This article is a comment in my post Math War Over Multiplication which is about the ongoing debate on whether or not multiplication can be defined as repeated addition. In the article, the author also raises a lot of other issues. Read, enjoy, and comment.
What started with a few words is now a feud. While Truth is not the exclusive possession of either side, neither are the misconceptions. It’s interesting that a debate over mathematical definitions, teaching tactics, and strategies is so hotly contested. Can we define multiplication as repeated addition? That is the question, and Opinions are not in short supply. Now, added to them is another. I will attempt to persuade you that multiplication can be described as repeated addition, look into the reasons the question is asked in the first place, and talk about what that answer means to our teaching.
Before proceeding, remember that arguments involve social standing. We argue to protect our selves and our social groups from embarrassment, and losing an argument can diminish our respect and damage our social ties. Everyone who has taken a sde on this issue has emotional influences that might motivate them to argue on less than logical grounds. After acknowledging those motives at work within us, let’s agree that both sides sincerely want to teach and help students. Valid points have been made on both sides. That being said, let the argument in favor of describing multiplication as repeated addition continue.
The addition concept has been around for at least as long as humans have been recording history. Repeated addition gets tiring, so someone comes up with a shorthand way of saying, “Hey, we’d just like to repeat the addition of a certain number this many times.” According to Professor Devlin, it was about 10,000 years ago. Then, as mathematical expertise developed we came to the realization that there were many facets to consider when describing quantities. People began using words like scaling to describe multiplication. This new word gave humans greater ability to describe increasingly abstract ideas, like adding halves, starting out with non-whole portions of an original base value, or changing the direction of counting repeated times. All these cases and more make description of multiplication a chore. It should be obvious that when you have a larger concept to describe it becomes easy to overlook details and there will be more room to miss important points. Teachers should hope their students will have enough number sense that they can imagine non-whole values described by ratios before they take on tasks as daunting as fully understanding an all encompassing definition of multiplication. If they don’t, there’s going to be trouble. That is our real problem. We have a huge population of students who just do not have the ability to visualize diverse quantities and changing directions.
Now, the only problem with defining multiplication as repeated addition comes when the definition of addition is incomplete. Anyone saying that multiplication as a scaling factor and multiplication as repeated addition are not the same thing seems to be making an arbitrary point. Why limit the definition of addition? It can be easily connected to scale because it is connected to scale. And whoever decided that fractions and negative numbers are counterexamples to the repeated addition explanation unfortunately does not seem to understand either. Since giving examples using the words “repeated addition” to describe the multiplication of positive whole numbers is unnecessary, here are some examples of the more abstract cases.
Assuming four times three is four plus four plus four, a question is “What is four times one?” Well, it’s four. So how can one use the word add or addition to describe a case when there is no, you know, addition? The answer is, “Start with one whole four and don’t add any more because you already have the one you need.” There is no need to get creative. What about the question, “What is four times two thirds?” Since it is normal to start out with one or more fours, it can be explained that we are also allowed to start out with less than one whole four. Then, again, don’t add any more to that. Sure, scaling is a good way to describe this. That doesn’t change the fact that the process be can described without ever using those words. Concepts are not limited by a definition. We can and often do fashion definitions to serve our purposes.
Now maybe two fractions would seem more of a bother. But we just want a portion of a portion. And the fact is, all rational numbers can be expressed as fractions. That has always included all of the whole numbers where the words repeated addition have worked so well. Teaching these concepts could be an opportunity to talk about the standard order of operations. The multiplication of fractions can be described using processes with whole numbers multiplied like normal. And negatives are not a problem either, if you define the negative sign as it is meant to be defined. It means do the opposite of what you were doing. If you were counting in one direction, then count in the opposite direction. Two negatives? Change directions again. Now you’re going back the way you started. Really the problem has never been the definition of multiplication, but rather the definitions of addition and, especially, subtraction.
The definition of the negative sign as opposite works for exponents too. Adding a one into the process and using repeated division is how it works because division is the opposite of multiplication. There is plenty of room here to talk about the connection between multiplication and division. These are concepts that can be introduced in elementary school, but will need continuous reinforcement up into college. Ratios and proportions should be given more attention anyways. Proportions that include exponents and radicals can describe an incredible array of natural processes, direct variation, inverse variation, nonlinear growth and decay, joint variations. This is the stuff of life. And really, these ideas preceded calculus by a long shot. The impulse to change the foundation upon which calculus was based on to serve the needs of higher mathematics is misguided at best. It shows a lack of understanding of said base, and it is also putting the cart before the horse.
Speaking of the calculus, one must remember that it is just a process used to discover the fact that certain aspects of certain phenomena can be described using certain other phenomena. One in particular is when we describe relationships between values using polynomials. The phenomena are often related in that they are at whole degree intervals of each other. It bears repeating that calculus is a process used to discover facts. The relations of forces and dynamics often follow beautiful patterns, and using the limit process and infinitesimals illuminates that wonderfully. There are other ways to describe these relationships between relationships, the overarching connectedness of reality. Some relationships are difficult to describe at all, but the processes used to discover these truths does not in any way control said truths.
It seems now necessary to attack the highly esteemed pioneers who took such significant steps in turning the focus of mathematical inquiry away from physical reality. Now Cauchy, God rest his soul, was a stubborn man. This served him in his meticulous and long suffering efforts to bring rigor to mathematics and calculus in particular. Those efforts were not in vain, but he began the mistake of trying to forcibly lead non-mathematicians to the mathematician’s abstract pursuits. As mathematicians began to delve farther into the intangible, it should have been apparent to them that society could not and would not be able to follow in sizable numbers. At present, we can only afford to support so many mathematicians per hundred people. After all, someone needs to get work done. I write partially in jest, but at least Galois was aware of the absurdity of the results he strove so hard for. Not to take away from either his or Cauchy’s magnificent works, but let them be kept them in their proper place. Also, it would be best for all to remember that, while for a time it seems that mathematics had its own structure apart from the physical world, in reality it was often just describing aspects of the universe that we had not yet discovered.
Now, some people enjoy spending time in purely mathematical pursuits. It is a fascinating world of thought. These people study the deep relations of consistent arguments and extend them as they are able. Like the rest of us, mathematicians prefer to earn a living doing what they enjoy. It is often beneficial to us all that a few people devote themselves to the study of numbers and such. Indeed, we are often served by such work. However, people who are skilled in working with numbers are not often described as equally equipped for dealing with other matters. We are a society of specialists. Since there are only so many paying jobs in the field of mathematics, we put mathematicians in teaching positions, knowing that we profit from their need to share their work with other humans, thus passing on knowledge to the next generation and furthering the accumulated body of knowledge. As we should all admit however, mathematicians can be inept with interpersonal communication, which unfortunately is rather critical in pursuits such as marketing, sales, counseling, labor and, obviously, education. As many of our math professors find out, it can be difficult to share knowledge in the classroom. When this happens, as it is happening, frustration, anger, blame, denial, ignorance, and fear can surface. Professors are having difficulties explaining concepts to their students. That often leads to job insecurity, especially when it is so difficult to prove competency and effectiveness as a teacher. Our work products have a tendency to move around and don’t even bother to send us evaluations of our efforts. And seeing how there are not a lot of options for people who get paid to talk about the cyclic cut width of the nth dimension, losing a teaching job can have rather weighty ramifications for a mathematician. Since our culture teaches many to believe that they are entitled to a decent wage and a career in their field of choice, they get rather indignant at the thought of living a life in which that is no longer true. So what is the normal response? I believe the answer is to pass the buck. It is always easier to blame someone else than to take responsibility for our own choices and predicaments.
The point is, it doesn’t help much for us to identify where someone else went wrong. The best chance for success includes figuring what it is going to take to be successful in our particular endeavors and thankfulness for what we have been given, no matter how many times other people mess things up. Whatever words used to describe multiplication, if students have a decent education, the idea of multiplication as a scaling provess should make sense to them. If they aren’t prepared for that task, chances are it wasn’t because someone didn’t use a certain word. And by the way, we can stretch the definitions back a step further by defining addition as repeated counting. Counting is one of the four basic skills of math. The rest are comparing, organizing and the ability to memorize definitions and algorithms. Counting is the main goal of number operations, including addition, subtraction, multiplication, and division. Students generally have problems with non-whole numbers. This is when comparison becomes the primary task, and value comes from the way two numbers compare. Even here, counting is the basic skill we rely upon to be able to compare.
And, division is taught as repeated subtraction, despite what some people think. It can even be described as counting how many times a number fits inside another number. When we do things in algebra like the division of polynomials, there is just another level of abstraction to deal with. Nothing done at the elementary level is going to adequately prepare students for that level of abstraction. Maybe our professors should not waste time telling students that multiplication is not repeated addition when their only goal is to teach them to think about multiplication as scaling. And students have brains that are functioning for about 24 hours a day. The hour they spend in math class should not be connected every one of a student’s beliefs about multiplication. Class is a small part of teaching brains how to think, meant to hone, connect, and direct. As it is written, one sows and the other waters, but it is God who causes the increase. Pray for wisdom, we all need it.
emphasizes – makes transparent – that it is a perfect square, but de-emphasizes – leaves opaque – that it is divisible by 98.