One day, I sat down to write a blog post titled “How to Teach Mathematics”. As I immersed myself in writing, I rapidly realised that this was not a single blog post, nor was it a series. Instead, I had started to write a book; a book that I had been meaning to write for some time. My book is about how to teach those cognitively demanding subjects that sit at the core of most school curricula; the subjects that employers value and that bring lifelong joy and interest.
So I’ve started about halfway through. You can follow my progress and see how it evolves as I work backward to an introduction, forward to potential solutions and outwards from a consideration of mathematics alone. I hope you enjoy it but be warned; these are book chapters; they are not pithy postings.
If I get to the end and have crafted…
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In 1967, President Lyndon Johnson rose to deliver the annual State of the Union address to the United States Congress and to announce a new front in the war on poverty1. The Head Start program for disadvantaged pre-school children was going to be expanded into the early grades of school. Johnson was planning a budget of over $100 million but Congress ultimately slashed this to just $15 million. This provoked a rethink and the project was reconceptualised as a research programme. “Follow Through” was born. Despite the circumstances of its inception, it grew to become the largest and most expensive education research project ever undertaken.
The study had a classic ‘horse-race’ design. Different sponsors were engaged to deliver programmes and the results of these programmes were to be evaluated against each other and against the results gained my similar students who were not part of the project i.e…
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Last year, I was asked to speak at the Year 12 leavers’ breakfast. I just found a copy of that speech and thought it might be of interest.
What do I know?
When I was asked to give this address, I thought:
“Why are they asking me?”
“What do I know?”
While a great honour, it is also a tremendous responsibility, and self-doubt in such circumstances is natural.
However, let’s just examine this a little. What do I know? What sits in the filing cabinet of my head that I could pull-out, dust down and pass on to you as you embark upon the next stage of your life? For knowledge is a precious thing; more valuable that gold.
Judith Shapiro is the former president of an American college. One day, she was chatting to a group of young people about the benefits of university; the benefits of an education. And this is what she said;
“You want the inside of your mind to be an interesting place to spend the rest of your life.”
When I first read that, I was struck by its simplicity and it reminded me of how I felt about education. Right now, for many of you, education is about performing in exams and getting in to the right courses. At other times, people talk about the skills that we develop at school that employers will find useful. These are all important, but the real essence of a good education; that is about deepening and enriching your knowledge. Such knowledge bestows reserves of curiosity and interest that will sustain you and nourish you throughout your life. Our most rewarding tastes are acquired tastes and intellectual tastes are acquired through knowledge.
Knowledge should be a right but in reality it is a privilege. This may seem like an odd thing to say in the age of the internet but let us not confuse the possession of knowledge with the ability to access information. Knowledge is knowing what is relevant; knowing what is valid; knowing what will make an illustrative point. Knowledge is how you know what to look for on the internet. Knowledge enables you to evaluate what you find. Knowledge is how you understand what you find.
When I was young, I went to a government school. My school was pretty tough and we didn’t always learn a lot. The schools in my area only went as far as Year 10 so, if anyone wanted to stay on past this, we had to go to a place called a ‘sixth form college’ in order to study for our A-Levels – the English equivalent of the VCE.
I arrived at my first chemistry lesson at sixth form college and I couldn’t understand what anyone was going on about. There were words that I had never heard. The teacher asked questions – I didn’t even understand the questions so I had no hope of giving an answer.
This is what a lack of knowledge feels like. It is abject. It is isolating.
This is how it feels if you pick up a copy of The Age and you cannot access the stories because you just don’t know enough about what they are talking about. It’s a miserable state that shuts you out from the discussion; that disempowers.
On the other hand, you are fortunate people. You and your family understand the value of knowledge and may have made sacrifices in order be here today. During your time at College, you have had knowledge carefully selected for you by some of the finest, most passionate educators that I have ever met. They have engaged you in this knowledge and they have concerned themselves with ensuring that you understand it.
Do not let it stop here.
The world is full of intellectual riches. Many of these will not yet interest you because you don’t know enough about them – don’t write anything off.
Two men crossed a meadow. The runner, raced across the meadow from one end to the other. When he arrived at the other side, he jumped up and down shouting, “I am the winner.” The wanderer wandered through the meadow, looking at the flowers, observing the insects and feeling the cool soft breeze upon his cheeks. If we see this as a metaphor for life then who is the winner? The one who can move most efficiently from birth to death without distraction, or the one who pauses to feel the breeze upon his cheeks?
With knowledge, comes wisdom; the understanding of the interconnectedness of all things. What start out as isolated pockets of information, grow, merge and interact. This is where creativity lies. Creativity is not the ability to think of numerous absurd uses for a paperclip; creativity is knowing the rules and then choosing which rules to break in interesting and strategic ways.
As a school, we are always trying to develop your ability to think clearly and rationally. However, thinking is difficult. You will all have experienced that point where you throw yourself back into some hard, mental work even though you really cannot be bothered. Relax. That’s quite normal. In the words of the psychologist, Daniel Willingham;
“Humans don’t think very often because our brains are designed not for thought but for the avoidance of thought. Thinking is not only effortful… it’s slow and unreliable.”
So what makes a successful thinker? Do successful thinkers find thinking easy? No. They find it just as hard as you do. But they keep throwing themselves back into it and, crucially, they have large reserves of knowledge to fall back upon so that they don’t need to work everything out from scratch.
Perhaps it is now time to open up that filing cabinet and pass on a few of the things that I know;
Over 90% of the mass of your body comes from stardust, formed billions of years ago in the centre of a star and then blasted out into space as the dying star exploded.
It is easy to overcook fish
It is always worth taking a hat to the MCG
There is no strategy that a man can use that will guarantee that he picks-up at a party
Bryan Adam’s song “(Everything I do) I do it for you” was at number one in the UK pop charts for 16 weeks in 1991. It was a difficult time for all of us.
If something is 90% fat-free then that means that it is 10% fat
You need to set clear ground rules before teaching a science unit with a Year 7 class on human reproduction
Australians think that they have the best coffee in the world
Most of the quotes that you find on the internet are wrong
My English teacher used to say, “Discretion is the better part of valour,” over and over again. If you don’t know what that means then Einstein said something similar, “If A is success in life, then A = x + y + z. Work is x, play is y and z is keeping your mouth shut.” Although we live in a free, liberal democracy, this does not mean that it is always wise to say exactly what you think, particularly to your boss.
And finally, I know of an excellent piece of advice that has come down to us from the philosopher Michael Philip Jagger, “You can’t always get what you want. But if you try sometime, you just might find, you get what you need.”
Given that you are about to embark upon the journey of your life – a journey towards adulthood and independence – then I think that’s pretty good advice to take with you.
But then, what do I know?
Mathematics can be pretty hard and this has to affect the way that we teach it. Contemporary maths teaching splits along two main lines; explicit teaching and constructivist teaching. These approaches attempt to deal with the cognitive demands in different ways. Before we can develop a model for effective explicit maths teaching – the model that I wish to promote – it is necessary to explain my scepticism about constructivist methods.
Constructivist mathematics teaching is popular with researchers and in schools of education. It received a significant boost by the publication in 1989 of the first set of NCTM standards in the US1 which took a broadly constructivist line and helped spark the ‘Math Wars’ which are still raging today. It is hard to tell just how far constructivism is a philosophy as opposed to an approach based upon scientific psychology. It has been argued2 that it is…
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