Sunday, August 2, 2015

"Why are we learning algebra?"

It had been several weeks since my Year 7 class had the discussion of why we were learning algebra, so I was taken off guard when the perennial question came up again: "Mr Zuber, why are we learning algebra (again)?" 

I have a whole range of answers I like to offer to this favourite question but this time something unexpected came out of my mind.  "Have you seen those amazing new pictures of Pluto that came in this week from the Horizon spacecraft?"  I was pleased to see many students in the class start to get excited - they certainly were inspired by those photos.  

Global mosaic of Pluto in true color (NASA) July 2015

"Well", I said, "that was algebra. Algebra brought us those pictures. Very complicated algebra, and physics and engineering worked out by smart people helped get that spacecraft just at the right place, at the right time above Pluto, millions of miles away from Earth, to get that photo and send it back to us. That's why we're doing algebra."

I think that was the best answer I gave in class all week - and the students seemed to like it. Thank you NASA!

Here are four reasons for learning algebra that I like to offer students when I start the introductory algebra topic.  

Firstly we have some utilitarian reasons:

Algebra is a tool to help solve problems.
We use it to find values of something we don't know.

Algebra allows us to record information about relationships between numbers in a formula.
We can then put values into those formulas to find related numbers. This could be the area of a triangle, or the dosage of medicine to give a child based on their weight.

At a deeper level, algebra has an important place in our discovery of the world:

Algebra allows us to describe how the world works.
Students like this image. The picture in the centre is a matter-antimatter collision and the formula is Heisenberg's Uncertainty Principle.

and in supporting our exploration and representation of mathematical ideas.

Algebra allows us to represent and explore mathematical ideas and mathematical objects.
At least some students in your class will have seen the Mandelbrot Set and know how complicated it is - they will be very surprised how 'simple' the algebra looks.

Putting all these ideas together, I like to summarise with the one big idea: algebra is a language.

So - for those people who say "but I will never use the quadratic formula in my future work", I would respond: "Wouldn't you like to learn this amazing language? It will open up so many career possibilities to you (a utilitarian argument) and it's also a fascinating and rich language that will let you access a whole new level of knowledge and ideas (a sheer pleasure argument)"

What's even more amazing about this language is that it's an international language. I can speak algebra with a Russian or a Chinese mathematician. Somewhere out there in space, a class of Year 7 students with green skin and three eyes is also learning algebra.  Can you think of another subject you're learning at school which is also being taught in Alpha Centuri?  (OK - science... but let's pretend that's the same as maths :-)

Postscript: Should I have mentioned that algebra helps us develop reasoning skills? Possibly.... but I'm not sure most students buy the "it's good for your thinking" argument. So I take the "what algebra will offer you" line, and make sure I give emphasis to its role in abstract thinking as well as in 'practical' applications.

Thursday, July 9, 2015

Wormholes and Tesseracts in the Classroom - Part 2

In Part A I looked at some inspiring ideas about teaching in the movie Interstellar. Here are some ideas for using some Interstellar content in the classroom.

Exploring Dimensions: whether you're doing lessons on 2D and 3D solids, or just having a discussion why we say "x-squared" and "x-cubed" but "x-to-the-fourth", it's time to bring out The Tesseract.  In the Interstellar version, it's an object that has spatial and time dimensions: as the main character Cooper moves through space, he's moving into different time "rooms".

"Time is represented here as a physical dimension"
Warning: SPOILER for the film!

My Year 7 students liked this, and were very fast on their internet devices to find more traditional mathematical representations of the tesseract (the hypercube) which made for a good discussion.

Permutations and Combinations: I haven't worked it out yet, but there's definitely a perms and combs activity to do with the CASE and TARS robots! Might link in well with a Quadrilaterals exploration too.

Watch CASE at work on Miller's Planet: 

Some good resources:

An exciting way to introduce circular motion:

A nice adjunct to the more classic and sedate circular motion sequences in my other all time favourite movie, "2001: A Space Odyssey".

And for something different, you may like to point your music teacher friends at this mini documentary about the making of the soundtrack:

Wormholes and Tesseracts in the Classroom - Part 1

"You have to go see the film Interstellar", I told my Year 12 class recently, "Maths saves the human race!"  They corrected me immediately:  "No Sir - that's not true, LOVE saved the human race".  Yeah - we have a lot of science fiction movie buffs in our school...

But quibbling aside, what a wonderful resource Christopher Nolan has given with this film. Here are a few highlights that inspired both my teaching and enriched the conversations in class.

Interstellar: Inspiration for Teachers

Need to describe the projection of 4 dimensions into 3 dimensions in under 30 seconds? This explanation of why a wormhole should be a sphere is astoundingly concise (start at 03:00)

Wow! Imagine if we could teach complex content this easily.

Probably the scariest conversation a maths/science teacher could ever hear takes place early in the film - a wake up call to all us! In a four minute sequence, Nolan describes the futility of assigning a single number to measure student success, text book wars and the battle we have on our hands to defend the scientific world view:

So why do I teach? In an astounding sequence, Cooper mumbles muses:
"We used to look up in the sky and wonder at our place in the stars. Now we just look down and worry about our place in the dirt."  
(sequence starts at 3:00)

We need more star gazing. If I can get even one student a year to decide to look at the stars, it's all been worth it.

Part 2: Curriculum links : ways to use Interstellar in your classroom.

(The videos are probably not going to be around for long ... watch them while you can!).

Sunday, April 26, 2015

Just change one word

Ever found yourself describing a student or a class to another colleague as "low ability"? It's a shortcut we use more frequently than we may realise, even if said in the most caring, well-intentioned way. Early in my teacher education at Sydney University, I was very fortunate to be given a very simple and powerful idea: change one word.  Replace the word "ability" with "achievement". 

The result may surprise you. Here's my favorite example: next time you hear yourself saying: "Let's save that (interesting, challenging) activity for the high ability class", change it to: "Let's save that (interesting, challenging) activity for the high achieving class". Wow! Would you really want to do that? Changing one word isn't about being politically correct - it's about altering our mindset from a fixed mindset to a growth mindset. When done with awareness, changing one word can make a real impact in your classroom and your school.

It's approaching five years now since I graduated from Sydney University, and this simple idea continues to pay dividends in my teaching. So it seemed only fitting to make it the topic for my presentation at our alumni conference, SUSMAC 2015. 

With thanks to Judy Anderson and Maria Quigley for organising the conference, and to Eddy Woo for his work to make the conference available on the internet.

For a full set of videos and notes from SUSMAC 2105, featuring a keynote from Andrew Martin, and over 20 short presentations from teachers and preservice teachers, see:


Tuesday, November 11, 2014

The Pi Collection : 50 maths enrichment books for your school libary

A little project I've been working on for the last year with my school librarians: we call it "The Pi Collection".  

We've built a carefully selected collection of around 50 books with maths themes - fiction and non-fiction, in an effort to entice our students to engage with mathematics beyond their regular classroom work.

Officially our idea is to encourage students to read more widely and deepen connections between maths and other subjects, but in truth, we just love having these great books available to share with our students. So, when, for example, we're exploring extra dimensions, we can ask "What if someone built a house in four dimensions?" - and point at the classic Robert Heinlein short story "And he built a crooked house" which just happens to be the library waiting for you to read.  Or when we're talking about equations, and wondering if we could use them to describe everything - point students at the wonderful Isaac Asimov "Foundation" series - every thirteen-year old nerdy boy's dream of running the universe through maths.   Or perhaps someone thinks girls don't math? Well have we got several books in the Pi Collection to show you otherwise!

Along the journey of building the collection, we've discovered books about people who think differently ("Born on a Blue Day"), a terrific manga-style book questioning the inner truth of logic in mathematics ("Logicomix") and short stories about what could happen if you were allowed to divide by zero ("Stories of your life"). And who could forget that the answer to the meaning of life is 42 ("Hitchhikers Guide to the Galaxy") - a book which also has great fun exploring probability - remember the "Improbability Drive"?  Doing permutations and combinations? Well that's just begging for a reference to Arthur C. Clarke's short story "The Nine Billion Names of God".

Nothing however quite had the impact of "The Cold Equations" - this flew off the shelf as soon as we posted it up:

Maths tells our hero he needs to eject his stowaway into space.
His heart tells him otherwise. What decision will he make? 

I was a little worried about the unsubtle appeal  to baser instincts - but hey - anything to get the students reading! And it turns out our English faculty teaches Science Fiction in Year 7 and The Detective Novel in Year 8 - so we made sure to select books in these genres that also had a maths element.  As we expand the collection, we're finding more connections to Geography, History, Science and Art.

We've also included maths extension and enrichment books for curious students who want to go beyond the official high school curriculum. Collections of classic puzzle books, short articles on maths topics as well as some more challenging books. We even snuck in a few that might encourage some students to consider a teaching career (thank your Mr Lockhart!).

The full list is available here. Do you know of any books we should add?

Looking for more maths themed books?  An invaluable resource is Alex Kasman's collation of titles. Some care is required though, because not every book here is suitable for high school students.

Updated: 2015 Semester 1 Pi Collection

Thursday, October 23, 2014

AB Quizzes in the mathematics classroom

Notes and resources for a presentation I gave to the North Sydney Region Maths Association this week:

PowerPoint template for an AB quiz

Author's copy of article in MANSW Reflections - 2013 Conference Issue

Thanks for the invite and feedback - some great ideas from the group to experiment further with the AB Quiz concept.

Saturday, May 10, 2014

Bringing "childishness" into the mathematics classroom

Why would you show a Sesame Street video to a senior Year 12 mathematics classroom? Is there any value is doing paper folding exercises with such a class?  Here's my thoughts on the value of bringing "childishness" into every mathematics classroom - no matter the age group or the current achievement level, presented at a recent conference.

Games played by children and mathematicians
April 23, 2014
University of Sydney, Mathematics Education Alumni 2014 Conference. 

Or just preview the presentation:
Click on the image for the Google Drive document