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.
Instead of reviewing the steps or combing through them for his mistake, I lent some context to the question, asking how he would manage if we had $544 to share equally between us. Immediately his energy and attitude changed – I could see in his face that he understood the task and had an idea of how to approach it. He began talking about what kinds of bills the money could be in, and I kept pace with his thoughts by drawing pictures of the bills as he talked. When he saw them in front of him, he drew loops to divide them into two groups, even deciding to go to the bank to make change when the bills couldn’t be evenly divided. When I asked him this time what reason he had to feel confident in his answer, he pointed to the two groups, showing me that they were both the same and that he had “given out” all the money. Using his conceptual understanding of division, he arrived at the correct answer with confidence and was proud of his hard work. I followed up with another “finding half” problem and then asked him to see if he could use his strategy to divide a number into three equal parts which he did with relative ease. By this time, the strategy was becoming a little more abstract – just numbers in boxes instead of pictures of bills– and eventually even the boxes fell away. It was especially fascinating to watch him because his approach evolved into something that I could not have taught him. It was his own invention born of his own understanding.
It is not just our students who suffer from the misconception that math is all about steps. Most of us have also been socialized to believe that the key to success in math is being able to follow directions. I remember my tenth grade math lesson on completing the square (a strategy for solving quadratic equations). My teacher drilled us over and over again on the sequence of steps and said, “Keep your pencil moving! Don’t think!” I was overwhelmed and lost and despite many other successes in math in the years following, I believed for a long time that I would never understand completing the square. I just couldn’t remember all those steps. When it came time to teach it in my own classroom (during my stint in K-12) I did only marginally better with my students than my teacher had done with me, giving lip service to explaining the algebraic steps and trying to be more gentle and compassionate with them as they struggled to memorize them. It was even longer before I made sense of the method for myself and saw that it was in no way beyond my ability.
As teachers, we need to stay strong when our students beg us to just give them the steps – when they say, “I know if you just show me how to do it, I’ll understand.” I have had students who have insisted that memorizing steps is “how they learn best,” but even those students who are good at memorizing benefit more from understanding. For all of us there is a limit to how much we can remember – but not to how much we can understand.
We have to take the long view and know that even getting a perfect score on a worksheet is not the same as understanding. Taking the time to develop conceptual understanding can feel slow and our students are usually in a hurry to move forward with their academic and career goals. But learning math takes time just like any other kind of meaningful learning. For our students, the difference between spending their time learning deeply and spending it memorizing steps may very well be the difference between progress toward their goals and one more failure in one more ABE class.
The misbegotten trifle came about because of a blind reliance on following steps. The consequences of such an approach for our students may be far worse than an unpalatable dessert!
Sarah Lonberg-Lew has been teaching and tutoring math in one form or another since college. She has worked with students ranging in age from 7 to 70, but currently focuses on adult basic education and high school equivalency. She teaches in adult education programs in both Gloucester and Danvers, MA. Sarah’s work with the SABES numeracy team includes facilitating trainings and assisting programs with curriculum development. She is also an actively involved member of the Adult Numeracy Network.