So, are the two rectangles from my last post the same or different? How do you perceive them in your head?
This is your mental model of rectangles. As I said last time though, it also gives some insight into your mental model of multiplication. We can use the sophisticated term commutativity (year 4 in the Australian Curriculum) to label the idea but the word is not something that can be grabbed hold of and manipulated or walked around like a garden bed. Big words have a tendency to slip past our consciousness and glossed over if there is nothing solid to nail them to.
To be useful in helping us feel comfortable with a concept our mental models, those constructs we hold that help us perceive the situation, have to be clear and easy to bring to mind. Images and stories are good for this. Multi-syllabic words, not so much.
So, how do we assist our students in developing robust mental models for multiplication? By repetition, play, and manipulation of the physical models and by asking searching questions and requiring them to communicate their answers in verbal, written and symbolic forms.
Figure 1: Three ways to imagine 3 lots of 2
The times tables are patterns formed by groups of objects. Three lots of two objects will always give us six objects, whatever the objects are. If we can individually satisfy ourselves that this is always true then we can make life easier by memorising the most common groups of values. This is what Hattie and Yates (2014) among others call reducing cognitive load and developing automaticity. We move from a reliance on physical models, to confidence in our understanding of mental ones and from there to automatic recall. This in turn frees up our minds to focus on more complex problems.
Think about this: What is 23 times 47? Work it out and then consider the method you used. What mental model did you use? Were you conscious of a mental model at all, or did you just launch straight into a mathematical process? If you did launch in, look back now and see what your mathematical model tells you about your understanding of multiplication.
When you have done this, work out 23 times 47 again using a different approach.
Until next time.
