Using pictures as a tool to justify

One of the things that is particularly interesting to me right now is how we help young students develop the ability to justify why things work in math. Often, kids know the correct rule but have no idea why the rule works. When I ask them to explain why a rule works, they wind up just listing the steps of the procedure.

Part of developing a conceptual understanding in math is being able to anticipate what procedure to use and why that procedure works.  For example, when I am trying to convert a mixed number into an improper fraction, I don’t simply need to know that I multiply the denominator by the number of wholes and add the numerator.  I also need to know why that rule works and why it will give me the correct number of parts in the mixed number.

I’ve found that children (and even adults) can have a really hard time writing or even explaining out loud their justification for why something works.  They often say “I know why but I can’t explain it.”

Recently, I have noticed that drawing diagrams or pictures is one way to help them begin to justify.  It’s as if the diagram allows them a way to make explicit what they are doing when they perform a calculation.

For example, the other day I was working with a group of elementary school teachers on fractions.  They worked in groups to draw the pictures like the one below to show why the calculations they performed worked.


It was an interesting experience for them because they had to think deeply about what it really means to say, multiply fractions, as opposed to just remembering the formula.  It also gave them something to refer to when explaining to the rest of the class.

I’m currently digging through the research on this, but I’m curious to hear what your experience has been. What do you do to help student justify in math?