Thursday, 28 March 2013

World Tour of Maths

Gentle Reader,

You may recall sometime last year when I was awarded the Winston Churchill Memorial Fellowship for my proposal to travel the world looking at maths education in primary schools.

Well, departure day is in zero minus 10 and the adrenalin is starting to kick in. Let me give you a broad outline of where I am going, what I am doing and who I am talking with while on the World Tour of Maths.

New York - I will be spending four days in four different schools - not sure if I can name them (I will check this out when I get there). The schools are diverse - one is a private school, one is a selective school and two are charter schools. I will also be visiting the Museum of Mathematics (MoMath) which should be great. Probably won't have time to fit in any shopping, baseball, sight-seeing etc.

Denver - In Denver I will be attending the National Council for Teachers of Mathematics (NCTM) Conference. This should be great - something we don't get to experience in Australia - at least not on this scale. I also hope to meet with a few special people - Dan "Math Class Needs a Makeover" Meyer and Marilyn "I Hate Mathematics" Burns. 

San Francisco - I am going to meet a few of the directors of the San Francisco Math Circle - no, not a secret cult studying arcane lore but a group that promotes after school maths activities to stimulate and challenge interested young mathematicians.

Tokyo - Well, not exactly Tokyo but close - I am going to spend three days in a school in Kawaguchi. This will be very exciting, particularly given my level of Japanese. However, my visit does coincide with Golden Week so I think I am going to have a good time.

Singapore - The final leg of the trip is Singapore where I get to visit six schools and catch up with a few ex-Radford staff who have escaped Canberra's chilly winters for some tropical heat. 

Canberra - And back again to Canberra, where I will unleash all my new-found passion for maths on my unsuspecting class 6BF and the rest of the Radford Junior School. 

So if you see a slightly bewildered-looking mid-aged Australian male clutching his i-Pad and mumbling greetings in less-than-fluent Japanese, come up and say hello.

Saturday, 23 March 2013

Subitising With Kindergarten

We were scheduled for our regular visit to see our Kindergarten buddy class. Their teacher, the wonderful Mrs M, and I had a discussion about how the previous visits had gone and decided that this time we would give the Year 6 kids a task - to design a game or activity that would teach their buddy something related to what they were doing in class.

As the kindies were about to launch into "subitising" - recognising numbers from a group of objects without needing to count each item - we agreed that this would be the focus for the buddy activity.

Well, I announced the plan to Year 6 and they got very excited. If you are ever short of creative ideas for teaching, just ask the kids for some suggestions.

Here's a few of the games that we took to our buddies, who were so happy with them that they wanted to play them, over and over and over and...

Know Your Shapes - the idea here was that if you could recognise shapes such as triangle and squares, when you saw a group of objects arranged in that shape you would know how many there were by knowing that a triangle has 3 corners, a square has 4 corners etc

Concentration - this was a game of Concentration that used dot patterns. The student had to lay out all the cards face down and then choose 2 to see if they matched. If they matched, they kept that pair. If they didn't they had to put them back face down and the other player had a turn.

Dinosaur Game - my Year 6 boy knew that his buddy loved dinosaurs, so he made a game with dinosaur pictures set out in patterns. If his buddy could say how many dinosaurs there were, he could take that number of dinosaurs from the tub

Lucky Hearts - In this board game, you moved around the board and when you landed on special squares, you got to pick up a card and say how many dots were in the pattern

Puppies - in this game you travel around the board, using the fantastic spinner, and when you land on a puppy card you need to say how many there are in the group, then you get to keep the card.

Puppet Show - this amazing student made a puppet show out of an old shoe box, painted in the background and even included a cut-out silhouette audience. Her puppets on coloured sticks popped up from the base and her buddy had to quickly say how many items there were in each group.

Snap - the traditional game of Snap using cards that had dot patterns as well as number cards.

Pop-up Easter Egg Chickens - not so much a game but certainly a very cute and enjoyable activity where the eggs pop open and you get to say how many chicks are inside.

My Amazing Class

My input in this activity was minimal - I gave my kids a few minutes of introduction as to what and why. They did the rest. They had 2 days to get it done and everyone produced something special. Very little class time was allocated to this - much of the work was done at home or in lesson breaks.

The kindy buddies loved it. As a bonus, they got to keep their games and take them home to play with their parents and siblings.

Mrs M loved it - she said she would never have had the time or imagination to put together such a great range of activities.

The Year 6 kids loved it - they were so proud of what they had done and were really challenged by having to make an activity at the right level that would be enjoyed by their buddies.

And I loved it because I was able to see my class engage in the learning process as learners and as teachers.

Moral of the Story

Let the kids be creative - they will surprise you.

Tuesday, 19 March 2013

Straight Lines Make a Curve

My son was in my classroom this morning.

Yes, he was finishing off an assignment.

Yes, it was due today.

Yes, he was given it weeks ago.

It's a boy thing, alright?

So, he was using one of those corrector tape things to draw lines and boxes on his poster.

Then he decided to write "The End" at the bottom using the tape as well.

"Hmm," thinks his wise old father, "I wonder how he will deal with the letter D. That one has curves..."

"Oh right!" I said. "A curve is a series of straight lines!"

"Well," he replied, "The Sydney Harbour Bridge is made up of all straight pieces of metal. There's no curved bits on it."

Of course! I knew this, didn't I? But to be reminded by my son was refreshing.

And then I did some searching and found a few images and things. Remember those drawings and string creations we used to make back in the 70s?

Here's a few examples to remind you...

Monday, 18 March 2013

When the teacher is the reason you can't learn maths


I was talking to a father last week who told me the old, familiar story. His teenager son came home from school and asked, "When will I ever need to use all this stuff they are trying to teach us in maths at school?"

Heard it before? I have - even from my own son! (shock, horror)

If the maths isn't taught as being relevant to daily life, then maybe the teacher doesn't know that it is

Before I am disbarred from the Australian Association of Mathematics Teachers and hung from the nearest lamp post, can I please make one observation?

Maybe when bad learning is happening, maybe there is also bad teaching happening.

There - I've said it.

And being a teacher, I share this responsibility. It is my responsibility to the kids I teach to be the best teacher I can be, to know my subject and to make the best decisions I can to help them to learn.

Part of my challenge to myself is to make my teaching relevant, comprehensible and interesting for my students. I need to engage them at the zone of proximal development but I also need to engage them at the zone of proximal interest and relevance.

The solution to bad teaching is teacher improvement, not teacher replacement. And then there is my role as a mentor and colleague to other teachers around me - what can I do to support them to be the best teachers they can be?

Sounds simple, or simplistic, doesn't it?

But maths doesn't have to be useful for daily life, I hear you say

Really? Then what is it's purpose? To keep mathematicians employed? To make school difficult for young children? If it doesn't serve a practical purpose then why do we bother?

I believe that Mathematics is a language we can use to make sense of the world around us. It helps us to understand who we are, where we are in place and time and how the world works. (You may recognise these as some of the transdisciplinary themes from the International Baccalaureate Primary Years Programme)

I don't think Mathematics was invented by Euclid, Archimedes and Pythagoras one day because they were bored and had nothing to do. They were interested because it was useful to them, it helped them to solve problems (real problems, not just made up ones from a text book) and because they understood the insights into the nature of reality they gained from this type of thinking.

Other purposes of maths

Now, I'm sure there are things that I was taught in senior high school maths that I have never had the need to refer to ever since. I don't do a lot of differential equations as a rule in the general course of a day. But some people do.

I have seen a TV police show where it seems that every crime can be solved by plugging in the correct mathematical formula. And there are probably many other (real) contexts in which (real) people employ higher-order maths skills each day in ways that I cannot begin to imagine. Even if I don't have a clue about how to launch a satellite into space, I pretty glad that someone else does, each and every time I turn on the TV, use my mobile phone or post a blog.

And having learnt all about calculus many years ago, even if I never use it again, I have benefitted from developing thinking processes and logic that I will use each day of my life.

Mathematics in the window factory

I attended the ACER research conference on teaching mathematics in 2010 in Melbourne. Ian Hunter, the aboriginal elder who performed the welcome to country ceremony, told a story of his own education.

He left school at the age of 15 and, like many young people, thought he would never have to do any more maths as long as he lived.

He got a job in a window factory.

And did more maths in 3 months in the factory than he had ever done in 10 years at school.

He said, "When I left school, I never thought I'd get a job as a mathematician."

Top 10 Geometry Songs


My Top 10 Geometry Songs


I didn't realise when I listed my Top 10 Mathematics Songs that I would alienate so many 
enthusiastic (and belligerent) geometers. The hate mail I received! 

 "Maths is more than numbers!"
"Give shapes a chance!"
And something unprintable about f(x) my own derivative.

To redress the balance and to restore harmony to the Math-iverse, I have listed for everyone, 
from all walks of Mathematics, my Top 10 Geometry Songs.

10. Another Pyramid - Elton John

Musical genius-come-mathematician EJ exploring the third dimension. 
And what do you buy for the man who has everything? 
Another pyramid...

9. Arc of a Diver - Steve Winwood

An exhibition of magestic beauty
 - unless of course it's Greg Louganis in the 1988 Olympics, 
in which case it would be the "#$@! of a diver"

8. I Walk the Line - Johnny Cash

Don't think that Johnny ever walked in many straight lines but it is much easier 
to sing about than "I walk in a random, irregular fashion 
following no single purposeful direction"

7. Come Up and Be a Kite - Kate Bush

Not sure what Kate was getting at here but I'm pretty sure there 
would have been a funky dance to go with the song

6. Parallel Lines - Kings of Convenience

I've never heard of these guys but they are probably part of some underground alternative Geometry network, pursuing a passion for all things linear - and the song's not bad either


5. Shine On You Crazy Diamond - Pink Floyd

A classic psychedelic 70's prog-rock tribute to quadrilaterals


4. Straight Lines - Silverchair

An arty homage to Piet Mondrian that has been ruined by 
too many covers on "Idol" 

3. Will the Circle be Unbroken? - Nitty Gritty Dirt Band

 Well, obviously it will be unbroken because otherwise it won't be a closed shape, 
ergo not a circle anymore.
Sheesh! State the bleedin' obvious and dress it up 
as homespun geometrical philosophy

2. Bizarre Love Triangle - New Order

 A classic 80's expression of passion for three-sided shapes. 
Nothing bizarre about that.

1.  It's Hip to be Square - Huey Lewis and the News

 The theme song from the sound track for my life

Friday, 8 March 2013

Women in Maths

Happy International Women's Day!

And in recognition of this day, I thought I might post a few links to some sites that present biographies of and information about many great women mathematicians.

A quick Google search will get you to some of these:

Biographies of Women Mathematicians  

Wikipedia List of Female Mathematicians

Five Historic Female Mathematicians You Should Know - The Smithsonian 

Famous Women Mathematicians

Famous Women Scientists and Mathematicians

What would you do to get an education in mathematics?

Sonya Kovalevsky - married for convenience to get out of Russia and then tried everything she could to get acceptance in universities in Germany.

Mary Somerville - her parents disapproved of her learning mathematics so she had to read her maths books by candlelight. When her parents confiscated the candles, she memorised her texts.

Keep reading some of the texts in the sites linked above.

And have a great International Women's Day

Sonya Kovalevsky - an inspiration to all of us

Thursday, 7 March 2013

The Password of Mathematics is "Pattern"

We've done a lot of inquiry-based activities in maths over the last year. It has been fascinating to watch the kids (and their teacher) develop their understanding of mathematical thinking.

But as I reflected back on what we have been doing, I started to wonder if my inquiry maths journey was more than just a series of fun activities. Shouldn't inquiry go a bit deeper?

Well, I had made an effort to link our maths to what our unit of inquiry was that we were investigating. We are a PYP school (International Baccalaureate Primary Years Programme) and we have 6 units of inquiry each year. Making authentic links is central to true transdisciplinary learning. 

But is this enough?

And what about those left-over bits of content that don't fit authentically into any of the 6 units of inquiry? You still have to teach them. We often refer to them as the "stand alone" bits of the curriculum.

Or for those who don't teach PYP, how would you approach teaching maths as an inquiry? Where would you start? Where would you go?

Numbers Patterns - A Reflection

Last week, we launched into an inquiry into number patterns. It was something we had to "cover" (don't you hate that expression - sounds like you're making a bed or tiling a bathroom floor rather than inspiring children to learn...).

Here's where we went on our journey.

Tuning In

We started with the provocation - "The Password of Mathematics is Pattern" - taken from one of my favourite books: "The I Hate Mathematics Book" by Marilyn Burns.

What did this statement mean? Does it apply for ALL mathematics? And what type of patterns are we talking about? And what is a pattern?

This was a very interesting discussion.

We followed up with the kids creating 10 of their own patterns. They shared their patterns with a partner. The partner was asked to add the next number to the pattern and explain what was happening.

Here's a few patterns they came up with. See if you can find the next number:

1, 2, 4, 8, 16, .....

8, 20, 32, 44, .....

2, 6, 7, 21, 22, 66, 67, ......

100, 50, 25, 12.5, 6.25, ......

50 000, 5 000, 500, 50, 5, .....

The debrief was interesting. Here's a few of the comments:

"Can this have 2 answers?"

"I know these two are the same but I don't know what they're called."

"Is it a pattern if you add and multiply?"

"It's a pattern if the change is regular."

"What does it look like?"

We also found one of my all-time favourite movies on Youtube:

Donald in Mathmagic Land - a 1959 Donald Duck classic. I remember watching this (several times) when I was in high school on a 16mm movie projector. 

Finding Out

Next step was to find out a bit more about patterns. Some independent research uncovered several different types of patterns. We found out about:

  • arithmetic sequences
  • geometric sequences
  • triangular numbers
  • square numbers
  • cubed numbers
  • Fibonacci sequence

Students were able to share what they found out about each type of number pattern and discussed how the patterns looked. came in handy here.

Sorting Out

Now we knew a bit about different sorts of patterns, the next step was to see if we could sort out a few examples. I provided about 20 examples of different patterns, some written numerically, some as pictures or diagrams. 

Using our 6 groups of patterns that we had identified, we proceeded to spend the next hour arguing about what went where and why. Some were easy - others less so.

Here is how it looked:

It's a bit hard to see properly but there were 22 patterns that the kids had to place in one of 6 groups.
There's a diamond-shaped pattern that is in the "Geometric Sequence" group that proved difficult.

Going Further

There are some great clips on Youtube that we had a look at, to see what other people had learnt about patterns.

Two favourites were:

Amazing Number Patterns - this we great - we watched it a few times and paused the video frequently to discuss the patterns that were presented

Nature by Numbers - an awesome visual exploration of patterns by Etereae Studios 

Then we grabbed the i-pads and headed outside to photograph any patterns that we found. Finally, we imported the photos into the Comicbook app and annotated our comic strip. This provided an opportunity for the kids to demonstrate their understanding of patterns.

Here's some examples of these:


This inquiry cycle was conducted over 7 days. After it was concluded, I had a few questions for myself:

  • Did we actually go any further than just doing a series of fun activities?
  • Did the final activity actually provide the opportunity to show an understanding of a diversity of number patterns?
  • If "patterns" are the "password of mathematics" then shouldn't this inquiry run for a bit longer than 7 days? In fact, shouldn't it underline everything we do throughout the year?
  • And why is it being taught outside the rest of the inquiries that we are pursuing in the classroom? 
  • And where were the kids in the process? Had I pushed them in directions that I chose for them rather than letting them have a voice in the process?

Ahhh....teaching is learning!