Friday, June 19, 2015

Balanced Math = Mastery

I must give a hat tip to the folks at Mastery Maths who always have a way of pushing my thinking. In particular,  this post raised a question that had been simmering away in my mind for a while: what do people mean when they talk about Mastery? 
There is often a lot of noise in various media that students should just focus on 'mastering the basics' and that problem solving is something that should be left to later on. The post above views Mastery as the ability to apply something in an unfamiliar situation. So if a student can do a hundred long multiplications or long divisions but cannot then see which of these to use when solving word problems, it would be illusory to say that the student has mastered multiplication and division. Sometimes I get asked if I think students should learn their number facts. I reply, of course they should but that I think that the wrong question has been asked: instead we should be asking 'How should students learn their facts?'. I like what the folks at Mastery Maths write: 

The challenge is developing skills and understanding concurrently...

In our Board, we have been doing a lot of work trying to get a better idea of what Balanced Numeracy is and what it looks like in a school. A few months ago, Dylan William tweeted this:

When I saw it, I immediately thought how this could be tweaked to give a Math perspective:
I think this still needs some work but I have shared this graphic with many colleagues in my Board and they have found it a useful way of helping to understand why a good Math program will not focus entirely on skills and drills nor entirely on problem solving: they must complement each other. 
In fact, I would argue that true Mastery can only come out of a Balanced Math program.
                                       *                                 *                                 *
Out of curiosity I gave the following computation (mentioned in the post above) to a group of my colleagues: 


Even though I gave this question to them individually with a cheeky but friendly grin, a lot of them admitted to feelings of anxiety: "Would I be able to remember how to do this? Could I use a calculator? What if I get it wrong?". Now these are adults who all told me that they learned Math in an algorithm-driven manner, yet they were not confident with this question. It turns out that they all multiplied the 36 and 175 first to get 6300 which they then divided mentally by the 63 to get the answer of 100. Some even chose not to use the algorithm that they learned at school but instead used a partial products-type method. 

I would argue that if we truly understand multiplication, we might use the commutative law to see that 36×175÷63 can be rewritten as 36÷63×175 or 4÷7×175. This can now be rewritten as 4×175÷7 which is 700÷7 or 100. Or, we might see that since 175÷7 is 25, then 4×175÷7 becomes 4×25 again giving 100. This ability to decompose and recompose computations and numbers is crucial in helping our students develop a mastery of Multiplication but will not happen if all they experience is the standard algorithm.
In fact, the whole notion of decomposing and recomposing objects is something that is crucial in getting students to master mathematics. 
But more on this in a future blog...

Thursday, June 4, 2015

Some of my Favourite Measurement Puzzles

In previous posts, I have shown how we can get students to understand various measurement formulas by developing these formulas with them. Once students understand these, it is important (in the context of a Balanced Numeracy approach) that students be given opportunities to practise using these formulas with straightforward, closed questions (e.g. find the area and circumference of a circle with diameter 10 cm). At the same time, it is also important that students are given opportunities to non-standard, open problems. This is where we can help students develop their problem solving skills and their perseverance. Over the past year or so, I've collected many great puzzles from the Math community on Twitter. Below, you will find some of my favourite measurement puzzles, mainly suitable for intermediate students. Try them yourself and then see if you can get each solution in a different, more beautiful way.  
Answers are in the back of the book.