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.
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.
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.