If you want to know how you can become better in math and rewire your brain to be math compatible I‘m afraid you will be none the wiser after reading this.
- Memorization and rote practice are important for learning, not just the current Zeitgeist (2014) of “understanding” without the former. This becomes the foundation that allows you to focus on higher-level things like understanding and applying formulas.
- Experts develop “memory chunks” which allows for example chess masters to draw on thousands of different past games, openings, variations.
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...
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!
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.
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.
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.
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.
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.
Personally I only accept „understanding“ once I can explain it and reuse it in a different context. But I am not self centric enough to deny that there absolutely are plenty of people who love to memorize without having an abstract understanding. And they are doing just fine.
The cool thing about area of a rectangle is you can just turn it into a multiplication array. Which is something they learned to help them understand multiplication.
If she just memorized multiplication, she wouldn't actually understand the formula.
Thoughts flow through my brain like electrons through wires, both at a speed I cannot truly comprehend. The paths of my thoughts, the way words connect together, the emotions they evoke, the feelings that are associated with - these are all malleable. I have been working to rewire myself away from pessimism and towards optimism for years now. It's not easy, and sometimes I fall back into old patterns. As years have passed, though, I've found the new pathways easier, the new roads getting more familiar. My thoughts have previously wanted to go one way, and I spent time yanking at them to go a different way. I spent enough time at it that I now on good days more naturally go they way I want to go.
It's not just sitting and repeating a single thing over and over again, it's working with your own natural inclinations so that you can recognize when you experience X and naturally reach for Y that perhaps Z is what your preference really is, upon reflection. If you can practice that enough then, in the moment, you can sometimes find yourself not following the old pathways but the new.
Rewiring seems like a pretty good analogy as when you rewire a house you work hard -- it is dirty, dusty work. Pulling wire is hard and thankless because when you're done you cover it all up and, if you're lucky, it works! Then at the end you're . . . back in a state where nobody but you knows any different what is happening behind the walls. Things work, and others might have no idea anything changed at all. But you put in the hard work and you know how things actually work on the inside now and it's exactly how you want it, not how it was before.
If you've got a better analogy, I'd love to hear it.
I would think that would be obvious to anyone that merely memorizing something like 'f=ma' would be meaningless without deliberate attempts at application (both theoretical and practical).
There was a kludgy attempt at tying the study of foreign languages to STEM, but it just amounted to, everything is ultimately a craft. You have to practice to perfect it.
The author, Barbara Oakley, has a free Coursera course that is pretty good:
... then you just have to do it. And keep working on it, even though it feels awful
Currently, the emphasis in training and education is on ensuring students understand the material, and rote memorization is viewed as a failure mode.
The author acknowledges this but introduces an argument that rote memorization is critical to achieving fluency.
I suspect that stating this position among typical education-focused circles will result in pushback.
To lend credibility, she adds her lived experience as a way to explain what she means—and to clarify that she isn’t saying everyone is wrong, just that we may be too harsh on memorization.
This is apparent to me because, frankly, if it weren't for the additional content she added, I wouldn’t have spent more than a few seconds before dismissing it.
The article even made me concerned about an internal project I am involved in, prompting me to verify that I hadn’t overlooked some issues.
If you want a TL;DR, the golf analogy matters: If you want to learn math, you need to understand it and then practice it so much that it becomes second nature.