I don't know, force of habit? Also, while I of course support universal pre-K, the leadership and funding mechanisms are very different so I'm not sure if it's usually apt to include.
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.
Just wanted to say, as a father trying to help my 4th and 7th grader with math, I really appreciated reading a piece like this which is informative and also open about the uncertainties.
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.
The great paradox of American K-12 schooling is that in some sense the distribution of power and authority across so many stakeholders is a great boon, but it also ensures that there are too many hands on the wheel, resulting in things like these cyclical math wars that pit academics and policy wonks against teachers and parents.
Yes, with that collective 'wonkiness' -- from my 'felt experience', the transmission of that kind of math teachings filters down as too impersonal and abstract vs. a more personal connection that I got from, eg., the nuns. I believe in energetic transmission of information and if you don't establish that 'presence' of connection for students in a grounded way, their little hearts cannot create 'attachment' to anchor to the information properly. Interestingly, my cousin is a world famous Economist and big time tenured professor in US - he can write strings of mathematical theories but he is very robotic and emotionally disconnected. Those guys will excel regardless. The problem is that people like me, in the psychology field, and I know many, get denied the higher degrees because of math limitations and then I hear 'all the talk' by so many clients how these 'graduated psychologists' in their practice lack emotional empathy and presence -which are most critical in healing people.
>"I believe in energetic transmission of information and if you don't establish that 'presence' of connection for students in a grounded way, their little hearts cannot create 'attachment' to anchor to the information properly."
I completely agree with this! And it's completely missing from any educational discussion. My version of it is "let teachers teach".
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?
You know I'm not aware of any specific literature on Khan Academy, but there is a general (and sensible) finding about all kinds of supplemental online education like the Great Courses Plus etc - the people who learn from and thrive at them are the people who already did well in education, because learning from them requires the prerequisite supplementary skills that help in education in general, such as time management, persistence, etc.
Yes, but wasn't the selling point for the Khan videos that they actually did a decent job of teaching mathematics compared to (meaning as a replacement for) most public school curricula? I just feel that mathematics instruction in the United States is completely broken in the sense of being opaque and confusing. It leads me to wonder about those alternatives that supposedly yield better results: the Khan videos, school instruction in places like Scandinavia and Asia, etc. Has anybody looked into the instruction methodology followed by Finnish teachers, for example? Is this problem really that hard?
I should probably look this up but when you match race for race the United States is pretty much the best educational performer in the world. In PISA scores our white kids beat every dominantly-white country except sometimes for Finland, often regarded as the best in the world. Our Asian students outperform every Asian country except for the affluent urban kids China selects to perform for them, and those are well-known and disqualifying confounds. American Hispanic schools kick the ass of every participating Latin American nation. There's a lack of data for Africa but students from say Trinidad and Tobaga lag far behind American Black students. So the notion that our math instruction, or any of our instruction, is a basket case just doesn't bear scrutiny. This argument doesn't get heard a lot because people are uncomfortable making race-based cross-country comparisons. But like it or not race is the most consistently consequential variable in education, so those comparisons are necessary. And they show that the United States is in fact very strong in education; it's just very racially diverse compared to many countries its frequently compared to.
Fascinating. I’m not “woke” but I do fall into the “race-as-we-know-it-is-a-construct” person. So you’re causing me some cognitive dissonance but I’d rather know the truth. Do you have links to this stuff? And also, do these studies take into account counterfactuals like poverty, social status in each country etc?
I'll try to dig up where I first found it - it was certainly a "race science" adjacent publication, I'm sorry to say, but they're typically the only people willing to even get close to some of these issues. These are just raw PISA result comparisons, which are meant to represent a cross-section of students from each country but often practically don't. The important point here though is that it's raw PISA comparisons and similar that generate these dynamics, which are appropriate because it's raw PISA comparisons that drive complaints about failing American education.
One ironic takeaway is that skin color, our most powerful proxy for race, is only 'accidentally' somewhat correlated with genetic groupings.
I think it's important to keep in mind that, even if (or tho) race is (largely) a 'social construct', that doesn't imply that there are no relatively distinct genetic groups of people, or that our social constructs don't correlate with those groups to some degree (which they do).
Right: this kind of "comparison" has to have been treating race as a sociological, not a genetic, category. The achievement gap results not from having African greeeeat grandparents but from having grown up in, whisper it softly, a less-than-privileged household. Unless someone had a very severe attack of sanity and a lot of help from 23andMe.
I relied on Kahn videos to help me through an online statistics course as a prerequisite for nursing school. At the same time, I have two school-aged kids asking me math homework questions.
From my perspective, some of the basics being taught Common Core-style are very helpful for people who 1) aren't fans of memorization or 2) need to understand "why" before the "how" makes sense. I'm one of those people. I think I would've potentially struggled less with math if I'd had different, more compartmentalized ways of solving problems.
The Kahn stuff I relied on (from my very limited perspective) seems to be a nice fusion of general memorization of basic arithmetic which then serves more conceptually advanced problems.
As a female, I never thought "oh that's why I'm bad at math;" I had many female friends who excelled at math. Me, though, well I took trigonometry in high school and again in college and got a C both times. I'd hit my upper limit (or at least the limit of what I was willing to suffer through to make progress).
I do think having diversified math classes in later years would be a boon. Once basic math is mastered, very few people need to be "pure math", and I think a lot of people would benefit from and maybe even enjoy math classes that were specifically teaching math skills alongside their real-world application.
Have a financial literacy class, a mechanics class that merges physics and math, a statistics class combined with media literacy, a computer science discrete math course. Math is a huge topic! There's lots you could do. One of my favorite college classes was a math-heavy physics class about music, breaking down how sound waves function and interact to produce something more than noise.
Of course, the problem is that this would be expensive and require a lot of curriculum writing.
The near-universal lack of classes on financial literacy is a great tragedy. My impression is that the creation of any such classes on a large scale would be fought tough-and-nail by the financial-services industry, which likely helps explain why they don't exist.
Parent here and my take is that it is too soon to conclude anything about the common core way of teaching math, but even though it was disorienting, it did seem like it was preparing from the beginning to think about math more abstractly and deeply -- and while Freddie's point about parents helping is relevant, in my experience Khan Academy is really good at this stuff and can be a good substitute for a parent assuming wifi and computer access at home.
Might have put this inartfully - the point is not that math is just arithmetic. The point is that there isn't a meaningful dividing line between quantitative reasoning and mathematical computation, that they are actually different ways of looking at what are in fact the same skills, and that thinking in a binary way creates problems like the ones described here (ie parents annoyed that process is elevated above results) and also doesn't facilitate transition into higher-order math because these distinctions collapse as things get more complex (ie calculus is often seen as another level of difficulty because the problems inherently contain heavy doses of what some would call both quantitative reasoning and calculation, although I never took calculus so I don't really know what I'm talking about).
I certainly don't want to ascribe skepticism about the existence/importance of numbers sense to him, but the math textbook author Caleb Gattegno was an example of someone whose approach did not emphasize numbers sense, although he died I think in 1988 so he preceded the current yen for it. Perhaps I should have emphasized instead that there are a lot of practitioners who think that time spent on number sense detracts from the kind of computational practice that, some feel, eventually leads to abstract understanding. Certainly the much-ballyhooed success of East Asian approaches to math, with their heavy emphasis on grinding endless numbers of equations, would be a point in the favor of a computation-heavy approach.
Here's a paper on the subject of numbers sense, although I can't say it provides strong evidence for either position: https://files.eric.ed.gov/fulltext/EJ1069023.pdf The correlation between number sense and mental calculation is pretty strong for education research - .54. But that's lower than I would have thought. On the one hand, you could say that a high correlation shows that a division between number sense and mental calculation skills show is unnecessary. On the other hand, you could say that a low correlation shows that number sense does not improve the kind of raw calculating ability many people emphasize. I'm not sure how to interpret the results myself.
Ah, I see. First and foremost, it’s absolutely true that you need arithmetic ability to perform calculus and any other higher math. It’s also true that without intuition, you will never be able to do any form of research or apply the math in novel situations. Also, I suspect both abilities are highly correlated and it may just be easier to teach arithmetic. Certainly, arithmetic is a less abstract skill that can be learned more easily with rote practice.
But I also don’t understand why we teach algebra or calculus in schools as anything but a elective for would be engineers and physicists. If it were up to me the standard curriculum would be application-focused statistics and probably classes.
Facility with some basic algebra is pretty much essential for learning stats as more than a set of rules to follow in specific situations. That doesn't mean you need to even solve quadratics or factorize polynomials, but you have to be able to translate simple formulas between different forms. E.g. if the formulas for odds in terms of probability and for probability in terms of odds seems like separate things to be memorized, it's just too hard to get much further.
To go on past the first course, you generally have to understand what a probability density is and what its integral means. That requires almost no facility with calculus, but does require a comfort with the basic terms.
My claim is based on my wife's experience teaching several tens of thousands of students beginning stats.
Also I feel compelled to note that I'm being sincere. People whose brains work this way are a marvel to me. But nothing on the internet is sincere, so now I have to say it so you don't think I'm trolling. Ugh.
Our brains are no more miraculous than anyone else's, it's just a particular set of mental skills that we happen to be good at. I'm terrible at plenty of other things like hand-eye coordination, languages, memorization, emotional intelligence (though I got a lot better in my 40s), patience, and a thousand other things that are at least as valuable to society.
You don't need "arithmetic ability" — or at least, not outstanding arithmetic ability — to ace calculus and higher subjects. Or to become an Ivy-League math prof, provided you picked the right specialty. Better arithmetic makes you more efficient at higher mathematics, so you'll hurt for not having it — but maybe not as much as the guy who's a computational whiz but deficient in geometric intuition.
It's best to have the whole package, obviously. If you don't quite have the whole package, though, you can still succeed, though with more hassle, if the parts of the package you do have are exceptional.
I've seen waitstaff laugh at the poor mathematician who's "too drunk" to calculate the tip quickly — the poor, completely sober, mathematician. Ordinary people's expectations of the computational proficiency of mathematicians can be inaccurately high.
I was first taught to believe my less-than-prodigious computational speed marked me as more mathematically deficient than it did. I gradually got faster at arithmetic after high school with practice, but only after playing to strengths I already had.
Maybe I should elaborate more. I am about 3 weeks away from having a PhD in Computer Science, and I can say that the idea that math is first and foremost about arithmetic is patently absurd, and it’s a rigidity in this belief by educators who I quite frankly believe could never perform graduate level higher math that causes so many problems with math education.
Math is first and foremost a language for describing structure. Understanding things like “x squared is the area of a square with width x” and more importantly understanding why that is true is the only way you can ever succeed at math. Equations are really just sentences in a language used to describe structure.
There's a famous result demonstrating that all mathematics is 'equivalent' to number theory and number theory is, in some very real sense, 'just' arithmetic. (Tho _doing_ number theory is very very different than performing an arithmetic calculation.)
That was a charitable interpretation of that claim that occurred to me anyways.
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've never understood why frequentist methods still are used so extensively. As far as I can tell it is due to a false belief that prior-free conclusions are possible, but there is always a hidden prior in any frequentist calculation. Much better to bring the prior to the front so that everyone can see it.
After I think 5? classes in traditional frequentist statistics (mostly regression and ANOVA etc.) I was signed up to take a Bayes class, but I told my stats prof and he said that Bayes requires calculus, so I dropped the class. And then I met the professor who taught the Bayes class and he told me that I would have been fine without a calculus background. Quite a regret of mine.
I agree that you would have been fine without a calculus background. The basic ideas of calculus (what a "derivative" is and what an "integral" is) are straightforward, the "hard" parts are evaluating these quantities in various situations, which is most of what you learn in a calculus class.
Largely it's from inertia. E.g. the medical residents who I help with stats have to read many NEJM etc. papers that use exclusively frequentist stats. So people wanting to understand the publications need to know them. Then they use them.
But to be fair, frequentist stats are in principle prior-free. It's just that they don't answer the questions we need answered, so the interpretations usually sneak in priors.
Exactly: "they don't answer the questions we need answered", and so are useless. Then, since no one wants to do something useless, there is inevitably a bait-and-switch where a prior is snuck in, but in a way that makes it difficult to see what it was.
I hate this *so* much. Why the field perpetuates this nonsense generation after generation is utterly beyond me.
Again, TBF, as Sander Greenland has pointed out, frequentist stats are good for evaluating whether a specific prediction has failed. There aren't good for initial explorations, for which they are unfortunately routinely used.
>"frequentist stats are good for evaluating whether a specific prediction has failed"
I don't agree. To decide whether a prediction has "failed", there has to be an alternative, and there has to be a prior on the set of alternatives. Example: I predict that a coin is fair. I flip it 100 times. It comes up heads every time. Has my prediction "failed"? Remember that, if the coin is fair, any particular sequence of heads and tails is equally likely, so all heads is no more or less likely than any other particular sequence, and so cannot be taken as evidence that the coin is not fair!
Bottom line, there is always always always a hidden prior somewhere, or there is no conclusion at all.
"Remember that, if the coin is fair, any particular sequence of heads and tails is equally likely, " Um, I did remember that. When I don't remember that, it'll be time to just put me away.
"so all heads is no more or less likely than any other particular sequence, and so cannot be taken as evidence that the coin is not fair!" Your argument that not even 100 straight heads can be "be taken as evidence that the coin is not fair!" is exactly why people distrust Bayesian reasoning.
You might also be interested in some curse-of-dimensionality problems for which Bayesian estimators don't converge but frequentist estimators do.
There are always hidden priors. But, once you admit there are, it can be tempting to use the fact that there will always be priors as an excuse to avoid updating your priors when you should: there's no clear distinction between confirmation bias and Bayesian rationality.
A moral commitment of trying to draw prior-free conclusions, even if impossible, is the sort of moral commitment that appeals to the mathematically inclined. That same kind of moral commitment *should* cause distress in those who realize the moral commitment is impossible to achieve, and the pretense we can is a lie. But I also understand the worry that, if we drop the pretense, we give sophists the excuse to get away with as much cognitive bias as they can: "I will always claim my priors are 'just strong enough' to justify my refusal to update my beliefs in light of the evidence you present me."
For example, Bryan Caplan argues his priors permit him to be unpersuaded by studies that don't find a disemployment effect from higher minimum wages. I think he makes a good point — only it's exactly the sort of point someone looking for an excuse to ignore empirical studies on this matter *would* make, which, even if you agree Caplan's approach is probably correct, should be pretty disturbing:
"Part of the reason is admittedly my strong prior. In the absence of any specific empirical evidence, I am 99%+ sure that a randomly selected demand curve will have a negative slope. I hew to this prior even in cases – like demand for illegal drugs or illegal immigration – where a downward-sloping demand curve is ideologically inconvenient for me."
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).
Yeah, I do not get the resistance to the boxes--I could never do mental math until someone explained the box method to me, and then it felt so stupidly simple I wondered how I hadn't figured it out myself.
I think the resistance is very understandable – it's unfamiliar, and most people don't have an intuitive understanding of the algorithms they learned either.
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".
Haha! Complex numbers are *much* easier to understand as permitting scalars to rotate, rather than as a ring, or so sez I. The ordered-pair notation is clumsy, the polar form, more intuitive. (Or so sez I.)
I did have one outstanding math teacher in high school whose solution to the "Sexual Healing" problem would be to return the next day with a math parody to belt out, "Conceptual Feeling!", if it happened again.
Yeah, I taught a couple of kids via introducing i and then mapping multiplications to rotations. It was sort of a side project to doing some group theory, which I chose to avoid making school more boring because it was far from the curriculum.
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.
"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')."
Right, but have you ever had really bad music lessons? I did, once, from an elderly family friend. She assigned technical exercises far in advance of the actual pieces she assigned pupils, and had a knack for picking pieces that failed to hold her pupils' interest. A better music teacher (and I had better ones later) is better at pairing real music and technical exercises to reveal the technical exercises have a point. It's more intrinsically motivating to do that endless repetition of your own accord when it clearly benefits the music you want to do.
I've taught STEM subjects before in underserved public schools. Part of that teaching included teaching test prep, and, at the time, I was stumped by how little even very smart students were motivated to pay any attention to test-taking skills that could easily improve their scores. They didn't trust that their scores could be improved with practice, or that, even if they could be, that those scores could matter to them. At the time, I didn't see the connection between their distrust and my childhood distrust that Hanon's virtuoso exercises were necessary to master introductory variations on "Mary had a little lamb". I had the nice, suburban background teaching me from the cradle that test-taking is an important skill. My music teacher, a former virtuoso herself, couldn't even remember a time when the advantages of mastering Hanon's exercises wouldn't have been obvious. Hindsight is 20/20, and it's a bit tough to blame inexperienced children for not having the foresight to trust their teachers that grinding, boring exercises with no cognitive reward beyond teacher approval will pay off someday.
To be fair, I'm not aware of any math instruction in the US that doesn't *try* to pair rote drills with the cognitive reward of higher-order thinking. But I do suspect that teachers can be either like my ex-virtuoso (but actually pretty bad) music teacher in making that pairing persuasive, or be bad enough at the higher-order thinking themselves that they also fail to make a persuasive pairing.
I think you may be projecting an adult attitude toward memorization onto children. My three year old likes learning new words and is excited by memorization, but trying to get him to think abstractly only makes him frustrated. And the baby absolutely loves practicing his counting over and over and over again. My sense is that kids really enjoy working on problems that match their current developmental stage.
Even the struggling fourth graders I tutored didn't dislike multiplication drills to nearly the same degree that they hated higher level skills such as story problems and fractions. It is the teachers and parents who get bored with repetition, not the kids.
I can see why you might say that, but I'm remembering what I found terrifying about music and math instruction when I was in elementary school. I remember the scared kid I was in the face of the speed tests they used to measure rote skill, and the shame of it.
On the other hand, I loved story problems and fractions in grade school — they were math I was effortlessly good at (but also the part least likely to count for credit). I was good enough at music despite lousy technical skills to make up stuff on the piano grownups thought I must have learned in lessons.
I would not wish my childhood terror of speed drills, mathematical or musical, on anyone. There's no advantage to doing speed drills poorly, and good advantage to doing them well. But does help to measure early excellence much more by rote skills than later excellence will be? Sure, I *liked* memorizing the multiplication table, but goodness at multiplication was still measured by... speed tests, and by that metric I was a failure.
For whatever reason, I did get bored with repetition unmoored from a larger, "more beautiful" structure. Bored, therefore inattentive, then scared by the "chaos", the "unmusicality" (literally, of Hanon exercises). I was bewildered by whiz kids who could blaze through speed tests, play pieces whose purpose wasn't musical expression, but showing off virtuosic speed. How could they make sense of it?
Later, after high school, on another instrument, I began mastering virtuosic passages at high tempos, but they were from real pieces, in order to *do* real pieces. I'm still the opposite of a great performer, but the few times I sprang for lessons from those who were, they recommended drilling by abstracting from real pieces. It helped. It made regular, lengthy drilling *possible* for me. Too late to change my life, but enough to bring mastery of a few things.
Erin Ethridge wrote in these comments, "some of the basics being taught Common Core-style are very helpful for people who 1) aren't fans of memorization or 2) need to understand 'why' before the 'how' makes sense. I'm one of those people." Some of us were these people as children.
Kids enjoy problems matching their developmental stage, but not all kids develop the same. Freddie laid out some pros and cons of making "number sense" an early goal of math ed, versus what's derided, unfairly, as "drill and kill". Freddie's theme of educating for skill, rather than generic "smarts", suggests children be given more chance to play to their strengths. For the many who won't need higher math, drilling may make them happy while higher demands make them unhappy, and they deserve better than years of torture over something they'll never use. That's fine. But hostility toward "drill and kill" didn't come from nowhere.
I'm going to make the hypothesis that kids who need to understand the 'why' before they're comfortable with the 'how' are a) a minority overall and b) generally the ones who will go on to do better anyway. A lot of higher ed is (or perhaps used to be) geared towards this kind of person. I'm one of them myself, and it took me a lot of time to understand that not everyone thinks like that; once I did I found that I "connected" a lot better with lots of my pupils. H Ann's "loves memorisation, gets frustrated by abstraction" seems to be the default human state of being, it's probably possible to some extent to train people out of this but starting too early is one of many ways of putting people off school for life.
I don't believe a difference between the demographics in a particular field and the demographics of the general population need be a sign of injustice, but "will go on to do better anyway" is something that works better for the family life of men than it does for women.
If we're trying to make STEM careers more feasible for women without lowering standards for them, one way may be to waste less of talented girls' time in earlier grades, supposing they'll catch up later anyhow. Yeah, bright kids often do, in time to make the choice between career and childbearing, a fraught choice even under good circumstances, more fraught.
The world will hardly end if talent identification, originally geared toward men, considers equality between the sexes achieved by putting girls on boys' timetable. It isn't ending now. Families with resources for extracurricular enrichment can already accelerate their daughters' timetables privately, anyhow. And, if it became popular to try acting on the belief that "they'll catch up eventually" is unfairly harder on girls than on boys, we'd have another "they're leaving our boys behind!!!" panic, to boot.
Selection effects doubtless explain much success of early childhood programs advertising something "better than" rote learning. From the extracurricular Russian School of Mathematics' advertising: "Young children are naturally curious, uninhibited, and can easily grasp very complex ideas. Our curriculum and methodology are built with this in mind. Our students learn to work with, and develop an appetite for, challenging mathematical concepts." Montessori schools use similar rhetoric. Perhaps the greatest accomplishment of these programs is to salve the conscience of the better-off parents who use them, so they can believe it wasn't SES, but the enrichment they bought with their SES, that explains their children's later successes. On the other hand, the RSM school around here seems to attract immigrant families of modest means. Anyhow, parents don't pay for these programs to immiserate their children and put them off school for life, but rather the opposite. The minority of children flourishing rather than crushed by this approach seems substantial.
Bertrand Russell can complain, “I was made to learn by heart: ‘The square of the sum of two numbers is equal to the sum of their squares increased by twice their product.’ I had not the vaguest idea what this meant and when I could not remember the words, my tutor threw the book at my head, which did not stimulate my intellect in any way," catch up later, burn through several marriages, and still be siring children at 65 (Conrad Russell). He was also a governess-hiring aristocrat. Those are hard conditions for American motherhood to meet.
"The bright ones catch up anyhow" may make equalitarian parenting for young fathers less appealing, too. "Sorry, can't parent now, I'm catching up anyhow."
That "they'll catch up anyhow" seems to make balancing talent and motherhood unnecessarily hard is a first-world problem, maybe the priority of least concern in providing American children a fair, non-miserable education. But since I've seen it happen, I'll say it happens.
I don't know, force of habit? Also, while I of course support universal pre-K, the leadership and funding mechanisms are very different so I'm not sure if it's usually apt to include.
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.
Just wanted to say, as a father trying to help my 4th and 7th grader with math, I really appreciated reading a piece like this which is informative and also open about the uncertainties.
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.
The great paradox of American K-12 schooling is that in some sense the distribution of power and authority across so many stakeholders is a great boon, but it also ensures that there are too many hands on the wheel, resulting in things like these cyclical math wars that pit academics and policy wonks against teachers and parents.
Yes, with that collective 'wonkiness' -- from my 'felt experience', the transmission of that kind of math teachings filters down as too impersonal and abstract vs. a more personal connection that I got from, eg., the nuns. I believe in energetic transmission of information and if you don't establish that 'presence' of connection for students in a grounded way, their little hearts cannot create 'attachment' to anchor to the information properly. Interestingly, my cousin is a world famous Economist and big time tenured professor in US - he can write strings of mathematical theories but he is very robotic and emotionally disconnected. Those guys will excel regardless. The problem is that people like me, in the psychology field, and I know many, get denied the higher degrees because of math limitations and then I hear 'all the talk' by so many clients how these 'graduated psychologists' in their practice lack emotional empathy and presence -which are most critical in healing people.
>"I believe in energetic transmission of information and if you don't establish that 'presence' of connection for students in a grounded way, their little hearts cannot create 'attachment' to anchor to the information properly."
I completely agree with this! And it's completely missing from any educational discussion. My version of it is "let teachers teach".
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.
The one thing I can promise you de-tracking math classes won't do is to eliminate differences in performance between students.
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?
You know I'm not aware of any specific literature on Khan Academy, but there is a general (and sensible) finding about all kinds of supplemental online education like the Great Courses Plus etc - the people who learn from and thrive at them are the people who already did well in education, because learning from them requires the prerequisite supplementary skills that help in education in general, such as time management, persistence, etc.
Yes, but wasn't the selling point for the Khan videos that they actually did a decent job of teaching mathematics compared to (meaning as a replacement for) most public school curricula? I just feel that mathematics instruction in the United States is completely broken in the sense of being opaque and confusing. It leads me to wonder about those alternatives that supposedly yield better results: the Khan videos, school instruction in places like Scandinavia and Asia, etc. Has anybody looked into the instruction methodology followed by Finnish teachers, for example? Is this problem really that hard?
I should probably look this up but when you match race for race the United States is pretty much the best educational performer in the world. In PISA scores our white kids beat every dominantly-white country except sometimes for Finland, often regarded as the best in the world. Our Asian students outperform every Asian country except for the affluent urban kids China selects to perform for them, and those are well-known and disqualifying confounds. American Hispanic schools kick the ass of every participating Latin American nation. There's a lack of data for Africa but students from say Trinidad and Tobaga lag far behind American Black students. So the notion that our math instruction, or any of our instruction, is a basket case just doesn't bear scrutiny. This argument doesn't get heard a lot because people are uncomfortable making race-based cross-country comparisons. But like it or not race is the most consistently consequential variable in education, so those comparisons are necessary. And they show that the United States is in fact very strong in education; it's just very racially diverse compared to many countries its frequently compared to.
*write this up
Given the current political climate I am not surprised that this point is ignored. It's the first time I have head it made.
Fascinating. I’m not “woke” but I do fall into the “race-as-we-know-it-is-a-construct” person. So you’re causing me some cognitive dissonance but I’d rather know the truth. Do you have links to this stuff? And also, do these studies take into account counterfactuals like poverty, social status in each country etc?
I have to think that the most prominent discussion of this issue in popular media is Murray's "The Bell Curve".
I'll try to dig up where I first found it - it was certainly a "race science" adjacent publication, I'm sorry to say, but they're typically the only people willing to even get close to some of these issues. These are just raw PISA result comparisons, which are meant to represent a cross-section of students from each country but often practically don't. The important point here though is that it's raw PISA comparisons and similar that generate these dynamics, which are appropriate because it's raw PISA comparisons that drive complaints about failing American education.
Here's a good overview of genetic diversity that I think is very informative for thinking about race: https://razib.substack.com/p/out-of-africas-midlife-crisis
One ironic takeaway is that skin color, our most powerful proxy for race, is only 'accidentally' somewhat correlated with genetic groupings.
I think it's important to keep in mind that, even if (or tho) race is (largely) a 'social construct', that doesn't imply that there are no relatively distinct genetic groups of people, or that our social constructs don't correlate with those groups to some degree (which they do).
Right: this kind of "comparison" has to have been treating race as a sociological, not a genetic, category. The achievement gap results not from having African greeeeat grandparents but from having grown up in, whisper it softly, a less-than-privileged household. Unless someone had a very severe attack of sanity and a lot of help from 23andMe.
I relied on Kahn videos to help me through an online statistics course as a prerequisite for nursing school. At the same time, I have two school-aged kids asking me math homework questions.
From my perspective, some of the basics being taught Common Core-style are very helpful for people who 1) aren't fans of memorization or 2) need to understand "why" before the "how" makes sense. I'm one of those people. I think I would've potentially struggled less with math if I'd had different, more compartmentalized ways of solving problems.
The Kahn stuff I relied on (from my very limited perspective) seems to be a nice fusion of general memorization of basic arithmetic which then serves more conceptually advanced problems.
As a female, I never thought "oh that's why I'm bad at math;" I had many female friends who excelled at math. Me, though, well I took trigonometry in high school and again in college and got a C both times. I'd hit my upper limit (or at least the limit of what I was willing to suffer through to make progress).
I do think having diversified math classes in later years would be a boon. Once basic math is mastered, very few people need to be "pure math", and I think a lot of people would benefit from and maybe even enjoy math classes that were specifically teaching math skills alongside their real-world application.
Have a financial literacy class, a mechanics class that merges physics and math, a statistics class combined with media literacy, a computer science discrete math course. Math is a huge topic! There's lots you could do. One of my favorite college classes was a math-heavy physics class about music, breaking down how sound waves function and interact to produce something more than noise.
Of course, the problem is that this would be expensive and require a lot of curriculum writing.
The near-universal lack of classes on financial literacy is a great tragedy. My impression is that the creation of any such classes on a large scale would be fought tough-and-nail by the financial-services industry, which likely helps explain why they don't exist.
* "tooth" not "tough": I think I have to figure out how to turn off autocorrect ...
Parent here and my take is that it is too soon to conclude anything about the common core way of teaching math, but even though it was disorienting, it did seem like it was preparing from the beginning to think about math more abstractly and deeply -- and while Freddie's point about parents helping is relevant, in my experience Khan Academy is really good at this stuff and can be a good substitute for a parent assuming wifi and computer access at home.
Who are these experts arguing that math is just arithmetic and how many of them actually have PhDs in mathematics?
Might have put this inartfully - the point is not that math is just arithmetic. The point is that there isn't a meaningful dividing line between quantitative reasoning and mathematical computation, that they are actually different ways of looking at what are in fact the same skills, and that thinking in a binary way creates problems like the ones described here (ie parents annoyed that process is elevated above results) and also doesn't facilitate transition into higher-order math because these distinctions collapse as things get more complex (ie calculus is often seen as another level of difficulty because the problems inherently contain heavy doses of what some would call both quantitative reasoning and calculation, although I never took calculus so I don't really know what I'm talking about).
I certainly don't want to ascribe skepticism about the existence/importance of numbers sense to him, but the math textbook author Caleb Gattegno was an example of someone whose approach did not emphasize numbers sense, although he died I think in 1988 so he preceded the current yen for it. Perhaps I should have emphasized instead that there are a lot of practitioners who think that time spent on number sense detracts from the kind of computational practice that, some feel, eventually leads to abstract understanding. Certainly the much-ballyhooed success of East Asian approaches to math, with their heavy emphasis on grinding endless numbers of equations, would be a point in the favor of a computation-heavy approach.
Here's a paper on the subject of numbers sense, although I can't say it provides strong evidence for either position: https://files.eric.ed.gov/fulltext/EJ1069023.pdf The correlation between number sense and mental calculation is pretty strong for education research - .54. But that's lower than I would have thought. On the one hand, you could say that a high correlation shows that a division between number sense and mental calculation skills show is unnecessary. On the other hand, you could say that a low correlation shows that number sense does not improve the kind of raw calculating ability many people emphasize. I'm not sure how to interpret the results myself.
Ah, I see. First and foremost, it’s absolutely true that you need arithmetic ability to perform calculus and any other higher math. It’s also true that without intuition, you will never be able to do any form of research or apply the math in novel situations. Also, I suspect both abilities are highly correlated and it may just be easier to teach arithmetic. Certainly, arithmetic is a less abstract skill that can be learned more easily with rote practice.
But I also don’t understand why we teach algebra or calculus in schools as anything but a elective for would be engineers and physicists. If it were up to me the standard curriculum would be application-focused statistics and probably classes.
Facility with some basic algebra is pretty much essential for learning stats as more than a set of rules to follow in specific situations. That doesn't mean you need to even solve quadratics or factorize polynomials, but you have to be able to translate simple formulas between different forms. E.g. if the formulas for odds in terms of probability and for probability in terms of odds seems like separate things to be memorized, it's just too hard to get much further.
To go on past the first course, you generally have to understand what a probability density is and what its integral means. That requires almost no facility with calculus, but does require a comfort with the basic terms.
My claim is based on my wife's experience teaching several tens of thousands of students beginning stats.
Oh absolutely. I was referring to things like polynomials when I mentioned algebra, but you definitely need the basics of abstract reasoning.
"How's your Polack-says-what index? Thanks Kowalski." --Michael Scott --me, who has no facility with calculus or comfort with the basic terms
I salute you people and your miracle brains.
Also I feel compelled to note that I'm being sincere. People whose brains work this way are a marvel to me. But nothing on the internet is sincere, so now I have to say it so you don't think I'm trolling. Ugh.
Our brains are no more miraculous than anyone else's, it's just a particular set of mental skills that we happen to be good at. I'm terrible at plenty of other things like hand-eye coordination, languages, memorization, emotional intelligence (though I got a lot better in my 40s), patience, and a thousand other things that are at least as valuable to society.
You don't need "arithmetic ability" — or at least, not outstanding arithmetic ability — to ace calculus and higher subjects. Or to become an Ivy-League math prof, provided you picked the right specialty. Better arithmetic makes you more efficient at higher mathematics, so you'll hurt for not having it — but maybe not as much as the guy who's a computational whiz but deficient in geometric intuition.
It's best to have the whole package, obviously. If you don't quite have the whole package, though, you can still succeed, though with more hassle, if the parts of the package you do have are exceptional.
I've seen waitstaff laugh at the poor mathematician who's "too drunk" to calculate the tip quickly — the poor, completely sober, mathematician. Ordinary people's expectations of the computational proficiency of mathematicians can be inaccurately high.
I was first taught to believe my less-than-prodigious computational speed marked me as more mathematically deficient than it did. I gradually got faster at arithmetic after high school with practice, but only after playing to strengths I already had.
Maybe I should elaborate more. I am about 3 weeks away from having a PhD in Computer Science, and I can say that the idea that math is first and foremost about arithmetic is patently absurd, and it’s a rigidity in this belief by educators who I quite frankly believe could never perform graduate level higher math that causes so many problems with math education.
Math is first and foremost a language for describing structure. Understanding things like “x squared is the area of a square with width x” and more importantly understanding why that is true is the only way you can ever succeed at math. Equations are really just sentences in a language used to describe structure.
There's a famous result demonstrating that all mathematics is 'equivalent' to number theory and number theory is, in some very real sense, 'just' arithmetic. (Tho _doing_ number theory is very very different than performing an arithmetic calculation.)
That was a charitable interpretation of that claim that occurred to me anyways.
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.)
I haven't read it! Sounds interesting.
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've never understood why frequentist methods still are used so extensively. As far as I can tell it is due to a false belief that prior-free conclusions are possible, but there is always a hidden prior in any frequentist calculation. Much better to bring the prior to the front so that everyone can see it.
After I think 5? classes in traditional frequentist statistics (mostly regression and ANOVA etc.) I was signed up to take a Bayes class, but I told my stats prof and he said that Bayes requires calculus, so I dropped the class. And then I met the professor who taught the Bayes class and he told me that I would have been fine without a calculus background. Quite a regret of mine.
I agree that you would have been fine without a calculus background. The basic ideas of calculus (what a "derivative" is and what an "integral" is) are straightforward, the "hard" parts are evaluating these quantities in various situations, which is most of what you learn in a calculus class.
Largely it's from inertia. E.g. the medical residents who I help with stats have to read many NEJM etc. papers that use exclusively frequentist stats. So people wanting to understand the publications need to know them. Then they use them.
But to be fair, frequentist stats are in principle prior-free. It's just that they don't answer the questions we need answered, so the interpretations usually sneak in priors.
Exactly: "they don't answer the questions we need answered", and so are useless. Then, since no one wants to do something useless, there is inevitably a bait-and-switch where a prior is snuck in, but in a way that makes it difficult to see what it was.
I hate this *so* much. Why the field perpetuates this nonsense generation after generation is utterly beyond me.
Again, TBF, as Sander Greenland has pointed out, frequentist stats are good for evaluating whether a specific prediction has failed. There aren't good for initial explorations, for which they are unfortunately routinely used.
>"frequentist stats are good for evaluating whether a specific prediction has failed"
I don't agree. To decide whether a prediction has "failed", there has to be an alternative, and there has to be a prior on the set of alternatives. Example: I predict that a coin is fair. I flip it 100 times. It comes up heads every time. Has my prediction "failed"? Remember that, if the coin is fair, any particular sequence of heads and tails is equally likely, so all heads is no more or less likely than any other particular sequence, and so cannot be taken as evidence that the coin is not fair!
Bottom line, there is always always always a hidden prior somewhere, or there is no conclusion at all.
"Remember that, if the coin is fair, any particular sequence of heads and tails is equally likely, " Um, I did remember that. When I don't remember that, it'll be time to just put me away.
"so all heads is no more or less likely than any other particular sequence, and so cannot be taken as evidence that the coin is not fair!" Your argument that not even 100 straight heads can be "be taken as evidence that the coin is not fair!" is exactly why people distrust Bayesian reasoning.
You might also be interested in some curse-of-dimensionality problems for which Bayesian estimators don't converge but frequentist estimators do.
https://normaldeviate.wordpress.com/2012/08/28/robins-and-wasserman-respond-to-a-nobel-prize-winner/
There are always hidden priors. But, once you admit there are, it can be tempting to use the fact that there will always be priors as an excuse to avoid updating your priors when you should: there's no clear distinction between confirmation bias and Bayesian rationality.
A moral commitment of trying to draw prior-free conclusions, even if impossible, is the sort of moral commitment that appeals to the mathematically inclined. That same kind of moral commitment *should* cause distress in those who realize the moral commitment is impossible to achieve, and the pretense we can is a lie. But I also understand the worry that, if we drop the pretense, we give sophists the excuse to get away with as much cognitive bias as they can: "I will always claim my priors are 'just strong enough' to justify my refusal to update my beliefs in light of the evidence you present me."
For example, Bryan Caplan argues his priors permit him to be unpersuaded by studies that don't find a disemployment effect from higher minimum wages. I think he makes a good point — only it's exactly the sort of point someone looking for an excuse to ignore empirical studies on this matter *would* make, which, even if you agree Caplan's approach is probably correct, should be pretty disturbing:
https://www.econlib.org/archives/2013/03/the_vice_of_sel.html
"Part of the reason is admittedly my strong prior. In the absence of any specific empirical evidence, I am 99%+ sure that a randomly selected demand curve will have a negative slope. I hew to this prior even in cases – like demand for illegal drugs or illegal immigration – where a downward-sloping demand curve is ideologically inconvenient for me."
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).
And they should be!
Yeah, I do not get the resistance to the boxes--I could never do mental math until someone explained the box method to me, and then it felt so stupidly simple I wondered how I hadn't figured it out myself.
This is actually the first time I've seen the boxes. It's just the method I reached for instinctively when doing mental multiplication.
I think the resistance is very understandable – it's unfamiliar, and most people don't have an intuitive understanding of the algorithms they learned either.
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".
Haha! Complex numbers are *much* easier to understand as permitting scalars to rotate, rather than as a ring, or so sez I. The ordered-pair notation is clumsy, the polar form, more intuitive. (Or so sez I.)
I did have one outstanding math teacher in high school whose solution to the "Sexual Healing" problem would be to return the next day with a math parody to belt out, "Conceptual Feeling!", if it happened again.
Yeah, I taught a couple of kids via introducing i and then mapping multiplications to rotations. It was sort of a side project to doing some group theory, which I chose to avoid making school more boring because it was far from the curriculum.
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.
"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')."
Right, but have you ever had really bad music lessons? I did, once, from an elderly family friend. She assigned technical exercises far in advance of the actual pieces she assigned pupils, and had a knack for picking pieces that failed to hold her pupils' interest. A better music teacher (and I had better ones later) is better at pairing real music and technical exercises to reveal the technical exercises have a point. It's more intrinsically motivating to do that endless repetition of your own accord when it clearly benefits the music you want to do.
I've taught STEM subjects before in underserved public schools. Part of that teaching included teaching test prep, and, at the time, I was stumped by how little even very smart students were motivated to pay any attention to test-taking skills that could easily improve their scores. They didn't trust that their scores could be improved with practice, or that, even if they could be, that those scores could matter to them. At the time, I didn't see the connection between their distrust and my childhood distrust that Hanon's virtuoso exercises were necessary to master introductory variations on "Mary had a little lamb". I had the nice, suburban background teaching me from the cradle that test-taking is an important skill. My music teacher, a former virtuoso herself, couldn't even remember a time when the advantages of mastering Hanon's exercises wouldn't have been obvious. Hindsight is 20/20, and it's a bit tough to blame inexperienced children for not having the foresight to trust their teachers that grinding, boring exercises with no cognitive reward beyond teacher approval will pay off someday.
To be fair, I'm not aware of any math instruction in the US that doesn't *try* to pair rote drills with the cognitive reward of higher-order thinking. But I do suspect that teachers can be either like my ex-virtuoso (but actually pretty bad) music teacher in making that pairing persuasive, or be bad enough at the higher-order thinking themselves that they also fail to make a persuasive pairing.
I think you may be projecting an adult attitude toward memorization onto children. My three year old likes learning new words and is excited by memorization, but trying to get him to think abstractly only makes him frustrated. And the baby absolutely loves practicing his counting over and over and over again. My sense is that kids really enjoy working on problems that match their current developmental stage.
Even the struggling fourth graders I tutored didn't dislike multiplication drills to nearly the same degree that they hated higher level skills such as story problems and fractions. It is the teachers and parents who get bored with repetition, not the kids.
I can see why you might say that, but I'm remembering what I found terrifying about music and math instruction when I was in elementary school. I remember the scared kid I was in the face of the speed tests they used to measure rote skill, and the shame of it.
On the other hand, I loved story problems and fractions in grade school — they were math I was effortlessly good at (but also the part least likely to count for credit). I was good enough at music despite lousy technical skills to make up stuff on the piano grownups thought I must have learned in lessons.
I would not wish my childhood terror of speed drills, mathematical or musical, on anyone. There's no advantage to doing speed drills poorly, and good advantage to doing them well. But does help to measure early excellence much more by rote skills than later excellence will be? Sure, I *liked* memorizing the multiplication table, but goodness at multiplication was still measured by... speed tests, and by that metric I was a failure.
For whatever reason, I did get bored with repetition unmoored from a larger, "more beautiful" structure. Bored, therefore inattentive, then scared by the "chaos", the "unmusicality" (literally, of Hanon exercises). I was bewildered by whiz kids who could blaze through speed tests, play pieces whose purpose wasn't musical expression, but showing off virtuosic speed. How could they make sense of it?
Later, after high school, on another instrument, I began mastering virtuosic passages at high tempos, but they were from real pieces, in order to *do* real pieces. I'm still the opposite of a great performer, but the few times I sprang for lessons from those who were, they recommended drilling by abstracting from real pieces. It helped. It made regular, lengthy drilling *possible* for me. Too late to change my life, but enough to bring mastery of a few things.
Erin Ethridge wrote in these comments, "some of the basics being taught Common Core-style are very helpful for people who 1) aren't fans of memorization or 2) need to understand 'why' before the 'how' makes sense. I'm one of those people." Some of us were these people as children.
Kids enjoy problems matching their developmental stage, but not all kids develop the same. Freddie laid out some pros and cons of making "number sense" an early goal of math ed, versus what's derided, unfairly, as "drill and kill". Freddie's theme of educating for skill, rather than generic "smarts", suggests children be given more chance to play to their strengths. For the many who won't need higher math, drilling may make them happy while higher demands make them unhappy, and they deserve better than years of torture over something they'll never use. That's fine. But hostility toward "drill and kill" didn't come from nowhere.
I'm going to make the hypothesis that kids who need to understand the 'why' before they're comfortable with the 'how' are a) a minority overall and b) generally the ones who will go on to do better anyway. A lot of higher ed is (or perhaps used to be) geared towards this kind of person. I'm one of them myself, and it took me a lot of time to understand that not everyone thinks like that; once I did I found that I "connected" a lot better with lots of my pupils. H Ann's "loves memorisation, gets frustrated by abstraction" seems to be the default human state of being, it's probably possible to some extent to train people out of this but starting too early is one of many ways of putting people off school for life.
I don't believe a difference between the demographics in a particular field and the demographics of the general population need be a sign of injustice, but "will go on to do better anyway" is something that works better for the family life of men than it does for women.
If we're trying to make STEM careers more feasible for women without lowering standards for them, one way may be to waste less of talented girls' time in earlier grades, supposing they'll catch up later anyhow. Yeah, bright kids often do, in time to make the choice between career and childbearing, a fraught choice even under good circumstances, more fraught.
The world will hardly end if talent identification, originally geared toward men, considers equality between the sexes achieved by putting girls on boys' timetable. It isn't ending now. Families with resources for extracurricular enrichment can already accelerate their daughters' timetables privately, anyhow. And, if it became popular to try acting on the belief that "they'll catch up eventually" is unfairly harder on girls than on boys, we'd have another "they're leaving our boys behind!!!" panic, to boot.
Selection effects doubtless explain much success of early childhood programs advertising something "better than" rote learning. From the extracurricular Russian School of Mathematics' advertising: "Young children are naturally curious, uninhibited, and can easily grasp very complex ideas. Our curriculum and methodology are built with this in mind. Our students learn to work with, and develop an appetite for, challenging mathematical concepts." Montessori schools use similar rhetoric. Perhaps the greatest accomplishment of these programs is to salve the conscience of the better-off parents who use them, so they can believe it wasn't SES, but the enrichment they bought with their SES, that explains their children's later successes. On the other hand, the RSM school around here seems to attract immigrant families of modest means. Anyhow, parents don't pay for these programs to immiserate their children and put them off school for life, but rather the opposite. The minority of children flourishing rather than crushed by this approach seems substantial.
Bertrand Russell can complain, “I was made to learn by heart: ‘The square of the sum of two numbers is equal to the sum of their squares increased by twice their product.’ I had not the vaguest idea what this meant and when I could not remember the words, my tutor threw the book at my head, which did not stimulate my intellect in any way," catch up later, burn through several marriages, and still be siring children at 65 (Conrad Russell). He was also a governess-hiring aristocrat. Those are hard conditions for American motherhood to meet.
"The bright ones catch up anyhow" may make equalitarian parenting for young fathers less appealing, too. "Sorry, can't parent now, I'm catching up anyhow."
That "they'll catch up anyhow" seems to make balancing talent and motherhood unnecessarily hard is a first-world problem, maybe the priority of least concern in providing American children a fair, non-miserable education. But since I've seen it happen, I'll say it happens.
I've taught math-heavy college STEM classes for three+ decades, and I completely agree.