This is my third year teaching 7th grade math at Success Academy, and every day my students blow me away with the quality of their work.
In schools, we often discuss the “quality of work” we see from students. From my perspective, it is easy to gauge mastery in the humanities. Zahir is a nimble writer because he uses complex-compound sentences. Destiny is a skilled reader because she can find the main idea in a challenging text. But what does it look like when the quality of a scholar’s mathematics work is truly excellent? In math, doesn’t it just matter whether or not they got the right answer? Mr. Green had six watermelons. You either got the answer or not – who cares how you got there.
I used to be a member of that camp, but this is not how my students think about mathematics, and in fact, most of them would be outraged if you even suggested it! I look at a standard ratio problem or one involving fractions, and I revert to my schooling; there is only one clear and routinized way to the find the answer, and it’s the way your teacher modeled for you – let’s say, in four clear procedural steps. But the 11-year old students in my class taught their math teacher that he was pretty ridiculous for thinking that way, and they put me to shame for it on a daily basis!
Most of the time, my students care less about the right answer (but they are confident that they will get there in due time). Instead, they take pleasure in the process of formulating clever strategies to solve math problems. They visualize each problem and wield their knowledge of number rules to master the challenge facing them (or Mr. Green). They can be creative and manipulate the rules of math to their advantage because they have a deep understanding of why numbers operate the way they do. My students know that the area of a triangle is not ½ x base x height because a teacher told them so; they know it because they discovered collaboratively that a triangle is half of a square, and if you multiply base times height (which is just length x width), you must take half of the product or else it’s just the area of a square. When you multiply a whole number by a fraction, the product doesn’t get smaller because that’s the rule, it gets smaller because you are actually repeatedly adding values less than one, thus your product will be smaller your original value. Just ask my students.
Finally, mastery comes from the conversations that students have after solving a difficult problem. We always take time in each class to discuss at least one of the problems we just solved, and scholars don’t feel satisfied moving on from a question until they can see the strategies that their peers used. They are certainly competitive but also love hearing different perspectives and learning from each other. Thus, mastery occurs when each scholar in the room can properly articulate why their strategy makes sense. During these discussions, scholars push each other to speak as mathematicians and to think about the deeper meaning behind each question. Yep, I said it, there is deep meaning in algebra and there is a universal philosophical truth within every percent question. I never used to think this way; my scholars taught me to.