MustBe Maths
Maths facts that must be so
Thursday, 18 February 2016
Saturday, 13 February 2016
Monday, 19 October 2015
Categorising Mathematical Etudes #mathsjournalclub
This post is in preparation for this evening's #mathsjournalclub twitter chat about the article "Mathematical études: 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 centrestage 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 overarching 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.)
Examples:
 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.
Examples:
 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.
Examples:
 Finding areas of successive polygons surrounded by circles, which in the limit will tend towards pi. Embedded practice of using trigonometry and finding areas.
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 centrestage 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 overarching 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.)
Examples:
 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.
Examples:
 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.
Examples:
 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:
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, mustbe 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 wellgritted when I see a question like this.
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, mustbe 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 wellgritted when I see a question like this.
Sunday, 27 September 2015
Reflections on #MathsConf5  The dangers of students over or undergerneralising
Yesterday, I attended #MathsConf5 in Sheffield. Amongst many
interesting and thoughtprovoking 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.
Overgeneralising
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 overgeneralised
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, problemsolving 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 overgeneralised 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 overgeneralisations from being embedded.
Undergeneralising
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 undergeneralising 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 overgeneralised rules (to find the mean, add the numbers and
divide by how many there are) or undergeneralised
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.
(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.)
You can find the two tasks I set up for students to work on independently on MathsPad at these links:
www.mathspad.co.uk/interactives/alevelfunctions/a.html  number sets
www.mathspad.co.uk/interactives/alevelfunctions/b.html  domain and range
(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 overreliance on video and the subsequent oneway transmissionofinformation 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 selfevaluate and see an overview of progress.
 at the end of each page, ask students to very briefly selfevaluate 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 asyetexperimental 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.
Labels:
A Level,
Flipped Learning,
Functions,
Number Sets
Tuesday, 21 April 2015
Mathematical Proof: the very essence of mustbe maths?!
Consider this proof problem:
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, mustbe 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, mustbe properties of the situation.
Prove that the sum of any 3 consecutive integers is always a multiple of 3.
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.
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, mustbe 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, mustbe properties of the situation.
Subscribe to:
Posts (Atom)