I came across this term from The Creativity Post.

The February 18 post attributes the original use of the term to Michael Ondaatje in his novel *The English Patient*.

In an educational sense, this post refers to the importance of allowing students time to ‘tinker’ with actual concrete objects or ideas. This appealed to me. I have advocated for a long time of the importance of giving students opportunities to be active learners by connecting to previous knowledge and construct their own frameworks. Neurologist, turned teacher, Judy Willis, also advocates giving students opportunities to ‘wake up their brains’ by providing something surprising so that students can make ‘bets’, to mentally engage their thinking networks to predict what could happen next. The act of trialling, taking notice of what worked and what didn’t and then trying something else and forming a conjecture and then testing it are also important facets of what some refer to as the ‘mathematical method’; working mathematically.

It is important to also note here that, as Dylan Wiliam pointed out in a recent tweet, students being active in their learning is not the same as the constructivist “active learning” that seemed to downplay the role of the teacher in actively directing the learning that was meant to be taking place in classrooms. Teachers have a very important role in knowing what learning intentions are involved in a lesson and designing, and directing, learning activities that lead purposefully to those learning intentions. See this article for further reading on this.

Let me tell you about two recent classes I had.

In the first, we were looking at surd arithmetic. Previously to this particular lesson, students had been given the homework task of looking up Google to discover what a surd was. We discussed this briefly at the start (MUCH better than me telling them!) then I set up four investigations using the students’ CAS calculators. In each investigation, students were given a list of surd objects to input into the calculator and they were to write down what the calculator gave them as output for each, notice any patterns and think about what the calculator could be doing to produce the result it did.

For example, the first investigation was all about simplifying surds and students were given things like rt(12), rt(24), rt(8), 3 x rt(20) etc. The answer was not the object, the calculator provided the answer. The task was to think about how the answer was obtained. What mathematics was happening?

In the second investigation, I gave them a number of addition and subtraction problems involving surds such as rt(5) + rt(3), rt(12) + rt (3) etc. My learning intention was for students to link what they had discovered in the first investigation about simplifying to the one involving adding and subtracting.

The next investigation involved multiplying and dividing surds that didn’t require rationalising.

The final one was about rationalising denominators.

During each investigation, students were free to talk to others about what they thought was happening, I wandered the room to check on progress, to ensure some weren’t heading down a misconception path, to ‘poke’ thinking and to give hints where required. At the end of each investigation, students had to write down their conclusions in their own words.

And it worked brilliantly. Taking away the necessity of getting an ‘answer’ meant they could focus on noticing patterns, making conjectures and testing them out on the next one to see if their method ‘worked’. I then held plenary sessions to ensure all took away the correct language and ways of describing the mathematics. Even those who didn’t ‘see’ the patterns were eager to know how and why the surd arithmetic worked as they had invested their time and thinking into it beforehand..they’d made a ‘bet’ and had a vested interest in finding these things out. And the risk-taking behaviour increased. Some students made significant improvement in believing in the importance of learning from mistakes made.

The second lesson I want to write about is one on matrices. Matrices can be a rather dry topic and quite challenging to students who have never seen anything like these mathematical objects before.

I started by grabbing attention using a clip (“Red pill, blue pill”)from the movie, *The Matrix*. For those who don’t know it, it is when Neo is offered a blue pill, which, if he takes it, will ensure he wakes up the next morning and go on in his world as he has before. If he chooses the red pill, however, he will be shown the reality of his world and he will learn more about it than he may have ever wanted to…the veil will be lifted. I had some red jelly snakes cut up into pill size and, when they’d watched that clip, they all got a ‘red pill’. I told them that they were going to enter a new world of learning, whether they wanted to or not…but it would be exciting and different!!

I then showed them another clip I’d found on YouTube about the basics of what a matrix was, what the dimensions of a matrix referred to and what the elements of a matrix was. They were instructed to listen carefully then complete a summary of matrices I’d given them. This summary was set out like a mind map with headings and connecting arrows but blank explanatory boxes. After watching the video, they completed the boxes on “What is a matrix”, “What is an element” in their own words.

We then went through how to define and enter a matrix on their CAS calculators.

I didn’t want to just tell them what matrix arithmetic looked like. I wanted to engage their imaginations, I wanted them to invest some thinking and predicting first, to get a surprise and take notice.

So I asked them to make any size matrix they wanted to, call it B and evaluate 2xB, 0xB, -3xB, 0.5xB etc

As I had with the surd investigations, they had to draw their own conclusions and write them down.

We then had a class discussion about what they discovered.

Then they were asked to make another matrix and try and add it to their matrix B.

Some got answers, most did not.

I challenged them to find a matrix that could be added to their matrix B. Some students noticed that the error message the calculator gave them (Incorrect dimensions) was a good clue and tried another matrix with the same order as B.

We, of course, concluded that only matrices of the same orders could be added/subtracted.

In future lessons, I will use a carefully guided investigation to investigate matrix multiplication, starting with square matrices, noticing patterns and then extending to other matrices.

I was so pleased with the way in which both of these lessons played out. I was not merely a ‘facilitator’ of their learning but I was never ‘sage on stage’ either. An awful amount of thinking, on my part, went into forming these lessons so that the learning was carefully and deliberately planned. It was not only my thinking that was important but also the way in which I wanted to develop and lead my students’ thinking.

Lots of thinkering.