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Notes on Pedagogy Prompts and Surprising Practice

Why Pedagogy Prompts?

I was so far from being a natural in the classroom on day one it’s laughable…nor was I on day 100, or day 200 for that matter. I was acutely aware that I had to engage in deliberate practice and deep reflection to improve as an educator, but when I look back at my first 5 years in teaching it seems now that I could have improved certain elements of my practice through more efficient and collaborative methods. In particularly, thousands of hours reading blogs, websites and journal articles in isolation may not have been the most fruitful way to increase pedagogical content knowledge (PCK). Don’t get me wrong, I personally loved – and still love – engaging with these forms of media, but for those who don’t love it – or more commonly don’t have the time to do this – is there another style of medium that we might draw upon to develop more attuned PCK for all educators at a faster rate…

So which medium might catalyse a more rapid increase in PCK?

Here I present pedagogical prompts. A pedagogical prompt contains 1-6 ideas designed to prompt thinking or induce critical debate between educators. They contain multiple proofs, visualisations, methods, connections between topics – essentially, many of the elements lacking in textbooks. Whether you’re an NQT, a teacher trainer, a head of maths or an passionate professional, I believe that pedagogical prompts help fast-track PCK, and provide a more accessible medium for positive critical debate.

How can a pedagogy prompt be used?

How about we amend location, location, location, to collaboration, collaboration, collaboration. Let’s learn from our colleagues, both new to the profession and those more experienced, to develop PCK in maths departments or teacher training. Using a pedagogy prompt for the first 10 minutes of a meeting, or in some cases for longer, may provide that lifting platform to generate rich discussion amongst colleagues.

If you’re not in a department that chooses to use meeting time to discuss pedagogy, I hope you can still benefit from discussions on twitter.

I’m a HoD, how might I introduce the use of pedagogy prompts in department meetings?

Surprising Practice: Mathematical Beauty and Unexpectedness

In writing Mathematical Beauty, I developed the mathematical aesthetics framework to improve on previously developed criteria for mathematical beauty. You’ll notice that I included an upper level of subjectivity, alongside a lower level of objective criteria. Each of the elements are explained in the book, but here I would like to focus specifically on unexpectedness.

Unexpectedness – or surprise – permeates the wide majority of beautiful mathematics….

Whether it’s an unexpected connection, unexpected application or unexpected elegance, surprises increase curiosity and deepen understanding, which leads to a more enlightened view of the wonderfully rich world of mathematics.

From a teaching perspective I’m not sure that I’ve done a good job at utilising unexpectedness to increase the curiosity and aesthetic pleasure of students…here’s where surprising practice can help.

What is Surprising Practice?

Surprising Practice is not mutually exclusive from Colin Foster’s notion of a mathematical etude, in which the aim is to develop fluency with a deeper, more stimulating mathematical goal in mind. Specifically, …

Surprising Practice is the development of fluency with an unexpected midpoint or endpoint.

There’s a plethora of concepts and ideas which lend themselves well to this style of task, such as exponential growth, the connection between Pascal’s triangle and the Binomial Theorem, buffon’s needle experiment, etc. but my hope is to also develop tasks – whether they’re whole class mini-whiteboard based, extension problems, or simply a question embedded within a sequence – which draw upon the element of surprise through appropriate sequencing. Ultimately, the sequencing of a question or explanation cannot be underestimated in the build up to a surprising moment.

For example, consider basic addition of two-digit numbers. Can we generate a surprise whilst developing addition fluency? How about drawing a triangle in Geogebra and asking students to add the angles utilising mini-whiteboards to assess responses, then drawing another, and another.

Students practice adding 2-digit numbers, but then noticing that the second answer is the same as the first and that the third also gives an answer of 180 degrees is a surprising moment to the majority of students in the room.

This acts as an integral reminder to one of John Mason’s thought-provoking comments:

“There are no rich tasks, only tasks used richly”

John Mason

Clearly there are tasks which have “greater potential” for richness, but Mason’s comment provides an integral reminder to us all that the pedagogy surrounding a task is at least as important as the task itself.

It will be clear to most reading this that we must actively remind ourselves not to impose our own “curse of knowledge” on possible moments of surprise. Increased knowledge and understanding has implications for aesthetic pleasure – at times increasing it – but also dulling it when over-familiarity tarnishes our enthusiasm and passion. This consequently results in sequencing and pedagogical decision making that can reduce awe, wonder and pleasure in classroom experiences for our students.

Submitting Surprising Practice or Pedagogy Prompts

It would be incredible if anyone would like to submit an idea for either surprising practice or a pedagogy prompt. I will happily feature ideas giving full credit where it is due. If you are keen to do so, please contact me at

A Note about Pedagogy

My personal stance as an educator is one of balance, although I agree wholeheartedly with Alan Wigley on the nature of balance in educational practise…

“The problem is not whether one should use a mix of methods (of that I have no doubt) but precisely how the blend should be achieved.”

Alan Wigley (1992)

I love to deeply analyse concepts, topics, myself as an educator, and my knowledge of the students in front of me as people and as learners, to determine how to approach any given concept. My hope is to maximise learning so that students build confidence through success (and that does mean that they feel safe and able to meet the demands of unit tests and external exams), but also get exposure to the wonderfully rich world of mathematics, through patient problem solving (Pearcy, 2015), interesting applications, investigations and inquiry. I enjoy utilising different educational theories independently, such as Engelmann’s theory of instruction during an episode of explicit instruction, or blending ideas such as considering cognitive load through clear phases of structured or guided inquiry based learning.

I have no issue with shifting my pedagogical position based on reading, reflection and new evidence. There are times when I feel that my bias in engaging with specific ideas might result in a tending towards a specific point (just as a sequence tends to a limit), but then taking a bat to those biases often results in unexpected ruptures, resulting in a somewhat frustrating yet pleasingly refreshing change in opinion.

How fortunate we are to play a part in such a rich, diverse, fascinating, infuriating, invigorating, important activity as helping students learn and love this beautiful subject.


Pearcy, D. (2015). Reflections on Patient Problem Solving. The Association of Teachers of Mathematics, MT247. Derby: UK

Wigley, A. (1992). Models for Teaching Mathematics. The Association of Teachers of Mathematics, MT141. Derby: UK