Too often, I feel that Math teaching rushes to the symbolic representation of the problem and downplays the importance of the diagrammatic and concrete representations. Which is strange really: if I get stuck on a problem, one of the first things I do, is draw a picture to help me better understand what is going on. And if this doesn't help, then I'll get some hands-on materials to help me figure out what to do.When solving problems, probably the most common question that students ask their math teachers is "What do I have to do?" Often they want us to tell them what they have to add, subtract, multiply, divide (or any combination thereof). Over the years, I have become more convinced that this is because students haven't developed a schema of what the operations look like in real-life. Instead they are reduced to looking for keywords that might (or might not) be in the question and that might (or might not) actually mean what they are supposed to mean.
This is why I have been recently singing the praises of bar models to the educators I work with. These are sometimes referred to as 'Singapore Bar Models' due to their extensive use by students in that country to solve problems. However, I saw bar models when I began teaching in 1990; they were used to illustrate how to visualise percentages in the SMP Red series texts. And, I suppose, my first concrete experience of bar models as a student, was when I worked with Cuisenaire rods as shown in this earlier post.
For those unfamiliar with bar models, see how this Grade 7 student used one for the very first time in solving the following problem:
A crowd of 2400 go to see the local hockey team play but as they are doing so poorly, three-quarters of the fans leave at the end of the first period. A further third of the remaining fans leave at the end of the second period. How many fans watch the third period?
Much as I love bar models, I don't think it is a method that students will come up with themselves independently. It will require a lot of co-ordinated effort and consistent modelling from teachers all the way from Kindergarten up. I will post some ideas on how we can develop students' efficiency in using bar models in future posts, but in the meantime, it is worth checking out these summaries here and here.