This is the fourth chapter in Lofty Lipservice.

- First I focussed on the issues I have been struggling with
- Then the department spoke about our visions more broadly
- I planned a lesson, to put my vision of student-choice into practice
- Now I will reflect

Last term I developed a lesson on Pythagoras’ Theorem, focussing on providing students with the choice of which question to answer, and how to answer it. I have been working, across Yrs 7,8 and 9, to give more productive freedom to students.

- In Year 7, freedom in ways of working
- In Year 8, freedom in interacting with objects in the classroom
- In Year 9, freedom in developing an individual question

In a (dream) mixed-ability class, students should be working on self-generated questions, in their own way. The teacher should be ensuring the questions are rigorous, the groups productive, and the methods mathematically useful.

**Year 7: Ways of working**

The class had a healthy competitive atmosphere. Students were practising substitution, trial and improvement, negative numbers, collaboration.

**Year 8: Interacting with objects**

Finding the curvy length around a circle is difficult. We scoured the classroom for circles, measured their diameter and curvy length, and tried to spot patterns. Students enjoyed thinking creatively, searching for doorknobs, clocks, watches, stools, buttons on their iPads… Useful to emphasise that Pi is, by definition, the ratio between circumference and diameter.

Recording ratios, sum and products…

**Year 9: Developing individual questions**

I told the students that a nice triangle, for me, is one with integer side lengths and one 90 degree angle. With-holding Pythagoras’ Theorem, I asked them to search for as many “nice” triangles as possible.

Next, students developed their own definition of nice.

Some student definitions include:

- Whole-number area
- Whole-number perimeter (even if individual side-lengths are not whole numbers…)
- “Any normal simple triangle” (the result of too much freedom is ridiculous definitions)
- One side is a multiple of the others
- “Area” (some students find it difficult to ask well-posed questions)

**Further questions**

- When everyone is answering a different question it is difficult to provide meaningful targets for quantity (and quality?) of work. A student with a very restricted definition might not find as many triangles as someone who is only searching for triangles with one line of symmetry.
- I have a fine-tuned notion of what makes a good question. How to train students in this more effectively?
- Searching for circles is tangible. What about topics that do not lend themselves to physical objects like this?