I’m working through a free online class from Jo Boaler, author of What’s Math Got to Do with It?, a recommended read, (registration may be open still, and the further in the class I get, the more I recommend it). In the session I finished yesterday–“Appreciating Algebra”–she differentiated between two different kinds of algebraic thinking:
First, there’s procedural algebra. That’s generally when a variable stands for just one number, and we ask students to find that number. You know:
2x – 7 = 7. Find x.
But there’s also structural algebra. This is when we use variables to abstract and generalize–where rather than having just one value, a variable is used because it allows for flexibility, so that we can apply the same rule various circumstances. So, for example, we could build a rule that says in the nth case with this pattern, there will be 1 + 2(n – 1) blocks.
That’s powerful for various reasons. One big one is that it allows the algebra to rest on top of other ways of understanding. Even before students wrap their head around variables, they can see what the pattern is here: that in each new instance, two new blocks are added. They can see that in the first case, there was just one block. We can move from that intuitive understanding to an idea of how algebra and variables allow us to generalize–and not just extend this pattern to the fifth, sixth, seventh case, but tell us exactly what these shapes will look like in the 100th case, or in the 1000th case.
One study showed, though, that 73% of 13-year-olds believed variables could only have one value. That is, they could only understand procedural algebra. That’s because we spend so long at the beginning of algebra classes (or units, in earlier grade levels) doing that kind of work.
So how do we fix that? Research shows that an introduction to algebra should start with exploring patterns–without any algebraic symbols. Rather than asking closed questions (“how many blocks will be in the 5th shape?” “how many blocks will be in the hundredth shape?”) ask open questions (“how do you see this pattern growing?”, which in visual terms could have multiple correct answers), that help them understand that this is not a matter of right and wrong answers–it’s a matter of thinking and visualizing. This can be done with interesting, real world problems–and we should ask students to show their thinking with diagrams and tables of values. Then work up to doing structural algebra, representing the general case with variables that can take on multiple values. Only after that should students see structural algebra.
What do you think of all that? How does it shape the way you’ll approach your first units? Share in the comments, or in the Facebook group!