Mathematics

TER Podcast follow-up: the big issues

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2014 has been a year full of firsts for me. First year teaching in a comprehensive school (student teacher placements notwithstanding). First year as a head teacher (which has produced a whole lot of firsts of its own). First time recognised in public because of the videos I make. And as I type this, I’m at my first MANSW Annual Conference (arguably the biggest gathering of maths teachers in the state all year), my head spinning from considering new ideas and meeting new people (or in some cases, seeing people face-to-face who I’ve been interacting with online for a long time now).

Another first happened last month, when I participated in a phone interview with Corinne Campbell (@corisel) for the TER Podcast. The topic was Is Maths Education Broken?, and I was there to provide a sort of foil to an interview that Corinne had with the luminary Conrad Wolfram (of Wolfram Alpha fame). You can listen to the entire episode (and if you’re in education and haven’t subscribed to the podcast, you really ought to). It was an interesting experience, not least because it was so unusual to actually interact with a voice that I was so used to just listening to passively through a podcast.

I’m the kind of person who thinks of the perfect witty comeback or joke ten minutes after the conversation is over. So even though I had prepared my own thoughts and notes before the interview and tried my best to cover everything that would be important, I found myself in the shower that evening thinking, “Oh, _this_ would have been the perfect answer to that question!” and “How on earth did I forget to say _that_?” So here are a few of the things that I should have said, but forgot to. In this post I’m going to talk about the large-scale issues that are related to the “STEM crisis” Australia is experiencing, and in a follow-up post I’ll talk about some of the smaller practical strategies that can be employed in the classroom to help our students from day to day.

How do we improve STEM skills in Australian schools?
There’s no simple solution to this one – so you can know with a fair degree of certainty that if someone tells you they have a straightforward way to fix this problem, they’re probably just oversimplifying the situation. The so-called STEM crisis is a perfect storm of different factors and so there won’t be a single actionable item to fix things.

But there are definitely many identifiable aspects of the challenge. For instance, the syllabus is by-and-large divorced from real mathematical practice (both in everyday life and in vocational contexts). Here’s a great little quote from Optimising the Future with Mathematics (via The Conversation):

Current mathematics education, in schools and universities, is satisfied with programming students to carry out certain mathematical processes, and assessment rewards students who can calculate everything even if they understand nothing.

So what can we work on at the ground level? Firstly, it’s vital to recognise the enormous continuity of learning in maths. All key learning areas exhibit a degree of intra-dependence within their skills and knowledge, but it seems to be especially noticeable in maths where a single “weak link” in the chain can be disastrous! Once student confidence is lost, it is hard (not impossible, but significantly challenging) to rebuild it.

Secondly, top-down (syllabus level) change is required, but we can’t wait for that to happen. Policy is always hard and slow to change (it must be in a democratic and bureaucratic environment), but we can push the envelope of our daily practices right now and see what works. We can undertake action research projects into what is effective and helpful. Another insightful quote:

We need mathematics “to be taught more like it is done” by those engaged in it, in both the innovations economy and research. This is a cultural change that involves the discipline itself, one that must be mainstreamed into school and university systems.

These cultural changes almost never come as mandates from above – they are typically born out of grassroots movements from below that are then recognised and ratified by authorities.

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Beauty & Mathematics continued (25 July 2014)

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Hello class! Apologies that I can’t be there with you today – but you have some very interesting material to cover today nonetheless.

VIDEO 1: Donald Duck in Mathmagic Land

This video is quite long (about 27 minutes) and covers a wide range of topics. As you watch it, take notes on the following:

  • What mathematical ideas are presented?
  • Select two that interest you in particular and research them further. How do they relate to the mathematics you already know? Where else in the world do these concepts reappear?

You may take notes on your laptop, but you will definitely need to have a pen and paper available anyway as most of the concepts addressed in the video are visual and you will need to illustrate them in some way. Now that you know what you’re looking for, here’s the video:

After the video is finished, take 15 minutes to look over the notes you have taken and reflect on them as usual. (You should use the questions I wrote about in this post to guide your thoughts.)

VIDEO 2: TedXEast – Matthew Cross

The next video you watch is a TED talk. Again, take some brief notes (but be aware that the presenter goes through the material very quickly). Similar to before, select the two most surprising examples that he talks about and briefly describe why you think these are so unusual. Here’s the video:

VIDEO 3: How to measure beauty

Here’s the last video you’ll be watching. As you watch it, you simply need to answer this question: “Is human beauty just about numbers? Why or why not?” Answer in some detail and try to justify your response with evidence and examples.

As usual, please compose your thoughts in a post and email it to me. Some of you have kept on top of each lesson’s assigned tasks, but others of you are slipping behind – use this opportunity to catch up! (And don’t forget to carefully follow the instructions I gave you in the very first lesson.)

See you again next week!

What is “Exploring Mathematics”?

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Exploring Mathematics is an elective semester course offered to Stage 5 (Year 9-10) students at Cherrybrook Technology High School.

To understand what this course is, you must first understand what it is not:

  • This course is not about acceleration (learning content from years 11-12 in advance so that you will be more familiar with it when you encounter it in the future). In fact, topics in the Stage 6 mathematics subjects have been intentionally avoided so that they can be given their proper introduction in the Preliminary and HSC courses.
  • This course is not like your regular mathematics class in its classroom activities or its assessments. In fact, there is a very conscious emphasis on branches of mathematics that are not understood through repetitive exercises, nor assessed in traditional examination formats.

By contrast, the goals of this course are:

Read the rest of this entry »

Extra teaching thought: will the real basic transformation please stand up?

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I’m a pretty slow thinker. In an argument, I’m that guy who comes up with the perfectly witty comeback… about two hours after the conversation is over. This doesn’t just happen to me in social settings – it also happens to me in the classroom! Often I’ll teach through a skill or concept only to realise, after the lesson is over, that there was something else I should have said, or some other analogy that would have been immensely helpful, which would have added valuable insight or made things clearer.

Something like that happened when I taught this lesson on congruence transformations:

In geometry, a congruence transformation is – roughly speaking – a way that we can change (“transform”) a figure in such a way that it is still the same shape and size (“congruent”). There are three main kinds of transformation that we cover in year 7: translation (sliding the shape to a different position), reflection (flipping a shape over) and rotation (spinning the shape around).

After introducing the idea of transformations and how they work, we start to think about “composite transformations” – what happens when you combine more than one transformation and consider the total effect from original to final image. An interesting fact emerges: that not all transformations are created equal. In fact, one of the transformations is more simple and basic than the others. One transformation can be used to create each of the other two. Which do you think it is?

It turns out that the most basic transformation is reflection. You can make translation and rotation out of a series of reflections, but not the other way around; though it’s counter-intuitive, reflection is the most basic kind of geometric transformation. Perhaps that’s why reflectional symmetry is so deeply ingrained into our natural sense of beauty.

 

Inaugural Alumni Conference

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I have fond memories of university. It was a time of growth and change, where I face immense personal and intellectual challenges that have proved to be definitive in my life. If people who know me well were asked to describe me in objective terms, I am confident that the major aspects of my character and personality (both positive and negative) can be traced back to things that happened to me during my time at uni.

It was such a formative stage for me, which is why when I received the invite from Judy Anderson (@JudyAnderson6) to attend and participate in the inaugural Alumni Conference for graduates of the Secondary Maths education degree at Sydney University, I was genuinely delighted by the prospect of returning to my alma mater and reliving some old times.

SUSMAC2014B

It was during the school holidays, which made me pause before I decided to commit to attending, but eventually I decided it would be a worthwhile thing to go to – and I don’t regret it. I haven’t felt so energised and excited by a gathering of people in quite a long time, and it would have been hard for anyone to walk away from the door with a renewed passion for mathematics education.

I wish I had time to write a bit more about my experiences on the day – perhaps once the school term settles down, I might (though it very likely will not) – but for now I just want to say that I was able to video every presentation during the day and I’ve made them all available on a single easy-to-browse page: SUSMAC 2014. Enjoy and share with any maths teachers you know!

Nemesis

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I enjoy doing maths and I spend a lot of time working on it, but I have a hard time calling myself a mathematician. It’s not because I dislike the label – on the contrary, I don’t feel as though I’m really worth of the title. Real mathematicians… well, they’re the kind of people who go to the National Mathematics Summer School (NMSS, affectionately pronounced as Nemesis).

Image
Some of the view during my walk from the bus stop to the part of the ANU campus where I stayed.

Perhaps you think you know some nerds. Do they chuckle with childish delight when considering the cyclical nature of inverses that exist in the set of Gaussian integers modulo the complex number (4 + i)? No? Then step aside and let the real nerds take the stage. These guys – and hence by extension, their tutors and lecturers (who are mostly NMSS alumni) – are the real deal.

I would never have attended NMSS as a student. I didn’t have anywhere near the mathematical chops to even be considered as a candidate (there are roughly 70 positions for the entirety of Australia). But I may well have enjoyed it if I had been invited. Since it’s a gathering of students from across the country, they try to assume very little prior knowledge – hence their focus on number theory, which is renowned as easily accessible and abundance of opportunities to “think deeply about simple things”, the motto of NMSS’s recently retired director.

It’s intentionally different from a school learning environment, which by its very nature emphasises assessment and competitiveness. No one hands in their problems and marks aren’t assigned for anything. The whole experience is crafted to encourage exploration, playfulness and creativity. If you’re not a maths teacher – or even if you are – and those words seem like the antithesis of mathematics to you, then that’s a sad testimony to just how different high school maths is to the actual maths that mathematicians do. (I’m not sure if that’s a gap that will ever be bridged, but there it is for whatever you want to make of it.)

But this week, I wasn’t there as a student – I was there as a teacher, to get a concentrated version of what the students were experiencing and then to think about how that would inform our practice as educators (particularly with regard to nurturing and encouraging gifted and talented mathematicians). It was a jam-packed couple of days and I found myself constantly thinking of new and awesome ideas that I would love to start implementing when I get back to the real world, but unfortunately I think I’ve just about maxed out (or exceeded) the number of new things I’ll be doing this year. So mostly I think I was mentally filing things away for the future, waiting for a time when I can act on them and give them the time and effort they deserve.

One thing that remains deeply impressed on my mind, though, is the importance of teaching mathematics in an engaging way (and, related to that, encouraging people who are capable of that into the profession rather than ushering them off into engineering or actuarial studies). Being exposed to so many passionate maths teachers (and I use that term broadly of anyone who teaches mathematics, not just people who work in high schools) was a vivid reminder of how important the delivery method is in shaping a students’ experience of a subject.

If someone teaches you how to cook by forcing you through lessons and explaining things in a bland way (see what I did there?), then who can blame you for disliking the kitchen? But if someone visibly enjoys the process of mashing food together in an awful mess, if they express genuine delight at the intriguing ways that foods can relate and be combined with one another, if they marvel with closed eyes at the smell of what they have just concocted, then who can help but feel inspired to try and master the same subject that brings so much joy? And I think that is a part of why Jamie Oliver rose to fame so rapidly (and subsequently kept it). I hate cooking, I hate the lengthy preparation, I hate the mess, and I hate the low-quality stuff that I usually produce. But when I watch Jamie Oliver at work, I want to get up and cook. I want to give it a go and learn how he does what he does. And that’s exactly the same vibe that the NMSS tutors and lecturers give off to the students who are privileged enough to attend.

Wouldn’t it be fantastic if ordinary students could experience some of that during their normal schooling? It shouldn’t just be for the elites. I’m under no illusion that the whole NMSS experience can be replicated on a large scale for the entirety of a school year, but the world deserves to know that mathematics is a fascinating and amazing subject – not the dry, boring thing that most people think maths is. And we’ll need passionate mathematicians and educators to accomplish that. Now there’s a long term goal worth working on!