This post describes a couple of things that happened during a couple of lessons with J and K, who are studying Higher maths. The first incident occurred when going over a question they found difficult from a homework:

We had worked for a while with dot products, but had not talked about expressions of the form

**a**.(**b**+**c**), had not yet discussed whether**a**.**b**=**b**.**a,**or that**a**.**a =**|**a**||**a**|. I wanted them to explore these ideas for themselves. Here is K's work on the problem:

I found this noteworthy because K managed to convince himself that

**p**.

**q**=

**q**.

**p**using a general algebraic form for

**p**and

**q**directly.

He then created a general algebraic expression for

**q**.

**q,**but switched from a purely algebraic approach to substituting

*possible*values for q1, q2 and q3 into his expression. What can not be seen from his work is that he then stated that,

*"It didn't matter what the values for q1, q2 and q3 were,*

He demonstrated this verbally by giving other examples for q and showing that

**q**.**q**would always be 16 because |**q**| = 4."

**q**.

**q**must always be |

**q**||

**q**|. We then discussed how

**a**.

**a**= |

**a**||

**a**| for any vector

**a**by returning to the algebraic structure. This is a different way of generalising about structure, from an example to the algebraic, rather than directly from algebra.

The second notable incident occurred when I presented J and K with this image from an exam question, and asked them to create a multi-part question about it.

Here are J's questions:

And here are K's:

They solved their own questions, before giving them to the rest of us to solve. This 'simple' act of asking J and K to create the questions had a number of beneficial effects:

- It gave J and K an opportunity to think about the structure of these types of multi-part questions.
- It gave them an opportunity to challenge themselves, and each other, which created an energy.
- I also wanted to solve the problems. John Mason calls this
*'being mathematical with, and in-front-of, learners'*. This too created an energy, an atmosphere in which we were doing mathematics together. It also revealed how I might solve these problems (i.e. creating worked examples) in a genuine/natural way, and how I might also make mistakes. - The values in the problems were unwieldy (they were not 'nice' numbers), which had the (paradoxical?) effect of drawing attention to the structure of the problem.
- On completing the problems, the only way of identifying whether answers were correct was to compare and discuss.

Here is our work on J's questions (click to view, then use left arrows to scroll through):

And here is our work on K's questions (click to view):