I think it's very similar in education. There are vast swaths of the education system that are doing absolutely terrible. Much of the problems are outside of school administrator's control (broken families, gangs, poverty, etc), so they reach for "process improvements" and hope that makes a difference.

> [a study looked at] the relationship between eighth grade enrollment [in Algebra] and National Assessment of Educational Progress math exam scores at the state level. In short, they didn't find one... [In a different study], they found that early exposure was actually associated with a net decrease in average student math score exams within a given district.

Maybe we're looking at the wrong metrics. Obviously introducing the average kid, whose average abilities change much more slowly than "process improvements," would have a decrease in today's math exam.

Maybe there's something wrong with the exam! And not in the sense of what questions it asks, but more like, who the hell cares?

I know what the answer is, I'm not stupid. But eventually, we'll have to square away the multitudes sitting inside of parent's brains: that they want their kids to actually learn something, while simultaneously having the highest scores / grades / metrics possible. The two cannot be true and honest at the same time.

But the tension isn't learning versus evaluation. It's honest versus dishonest. Under the early algebra scheme, you get a more honest evaluation (students perform worse at harder, more realistic material) and more learning (they're not rehearsing stuff that's easy for them).

The real dynamic is that no one is going to enroll their kids in a school with low test scores.

0: https://www.businessinsider.com/poorly-timed-headquarters-20...

In tech, we have many examples: Agile, Ruby on Rails, React, RISC, OOP, microservices, and so on and so on. All have some real benefits, but also have or have had huge hype cycles.

You can make the same argument for most other industries (if not all of them). And then there are things like diets, religions, etc.

Nope. But businesses fail when they succumb to fads, while government just raises taxes.

I wasn't particularly smart growing up, but my parents always kept me up to date with the math and science curriculum they had at their respective ages in their own country. At least up until the end of middle school, the difference was astounding and had a dramatic effect on my academic (and thus future) success.

My grandfather never advanced beyond trigonometry in high school, but went on to become a moderately famous theoretical physicist.

So if we're looking at anecdotes, it's hard to form a clear conclusion. Today I teach high school mathematics and I see accelerated kids who fall to pieces their junior year, and also kids like me who arrive in the 9th grade knowing nothing, but somehow end up in an accelerated track by the time they graduate.

In my case I started getting serious in grade 10, entered accelerated track for grade 11 & 12. Unfortunately I sorta peaked in grade 11 and started coasting in grade 12. I probably shouldn't have gotten serious until grade 11.

I wonder how many people grow up hating math, like I did in elementary school, because they just don't like how rote arithmetic is and how trite simple word problems are because they steer away from algebra. It wasn't until I got to Algebra that I really, really started to love math.

It is vital not just to teach to the curriculum, but to teach to the student.

And that's where a lot of modern education falls apart.

It's not just algebra though. History. English (from literature to grammar). We're just failing to connect.

Conveying that there's a deeper conceptual "mathematical thinking" seems much more important than memorizing the algorithm for "completing the square" or whatever.

Because it's something kids do really well and it's something to take advantage of when learning at that age. In other words, it's better to memorize the multiplication table then find insights into how it work later, and do proofs and make connections with calculating an area of a rectangle etc. A lot of the tricks and insights don't mean anything to the children if they didn't already do many examples the rote, repetitive way and in a lot of cases the "insights" are often a distraction as well. Those should come later. Even more interesting is when children see or discover these tricks or rules on their own, then they become really memorable to them.

I don't have many samples work with, but I have observed this over the years based on my own experience, my kids and my extended family members and I have noticed the same patterns.

Me too! I thought I was so smart as a kid, realizing that stuff I didn't want to do was useless.

However, I was wrong. An over-dependence on calculators due to a lack of arithmetic fluency has sabotaged all my encounters with mathematics ever since. I had a hard time understanding and developing any kind of fluency because I was forced to push everything through a black box that also distracted me.

I now realize that memorized facts are the foundation for knowledge and advanced thinking. Dismissing memorization, especially of fundamental facts, is like choosing not to use RAM because you can always swap to disk.

Making it competitive between the two of the kids helps too... after all, sports is “just” being better at getting the ball in the goal better than the other team, right?

That said, I think your statement is absolutely incorrect in the context of higher math (and experimental science). The more facts you know -- whether literal facts, axioms, proofs, applied example or theories -- the more facile problem solving will be. This holds true in all disciplines (and not just math & hard sciences, but also engineering, social sciences & business).

A linear algebra course, on the other hand, is going to make mincemeat out of any student who can't do basic arithmetic in their head at high speed. Gaussian elimination, matrix multiplication, determinants, diagonalization; all of these tasks are extremely arithmetic-intensive. If someone needs a calculator to multiply 8x6 then they are not going to be able to solve a linear system in 4 unknowns on an exam that disallows calculators (which is all math courses at my university, besides stats and act sci).

IMHO, that's wrong in many cases. For instance: there's arithmetic in algebraic manipulations, and a lot of that is in the times-table memorization range. I definitely had to rely on my calculator in calculus class (and specifically swapped professors to one that would allow the use of one for that reason).

There's also the fact that neglecting memorization as a "waste of time" at the start of your math career sets up a bad precedent and bad habits that will continue throughout it, barring substantial corrective effort.

I, personally, only stopped having problems due to my deficit when I got to discrete math, which was the one math class I took that didn't involve numbers, like you said.

That's narrowing the argument I made, I said "harder problems", of which higher math is just one aspect.

Even sticking to just higher math; you're going straight from arithmetic to... calculus? What about algebra, trig, geometry; all much easier when your arithmetic chops are blazing?

(Which is what most people ought to be taking anyway instead of calculus.)

Additionally to your point, I think forcing arithmetic memorization on students is harmful because it makes math seem boring and hard.

But my point wasn't that it's foundation in general, category theory, set theory, etc are foundational, but that it's important to start from it. Just because something is foundational doesn't mean it's a good starting point.

> it doesn't mean you need to know an entire table.

Kids at an young age are usually very good at memorizing things and once they have memorized then they can build on that and learn new tricks, relationships, abstract and foundational principles.

> Not knowing 7x8 never held me back in higher math/math-relevant classes like Calculus I-III,

Neither did me either. Embarrassed to say, I had forgotten a lot of it. But I would have been struggling early on in the 4-9th grades if I didn't know how to add or multiply numbers then. I would have been wasting time on tests doing it the slow way, instead of thinking of more interesting problems.

So to summarized, yes, it is not foundational but it all depends on age, it's better to take advantage of what the brain already knows how to do well at that particular state in time.

> In fact I'm pretty sure mathematicians are notorious for poor arithmetic.

Could be but if they are poor they would be left behind in early grades and might never becomes the mathematicians they are. There is also a bit survivorship bias in the sense that if there mathematicians who are poor ar arithmetic, they'd stand out and become memorable, but those that are good are not noticeable because they are expected to be good with numbers.

Also, smart kids who hate memorizing are often really bad at languages. Which are mostly memorization; there is no way to derive the language from the first principles. But I guess if one's first language is English, this is not a big problem.

I get it: memorizing stuff doesn't signal how smart you are. But at some moment, the lack of factual knowledge is going to overcome the advantage of being smart. You will start making logically correct conclusions from factually wrong premises, and you will be proud that you didn't fill your brain with garbage.

Mandarin has 5 tones and having a keen ear to identify each tone and then recognizing it and memorizing it and matching sound to meaning is an enormous advantage.

Also...teaching music..to recognize diff sounds/tones..musical notes(Bach would do) also gives mandarain/language/math students an edge.

If you do common ones like 2, 3, 5, and squares, that already covers most of them and you only have 10 left to remember them all.

Being able to do mental arithmetic for small numbers is actually a very useful life skill. Sure, everyone has a calculator in their pocket these days (despite what my maths teacher told me), but that overhead of having to pull your phone out does slow you down.

But I don't believe that rote learning your times tables up to 12 is the way to do it. Mental arithmetic should be incorporated into the curriculum in a more integrated way. The problem is that school systems love testing and grading people. And it's quick and easy to grade people on their times tables by giving them a speed test to see how many equations they can complete in 5 minutes. You can say that little Jimbo is doing his times tables at a third grade level because he knows his 2, 3, 5 and 10 times tables. It's a terrible method of testing student progress though, I managed to get all the way to high school and was in the top stream class and never learned my times tables.

The actual concept of multiplication tables is an important concept for children to learn though, as it does help solidify fundamental mathematical concepts.

Tip is another thing. And I like to "min/max" grocery shopping so I typically do mental math when shopping to maximize things like grams of protein/$.

It's just a general life skill.

120 x 500 gives a much better approximation, and just needs you to know 12 x 5 = 60. Better still would be 125 x 460, which is 1/8 x 46 x 100000. That's within 3% of the true answer, and all it takes is quickly dividing 46 by 8.

It's very helpful to be on good terms with numbers less than about 100, which is why the multiplication table is taught. It's useful for reverse calculations (like 46 / 8) as well as forward ones. Decimal equivalents (1/8 = 0.125) are worth learning as well.

The key to being able to quickly approximate things is to populate your mind with memorized values. They work as landmarks (so the 7x8 example, in conjunction with the decimal equivalent of 1/8, means that you know almost immediately that, e.g. 55 x 125 is close to 7000).

Ideally, we teach children the intuition for mathematics. Just memorizing arithmetic algorithms isn't ideal, but it's surely better than memorizing finite arithmetic tables.

First memorize the times tables. Then learn the long method. Then learn the shortcuts. IMO, of course.

Now, that is not to say that you should forego understanding multiplication. But forcing student to memorize the basics (like forcing them to memorize verb forms) (1) makes their life easier later on and (2) gives them the chance to spot patterns themselves.

I was mainly talking about addition and multiplication up to 10x10. Looking through all the tricks and hacks they teach the kids in an attempt to simplify ideas seems very wasteful as the brain at that age is so good at memorizing things. ut the schools don't seem to be taking advantage of that.

They think like adults in the sense of "oh let's introduce this abstract idea and prove it and derive the specific examples from it". Yes, a college kid or high school would do well there, but in earlier grades I haven't seen it work. It's a bit like learning a language. An adult might do well to start learning about grammar rules, tenses, cases etc, a kid might do better hearing a lot of example and deriving the rules on their own. Later they might find out that there are 8 tenses, 4 cases, predicates, adjectives etc.

To perform well at algebra, you can't be stopping to punch numbers into a calculator. You can't be constantly making mistakes. The slowness and errors will make you stumble and lose your train of thought, preventing you from reaching any deeper conceptual "mathematical thinking".

Those old-fashioned methods for doing math are reliable and general. If you follow the rules, you get the answer.

> the same knowledge will come naturally as students learn other aspects of math

The key difference here is that being slow at typing for twenty years didn't discourage you from typing. Being slow at math can discourage kids from doing math, and the "twenty years of regular practice later" moment will never arrive.

To bring it back to the article: It's similar with a lot of things - you sort of need to memorize aspects of it to do it fluently. As part of calculus, you sort of need to know how to do polynomial factorization. Intuition is great and helpful, but you also need to be able to proceed methodically in order to produce or understand proofs. Scales on piano are boring, but you don't just tell kids to feel like they're Bach and they start playing Goldberg Variations. Why on earth we think the intuition is not only necessary but sufficient is unfathomable to me.

Practice/muscle memory in other fields, like the physical sciences, can not come from a page out of the book, since it requires interacting with nature in a multi-sensory way.

Although, this isn't entirely true, I remember doing proofs in Geometry class in Jr High School. So I was at least introduced to it, though we never really used it again.

People who despise rot learning are either misunderstanding it or are simply lazy.

In other words, you can only push the mathematical thinking as far as the developmental stage of the child allows you to. Memorization helps though.

My son is 6. He doesn't know the multiplication algorithm, but he does know that 2 x 2 = 4 or that 3 x 4 = 12. That's sufficient for him to do discoveries of his own. 3 x 4 = 3 x (2 x 2) ! Picture that!

[1] https://en.wikipedia.org/wiki/Piaget's_theory_of_cognitive_d...

-John Von Neumann

[1] http://www.corestandards.org/Math/Content/NBT/

The issue is one of cognitive load. You can't get to mathematical thinking and appeal to both a higher-level "intuition" or "key idea" if you aren't fluent in the details. Else it becomes hand-waving that doesn't cash out to proof-ready reasoning. Likewise, you can't internalize the details if you don't have a principle which compresses them and lets you unpack them appropriately for the circumstance of the problem. This is fair, but first there has to be something to compress.

The question is, "How much detail can be advanced at this point in the student's growth?" If the key issue is to dispel the notion that mathematics is arcane, you should narrow down the scale of the problem so that you can quickly see how the details contribute to generalizations, and how a procedure can be taken from it. This achieves what you are looking for with less effort and less risk.

With that point of view illustrated with a simpler case, it can be developed or built upon with bigger cases. Bigger cases inevitably means more details to internalize. This protracts the phase where memorization would be appropriate. But the student, having seen how mathematics is developed in a simpler case, would understand that these details have a context that will be more adequately realized at a second or third pass of the material.

Eschewing the internalization of detail as you are purporting would encourage a sense that math is subjective, because the high-level impressions of mathematics would not have the memories of detail to back them up, allowing for more to be said, with less control.

Granted, she was also a programmer, and my grandfather was a mathematician, and both my parents were engineers - so maybe my baseline for fun and education was set differently.

I was raised by my grandparents, and I had enormous respect for my grandfather, who was the most badass human being I've ever known, though you would never know it to see him. He was very quiet, he spoke little, always watching and listening.

But he grew up in the https://en.wikipedia.org/wiki/Great_Depression and fought in https://en.wikipedia.org/wiki/Guadalcanal_Campaign , with https://en.wikipedia.org/wiki/Merrill%27s_Marauders among others.

He was in his 50s when I was born and lived with them permanently when I was 5 years old. He and my grandmother were my parents.

They were always patient and kind, and always stoic about everything. One day while camping my grandpa was sitting next to a dart board, and one of the darts I threw went crazy wild; it hit and stuck deeply into his left calf. I just stood there with my mouth open while he puffed his pipe, staring at me. After a couple of moments, he calmly pulled it out, wiped the blood off of it, and carefully tossed it back to me, saying, "Careful where you throw those, boy."

I always had a lot of respect for them, though I didn't show it. In fact, by my actions, I took them for granted.

They both died of old age when I was in my 30s, even as I was just starting to truly appreciate them.

Now, many years later, I would give almost anything to be able to spend another day with them. There's so much I'd ask, so much to learn.

In summary: spend as much time as you can talking to your grandparents while you still have them, because once you're old enough to really appreciate their value, they might be gone.

It's like giving a jogger a car. He arrives at his destination faster and easier, but didn't get any stronger.

In the Los Gatos school district my daughter completed her first year of college level calculous and some second year calculous by the time she graduated from high school. This was better than my education as I only had access to pre calculus in my high school.

As an aside - giving kids math puzzles is a much better preparation for math than memorization. The only thing I had to memorize was the multiplication table, everything else I could derive. I could always derive a formula faster than memorizing it even at a young age. I went to school in the era of "new math" where we were introduced to functions and variables in 2nd grade.

My grandsons are between 3 and 8 - so they are in this age group. They are amazing at solving problems when it leads to something they want like winning a game against their brother. Puzzles are great for teaching math to this age group.

https://www.teachwire.net/news/ever-increasing-circles-what-...

To some extent? Sometimes I feel like that's the only thing that developers do.

Some are good at it, some are not.

This curriculum is aimed at equalizing things for those who are not.

If I had to design a system that helps everyone with repetition, it would go like this:

- each concept must be explained and understood first (never progress to repetition unless you are sure the child groks it);

- afterwards, add the concept to your "spaced repetition" software (conveniently provided by the teaching institution);

- give kids some time at school when they can do the spaced repetition exercises (so it doesn't become an extra homework).

The first step avoids the kind of stupid memorization that people do when they don't have enough time, and that gives them problem (and even greater time pressure) in the future. This makes the entire thing more meaningful that current education.

The second step avoids the situations like "I felt pretty sure I understood it yesterday, but now I have no idea where to begin". This makes the entire thing more efficient.

> But don't do it in a way that makes things worse for the kids that are good at memorization.

Uhm, if you complete the spaced repetition exercise and there is still some time left, you can go play or read something. Or perhaps take a bonus lesson that is not a part of the standard curriculum.

Also, a large part of Soviet mathematical output came from people who were politically forbidden from getting an official scientific job (and then the Academy of Science was filled with people like Lysenko), and sometimes had to get a manual job. So they did math in their free time, corresponding with each other. Which, paradoxically, probably provided more freedom than they would get in an academic institution: no pressure to "publish or perish", freedom to spend years solving some obscure problem with no impressive intermediate results, true cooperation between the outcasts instead of fighting for a limited number of academic job.

A large part of Soviet mathematical progress was made in opposition to the regime. Or perhaps, to put it more mildly, the system was so hopeless that people didn't waste their time, and organized better mathematical education outside the system. Then they created things like the International Mathematical Olympiad (the first twenty years, it was an Eastern Bloc thing), various correspondence seminars, etc. But this all happened outside the official educational system.

Yup, suspicions confirmed, all around the world, no matter what, bureaucracies have a lot in common: If have something good and want to kill it off, then set up a bureaucracy to help it!!!!! That's partly a black joke; likely the US NSF and NIH and lots of private efforts in math, physical science, medical science do terrifically good things. And, no joke that the LHC is a big bureaucracy and found the Higgs Boson, a darned good thing.

Still there can be some truth to the joke. E.g., in the US I've seen some of what you said about publish or perish: While I never really wanted such an academic research career (I just wanted a JOB and to support a family and looked at my math Ph.D. as vocational education), eventually it dawned on me that actually in some fields the publication standards sufficient for publish or perish are low and that a good researcher should be able to toss out dozens of such papers, if only as side efforts, while otherwise doing the important stuff.

Bluntly when I was considering academic math research, the main obstacle, and it was SEVERE, was that I didn't quite yet have access to a computer that could run Knuth's TeX math word processing software. Bluntly the word processing for typing in the math research was MUCH more challenging than doing the math research!

There's another relevant rule: In social, political, bureaucratic, etc. systems, when want to do better, measure it, and, for that, create, design, implement, and setup measures.

Then the system responds, does well on the measures, just the measures -- games, hacks, manipulates, tricks, fools, takes advantage of weaknesses in the measures -- with harm to what was being measured! Partly another black joke but, still, with some sad truth.

With bad measures, the Soviets were not the only ones with the problem of "They pretend to pay us, and we pretend to work.".

Then we get reminded that democracy and free enterprise are the worst systems ever invented except for all the other systems that have ever been tried. Another black joke; sometimes democracy and free enterprise do really well. E.g., in a democracy, we can elect liars, crooks, incompetents, etc. but eventually the disaster becomes obvious to enough voters, and then we get some better candidates and an ELECTION to throw the rascals out -- been known to happen, sometimes slowly, but, still, happens.

I know next to nothing about the Soviet system, but from what little I know it looks like it was an astounding juxtaposition of just horrible suffering and lost opportunities, waste, e.g., on military efforts, lots of just horrible inefficiencies, yet at times some, maybe not enough, maybe with the movie remark "But do you know what it cost?", astoundingly good things -- Kolmogorov, Oistrakh in violin, Giles in piano, Garanca in voice (grew up in one of the Baltic states when they were part of the USSR), the Mariinsky Ballet, etc.

I'm trying to get my startup live on the Internet; there are some ways it could help civilization. Otherwise I can't solve all the problems. But democracy and free enterprise when they work well -- in the US from 1929 to 1942 they didn't work so well -- can be amazingly good things.