Improving instruction one school (your local one) at a time is a great idea.
Early exposure to programming in connection with math ("write a C code for the Euclidean algorithm") is a great idea - too bad these days the task can be outsourced to an AI.
As about the rest, let me sketch an alternative picture.
1. Any revolutionary reform in education ends in a disaster (Examples: "New Math" of the 60-ies, "Fuzzy Math" in the 90-ies). The reason is: education = passing culture from one generation to another, while any reform makes teachers unable to teach and parents unable to help. Only gradual evolution of curriculum can be successful.
2. Totalitarian countries (where individuals exists for the benefits of the society, and not the other way around) have stronger math curricula (the society needs engineers) than democratic ones - where it is easy to make an argument that any particular item should not be taught "because very few people use it in real life". The proposal to replace "pre-calculus" (which is the only common-sense math in the US curriculum) with something "more useful" (such as probability and linear algebra) is, I am afraid, of this nature.
3. Those who think that math of early grades is "easy" and there is not much to learn there are gravely mistaken - read the book "Arithmetic for Parents" by Ron Aharoni of Technion.
4. Problem in the US math education begin in elementary school (by grade 2 former English majors successfully pass their hatred for the subject to their students). Even the excellent (grade 1-6) math curriculum from Singapore turns in the US soil in the absurd of solving 2-step arithmetic problems by labeling "bar diagrams" following an 8-step plan devised by devilish US educators (if you don't believe this, check "Signapore math" in youtube).
5. The mere discussion of how much should be memorized in math indicates the problem: in Russia, kids who prefer math to history say that they like math because "if you understand it, you don't need to memorize anything". And they are right!
6. "Calculus" = "Mathematical Analysis without proofs". It better not be taught at all. Especially in high school, because many will have to spend even more time with it when they retake it in college.
7. The main grade-school level math subject which, if taught properly, teaches theoretical understanding and creative thinking is geometry. As examples for the readers of this post, enjoy the following three math challenges: Using straightedge and compass, construct a triangle given (a) its three medians, (b) its three altitudes, (c) its altitude, median, and the angle bisector drawn from the same vertex.
8. If any curriculum change can be beneficial in the US landscape, it should be replacing the current "consecutive" curriculum structure (algebra-1 - geometry - trigonometry - algebra-2 - precalculus - calculus) with a "concurrent" one, where geometry is taught in parallel with everything else in a span of several years.
Finally, let me comment on math per se: the Pythagorean Theorem. Its proof illustrated in this post (apparently due to Martin Gardner) is indeed the most popular one, but just as most of the other 300+ proofs it is misleading about the nature of the theorem. It is presented as a clever way of cutting-and-tiling geometric shapes, while the theorem is actually about similarity. In the book 5 of his "Elements", Euclid revisits the Pythagorean Theorem (first proved in book 1) and generalizes it this way:
Suppose three figures of the same but arbitrary shape (not necessarily squares) are bult on the sides of a right triangle. Then the areas of the two figures built on the legs add up to the area of the figure built on the hypothenuse.
Since the shapes are arbitrary, no cut-and-tile argument would help. On the other hand, this statement for any one shape (e.g. squares) will imply it for any other shape - for proportionality reasons.
So, why take squares? The altitude dropped from the vertex of the right triangle to its hypotenuse DIVIDES the triangle into two, of the same shape as (i.e. similar to) the whole triangle and built on its legs (as their hypothenuses). That's it: the area of the whole triangle is the sum of the areas of these two parts.
If this exposition sounds too terse, check my article https://sumizdat.com/homepage/papers/eu.pdf - it also contains a Kindergarten-level proof of the standard Pythagorean Theorem which turns out to be about the three similar houses built by the Three Little Pigs.
There's a lot of depth here, and while I disagree with parts I so much admire (7), (8) and especially your comments on the Pythagorean theorem that I will start by responding only to those
I had no idea that Euclid had generalized the Pythagorean theorem, and my first reaction was "aha, basic Pythagoras plus Cavalieri's principle". But reading your article -- you clearly attended a much more sophisticated Kindergarten than I did :-) -- I see that in a way you're reducing everything to symmetries and the squaring of sides inherent in the area of squares. Imo that's a brilliantly useful simplification.
That brings me to (8) where you propose teaching geometry concurrently. I consider the separation the weakest part of my proposal. I was aware of the close relation between shapes and numbers, with shapes in some ways preceding numbers, or perhaps it is symmetries that underly both, but I'm not very comfortable with it so took a types-of-numbers approach that I adapted from Feynman. I'm happy to defer to a world-class topologist like you who tells me this can and should be integrated and has ideas on how.
Re (7), I like any math teaching that encourages intuition and creativity. I'm not qualified to say whether this particular activity should be included. Again, my hope is that national tests for core competencies at various levels (which should include mathematicians like you in the blue ribbon commissions that decide these) will encourage experimentation, which emulation and competition will quickly whittle down to 2-4 approaches that seem to work best.
I wholly agree with you on (3)-(5). On (1), you're right about danger but I am hoping to avoid it by not messing with elementary and middle school math pedagogy other than the neater separation of topics (even if geometry is folded in as you suggest).
Where we differ most is on my dumping pre-calculus in favor of a direct jump to calculus from algebra. That covers (6), (2), and part of (1). Instead of arguing my case, I'll send you my short book Calculus for the Curious and invite your comments.
One point I will make here is that I'm not nearly as averse to theory-for-the-sake-of-wonder-and-awe as my comments on usefulness might have lead you to think. My short 140-page book, which presumes no prior knowledge beyond basic algebra and geometry, concludes with proofs of all three Kepler's Laws, which is well beyond Calculus BC. And the Python coding exercises I have developed to accompany them aren't endangered by AI as the code already works; I just ask students to explain why it works and play with the parameters. I have offered this to various schools for free but evidently the price in perceived disruption was too high.
Thanks - I tried to take a look at your book, but there is very little in the sample pages at amazon.com. So, I'll look forward to seeing your book (especially Kepler's problem - that was the subject of my high school astronomy term paper, https://sumizdat.com/homepage/kepler_091615.pdf )
Interesting ideas. I am not sure I agree (I definitely disagree about reducing memorization, if anything we need to increase it) but I do agree that introducing advanced concepts sooner is a good idea, especially to give more perspective on how math is used in real life (its everywhere) and also expanding more on the history of it and its concepts, which help to explain why they exist. Also the integration with computer programming is maybe good, I see some benefits, especially the idea of learning by teaching.
Thanks for commenting. Could you give me 1 or 2 examples of the memorization you think we need to retain and have more of? For an example of what I think we could dispense with, I'll submit pi=3.14. But seeing that pi>3 and pi<4 is important, which we can do by inscribing a hexagon inside the circle and a square outside it.
It's easier for people to understand what pi=3.14 means than 3<pi<4. If they KNOW the value of pi, it serves as a touchstone for the more complex idea of the inequality.
If people know their times tables and area formula then it makes it easier for them to learn the ideas of calculus. If they don't have them down, then there is no recognition, no AHA! when you draw little rectangles with height of f(x) and widths dx or breaking a circle into triangles of height r and base rd\theta.
My main aim is to seriously raise standards in middle and high school math. How one gets there is less important and might differ by student or teacher; they're welcome to reject my "memorize less, probe why more" advice as you, Stephen Lightfoot, and many others do.
However, your pi comment does not support your case. Pi does not equal 3.14. No ratio of integers can fully capture it; no finite algebraic equation defines it. Your statement needs correction to 3.14<pi<3.145. That cannot be easier to understand than 3<pi<4, as the latter has fewer digits and simpler geometric backing. Perhaps you meant that pi=3.14 gives better answers than pi=3 or pi=4, since it is accurate to 1 part in 2000. Yes it does, but no modern engineering would work at that limited degree of precision. Since even pi=3.14159 can fail, I personally recommend less attention to memorizing the digits and more attention to the various ways mathematicians in various cultures learned to devise sophisticated approximations. In their calculus-plus-coding classes I would have students run various Taylor series approximations and compare them.
I think one should memorize the 12x12 times tables cold, and know all the constants to several decimal places and all the basic formulae up to high school (to this day I can say in an instant sin=opp over hyp). Some years ago I ran across an article by Barbara Oakley titled “How I Rewired My Brain to Become Fluent in Math” (Nautilus, published September 11, 2014) and agree with its basic premise, that its only by memorizing and having fast access to the basics that you can properly appreciate more advanced concepts. You may wish to read it. I think she ended with a PHD in EE - only started serious math in her 30s after being a math-phobe prior. Personally I have a Mech Eng degree from McGill and wouldn't have gotten where I have without a lot of rote memorization in the early days.
Well, we disagree on that. But you might be right, and almost surely your view commands more support than mine among people genuinely committed to higher math standards.
I strongly agree with you. I increasingly believe that the math that is most used by people in their lives should be taught rote. It serves as a foundation for future study! If the basics aren't mastered then they have to be rehashed every time you try to build on them. This becomes untenable pretty quickly.
I dont how it could be otherwise. Its no different from language, if you don't have a vocabulary memorized you cant make sentences. It all starts with a minimum of rote memorization.
We're not disagreeing over "a minus of rote memorization" That includes definitions like "8 is one more than 7" and "sin=opp over hyp". They should know the Pythagorean theorem and realize that "sin^2+cos^2=1" is a restatement. I would not insist they memorize "tan^2+1=sec^2" as that's needless clutter, better that they can confirm the reduction to the Pythagorean theorem.
I agree. Despite my phrasing, I didn't mean to deny likely genetic influences in math, any more than I deny likely genetic influences in sports proficiency. I just don't want them used as an excuse to justify low standards and crap teaching. Math is a sport of the mind. Nearly all humans can learn to play it, even though only a few are capable of playing it really well.
Agreed, education matters for all, including the less able. Indeed, raising the level of a less-able child might be hard, but can still lead to a significant improvement in their life trajectory and thence societal impact.
Early learning is key. Bad habits begin early. Teach counting by twos, tens, twos, threes in kindertarten and get beyond counting by ones a quickly as possible in order to conceptualize number series. But, good luck in the USA with any new curriculum because you have to convince curriculum specialists in 10K districts in 51+ state/territory departments of education, and train the college professors to integrate math into teacher training. America is the land of 10K different curriculum decisions. Begin with early childhood for all aspects of curriculum and require mastery of basic skills by the end of 4th grade. Remember also that Germany and some other countries begin academics at age 7 and neuro-developmental school readiness (including physical aspects of vision and auditory aspects) is important for a solid foundation.
England has a national math curriculum that imo is quite good. For students heading to university, calculus gets introduced about 10th grade and is well integrated with mechanics (Newtonian physics). So I'm not against a centralized approach when standards are high. And the US, despite generally lower standards in math, does have pockets of excellence in some prep schools and richer suburbs. But the US has centralized a lot since the Dept of Education was formed, and at the state and big-city levels too, partly thru pressures from teachers' unions and civil rights groups. Some centralization also results from US-wide pressures to prepare for SAT and ACT tests.
But that's not the worst part. The worst part is that standards have been dumbed down a lot in the name of generating more equal results. Now in the name of DEI or whatever it's renaming itself as. Before in the name of "No Child Left Behind", which in practice often amounts to "No Child Run Ahead". Read about NY Regent's Exam in Wikipedia to see how a once banner leader in higher school standards has sunk. The SAT dumbed down to come with the once-lower ACT. Race to the bottom...
“And the US, despite generally lower standards in math, does have pockets of excellence …”.
Math standards for Asian American students are fully up to the best international standards and white Americans also do comparably well. The “generally lower standards” is because Hispanic Americans are slightly behind international standards and black Americans are a long way behind. The difference is not because of school systems or funding differentials (for example the US spends more per black child than per white child). And indeed different school systems and different curricula matter much less than commonly supposed (for example twin studies show that “shared environment” is only a minor factor in outcomes).
While I am reluctant to judge the causes, the longer-term persistence of significant group disparities in STEM performance does need to be more clearly acknowledged. Several more HxSTEM articles on this topic are forthcoming. The biggest policy implication is that group parity in higher STEM employment is incompatible with merit-based hiring.
The emphasis on calculus is misplaced, I think, especially for early high school. Combinatorics - ways of counting - is more useful to more people, but gets shoved aside because of overbearing parents' insistence on a calculus track for their precious children.
"Every high school graduate in the US [who plans pursuing college degree in the US] should be familiar with core elements of calculus, linear algebra, and probability theory."
I definitely agree with you that those core elements are vital in our intellectual development, allowing all university students comprehend limits, multiple dimensions, and stochastic processes (as well as various other aspects of 'Nature' that are not necessarily described by mathematical formulas).
However for most of the population, it is sufficient just achieving the initial 4 parts of the total 7 stages in your suggested curriculum.
I'm fine with your modification provided that anyone who doesn't take the college path switches to vocational study. This is basically the German system: Gymnasium steers toward university, Realschule and Hauptshule provide vocational schools with and without apprenticeship, and Gesamtschule provides mixes for those who aren't sure. That way everyone learns something useful to be proud of and learns it sooner. The main criticism is that it tends to openly congeal divisions by socioeconomic class (kids tend to take their parents' track), which most Americans prefer to hide.
First I think that we need to question how important the topic of mathematics is in the first place.
I am not convinced that mathematics beyond basic arithmetic is very useful for preparing youths for adulthood. Very few adults use the vast majority of the math that they learned in school and a high percentage hate it. I think a focus on personal finance, investing, quantitative reasoning, and entrepreneurship would be much more useful than the more advanced mathematics that you list in his article.
I believe that K-12 should be much more focused on character-building, life skills, job skills, and physical fitness and less focused on academic subjects that are primarily designed to get a student into college. High school grads should be competent adults with real job prospects regardless of whether they go to college. Advanced mathematics does not help in that regard and takes up valuable class time that could used for other more relevant topics.
Good investing requires attention to marginal returns, compounding, risk, and diversification. Basic calculus makes it easier to reason quantitatively about marginal returns and compounding. Basic probability theory is synonymous with quantitative reasoning about risk. Diversification is best understood using basic linear algebra. So what you mentioned is exactly the sort of topics I want the revised math curriculum to focus on.
You seem to be proposing MUCH more emphasis on mathematics beyond basic arithmetic than I am.
I think that it is easily possible to teach investing without a prior knowledge of advanced mathematics. Marginal returns, compounding, risk, and diversification only require basic arithmetic.
Just look at any typical book on investing. No geometry, trig, algebra, or calculus is necessary.
Yes, I am. I'd like to better understand your point of view. If math beyond basic arithmetic doesn't contribute to the life skills and job skills you'd like 13 school years to impart, can you tell me pls what level of knowledge you consider adequate for high school graduates in other academic subjects? And what are the core job skillsyou expect high school grads to have?
I can’t realistically write out an entire K-12 education curriculum in the comments. I already mentioned a few topics that I believe are under-taught to give you a basic idea. I also wrote this article on vocational training:
My point is that mathematics beyond basic arithmetic is rarely used by more than a tiny minority of adults in their jobs or elsewhere, so it does not make sense to teach even more advanced mathematics that even fewer adults will ever use.
For evidence as to why I believe that there is already too much mathematics in the K-12 curriculum, I would recommend:
Reading this, I was wondering when you’d get to the teachers. Your list of big barriers at the beginning didn’t include them, and I was wondering from the beginning why not. The whole time I was reading the essay, the old saying “if wishes were horses…” kept rolling through my mind. Do I think deficiencies in math instruction exist? Heck yeah. Do I see things changing? Not until there’s a “Sputnik” moment (e.g., a massive and visible cyberattack) that exposes our weakness in this area. I just hope it is not by then too late. By the way, have you run this by the new SecEd? It would be interesting to see her response.
Between some great YouTube channels on math, online courses through Coursera and the like, and AI agents like ChatGPT, Grok, or Gemini, it is far easier than ever before to get access to engaging, high-quality math instruction. Most K-12 math teachers need to shift from explainer and tester to concierge and mentor - pointing students to what they should work on and helping them find good platforms for learning. As for SecEd, I have no connections and imagine that it is hard to consider new approaches given the current focus on cutting back. Some states might be receptive but I have no connections there either.
Improving instruction one school (your local one) at a time is a great idea.
Early exposure to programming in connection with math ("write a C code for the Euclidean algorithm") is a great idea - too bad these days the task can be outsourced to an AI.
As about the rest, let me sketch an alternative picture.
1. Any revolutionary reform in education ends in a disaster (Examples: "New Math" of the 60-ies, "Fuzzy Math" in the 90-ies). The reason is: education = passing culture from one generation to another, while any reform makes teachers unable to teach and parents unable to help. Only gradual evolution of curriculum can be successful.
2. Totalitarian countries (where individuals exists for the benefits of the society, and not the other way around) have stronger math curricula (the society needs engineers) than democratic ones - where it is easy to make an argument that any particular item should not be taught "because very few people use it in real life". The proposal to replace "pre-calculus" (which is the only common-sense math in the US curriculum) with something "more useful" (such as probability and linear algebra) is, I am afraid, of this nature.
3. Those who think that math of early grades is "easy" and there is not much to learn there are gravely mistaken - read the book "Arithmetic for Parents" by Ron Aharoni of Technion.
4. Problem in the US math education begin in elementary school (by grade 2 former English majors successfully pass their hatred for the subject to their students). Even the excellent (grade 1-6) math curriculum from Singapore turns in the US soil in the absurd of solving 2-step arithmetic problems by labeling "bar diagrams" following an 8-step plan devised by devilish US educators (if you don't believe this, check "Signapore math" in youtube).
5. The mere discussion of how much should be memorized in math indicates the problem: in Russia, kids who prefer math to history say that they like math because "if you understand it, you don't need to memorize anything". And they are right!
6. "Calculus" = "Mathematical Analysis without proofs". It better not be taught at all. Especially in high school, because many will have to spend even more time with it when they retake it in college.
7. The main grade-school level math subject which, if taught properly, teaches theoretical understanding and creative thinking is geometry. As examples for the readers of this post, enjoy the following three math challenges: Using straightedge and compass, construct a triangle given (a) its three medians, (b) its three altitudes, (c) its altitude, median, and the angle bisector drawn from the same vertex.
8. If any curriculum change can be beneficial in the US landscape, it should be replacing the current "consecutive" curriculum structure (algebra-1 - geometry - trigonometry - algebra-2 - precalculus - calculus) with a "concurrent" one, where geometry is taught in parallel with everything else in a span of several years.
Finally, let me comment on math per se: the Pythagorean Theorem. Its proof illustrated in this post (apparently due to Martin Gardner) is indeed the most popular one, but just as most of the other 300+ proofs it is misleading about the nature of the theorem. It is presented as a clever way of cutting-and-tiling geometric shapes, while the theorem is actually about similarity. In the book 5 of his "Elements", Euclid revisits the Pythagorean Theorem (first proved in book 1) and generalizes it this way:
Suppose three figures of the same but arbitrary shape (not necessarily squares) are bult on the sides of a right triangle. Then the areas of the two figures built on the legs add up to the area of the figure built on the hypothenuse.
Since the shapes are arbitrary, no cut-and-tile argument would help. On the other hand, this statement for any one shape (e.g. squares) will imply it for any other shape - for proportionality reasons.
So, why take squares? The altitude dropped from the vertex of the right triangle to its hypotenuse DIVIDES the triangle into two, of the same shape as (i.e. similar to) the whole triangle and built on its legs (as their hypothenuses). That's it: the area of the whole triangle is the sum of the areas of these two parts.
If this exposition sounds too terse, check my article https://sumizdat.com/homepage/papers/eu.pdf - it also contains a Kindergarten-level proof of the standard Pythagorean Theorem which turns out to be about the three similar houses built by the Three Little Pigs.
There's a lot of depth here, and while I disagree with parts I so much admire (7), (8) and especially your comments on the Pythagorean theorem that I will start by responding only to those
I had no idea that Euclid had generalized the Pythagorean theorem, and my first reaction was "aha, basic Pythagoras plus Cavalieri's principle". But reading your article -- you clearly attended a much more sophisticated Kindergarten than I did :-) -- I see that in a way you're reducing everything to symmetries and the squaring of sides inherent in the area of squares. Imo that's a brilliantly useful simplification.
That brings me to (8) where you propose teaching geometry concurrently. I consider the separation the weakest part of my proposal. I was aware of the close relation between shapes and numbers, with shapes in some ways preceding numbers, or perhaps it is symmetries that underly both, but I'm not very comfortable with it so took a types-of-numbers approach that I adapted from Feynman. I'm happy to defer to a world-class topologist like you who tells me this can and should be integrated and has ideas on how.
Re (7), I like any math teaching that encourages intuition and creativity. I'm not qualified to say whether this particular activity should be included. Again, my hope is that national tests for core competencies at various levels (which should include mathematicians like you in the blue ribbon commissions that decide these) will encourage experimentation, which emulation and competition will quickly whittle down to 2-4 approaches that seem to work best.
Thanks for taking the time to weigh in.
I wholly agree with you on (3)-(5). On (1), you're right about danger but I am hoping to avoid it by not messing with elementary and middle school math pedagogy other than the neater separation of topics (even if geometry is folded in as you suggest).
Where we differ most is on my dumping pre-calculus in favor of a direct jump to calculus from algebra. That covers (6), (2), and part of (1). Instead of arguing my case, I'll send you my short book Calculus for the Curious and invite your comments.
One point I will make here is that I'm not nearly as averse to theory-for-the-sake-of-wonder-and-awe as my comments on usefulness might have lead you to think. My short 140-page book, which presumes no prior knowledge beyond basic algebra and geometry, concludes with proofs of all three Kepler's Laws, which is well beyond Calculus BC. And the Python coding exercises I have developed to accompany them aren't endangered by AI as the code already works; I just ask students to explain why it works and play with the parameters. I have offered this to various schools for free but evidently the price in perceived disruption was too high.
Thanks - I tried to take a look at your book, but there is very little in the sample pages at amazon.com. So, I'll look forward to seeing your book (especially Kepler's problem - that was the subject of my high school astronomy term paper, https://sumizdat.com/homepage/kepler_091615.pdf )
Interesting ideas. I am not sure I agree (I definitely disagree about reducing memorization, if anything we need to increase it) but I do agree that introducing advanced concepts sooner is a good idea, especially to give more perspective on how math is used in real life (its everywhere) and also expanding more on the history of it and its concepts, which help to explain why they exist. Also the integration with computer programming is maybe good, I see some benefits, especially the idea of learning by teaching.
Thanks for commenting. Could you give me 1 or 2 examples of the memorization you think we need to retain and have more of? For an example of what I think we could dispense with, I'll submit pi=3.14. But seeing that pi>3 and pi<4 is important, which we can do by inscribing a hexagon inside the circle and a square outside it.
It's easier for people to understand what pi=3.14 means than 3<pi<4. If they KNOW the value of pi, it serves as a touchstone for the more complex idea of the inequality.
If people know their times tables and area formula then it makes it easier for them to learn the ideas of calculus. If they don't have them down, then there is no recognition, no AHA! when you draw little rectangles with height of f(x) and widths dx or breaking a circle into triangles of height r and base rd\theta.
My main aim is to seriously raise standards in middle and high school math. How one gets there is less important and might differ by student or teacher; they're welcome to reject my "memorize less, probe why more" advice as you, Stephen Lightfoot, and many others do.
However, your pi comment does not support your case. Pi does not equal 3.14. No ratio of integers can fully capture it; no finite algebraic equation defines it. Your statement needs correction to 3.14<pi<3.145. That cannot be easier to understand than 3<pi<4, as the latter has fewer digits and simpler geometric backing. Perhaps you meant that pi=3.14 gives better answers than pi=3 or pi=4, since it is accurate to 1 part in 2000. Yes it does, but no modern engineering would work at that limited degree of precision. Since even pi=3.14159 can fail, I personally recommend less attention to memorizing the digits and more attention to the various ways mathematicians in various cultures learned to devise sophisticated approximations. In their calculus-plus-coding classes I would have students run various Taylor series approximations and compare them.
I think one should memorize the 12x12 times tables cold, and know all the constants to several decimal places and all the basic formulae up to high school (to this day I can say in an instant sin=opp over hyp). Some years ago I ran across an article by Barbara Oakley titled “How I Rewired My Brain to Become Fluent in Math” (Nautilus, published September 11, 2014) and agree with its basic premise, that its only by memorizing and having fast access to the basics that you can properly appreciate more advanced concepts. You may wish to read it. I think she ended with a PHD in EE - only started serious math in her 30s after being a math-phobe prior. Personally I have a Mech Eng degree from McGill and wouldn't have gotten where I have without a lot of rote memorization in the early days.
Well, we disagree on that. But you might be right, and almost surely your view commands more support than mine among people genuinely committed to higher math standards.
I strongly agree with you. I increasingly believe that the math that is most used by people in their lives should be taught rote. It serves as a foundation for future study! If the basics aren't mastered then they have to be rehashed every time you try to build on them. This becomes untenable pretty quickly.
I dont how it could be otherwise. Its no different from language, if you don't have a vocabulary memorized you cant make sentences. It all starts with a minimum of rote memorization.
We're not disagreeing over "a minus of rote memorization" That includes definitions like "8 is one more than 7" and "sin=opp over hyp". They should know the Pythagorean theorem and realize that "sin^2+cos^2=1" is a restatement. I would not insist they memorize "tan^2+1=sec^2" as that's needless clutter, better that they can confirm the reduction to the Pythagorean theorem.
“Rather, it suggests cultural and class influences that act like genetic factors even if they are wholly environmental.”
At some point America will need to stop studiously ignoring the possibility of genetic factors themselves.
I agree. Despite my phrasing, I didn't mean to deny likely genetic influences in math, any more than I deny likely genetic influences in sports proficiency. I just don't want them used as an excuse to justify low standards and crap teaching. Math is a sport of the mind. Nearly all humans can learn to play it, even though only a few are capable of playing it really well.
Agreed, education matters for all, including the less able. Indeed, raising the level of a less-able child might be hard, but can still lead to a significant improvement in their life trajectory and thence societal impact.
Open an academy and I will send my children there.
Early learning is key. Bad habits begin early. Teach counting by twos, tens, twos, threes in kindertarten and get beyond counting by ones a quickly as possible in order to conceptualize number series. But, good luck in the USA with any new curriculum because you have to convince curriculum specialists in 10K districts in 51+ state/territory departments of education, and train the college professors to integrate math into teacher training. America is the land of 10K different curriculum decisions. Begin with early childhood for all aspects of curriculum and require mastery of basic skills by the end of 4th grade. Remember also that Germany and some other countries begin academics at age 7 and neuro-developmental school readiness (including physical aspects of vision and auditory aspects) is important for a solid foundation.
England has a national math curriculum that imo is quite good. For students heading to university, calculus gets introduced about 10th grade and is well integrated with mechanics (Newtonian physics). So I'm not against a centralized approach when standards are high. And the US, despite generally lower standards in math, does have pockets of excellence in some prep schools and richer suburbs. But the US has centralized a lot since the Dept of Education was formed, and at the state and big-city levels too, partly thru pressures from teachers' unions and civil rights groups. Some centralization also results from US-wide pressures to prepare for SAT and ACT tests.
But that's not the worst part. The worst part is that standards have been dumbed down a lot in the name of generating more equal results. Now in the name of DEI or whatever it's renaming itself as. Before in the name of "No Child Left Behind", which in practice often amounts to "No Child Run Ahead". Read about NY Regent's Exam in Wikipedia to see how a once banner leader in higher school standards has sunk. The SAT dumbed down to come with the once-lower ACT. Race to the bottom...
“And the US, despite generally lower standards in math, does have pockets of excellence …”.
Math standards for Asian American students are fully up to the best international standards and white Americans also do comparably well. The “generally lower standards” is because Hispanic Americans are slightly behind international standards and black Americans are a long way behind. The difference is not because of school systems or funding differentials (for example the US spends more per black child than per white child). And indeed different school systems and different curricula matter much less than commonly supposed (for example twin studies show that “shared environment” is only a minor factor in outcomes).
While I am reluctant to judge the causes, the longer-term persistence of significant group disparities in STEM performance does need to be more clearly acknowledged. Several more HxSTEM articles on this topic are forthcoming. The biggest policy implication is that group parity in higher STEM employment is incompatible with merit-based hiring.
The emphasis on calculus is misplaced, I think, especially for early high school. Combinatorics - ways of counting - is more useful to more people, but gets shoved aside because of overbearing parents' insistence on a calculus track for their precious children.
I propose slight modification in your bold claim:
"Every high school graduate in the US [who plans pursuing college degree in the US] should be familiar with core elements of calculus, linear algebra, and probability theory."
I definitely agree with you that those core elements are vital in our intellectual development, allowing all university students comprehend limits, multiple dimensions, and stochastic processes (as well as various other aspects of 'Nature' that are not necessarily described by mathematical formulas).
However for most of the population, it is sufficient just achieving the initial 4 parts of the total 7 stages in your suggested curriculum.
I'm fine with your modification provided that anyone who doesn't take the college path switches to vocational study. This is basically the German system: Gymnasium steers toward university, Realschule and Hauptshule provide vocational schools with and without apprenticeship, and Gesamtschule provides mixes for those who aren't sure. That way everyone learns something useful to be proud of and learns it sooner. The main criticism is that it tends to openly congeal divisions by socioeconomic class (kids tend to take their parents' track), which most Americans prefer to hide.
I'm somewhat familiar with the German educational system; we could surely use it as basis while innovating the American educational system.
Vocational schools can be quite beneficial; the degrees which they provide can be much more useful than many degrees of humanity departments.
First I think that we need to question how important the topic of mathematics is in the first place.
I am not convinced that mathematics beyond basic arithmetic is very useful for preparing youths for adulthood. Very few adults use the vast majority of the math that they learned in school and a high percentage hate it. I think a focus on personal finance, investing, quantitative reasoning, and entrepreneurship would be much more useful than the more advanced mathematics that you list in his article.
I believe that K-12 should be much more focused on character-building, life skills, job skills, and physical fitness and less focused on academic subjects that are primarily designed to get a student into college. High school grads should be competent adults with real job prospects regardless of whether they go to college. Advanced mathematics does not help in that regard and takes up valuable class time that could used for other more relevant topics.
Good investing requires attention to marginal returns, compounding, risk, and diversification. Basic calculus makes it easier to reason quantitatively about marginal returns and compounding. Basic probability theory is synonymous with quantitative reasoning about risk. Diversification is best understood using basic linear algebra. So what you mentioned is exactly the sort of topics I want the revised math curriculum to focus on.
You seem to be proposing MUCH more emphasis on mathematics beyond basic arithmetic than I am.
I think that it is easily possible to teach investing without a prior knowledge of advanced mathematics. Marginal returns, compounding, risk, and diversification only require basic arithmetic.
Just look at any typical book on investing. No geometry, trig, algebra, or calculus is necessary.
Yes, I am. I'd like to better understand your point of view. If math beyond basic arithmetic doesn't contribute to the life skills and job skills you'd like 13 school years to impart, can you tell me pls what level of knowledge you consider adequate for high school graduates in other academic subjects? And what are the core job skillsyou expect high school grads to have?
I can’t realistically write out an entire K-12 education curriculum in the comments. I already mentioned a few topics that I believe are under-taught to give you a basic idea. I also wrote this article on vocational training:
https://frompovertytoprogress.substack.com/p/why-we-need-more-vocational-education
My point is that mathematics beyond basic arithmetic is rarely used by more than a tiny minority of adults in their jobs or elsewhere, so it does not make sense to teach even more advanced mathematics that even fewer adults will ever use.
For evidence as to why I believe that there is already too much mathematics in the K-12 curriculum, I would recommend:
https://www.amazon.com/Math-Myth-Other-STEM-Delusions/dp/162097391X/
Reading this, I was wondering when you’d get to the teachers. Your list of big barriers at the beginning didn’t include them, and I was wondering from the beginning why not. The whole time I was reading the essay, the old saying “if wishes were horses…” kept rolling through my mind. Do I think deficiencies in math instruction exist? Heck yeah. Do I see things changing? Not until there’s a “Sputnik” moment (e.g., a massive and visible cyberattack) that exposes our weakness in this area. I just hope it is not by then too late. By the way, have you run this by the new SecEd? It would be interesting to see her response.
Between some great YouTube channels on math, online courses through Coursera and the like, and AI agents like ChatGPT, Grok, or Gemini, it is far easier than ever before to get access to engaging, high-quality math instruction. Most K-12 math teachers need to shift from explainer and tester to concierge and mentor - pointing students to what they should work on and helping them find good platforms for learning. As for SecEd, I have no connections and imagine that it is hard to consider new approaches given the current focus on cutting back. Some states might be receptive but I have no connections there either.