Republic of Mathematics blog

5 great things about being a maths teacher

Posted by: Gary Ernest Davis on: May 1, 2012

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!

 

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A lovely observation

Posted by: Gary Ernest Davis on: January 13, 2012

Ben Vitale (@BenVitale on Twitter) recently made the elementary and lovely observation that

\frac{1+3}{5+7}=\frac{1+3+5}{7+9+11}=\frac{1+3+5+7}{9+11+13+15}= \ldots = \frac{1}{3}

x

Some people asked why this is so, and the answer, simple as it is, makes a nice middle or high school problem

First let’s think about what’s happening here.

The numerators of these fractions are sums of the first few odd numbers.

The denominators are sums of the same number of odd numbers, starting where the numerator leaves off.

To use variable notation – something some middle schoolers are still struggling with – the numerators look like:

1+3+5=\ldots + 2k-1

as the positive integer k increases by 1.

The denominators, notationally a little more complicated, look like:

(2k+1)+(2k+3)+\ldots + (2k+2k-1)

We can find short algebraic expressions for these numerators and denominators, and it is these simple expressions that will show us where the fraction \frac{1}{3} comes from.

Let’s write S(k)=1+3+5+\ldots +2k-1

and

T(k)=2+4+6+\ldots +2k-2

x

Then

S(k)+T(k)= 1+2+3+\ldots + 2k-2 + 2k-1= \frac{1}{2}(2k-1)\times 2k=k(2k-1)=2k^2-k

and

T(k)=2+4+6+\ldots +2k-2= 2(1+2+3+\ldots + (k-1))=2\times \frac{1}{2}(k-1)k=k^2-k

Therefore, S(k)=(2k^2-k)-(k^2-k)=k^2.

For the denominator we have:

(2k+1) + (2k+3) + \ldots + (2k + 2k-1) = (2k+2k+\ldots +2k) + (1+3+\ldots +2k-1)= 2k^2+k^2=3k^2

Therefore,

\frac{1+3+5+\ldots +2k-1}{(2k+1)+(2k+3)+\ldots + (2k+2k-1)}=\frac{k^2}{3k^2}=\frac{1}{3}

independent of k.

The situation for similar sums of even numbers is not quite so simple.

2+4+6+\ldots + 2k = 2(1+2+3+\ldots + k)=2\times \frac{1}{2}k(k+1)=k^2+k

while

(2k+2)+(2k+4)+(2k+6)+\ldots +(2k+2k) = (2k+2k+2k+\ldots +2k)+(2+4+6+\ldots +2k)= 2k^2+k^2+k=3k^2+k

x

Therefore,

\frac{2+4+6+\ldots +2k}{(2k+2)+(2k+4)+\ldots + (2k+2k)}=\frac{k^2+k}{3k^2+k}=\frac{1+1/k}{3+1/k}\to \frac{1}{3}

x

as k increases.

A lovely observation, some simple algebra, and a challenging yet rewarding problem for middle and high school students.

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