# Critical values with... Sam Hoggard

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- Critical values with... Sam Hoggard

## Critical values with... Sam Hoggard

Sam Hoggard is the Head of Mathematics at Wood Green School. He completed a Master’s degree in Teachers’ Professional Development from Cambridge and is a curriculum tutor on the Oxford PGCE course.

**1. If you had to sum up your school’s view of mathematics as a subject in one sentence, what would it be?**

This quote from John Mason sums up our view of mathematics lessons:

“A lesson without the opportunity to generalise mathematically, is not a mathematics lesson.”

“Generalise” does not mean formal algebra. I take this to be more akin to something I can use in another situation. I’ve found Dave Hewitt’s *Approaching Arithmetic Algebraically* article to be a good framing for this. In this article Dave argues that arithmetic requires one to think in generality; for example, understanding the number system so that 1089 is understood to mean one thousand, zero hundreds, eight tens and nine units.

**2. How does your school embed numeracy across the curriculum?**

The teaching in maths encourages students to develop a deep relational understanding of number so that students are flexible in their approaches. We do not currently exploit links between departments very much although this is something we would like to do more of.

**3. What is your view of mathematical ability and how is that supported by your school policies?**

I believe that all students have the capacity to reason mathematically and should be given opportunities to do so regardless of where they are at on their mathematical journey. I have taught in mixed attainment groupings at previous schools and found that students can still focus on the same ideas at different levels of depth. I also worked with Mike Ollerton on writing *Planning for Teaching GCSE Mathematics with Mixed Attainment Groups*, an ATM publication. During the writing of this we focussed on how key topics at GCSE can be accessed by all students.

**4. If I came into a maths lesson at your school, what would I always see? Sometimes see? Never see?**

You would always see care for students’ development, both as mathematicians and as people. If I feel it is appropriate, I will happily take time away from mathematics in a lesson to encourage students to be kind to one another. I think you would also see care for the mathematics itself, by which I mean exposing students to what (at least I perceive) real mathematics is. I also hope you would see where these two intersect – a place where students are being cognitively cared for, where their mathematical journey is considered. This involves not teaching for short-term wins or goals being focused on certain questions being answered. For example, in KS3 I would not see the goal of a lesson to be to expand single brackets – a skill that is easily forgotten if it holds no meaning to the student – but rather I would see the lesson to be about the distributive law and different places that arrives, and why it holds true in certain instances.

Another thing you would see is a focus on problem solving heuristics and metacognitive strategies, so rather than the goal being to answer a set of questions, the goal is to learn some strategies that would allow us to solve problems which are similar to the ones we do.

An example of these two things together is in the teaching of solving linear equations, an approach based on the work of Susan Pirie and Lyndon Martin. The first question students see is this one.

Students are then encouraged to come up with a strategy for solving problems like this and to justify why this works. The nature of listening to students and developing their narrative of equations is the cognitive care. The focus on what is possible and when that may be useful is about a tool students can use elsewhere in mathematics. And starting with two-sided equations, which leads to a balancing approach, is about developing a strategy that will be useful in the long term.

**5. What are your favourite mathematical tools, objects or manipulatives and why do you like them?**

My favourite tools are the heuristics for solving problems. These include conjecturing, specialising, generalising, and shifting representations. I like using different models and exploring them in class; for example, subtraction as difference versus taking away. Or multiplication as stretch versus area.

**6. How does your school support professional development of teachers in mathematics?**

Department meeting are mostly focussed on developing practice. This involves doing mathematics as an exercise in empathy, not just to feel the experiences of joy, surprise, frustration and anger. But also, mathematical sensitivities such as what might we choose to do when working on a task. This means trying different approaches, reflecting on why we and students may choose different approaches, considering what prior experiences would allow students to access the task, and what the task can illuminate you to.

**7. What is your favourite quote about mathematics?**

“Mathematics is not about numbers, equations, computations or algorithms; it’s about understanding” – William Thurston. I think this quote is useful to shift focus in a mathematics lesson towards something beyond the obvious content. For example, understanding the distributive law versus the process of expanding brackets, or what an equation represents and what a solution is versus how to solve. I would also include problem solving heuristics and metacognitive strategies as beyond the obvious content.

**References:**

- Hewitt, D. (1998). Approaching arithmetic algebraically.
*Mathematics Teaching, 163*, 19–30. - Ollerton, M., & Hoggard, S. (2019).
*Planning for teaching GCSE mathematics with mixed attainment groups*. Association of Teachers of Mathematics. - Pirie, S. E., & Martin, L. (1997). The equation, the whole equation and nothing but the equation! One approach to the teaching of linear equations.
*Educational Studies in Mathematics, 34*, 159–181.

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