Slide 1: Hello. This is Marged Dudek, your presenter for this webinar about math facts, memorization, and practice. I work for WordFarmers Associates on behalf of the Ohio Partnership for Excellence in Paraprofessional Preparation at the University of Dayton School of Education and Health Sciences Grant Center.
Slide 2: Earlier in this module we have already admitted that learning math facts is important. The question is how students might develop this knowledge. This question is very important in elementary school because arithmetic is the usual content. And math facts, memorization, and lots of practice is what “arithmetic” means to many of those involved, including students, parents, and teachers.
Slide 3: The traditional method for learning math facts is one we probably all know! And we remember it all too well. It’s a rather long sequence of memorization exercises, timed tests (again and again), and a lot of repetitious practice. As a result of this practice, some of us wind up knowing what 7 x 8 and 9 x 12 equal. And all the rest.
Slide 4: But many of us do not. That’s a shame, but it’s not the real shame. The real shame is that this well-established routine of fear and failure (sometimes called, “drill and kill”) is the sign of math instruction that lacks substance. Remember that mathematics, even arithmetic, is about ideas, meaning, and the sorts of connections among ideas and facts that we call “logic.” How does all of that apply to something as basic as arithmetic and math facts? We’ll be thinking about that question throughout this webinar.
Slide 5: But first, let’s think about traditional views about math facts. Don’t students have to know the math facts backwards and forwards, perfectly, and on the tip of their tongues, before they are prepared to take algebra? Well, it is important that we do know many facts in a routine way: absolutely. But we don’t need to know them all. In fact, we can’t know them all, as you will see! So the answer is a definite no. And it’s “no,” even though facts, relentless memorization, and endless practice was and remains so common in many schools! Just because it’s always been done, doesn’t make it good. This might sound strange, so we will explain.
Slide 6: Let’s look at an idea that can help you understand why memorizing number facts can sometimes—too often—get in the way of math learning. That idea is “number sense.” Number sense is being able to move numbers around logically to help make problem solving easier. Yes, thinking makes things easier. Surprise!
Slide 7: With number sense, the point of arithmetic changes from beating the facts into students’ heads to getting them to use numbers well by thinking with numbers. Number sense includes facts but goes way beyond them. Here’s an example of number sense in use. Let’s take the subtraction problem 73 minus 15. It’s a math fact, right? Have you memorized the answer? Not at all. Oops. Apparently there are lots of math facts that the traditional method of “drill and kill” doesn’t ask us to know. So what can we do to find the answer? We can make use of number sense.
Slide 8: Remember, though, that the usual approach to “getting the answer” is to use a step-by-step method. Here’s how that goes: Set up the problem on paper. “Borrow” 1 from 7 to make 13 in the ones’ place. Cross out the 7 and write 6. And so forth. It works. But one doesn’t have to think much, except to remember the rules. Using the calculator requires even less thought. What all the “crossing out” and “borrowing” means is not usually clear. Just do it and get the answer. This approach hardly requires any thinking at all. Number sense, by contrast, involves thinking.
Slide 9: With number sense, we don’t grab a pencil or a calculator. Instead, we move things around a bit to make the calculation easy enough to do in our heads! Here’s how this goes. In this case—remember it’s 73 minus 15. We can make some changes to the problem that make the mental calculations easier. For example, if we subtract 3 from 73 and also from 15, we’ve altered the terms of the problem but not the difference between those terms. The right answer remains the same, even though the numbers in the problem are now different! 70 – 12 is still 58, just like 73 – 15, but it’s easier to do in your head. That’s using number sense. And you can do this sort of thing with almost any computation problem. Even easier would be to subtract 5 from 73 and 5 from 15, turning the problem into 68 minus 10, which makes the answer even more obvious. What’s going on here? We subtract (in our heads) the same amount from both numbers. Why?
Slide 10: Where’s the sense? Let us explain. Subtraction has to do with the difference (or distance, to use a geometric idea) between numbers. And when we subtract 5 from each number in the problem, we keep that distance the same, even though the numbers are different. That’s number sense. Maybe hearing me explain this isn’t enough, so let’s put it on a number line in the next slide.
Slide 11: Again, the subtraction problem is 73 minus 15. With number sense, we realize that we can adjust the problem slightly to make it easier to see difference. We just subtract 5 from each number, so that we’re just subtracting 10 from something else—since subtracting 10 from anything at all is pretty easy!
Slide 12: But why can we subtract 5 from each number without changing the problem? It’s because we are simply rearranging the sequence of calculations. We’re rearranging them in the way that’s shown on the slide.
Slide 13: You can probably imagine that this example can be extended to all the math facts, including the few that students are given to memorize. Let’s take a look at a couple of the familiar, but, for some of us, difficult-to-remember facts from the times tables.
Slide 14: Rather than going through these steps, is it “easier” just to remember that 8 x 7 = 56 and that 7 x 9 = 63? Yes. But it is far, far more important to be able to do the mental math written out in the equations on the slide. It works with all multiplication “facts” even those with very large numbers. And it prepares someone for the sort of thinking that’s needed for math beyond arithmetic. But students can—and should—develop number sense as part of arithmetic; they shouldn’t have to wait until they get to algebra. This sort of thinking is useful for everyone, and it doesn’t teach students who have trouble memorizing that math is stupid, or that they are stupid. That sort of instruction is terrible.
Slide 15: So in this webinar we have actually seen how thinking applies to arithmetic—which so many people think of as the “drill-n-kill” memorization of math facts. We have seen that these simple math facts also contain powerful ideas.
Slide 16: Sure: practice is important for any developing skill. And it is handy to know many math facts, but in a way, how you learn them is much, much more important than knowing them all perfectly. It’s enough to know many facts. But you can’t know them all. There are an infinite number of math facts that no one is ever expected to memorize. For many students, endless unsuccessful repetition just makes them feel bad and hate math. There are other, and much better, ways to deal with “getting the right answer.”