Multiplication is Much More than Repeated Addition

When we teach multiplication, it’s easy to fall into the trap of treating multiplication only as repeated addition. This mental model works as a sufficient bridge for bringing students familiar with addition facts into the world of integer multiplication and the familiar multiplication chart, but it is an incomplete concept that only captures the most surface aspects of multiplication.

We introduce division to grade school students by explaining how multiplying can be thought of as process of adding up one of the multiplicands some number of times. For example, posing the question of how many eggs are in four cartons, a student can readily visualize four rows of 12 eggs as 12+12+12+12, and quickly reach 48 eggs as an answer.

The commutative property of multiplication starts to complicate things, and for some students understanding that 4 x 12 is the same as 12 x 4 suddenly seems daunting. While adding up a sequence of 12+12+12+12 = 48, the same approach used to sum a sequence of 12 instances of the number four as in 4+4+4+ … 4 = 48 is more strenuous. As adults we quickly fall back to commutation to simplify obvious multiplication problems, but students who are stuck in the land of repeated addition might become mired in long sequences of trivial (and possibly error prone) addition if this is their only tool.

Using addition in this way to calculate multiplication products as a further conceptual failing in that multiplication very often results in values smaller than the problems constituent multiplicands. Consider the following word problem:

A jacket has a normal retail price of $100, but it is on sale for 25% off. What is the sale price of the jacket?

This problem of multiplying percentages involves a fraction calculation and the product will be a value less than the starting multiplicand. This concept is completely divorced from any sort of grid representation or repeated addition model of multiplication. While familiar to adults (especially us sale shoppers!) this requires a deeper conceptually understanding of multiplication as a process for increasing or diminishing a value.

Finally, multiplication of negative numbers creates another set of distinct conceptual challenges. Attempting to come up with a reasonable physical model of a multiplication problem using negatives such as (-10) x 5 = -50 is extremely challenging, yet exactly these sorts of problems appear regularly in elementary arithmetic and algebra.

When we teach multiplication, even if we start out with repeated addition, it’s useful to keep these other more complicated behaviors of multiplication in mind. Many of the assumptions we make as adults with years of experience multiplying both real-world problems and abstract expressions inure us to the points of confusion early learners traverse in their mathematical journey. By keeping these subtleties in mind, as teachers we have a chance to gradually introduce more nuanced understanding to our pupils and pave the path for their enjoyment of the richness of mathematics.

Photo credit Erol Ahmed via Unsplash.com