When it comes to education, there are two kinds of people: those who are just good at math, and those who aren’t. Right? WRONG! Of course, it’s easy to fall into this line of thinking. We’ve all had experiences when we were young where math either just clicked for us or—just didn’t. And those early educational experiences can make us feel like we either are or aren’t “math people.”
But the truth is that mathematical understanding, like any skill, can be improved through guided practice, regardless of your natural inclination. We’ve written here before about how math practice can be seen as the same thing as practice for a sport, such as basketball. Through the guided investigation of key elements, we see improvement. The trick is to have a good guide, or coach, using the basketball metaphor, and to study the right elements consistently. Regular instruction in a guided system, with one-on-one attention, is a definite way to make progress in learning math.
Also, it’s important to believe you can get better at understanding math. Because you can! Just ask Albert Einstein. He said: “Do not worry too much about your difficulties in mathematics, I can assure you that mine are still greater." Despite his troubles, he went on to accomplish some really amazing things using mathematics.
So how does math learning work? The first step in understanding how math learning works is about language. This picture is a good example:
This example is funny, but also demonstrates a good point about learning math. In this instance, the word “find” means something very different from what it usually means. Specifically, it means: Using your understanding of the relationship of sides of a triangle and the values given for these two sides, what must the value be for the third side? Or, more simply, what is the relationship between three sides of a triangle? Because, if a student understands this underlying geometric concept (i.e., the relationship between three sides of a triangle), then the answer to this specific instance of that general understanding would come easily.
So, when a student is asked to “find” something, we are actually assuming a great deal of mutual understanding that may in fact never have been stated explicitly. While we don’t want to go too far into explanations—at some point, we have to assume that we’re simply “speaking the same language”—it’s important to step back occasionally and make sure that all of our terms are understood.
After defining our terms, the next step in how math learning works is making sure students move from specific facts to general concepts. As in the example about triangles, math learning often moves from a specific example to the understanding of a general concept. Each specific instance of a triangle is only one example of a principle that holds true for all triangles everywhere. This reality is what makes math a universal language, and one of the reasons so many people call math beautiful. When the poet Edna St. Vincent Millay writes that “Euclid alone has looked on Beauty bare,” we can’t help but think that she’s talking about the universality of geometry, and of math in general.
So how do you move from the specific to the universal? How can you move beyond memorization—which, we all agree, can grow tedious—and open your eyes to the possibility of beauty in mathematics? Because that is where a very special kind of learning begins—the kind that lights a fire in students that may well last the rest of their lives, and the kind that takes students who think they “aren’t good at math” and shows them that they are.
As we said at the start of this blog, there is no getting around practice. But practice has to be done with a good teacher. Steady work is an important component to math learning, and some memorization is required. However, continuing a conversation with students about why you are doing different operations, and the ideas behind the math, can allow them a way to access the beauty once they’ve developed enough of a basic understanding to do so. Michael Jordan didn’t get to float across the court on his first day of practice, and we have to imagine that Edna St. Vincent Millay put in a little work in Euclid’s Elements herself before writing her poem.
The move from specific practice problems to an understanding of universal concepts is related to the first step of math learning—defining our terms—because without an explicit explanation of what we are trying to achieve, students won’t understand what the goal is.
Many math students may not see a connection between one problem and the next, even if the teacher continues to say, “OK, we’re working on fractions now. Please do problems one through ten so we can understand fractions.” In fact, this inability to immediately grasp mathematical connections—to speak the “math language”—may be the start of students dividing themselves up into “math learners” and “non-math learners.” An attentive teacher is the crucial piece for students to turn problem-solving into general math knowledge—the kind of knowledge that is universally true, and will stay with them the rest of their lives.
Because saying the word “fractions” may not be enough for students to understand what the word means, even if the word has been defined using mathematical terms. To make sure all students are on the same page, it’s good to go over every term many times, using different mediums. Providing examples is important, and so is defining your terms in new and different ways. As we know, the more types of approaches we can use to present a topic, the more likely that topic will sink into the minds of our students. Present visuals, talk through definitions, and have students handle objects that demonstrate the concepts and skills you’re working on.
Of course, this is just the beginning of how math learning works. But carefully defining your terms, and then moving from specific examples to universal concepts, are two early and important steps in how learning math. Because math is for everyone—now go spread the word!
This article relied in part on material presented by Dr. Marsha Lovett in her lecture “How Learning Works.” Click here to see Dr. Lovett’s presentation.
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