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Nicholas Stone: February 2012 Archives

February 2012 Archives

Math Smarts

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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.

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Culture certainly may have impelled the change in my students. Our understanding of “growing up” is deeply entwined with the passage through school. As my students ascend the school hierarchy, they accommodate themselves to changing models of a child their age. The archetypical first grader is intrinsically more sophisticated than a kindergartner and accordingly, they became more sophisticated. The presentation of a new self-concept facilitated change.

Neither culture’s deep impact on development nor the principality of self-concept in a child’s developmental tendencies are radical ideas in the educational community.  Lev Vygotsky posited eighty years ago that cognitive development was an inherently socio-cultural phenomenon.  He pointed out that culture, not biology, creates most of the tools we use to develop and the impulses that drive development. “Self-concept,” as far as I can tell, entered common educational parlance more recently, but nowadays research abounds on academic self-concept and how schools can influence it. The cultural construction of childhood and education may dictate a developmental course.  

However, I’ve come to appreciate the beguiling simplicity of the change in my students. Who deserves credit for their growth? Not me, not school, not biology.  “Culture”? That’s a useless answer and a cop out on my part.  They do. Each of them has set new expectations for themselves and risen to meet them. School, culture and I only contributed through suggesting standards. Development might be fluid, but children govern themselves autonomously enough to consciously adjust to the structure of school, to reconsider their self-image and to create new ambitions. Despite spending most of my day in the classroom, I run abreast of dividing my critical thought about educational theory (to which I have only moderate exposure) from that about the needs of my students. I hope this is a common pitfall for fledgling teachers. The students’ needs are immediate; theory abstruse and ever changing, both are new and the two demand different sorts of consideration. Yet, ultimately, children’s decisions and goals have the biggest impact on how they grow. Theory, of course, only serves them if it helps us provide guidance in shaping and actuating those decisions. 

In the first period of first grade, we set down to work on weekend news, a weekly routine continued from kindergarten. The kids’ handwriting was tidy, their accounts thoughtful. They spelled sight-words correctly and each wrote several sentences. Some of their best writing to date – even from students who struggled with writing last year – completely self-motivated. They intended to realize their standard for a first grader. With a confounding array of theories and concepts to inform my practice, all of which are thrilling at this point in my career, I keep reminding myself what really has the biggest influence on my students’ development: my students.

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