## The Misbegotten Trifle: What Cooking Fails Have in Common with Math Learning

by Sarah Lonberg-Lew

In a Thanksgiving episode of the popular nineties sitcom Friends, the character Rachel attempts to make a traditional English trifle. The layers include: ladyfingers, jam, custard, raspberries, beef sautéed with peas and onions, bananas, and whipped cream. As she lists them, the other characters’ reactions go from appreciative to incredulous to disgusted, and the mention of the beef layer gets a big laugh from the studio audience. Upon investigating, her friends discover that the pages of the magazine from which she got the recipe were stuck together and she had in fact made half a trifle and half a shepherd’s pie.

The scenario gets such a big laugh because most of us know it is absurd to have beef sautéed with peas and onions in the middle of a dessert, but Rachel clearly did not. If she had, she might have questioned the recipe or even dared to modify it on her own. As it is, it turns out she did find it a bit odd, but she had more faith in the recipe than in her own intuitions. (Click here to see the “trifle scene.” Warning: The clip contains some PG dialog.)

Sadly, this is exactly the situation many of our students are in. Lacking conceptual understanding or a desire to seek it, they rely almost exclusively on procedures, believing that faithfully executing steps, practicing and memorizing them, is the only way to reach their math goals. They don’t even pause when they achieve an absurd result because they have no concept of what kinds of answers make sense.

The reasons they believe so strongly in steps are not insignificant. For one thing, many have never known that there can be any more to math than following steps. For another, it really feels like it’s working in the short term. By teaching students steps to memorize and then giving them practice problems that require nothing more than applying them, both the teacher and the students can feel successful in the moment. However, when it comes to long term retention, knowing when an answer is reasonable, or the ability to solve non-routine problems or apply their learning outside of class, the time and energy invested in memorizing steps fails to serve our students in any useful way. In fact it is counter-productive because they have missed the chance to use their time to develop genuine understanding and they end up having to “learn” the same material again and again. (See “What Community College Students Understand about Mathematics” by Stigler et. al. for an in depth explanation of research supporting this.)

I recently had an opportunity to observe a student struggling through finding half of 544. He knew the steps and diligently used long division to divide by 2, even going through the procedure twice to be sure. Both times he arrived at an answer of 322 (having made the same mistake twice) and said he felt confident about the answer because he had followed the steps (he really said that!). We then talked a little about what half means and I asked him if he could find a way to check whether 322 was really half of 544. He multiplied it by 2 and was completely stymied when it did not come out to 544. His steps had failed him and he had nowhere else to go.