Multiplication at NCCL
“But that’s not the way I learned to do multiplication.”
“Is my child ever going to learn the real way?”
By Marilynn Magnani, Previous Educational Director and Group 3 Teacher
These are concerns I’ve heard many times from parents who watch their children solve multi-digit multiplication in a way that seems so unfamiliar. I’ll try to shed some light on how and why at NCCL we teach multiplication the way we do and how it helps build number sense.
I have watched children do mathematics closely during the past twenty years. I have seen, over and over again, the confusion that children experience when they are taught procedures that make little sense before they have a solid understanding of what is happening. Teaching a procedure before there is understanding interferes with number sense.
For many, myself included, the algorithm or procedure we learned for multiplication is as follows:
Now, when we traditionally begin to teach this type of multiplication, at about age 9, we take for granted many things about what children know. First of all, we assume they have an idea of the magnitude of the answer they are to get. Second, we expect they can hold several abstract concepts in their mind at once. Third, and most critical, we assume that children have a solid understanding of place value so when they multiply the 3 by the 7, for example, they understand that it’s really a 30, not a 3 they are multiplying.
The fact that it is a 30, not a 3, is obscured by this algorithm to the point that most children, and some adults, lose track of any number sense and go on pure memorization of procedure. The problem is, of course, that if memory fails and the procedure makes no sense, we are left with nothing to fall back on.
Yes, yes, I know, many of us survived this senselessness and learned to understand despite it. But it took a lot of us years to put it together and caused us to feel that math was a bag of tricks and procedures that made little sense.
I would love to have a dollar for every parent, over the years, who said, “I was never good at math.” Some got lost during multiplication, others when they hit long division and many when faced with fractions. At NCCL we believe that most of these casualties could have been avoided if number sense was built before rote procedures were taught.
On the surface, teaching algorithms seems like a good idea. It appears efficient, usually it can be memorized after sufficient practice, and it seems faster and easier than teaching for number sense. Unfortunately, memorization is not understanding and when we don’t understand something which we are required to do over and over, frustration and a feeling of inadequacy is the result.
So, let’s take a look at how we tackle this problem at NCCL. First, let me state that teaching for understanding takes time; there is no way around it. I like to look at it as paying up front for quality and reaping the benefits over the long haul.
In the youngest classes, the concept of multiplication is introduced with objects and using real-life situations.
There are four children and each has two sneakers. How many sneakers are there altogether? There are five tables and each has four chairs. How many chairs are there altogether?
Children count by 2’s, and 3’s and so on looking for patterns. They group and regroup bundles of unifix cubes or dinosaurs. By the time they reach 7 or 8 they are beginning to record multiplication problems with symbols and some begin memorizing facts. By this age, many students know the 2 and 10 times tables and individual common facts.
As they memorize the multiplication facts, work goes on with building an understanding of place value and what happens when you multiply by 10 or 100. This is all done many times with models and pictures and much, much talking together.
Children spend a lot of time building models of multiplication such as 3 x 4, 30 x 4, 40 x 3, and 40 x 30 and comparing them. How does 4 relate to 40? How does 3 times 4 relate to 30 times 4?
At the same time, complex multi-digit multiplication problems are being solved by the students. Some stick with repeated addition for a while, others use a combination of addition and multiplication. Many different strategies are used depending on the problem being solved. A child might solve 8 x 40 something like this:
Two 40’s are 80 so four are 160, so eight must be 320. Later on, she will see that 8 x 40 is ten times larger than 8 x 4 and be able to solve the problem in one step.
That same child might solve the problem 12 x 15 like this:
Ten 15’s are 150 and two more 15’s are 30, add that to 150 and get 180.
When a child can think like that, you know she has great understanding of numbers and computation. By the way, for this problem, her strategy makes much more sense than setting the problem up in the standard form.
When one and only one way or algorithm is taught for all problems, children will use it even when it would be much simpler to figure the answer another way. This is because if shown only one way they come to believe that it is the only way to solve a problem. They overgeneralize and lose a sense of other strategies.
As the children begin solving larger problems they find ways of recording so they can keep track of the steps. One way that we encourage is shown below.
86 x 45 = (80 + 6) x (40 + 5)
80 x 40 = 3200
80 x 5 = 400
6 x 40 = 240
6 x 5 = 30
When I talk with the students I am always checking to see if they are making sense of what they are doing. If they can explain their work and answer my guiding questions I can be pretty sure they understand. Too often I have asked a child who uses the standard multiplication algorithm why he has put down 0 and he answers, “I don’t know, that’s just how you do it.” He’s learned not to question but just accept and that is not good for continued mathematical development.
So, do we ever allow a student to use the standard form? Yes, but only if he or she can explain what is happening. We believe that procedures which a student truly understands are useful tools; procedures which a student doesn’t understand are a hindrance.
But, can’t we just tell them what the algorithm means and they will understand? This is one of the great myths of teaching and learning. Telling rarely results in understanding. It is through experience and reasoning about that experience that understanding develops.
As we say in our brochure, “Tell me, I forget. Show me I remember. Involve me, I understand.”