For much of my academic career, I failed to understand what mathematics is really about. I always approached math as the pursuit of skill and concept mastery (or in my case, competence). Teaching revealed that skills and concepts are, in fact, only means to an end. The real aim of math education isn’t building a mathematical toolbox, but promoting flexible, abstract thinking. The most successful math students aren’t necessarily those who can use the tools teachers provide, but those who can think beyond them. Many students can develop astute problem-solving strategies, but lack the flexibility to apprehend multiple modes of representation or variable lines of reasoning. I’ve often misidentified keen application of taught skills as real understanding, but I’m coming to appreciate who, in my class, are really growing as thinkers.
It was easy to loosely sort my students into four basic categories as mathematicians: those who struggle with basic mathematical tools (counting skills, addition and subtraction facts, number families), those who rely on strong memory to implement and integrate mathematical tools, those who use higher reasoning to develop original problem-solving strategies and those who can truly think flexibly. Students in the last category look at problems from multiple angles and take a truly experimental approach to mathematics. I’d like all of my students to be experimenters, although it seems that even among adults, thinkers like this are rare. What initially surprised me, but now seems obvious, is that the kids who struggle the most in math also have the most experimental outlook.
Early on, I fell prey to the common temptation to treat classifications as hierarchical and fixed. I saw the creative problem-solvers, the abstract thinkers, as simply superior mathematicians to the kids who were still approaching math through concrete and semi-concrete processes. The child who quickly wrote an equation to solve a problem simply looked like a more sophisticated thinker than the one who painstakingly worked through a drawing. I perceived my role as helping students move onward and upward through a progression without considering the unique challenges and opportunities of each mode of thinking.
Of course, as with many ways we classify students, these categories are fluid, and students move between them as the demands of our math curriculum change. As our curriculum has come to place greater emphasis on word problems, it’s become clear that many kids who manipulate numbers with great facility lack the analytic skill to deconstruct a problem. They pick out the salient numbers and identify the operation they need, but may fail to apprehend what their answer means. These are kids who, when asked how many gloves are in the room if five kids have blue gloves and three have red gloves immediately declare eight, without considering the fact that gloves come in pairs. Many students even arrived at an incorrect answer after carefully organizing their work into the tables and tally marks. Students who took a more literal approach - actually drawing each kid and each glove - didn’t overlook this detail. They paid attention to the problem, not just the numbers. These are the children who solve problems by trial and error and who may start out working with little sense of exactly where they’re headed. They’re not sure how to solve the problem, so they just start drawing gloves. Their experiences struggling in math make them amenable to open-ended thinking.
We also just finished a delightful unit on surveys and data analysis in which the students surveyed one another. They had to develop both a method of recording their survey data and of displaying it for others. The catch was that they had to display it in a different way than they recorded it; one sheet would be used to take down survey answers, a different one to present the spread after the survey had been completed. I hoped that the students would come away with the all-important understanding for statisticians that the way data is presented influences how it is interpreted. Many of my most efficient mathematicians simply couldn’t do it. I was surprised to see kids who can develop highly reasoned problem-solving strategies staring in frustration at a display and a recording that looked exactly the same. Many had developed tidy and effective devices for recording their peers’ answers: careful tables, class lists, labels, and check marks. These children could clearly see the procedure for the taking the survey from start to finish, but the comprehensiveness of their vision hindered them. Their recordings lacked errors, so they couldn’t envision a different way of presenting their data. Their process didn’t leave room for experimentation; they got it too right the first time.
Meanwhile, the kids who struggled to develop a clear recording strategy thrived on the opportunity reconstruct their data in a new way. They saw the errors and limitations of the strategies that they had developed and took them as inspiration for alternatives. Hence, kids who had made discombobulated lists or written out cumbersome charts translated their data into clear and original displays. They worked flexibly because they’re used to having to change course. They don’t expect things to work out the first time. In the end, they experiment the most and are most receptive to different perspectives and approaches. Number sense is plenty helpful in life, but this is the real ability that we teach for in math. Math is more about flexibility than efficiency, which early aptitude for numbers can get in the way of. We want kids (and grown-ups) to learn to see problems from different angles. It’s easy to conflate mathematics with computation, but really math teachers are trying to create open-mindedness, not procedural knowledge.