Supporting Communication Through Understanding A Child's Process

Earlier I wrote about the importance of supporting the development of mathematical communication in a child.  I thought break my thoughts up into chunks.  Today, I'd like to talk about listening to understand a visual thinker.

One of the difficulties in communicating with a visual-spatial person, especially a child, about how a solution is reached is that they often understand things in pictures.  For one thing, there isn't really a sequence of steps they took to arrive at "an answer."  Rather, everything is there all at once in the picture.  For another, since we converse using language, ask children to explain themselves using language, and often model using language to explain, children expect to explain themselves using language.  As children get on in Math, this might evolve into a habit of trying to explain steps in a process using numbers or equations.  We need to break out of these two boxes. 


"Every behaviour makes sense to the person doing it."  Fifteen years ago, my very wise teaching faculty advisor embedded these words in my brain.  If we can understand what a child is seeking, we can help them to solve the problem they see (in an acceptable way).  Similarly, every child has the ability to reason.  When the line of reasoning is not immediately apparent, in order to connect with a child, the first order of business is to understand their thought process so that you can meet them where they are.


For a child who sees in pictures, invite them to draw what they were seeing when they got the answer and use the picture to explain.  If they are having difficulty with the language, as many children often will since they haven't yet had much practice translating between pictures and words, offer your own interpretation and ask if it agrees with what they see or what they were thinking.  Ask specific questions to aid your understanding of how they came to a result (correct or not). 

One example is this image representing the multiplication of 32 x 43, and similar images, that have been floating around the internet in the last couple of years.  At first it seems like magic.  Why does this process work?  Well, we know that there are forty-three 32's, but that can be broken down into forty 32's and three 32's.  So how much is three 32's?  Well, 32 is 3 tens and 2 ones, right?  So three times 2 ones - that's the number of intersections on the far right.  And three times 3 tens - that's the number of intersections on the bottom.  So, 96 so far.  And then we go through a similar process with forty times 32.  

Writing the explanation into words, this process seems lengthy and complicated, but no more so than what we usually learn in school.

Here, we have 12 instead of 32, but the process is the same.  Multiply the "ones," multiply each "tens" value with each "ones" value and add, multiply the "tens," and put them all together to get an answer.  In fact, if we take the line drawing above and twist it so that the intersections in the far-right corner move up and the far-left corner moves down, we have exactly what's happening in the middle step of the numerical method here.

The lines method may be new and therefore take a little work to understand, but it's not as unfamiliar as we might first expect.  The first three methods on Bruce Ferrington's blog page, "5 Ways With Multiplication," are the same.  But look at method #5, "Russian Multiplication."  The process is completely different, isn't it?  At first glance, it looks like a scramble of numbers on a page, magic.  If we read the explanation though, it becomes understandable.

There's more than one way to multiply, and indeed, more than one way to approach any math problem.  The look of satisfaction on a child's face when an adult understands their approach is priceless.  We can always make an argument for our method:  "My way is faster; it takes less room on the paper; it's easier; it works with more numbers."  Sometimes, there is room to introduce our method.  First find something to appreciate in theirs:  "That's a neat way to look at it; I hadn't thought of it that way before; I like how visual it is; that's a powerful method; that's an elegant solution; that's similar to <this other concept we were talking about the other day>; that's a simple solution!"  What your child will hear is, "I did well.  My thinking is valid.  Mom or Dad or teacher values what I think.  I can 'do Math'."