Anyone who has ever made his way through a third-grade math workbook will be familiar with the standards of textbook language and the countless examples involving pizza pies, marbles, livestock, and bake sales. For some, it may even be impossible to conceive of certain mathematical concepts without the use of such imagery—think pizza pies and fractions. Whether or not one perceives these approaches to be incontrovertible, it is productive to imagine alternatives.
While designing the Knewton college readiness course, we knew that we wanted the content to transmit skills, awaken curiosity, and promote a love for learning. To accomplish this, we conducted student engagement research, read pedagogical essays on the subject, and scrutinized the conventional methods we use to teach math.
What are we implicitly telling students about math when we employ babyish examples (cupcakes and chu-chu trains for seventeen year olds), endless acronyms (PEMDAS, Please Excuse My Dear Aunt Sally), and nursery-style stories (Mr. C and Mrs. A, for circumference and area), and emphasize “workbook” problems and gold stars to the exclusion of all else? Does the over-simplified, pre-digested style and childish language of math instruction subtly suggest to students that school is in fact for babies, that real complexity and excitement is only to be found outside the classroom? And if so, wouldn’t feeling that way make anyone prone to dislike a subject?
It has long been said that popular children’s books are successful because they are not in fact childish; beneath the simple language lie real characters, real danger, and thus real wisdom (imagine Matilda without its anti-authoritarian themes or The Little House on the Prairie series without prairie fires, clouds of grasshoppers, and the constant threat of the elements). No matter what the skill or age level, human beings are attracted to that which has high stakes. Want a student to take school seriously? Convince him he’s doing mature work that requires creativity and focus, and watch a sense of purpose take over.
So what would a more mature math textbook look like?
To answer this question, we looked to Paul Lockhart’s essay, “A Mathematician’s Lament.” In the piece, Lockhart argues that if you ask anyone why we learn math in schools, there are two predominant responses:
- To prepare students to do everyday tasks like balancing the checkbook or calculating the
- To prepare those who are gifted at math to become accountants and engineers
But most students do not go on to become engineers or accountants. Therefore, according to this line of thinking, the only reason left to study math is to be able to be competent at everyday tasks. Why, then, take trigonometry or calculus or logic or anything beyond seventh grade?
Most adults aren’t sure how to answer this question, and students instinctively know this. (One of the things which surprised me about teaching was the fact that my students could always sense when I was fuzzy about a subject or why it was important to master it.) But if you think about it a bit, the answer becomes clear. At its core, the study of math is not about everyday tasks or engineering; it’s about developing an appreciation for pattern and symmetry, as well as the process of problem solving and discovery. It’s about learning to speak the language of ideas, shapes, and space.
For instance, the following is what Paul Lockhart would teach instead of “Mr. C and Mrs. A and her two pies”:
“What about the real story? The one about mankind’s struggle with the problem of measuring curves; about Eudoxus and Archimedes and the method of exhaustion; about the transcendence of pi? Which is more interesting–measuring the rough dimensions of a circular piece of graph paper, using a formula that someone handed you without explanation… or hearing the story of one of the most beautiful, fascinating problems, and one of the most brilliant and powerful ideas in human history?”
Ultimately, we simply need to teach math in a way we would want to be taught math–that is, maturely.
Here are some general principles we came up for in the process of creating our college readiness course.
1. When writing problems and creating scenarios for math to enter, use genuinely interesting, illustrative ones. Try to minimize infantalizing language.
2. It is ok to use rhymes and acronyms (it is natural for the human brain to respond to the sonic patterns of rhyme and alliteration), but understand that one need not rely on “funness” and “cutesyness.” The questions at the heart of math are fascinating enough.
3. Explain why, or rather, assist students in discovering why we use certain formulas. Don’t ask them to just take your word for it. If you’re teaching students about the area of triangles, for instance, have them discover that you can break a triangle into two right triangles, each of which is one-half of a full rectangle (hence the area formula).
4. This one is my favorite: contextualize the study of math with history, philosophy, and current events. Let students know that what they’re doing connects to the work of mathematicians at the forefront of their field.
5. Understand the importance of drilling and technique. The act of doing often creates understanding. Some people learn by grasping ideas before seeing them in action. Others only understand when they put pencil to paper and “wrestle” with the problem. It could very well be that some people need to go through the motions of dividing fractions before actually understanding why they “multiply by the reciprocal when dividing.” Far from being the province of unimaginative minds, drilling can create a familiarity with numbers that stimulates curiosity.
6. If you want to let students know that they need math to function in the “real world,” make sure you’re making sense. Think of projects that might help students grasp this idea in a visceral and engaging way. Have them submit a business plan, participate in a stock market game, conduct research on their physical environment, etc. Don’t tell them to learn trig because they need to balance their checkbook later.