Mathematics is not a spectator sport

In the video below, Aziz Belhassane Sensei is demonstrating Aikido at the Sugano Shihan Memorial Summer School 2011.

Whether you know Aikido or not, the thing you will notice, especially if I point it out to you, is that Aikido is not a spectator activity. You cannot learn Aikido simply by watching someone else. Watching is critical, paying attention is critical, but doing is the essence.

Currently I am teaching a small class – 11 students – what is known in the U.S. as Advanced Calculus. This is their second semester of Advanced Calculus, and I am conceiving it as an introduction to real analysis. This past week we have been looking at the Cantor function (also known as the Devil’s Staircase).

The Cantor function can be described as the limit of a sequence of piecewise linear functions. The first 3 are shown below:

Cantor_function_first_three

The piecewise linear functions f_n can be defined on the interval [0,1] recursively as follows:

f_0(x)=x

f_n(x) =    \begin{cases}  f_{n-1}(3x), & \text{if }0\leq x \leq 1/3 \\  1/2, & \text{if }1/3 \leq x \leq 2/3\\    (1+f_{n-1}(3x-2))/2, & \text{if }2/3 \leq x \leq1\\  \end{cases}

So we have a potential pictorial and recursive understanding of the sequence of approximations to the Cantor function.

These do not make it immediately obvious how to calculate the value of the Cantor function at a given point in the interval [0,1], particularly at points of the Cantor set, such as x=1/4.

A simple recipe remedies this:

Write x = 0.d_1d_2d_3\ldots base 3 (with no all 2’s from some point on ). If one of the d_i=1 replace all digits after the first such occurrence by 0. Then convert all occurrences of 2 to 1 and read the new number base 2.

For example, 1/4 = 0.020202\ldots base 3, so, according to the above recipe,  the value of the Cantor function at 1/4 is 0.010101\ldots base 2, which is 1/3.

The issue for the students was to reconcile the pictorial and recursive approximations to the Cantor function with this recipe.

Sadly, as was the case for most of the semester, they sat in silence waiting  for me to do something, even though the course is seminar style in which they present their efforts at problem solving. So I gave them all marker pens, got them up to the board, and got them working and talking. Eventually most of them started thinking.

In my view, mathematics is best done on your feet, actively moving.

What do you think?

5 great things about being a maths teacher

This is a guest post written by Kimberley McCosh (@spyanki_apso on Twitter)

Kim McCosh

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5 great things about being a maths teacher

Kimberley McCosh

I love maths.  I have had a few jobs before becoming a maths teacher but the urge to teach was always there.  I am a self confessed maths geek and I love nothing more than converting some of my students to math lovers too!  I teach 12 to 18 year olds in a secondary school in Scotland.

1.  The interaction with pupils and knowing when you’ve really got through to them with maths.  One particular highlight was when my class cut out triangles then stuck the angles from this down in a line to prove that the angles in a triangle sum to 180 degrees.  The next day one of the boys (aprox 13 years old) was so eager to tell me that after the class he went home and searched the internet and found that all the angles did in fact always add up to 180 degrees.  I know I had got through to him since he was choosing to look up maths in his own time.

2.  Getting pupils interested in maths.  I always try not to give “just a maths lesson” but also giving some background too.  Ask any of my S3 class and they will be able to tell you more interesting facts about Pythagoras and his life than they can about the latest boy band!  I always try to make my lessons interesting, different but still always relevant.  When the pupils are interested, they are engaged and I have achieved my goal of sparking their interest in maths.

3. Helping pupils to think for themselves.  Whether it be problem solving or applications of maths, whenever the pupils make the links for themselves it is always a real fantastic moment for me as a teacher.  They have learned the building blocks and are piecing them together and starting to see the big picture.

4.  The feeling of achievement when the penny drops and the class “get it”.  It’s all in that moment when the pupils say “Ahhh!  So that means…”.  Or even better, when the pupil who has been struggling but working hard turns round and says “This is really easy!”.  To know you have taught something which the pupils can now use in future years is what it’s all about.

5.  Although not specific to maths, it is fabulous to make a difference in someone’s life.  As a teacher you have daily interaction with pupils who may not always have the perfect home life but when they come into your class they are praised, encouraged, challenged and motivated to be the best they can be.  To see a whole class strive to be the very best they can is the biggest reward you can ever receive.

I could go on – I just love my job!  As a maths teacher you really make a difference.  From teaching basic numeracy skills to complicated calculus, each lesson is important.  I always try to remember that we are preparing pupils for jobs that haven’t been invented yet so who knows what level of maths they will require in later life.  As a teacher, you can get an amazing high from something as simple as a pupil finally mastering percentages or cracking vector calculus.  Each pupil, each class, and each lesson has highlights and I wouldn’t change my career for anything!