Monday, 19 October 2015

Categorising Mathematical Etudes #mathsjournalclub

This post is in preparation for this evening's #mathsjournalclub twitter chat about the article "Mathematical ├ętudes: embedding opportunities for developing procedural fluency within rich mathematical contexts" by Colin Foster, 2013 (available here.)

I have really liked the idea of etudes ever since I came across this article about a year ago. Embedding practice into a mathematically rich context seems to perfectly balance the need for practice with the desire that students be working productively towards something meaningful - as well as ensuring that mathematics stays centre-stage in the lesson.

My aim in this post is to link to various resources I believe fulfill the requirement of an etude and to find commonalities in the types of over-arching mathematical goal involved in each case - that is to categorise them!

Category 1
Students follow a rule leading to a pattern of results - can the pattern be justified?
(ie. the teacher taps into a seam of related examples that follow some rule and students investigate this seam.)

- At MathsPad, we developed this task on factorising and expanding (this is still in flash at the time of writing), inspired by an interesting book on productively using algebra by Martin Kindt (available as pdf here.)
- Calculations with fractions in this task by Don Steward.

Category 2
Students are given a number of criteria and examples must be found that fulfil different combinations of these criteria.

- Johnny Griffith's Risps are good examples of these - such as Risp 10 on coordinate geometry, where for each section of a Venn Diagram, an example must be found.

(Also, students must combine given elements to form as many examples as possible - such as RISP 3)

Category 3
Students work on examples where the results converge towards a limit.

- Finding areas of successive polygons surrounded by circles, which in the limit will tend towards pi. Embedded practice of using trigonometry and finding areas.

Thursday, 15 October 2015

A point of pendentry? Explain why sin(60) = sqrt(3) / 2.

Very recently, I came across this question from a reputable source:

I have a real problem with this question, which may perhaps be perceived as pedantry, but to me is rather a bugbear. (Actually, it wasn't exactly this question, but the idea is equivalent.)

Clearly, the intent of the questioner is to elicit a response involving one of the 'special triangles' - half of an equilateral triangle, some Pythagoras and the definition of sin(x) as opposite / hypotenuse:

It is not the intent of the question I have a problem with, but the specific wording. The 'solution' shown above does not explain why. In fact, I do not believe it is possible to explain why this fact is true, it simply is true. Beyond the arbitrary definition of degrees and the arbitrary symbols used to represent the numbers and functions involved, the relationship in question is a necessary, must-be maths fact - a universal constant, if you will.

It is no more possible to explain why this fact is true than it is to explain why pi equals 3.14159...

It just does. Indeed this question is more philosophical than it is mathematical - why does dividing the circumference of a circle by it's diameter always necessarily yield this particular constant? And then perhaps, could a universe exist with pi equal to (say) 4?

 As I said, I have no problem with the intent of the question, it's just that the questioner did not mean 'explain why', but rather 'demonstrate that'.

Perhaps this is a pedantic point. As long as the intention of the questioner is communicated to the student, they will be able to 'answer' it. Nevertheless, my teeth will remain well-gritted when I see a question like this.

Sunday, 27 September 2015

Reflections on #MathsConf5 - The dangers of students over- or under-gerneralising

Yesterday, I attended #MathsConf5 in Sheffield. Amongst many interesting and thought-provoking moments, two related things particularly stood out for me, both about the dangers of the ways in which students form incomplete generalisations of mathematical concepts, how this can lead to later misconceptions and how this can be avoided in designing the learning experience.


Firstly, Peter Mattock (@MrMattock) ran a session on ‘Concrete Approaches to Abstract Maths.’ At one point in this session, he made an impassioned plea… that students should not learn that the mean average is ‘all the numbers added together divided by how many there are’, as this causes confusion when students encounter frequency tables and are required to find the mean – adding all the visible numbers in this case is simply nonsense.

The trouble here is that students have over-generalised their entire concept of mean from a subset of similar cases. Are they aware that there are cases outside of this subset where this rule does not apply?

Peter proceeded to demonstrate how he introduces the mean average to students, by first weighing a bag of apples and asking for the mean weight of an apple, then weighing a bag of oranges and asking for the mean weight of an orange, then weighing a bunch of bananas and asking for the mean weight of a banana. Does the rule apply here? There wasn’t an addition sign in sight! Rather, he contended, the rule should be understood as something akin to ‘the total divided by the number of things.’

It strikes me that the early examples students encounter when first learning about a new concept have a great impact on the way they embed a generalised rule. If students are solely presented with lists of numbers where the mean is to be found (using exactly the same steps each time), it is no surprise that they internalise a rule which optimises solving those cases – but this can cause problems later.

I can see exactly why for students, remembering the concept of mean as ‘all the numbers added together divided by how many there are’ is appealing. This algorithmic definition leaves little thinking for the students to do – simply put in the ingredients and follow the simple recipe. However, this makes the student passive in the process of doing mathematics, a slave to the algorithm – not the active, thinking, problem-solving mathematics student we are trying to develop. Generalising a concept as an algorithm that solves particular cases can limit students’ flexibility of thought when faced with a problem in which a different, more efficient approach can be used. For example:

I have written before about the mean and the idea of ‘levelling out’ the numbers.

If we want students to be good problem solvers, they need to be able to think flexibly about concepts, rather than being constrained by an over-generalised rule. To this end it is important that students:
  •         Focus on the ‘idea(s) behind a concept’ rather than an algorithm for solving a particular subset of cases. (See previous blogpost.)
  •        Encounter different types of cases as soon as possible in learning about a concept, to avoid incorrect over-generalisations from being embedded.


Secondly, Bruno Reddy (@Mrreddymaths), Craig Jeavons (@craigos87) and Matt Fox (@MFx15) ran a session called ‘Sh****ai is not a dirty word’, in which they shared their recent experiences observing maths lessons in Shanghai.

One of the things explored was the different approaches taken to introducing the law of indices for multiplication in the UK and in Shanghai.

Typically in the UK, students will see the rule applied in a simple case, and practise using the rule in many similar cases – here is an example of the sort of questions students might initially encounter on this rule:

“Fair enough, students are getting practise of applying the rule” one might think. However, comparing this to the first few questions students would encounter on this concept in a Shanghai classroom gives pause for thought:


Right from the outset, students are exploring a much broader example space. The difference in how students are likely to mentally generalise the rules are implicit:

From this, we can see that in this scenario UK students are under-generalising the rule. They are only aware of its application to a subset of the example space:


Would UK students even be aware that there are cases outside of the narrow subset they have encountered? When they come across a case from outside the subset at a later point in their learning, will they be able to transfer the idea?

The human mind is keen to generalise ideas. When a pattern is spotted, the tendency to find a generalised rule for it is compelling – indeed, this may be said to be at the very heart of mathematics. However, we as teachers need to think carefully about how we introduce concepts to students in such a way that they do not internalise either over-generalised rules (to find the mean, add the numbers and divide by how many there are) or under-generalised rules (the multiplication law of indices applies where a, m, and n are single numbers or letters.) Once these rules are embedded, it can be hard to change ideas later on, and misconceptions often arise. Exploring the whole example space as fully as possible and as early as possible may be the key to this.

Sunday, 28 June 2015

My first experiment with flipped learning

I have recently started experimenting with a flipped learning model with my A Level classes - with the topic of Functions (in Edexcel C3) they are currently studying having finished their AS exams. I am keen to develop a model and am hoping to implement this model in the new academic year with the new cohort of Year 12 classes.

You can find the two tasks I set up for students to work on independently on MathsPad at these links:

(Disclaimer: There are still potentially some minor bugs with the programming on these pages, particularly if you are using Internet Explorer (nothing dangerous to your computer though!) Also, they are not linked on the main MathsPad menu system.)

One of the concerns I had about a flipped classroom model is the potential over-reliance on video and the subsequent one-way transmission-of-information pedagogical style that would ensue. This does not sit comfortably with me at all. I believe deep learning will occur more readily when students are actively engaged with thinking about ideas, and developing their own understanding of it - a broadly constructivist perspective. So it was that although I did want to use video as one of the features of the flipped learning task, I intended to keep to very short clips that fitted alongside other interactive elements,

A basic model I tried to emulate in these tasks is one that I often use in my classroom teaching:
Stimulus - student thought (and often discussion in classrooms) - teacher validation.

Here are some screenshots of the ways in which I have sought to actively engage the student in thinking about the task:

Number Sets task
1. Stimulus question:

2. Finding out about the number sets, their terminology and definitions:

3. Active engagement:

...with feedback that prompts further thinking where necessary:

Domain and Range Task
1. Finding out about domain and range:

2. Applying understanding of domain and range to functions:

Improvements I hope to make:
- for the domains task, add in a simple modelling situation to illustrate why a function might have a restricted domain (eg. x is a length)
- create an individualised menu system for students that allows them to self-evaluate and see an overview of progress.
- at the end of each page, ask students to very briefly self-evaluate their understanding (with categories such as 'I will need to return to this page.') - this will be fed back to the teacher and available for students to see an overview of in the menu system.

I am very keen to hear your thoughts on this as-yet-experimental model of flipped learning, whether positive or negative - please do let me know through the comments or via twitter or email. Also, do feel free to share these links and trial this with your own students if you wish.

Tuesday, 21 April 2015

Mathematical Proof: the very essence of must-be maths?!

Consider this proof problem:

Prove that the sum of any 3 consecutive integers is always a multiple of 3.

Many students' first attempts may involve starting with three unknowns: a, b and c. However, this approach clearly falters immediately, as summing these gives no indication as to their factors.

The required insight is to use the way the three numbers are related to each other - ie. that they are consecutive. Since the three numbers are consecutive, the second is necessarily one more than the first and the third necessarily two more than the first.

This means starting with as few arbitrary elements as possible.

From this can follow a series necessary properties of algebraic equivalence - that repeated addition can necessarily be written as multiplication (ie, terms can be collected) and that an expression with a common factor can necessarily be factorised (ie. the distributivity law) thus proving that the sum is a multiple of 3.

The lens of arbitrary and necessary thinking is important here: to successfully build a proof through a chain of deductive reasoning, the minimum number of arbitrary elements should be used. Much more importantly, necessary, consequential, must-be mathematical properties must be teased out and used to produce the argument. In the arbitrary there is doubt, but in the necessary there is certainty.

Thus it is to me, that to think mathematically (and proof is surely the ultimate form of mathematical thinking for the pure mathematician) is to use arbitrary elements sparingly and with caution, and always to look for the necessary, must-be properties of the situation.