So, what does 23 times 47 equal? Of the two methods you used, did one feel easier than the other? Did you use the easier one first or second? How automatic was your recall of 2 times 4, 3 times 7 and the like?
When asking our students to work out similar questions, what methods and strategies do we tell them are acceptable, both implicitly and explicitly? How many strategies do we accept?
The answers to these questions should vary depending on the year level we are teaching. Older students should have a greater number and a wider range of sophistication in their strategies than younger students. What I want you to consider though is how many different ways to multiply numbers together are students actually taught and why are these taught and not others?
At the end of this post are two pages with twelve different written methods for multiplying numbers together. Take some time now to look at each one. The Western Written Method is first and the Italian Lattice Method (year 5 in the Australian Curriculum) is second, followed by two variations of the Grid form. I will wager that most readers will have used some combination of two of these four in the challenge from the previous post. Have a look at these four as a group and place them in order of most user friendly and intuitive to least. Where does the Western method fit? Why is it that many text books and teachers still insist on this being the “acceptable” method for “doing” multiplication?
In terms of developing understanding, in what contexts and at what year levels could the other eight methods be incorporated? How can these other methods be used to introduce or explore other topics such as Algebra, Quadratics, Indices and Logarithms, Bases other than ten and Place Value?
To illustrate the point a little here is an experience from my own teaching. One year I had a rather weak year nine class. It was toward the end of second term in Melbourne.
It was a Friday. It was wet. It was the last period after lunch, a full 75 minutes. The topic we were working on at the time was expansion and factorisation of binomial expressions. Happy days.
I started the class by multiplying numbers together, a one digit and a two digit and then two two digit numbers. I introduced them to the Grid addition method which none of them seemed to have seen before even though it is a lovely approach for year 5 alongside the Lattice approach.
I had one boy in particular, a tall, thin Sudanese who worked harder at not doing work in class than doing it. Most lessons he would have to borrow a pen and paper. I’ll name him Marley for the purpose of the story.
After demonstrating the Grid method a few times I then picked “volunteers” to work through their own examples on the board. Next, we replaced the two values in the tens positions with letters and continued as before.
After about an hour of this Marley asks if he could try the next example. Sure. He came up, worked through the entire problem perfectly first time, and went back to his seat with one of the biggest grins I have ever seen.
At the end of the lesson I asked the class what they thought and the answer I was given was “Best lesson ever!” This is why it is so important to have more than one way of being able to multiply.
Think about this: What is the connection between times tables and straight lines?
Until next time.