One thing I’m trying to do more of here is pieces of the type “here are some facts about the world that many people don’t know along with some thoughts from me that aren’t supposed to cohere into a big overarching argument.” My post about how power is wielded in education was meant to be the first major effort in that genre. The fact is that there are often complicated topics we interact with in politics that the average person can’t possibly be fully informed about; society is too complex. We naturally want to appear savvy and informed, so we sometimes find ourselves bluffing about how much we really understand a subject, even bluffing to ourselves. Certainly I’m guilty at times. So I’d like to do things that are not “explainers” in the condescending sense, nor the “here’s a standard-issue big-thesis op-ed piece I’m smuggling under the guise of a factual informer” sense. Just information that I think might be relevant and valuable to my readers in a format they won’t get at Wikipedia. It happens that educational assessment and the broader worlds of education research and policy are areas where I can claim the most expertise, so naturally I’ll spend some time in that arena. I was very gratified to hear from readers who had precisely the experience I wanted them to have with the previously-mentioned post; several people shared some version of “I feel like I care a lot about education, but I never considered how X works before.” That’s the whole idea.
Freddie --- Header: " I wish you could be on some influential panel or board where you could have a lot of influence on school policy shaping." I feel that you've specifically 'nailed' this in ways that most people have not grasped. As I shared with you in past, I did very well in school and post graduate degrees, etc., but I was lost by the time I was in Grade 9/10 in math. It was the way they were teaching it (too abstract) and they went waaaayyyyy too fast for my 'math learning curve.' In grade 8 and 9, I went to private schools run by a bunch of nuns and they taught math the old way - it was great, I could follow it. Then I switched to the a public high school with all the new modern math - that was it, I was done. I slipped thru the cracks, nobody was there to help me, certainly not my parents. I just ended up dropping math which was so sad. Thanks again for your contribution. You are great on TV as well and your book is so necessary right now.
This is also interesting in the context of plans to de-track math classes since we also have no idea how that’s going to turn out. Probably that natural ability curve is going to hit even harder.
If it was up to me I would teach one combined trig/physics/calculus super course over two hours. Math for math's sake is ridiculous for 90% of the population and teaching it divorced from any real world application is only going to lead to confusion and boredom.
I also wonder what counts as "advanced math" and whether or not most students will benefit from exposure to it. And I heard so much about the Khan Academy videos a while back--were they actually effective? And if so what methodology did they use?
Question: what sorts of methods exist to disentangle the effects of external-to-the-classroom effects (viz., parents helping their kids with homework, individual tutoring, etc.) from what might be called talent?
I ask because I've been reading a lot about the science of music practice lately. Music is one of those areas where performance is largely driven by quantity of practice as well as individualized attention by private music teachers not available in the classroom—the extent to which "talent" is a significant differentiator is hard to determine because the number of hours needed to invest in an instrument for talent to matter is astronomically high. ("Talent" is anathema in the music world too, but in addition to the "practice" aspect, I suspect it's less because talent offends our sensibilities but rather that belief in talent is a uniquely destructive belief to music students at any skill level, even if it talent really exists.)
When it comes to math, it strikes me that in addition to the quantity of at-home instruction, there's also a problem with highly variable quality—even a child with a stay-at-home parent with all the time in the world to check their homework won't get much better if their parent can't do algebra themselves.
Another great piece. I have a 7-year old that just got through 1st-grade virtual schooling at home. Nice to understand where these groups of ten, and various strategies to addition are coming from.
I happen to have a minor in math that happened kind of by accident, (I had to take so much of it for my major Comp Sci, figured might as well) but all it did was let me know that Math gets really hard.
I've been reading "In Over Our Heads: The Cognitive Demands Of Modern Life" by Robert Kegan.
I wonder if the successful point around fourth grade might involve having left behind the painful and disruptive transition from stage 1 to stage 2 of cognitive development, and being solidly established in stage 2. The eighth-grade inflection point might involve the painful transition to enter stage 3.
Each stage involves more abstract reasoning and layers of complexity, which is relevant to a lot of things, one of which is mathematical abstraction.
Have you read it?
(P.S. The stage 3 to 4 transition, by the way, used to happen in college, but that transition has been disrupted by teaching a crude misunderstanding of stage-5-- known as postmodernism-- to undergraduates who have not attained stage 4 systematicity and therefore cannot transcend stage 4. They just assume systems are nothing but illicit power and stay stuck in a teenage stage 3 mode. For example, SATs are formulaic stage 4 systems, but a person in stage 3 can only see them as corrupt gatekeeping.)
Is math an important skill to learn for its own sake, or is it only valuable insofar as doing well on tests/mental math at the grocery store/other applied methods? If the former, then getting students to understand it at a reasoning level is clearly essential. Like English: we teach the language at many levels, some rote and others sophisticated.
I'm a former teacher and current data professional, for context: the problem fundamentally is that we're looking at math in different contexts. Policy-makers, parents, and journalists alike (no offense) aren't very good at math and don't understand it, so to them there's no underlying value to the subject beyond rote test scores. It is an extremely valuable subject in its own right, though; thinking and reasoning skills are very important, and math where properly utilized can develop these.
It's just that learning math by rote algorithm does nothing for these skills. It's good for the perspective that math has no innate value, but it ruins the innate value math has.
I have little to add to the general discussion, other than to re-emphasize that even very good ideas for making the important parts of math more accessible often turn out to be counter-productive when they are turned over to math-averse teachers for implementation.
Meanwhile, if anybody would like to learn basic stats in a way that combines intuition with careful reasoning, here's some free online lectures available to anyone.
The best student one semester was a high school kid who took the course online.
Disclosure: the course creator and lecturer is my wife.
An online series of a less mathematical version should, I hope, roll out in the Fall 2021 semester.
Someday I hope that a similar series of lectures will be made with more Bayesian and causal material, although the causal content here is distinctly better than the fuzz in most stats courses and the Bayesian bit is more intuitive than usual.
I don't know much about the new new math, but I can tell you that those Common Core boxes for multiplication are _way_ closer to how I do multiplication in my head than the "carry the 2" method I was taught as a kid (which is _much_ harder to do in one's head).
A couple of anecdotes: At one point a group of mathematicians tried designing a general curriculum. Complex numbers were to be learned at around age 12 as the ring of polynomials mod(x^2 + 1). Slower kids could learn them as ordered pairs with a special multiplication rule.
Some math colleagues had some helpful advice for my wife (then teaching middle school) on how to teach some algebra. She thanked them and asked how that would work if four kids in the back row were singing "Sexual Healing".
I had a wonderful teacher who identified my mathematical talent when I was 9, but math was still my worst subject till calculus (which, in my high school, at least, no one needed "algebra 2" to understand — algebra 2's purpose was to gatekeep calculus to keep AP scores high). When I began an undergraduate degree in a related major, I flatly disbelieved the math profs who told me I ought to become a mathematician. Biographical confounds of my own derailed me from the mathematics PhD track after I was finally on it, but the profs weren't wrong about identifying mathematical talent.
If your goal is cultivating mathematical talent in children who have it, and cultivating "number sense" in children more generally, introducing proof-like reasoning in elementary school make sense. More importantly, there's *no* excuse for telling middle-schoolers (as I was told) that proofs of matters like Pythagorean theorem, the quadratic formula, and the area of a circle given pi, are "too abstract" for them. It's lying. Sure, some may not get those (then, or maybe ever), but telling those who can they're not developmentally ready because of their age only hurts them. Not telling kids about complex numbers when they first learn to factor polynomials is a lie, too. Teachers saying one year, "This has no solution," and the very next, "Haha! we were kidding, it does have a solution!" *ought* to piss off anyone of mathematical integrity. Math isn't supposed to work like that! If it worked like that, why trust it to be math?
At the same time, kids whose talents lie elsewhere would benefit from off-ramps sooner than "algebra 2" or calculus.
"Binning" is useful preparation for algebra. Long division is most useful for polynomial long division, not plain ol' numerals in the age of calculators. (Fractions are more important than long division for most understanding.) I was an artsy-fartsy kid who kept a sketchbook — that also contained many proof I began on my own, but rarely completed, since American kids are strongly discouraged from specializing early, and why keep working on a proof you won't even get graded for when your math grades are already your worst?
Girls especially are expected to be people-pleasers, more likely to believe if their grades in a subject aren't already excellent, they're probably untalented. Many girls will eventually become mothers, which makes delaying the development of girls' mathematical talent more costly than delaying that of boys'. Guys who want families can afford to be late bloomers, either by nature or because their prior schooling didn't serve them well, somewhat better than gals who want families.
Expecting "smart" kids to be straight-A students "in everything" is a prodigious waste. It's a prodigious waste of both STEM talent and non-STEM talent. Not everyone with design talent has the mathematical chops for an engineering degree, for example. I know someone who's a successful designer now for whom guidance to enroll in engineering was a waste (of both resources and self-worth) — a predictable waste for anyone distinguishing "smart" from "can do all the maths if only tries hard enough".
Math really is beautiful. Curricula that give kids a glimpse of what math "really looks like" early can help both the talented, who'd find it a natural fit, and the untalented, who could then be able to decide earlier math might not be for them — if the system let them decide. Torturing those who won't become mathematicians year after year with "the math" makes life worse for them, and for the talented, since a curriculum geared toward grinding a subject into those with little aptitude or interest is rather different from one drawing out aptitude and interest.
I had a middle school teacher who bet me I couldn't prove the quadratic formula. I won. That didn't mean I could easily remember the formula, though. And I never had to in college, while getting an actual math degree.
I was taught New Math in elementary school. It was basically set theory tending towards groups and Boolean logic. Teachers tried their best, I think, but this conceptual stuff maybe came at the expense of drilling multiplication tables or whatever....at least in the minds of parents. And when the curriculum took a turn towards algebra and calculus, those subjects were sufficiently well established that the New Math didn't influence their pedagogy, so what was the point? The "top down" part you mention, Freddie, is a result of "Sputnik panic" that got the government and big foundations interested in revamping STEM instruction at the time.
Having said all this, New Math did prime a bunch of 1960s kids to think like computer scientists--to be comfortable with binary and modular arithmetics, non-base10 systems, and the like. I think it may have been more influential than is generally acknowledged.
I teach math for a living, and I have strong opinions on this. With the caveat that opinions are not the same epistemic status as peer-reviewed research:
Point one: a lot of "innovation" in pedagogy over the past couple of generations has been to move away from rote learning, or "drill and kill" - school needs to focus on being exciting, Real World applications etc. etc.
And to a certain extent, it does.
But there's some very well validated and replicated research into learning from psychology under the general heading of Cognitive Load Theory, which says that to teach students more advanced skills, one of the best things you can do is practice subskills to the point of automation, or "overlearning" as it's called in the literature. This is completely uncontroversial outside the modern, western classroom - musicians practice their scales, sportspeople practice ball control and the like, and martial artists spend hours practicing their Kata (or other names for "forms").
The idea is that for example to be able to solve the more advanced problem of computing 36x57, whether longhand or mentally, you need to be able to do 6x7 without "context switching" because you've run out of working memory. Similarly if you're working a word problem about the cost of tiles to lay a 6x7 tile patio at $12 a tile, then if 6x7 is overlearnt you're less likely to lose track of what you're meant to be doing in the first place because you're counting on your fingers or something. Another advantage of keeping the big picture in mind is that if you get a negative result due to a calculation error, that might ring some alarm bells.
When you encounter something like 12x + 17 = 21 for the first time, it's hard enough figuring out what "x" means and how you need to isolate it; it's just that bit easier if once you've decided to go for a subtraction, your brain isn't too taxed by calculating 21-17 while hanging on to that "x".
For the same reason, when you're trying to do partial fraction decomposition to deal with a certain kind of integral, you really don't want to be expending mental effort on subproblems like a(b+c)=ab+ac or whether a/b + a/c = a/(b+c) or not.
So if the destination is being able to solve advanced integrals, or whichever flavor of "higher order skills" is in fashion this month, then the best path there in my opinion is to make sure the fundamentals are absolutely solid. If you can't do 11x12-36 in your head without going red in the face, then you absolutely do need to practice your arithmetic first. Times tables, number bonds and so on - these things were all added to the curriculum at some point for a reason, and for all the faults of Common Core or "back to basics" type approaches, a lot of this stuff in math is actually doing work for you.
One thing I’m trying to do more of here is pieces of the type “here are some facts about the world that many people don’t know along with some thoughts from me that aren’t supposed to cohere into a big overarching argument.” My post about how power is wielded in education was meant to be the first major effort in that genre. The fact is that there are often complicated topics we interact with in politics that the average person can’t possibly be fully informed about; society is too complex. We naturally want to appear savvy and informed, so we sometimes find ourselves bluffing about how much we really understand a subject, even bluffing to ourselves. Certainly I’m guilty at times. So I’d like to do things that are not “explainers” in the condescending sense, nor the “here’s a standard-issue big-thesis op-ed piece I’m smuggling under the guise of a factual informer” sense. Just information that I think might be relevant and valuable to my readers in a format they won’t get at Wikipedia. It happens that educational assessment and the broader worlds of education research and policy are areas where I can claim the most expertise, so naturally I’ll spend some time in that arena. I was very gratified to hear from readers who had precisely the experience I wanted them to have with the previously-mentioned post; several people shared some version of “I feel like I care a lot about education, but I never considered how X works before.” That’s the whole idea.
Freddie --- Header: " I wish you could be on some influential panel or board where you could have a lot of influence on school policy shaping." I feel that you've specifically 'nailed' this in ways that most people have not grasped. As I shared with you in past, I did very well in school and post graduate degrees, etc., but I was lost by the time I was in Grade 9/10 in math. It was the way they were teaching it (too abstract) and they went waaaayyyyy too fast for my 'math learning curve.' In grade 8 and 9, I went to private schools run by a bunch of nuns and they taught math the old way - it was great, I could follow it. Then I switched to the a public high school with all the new modern math - that was it, I was done. I slipped thru the cracks, nobody was there to help me, certainly not my parents. I just ended up dropping math which was so sad. Thanks again for your contribution. You are great on TV as well and your book is so necessary right now.
This is also interesting in the context of plans to de-track math classes since we also have no idea how that’s going to turn out. Probably that natural ability curve is going to hit even harder.
If it was up to me I would teach one combined trig/physics/calculus super course over two hours. Math for math's sake is ridiculous for 90% of the population and teaching it divorced from any real world application is only going to lead to confusion and boredom.
I also wonder what counts as "advanced math" and whether or not most students will benefit from exposure to it. And I heard so much about the Khan Academy videos a while back--were they actually effective? And if so what methodology did they use?
Who are these experts arguing that math is just arithmetic and how many of them actually have PhDs in mathematics?
Question: what sorts of methods exist to disentangle the effects of external-to-the-classroom effects (viz., parents helping their kids with homework, individual tutoring, etc.) from what might be called talent?
I ask because I've been reading a lot about the science of music practice lately. Music is one of those areas where performance is largely driven by quantity of practice as well as individualized attention by private music teachers not available in the classroom—the extent to which "talent" is a significant differentiator is hard to determine because the number of hours needed to invest in an instrument for talent to matter is astronomically high. ("Talent" is anathema in the music world too, but in addition to the "practice" aspect, I suspect it's less because talent offends our sensibilities but rather that belief in talent is a uniquely destructive belief to music students at any skill level, even if it talent really exists.)
When it comes to math, it strikes me that in addition to the quantity of at-home instruction, there's also a problem with highly variable quality—even a child with a stay-at-home parent with all the time in the world to check their homework won't get much better if their parent can't do algebra themselves.
Another great piece. I have a 7-year old that just got through 1st-grade virtual schooling at home. Nice to understand where these groups of ten, and various strategies to addition are coming from.
I happen to have a minor in math that happened kind of by accident, (I had to take so much of it for my major Comp Sci, figured might as well) but all it did was let me know that Math gets really hard.
I've been reading "In Over Our Heads: The Cognitive Demands Of Modern Life" by Robert Kegan.
I wonder if the successful point around fourth grade might involve having left behind the painful and disruptive transition from stage 1 to stage 2 of cognitive development, and being solidly established in stage 2. The eighth-grade inflection point might involve the painful transition to enter stage 3.
Each stage involves more abstract reasoning and layers of complexity, which is relevant to a lot of things, one of which is mathematical abstraction.
Have you read it?
(P.S. The stage 3 to 4 transition, by the way, used to happen in college, but that transition has been disrupted by teaching a crude misunderstanding of stage-5-- known as postmodernism-- to undergraduates who have not attained stage 4 systematicity and therefore cannot transcend stage 4. They just assume systems are nothing but illicit power and stay stuck in a teenage stage 3 mode. For example, SATs are formulaic stage 4 systems, but a person in stage 3 can only see them as corrupt gatekeeping.)
Is math an important skill to learn for its own sake, or is it only valuable insofar as doing well on tests/mental math at the grocery store/other applied methods? If the former, then getting students to understand it at a reasoning level is clearly essential. Like English: we teach the language at many levels, some rote and others sophisticated.
I'm a former teacher and current data professional, for context: the problem fundamentally is that we're looking at math in different contexts. Policy-makers, parents, and journalists alike (no offense) aren't very good at math and don't understand it, so to them there's no underlying value to the subject beyond rote test scores. It is an extremely valuable subject in its own right, though; thinking and reasoning skills are very important, and math where properly utilized can develop these.
It's just that learning math by rote algorithm does nothing for these skills. It's good for the perspective that math has no innate value, but it ruins the innate value math has.
I have little to add to the general discussion, other than to re-emphasize that even very good ideas for making the important parts of math more accessible often turn out to be counter-productive when they are turned over to math-averse teachers for implementation.
Meanwhile, if anybody would like to learn basic stats in a way that combines intuition with careful reasoning, here's some free online lectures available to anyone.
http://courses.atlas.illinois.edu/fall2020/STAT200/
http://courses.atlas.illinois.edu/fall2020/STAT200/calendar.html
The best student one semester was a high school kid who took the course online.
Disclosure: the course creator and lecturer is my wife.
An online series of a less mathematical version should, I hope, roll out in the Fall 2021 semester.
Someday I hope that a similar series of lectures will be made with more Bayesian and causal material, although the causal content here is distinctly better than the fuzz in most stats courses and the Bayesian bit is more intuitive than usual.
I don't know much about the new new math, but I can tell you that those Common Core boxes for multiplication are _way_ closer to how I do multiplication in my head than the "carry the 2" method I was taught as a kid (which is _much_ harder to do in one's head).
A couple of anecdotes: At one point a group of mathematicians tried designing a general curriculum. Complex numbers were to be learned at around age 12 as the ring of polynomials mod(x^2 + 1). Slower kids could learn them as ordered pairs with a special multiplication rule.
Some math colleagues had some helpful advice for my wife (then teaching middle school) on how to teach some algebra. She thanked them and asked how that would work if four kids in the back row were singing "Sexual Healing".
I had a wonderful teacher who identified my mathematical talent when I was 9, but math was still my worst subject till calculus (which, in my high school, at least, no one needed "algebra 2" to understand — algebra 2's purpose was to gatekeep calculus to keep AP scores high). When I began an undergraduate degree in a related major, I flatly disbelieved the math profs who told me I ought to become a mathematician. Biographical confounds of my own derailed me from the mathematics PhD track after I was finally on it, but the profs weren't wrong about identifying mathematical talent.
If your goal is cultivating mathematical talent in children who have it, and cultivating "number sense" in children more generally, introducing proof-like reasoning in elementary school make sense. More importantly, there's *no* excuse for telling middle-schoolers (as I was told) that proofs of matters like Pythagorean theorem, the quadratic formula, and the area of a circle given pi, are "too abstract" for them. It's lying. Sure, some may not get those (then, or maybe ever), but telling those who can they're not developmentally ready because of their age only hurts them. Not telling kids about complex numbers when they first learn to factor polynomials is a lie, too. Teachers saying one year, "This has no solution," and the very next, "Haha! we were kidding, it does have a solution!" *ought* to piss off anyone of mathematical integrity. Math isn't supposed to work like that! If it worked like that, why trust it to be math?
At the same time, kids whose talents lie elsewhere would benefit from off-ramps sooner than "algebra 2" or calculus.
"Binning" is useful preparation for algebra. Long division is most useful for polynomial long division, not plain ol' numerals in the age of calculators. (Fractions are more important than long division for most understanding.) I was an artsy-fartsy kid who kept a sketchbook — that also contained many proof I began on my own, but rarely completed, since American kids are strongly discouraged from specializing early, and why keep working on a proof you won't even get graded for when your math grades are already your worst?
Girls especially are expected to be people-pleasers, more likely to believe if their grades in a subject aren't already excellent, they're probably untalented. Many girls will eventually become mothers, which makes delaying the development of girls' mathematical talent more costly than delaying that of boys'. Guys who want families can afford to be late bloomers, either by nature or because their prior schooling didn't serve them well, somewhat better than gals who want families.
Expecting "smart" kids to be straight-A students "in everything" is a prodigious waste. It's a prodigious waste of both STEM talent and non-STEM talent. Not everyone with design talent has the mathematical chops for an engineering degree, for example. I know someone who's a successful designer now for whom guidance to enroll in engineering was a waste (of both resources and self-worth) — a predictable waste for anyone distinguishing "smart" from "can do all the maths if only tries hard enough".
Math really is beautiful. Curricula that give kids a glimpse of what math "really looks like" early can help both the talented, who'd find it a natural fit, and the untalented, who could then be able to decide earlier math might not be for them — if the system let them decide. Torturing those who won't become mathematicians year after year with "the math" makes life worse for them, and for the talented, since a curriculum geared toward grinding a subject into those with little aptitude or interest is rather different from one drawing out aptitude and interest.
I had a middle school teacher who bet me I couldn't prove the quadratic formula. I won. That didn't mean I could easily remember the formula, though. And I never had to in college, while getting an actual math degree.
I was taught New Math in elementary school. It was basically set theory tending towards groups and Boolean logic. Teachers tried their best, I think, but this conceptual stuff maybe came at the expense of drilling multiplication tables or whatever....at least in the minds of parents. And when the curriculum took a turn towards algebra and calculus, those subjects were sufficiently well established that the New Math didn't influence their pedagogy, so what was the point? The "top down" part you mention, Freddie, is a result of "Sputnik panic" that got the government and big foundations interested in revamping STEM instruction at the time.
Having said all this, New Math did prime a bunch of 1960s kids to think like computer scientists--to be comfortable with binary and modular arithmetics, non-base10 systems, and the like. I think it may have been more influential than is generally acknowledged.
I teach math for a living, and I have strong opinions on this. With the caveat that opinions are not the same epistemic status as peer-reviewed research:
Point one: a lot of "innovation" in pedagogy over the past couple of generations has been to move away from rote learning, or "drill and kill" - school needs to focus on being exciting, Real World applications etc. etc.
And to a certain extent, it does.
But there's some very well validated and replicated research into learning from psychology under the general heading of Cognitive Load Theory, which says that to teach students more advanced skills, one of the best things you can do is practice subskills to the point of automation, or "overlearning" as it's called in the literature. This is completely uncontroversial outside the modern, western classroom - musicians practice their scales, sportspeople practice ball control and the like, and martial artists spend hours practicing their Kata (or other names for "forms").
The idea is that for example to be able to solve the more advanced problem of computing 36x57, whether longhand or mentally, you need to be able to do 6x7 without "context switching" because you've run out of working memory. Similarly if you're working a word problem about the cost of tiles to lay a 6x7 tile patio at $12 a tile, then if 6x7 is overlearnt you're less likely to lose track of what you're meant to be doing in the first place because you're counting on your fingers or something. Another advantage of keeping the big picture in mind is that if you get a negative result due to a calculation error, that might ring some alarm bells.
When you encounter something like 12x + 17 = 21 for the first time, it's hard enough figuring out what "x" means and how you need to isolate it; it's just that bit easier if once you've decided to go for a subtraction, your brain isn't too taxed by calculating 21-17 while hanging on to that "x".
For the same reason, when you're trying to do partial fraction decomposition to deal with a certain kind of integral, you really don't want to be expending mental effort on subproblems like a(b+c)=ab+ac or whether a/b + a/c = a/(b+c) or not.
So if the destination is being able to solve advanced integrals, or whichever flavor of "higher order skills" is in fashion this month, then the best path there in my opinion is to make sure the fundamentals are absolutely solid. If you can't do 11x12-36 in your head without going red in the face, then you absolutely do need to practice your arithmetic first. Times tables, number bonds and so on - these things were all added to the curriculum at some point for a reason, and for all the faults of Common Core or "back to basics" type approaches, a lot of this stuff in math is actually doing work for you.