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Comment by rdtsc | Hacker News Reader

> Why are you advocating memorization?

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.

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Memorizing the multiplication table is a useless waste of time and I recognized as much when they tried to make me do it and I refused. Nobody needs to know 8x7 off the top of their head. What's really important is estimation techniques to tell if something passes a sanity check, like reducing things to powers of 10/etc and seeing if they're close to the result you got on a calculator, like how 123x456 should be close to 50000 (100x500)

How in the world are you supposed to estimate products of large numbers if you don't know the products of small ones?

You'll pick up on a few as you go along and that's all you need.

So, you'll memorize them.

A few that you need, on an ad-hoc basis, instead of front-loading a bunch of calculations you're probably never going to use.

Memorizing every combo of 2-9 x 2-9 is only 36 numbers. That doesn't seem like a big waste of time to me.

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.

> Memorizing the multiplication table is a useless waste of time and I recognized as much when they tried to make me do it and I refused. Nobody needs to know 8x7 off the top of their head. )

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.

Except memorizing multiplied numbers is not foundational knowledge for any higher math, knowing what the concept means is. Learning this involves working through some examples but it doesn't mean you need to know an entire table. Not knowing 7x8 never held me back in higher math/math-relevant classes like Calculus I-III, Discrete Math, Linear Algebra, Physics I-II, Chemistry, Macroeconomics, Computer Science (anything from basic programming to language processing or algorithm proof), Music Theory. Had a fun and mostly easy time in all those. Once in a blue moon on a test I'd have to do a multiplication and count it out (8x5, add 5 8 times) because it was a product in a derivative or something and that class had an (unrealistic for the real world) no-calculator policy. In fact I'm pretty sure mathematicians are notorious for poor arithmetic.

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

Do the memorisation by spaced repetition and make it competitive, and it takes almost no time to learn them. Of course if the kids begrudge it, they’ll hate it and fail to work at it.

> Except memorizing multiplied numbers is not foundational knowledge for any higher math

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.

This is exactly what I tell my kids right now as they learn the tables by rote: knowing the answers instantly by drawing from your subconscious makes it vastly easier to solve harder problems later on.

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?

Higher math isn't harder arithmetic, it's increasingly more abstract and sometimes doesn't even involve numbers. A student with their times tables down isn't going to do any better in a calculus course than one who relies on a calculator.

Statistics, on the other hand...

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

What about statistics exactly? Bayes' theorem etc are enlightening and the numbers never gave me a problem.

I don't believe you're correct. As my 2nd & 4th graders' teachers tell them, they need to learn a set of level-appropriate "math facts" ... but they're not forced to start by memorizing those facts (my son, for example, has a printed multiplication table up to 15x15 to reference while practicing decomposition of 3 & 4 digit multiplication problems). I truly appreciate the Common Core principle of focusing on teaching methods over facts, and -- at least in my children -- it is apparent that it sets them up for problem solving success IRL. YMMV. It's not a perfect curriculum and is still tailored to the lowest common denominator student, but I do think there's value there. And again, in a classroom with a strong teacher who can take time for differentiated instruction, there are ample opportunities for advanced students to go beyond or to practice their learned skills by helping their classmates.

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).

> Higher math isn't harder arithmetic, it's increasingly more abstract and sometimes doesn't even involve numbers. A student with their times tables down isn't going to do any better in a calculus course than one who relies on a calculator.

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.

A student with their times tables down isn't going to do any better in a calculus course than one who relies on a calculator.

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).

Such a math course is being taught wrong if passing a test relies on churning through a million arithmetic steps instead of demonstrating understanding. I did fine in my course. Had to use the technique of mentally placing objects around a familiar path to memorize a silly list of matrix properties though.

The computational part was only 60% of the exams. The rest was proofs. For most of us, we needed every mark we could get in the computations because the proofs were really hard.

> Higher math isn't harder arithmetic

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?

I do mental math all the time. Nobody needs to memorize tons of math but it's laughable that you would take that to such an extreme that you can't do basic multiplication without a calculator.

Can you give the example of the last mental math calculation you did? I almost never do it and I still almost never did it even when I was in college taking higher math. Tip calculation is all I can think of.

I work in big data/cloud stuff so I do estimates all the time to figure out what size resources I need/how long something will take. Less than 5 minutes ago I did mental math to see how long a job would take (I knew how long it would take for one read, and I had 2 days of reads over 3 data sources each with 8 data partitions => 48 reads each taking about 90 seconds => about 1.25 hours).

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/$.

I do rough division all the time in my head, for instance figuring out how much each can of beer costs if a slab of 24 is $60. Or on the flip side, how much it's going to cost me to get a round of 4 beers at $8 each.

It's just a general life skill.

I can be claustrophobic..I don’t like being in a closed car..so when I am at traffic stops..I convert alphabets to numbers and numbers to alphabets of the license plates..and sometimes I add the numbers together to check if it’s a prime number. Calms me the fuck down.

in farming..construction..cooking..I use it all the time in my everyday life.

I may chime in with another story. I‘m teaching operating systems at university and use these skills (for small numbers, and written multiplication/division for slightly larger ones) all the time. My general consensus after overseeing a few hundred students is that those that are unable to quickly do simple math in their head also struggle with the rest of the curriculum. Especially if they need a calculator for computing 57:2 everybody gets distracted and we have like 2 minutes idle time while everybody starts entering the equation. I’d rather spend my time teaching actual os stuff, it’s a shame enough that I have to teach how to decimal <-> binary <-> hexadecimal base conversion.

You may be a bit embarrassed when you get older and a medical practitioner gives you a cognition test. Start with 100 and repeatedly subtract seven, announcing the results.

I'm certain they have different tests for people who don't have such arithmetic down.

Actually, that's on a standard cognition test. Other questions include identify pictures of animals, "what date is today", "Remember these five words (and get asked them at the end of the test), an "a is to b as c is to" question, etc. The assumption is that people over 50 in the USA can do simple arithmetic, because they all went to elementary school. Sadly, that assumption may have to change.

The opposite extreme is when a smart child cannot solve a math problem, because despite having interest and some cool insights, and making a few hopeful steps, they predictably fail at one of the "2 + 3 = ?" steps.

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.

I had a hard time getting in the habit of learning French but I adapted to it when I had to. However, as I've mentioned in another comment, shoddy arithmetic in particular was never an issue for me in real math contexts.

I agree. That’s why there I some truth to the stereotype of Chinese kids being better in Math.

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.

I sort of both agree and disagree with you.

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.

Addendum: I highly recommend this reading material, A Mathematician's Lament: https://www.maa.org/external_archive/devlin/LockhartsLament....

The example of approximating 123 x 456 with 100 x 500 tends to undermine your point a little. It's too simplistic and the answer isn't even within 10%.

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).

Although I tend to mostly agree with your main argument, the example you presented ( memorizing multiplication tables ) is a bad one. My mind was blown when I found out how they teach multiplication to children in Japan. Link: ( Vedic method ) https://www.youtube.com/watch?v=z4X98Mnj0tc The superiority of the teaching method shows to be significant when compared to other teaching methods elsewhere and even remains noticeable when children encounter higher levels of math.

This is similar to the "Lattice Method" that they taught (still teach?) here in the US. I can't stand it. The students simply learn an algorithm, without developing a sense of place value and an intuition for "how numbers work." Sure, they can quickly multiply large numbers and it's a cool "party trick." But they aren't learning math. Might as well use an iPhone.

Fair point that memorizing an algorithm doesn't necessarily advance the student's intuition for how numbers work, but keep in mind that you're replying on a subthread about how learning a general algorithm is an improvement over memorizing multiplication tables.

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.

I don't think it is better to learn some shortcut algo at a young age. Memorizing your times tables up to say 10 * 10 or 12 * 12 is fundamental to understanding multiplication. Remember, the brain is not a turing machine, it's an associative memory machine. Facts are the foundation of knowledge.

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

I agree with you. To do multiplication by any non-parlor-trick method you have to be able to multiply single-digit numbers. If you don't memorize the table then every time you do larger numbers the process becomes that much more of a chore.

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.

> The superiority of the teaching method shows to be significant when compared to other teaching methods elsewhere and even remains noticeable when children encounter higher levels of math.

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.

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