"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 :-)

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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.

IWB Tips: Making invisible algebra visible

Second in a series on quick easy tips to get more from your Interactive Whiteboard + SMART Notebook software - mostly for maths teachers but might apply to other subjects too.

A perennial challenge when teaching algebra is getting students used to the conventions - and in particular, the conventions of what we don't show.  When I look at the algebraic expression "3x", I don't just see a '3' and an 'x' - I see, or at least I know, this is 3 times x.  And when I see x on its own, I know it's the same as 1x.  The challenge is to help our students see and understand the presence of these implied algebraic ideas amongst the more visible characters.

In my class, we have names for these algebra conventions:

If you're wondering about that hat ... see my post
on how the royal wedding helps teach algebra.

The joy with an IWB + SMART Notebook is you can show the invisible symbols. Here's how....

Mr Zuber:  " ...and don't forget the invisible one. Can you see it? I can see it". 

Step 1: Can you see the "invisible 1" ? It's there! Really!

Student#1: "I can't see it! Where is it?"

Student#2: "Show it! Show it!"  (they all know what is coming now!)

I switch my SMART Notebook pen to the Magic Pen, 

and write in the 'invisible' part of the expression.

Step 2: Use the Magic Pen to write the 'invisible 1'. 

The whole class holds their breath in anticipation .... waiting, waiting ....  and sure enough, five seconds later, the Magic Pen marking fades - and my invisible 1 is gone.

Step 3: Five seconds later - the 'invisible 1' has disappeared.
Back to where we were - but with the 'invisible 1' in place!

But I can now quite reasonably continue talking as if it's really there - it was there wasn't it? Did you miss it? Maybe you weren't watching? :-)

Any time I want to show 'invisible' elements, or hidden, assumed conventions, I use the Magic Pen to temporarily write them in. It really grabs the class attention and drives the point home.  OK - it's only a little gimmick - but it seems to have a real impact. Something about the anticipation of waiting for the fade, and seeing it fade automatically, combined with just the sheer fun of the trick really does seem to drive the point home.  I've been using the Magic Pen this way for six months now and my Year 8 class still  hasn't tired of it - they still watch, wait, and then ooh and ah and laugh when the text disappears. Indeed, whenever I mention the invisible one or the invisible multiple sign, they usually insist I demonstrate it. Funny thing is, even Year 11 students, who are "way beyond childish things" still get a chuckle from the Magic Pen.

My thanks to my wonderful colleague Ms Tran who gave me some early lessons on using SMART Notebook and showed me the power of the Magic Pen. The Magic Pen changes function depending how you use it: if you write with it, you get disappearing ink. Try drawing a circle or a rectangle to see some other fun tricks.

No geometry, no banana!

One of my math heroes, Michael Atiyah describes how mathematics is often partitioned between algebra and geometry, but warns about the seductive power of relying exclusively on algebra. In that colourful tone of his, he writes:

With apologies to South Park.

"Algebra is the offer made by the devil to the mathematician. The devil says: `I will give you this powerful machine, it will answer any question you like. All you need to do is give me your soul: give up geometry and you will have this marvellous machine.’”
Atiyah (2001)

When I was a young student, while I found Euclidean geometric proofs fascinating (yes - I was a math nerd), the glory of Algebra - the revelation of being able to work with symbols and expressions blinded me to anything else. Perhaps it was because I'm not that spatially aware (as my long suffering World of Warcraft guild mates found out this weekend when I kept running into the fire breathing dragons despite repeated deaths...) - but I just took to algebra like a duck to water. And then I discovered coordinate geometry - who needed shapes anymore? I could turn every geometry problem I met into an algebra problem .. even it if was a bit messy sometimes!

However while relearning mathematics 20(ish) years later as part of my post-graduate teaching degree, I discovered the joy and elegance of the geometric view. So many ideas, including algebraic ideas, are clearer, easier to learn and remember when seen geometrically. As Lockhart points out - how much more interesting is the area of a triangle when seen geometrically as a half rectangle, as opposed to a formula learnt by rote? 

Now that I'm teaching maths, I'm exploring what happens when we use geometry to help understand algebra. It was interesting to observe my top Year 9 class when set the task to explain the "difference of squares" using a geometric argument:

I handed out sheets of coloured paper and scissors, but how they resisted! Some drew small sketches of the idea, but just refused to cut out accurate representations and manipulate the shapes. I chuckled at their resistance, because I knew that's what I would have done in my school days - the algebra is 'obvious' - what's the point? We got there in the end - no doubt that will be recorded as one of those 'weird activities' I made them do :-)

Does using a geometric demonstration help learn ideas? Absolutely! I have no difficulty remembering the formula for the sum of angles in a polygon now, because I can see the triangles arranged inside it; I feel the triangle area formula - and now even the odd looking rhombus area formula is in my bones - because I can see the triangles; and the difference of squares - well it's just so clear; and you absolutely know that  because you can see it in the diagram - those two extra bits are in there!

So many ways understanding the geometry helps understanding the forumulae!
Click for a larger view

Michael Atiyah was right when he said that having to choose between algebra and geometry is like asking someone if they would rather be blind or deaf : "On the whole we prefer to have both faculties" (Atiyah, 2001).

Atiyah, M. (2001). Mathematics in the 20th Century: geometry versus algebra, Mathematics Today, 37(2), 46- 53.

Illustrations made in PowerPoint, GeoGebra and Paint. GeoGebra is an important part of the solution to breaking down the false dichotomy between algebra and geometry - a very good reason to use it in class!

Answer to difference of squares problem: Cut the top green rectangle into the two obvious pieces, rotate the narrow green piece to line it up below the larger green piece. The area covered is very clearly

Algebra with style, or, how to get the royal wedding into algebra class

This week we've been laughing way too much in Year 8 algebra class - and I put part of the blame on the royal wedding.  But let's backtrack a bit ....

While preparing to teach my first ever full unit of Algebra to junior high school students, I was amazed at the number of ideas and skills a student has to master in the first few weeks of algebra. For those of us who "just get it", it really is an eye opener to list each individual idea and skill. I counted twelve logical steps, seven starting definitions and six little 'gotchas'. Would you believe some textbooks combine most of these ideas in the first one or two sections of their algebra chapter? No wonder so many students struggle with algebra! 

So what do I mean by the 'gotchas'? Take for example whether we should write 1x or just x.  Is it wrong to write 1x? Of course not - it's just not the way we usually write it.  Maths teachers often use the word convention at this stage - and discuss why we have conventions. But as I was writing up the list of the algebra conventions, I found myself thinking that really it's about style. If you write 1x or 0x, you are sort of lacking a certain style - but you're not wrong. I teach at a girls school, so style really is something we all want to have! I found that by talking about style, my students understood the idea immediately: it's a matter of taste and "class" - not that you are wrong. My students, like most I suspect, are terrified of being "wrong" in maths class - so talking about "style" helps work around that. We have similar style discussions as to whether we should write 5xy or 5yx, and if it really matters if we write 5x +2y  or 2y + 5x.

And now, courtesy of Princess Beatrice, I have a wonderful image to use with my class about what it means to have good style.

For the record - here is the full set of "Algebra Style" elements we cover in our Year 8 class. The numbers in the circles relate to the step number in the teaching sequence. As a class we build up a table of "Algebra Style" elements and then take a moment to consider why we have these conventions.

Click on the image for a larger view.

Speaking of the royal wedding - don't miss this wonderful post at MathsPig about the forces at work in the wedding dress train. I'm intending on building a short sequence in my algebra class next week using this idea - see if the students can model the problem.

If y = 3, then 6y = 63 ... or is it?

Calvin & Hobbes is copyright
Bill Watterson and UPS

Here is a little worksheet I made for students to explore common algebra misconceptions. I'm planning to use it as part of a small-group activity, as a way to elicit discussion about algebra concepts.

https://drive.google.com/file/d/0ByVkChxwrC4DSUl3OGV3Q210S0U/view?usp=sharing

Can you see why someone might think (Calvin in the last panel) that

n + n = 28

even though they have been given no value for n ?

I sourced the algebra misconceptions from

MacGregor, M., & Stacey, K. (1993). What is x? The Australian Mathematics Teacher, 49(4), 28-30.

and Calvin just forced his way onto the page (apologies to Bill Watterson!).