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/trav...
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!
> 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.
> 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):
> 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/) that go completely unstated. Understanding English and Maths aren't enough to answer this question: you also have to understand Maths Exams.
> "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.
I think we are in violent agreement. Your use of language comes from a theoretical perspective (what word problem should mean), and mine from a practical perspective (the things that were called "word problems" throughout my schooling, and throughout all the curriculum-based teaching materials I've read).
In my experience, the first step to solving a "word problem" is always to reverse the find-and-replace performed by the question setter, to yield the original algebra (e.g. the first part of the Hannah's sweets question). This is what we're taught to do in schools. You're right to object from a pedagogical perspective, because this is a horrible thing to make children do, and completely defeats the purpose of word problems.
The trouble comes when schools teach people how to pass exams. But, given how hard exams are, and how little they reward understanding beyond that stage of the curriculum, the optimal strategy is to learn no more than your current stage (plus exam technique), then learn no more than the next stage (plus exam technique), and never get around to actually learning maths.
This is why I say that most US primary/secondary education to first approximation doesn’t have any nontrivial word problems. The single-step formulaic problems called “word problems” are not remotely the same thing, and miss the point.
I think there's a nice in-between where the problems are both generally useful and also actually do test some concept relatively directly.
Kind of like what FizzBuzz can be like for testing for basic pseudocode programming. Where someone who isn't really thinking about the problem will mess up and go for what seems like the obvious solution on the surface, but someone who understands how to code will eventually realize that the obvious "clever" seeming solution is a trap and they have to do it the plain way.
I've found those kinds of problems to be the most fun to deal with. A math specific example I vaguely recall from middle school involved calculating the number of handshakes that would happen in total if everyone in the class shook everyone else's hand once. The path to the solution isn't explicit in the question, and for someone who doesn't already know of combinations/permutations, it takes a bit of abstract logic to construct the necessary expression. Yet it doesn't require particularly advanced math to figure out and tests the student's understanding of how to translate real problems into math.
Problems where the word problem is just a literal transcription of an equation are not that useful or fun, but also, I think that it takes getting to a fairly high level of education before the math gets advanced enough to more frequently cover meaningful word problem solving (probabilities and statistics, linear algebra, or differential equations), yet the tools for interpreting such problems need to be taught earlier.
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/trav...