Category Archives: presentations

#SciPubMath

I felt privileged to be one of three speakers at last week’s Science in the Pub. Held monthly on Friday evenings, the aim is to promote understanding of and enthusiasm for science. Science in the Pub Adelaide celebrated their second birthday with their first maths-focused event: ‘SciPub Math: I got 99 problems but math ain’t one’. 

The format is intentionally informal; people leave their seat at any time for another drink, or simply mill at the bar while the three panellists give short 10-15 minute presentations, followed by a 30-minute Q+A. The aim of holding it in a pub is to encourage attendance by people who may be intimidated by the traditional academic setting. This was going to be my first Science in the Pub. (The Youtube clip of my talk is at the end of this post.)

The invitation asked me to talk about current work with my colleagues in the Scheduling and Control Group at UniSA: ‘… we thought your research in efficient transport and railway operations would be a great demonstration of some of the ways in which maths can be applied to solve real-world problems‘. I was delighted to discover that my co-panellists were my good friend Jono Tuke (The University of Adelaide), and my collaborator Jerzy Filar (Flinders University).

I’ve given many talks about my research in the past, and to a wide range of audiences, so I was surprised to find myself quite apprehensive about this one. I thought the problem was that I couldn’t carve out the time to prepare the talk well in advance. I spent the month beforehand with ideas rattling around in my head, while dealing with a whole bunch of other work and illness. I wanted the first maths-focused night to be a resounding success, so that the organisers would include more in their regular program.

When I finally sat down the afternoon before (!) to properly prepare, I realised that I was mainly anxious about these people. The audience.

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These people were choosing to come to a Friday night event, likely at the end of a busy week. They were looking to unwind and be entertained, and I was the one who was meant to provide it by talking about two topics that can be fairly dry to others: maths and trains. Some of them would be mathematicians, or research scientists. Some would be interested members of the ‘general public’. I know how to give a short non-technical talk about my research, and I can give a highly technical conference-style presentation. But I wasn’t confident I could give a moderately substantial, 15-minute talk to a diverse audience that was also appropriate for a Friday night at the pub.

I spent the first couple of hours of preparation by crafting an analogy and creating an elaborate title slide that I thought would be a hook. From there, I roughly knew how the talk would pan out. (After all, I’ve given versions of it plenty of times.) That night, I tried the start and the end of the talk on my partner. He told me what I needed to hear; he hated it. The analogy was too forced. Start again.

I decided to focus on the basics, and what I knew to be true.

  1. Substance. It’s ‘Science in the Pub’ not ‘Science Jokes in the Pub’. The aim is to give people an insight into scientific research; to do that we might need to dig into the big ideas which can get a little technical. As a side note, it is important to me for female mathematicians — particularly when there is only one on a panel —  to show ‘mathematical muscle’ (as my colleague Lesley Ward calls it). I know that there is no difference between the work done by female and male mathematicians, but not everyone subconsciously or consciously holds that view.
  2. Be myself. I can’t script entertaining, but I can script engaging. I know how to tell a good story about my research, it just might not be intentionally funny. (I can do unintentionally funny.) Luckily, I knew that my talk was going to be the second of three, and that my co-panellists are both entertaining presenters who would give good opening and closing talks.
  3. I mostly give good talks. Whenever I prepare a talk (from keynotes to weekly classes), I try to incorporate the six principles from Chip and Dan Heath’s ‘Made to Stick’ SUCCESs model: Simple, Unexpected, Concrete, Credible, Emotion, Story. My aim was to give the audience an intuitive insight into the mathematical ideas that help save energy on trains. So I tried to tell them a simple story. I used a concrete experience for most people (riding a bike) to help tell my story. There was an unexpected (for them) moment in my talk, too.

I finished putting my talk together on the day, in between teaching a two-hour class and having a research meeting. I managed to run through it a couple of times beforehand. I’m pleased with how it went, although I can recall plenty of places to improve. You can judge for yourself below. (I haven’t watched it all the way through.) Many thanks to Matt Skoss for filming on his iPhone.

The whole night was absolutely fabulous and very enjoyable. The three talks complemented each other beautifully. It was especially fun to co-present with two people I know very well; we were joking around for much of the evening. The Q+A was stimulating and particularly in my corner, with much discussion about how best to engage students with mathematics. I was pleased to make a few points around the importance of playing with maths, the value of making mistakes, respecting students’ ideas and previous mathematical experiences, and displaying maths visually. And I’ve been delighting in the feedback, like this:

My 14 year old son, in particular, was so interested to hear of your work. He loves science and maths, so learning of the work you are involved with helped him see his high school maths in perspective – ‘real life’ practical application!! Plus a real mathematician!

No dead white male mathematicians here :).

 

 

 

Counting in unexpected ways

It was a delight to recently spend five days working with students and teachers in Alice Springs at the invitation of MTANT, the Mathematics Teachers Association of the Northern Territory. I then spent a week in bed with the flu, which is one reason I’ve recently lost my voice (both physically and online).

The main purpose of the visit was to join the 8th Annual Maths Enrichment Camp at Ti Tree School, in a small remote town 200 km north of Alice Springs. Students travelled from all over Central Australia, some from as far as the Aboriginal community of Hermannsburg, 320 km south (that’s a short distance in the Northern Territory!). The camp runs Friday night to Sunday morning, and is full of fun activities (mathematical and non-mathematical) for kids, and teachers, to engage with. This was my first Maths Camp, and I was thrilled to be invited; thanks @matt_skoss!

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This year the Ti Tree Maths Camp attracted around 35 students from Years 4 to 10. Students were divided into three groups and on Saturday rotated through four activities, called ‘Worlds’. Thus, these activities needed to accommodate a broad range of mathematical expertise. To add extra challenge, I rarely work with students in the lower years, so I relied on a couple of trusted friends to help determine whether my planned modifications would be appropriate.

In this post I briefly describe how younger students responded to two of my favourite activities, which I’ve previously written about: The Game of SET and Domino Circles. I doubt that this is going to be revelationary to most teachers, but I am always learning how students make sense of mathematics (younger students, in particular), so I want to record my observations for the future.

Counting Dominoes

I worked on this problem with a combined group of Years 5 and 6 girls from Bradshaw Primary School and Araluen Christian College in Alice Springs.

Display this image. What do you notice? What do you wonder?

Dominoes

Responses include:

  • I notice: that there different numbers of dots on a domino.
  • I notice: that there are two groups of dots on each domino.
  • I notice: that the dots are different colours.
  • I wonder: what is the highest number of dots on a domino? (A fascinating side discussion commenced as we had to resolve whether we meant in total or on one half of the domino. We decided that in a Double 6 set, the highest number is six. What do you notice and wonder now?)
  • I wonder: what is the lowest number of dots on a domino? (Zero.)
  • I wonder: how many dominoes are there in the set? (My response is usually ‘Good question! I wonder if we can work that out?’ :))
  • I wonder: can a domino have more than one instance of the same number of dots? (Yes — I show a ‘double domino’, like 2|2.)
  • I wonder: is there exactly one of every combination of numbers of dots? (Yes.)

Usually my next question is to ask students to calculate how many dominoes there will be in a set. Some students start by drawing them all out. For this students I might show an image of a Double 18 set—too many to draw, right? This encourages students to find, and then explain, a formula for the number of dominoes in a Double ‘n’ set.

However, for younger students that I hadn’t met before, I was concerned that this question might be too challenging. Instead, I handed out sets of dominoes and asked students to have a look at them. Then I revealed that each set was missing a domino. Could they work out which domino was missing?

As you might expect, students needed to find a way to organise their dominoes so that they could identify the missing one. Several groups made arrangements like this. It’s easy to spot the missing domino now, right? How else could you have arranged the dominoes to make this discovery?

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I gave one large group two sets of dominoes (one paper, one physical) in case they wanted to work in smaller groups. I was delighted to find that they instead used both sets of dominoes in tandem. It looked something like I’ve reproduced in the photo below. Can you spot the missing domino from each set?

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There was an interesting moment in the middle of this activity as we discovered that some sets had more than one domino missing, and some sets had duplicates. (Guess who didn’t double-check the domino sets before starting the activity? <blush>) This could have been a disaster, but I took it as a true problem-solving experience for the group. We sorted out our sets eventually!

The rest of the session was largely spent exploring this question:

Is it possible to arrange an entire set of dominoes in a circle so that touching dominoes have adjacent squares with identical numbers?

Once you’ve experimented with a set of dominoes in which the highest number is six, explore whether it is possible for sets of dominoes where the highest number is different.

You can read more about this problem here.

We finished with a quick ‘Notice and Wonder’ with this short promotional video by Cadbury, in which they set up blocks of chocolate in a suburban street, and knocked them over like dominoes. I wish I could remember all the rich wonderings the students had — they were fabulous!

Counting SET cards

My chosen activity for my ‘World’ at the Ti Tree Maths Camp was The Game of SET. I’ll briefly recap the game, before talking about how students counted their set cards.

SET is a card game. Three of the cards are shown below. What do you notice?

SET Cards 01

Students eventually identify a number of attributes of the cards. Sometimes (but not often) they generate more than we need for the game. I acknowledge them and ask if we can focus on four particular attributes: number, shape, colour and shading. We notice that each attribute has three different values that it can take. For example, shape can be ‘oval’, diamond’ or ‘the squiggly thing’.

I confirm that these are all the possibilities of values of attributes. When I work with students in higher grades, my next question is usually as follows.

If a SET deck contains exactly one card of every possible combination of attributes, how many cards are in a deck?

To adapt this question to lower years, I did something similar to what I did with counting dominoes. But instead of removing a card, I asked them to find a way to be sure that they had exactly one card for every possible combination of attributes in their deck.

Their natural ability to group by features that were the same, and to organise in a systematic way, was not unexpected. But I enjoyed seeing the varied ways they went about this.

For example, these two girls made three rows (shown vertically in the photo). Each row corresponds to a colour. Within each row, they grouped first by shape. For example, all the red diamonds, then all the red squiggles, then all the red ovals. Within each shape they grouped by shading. Within each shading they organised by number. One explanation was that, for a particular colour, they knew that there were nine cards for a particular shape. There were three different shapes. So there were 27 cards in one row. There were three rows of different colours. So there were 27 × 3 = 81 cards.

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Some students started grouping by colour, but in a different way. In this grid, each row corresponds to one particular shape. Each column corresponds to one particular shading. Each ‘entry’ in the grid contains three cards, grouped by number. There were two other grids like this, each for a particular colour. One explanation was that, for a particular colour, we get a 3 × 3 grid where each entry contains 3 cards. So, each grid has 27 cards. There are three different grids, each corresponding to a different colour. So there are 27 × 3 = 81 cards.

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There are four different ways of organising shown in the photo below. In the left bottom half, a student is organising in a way similar to the grid method. Focus on the larger cards in the far right. These girls have nine columns. Each column corresponds to a particular colour and shading combination. For example, the far left column are cards that have purple shapes that are completely filled in. Within each column, they organised the cards in groups of three. The three groups are organised first by shape. Within each group the cards are organised by number. Their explanation is that there are nine columns, each with nine cards. So they have 9 × 9 = 81 cards.

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I loved all these ways — and more not described — that students found to count the number of cards they had in their deck. After students had completed their work, we congregated together and went on a tour of the room. Each smaller group explained to the whole group how they had organised their cards and confirmed that there were 81 cards in the deck.

An unexpected advantage of this approach is that students discovered for themselves how to make a SET, because of the natural ways that they grouped cards. In the game, a SET is a group of three cards where, for each of the four attributes, the features are the same across all three cards or different across all three cards. For example, the three cards below are a SET because shape is all-same, fill is all-same, number is all-different, colour is all-different.

SET Cards 02

Once students understood how to make a SET, we made a new discovery within their work. Consider again the top cards of the 3 × 3 grid shown below. Each row, column and diagonal forms a SET. It’s like a magic square. A magic SET square.

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Meanwhile, the students who organised their cards into a 9 × 9 grid decided to keep their columns the same, but rearrange it so that each row corresponded to a particular number and shading combination. I’ve reproduced it below. With a little bit of prompting from me, they discovered that they had a kind of super magic SET square. Can you see what I mean? So cool!

SET cards

The rest of the session was spent playing the game, and talking over some of our SET-finding and problem-solving strategies. A rough description is here.

What struck me is that these students’ understanding of how to form a SET was much more solid, and developed so much quicker, than many other older students I work with. This is because usually I explain, rather than have them explore. At Maths Camp I was reminded — again — that even in an activity full of moments for discovery, there are still more opportunities to slow down and let students construct knowledge for themselves.

Tracy Zager’s word clouds

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Over the past year I have held this image in my head as a reminder and a motivation. It comes from Tracy Zager‘s 2015 NCTM ShadowCon talk ‘Breaking the Cycle‘, which is mandatory viewing. (If you have limited time, stop reading this post and go watch Tracy’s talk instead.)

At the same time that Tracy was giving her talk in Boston in April 2015, I was doing some last-minute preparation for my own talk (that same day!) for maths teachers in Adelaide, ~17,000 km away. But I was procrastinating by looking at Twitter. Fawn Nguyen, the live-tweeter for Tracy’s talk, tweeted out an image of these word clouds that stopped me in my tracks. Tracy had articulated so well what I felt but hadn’t been able to put into words. I grabbed a copy of the image, worked it into my presentation, and was talking about it that afternoon. I had no idea at that stage who Tracy was; I hadn’t yet heard her say a single word, but her message was resonating with me loud and clear through the flurry of ShadowCon tweets.

In July 2015, I was honoured to give the Hanna Neumann keynote at the Australian Association of Mathematics Teachers (AAMT) biennial conference (the Australian equivalent of NCTM but with far fewer people). In my talk ‘More than mathematics: developing effective problem solvers’, I set out a case for incorporating into our classrooms the creative, active and collaborative ways in which professional mathematicians work, with examples from my own experience. In the middle of my talk, I said something like the following:

This brings me back to the cat in the dark room. Andrew Wiles, the British mathematician famed for resolving Fermat’s Last Theorem, describes mathematical research like exploring a completely dark enormous mansion. You stumble around bumping into the furniture but gradually you learn where the furniture is. After a while — perhaps six months or so — you find the lightswitch, you turn it on, and it’s all illuminated. Then you move into the next room and spend another six months in the dark.

Mathematicians are chronically lost and confused, and that is how it is supposed to be. It would be ridiculous to think of mathematicians spending their days solving problems that they already know how to solve. Instead, we spend a lot of time uncertain about whether something will work, or uncertain about what to do next.

Mathematicians grow to feel quite comfortable with this kind of uncertainty, but I suspect that most of our students do not. So, let’s shift to thinking about our students. Put yourself in the mind of your typical student. What words would they use to describe maths?

Cue, from Tracy’s talk, the word cloud about school maths (on the left of the above image), the word cloud from mathematicians (on the right), and the slide that defines my questions: ‘How do we, in our classrooms, shift from here to here? To help our students experience mathematics as a curiosity-driven, joyful, beautiful, endeavour?’ I spent perhaps five minutes in the middle of a 50-minute talk on this slide, but to me, it is one of the cornerstones.

For some reason my talk struck a chord. I’ve since been delighted by invitations to share the message — and Tracy’s slide — with hundreds of teachers at conferences around Australia. And I was amazed earlier this year to have a senior mathematics professor stop me in the corridor to say something about ‘that word cloud’. Turns out that one photo of my AAMT talk was shared at the Australian Council of Heads of Mathematical Sciences — and it was me standing in front of Tracy’s word clouds. I am beyond ecstatic that, even for the briefest moment, this question was in the minds of the leaders of Australia’s university mathematics departments and the Australian mathematics community.

These word clouds remind me what my purpose is. It is to orient my own students towards the creative, active and collaborative ways in which professional mathematics work, and to help them experience mathematics as a curiosity-driven, joyful, beautiful, endeavour. And, it is to help others position their own thinking and teaching towards this goal.

These word clouds prompted a ‘fourth-grade teacher at heart’ maths coach from Maine and a university mathematician from Australia to become friends and collaborators. Being able to meet Tracy at NCTM 2016 is one of the reasons that I finally decided to make the trip, and it opened up an abundance of other friendships, cemented mainly at #MTBoSGameNight. (That’s Tracy’s work again; she is the master of weaving together and strengthening the threads of this community.)

It doesn’t feel right to finish this post without mentioning Tracy’s new book, due out in December 2016. I’ve been fortunate enough to have a sneak peak at a couple of chapters, and it is good. Put it on your Christmas wish list. (The Australian distributor of Stenhouse Publishers is Hawker Brownlow Education.) If you can’t wait for December, you can read her blog now.

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Friendships like this are why I advocate Twitter and the #MTBoS to every teacher I meet. We have so much to learn from one another. All it takes sometimes is one tweet to get it started.

MAT 2016

Tomorrow’s alarm is set for a rather ridiculous 4:00am. I’m heading down to the Annual Conference of the Mathematical Association of Tasmania in Burnie. I can’t believe it’s taken me 36 years to get to Tasmania, but I can’t think of a better reason than maths.

I’m giving the opening keynote on Friday, and a couple of workshops on Saturday. (I couldn’t decide between the workshops, so I volunteered both. Sucker.) I don’t know the times for all of them yet, but if you are at the conference, you’ll work it out.

If you’ll be at MAT 2016, I’d love to meet you! If you won’t be there, follow along with the hashtag (#MATConf16), or join the live stream of my session (see below). I’ll post workshop slides, and links to videos, eventually.

FRIDAY: ‘More than mathematics: developing effective problem solvers’

Apparently I’m talking just before the conference dinner at a whisky distillery. That right there is an incentive to finish on time, or risk being walked out on.

Complex, loosely-defined problems encountered in both the workplace and everyday life demand more than technical proficiency in mathematics. They also require broader capabilities including formulating problems, devising and implementing solution approaches, creativity, teamwork, project management, and communication skills. Significantly, these skills are often needed for any challenging mathematical problem — independent of whether it originates in the ‘real world’ or not.

So, how do we prepare our students with these skills in a mathematical setting? How can we develop and broaden their abilities and confidence in posing and solving mathematical problems? To address these questions, I’ll draw on my experiences as an industrial mathematician, training workplace-ready students, and teaching a new course designed to build mathematical thinking and problem-solving skills in pre-service teachers through games and puzzles.

SATURDAY: ‘Prompting productive mathematical discussions’

This workshop will be streamed out live through the Connect with Mathematics initiative of the Australian Association of Mathematics Teachers. Talk about upping the difficulty level!

The livestream is Saturday 14 May at 9am (WA), 10:30am (SA, NT), 11am (ACT, VIC, TAS, NSW, QLD). The free registration and access link is here. It should also be available afterwards on Vimeo.

Mathematical conversations are crucial to mathematics learning. By trying to explain their problem-solving approaches and solution strategies to convince others that they are right, students refine their thinking and improve their problem-solving skills. Appropriate tasks, in which everyone can meaningfully contribute ideas, also help students feel valued and mathematically competent. This workshop will be packed full of tasks, ideas and structures to encourage rich mathematical discussions with links to a range of resources.

SATURDAY: ‘Building (and rejecting!) mathematical intuition’

With a big hat-tip to Tracy Zager for prompting me to think more about this. Tracy’s upcoming book ‘Becoming the math teacher you wish you’d had‘ has a thoughtful chapter on intuition and a beautiful cover. Out December 2016. Put it on your Christmas list. (I’ll remind you about it when it gets closer. I think there will be an Australian distributor by then, too.)

Mathematical reasoning often relies on intuition — an instinct for what `feels right’ or `makes sense’. Problem solving can zigzag between logical reasoning and intuitive understanding, right up to the point where intuition is either confirmed or totally rejected. This workshop will explore ways in which mathematical intuition can be developed. We will also look at delightful non-intuitive mathematical problems — from probability to geometry — and discuss how to bridge the gap between wrong intuitive thinking and correct but counter-intuitive analytical calculations.