For the most part we all know how to do arithmetic, right? We learned it years ago in elementary school. You have to learn the multiplication tables and then use that knowledge in specific ways to extrapolate the answers to arithmetic problems.
Take the problem 547 x 9.
If you learned math the way I did, you start at the right with 9 x 7 = 63. You write the 3 below the line under the 9 and carry the 6. Then you move on to 9 x 4 = 36 + 6 = 42. Write the 2 below the line under the 4 and carry 4. Then 9 x 5 = 45 + 4 = 49. Write the 49 under the line for a final answer of 4923.
The finished problem would look something like this:
A different approach to the problem is from left to right. You look at the 5 and notice that it is 500. So you write two zeros under the line. Then you multiply 5 x 9 = 45. So you write 45 to the left of the zeros to make 4500. Then you go to the 4, and notice that it is actually 40. So you write a zero under the last zero in 4500. Then multiply 9 x 4 = 36. So you write 36 in front of the zero for 360. Then you multiply 7 x 9 = 63, which you write under the 360 lining up the 3 in 63 with the 0 in 360.
This finished problem would look something like this:
Then you add them, 4500 + 360 = 4860 + 63 = 4923
Try both of these methods head to head with each other on paper. You may prefer the first way or you may prefer the second. But they each have a unique feel of problem solving, and both are equally effective.
Now try them without the paper. It has been my experience that the second method is much easier in my head than the first. It may help to say the numbers out loud as you work with them.
Or how about managing the problem this way? 547 x 9 = 9 x 500 equals 4500 plus 9 x 40 equals 360 plus 4500 equals 4860 plus 9 x 7 equals 63 plus 4860 equals 4923.
There is also another approach the problem, which may be the easiest of all.
547 = 550 - 3. So we could call this problem (550 - 3) x 9, or (550 x 9) - (3 x 9). Try that one. You probably won't even bother writing down the numbers. 550 x 9 = 4500 + 450 = 4950 minus 3 x 9 = 27. 4950 - 27 = 4930 - 7 = 4923.
Whichever way you find works for you is the way for you to do the problem. Each way is unique, and each gets you to the same answer, so we will assume it is the "right" answer for the sake of this paper.
But what is more interesting than precisely which method to use is the fact that there are at least four ways to work it, and many more once you really look at it. Every arithmetic problem can be solved in literally an infinite number of ways.
I was never taught that there could be any way to work a math problem other than what I was being shown. Because of this myopic approach, I was completely cut out of the process of problem solving other than to learn the steps by rote and repeat them over and over again. The numbers in the problem were set in stone, as was the method of its solution. I became nothing more than a calculator. If I learned the process and could repeat it, I succeeded at math. If I couldn't or wouldn't do it properly, I failed at math. My thinking ability had nothing to do with it. Is it any wonder that many of our young people lose interest in math at a very early age?
By offering a variety of possible methods to use to solve a math problem, we include the person doing the calculation as part of the process. They are not dehumanized in the name of learning. Every problem becomes touchable, changeable, malleable, alive. The first part of finding the answer to the problem becomes one of evaluating the numbers involved so as to find the most amenable route toward the solution. Simple multiplication becomes an active process of the mind that includes reasoning and discernment. For the students who find all of this confusing, they can find one method, learn it, and repeat it. But for the curious and engaged student, what was once a very dry, boring process where the student was primarily an observer becomes an activity in which the student plays a major role.
My personal preference on how to work this problem would be the fourth, but that would change with every individual problem. My favorite underlying method is the second one. But when we get to numbers over 4 digits long, it can get cumbersome, and I prefer the first method, as it is more compact. But which method I prefer is of much less importance to me than the fact that I have a choice. By having options, I become actively involved and the problem becomes a living entity with which I engage.
I showed this method of multiplication to a fifth grader the other day. She is my partner's niece, has memorized her multiplacation tables and knows the "right way" to work arithmetic. That's how she got the job of guinea pig. She is neither enamored nor dismayed by math or science, and is a B student. She agreed to let me show her this math because we promised her a pizza dinner alone with her aunt and myself. Her brother was not invited. After dinner, when we sat down to do some problems I made sure she knew she could stop this at any time. She said ok, and we dove into the math.
An hour and a half later, my partner broke up the math session because it was time to take my young colleague home. And she was disappointed! She had become totally engaged in the process of working with the numbers and finding the solutions in whatever way she wanted to use. We solved tens of 3-digit by 1-digit multiplication problems during that time, and I only corrected her answers twice. All the rest she got right on the first try.
Was this because she is a mathematical genius? I don't think so. It was because she was engaged and having fun. She didn't want to leave, and wants to work some more problems at the family BBQ coming up. I don't think I have ever had such an invitation before in my life. You can bet, I won't be missing that BBQ.
Maybe the invitation from my young colleague and my desire to not miss the BBQ is because learning and teaching math, or any other subject, is more fun, (and effective?) for both teacher and student if everyone gets to explore themselves through their different ways of doing things/thinking of the subject matter. In other words, more learning may take place for everyone when they are personally engaged in the activity, and different, effective approaches are welcomed, not made "wrong".
I want to acknowledge Arthur Benjamin and his "Mathmagician" show. It was in seeing him that I first was exposed to the options available in working arithmetic problems. An excerpt from the introduction of Arthur Benjamin's new book, "Secrets of Mental Math", scheduled to be published in August 2006, seems an appropriate way to end this essay ...
|"Too often, math is taught as a set of rigid rules, leaving little room for creative thinking. But as you will learn from this book, problems can be broken down into smaller, more manageable components. We look for special features to make our problems easier to solve. These strike me as being valuable "life lessons" that we can use in approaching all kinds of problems, mathematical and otherwise." -- Arthur Benjamin|