On Cognitive Load Theory and Story Problems

I’m currently reading “Sweller’s Cognitive Load Theory in Action” by Oliver Lovell, specifically the section on reducing extraneous load during education (ca. p. 32; I’ve got the e-book). This leads me think about story problems, such as those on the SAT, which often contain information that’s irrelevant to the problem. For example:

Carrie invites some friends to a party. For every two friends who bring snacks, there are five who bring nothing with them. If the number of friends who bring nothing is 15 more than the number of friends who contribute snacks, how many friends in total arrive at the party?

SAT Math: Word Problems, Anika Manzoor

The first sentence has nothing to do with the underlying math; it’s meant to contextualize the problem, to make it less abstract. But this puts information in our working memory, which is finite, and hence takes up room that we could be using for other things (such as actually solving the problem).

The second sentence has important information, but it also has some irrelevant information: We are going to have two groups of friends, “snack-bringers” and “non-snack-bringers”. The nature of these groups is irrelevant; this could be “people who watch TV” and “people who don’t watch TV”, and they could be gathering at a shopping mall. This could be “Democrats” and “Republicans” and they could be filling out a survey. But “snack-bringers” and “non-snack-bringers” are particularly clunky groups to have, cluttering our memory more.

The mathematically relevant part of this is just:

\[A + B = C \\ B/A = 5/2\]

For a mathematician, that’s where we are. For a non-mathematician, we’re also at: Who is Carrie? What’s the party about? What kind of snacks? Was everyone supposed to bring a snack? Are we to infer that there are no friends who brought, say, a game but no snacks? And the first two questions (completely irrelevant!) are loaded into working memory before any math can happen, taking up room.

Then we have the painfully confusing “if” clause. In mathematical notation, it’s:

\[B = A+15\]

with the “how” clause being:

\[\text{Find } C\]

But we have all this other language, including the unnatural structure of the “if” clause, to muddle through. So teaching a student to solve a story problem means teaching them to FIRST find the mathematical parts, THEN clear out working memory to focus just on the math, THEN solve the math.

The underlying math, meanwhile, is not as straightforward as it seems to a fluent mathematician, and requires checking back against the convoluted scenario it’s provided in. So it’s little wonder that many students, once they’ve gotten some sort of mathematical problem (right or wrong) out of a story, simply discard the story entirely and go with whatever number they wind up with as “the answer”.

Let’s solve this problem according to how we’d usually teach it in Algebra class.

\[A + B = C \\ B/A = 5/2 \\ B = A+15 \\ — \\ (A+15)/A = 5/2 \\ 5A = 2(A+15) \\ 5A = 2A+30 \\ 3A = 30 \\ — \\ A=10 \\ B=25 \\ C=35 \]

This is likely not how a fluent mathematician would solve it: More likely, we’d hazard a guess for the snack-bringers, noticing that for every two, there are “three more” non-snack-bringers. We might jump immediately to that “three more” and realize that every 7 people represent three “extra” non-snack-bringers (and hence there are 15/3 = 5 groups of 7 people involved).

So first we overload working memory with irrelevancies, then encourage students to discard those irrelevancies when solving, then encourage an inefficient-but-universal algorithm for solving, all while the clock is running.

I’m not saying that we shouldn’t have story problems. The common mantra in mathematics education is that the purpose of what we’re learning is so that we can apply it to real world scenarios.

The underlying process is an important one:

  1. Convert from natural language to mathematical notation
  2. Solve the mathematical problem
  3. Convert back to natural language

However, the first step requires extra time, and importantly, there is the need to release working memory between the first and second steps and then recall between the second and third steps, something which students often struggle with.

Just some thoughts as I’m reading through the book.

Addendum: I asked my 13-year-old about this question, and he wondered whether Carrie was meant to be counted at any point. I had given him a paraphrased version, and I think the question itself is phrased in a manner that says she’s not included at all. But given that many of my students have been trained to look for trick questions, this is a fair observation, something else that muddles up cognitive load.

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