Slide 1: Your presenters for this module are Dan Showalter, a mathematics faculty member—the young guy—and Craig Howley—the old guy. Dan teaches math education at Eastern Mennonite University in Virginia and Craig is retired from Ohio University. Craig studied and taught about rural education. We both like math, but we understand why most people don’t. This first webinar considers what math is.
Slide 2: Most people never ask, including lots of math teachers. For nearly everyone, math is just that stuff that happens in math class. Math teachers like that: most of the rest of the world, not so much!
Slide 3: So what is math? There are lots of good answers, but the one that I like is that math is a particular way of thinking logically … and logic is just a way to put things together so they make good sense. Math does this with numbers (algebra) and with shapes (geometry). So math is really making good sense with numbers and shapes.
Slide 4: Do the times tables make good sense with numbers? No. Do the addition and subtraction facts make good sense with numbers? No. Making good sense has to do with thinking. But where is the thinking with times tables and addition and subtraction facts? Maybe you’d answer, that the thinking is the memorization. Sure, memory is a kind of thinking, and maybe following instructions is a kind of thinking, too.
Slide 5: A lot of what happens in math class seems to be about following instructions. For instance, following the instructions for how to divide a three-digit number by a two digit number. Yikes: how do you do that? First you do this, and then you do that, and then you get the right answer. But what does it all mean?
Slide 6: No wonder most people are bored to tears by math! Sure, helping students memorize math facts is important.
But understanding the related ideas—helping students figure out what the ideas mean and how they connect with each other—is even more important. Understanding and logic connect ideas.
Slide 7: This probably sounds pretty vague, so here it goes: When we understand something, we realize what it means. So 3 + 4 = 7 is a fact, but it must mean something, too! What on earth could it mean, you might ask?
Well, what something means comes from the context in which it exists. Take 3 + 4 = 7. If you see 7 as seven days in a week, with 4 workdays and a long (3-day) weekend, the facts look very different.
We need to work with a couple of ideas in math. Zero is one of the most important ideas in all of math, and so is the idea of equality (the equals sign in 3 + 4 =7). What do these ideas mean? How are they related?
Slide 8: Perhaps you’ve never thought of “0” and “=” as ideas.
Why would you? But they allowed people to think in radically new ways, in fact, to start to think mathematically. Zero enabled people to write—and think with—large numbers: something that was far more difficult before. A new idea! The idea of equality (written as “=”) means that one thing is the same as the other. It’s a profound idea in mathematics. It makes mathematical logic possible, together with the idea of zero and a few other things.
Slide 9: The important thing in all of these math facts, whether it’s 3 + 4 = 7 or 3 x 4 = 12 is the equals sign. It makes a statement: the thing on one side is the exactly the same as the thing on the other side. That’s where the meaning is!
Almost everything that math does in the world and discovers about the world, and makes good sense of it depends on the equals sign! And we’re talking about a thousand years of math history, about rocket science, about keeping the bills paid, about moving big furniture though tight doorways. Equality: it’s a huge mathematical idea.
Slide 10: In both math and science, if you can establish that two things are the same—or that they “equal” each other—you can use the information logically to find out things. Things like whether or not the refrigerator will fit through the door. Things like where your property begins and where your neighbors’ ends. Things like whether the earth moves around the sun or the sun moves around the earth. And on and on. Scientists, mathematicians, and engineers have done this kind of thinking for centuries. So when kids learn math, they need to get a sense of the huge world of ideas and usefulness that they are standing on the edge of. It should be exciting, not boring.
Slide 11: Once we realize that the original math fact is a statement, and then rephrase the statement just a little bit, like this:
3 + x = 7
we have stepped over the line separating lifeless fact from a lively logic. Now we have a “variable” (x). And we need logic and understanding to discover what x is. It’s lively because what becomes important is not knowing the fact, but what the fact means in the presence of the equals sign.
If the expression on the left (3 + x) is the same as 7, we can apply logic to discover what “x” is. Sure, this isn’t “sophisticated” logic and it’s something everyone can understand. That’s the point. One can learn a lot of math this way.
Slide 12: In math, little things like the equals sign help to convey the meaning of a mathematical statement and become ways to make good sense with numbers and space. That particular kind of thinking is called “math.”
Slide 13: So what? The most important context for helping students learn math is for those teaching them to understand more math than the students do.
And that means knowing more about math as a way to think logically. Even for many elementary teachers, this can be a scary idea. How much more do you need to know to be helpful to elementary students, for instance?
Slide 14: Not really so much more in our experiences teaching math. But you really need to understand that little bit “more.” Many students finish high school without really understanding ratios and proportions (the meaning and use of equations with fractions, percentages, and decimals). So, for elementary instruction—and for some high school instruction—really understanding ratios and proportions is essential. It’s easy to find out how well you understand ratios and proportions, however, and to learn that little bit more you need to help students make good sense with numbers and space.
Slide 15: Most of what you will probably do with math instruction will be in the realm of arithmetic. But even at that level, math instruction is much more than facts. At every school level from kindergarten through high school, state and national standards now ask teachers to address concepts and procedures related to such forms of math as measurement and data, algebra, geometry, and probability.
Slide 16: If you keep in mind that math isn’t the giving of right and wrong answers, but rather a way of thinking logically about the world with numbers and shapes, you will have the right framework to learn more about how to help students learn—and like—math. Fortunately, there are lots of ways to learn more about this way of thinking. We’re going to be talking about many of them in other units in this module.
Slide 17 : That’s about it! Math is a way of thinking logically about the world. Math is more about ideas than facts—and in math, simple things like 0 and the equals sign are big ideas. Math is also in play outside of school, in the real world, in very ordinary ways—as with the example about moving furniture.