Solving word problems with models vs. direct translationI skimmed this interesting study this morning. They were testing an idea about how people solve mathematical word problems that are expressing a less-than/more-than relation between two numbers, and then asking for a solution number that makes use of that relational information. They phrased each problem in one of two ways, "consistent" and "inconsistent", and measured successful and unsuccessful solvers. They also measured through eye-tracking what parts of the problem the subjects returned to and what their recall of the problem was, with results that strengthen the interpretation of the successful/unsuccessful results. In the end they found evidence that successful solvers seem to construct a model of the problem internally which they then use to get at the solution, whereas unsuccessful solvers seem to try a direct-translation sort of approach where they take the items in the problem, directly translate them to their most intuitive mathematical operations, and compute.
The way a problem is phrased (consistent vs. inconsistent) is key to getting evidence for this idea. An example problem, phrased consistently, is this:
At Lucky, butter costs 65 cents per stick.
Butter at Vons costs 2 cents more per stick than butter at Lucky.
If you need to buy 4 sticks of butter,
how much will you pay at Vons?
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Why write?I haven't posted anything of substance here in a long time. Life has been weird since a couple days after my September 2014 post. I'm in a mind state I've been in before, and I'll work through it eventually, but during this time I'm just not capable of caring about much of anything beyond what I deem necessary in order to get to tomorrow, so my actions are limited.
I'd like to write more. I enjoy playing with ideas, not necessarily taking them seriously, but sometimes updating to some conclusion that results from playing. For example, it's kind of fun to think about ideas that would make big democratic governments work more efficiently, And by exploring that thought-space, one can arrive at conclusions where doing X is significantly better than doing Y, even if this other totally different and opposing thing Z is better than all of them. It's like noticing that a flu is strictly a lot better than stomach cancer, even though they're both terrible and being healthy is totally superior. Except it's funner to work with more abstract things than sicknesses, and it becomes more exploratory that way (since things aren't self-evident, or "trivially derived from some framework but I'll never show you a proof"... you have to do work to explore), and in the end sometimes you'll have something you can reduce back to a less abstract belief or suggestion.
What little idea playing I've been doing lately has taken place mostly in the context of my head. And unsurprisingly, it has less quality. Writing (and the concurrent and following process of editing) creates a better environment for exploring, because you have to crystallize what exactly you're thinking and not just rely on your brain's intuition. I have many thoughts that are interesting, but not expressed in words, but then I forget about the thought because I didn't write it down.
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