In reading through the multiplication-related blog postings of others while pondering multiplication and division as inverse operations, I came across Keith Devlin’s articles (original, follow-up, more, most recent), which led me to wonder about my own concept (or lack thereof) of multiplication.
I have a vague recollection of learning multiplication tables from flash-cards at home. When I could not remember a particular product, I would figure it out via the repeated addition model. So, I think my primary concept of multiplication (even today) uses the repeated addition model. This is probably due to two factors: being taught that way and the inherent appeal of, as Mr. Devlin put it, “reducing all of arithmetic to addition” (describing all arithmetic operations as being tied back to addition in a strong way).
Yet, in reading Mr. Devlin’s postings, I realized that I am also comfortable with the idea of multiplication as scaling. I may not rely on it as often, but use it without hesitation when needed. This led me to wonder when I first gained this perspective… my guess is that it is one I gradually became comfortable with while multiplying by fractions and solving similar triangle problems in Geometry. Thus I now have two models of multiplication rattling around in my brain, and while there are some links between them, they remain largely distinct and unreconciled.
Taking Mr. Devlin’s perspective, I should not “reconcile” my two conceptions… I should rebuild my primary conception of multiplication from the ground up. However, he also states that “the ‘What is it?’ question is simply not appropriate for the basic objects and operations of mathematics” since they are all abstractions to begin with. While I can accept this from a mathematical perspective, I firmly believe that analogy, metaphor, and concrete experience are all critical to building confidence and understanding in most students.
So how can multiplication be introduced in a concrete or metaphorical way without relying on repeated addition? Few of the examples I have read to date in various blog postings or articles grab me, so this posting represents my first attempt at thinking this through.
Since I wish to define multiplication on its own, without reference to a simpler precursor (addition), I thought I might “warm up” by describing addition in a similar way.
Addition Model
The mathematical operation of Addition applies to both abstract and real-life quantities. Addition combines two quantities to produce their sum: both quantities taken “together”.
Zero is the additive identity, the number which has no effect on a sum. If you wish, you are welcome to assume that all addition problems have a starting value of zero.
Addition only makes sense when applied to like quantities:
Adding unlike objects does not make sense unless I can somehow generalize them into some sort of like objects:
A student’s understanding of addition (from my experience) seems based almost entirely upon their life-experience in working with like objects: building blocks, books, pieces of fruit, shoes, toys, etc. Thus, geometric models can also work nicely:
– a line segment five units long added to (put end-to-end with) a line segment seven units long produces a line segment 12 units long
– five 1×1 squares on a plane added to seven more such squares result in 12 squares covering part(s) of the plane. Note that the addition result tells me nothing about what portion(s) of the plane is/are covered, or the shape (if any) formed by the squares… it only tells me that I have 12 of them.
Multiplication Model
The mathematical operation of multiplication also applies to both abstract and real-life quantities. Multiplication scales or stretches/shrinks one quantity by the other to produce the product of the two quantities: the quantity obtained by scaling one quantity by the other.
The number one (1) is the multiplicative identity, the number which has no effect on a product. If you wish, you are welcome to assume that all multiplication problems have a starting value of one.
Unlike addition, multiplication can work between like quantities, between a quantity and a scalar (a scaling quantity with no units), or between quantities with different units:
While multiplying many units of measurement can make sense, particularly when solving physics problems, there will also be times when multiplying different units do not make sense:
When multiplying a dimensional quantity like a line segment by a scalar, the “groups of” approach can seem to make it dimensional… yet the line segment that is scaled remains within its original “parent line”. The following illustration shows a three unit line segment within a line, followed by how it would appear after being multiplied by scalar quantity of two. Note that it remains within its original dimension (the line).
Multiplying a line segment by a negative scalar quantity will cause it to rotate by 180 degrees about the origin while scaling.
The example above shows that multiplying by a scalar does not change the units of the answer. The original quantity is stretched or shrunk, and keeps its original unit of measurement.
Another way that has been suggested to think of this example, or indeed any example that involves multiplying by a scalar, is by using groups. While this approach is certainly a valid view of multiplication, it seems to risk leading students into repeated addition mode:
Its usefulness also seems to depend on what you are dealing with. Oranges lend themselves to being thought of in groups nicely… they aren’t really measured in an end-to-end way. Inches, however would seem rather odd when treated this way because they are usually measured in an end-to-end way… is the answer a pile of inches or a length?
When multiplying a dimensional quantity by another dimensional quantity, things get interesting. Multiplication pulls the initial dimensional quantity into one or more new dimension(s) that is/are at right angle(s) to the original. This can often be readily visualized while we are still within our familiar three dimensions, but can be tough to visualize beyond three dimensions.
Multiplication can create a quantity that has more dimensions than the starting quantity. Starting with the same three unit line as in the previous illustration, if we multiply it by “two units”, instead of just “two:
we end up with a rectangle that has a 6 square-unit surface area. If we were to multiply 3 square inches (a two dimensional quantity) by 2 inches (a one dimensional quantity), we would end up with 6 cubic inches (a three dimensional quantity).
This visual model of multiplication seems to work fairly nicely, and is appealing because it seems it should generalize nicely to handle multiplication by scalar imaginary quantities (rotation by 90 degrees about the origin into the imaginary dimension while scaling). However, multiplying one scalar by another, which is what we all did as we learned our multiplication tables, is a completely abstract concept… and is not well represented by any visualization (unless units are attached to the quantities being multiplied).
To any who read this and wish to offer further improvements or alternatives to the above, I will be most grateful.
MathMaine 
Killer thought-provoking stuff… about the “repeated addition” thingy – I’m quite the same way, m’self. Funny, back when I was school-age, I remember learning that calculators and such actually use the ‘repeated addition’ technique, at the speed of electrons, to do multiplication problems. I suppose it’s just the Path of Least Resistance at work hehe ^_^
I’m inclined to suggest that in the 5 oranges times 7 example there’s something missing, namely that the 7 is still an abstraction. Hence, to make this sort of example concrete, you might consider the following as more ‘real’:
5 oranges per group x 7 groups = 35 oranges. I’m not sure we’re scaling oranges, but rather groups of oranges.
While the concept of a “scalar” seems almost the ultimate abstraction, it also dovetails very nicely (to my way of thinking) with the goal of conveying the abstract concept of “scaling”. Thus the appeal I saw in that example. It also allowed me to distinguish between situations that involve units or dimensions and those that may not.
Taking your lead, I could eliminate all “dimensionless” situations by substituting “groups” in as the dimension/unit. However, this somehow feels more awkward to me when/if the factor is not an integer (5 oranges per group x 7.2 groups) than it would if I left out all mention of “groups”. Obviously, I am still wrestling with this.
The “per group” approach, which I have been reading about in a variety of places, feels to me as though it pre-supposes (or foreshadows?) a familiarity with division… which has presumably not been introduced yet. Once students understand both multiplication and division, I think it is a great approach, which allows “unit cancellation” or “dimensional analysis” to be introduced as a solution-checking approach. But until then, I am wallowing around in search of an appealing approach to introducing the concept of multiplication (not the processes) that does not invite repeated addition to the party, and keeps division at arm’s length for a while too…
A wonderfully clear “thinking through” of the concept of multiplication!
When investigating multiplication with third and fourth graders, I found it helpful to have students build arrays of unit blocks (centimeter cubes) and then to represent what they’d built on centimeter graph paper, labeling and describing their arrays. Their descriptions included the number of cubes on the vertical and horizontal sides of the rectangles they created (2 by 3, or 2×3) and the area (2×3=6). Discussion of 2cmx3cm=6 cm sq came later.
It was a fairly easy step from that array practice to labeling factors and product and to understanding the relationship between multiplication and division (as a “missing factor” problem).
Does this approach lead them to understand multiplication as repeated addition or as scaling?
Does using an array model with integral factors, or indeed any model with a series of discrete equal pieces, risk introducing/confusing repeated addition as the foundational concept behind multiplication? While my example did use integral factors, it appealed to me because it did not contain multiple discrete and equal pieces, and could easily be modified to use either rational or irrational factors.
The “repeated addition” concept seems easy to grasp compared to “scaling”. As a result, I suspect many students (of any age) will “discover” the repeated addition link for themselves and call it a day (instead of grappling with the concept of scaling) because it offers such an appealing connection to the first arithmetic operation they learned. Having been there myself, I am trying to find a model that provides a solid foundation for a scaling conceptualization of multiplication as a student’s first (or later) exposure to multiplication, one which does not make it “too easy” for students to see a repeated addition link.
I’m so impressed with your insistence of children understanding multiplication. If only memorizing correct answers, they are likely to have trouble with all future math.
I’m retired now but taught for many years. Please see my four models of multiplication that I taught to my advanced first and second graders. The models promote a comprehension of various ways of understanding of multiplication. They loved the exercises and did very well. These models could be used for any age involved in learning multiplication.
http://peggybroadbent.com/blog/index.php?s=Models+of+Multiplication