Procedure vs. Meaning
Math is often taught in a way that focuses on procedures rather than meaning. A child only learns the process, but not the meaning behind it. They must then practice this type of problem over and over to cement it in memory. This can be quite boring for most children and is likely the reason that many children, especially those with attention issues, dislike or do poorly in math.
The psychology of learning and memory tells us that this type of approach only leads to surface processing. A child is unlikely to remember how to do a procedure that seems like nonsense. The only way they remember it is by practicing it over and over until it is learned by rote.
A better approach is to teach the meaning behind what you are doing. This allows for true understanding. It is easier to remember something you understand and can relate to other things you already know.
I frequently see homeschooling parents post on Facebook stating that their children are struggling with math. These struggles seem to start around 4th grade. Not coincidentally, this is the same age that children are expected to learn multi-digit multiplication, long division, various operations involving fractions, and other skills.
In 4th grade, math suddenly becomes much more complex. Gone are the days of simply adding and subtracting. Math is no longer something children can easily picture in their heads.
Proceed to the Answer
Children are usually taught the procedures for doing more complex math problems. You line the numbers up like this, you space over one space or add a 0 placeholder here, you multiply this and add that.
Very often, children are not told WHY they are doing those procedures. They are just told do this, do that. Voila! You get the right answer!
This is how I was taught when I was in school and likely how you were taught as well. It works. For some kids.
It doesn’t work for a large group of kids, though.
It really doesn’t work for kids who have ADHD or other attention problems. For these kids, doing endless practice problems to commit these procedures to memory is torture.
How Do Kids Learn Best?
I’m going to come back here to the psychology of learning since my background is in psychology. We all learn best when what we’re learning has MEANING to us.
Sure, we can memorize a set of seemingly random procedures. Usually this is accomplished by doing endless practice problems where we do those procedures over and over.
We can learn things by repetition. This is how we usually learn things like phone numbers or addresses.
It is not the most efficient or effective way to learn, though.
Meaning is the Key
We learn things much more easily if those things have meaning attached to them. If we are able to tie that new learning to something we already know and understand.
For math, that means tying in the new concepts to the old concepts and showing children why we calculate the way we do.
How to Teach Multi-Digit Multiplication
For example, let’s take multi-digit multiplication. Let’s assume that your child has already learned addition with no problem. They’ve also learned that repeated addition is multiplication. They’ve learned how to do single-digit multiplication.
Next, they are shown a multi-digit multiplication problem and they are told to use this procedure:
This can be very confusing for a child. Why am I multiplying the 2 times the 4 and the 1? Why am I spacing over or adding a 0 placeholder? Why am I adding? I thought I was doing multiplication?
The Math with Meaning Method
It can be very helpful to teach children the logic or meaning behind the math problem. Let them see WHY they are doing the problem the way they are.
Here’s one way to do that for the example above:
First, teach area of rectangles and how to calculate this. Geometry is much more concrete and children can easily see what they are doing with geometry. If you teach this first, you will have an easier time teaching multi-digit multiplication.
Then, draw the rectangle as show above. Divide each side into its tens and ones components as shown. Then, draw lines to create 4 rectangles inside the larger one.
It is clear and logical that calculating the area of each one and adding them would lead to the area of the larger rectangle.
Then, show the child that this is the exact same procedure they are doing when they do multi-digit multiplication using the original method.
Voila! Now, they understand what they are doing. It has meaning attached to it. And, they have a new way to picture it in their minds and calculate the answer if they ever get confused about the procedures.
For more tips on adding meaning to other elementary math procedures, I’ve created this helpful guide, Teaching Math with Meaning for the Elementary Grades.