Devlin's Angle

July-August 2008

It's Still Not Repeated Addition

Well, my previous column, It ain't no repeated addition, certainly generated some interest, both in the form of emails directly to me and a thread on a popular teachers blogsite Let's play math.

The thrust of my earlier column was a plea to mathematics teachers to stop telling students that multiplication is repeated addition. Doing that not only sets up the hapless student for later confusion when they encounter situations where that definition plainly makes no sense (negative numbers, fractions, irrational numbers), it also leaves them with a fundamental lack of understanding of basic arithmetic that those of us who teach at university level encounter with every year's new intake.

I was delighted to see the blog thread. As I said in my original column, I know about mathematics and mathematics instruction at the college level, but have no experience teaching math at the K-12 levels. Thus, my only hope of influencing the way mathematics is taught - and does anyone doubt that our (US) public mathematics education system is in dire need of a major makeover - is to persuade or provoke (and then help, if I can) teachers to initiate a change. Seeing practicing teachers discuss my article was a true joy, though I have walked this earth long enough to know that one article and one discussion thread is unlikely to achieve much if anything on its own.

What worried me were some blog entries and a number of emails I received from teachers who said, in a nutshell, that I was making much ado about nothing. That it was okay to tell students something that is totally and utterly false, and then keep modifying it each time the students subsequently encountered a situation where what they were taught plainly does not work. If ever there were a case of passing the buck, that is it. Not only is that educationally unwise to keep changing the rules, I personally resent it because that buck eventually ends up in my college classroom, where I discover that I am expected to teach university-level mathematics to students who do not properly understand basic arithmetic, and have formed deep-rooted, but erroneous conceptions that get in the way of progressing in mathematics.

Of course, those correspondents were not consciously doing that. Indeed, it seemed clear from what they said that they themselves simply had not grasped the basics of arithmetic! In some cases I believe they honestly feel that multiplication really is, at heart, just a generalization of repeated addition.

Because I want to address a number of issues my correspondents brought up, this column will be a bit longer than usual, and some of the points I make will overlap with others. I did not contribute to the blog thread, nor do I respond here to posts to that blog, because I am not a professional K-12 teacher and have no experience in that, and thus have no business on their discussion blog. As I said last time, my expertise is captured by the first word in the term "mathematics teacher". I am simply trying to raise awareness among teachers of some issues in contemporary mathematics that I suspect they may be unaware of (I may be wrong). How anything I say is factored into actual mathematics teaching practice is something outside my area of expertise. I strongly believe that the future lies in both groups, the mathematicians and the teachers, working together. (Not all my teacher correspondents agree with that approach, by the way, and some seem to resent an intrusion into what they regard as their domain and theirs alone. Ah well. I also know of many mathematicians who do not want to get involved in K-12 education either.)

The driving lesson

And so down to business. Taking the "it's okay to introduce multiplication as repeated addition" argument and applying it to another domain, here is what its advocates are suggesting. Imagine your child wants to learn to drive a car. You pay for a course of lessons, and at the end of the first week you ask your child how it's going.

"What do you mean you are learning how to hitch the horse to the wagon. I thought you were learning to drive?"

"Yes, we are. But the teacher says we should take account of the fact that horse-drawn carriages came before automobiles. The two forms of transportation have a lot in common, four wheels, a chassis, seats, and so on, but the earlier form is more basic since both the "engine" and the fuel are naturally occurring (horses and hay) and do not have to be designed and built by people. So my teacher thinks the best way to learn how to drive is to first master the more basic, earlier method, that people used for hundreds of years, and then we'll be shown how to modify what we've learned to the case of automobiles. We'll learn to understand cars and how to drive them by interpreting them in terms of horse-drawn carriages, which are more basic."

Now, both forms of transport do fulfill the purpose of getting you from point A to point B in a sitting position, without having to provide the power yourself. And in some circumstances the two forms of transportation are in fact pretty well equivalent. Indeed, on some occasions, the horse-drawn carriage may do a better job. Nevertheless, by this stage in your discussion with your child I suspect you are likely to lose your cool. After all, for all the similarities, and despite the fact that one was a direct precursor of the other, cars and horse-drawn carriages are simply different forms of transportation, and your child will become a much better driver - and perhaps a good carriage driver as well - and achieve mastery much quicker, if each form of transport were taught in its own right, not one in terms of the other.

To be sure, a driving instructor might occasionally find it helpful to point out the similarities between cars and horse-drawn carriages, particularly in an age of diminishing fossil fuels. Seeing similarities (and differences) usually aids learning. But since they really are different forms of transport, used mostly for different purposes, most of us would think it better to approach them as completely different activities that just happen to have some similarities. Likewise for addition and multiplication. It might once have been okay to view multiplication as repeated addition (though I suspect not), but in today's world, that is definitely not the case.

Arithmetic for today's world

Let me be plain about it. Addition and multiplication are different operations on numbers. There are, to be sure, connections. One such is that multiplication does provide a quick way of finding the answer to a repeated addition sum. Indeed, if the only thing anyone ever needed to do with numbers is add them, either once or repeatedly, then there would be no need to have something called multiplication; there would simply be a clever shortcut to find the answer to a repeated addition.

But the world has a habit of presenting us with situations where addition simply is not enough. This happens in business, commerce, finance, science, engineering, all over the place. For instance, there is no way to understand a (continuous) volume control on a radio in terms of addition, either singly or repeated. A volume control is not an additive device, it's multiplicative. Indeed, the entire domain of scaling (of which a volume control is just one simple example) is inherently multiplicative, just as combining collections is fundamentally additive.

Addition and multiplication aren't enough for our world either, as it turns out. Biological growth and population growth are inherently exponential and cannot be understood as "repeated multiplication" (which would cash out as "repeated repeated addition" for those who advocate reducing all of arithmetic to addition).

Folks, we are living in the twenty-first century. Look around at the world our children are living in. If they do not understand addition, multiplication, and exponentiation, and are not deeply aware of the HUGE differences between those operations, there is no way they can lead informed lives and contribute adequately to society. A person who does not know the difference between multiplication and exponentiation is not going to appreciate why global warming is so very, very dangerous and will almost certainly affect us much sooner than any of our intuitions tell us.

Mathematicians sorted out the basics of arithmetic several centuries ago, and finished the job during the nineteenth century. It took a considerable effort, and changed the nature of mathematics considerably, setting the stage for the highly technological and scientific age we live in today. Just as Henry Ford perfected the family automobile, your civic-minded mathematicians figured out that today's world (yesterday's, actually) requires a number system that has three basic, but very different operations: addition (and its inverse subtraction), multiplication (and its inverse division), and exponentiation (with its inverse, logarithms), having certain, specified properties.

Now, if there were no way to teach arithmetic other than to follow the historical path, maybe there would be no alternative than to introduce multiplication as repeated addition and exponentiation as repeated multiplication, and then face the inevitable problems that come later as and when they arise. (Just as we do now, it appears!) After all, if you were faced with introducing the automobile to a remote tribe that had hitherto known only horse-drawn transportation, you might go the "horseless carriage" route. But children growing up in today's world are surrounded by examples of collecting together, of scaling, and of biological or population growth, so there is surely no reason whatsoever why we don't teach arithmetic correctly from the very start. Why introduce multiplication as repeated addition when we encounter scaling every day? Why introduce exponentiation as repeated multiplication when every day we see exponential growth in action?

Note that I am not saying we introduce addition as collecting together, multiplication as scaling, and exponentiation as growth. The mathematical operations are all abstract mathematical notions. My point is that those real world examples can be used to motivate and illustrate the basic operations of arithmetic. The real world examples all fail at some point or another - negative numbers for instance. (Incidentally, you avoid cognitive problems with negative numbers when you say from the start that number systems are things people invented to do things in the world - see later for more on this.)

If teaching arithmetic in the incremental manner some of my correspondents advocate (starting with addition of positive whole numbers and building up through multiplication as repeated addition and exponentiation as repeated multiplication) did not have any major downside, I would not be writing this particular column. But there is a downside. Remember that old adage, "first impressions count"? As most math teachers are probably aware, when you teach a new mathematical concept to someone, the way you first introduce it is almost certainly going to be the one the student retains. No matter how much you stress that the concept will later be changed in some way. Hence, every year, university professors are faced with students in their class who, deep down, believe multiplication is repeated addition and exponentiation is repeated multiplication. So powerful and long lasting is this first-model phenomenon, that several of the math teachers who emailed me clearly harbored that view! It's a view of basic arithmetical operations that causes immense problems when students start to learn calculus.

So what should be done? How about taking a look at how mathematics is actually developed and used in the world.

Core mathematics - arithmetic in particular - is not developed in order to produce more complicated or more general variations of existing math. It is developed and expanded - with new operations introduced - to do new things in the world and to meet new needs in the way we live our lives. The world we live in - and even more so the world our children are living in and will live in - provides more than enough examples to motivate and explain the three basic operations of arithmetic. Why even try to motivate, justify and explain one arithmetic operation in terms of another (something that leads to later problems) when the world we live in provides all the motivation, justification, and explanatory power anyone could possibly need? No wonder children arrive at college not only having little or no genuine understanding of elementary arithmetic, they have long ago formed the view that math has nothing to do with the world they live in - that new math simply comes from old math, not from trying to do things in the world we live in.

The need for the concrete

Part of the problem, I suspect, is that many people feel a need to make things concrete. But mathematics is abstract. That is where it gets its strength. Multiplication simply IS NOT a generalized addition, and exponentiation IS NOT a generalized multiplication. Just as you can't really say what the number 7 IS in concrete terms - it's a pure abstraction - so too you can't say what addition and multiplication and exponentiation ARE. They are BASIC, not derived. A significant part of mastering mathematics is coming to terms with that.

I personally find it odd that people feel comfortable with addition as a basic operation, not reducible to anything "more basic," but not equally comfortable with multiplication or exponentiation as basic operations. After all the "putting together collections" application of addition only provides an interpretation in the case of positive whole numbers, and examples of putting together fractions of pies only gets you addition for positive fractions. What do you do to gain understanding when the numbers are negative or irrational?

The hard step is the one that takes you into the abstract realm of numbers in the first place. If a child has bought into addition (including negative numbers and irrationals), he or she has made the leap to accepting an arithmetic operation as basic and not reducible to anything "simpler". Take advantage of that.

On the other hand, I sympathize with those people who emailed me or contributed to that blog thread I mentioned, who are clearly struggling to find answers to the "What is it?" question. It's a natural question. Unfortunately, trying to find an answer holds back mastery of mathematics, which largely depends on getting beyond the concrete and into the realm of the abstract - on recognizing that the "What is it?" question is simply not appropriate for the basic objects and operations of mathematics. "It" is what "it" is. What is important is what "it" does.

Over a century ago, mathematicians finally learned to sidestep that unanswerable "What is it?" question by adopting the axiomatic approach, where you simply specify the properties of numbers and the arithmetical operations, and concentrate on manipulating them according to those rules. As the great mathematician David Hilbert put it so evocatively at the end of the nineteenth century, the objects of mathematics may as well be bar tables and beer mugs, and the operations on them equally bar-like, provided you specify the properties of those operations appropriately. (Hilbert was focusing on geometry at the time, but his remarks hold for any area of mathematics, and he advocated adopting that approach to all of mathematics.) Hlbert, by the way, was viewed as the best mathematician in the world at the turn of the twentieth century. Among the many things he did was find (and correct) many fundamental errors in Euclid's axiomatic geometry, something that no one else had done in two thousand years of studying Euclid's classic book Elements.

The system view

In the case of arithmetic, mathematicians since Hilbert have approached arithmetic as an activity that is done within a number system. The starting point - what you are given, as basic - comprises the numbers (be they whole numbers, rational numbers, real numbers, or whatever - and there are others) together with certain operations on them. (Usually these are addition and multiplication, but you could include exponentiation if you want.) This is what the word "system" means here. The basic properties of the system are specified by a set or rules, usually called axioms. The question of what the "numbers" are or what the "operations" are does not arise - that is to say, mathematicians learned long ago that it was fruitless to ask that question, and actually unnecessary to have an answer.

I should point out that this was not done as some irrelevant academic exercise. Rather, the rapidly changing world was throwing up new problems for which the old mathematics was (demonstrably) not adequate. True, the inadequacies never show up in the school curriculum, which focuses almost entirely on mathematics much of which was done two thousand years ago, and virtually nothing less than three hundred years ago. Thus many teachers (and my main focus is on the K-8 range) are totally unaware of the fundamental changes that took place in mathematics in the nineteenth and twentieth centuries. But some of their students will likely go on to pursue careers for which they very definitely need modern mathematics (the automobile rather than the horse-drawn carriage), and embedding false initial concepts in their minds does them a grave disservice.

One feature of the change in approach that Hilbert commented on (and advocated with great passion) is that it turned the historical development on its head. The familiar path from positive whole number arithmetic all the way to arithmetic on the real numbers (as used in calculus) that many teachers (for the most enticing of reasons!) find so attractive - which is actually not the same as the actual historical development, but never mind for now - goes completely counter to the way arithmetic systems should be developed to be able to meet today's societal needs. Arithmetic (with exponentiation) on the real numbers system is the most fundamental. All other arithmetics are special cases of that. (Full disclosure: the complex number system is the one you need to start with to really do everything you need in the world. But in this column I'm focusing only on the number systems that typically arise in K-8 or perhaps K-12 mathematics. Complex numbers have no good intuitive conception other than a geometric one that is only partially effective, and so have to be developed axiomatically. But they come so late in the educational process, and the step is restricted to students who have already mastered a lot of other mathematics, that there really isn't any need for a "conceptual prop", nor is there any danger of something having to be undone later.)

One consequence of the 180 degree turnaround is that addition on the positive whole numbers is a special case of addition on the real numbers, and multiplication on the positive whole numbers is a special case of multiplication on the real numbers. And make no mistake about it, real number multiplication most definitely is not repeated real addition.

The point is, the needs of society made it necessary to produce a single number system that works for all possible purposes. The real number system (strictly, the complex number system) is that system. All other number systems are subsystems. Since the real number system is the one that connects to the real world in the most significant way, it is the one that must be taken as the default.

Of course, since this all happened just a hundred years ago, many, perhaps most school mathematics teachers are never exposed to it. (If so, it's not their fault, the problem lies with the system that educated them.) As a result, it is not surprising that some of the teachers who contacted me and wrote on that blog really did not know what my point was. They did not know that a more-or-less historical-based approach, based on reducing successive operations to ones previously introduced, is no longer the way to go. (Actually, I don't think anyone can say that for certain that a particular way to teach is not going to work. But for the reasons I indicated above, it is going to be tricky to pull off without harmful fallout for the students down the line. But now we are into the teachers' domain of expertise, not mine.)

Interpreting the new and strange in terms of the old and familiar is a natural human way of coming to terms with change. For instance, when cars first came along, people did initially interpret them as a variant of what they were familiar with. Early cars were called "horseless carriages." But once people were familiar with cars, and in particular once cars had established themselves as a primary means of personal transportation, cars were viewed entirely in their own right. They had to be in order for society to advance. It's exactly the same with number systems.

What to do?

As I remarked in my last column, I am certainly not advocating we teach arithmetic to pupils in the K-8 grades using the axiomatic method in all its formal glory. This is pretty sophisticated stuff. But, since that is how mathematicians finally resolved the issue, it makes sense to take that as our guideline, if at all possible. Over a hundred years has passed since the development Hilbert referred to gained general acceptance among the mathematical community. In almost any other walk of life but mathematics education, that would be more than long enough for the message to filter through.

[Actually, the mathematics education people did try it once, but they bungled it. That was the disastrous "New Math" movement of the 1960s. As I said last time, I am definitely not suggesting a return to that debacle. But just because the execution was so poor does not mean there were not some good ideas floating around. A principal driving force was the recognition by universities that incoming students were not equipped to learn the kind of mathematics needed in today's world. They still are not.]

By the way, adopting an axiomatic approach does not mean that the familiar numbers systems (whole numbers, rational numbers, and reals) are arbitrary inventions. You are certainly free to come up with your own system, and it may lead to some intriguing mathematics, but it is unlikely to draw much attention unless it turns out to be useful. The purpose of the process Hilbert was advocating was to make precise, and help people to understand, systems that are useful in the real world. Numbers are used for counting and measuring, in particular. The axioms mathematicians formulated for them were carefully chosen to capture all the properties of numbers that you need when you use numbers in the world - including the useful fact that when you do a repeated addition, you can get the answer by multiplication.

Yes, that useful fact is built in to the system; but as something that can be done, not as a defining principle. It is simply a property of those two operations addition and multiplication that is a consequence of the axioms, not a recipe for building multiplication from, or reducing it to, addition. Which is just as well, since in many real world instances, multiplication is not repeated addition (e.g. when you multiply any two negative numbers or multiply pi by the square root of 2).

In the absence of being able to provide a concrete answer to the "What are these?" questions, what mathematics teachers surely can do (and I would say definitely should do) is (1) use real world examples, such as collecting together, scaling, and growth, to motivate, introduce, and exemplify the basic operations of arithmetic; and (2) provide and demonstrate the use (both in a "pure" setting and with applications) of the rules needed in order to do arithmetic.

And why not let the students have the thrill of discovering for themselves (perhaps with a hint or two) that multiplication provides a quick way to get the answer to a repeated addition. Historically, multiplication may have first arisen out of addition. But as with the automobile and the horse-drawn carriage, things have moved on since then, and we've developed our numbers systems to meet the needs of today's world. Yesterday's foundational ideas have become just useful tricks today.

Let me stress again that I am not suggesting we teach children arithmetic the way professional mathematicians view it. Rather, my point is that, however you teach it (and I defer to professional teachers in figuring out the how), don't do anything that is counter to the way the mathematicians do it. Remember we are in a mathematical age equivalent to the automobile, not the horse-drawn carriage. One reason to avoid running counter to the way mathematics is used in the real world is that some of your pupils may well end up in universities where they will HAVE to do it the right way (i.e., the way we have found works for all purposes in today's world), so that they can go on to actually use it in the world. And that means the student has to realize that addition and multiplication, and exponentiation if you get to it, are different basic arithmetical operations, with no one reducible to either of the others. (Technically, exponentiation is not "arithmetic," it's what is called "analytic," but that's a distinction outside my present scope.)

If, as I strongly suspect and have suggested elsewhere, understanding mathematics can come only after mastery of technique, then that is simply a part of learning mathematics we have to live with. The "learn the technique first and understand later" approach is very definitely the only way to learn chess, and millions of children around the world manage that each year, so we know it is a viable approach. Why not accept that math has to be learned the same way? (At least for now, if you believe a better way will eventually be found.)

[There is actually some evidence that students learn faster and with a more robust outcome when it is learned abstractly, by the rules, but that's another story and not a factor I'm basing my plea on here.]

The bottom line for me is that having students get to university without a proper understanding of arithmetic is simply not acceptable in a major developed country like ours. I see absolutely no reason not to do this right - or at least to avoid doing it wrong.

For the record

For the benefit of those readers who want to see the details, the axiom systems for the different number systems are: the axioms for complete ordered fields describe the real number system, the axioms for fields describe the rational number system, and the axioms for integral domains describe the whole numbers. You can find discussions of these systems in any contemporary college-level algebra textbook.

Starting with the reals, which are a complete ordered field, if you restrict to the rational numbers you get a field (which, though ordered, is not complete), and if you restrict further to the whole numbers you get an integral domain (which is not a field). The positive whole numbers do not really constitute a number system, and so mathematicians have had no reason to write down axioms to describe them as such. At the turn of the twentieth century, an Italian mathematician called Peano did formulate what are often called the Peano axioms, but their purpose is to show how the positive whole numbers can be defined from first-order logic; they are not a descriptive axiom system that tells you how to work in the system, as are the other axiom systems I just listed.

The point to bear in mind is that, once you have specified the real number system, everything else follows, whole number arithmetic, rational number arithmetic, and all the relationships between the different subsystems. In particular, there is just one kind of number, real numbers, one addition operation, one multiplication operation, and one exponentiation operator (where the exponent may itself be any real number). You get everything else by restricting to particular subsets of numbers. The axioms do not tell you what the real numbers are or what the addition and multiplication operations are; they simply describe their properties vis a vis arithmetic. The axioms for a complete ordered field describe the properties those operations have when applied to all real numbers, the axioms for a field describe the properties the operations have when restricted to the rational numbers, and the axioms for an integral domain tell you how the operations behave when you restrict them to whole numbers.

As I said earlier, I don't think it would be a sensible thing to teach arithmetic by starting with the real number system; indeed, I find it hard to imagine how that could possibly succeed. But since that is the culmination of the arithmetic learning journey, it would be wise to avoid doing anything that runs counter to that final goal system.


Devlin's Angle is updated at the beginning of each month.
Mathematician Keith Devlin (email: devlin@csli.stanford.edu) is the Executive Director of the Center for the Study of Language and Information at Stanford University and The Math Guy on NPR's Weekend Edition. Devlin's most recent book, Solving Crimes with Mathematics: THE NUMBERS BEHIND NUMB3RS, is the companion book to the hit television crime series NUMB3RS, and is co-written with Professor Gary Lorden of Caltech, the lead mathematics adviser on the series. It was published last September by Plume.