# Solving word problems with models vs. direct translation

I 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?

The same problem phrased inconsistently is this:

At Lucky, butter costs 65 cents per stick.
This is 2 cents less per stick than butter at Vons.
If you need to buy 4 sticks of butter,
how much will you pay at Vons?

Intuitively for these problems, "more" means "+", and "less" means "-". The 'consistent' phrasing of the problem is named so because it uses the word "more", and the solution requires figuring out that Vons = 2 + Lucky. The 'inconsistent' phrasing of the problem is named so because it uses the word "less", which intuitively primes the solvers to think a subtraction is going to be needed somewhere, when that's not the case here.

Testing against the 'inconsistent' phrasing of the problem resulted in unsuccessful solvers being wrong in the manner predicted by a direct-translation approach, that is they took "Lucky is 65; this is 2 less than Vons", and translated that to "Lucky is 65; Vons is 65 less 2, is 63". This is obviously wrong, and to actually solve the problem requires understanding the full meaning of the English sentences.

Policy-wise, we'd like more people to construct meaningful representations of problems instead of a blind keyword and number mapping. The problem with the direct-translation approach is that it's actually not terribly unsuccessful on 'consistently' phrased problems, because they're written such that a direct translation happens to also get the right answer, and so in order to get young students from relying on that model all the way into adulthood, it's probably necessary to use "tricky" phrasings like the inconsistent ones that they're more likely to mess up.

There are other interesting things they found that support this differing model approach idea, go check out the paper for them.

#### Posted on 2015-04-25 by Jach

Tags: math

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