform so best to do so. There are a ton of advantages for using the scale of government, at all the different levels. No reason to waste them. In other parts of the web many people wrongly equate it with charter schools although it explicitly isn’t, but it does have some aspects that are comparable to what the effective charters do when such ideas are effective. The formatting is bad since it is a blog comment sadly.

You can have a very customizable semi-unschooling style public education system if you choose to. Just have to get voter support. Just like out current global trade treaty system is garbage but that is almost purely because of policy choices and not a law of the univserse.

132. Scott Says:

Timothy Chow #130: On the one hand, you could be an amazing, successful computer scientist without ever learning calculus … much like you could be one without learning LaTeX, or how to sort in O(n log n). It’s just that you’d constantly be stepping around this huge, relevant thing that you didn’t know—to the point where eventually, learning it would be less effort than preserving your ignorance! 🙂

Maybe I should add: I completely agree that a rigorous high-school course on statistics, or combinatorics, or algorithms, could do just as much to prepare students for STEM careers as a rigorous calculus course. The problem is that they’re not planning to replace rigorous calculus courses by rigorous courses in those other subjects, but by non-rigorous courses, ones with very little mathematical thinking as you or I would understand it!

133. Jair Says:

@Timothy Chow #130, this seems like an odd proposal. Continuous calculus is generally easier than discrete calculus in my experience. Like, what’s the integral of x^n with respect to x? x^(n+1) / (n+1) + C. How do you find the sum of terms like $k^n$ from k = 0 to N? Well, prepare for a very long lecture on Bernoulli numbers and Faulhaber’s formula…

But I agree in general that discrete math/statistics courses, etc., would be fine as an alternative to calculus at the high school level.

134. DR Says:

arcana #131, others…

This might be pretty controversial, but I care deeply about the education of poor children, and gifted education of poor children is part of that. I hope some people see parallels in this to the American K-12 system.

The right to education law (RTE) in India and how it works:

Govt there has put the parents in jail for sending kids to small private schools in slums, saying that a small black board or lack of playground is hurting children. Note that the alternative is kids going to a govt school where the teacher with a job for life doesnt even show up for work. The kids are pulled out of the small private schools that cost Rupees 5 a month, where they are thriving, and the parents jailed, and the kids forced to return to the failed govt school.

Government everywhere is threatened by private and home schools.

Parents of the poor in India say, this works for their kids. The govt says no, we know better what works for your kids. Why is the govt doing this to the poor in India? Free govt schools are easy way for corrupt govt officials to siphon tax money into their own pockets. They don’t actually care about kids’ education.

So, the law is not about “right to education” in reality. It’s just called that.

It’s people opposed to this law who are actually supporting children’s right to education.

James Tooley lived in slums in India for 25 years, to understand how the poor are educating their children. He wrote a moving book about it. Here he is talking about it:

https://youtu.be/XuYFgkYZfvU

P.S : someone has copied the name of his book, so please don’t read the wrong one. Please do read his book though! It is life-changing, eye-opening…

135. DR Says:

Correction : Tooley lived in the slums of India AND Africa, for 25 years in all,to study how the poor educated their kids. The very same situation is seen in Africa too.

136. Boaz Barak Says:

I think we all agree that there are other ways than calculus to get the mathematical maturity needed for college.

However:

1) These other courses (discrete math, probability) are not any easier and they are not a solution for students that are not ready for calculus

2) It is much harder to find K-12 teachers than can teach those courses

Again I don’t think that all students need to take calculus. I think that every student should learn as much as they can, but whether some students finish in Algebra II, precalc or calculus, the important thing is that they actually know the material in these courses. So acceleration is a good choice for some students and a bad choice for others. Generally I would be in favor of letting students try out acceleration and then drop to an easier course if that doesn’t work out. In college (even highly selective ones like Harvard) we do it all the time, and students are fine with it and don’t take it as a personal failure. They treat it more as a choice of how much time they are willing to invest and what’s the best use of their limited time budget.

137. fred Says:

Calculus is really important for ballistic, which should be easy to prioritize given America’s obsession with guns.

138. Timothy Chow Says:

@Jair #133: We’re now getting rather far off topic, but your comment suggests that the following beautiful fact is not as well known as it should be, even though (for example) it is discussed in Chapter 2 of Concrete Mathematics by Graham, Knuth, and Patashnik. Define $$\Delta(f(x)) := f(x+1) – f(x)$$ and define $$x^{\underline{n}} := x(x-1)\cdots (x-n+1)$$. Then $$\Delta(x^{\underline{n}}) = nx^{\underline{n-1}}$$. Similarly, if $$x$$ is a nonnegative integer, then
$$\sum_{0\le k < x} k^{\underline{n}} = {x^{\underline{n+1}}\over n+1}.$$
So for polynomials at least, the formulas aren't any harder, as long as you use the correct basis.

139. arcana Says:

@DR #134:
Well I’m not sure you can use the slum school method in the US. Are the kids at those schools learning calculus or even getting to something equivalent to Algebra 2/Trig? Probably not right? The knowledge and skills from school you need somewhere like India are much different from those you need in advanced western democracies. Nor is the US as corrupt as the places where that method works.

If you read through my comment upthread about possible and effective reform you’ll find that in this country there is a lot you can do to improve public schools in ways that would benefit kids without having to have a knock down drag out fight with the government or the unions or various special interests.

India is an example of, at least recently, massive governmental failure. Modi did some brilliant things in his quest to get elected but once he was there I think we can all see how he doesn’t bring that same energy to helping India as a whole.

In America if a teacher was capable of providing effective education ahead of public schools and especially on the cheap they wouldn’t need to run a secret unacredited private school to help kids. Now they might not get the kind of administrative support that was necessary for something like what Jaime Escalante was doing for Hispanic kids in California way back when but we haven’t quite reached the level of corrupt developing countries in Africa just yet.

Also in America the financial corruption tends to center around the military and also kickbacks from land developers and such rather than education.

140. Jair Says:

Timothy Chow #138, That is indeed a very beautiful formula! I’ve seen that at some point but had forgotten. It still strikes me as a bit more complicated than the continuous case though.

141. Michael Weissman Says:

Boaz #89- Yes, I think we’re in complete agreement. Even at the college level the typical “data science” course seems to be training in data entry and display presentation, perhaps decorated with some training in bureaucratic ethical verbiage. It’s not on the path to real STEM work.
Again, my feeling is that some real stats, especially on the basic logic of causal inference, is of more use to more students than any other post-algebra-1 math. Understanding the difference between irrelevant variables, confounders, causal links (mediators) and maybe even a bit of collider bias can help people navigate through life even if they aren’t great at calculating. Plus a general sqrt(N) sense helps. The problem is that if taught poorly it becomes worse than useless, unlike the standard curriculum. That’s why I’m trying to help people access Ellen’s stuff.

142. Scott Says:

Michael Weissman #142: As I see it, the fundamental problem is that, much like in Russia in 1917, people can see that the existing system of math education is failing … but they are not putting sufficient thought into what actually stands ready to replace it if and when that system is burned to the ground. In the part of their minds that should have a clear perception of the anti-math educrats who are now salivating at the imminent prospect of taking complete control, there’s instead this utopian fantasy of bunnies and rainbows and equity and data science. Meanwhile calculus, kind of like the free market in 1917, is maligned as a source of inequality … but what gets ignored is its enormous virtue: namely, that the clarity of what it entails puts a floor on how bad things could possibly become. With something as vague as “integrated data science,” by contrast, much like with “the dictatorship of proletariat,” there’s no floor on how bad things could become.

143. Timothy Chow Says:

@Jair #140 Even if you think that the formula looks a little more complicated, one advantage is that the proof does not require taking limits. If you want the students to understand the proofs (and I realize that’s a big “if”), then I would argue that the discrete setting is much more favorable. For example, suppose you wanted to prove that $$\int_0^1 x^n\, dx = {1\over n+1}$$. How would you do it? Derive it directly from the definition of the Riemann integral? Prove the fundamental theorem of calculus first? Either approach is IMO far more laborious than proving the discrete summation formula.

144. Michael Weissman Says:

Boaz#125. Your point about calculus keeping students honest about algebra sounds like it was recorded at my dinner table. My wife nominally has a calculus co-requisite for high school students to take her online stats for exactly that reason. They don’t actually need the calculus, but they do need the algebra.

145. Michael Weissman Says:

Scott #142 Yes, that’s why I’m pushing links to Ellen’s stats courses, because she’s put years into developing real alternatives, not some phony phrase-mongering.
E.g. the lectures and some other material for the more mathematical one are here:
http://courses.atlas.illinois.edu/fall2020/STAT200/

146. Jair Says:

@Timothy Chow #143, I agree, if you are trying to be rigorous then the discrete version is far simpler in that case. I don’t feel that introductory calculus needs to be especially rigorous, though.

147. Raoul Ohio Says:

A couple thoughts on teaching calculus level math.

1. Using calculus is highly useful at all levels of STEM. The key idea in calculus is linear approximation:

f(x + h) = f(x) + f'(x)h + e,

where e is the error. Nothing more and nothing less. This puts all the hard stuff into e, which usually we hope is small, and can be ignored.

2. Learning to do proofs and “rigorous math” is essential for more advanced math. Is the standard calculus course the place to introduce it? Or is this tradition perpetrated by those of us who have mastered it think that is the right way to do it? However, limit arguments with deltas and epsilons are inherently slippery. I vote for linear algebra as the best place to start learning to prove things.

3. Do calculus books still present multiple variables without matrices? (I haven’t checked lately). Doing so is insane AF. The key idea in higher dimensions is

f(x + h) = f(x) + df(x)h + e

and everything works exactly as in one dimension. What’s not to like?

4. In arguing point 3 over the last half century, a common rebuttal is that “then you have to teach matrix multiplication” first. When did this get hard?

148. Bill J Says:

@Timothy Chow #143

1) The volume of an n-dimensional simplex is 1/n!, by symmetry (0<=x[P[1]]<=x[P[2]]<=…<=x[P[n]]<=1 for permutations P partition the unit cube).

2) The volume of an n+1-dimensional simplex is given by volume of an n-dimensional simplex (base) times integral of x^n.

149. Timothy Chow Says:

@Jair #146 Yes, I agree that introductory calculus doesn’t have to be rigorous. However, even if all you want to convey is an intuitive idea, the concept that taking differences and taking sums are inverse operations is much easier to grasp than the fundamental theorem of calculus. IMO the discrete case really is conceptually simpler than the continuous case; you don’t have to sweep as much under the rug. Less rug-sweeping is good if we want to convey that math is true not because the teacher says so, but because of its intrinsic logic.

Of course, there are legitimate reasons to choose to teach continuous calculus rather than discrete calculus, but I don’t think that being “easier” is one of those reasons.

150. krishna Says:

The bottom line of the problem with CMF as I see is that the draft is trying to cut off the legs who are running faster than their peers in the name of equity. Any reforms are supported that catered towards weaker students and make math more enjoyable. Stalling the progress of smart student is punishing or insulting their talent. If California think it’s important to uplift the weaker students, so is equally important to nurture the inborn talent and provide an opportunity to excel which the current system accommodate to some extent for talented students. Can we uplift the weaker students through various approaches simultaneously providing a stimulating environment for smarter kids to progress. This is not an uphill task if the politics of equality or pedagogic appeasement is taken out of the equation. I support many liberal policies like health insurance to all, Government intervention on climate change and controlled welfare programs. The liberals are not careful where to apply the liberal ideologies one ideal cannot fit all. My rant is that politics involved in reforming the education system. California being a ultra liberal state is liberalizing the educational policies by first dropping of the SAT requirement for college admission and now touching their hands on k12 education. Can’t we stop the Government from intervening into the education system like separation of church and the state?
Learning Algebra I is not that difficult for the middle school students I believe 90% of the students have the capability to learn and pass calculus.
The number of classes in the school can be reduced only to reading and math as compulsory and other subjects as optional classes to choose from. Reading and problem solving are like two eyes to human existence. “Letters and numbers are the two eyes of a man.” Students with the consent of the parents can explore the options offered in the school to find out their interests. There are too many subjects to learn in school and kids are over whelmed. If I know how to read and solve math problems I can read history, government, biology as options or teach myself. If Interested in further exploration of a subject school should provide an environment for deeper studies. The fundamental building block of well defined literacy is to read, think critically and decent problem solving skills. If we make the kids to write and solve problem at the most 2 hours in school(include classes) and HW not exceeding one hour approximately on an average that will go a long way in realizing their full potential. SAT should be a requirement for college admissions. If we focus on reading, writing and solving problems then that will be a byproduct preparing for SAT. Insane amount of hours are spent on boring stuffs in the school. As Scott said we need to introduce algebra very early in the curriculum instead of teaching them shapes or other useless stuffs. I don’t know whether the children brain are capable of abstractions but that will be a worth pursuit if introduced early in their life. Data science is built on the foundations of math. Without the proper foundation there will be superficial understanding just plugging the values and generating the output without understanding the mechanism of the output. No doubt The taste of math can be acquired through applications first and kindling the curiosity to further explore the subject but it seldom happens. The question can California have a balance of uplifting the weaker students also challenging the smarter students? Time will tell the answer.

151. Pedro Says:

Why do we make such a fuss about calculus when 99.99% ir jobs involve none of it? I have a master’s degree in math and in the job market. Interviewers keep asking me about programming experience and somehow never ask me to compute integrals.

152. Martin Mertens Says:

Raoul Ohio: I completely agree with your summary of calculus but unfortunately I don’t think most people see it that way. They’ll assume f(x+h) = f(x) + f'(x)h + e uses Taylor’s theorem (of course it might, depending on the nature of e). It was only in the past year that I realized you could algebraically rearrange the limit definition of f'(x) into this form.

In fact, I say let’s go further and define every order of derivative from 0 to infinity this way. Instead of saying f”(x) is the derivative of f'(x) say it’s the number satisfying f(x+h) = f(x) + f'(x)h + f”(x)/2 h^2 + e. Et cetera.

Also, introduce asymptotic notation in precalculus so the remainder terms make sense.

153. OhMyGoodness Says:

I know someone that always used to argue that the poor could have improved their economic standard, say by, becoming engineers. I knew that it was senseless to argue but always thought that if you handed out certificates to everyone below the poverty line proclaiming that they were engineers nothing would change. Educators (this is my belief and understandable) think of vast numbers of STEM students and then graduates as a desirable end point. It really isn’t an endpoint but a beginning point for contribution of something to society and receiving compensation in return. In the post-university STEM professions there is still the usual distribution of talents and abilities with many at or below the mean and very few at exceptional level. I don’t see how suddenly increasing enormously the number of STEM students and then STEM graduates would necessarily be of benefit to society. I doubt that the population of students that are transported by new teaching methods to a STEM diploma have the same distribution of abilities as the current population of STEM graduates that suffered through the current teaching methods. Many of the readers here likely primarily taught themselves through calculus or differential equations by going through textbooks by themselves. In general people are compensated by society commensurate with how common their abilities are and those abilities are not identical to what is learned in school.

Maybe a few more STEM graduates could be of benefit to society but there must be reasonable limits. If new advanced teaching methods suddenly resulted in the entire population having STEM degrees then nothing would change because the distribution of abilities would remain the same.

154. OhMyGoodness Says:

BTW-I use the Singapore Math curriculum for supplemental home schooling my daughters. In this curriculum students are doing single variable algebra problems in fourth grade. The curriculum uses block diagrams but I use a variable instead. One of my daughters says she doesn’t like using x so I asked her to pick a letter and she chose n. I never mentioned algebra nor the conventions for integer variables-n it is. 🙂

155. OhMyGoodness Says:

Boaz Barak #125

I have never been a teacher except to my children and continually question if I am presenting things in the best way. My observation based on that very small group is that it is easy to teach mechanical things like arithmetic operations but more difficult to impart how to actually think about real problems that can be broken down into small steps and solved using those simple mechanistic arithmetical tools. It seems to me the true focus for my children should be mathematical reasoning and the mechanical things are just not an issue at all. Some deeper skill is required to actually use math rather than just to parrot memorized mathematical facts. I hope I am fostering this skill in my children-good general thinking skills rather than just parroting. Even in the STEM professions many are good at parroting but not so good at thinking.

When I read through these posts I conclude there are not many engineers here considering the widespread questions about the need for calculus in education at all.

156. Timothy Chow Says:

@Bill J #148: That’s a clever ad hoc argument! But of course, the point is that we don’t expect first-year calculus students to come up with, or even reproduce, such a clever argument, but we do expect them to be able to write down
$$\int_0^1 x^n\,dx = \biggl[{x^{n+1}\over n+1}\biggr]_0^1 = {1\over n+1}$$
with some level of understanding of what they’re doing. I’m just saying that if we want students to understand the above computation (as opposed to just regurgitating it mechanically), then the corresponding discrete calculation is easier to prove or even to justify intuitively.

157. Will Orrick Says:

@Bill J #148, Timothy Chow #156 You can also show pictures like these, which, I think, help visualize the points you are making and simultaneously make both the Riemann sum and the fundamental theorem of calculus intuitively clear (in the case of the functions $$x\mapsto x^n$$, $$x=1,2$$). Using pictures, of course, does not eliminate the need to discuss limits, but the pictures also make it obvious why the error terms go to zero.

158. asdf Says:

Raoul #147, it has seemed to me for a while that some introductory mathematical logic should be part of the “standard cirriculum”. Maybe that is the right place to introduce “rigorous math”.

159. Boaz Barak Says:

Pedro #151: I don’t know about integrals, but derivatives (aka backprop) are quite common these days in coding interviews…

160. OhMyGoodness Says:

My last comment on this post and sorry if I am not using this digital space effectively-

I am still non-plussed by the anti calculus sentiment expressed by some here. I must be stuck in the outdated continuum but the Fundamental Theorem of Calculus was my first exposure to a mathematical result that I found beautiful and unexpected. My thought always was that if you could get someone to the level of appreciating this Theorem then mission accomplished. The realization that the mechanical math that preceded it was a very worthwhile investment and that further gems would likely be uncovered with additional study.

I hope the professional teachers/professors here are not insulted by my simplistic observations based on my limited personal experiences. If anyone has been involved with mathematical education of children around nine years of age and has observations or suggestions then certainly of interest to me.

161. Boaz Barak Says:

OMG #155: I didn’t follow all that you wrote but while deeper understanding is more important than competence with mechanistic procedures, the latter is often very important for achieving the former

162. OhMyGoodness Says:

I read back through this thread and fully understand why you didn’t fully follow what I wrote (poorly written and doesn’t flow with the discussion). I know from previous discussions, as well as his comments in this thread, that my views on secondary education are nearly identical to Dr. Aaronson’s. As I have described here previously, every day of my education from fifth grade to high school graduation (poor rural area) was like a scene from a Munch painting-the school clocks suffered from an obscene amount of time dilation. I taught myself through ordinary differential equations by sitting with textbooks and when I entered a selective engineering school it was like entering the beautiful light from the long dark.

Any of your efforts to assist those that are capable and want to proceed at a faster pace are of benefit not only on a personal basis to those involved but high probability those same students will eventually provide an inordinate benefit to society. I wish you every possible success and I know your goal is noble.

When I looked at home schooling math for my daughters the Singapore Math curriculum was the best that I could identify for a go-by. My daughters respond pretty well thus far but are getting ahead of the usual Singapore age group material and far ahead of their classmates. We are not currently in the US and the local system has flexibility and we will likely make some sort of adjustment after next school year. I will do anything to save them from what I went through and your efforts may save many others from the same experience.

163. OhMyGoodness Says:

JimV #116 and #120

Compressible fluids may have flow discontinuities but not incompressible fluids and incompressible fluids in laminar flow are especially well behaved. There are huge engineering dollars spent each and every year that fundamentally depend on well behaved flow of incompressible fluids.

164. JimV Says:

#163: I don’t think one or both of us is understanding the other (not that there is anything unusual in that) because your reply seems like a non-sequitur to me. Incompressible fluids may be well-behaved, but they are discrete systems (composed of discrete molecules) not continuous systems. I do not recall claiming that discrete systems, or continuous systems (if any exist) for that matter, are not well-behaved.

Engineers like myself also sometimes apply continuous Theory of Elasticity formulae to alloy materials whose grain structure is visible to the naked eye. More likely now though, they used discrete finite-element models. Such models are also used for fluid flow. Any computer calculation (except perhaps for some quantum-computer algorithms–I don’t know) is of course a discrete calculation.

I guarantee that the ability of anyone to sum series in their heads is through the use of tricks like the Geometric Series formula, or by previous memorization of mathematical tables, not by actually summing all the infinite number of terms. Such a mechanical summation is impossible for us, leaving no explanation as to how a natural process could do it.

As a last beat of this dead, off-topic horse, I note that is if there is a minimum increment of motion, ds, of which all motions are composed, and a minimum time increment dt in which that motion can take place, then the ratio c=ds/dt is a natural speed limit. Whereas it is not necessary for a continuous system to have a speed limit. (Yes, I know that no such minimum increments have been detected by our most precise experiments so far, so Zeno might turn out to have been wrong, but that does not mean he did not raise a worthwhile point, in my opinion.)

Calculus of course is very useful because it is the limit of discrete systems as the discrete increment goes to zero, and therefore is a good approximation to many engineering and scientific calculations in which the minimums are very small.

165. OhMyGoodness Says:

JimV #163

The smiley face was a clue. The following is a well known story about von Neumann and an infinite series-

“A famous mathematical puzzle problem involves the following: two trains on the same track begin a mile apart and head towards each other at 60 miles an hour. A fly on one train flies at 120 mph to the other train, and when it lands there, it flies back to the other train, and so on, flying back and forth between the two trains until it gets squashed in the middle. How far does the fly travel?

A student once asked the great mathematician John von Neumann the above problem. von Neumann thought about it a moment and said, “One mile”, the correct answer. “You know, Professor von Neumann, most people don’t realize that the problem’s really easy to figure out: the trains meet in half a minute, and the fly can travel 1 mile in half a minute. They think they have to add up the infinite series to find out how far the fly travels!” von Neumann looked stunned and after a pause said, “But that’s how I did it.””

Well behaved was intended in the mathematical sense-smooth without discontinuities or pathological behavior-differentiable at all points.

Thank you for clarifying and I understand your point. I thought you were questioning the inclusion of calculus in education. It’s funny that the finite difference and finite element models used in various engineering applications also approach real results as the mesh size decreases. At some point the approximation is close enough and the mesh size not decreased further due often to computing constraints or simply that the deltas are no longer significant. The results though continue to improve with respect to real measurement as the models approach the continuum to as fine a scale as measurement allows. As you note their may be some granularity underneath it all. Granular model deltas improve as granularity decreases just as infinite sums deltas decrease as the limit is approached but yeah there may be some granularity under it all (say at the Planck scale) that is not pertinent to current engineering that use granular computational models.

166. OhMyGoodness Says:

A couple times on old roller coasters my neck and back sent my brain an existence proof for the higher order derivatives of motion. 🙂

I couldn’t agree more that Political Correctness should stay out of (especially) STEM education, but in my opinion I view this as the more-or-less inevitable consequence of equity-obsessed progressivism and Social Justice movements. Progressivism has the dogma of “everyone’s the same” blank-slatism woven inextricably into its DNA. Dogmas are fine when they’re true, but in a world where blank-slatism is actually false, that dogma will be under constant threat from things like objective standards and accurate academic testing. Political causes devoted to that dogma will therefore distort logic to protect the dogma from contradictory evidence: eliminating testing or promulgating some nebulous but unsubstantiated alternate explanation like ‘structural racism’, ‘stereotype threat,’ or ‘unconscious bias.’ Well, look out at the world and what do you see? Overwhelmingly strong evidence for the power of IQ and a progressive left obsessed with racist boogeymen and eliminating objective standards. This is not unlike the Soviets’ rejection of economic theory, or (ironically) the Nazi’s rejection of IQ (since it revealed that Jews were actually smarter).

So what say you, Scott? You appear to count yourself among the ranks of the progressive left. Do you actually believe that IQ isn’t genetic and doesn’t vary substantially among ethnic groups (as decades of research indicates it does)? Because it’s very obvious to me that this anti-STEM nonsense is the direct consequence of those false beliefs. If you want to protect the integrity of science education, then protest the root of the threat against it. Free science can’t exist when false beliefs are encoded in political dogma. It’s completely unrealistic to strive for equality of outcomes in g-loaded endeavors. Some people simply can’t cut it, and no amount of political buck-passing will ever change that.

168. OhMyGoodness Says:

One addition to the above. If you are using a numerical engineering model for some dynamic process that includes some partial differential equation that can’t be solved analytically-then the smaller the mesh and the smaller the time step the better the agreement with actual measurement. The model in the limit approaches the continuum with monotonically decreasing error implying that in fact the continuum is an accurate of our macroscopic reality. As for what happens at scales say approaching Planck scales I don’t believe any knowledge exists and so not sure I can even determine meaningful questions about those scales. I am sure many readers here have thought about it much deeper than I have. I know its not turtles all the way down but after that I personally am clueless.

169. STEM Caveman Says:

@Bill 148

the same simplex argument exists in the discrete case, but a bit simpler (look at integer sequences of 0 < x(1) < x(2) < … x(k) <= N), and is close to the combinatorics already done in high school. The practical difficulty with "discrete calculus" is not the proofs but the extra algebraic complexity when lower order terms are not dropped.

170. Scott Says:

Wanda Tinasky #167: People sometimes use phrases like “Darwinian left” or “hereditarian left” for what I regard as just the obviously correct position—namely, that people differ in their interests and abilities, partly for inborn reasons, but do not differ (or at least, should not be treated by policy as differing) in their moral worth. And that it’s generally in society’s best interest to let each person develop to the fullest whatever abilities they have.

What do you call someone who’s terrified about global warming, but thinks the best response would’ve been a surge in nuclear power plants? Who wants to end world hunger and do it using GMO crops? Who wants to smash systems of entrenched privilege in college admissions, and do it using the SAT and other standardized tests? Who thinks the response to covid by the CDC, FDA, and other authorities was a historic disgrace—not because it was “authoritarian,” but on the contrary, because it was too weak, timid, and slow? What do you call someone who’s disgusted to the core by Trump and everything he represents, but also disgusted by the elite virtue signalling that made the rise of a Trump-like backlash figure predictable if not inevitable?

Do you call such a person “liberal,” “progressive,” “centrist,” “classical liberal”? Why not simply “correct”? 😀

171. asdf Says:

Scott #170, I think the word you want is “technocrat”. Like most other approaches, it has both good points and bad.

172. Shtetl-Optimized » Blog Archive » My values, howled into the wind Says:

[…] The Blog of Scott Aaronson If you take nothing else from this blog: quantum computers won't solve hard problems instantly by just trying all solutions in parallel. Also, next pandemic, let's approve the vaccines faster! « An alarming trend in K-12 math education: a guest post and an open letter […]

173. My Values, Howled into the Wind - Pentest-dB Says:

[…] without having clearly set out my values for posterity. So with that in mind: in the comments of my previous post, someone asked me why I identify as a liberal or a progressive, if I passionately support […]

174. Will Orrick Says:

From the other side of the country: it turns out that progressive education reformers may not be the most formidable foes that Algebra I faces. The Republican-controlled Pennsylvania legislature finds itself in court defending against a lawsuit claiming that the state’s school funding formula violates the state constitution. The lawyer representing Senate President Pro Tempore Jake Corman pursued this line of questioning:

“What use would a carpenter have for biology?” asked John Krill of Matthew Splain, superintendent of the Otto-Eldred School District in McKean County and president of the board of directors of the Pennsylvania Association of Rural and Small Schools, one of the plaintiffs. Splain had said his district’s scores on state standardized tests in biology and other subjects were not acceptable.

“What use would someone on the McDonald’s career track have for Algebra 1?” Krill continued.

As lawyers for the plaintiffs objected, asking what the relevance was, Krill said that the trial was about whether Pennsylvania was meeting its constitutional obligation to provide a “thorough and efficient” system of education.

“The question in my mind is, thorough and efficient to what end? To serve the needs of the Commonwealth,” Krill said. “Lest we forget, the Commonwealth has many needs. There’s a need for retail workers, for people who know how to flip a pizza crust.”

(From the Philadelphia Inquirer, December 28, 2021. I became aware of this via a blog post by Peter Greene.)

175. STEM Caveman Says:

@Scott 44,

> why was it Athens that produced Socrates, Plato, and Aristotle, and why was it Budapest that produced von Neumann, Erdös, Teller, Wigner, and Szilard?

There is no reason to presuppose any such outlier is replicable without genetic engineering or reverting lots of “social progress”. I don’t agree that Athenians in general and certainly not Plato and Socrates are specially worthy of emulation. For those who do believe in Greek marquee idols, they benefited from an academic environment built on slavery, patriarchy, aristocracy, race/sex/ethnic uniformity, and for all we know could have been helped by some genetic oddity like a bottleneck or an unusual level of Neanderthal ancestry. The days of low hanging fruit and first mover advantage are not coming back either.

Science education in the Budapest Jewish bourgeoisie, or in 1950’s Bronx Science, was nothing amazing compared to today. Jewish “early adopter” advantages have been diluted by all cognitive pursuits becoming far more competitive at the top (we’re all Jews now, educationally) and by Jews diluting their cultural, genetic, social, economic and ethnic distinction from Gentiles (Jews are Gentiles now, practically). The parade of Jewish world chess champions ain’t coming back because chess got harder. To the extent that Israel does well in science it’s from maintaining some distinctness of culture and system, from having an ethnostate, from support of Jews outside Israel, and the small pond effect that protects local talents from early exposure to excessive competition.

So in essence you are invoking ahistorical stories to propose unlikely and totally hypothetical social interventions that protect the Precious (segregating the smartest nerds into “separate but superior” programs lavishly supported by the state) rather than relying on mundane stuff that is straightforwardly known to work such as early specialization, small pond effects, and the Internet.

176. Shtetl-Optimized » Blog Archive » Scott Aaronson Speculation Grant WINNERS! Says:

[…] pushback from parents and STEM professionals, pushback in which I’m proud that this blog played a small role.) We live in a time when elite universities are racing to eliminate the SAT—and thus, for all […]

177. Pollux Says:

Of course, broadly, math in K-12, in particular, in 6-12, with algebra, plane geometry, trigonometry, solid geometry, calculus, linear algebra, …, are important to crucial for college education in nearly all the STEM material.

That said, it is too easy for students, their parents, and the K-12 educators to be misled: Here are some simple, blunt facts, facts of life for careers, as of now (tough to say what will be the case in 30 years): A bachelor’s degree in math can be good for (a) more time in school, (b) teaching math in K-12 if combined with a teaching certificate, (c) a job in the US Civil Service, or (D) some jobs in US national security. Otherwise, there are essentially no jobs and is essentially no job market for bachelor’s degrees in math. Same for a master’s in math. For a Ph.D. in math, might get a university job. That’s about the situation for math majors.

For more, the hot job market is for computer programmers, and there a computer science degree can help. A math major might get hired in this hot job market if they can show that they have some good skills with some recent topics in practical computing. To heck with the inverse function theorem; a JavaScript framework stands to be much more valuable; same for C, C++, C#, etc.

Generally for a career, a young person should concentrate their attention on business, in particular, owning their own business. For that, a trade school or junior college education might be helpful. A college degree as a math major might be helpful a few times in their career that they really got started via trade school, family members, or friends in business, etc.

E.g., in the suburbs of essentially any US city, go to the neighborhoods of single family detached houses with 2+ car garages, late model cars, stay at home mothers, and private K-12 schools and see what the careers are. You will find essentially no college math majors and essentially no use of math at all. You will find essentially no one who knows Fubini’s theorem and will seriously embarrass yourself looking.

Maybe someday some math might be a crucial advantage for a business career (I am hoping that, have some original math I derived and also am using a neglected advanced result in Rudin’s ‘Real and Complex Analysis’), but my use of math for business is extremely speculative.

Broadly normal people in good jobs with good families don’t like math, regard it as challenging, obscure, and irrelevant. That’s just what they think, and they have a lot of good supporting evidence. Bluntly, for US business and good careers, college math degrees are a waste of a likely expensive education. For anyone that wants to do well at family formation, a career in math is very likely financially irresponsible.

So, be very careful in urging young people to study much in math beyond, say, calculus and linear algebra or to get college degrees in math.

178. ACARA is More Dangerous than Boaler – BAD MATHEMATICS Says:

[…] Slow Boaler, and seemingly without a mathematician within cooee, the draft was, of course, very bad and consequently it was very hammered. The draft was rewritten and, just recently, rewritten again. […]

New Zealand manages to cover the entire American high school curriculum within New Zealand’s primary school:

By Level 8/Form 2 (New Zealand’s equivalent of Grade 8 in the United States), students are expected to recognise and use models of logic such as Boolean logic, use and manipulate vectors to prove theorems in Euclidean geometry, recognise and use De Moivre’s Theorem in manipulating trigonometric expressions of complex numbers, test to see whether a given series of real numbers converges or diverges, find the tangent and normal lines of various conic sections in the Euclidean plane, and write a paper on the various numerical methods of solving equations and finding roots of functions (such as the Bisection method).

In contrast, by Grade 8 many American students still have not yet learned to solve linear equations in one variable.

Never mind about the previous comment, it seems that the 8 levels in the New Zealand curriculum divide the entire curriculum (primary + secondary) rather than represent one year.

181. Shtetl-Optimized » Blog Archive » An update on the campaign to defend serious math education in California Says:

[…] you might remember, last December I hosted a guest post about the “California Mathematics Framework” (CMF), which was set to cause radical […]

182. An update on the campaign to defend serious math education in California by Tomte - HackTech News Says:

[…] you might remember, last December I hosted a guest post about the “California Mathematics Framework” (CMF), which was set to cause radical changes to […]

183. Shtetl-Optimized » Blog Archive » An update on the campaign to defend serious math education in California – OasisNews Says:

[…] you might remember, last December I hosted a guest post about the “California Mathematics Framework” (CMF), which was set to cause radical changes to […]

You can use rich HTML in comments! You can also use basic TeX, by enclosing it within  for displayed equations or  for inline equations.

Comment Policies:

1. All comments are placed in moderation and reviewed prior to appearing.
2. You'll also be sent a verification email to the email address you provided.
YOU MUST CLICK THE LINK IN YOUR VERIFICATION EMAIL BEFORE YOUR COMMENT CAN APPEAR. WHY IS THIS BOLD, UNDERLINED, ALL-CAPS, AND IN RED? BECAUSE PEOPLE ARE STILL FORGETTING TO DO IT.
3. Comments can be left in moderation for any reason, but in particular, for ad-hominem attacks, hatred of groups of people, snide and patronizing tone, trollishness, disingenuousness, or presumptuousness (e.g., linking to a long paper or article and challenging me to respond to it).
4. Even when no individual comment violates policy, when there are dozens of comments from a single commenter hammering home the same few themes, and the commenter shows no interest in changing their views or learning from anyone else, the commenter will receive a warning followed by a 3-month ban.
5. Whenever I'm in doubt, I'll forward comments to Shtetl-Optimized Committee of Guardians, and respect SOCG's judgments on whether those comments should appear.
6. I sometimes accidentally miss perfectly reasonable comments in the moderation queue, or they get caught in the spam filter. If you feel this may have been the case with your comment, shoot me an email.