An Introduction, Part 2: Training, and meeting “Arbitrary and Necessary”

"Arbitrary and Necessary" by Dave Hewitt, published

in the journal 'for the learning of mathematics'

Part 1 

When I entered teacher training, even though I was far from possessing the necessary skill set to achieve it, I had a firm idea of what I wanted to do as a teacher, or more specifically what I did not want to do: I articulated this at the time as not “telling” mathematics to students. I would now rephrase this as not generally wanting to share that which could be discovered, but instead working in such a way where students had the potential to form and reach mathematical conclusions themselves.

In 2007 my PGCE tutors at the University of Birmingham were Steph Prestage, Pat Perks, and Dave Hewitt, three people who were very influential in different ways in both my development and my practice to this day. I had no idea when I started what I was going to be faced with and did not even contemplate it too much: before starting, my training was about gaining confidence in myself and leaving with a certificate that would allow me to teach in state schools. I knew my views of what I wanted in my future classroom were very different than I had experienced: perhaps I was just out of line with everyone else and indeed reality. 

Thankfully that turned out to be far from the truth, as I was exposed to different thinking about mathematics and its education than I had previously encountered. Hewitt, in particular, approached the teaching of mathematics in a way that seemed to align with what I hoped to achieve, but to such an advanced level that I realised my grand philosophy was personal hubris and I had much work to do before I would be even remotely happy with my thinking and future teaching practice.

If I went back through my notes from the time, I am sure I would find many lightning-bolt moments in the course that influenced the professional and indeed man I have become. But I do not need to go to my notes to be reminded of writing I first met then and have revisited many times since, and what will become a focus of my writings here.

I cannot remember if Hewitt explicitly shared his educational writings or if I found them myself, as I wanted to learn more from this important figure in my life. I realised that within my idea of not “telling” there would be some elements of mathematics that would have to be shared; whilst much can be reasoned, I knew then that there were some parts of mathematics that were decisions made at some point in history or parts that were merely definitions. It turned out Hewitt had already divided mathematics along those lines, into what he termed the “arbitrary” and the “necessary”, in a three-part series published in the journal for the learning of mathematics. I immediately felt that I had a theory to latch my thinking onto and also to develop it; almost eighteen years later, this continues to be the case as it is my main philosophy for how I approach mathematics. This concept of arbitrary and necessary will be explored in many pieces to come, but ultimately will be behind everything I write.

Next: in the final autobiographical section of this introduction, further influences in training and beyond.

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Comments, questions and concerns welcome

at davidmlawrence@gmail.com

Bluesky account: @dmlawrence.bsky.social‬

©2025 David M. Lawrence.

All rights reserved.

An Introduction, Part 1: From being a student to becoming a teacher of mathematics

A student of mathematics

When I was young, there were two key ongoing experiences that would influence my thinking and philosophies when I entered the teaching profession.

The first was of myself as a learner of mathematics. Out of necessity and also teacher choice, in secondary school I largely learned mathematics by working at my own pace through text books, where I would follow along with examples given and then apply this mathematics to the questions in the exercises. Not only would I want to answer every question, I had a need to answer every question correctly. For a large portion of these questions I got the answers in the back on the first attempt, for the rest I would repeat until I did. My brain seemed to soak up this mathematical learning, apparently far more easily than those around me. I had no idea why. I was one of only three people in my year group who worked through the extra text books covering the mathematics required to obtain an A* at GCSE. Most of my peers did not seem to have a positive view of mathematics nor themselves as learners of the subject.

The second was regarding my mother and her reaction to mathematics. She was completely different to me; she would recoil in dread at the notion of being expected to perform anything she viewed as mathematics. Her story was hers to tell, but suffice to say that the root of her feelings lay in her formative years at school, where she had been made to feel inadequate and ‘stupid’ because she did not achieve highly in the subject: indeed, she had left school firmly believing she could not do mathematics. We thus approached mathematical problems with completely different mindsets.

I did not train to become a teacher immediately after my undergraduate studies. In the years betweenI helped several people with mathematics, most notably my mother. I saw bravery when she worked to overcome her anguish to want to improve. I found myself wanting her to understand how to answer a problem, which I realised was not the same as answering it. This was not how I recalled being taught, or at least not often. The development I witnessed was less rapid than the apparent success I had working with text in books but it was evident, especially and most importantly to my mother.

If my brain was different to my mother’s, I could not account for why. But I also realised that this was irrelevant to me; if everyone could achieve exam success in mathematics in the way I did, they would have done so already. So how else could mathematics be learnt and as a result taught, in a way to better ensure that all students access the underlying mathematics? And what is that underlying mathematics, and what does its learning look like?

These questions were at the forefront of my mind as I worked towards and entered teacher training in 2007. In 2025, they have and continue to shape my professional identity, practice and the ensuing writing with which you may choose to engage. 

Next: teacher training, and beginning to formalise my thinking through meeting the work of Dave Hewitt and “Arbitrary and Necessary”.

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Comments, questions and concerns welcome

at davidmlawrence@gmail.com

Bluesky account: @dmlawrence.bsky.social‬

©2025 David M. Lawrence.

All rights reserved.