Sunday, 15 March 2015

Arbitrary and Necessary and Must-Be Maths

Thanks for reading my first post! Here I will aim to explain a bit about the rationale I have for much of what I hope will follow on the blog - a focus on 'Must-Be' Maths. This has been inspired by Dave Hewitt's articles on Arbitrary and Necessary, and the example that follows is discussed in the first of these.

Students are often presented with information such as the following:

However, there is much more to this seemingly innocuous statement than may first meet the eye...

Abitrary Facts

Why are there 360 degrees in a full turn?

Show answer

This fact didn't have to be so - somebody decided on this number at some point. It could have been another number. Indeed angles can be measured in a variety of different ways - in a full turn we also have 2π radians or 400 gradians (an unsuccessful attempt to adopt a metric unit.) More intuitively, we may simply talk about fractions of a full turn.

360° in a full turn is an arbitrary fact or convention. Students could not come to know this fact unless they were told (or read) about it.

Necessary Facts

Now, what do the angles in the following triangle add up to?

Following the above discussion, you may be tempted to answer in a variety of ways - 180°, half a turn? However you say it, the important distinction between this fact and the previous one is that this is not arbitrary or conventional - nobody invented this, it is simply a fact that is true of all triangles. If you start to consider this simple fact, you may perhaps start to think that it's actually quite startling and unintuitive. Dave Hewitt characterised this as a necessary fact, and I am calling it a 'Must-Be' maths fact. If students were never told this fact, it would still be possible to deduce it for themselves.

How should students come to know these facts?

I believe the distinction between arbitrary and necessary facts has implications for the way we should design the learning experience of students encountering this knowledge. For arbitrary knowledge, students must receive the knowledge from another (or by reading) and it is important they learn this well to be able to engage with the accepted language of mathematics. In contrast, students should become aware of necessary knowledge through investigation, inquiry and deduction. Through doing so they will be adopting mathematical behaviours, will have a more powerful learning experience and are more likely to be find the result (and the journey of discovering it) interesting and intriguing.

When viewed through the lens of arbitrary and necessary, I now see the answer to the question above in three parts:

By simply stating that angles in a triangle sum to 180°, the necessary facts become obscured by the arbitrary.

In the classroom

Here are two contrasting approaches as to how students may first encounter this fact:

Lesson approach 1:

Students are presented with the rule and use it to perform a series of calculations.

I did not have to search for long to find this - it was the first result on a Google Image search at

Students have been presented with the necessary fact in the same way as they would be presented with arbitrary knowledge. The task for students actually has more to do with performing basic arithmetic than it has to do with exploring or coming to understand this necessary property of triangles.

Lesson approach 2:

Students are given the following instruction: "Draw any five different triangles. For each triangle, measure the three angles and add them up."

Through seeing the property for themselves, students gain a sense of the necessity of the fact – it is not because the teacher said so, or because the examples were carefully chosen to convince them of the fact. This approach fosters further curiosity in students and they are more likely to feel inspired and empowered. (Note that I am not claiming that all students would be able to deduce a proof for this property. Typically, students would encounter this fact before angle properties of parallel lines, from which a proof can be deduced.)

Can all mathematical facts that can be categorised as arbitrary or necessary ('Must-Be' Maths)?

Much more of my thinking in terms of arbitrary and necessary knowledge and how it has impacted my classroom practice will follow in future blog posts.

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