4 ms·

Memorization happens naturally through repeat exposure, and works better if this is exposure in some meaningful context rather than through cramming via flash c

Memorization happens naturally through repeat exposure, and works better if this is exposure in some meaningful context rather than through cramming via flash cards or whatever. The best kind of practice is practice that you are motivated to do because is inherently interesting. The less "rote" you can make this the better. For a primary school example, if you can solve basic arithmetic problems in service of a fun and challenging logic puzzle, that is more motivating than solving a page of arithmetic problems one after the other. More generally, while mathematics certainly requires putting in time and actually doing the work of thinking a whole lot about a variety of hard things in the service of solving hard problems, very little of that is memorization per se. > In the United States, the emphasis on understanding sometimes seems to have replaced rather than complemented older teaching methods that scientists are—and have been—telling us work with the brain’s natural process to learn complex subjects like math and science. The older teaching method also sucked. In my opinion, the single most important thing primary school math education should be teaching is how to attack and solve nontrivial word problems. Unfortunately we did not have any of that before, and still do not have any now. Cf. https://cs-web.bu.edu/faculty/gacs/toomandre-com-backup/travel/sweden05/WP-SWEDEN-NEW.pdf https://cs-web.bu.edu/faculty/gacs/toomandre-com-backup/trav...

kaitai 2y ago
It's important that we marry viewpoints when looking at what we call "meaningful". I spend a lot of time right now with a 6/7 yr old. This child does not care about meaning. This child likes to solve arithmetic problems one after another. It's motivating -- it's like a game -- the kid gets them right and gets a thrill. Word problems? Well, the math ability is outstripping the reading ability right now so 2000 x 2000 x 2000 is way more fun than reading some stupid sentence about tulips. I've spent plenty of time on math (PhD in algebraic geometry) and educating people, and for sure when I taught college freshmen and master's students I spent a lot of time challenging folks to engage their minds, spirits, and intellect. At the same time, we have to admit there is a stage of childhood where kids just love memorization and facts. Dino facts, shark facts, math facts, Pokemon facts, My Little Pony facts, whatever. Let's not force kids to reckon too much with meaning when they're in the facts for facts sake stage -- and once they've got their impressive facts list, they'll make sense of the meaning much more easily, as discussed in the article and here!
- andoando 2y ago
  I think you're highly generalizing. I hated any kind of memorization as a child for example.
- jacobolus 2y ago
  > This child likes to solve arithmetic problems one after another. Cool. If I try that with my 5 and 7 year old, they shout "boring" and run to the far side of the house. > math ability is outstripping the reading ability Presumably you can read problems aloud to the 6 year old. But also, kids can (if taught; this is by no means a necessity) learn to read much earlier than they can learn to write numbers with a pen. My two kids both could read very well by about the time they turned 4.
wizzwizz4 2y ago
> how to attack and solve nontrivial word problems. Please no. Non-trivial problems that are expressed in natural language, sure: I'm in favour. But once you get past basic arithmetic, "word problems" are just algebra obfuscated with a prefabricated template. A non-trivial problem, written in natural language, will usually admit multiple solutions: cutting that down to something you can fit into a mark scheme, without making the problem convoluted or forced, is hard. Example: > Jacob and Sally want to split a rectangular cake between them, but neither is very good at cutting. Jacob can cut precisely (wasting no cake), but will miss the middle by 10% of the distance to the edge of the cake. Sally can cut accurately (exactly in the middle), but will obliterate 10% of the cake in the process. (Neither Jacob nor Sally want to eat crumbs.) Who should cut the cake, and why? This is a fun problem, but a rubbish exam question! Word problems completely sidestep this issue by starting with the algebra, and then replacing symbols with words until it's almost prose. For example, consider the Hannah's sweets problem (solution: http://www.murderousmaths.co.uk/hsweets.htm http://www.murderousmaths.co.uk/hsweets.htm): > There are n sweets in a bag. 6 of the sweets are orange. The rest of the sweets are yellow. Hannah takes at random a sweet from the bag. She eats the sweet. Hannah then takes at random another sweet from the bag. She eats the sweet. The probability that Hannah eats two orange sweets is 1/3. What is the value of n? While nominally a "good" exam question, it only makes sense within a rigid and rigorous context that's quite alien to an untrained person's understanding of English prose. And it requires bold assumptions about the fundamental nature of statistics (see https://plato.stanford.edu/entries/statistics/ https://plato.stanford.edu/entries/statistics/) that go completely unstated. Understanding English and Maths aren't enough to answer this question: you also have to understand Maths Exams.
- jacobolus 2y ago
  > "word problems" are just algebra obfuscated with a prefabricated template. This is absolutely not true. It's worth throwing some "trick questions" at kids from time to time to make sure they read carefully, but the best word problems, while non-obvious to solve, are not obfuscated in their setup. Young children should not be taught to solve word problems using algebra, but should be helped to try a wide variety of their own methods (guessing and checking, making a table, drawing a picture, breaking the problem down into several steps, solving a simpler problem, working backwards, using physical props, etc. A rush to turn everything into algebra is harmful to children's mathematical development, and many if not most types of word problems can be more profitably attacked with a variety of other tools/ideas. Word problems are infinitely varied, and can get as difficult as you like, from 1st grade arithmetic up through unsolved professional math research problems. > cutting that down to something you can fit into a mark scheme This has nothing to do with the fundamental purposes of word problems. We're talking here about learning mathematics per se, not busywork or arbitrary ranking systems done for some bureaucratic purpose.

kaitai 2y ago

It's important that we marry viewpoints when looking at what we call "meaningful". I spend a lot of time right now with a 6/7 yr old. This child does not care about meaning. This child likes to solve arithmetic problems one after another. It's motivating -- it's like a game -- the kid gets them right and gets a thrill. Word problems? Well, the math ability is outstripping the reading ability right now so 2000 x 2000 x 2000 is way more fun than reading some stupid sentence about tulips. I've spent plenty of time on math (PhD in algebraic geometry) and educating people, and for sure when I taught college freshmen and master's students I spent a lot of time challenging folks to engage their minds, spirits, and intellect. At the same time, we have to admit there is a stage of childhood where kids just love memorization and facts. Dino facts, shark facts, math facts, Pokemon facts, My Little Pony facts, whatever. Let's not force kids to reckon too much with meaning when they're in the facts for facts sake stage -- and once they've got their impressive facts list, they'll make sense of the meaning much more easily, as discussed in the article and here!

andoando 2y ago
I think you're highly generalizing. I hated any kind of memorization as a child for example.
jacobolus 2y ago
> This child likes to solve arithmetic problems one after another. Cool. If I try that with my 5 and 7 year old, they shout "boring" and run to the far side of the house. > math ability is outstripping the reading ability Presumably you can read problems aloud to the 6 year old. But also, kids can (if taught; this is by no means a necessity) learn to read much earlier than they can learn to write numbers with a pen. My two kids both could read very well by about the time they turned 4.

wizzwizz4 2y ago

> how to attack and solve nontrivial word problems. Please no. Non-trivial problems that are expressed in natural language, sure: I'm in favour. But once you get past basic arithmetic, "word problems" are just algebra obfuscated with a prefabricated template. A non-trivial problem, written in natural language, will usually admit multiple solutions: cutting that down to something you can fit into a mark scheme, without making the problem convoluted or forced, is hard. Example: > Jacob and Sally want to split a rectangular cake between them, but neither is very good at cutting. Jacob can cut precisely (wasting no cake), but will miss the middle by 10% of the distance to the edge of the cake. Sally can cut accurately (exactly in the middle), but will obliterate 10% of the cake in the process. (Neither Jacob nor Sally want to eat crumbs.) Who should cut the cake, and why? This is a fun problem, but a rubbish exam question! Word problems completely sidestep this issue by starting with the algebra, and then replacing symbols with words until it's almost prose. For example, consider the Hannah's sweets problem (solution: http://www.murderousmaths.co.uk/hsweets.htm http://www.murderousmaths.co.uk/hsweets.htm): > There are n sweets in a bag. 6 of the sweets are orange. The rest of the sweets are yellow. Hannah takes at random a sweet from the bag. She eats the sweet. Hannah then takes at random another sweet from the bag. She eats the sweet. The probability that Hannah eats two orange sweets is 1/3. What is the value of n? While nominally a "good" exam question, it only makes sense within a rigid and rigorous context that's quite alien to an untrained person's understanding of English prose. And it requires bold assumptions about the fundamental nature of statistics (see https://plato.stanford.edu/entries/statistics/ https://plato.stanford.edu/entries/statistics/) that go completely unstated. Understanding English and Maths aren't enough to answer this question: you also have to understand Maths Exams.

jacobolus 2y ago
> "word problems" are just algebra obfuscated with a prefabricated template. This is absolutely not true. It's worth throwing some "trick questions" at kids from time to time to make sure they read carefully, but the best word problems, while non-obvious to solve, are not obfuscated in their setup. Young children should not be taught to solve word problems using algebra, but should be helped to try a wide variety of their own methods (guessing and checking, making a table, drawing a picture, breaking the problem down into several steps, solving a simpler problem, working backwards, using physical props, etc. A rush to turn everything into algebra is harmful to children's mathematical development, and many if not most types of word problems can be more profitably attacked with a variety of other tools/ideas. Word problems are infinitely varied, and can get as difficult as you like, from 1st grade arithmetic up through unsolved professional math research problems. > cutting that down to something you can fit into a mark scheme This has nothing to do with the fundamental purposes of word problems. We're talking here about learning mathematics per se, not busywork or arbitrary ranking systems done for some bureaucratic purpose.