You can do maths by:
- Picturing structures and using them to support your numerical calculations
- Using learned results
- Using processes (algorithms) which were at one time derived from mathematical structures (often by someone else).
Most people use a mixtures of these strategies, rapidly switching between them depending on the particular calculation.
Mathematical structures were clearly more evident in the Chinese classrooms than in most western classrooms. We spend a great deal of time teachign our students processes (algorithms) which were written down long ago and most mathematics teachers are not good at demanding students give structural justifications for their answers.
So given that both teachers and students tend to be lacking these structures, how do we 'put them back in' to mathematical thinking? As a teacher who's unsure of them with students who don't want to bother to try, where do we begin? If you're thinking along these lines then the following may work for you as it did for me.
What is the exercise? - It's just a quick starter I use often.
Write on the board: SDPQ 2, 8 (it doesn't have to be 2&8 but you always need two quantities and it makes sense to start with small integers).
Tell the students that when they see SDPQ they have to write down:
then they have to try to add, take, multiply and divide the numbers.
But there's a catch. They only get a mark for an answer if they can draw or fully explain their answer. No tricks allowed! If they get the righ answer but have used a method they can't fully justify they only get half a mark. =D
Some students will quickly write down four answers. Have they found them all? Are they sure?
Students will come to understand that addition and multiplication are commutative. You can use this word as it will make sense to them as giving vocabulary to something they can clearly see.
What about difference? Is that commutative? Division clearly isn't (4 and 0.25 are both possible answers). But can we have a difference of -6 or is difference always positive? I tend to deliberately allow this to be a moot point. We usually have a vote on it and the class splits. Why does it matter? We can decide. It's only vocabularly. I tend to give a mark for 6 and a mark for -6 but I don't clearly define what difference is. I want my students to learn to fight for what they instinctively feel is correct. I want them to experience how arbitrary mathematical vocabulary is. Once we decide that answer that fight is curtailed so we never do. I'm making a point. "Who cares what anyone else says? - If it's not true to you don't accept it."
Then we mark the exercise out of 6. To get a full mark a student has to be able to draw a picture of the calculation they did. Can they? If they can't but they got the right answer it's only half a mark remember. Being a bit of a pedant I also knock off half a mark if they didn't write down the words - sum, difference, product and quotient because I find that unless there's marks in it they don't bother and if one doesn't bother they all stop and that's a shame.
What structures did they use? Did they use number lines for addition and subtraction? Did they use money? For division did they use splitting (8 circles split into two groups, 2 cakes split into 8 parts) or chunking (how many 2s in 8, how many 8s in 2) or did they use another structure? How did they 'see' multiplication? Cherish variety, cherish the student who tries to explain but can't quite. Take time to make sure that everyone can see that you value all attempts students make to express their own thinking and to innovate.
Enough for today. We've covered one 5-10 minute starter so far. There will be more and they will be similar but each will challenge and develop students' and teachers' structural thinking.