Monthly Archives: May 2016

Better student talks

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Having a mathematical idea is only one part of the equation (excuse the pun!); we also need to be able to communicate it. A good example is Fermat writing next to his conjecture that he (purportedly) had a proof that the margin was too small to contain. If Fermat really did have a proof, then he could have saved mathematicians 350 years of frustration by going to the effort of communicating it with us!

There are many different dimensions to how we want our students to be able to communicate. Is it in written or oral form? It is to be polished, or in intentionally draft form for revision? Does it present the resolution of mathematical work, or is it work in progress? Who is the intended audience — peers, novices, experts, themselves?

The way we write or talk, and the skills we need for the process, depends on which of these dimensions are at play. I think it is unreasonable to expect that students can produce polished writing or give engaging talks without explicit coaching on how to do this — particularly in a mathematical context.

In this post I want to focus on what I’ve done to improve students’ skills in presenting completed work. These ideas build on and adapt (in places, minimally) the great structures that my colleague A/Prof Lesley Ward has put in place in the Mathematical Communication course in our undergraduate mathematics degree.

Brainstorming characteristics of good and bad talks

Students have heard many talks, likely more bad than good! I start by asking them to brainstorm in their groups the features of talks they’ve heard. They write them on the whiteboard under the following headings:

  • Delivery: volume, rate, articulation — how the message is transmitted.
  • Language: choice of words, grammar, using highly effective phrases — how the message is conveyed.
  • Organisation: sequences and relationships amongst the ideas — how the content is structured.
  • Content: what is said and how it is adapted to the listener and the situation.
  • Visual aids: slides, physical props — how the aids reinforce the major points and stimulate the audience.
  • Other: any ideas that don’t fit elsewhere.

There is now a collective understanding (or at least appreciation) of what to aim for and what to avoid. After this, I hand out the final presentation rubric which will be used to score the final project talks. This handout is minimally adapted from one used at Harvey Mudd College.

(Large whiteboard images here and here.)

2015 SP5 MATH 1070 Good Talks (Group A)

2015 SP5 MATH 1070 Good Talks (Group B)

Structured skill-building

Each week we start one class with a mini-talk (the other class starts with a Visual Pattern; thanks Fawn!). The aim is to slowly build skills, and to gradually improve. Talks increase in duration, and in other requirements.

  • Mini-talk 1: Getting started! A week in advance of the first talk, students choose their talk partners (usually a friend). I also hand out a mathematical fact (Collatz Conjecture or Koch Snowflake) with a few ideas for what to talk about, but they need to research some more themselves. They deliver their three-minute talks at the desks. After the talk, the listener thinks of one thing the other person did well and tells them. All elements are designed to be the least threatening option that I can think of.
  • Mini-talk 2: Build confidence; make improvements. Students choose different talk partners a week in advance. They re-present their fact; the aim is to revisit what went well last time and what could be improved.
  • Mini-talk 3: Standing up; using whiteboards. I select student pairs from each student’s usual work group. Students give three-minute talks about their project topics at the whiteboard.
  • Mini-talk 4: Present a new topic; listener steps up. I select student pairs from among the class. I give out new mathematical facts (Prime Numbers or Fibonacci Sequence). The only other change from Talk 3 is that the listener must ask at least one question.
  • Mini-talk 5: Preparing visual aids. I select student pairs from among the class. Students need to prepare something beforehand to write, draw or show. They present the ‘other fact’ from Mini-talk 4.
  • Mini-talk 6: Putting it all together. Self-selected groups of three. Students give four-minute talks about any mathematical topic of their choosing (not their project!). Must use visual aids to engage their audience. Students self-evaluate afterwards according to the presentation rubric. Students also practice time-keeping and ‘not panicking’ when shown the ’30 seconds to go’ yellow card.

I haven’t talked here about guiding students in selecting which parts of the mathematics to draw out and which details don’t need to be told; how to tell the mathematical story is an important skill to develop that I’m not going to cover in this post.

You can get a PDF of my guidelines here. What I’ve set out can of course be adapted and expanded; the key is continual skill development and exposure to giving talks. By the time students get to the final 10-minute presentation they are fairly comfortable with giving talks about mathematical ideas to their peers. The final project talks are also far more enjoyable to listen to because of it!

What tips do you have for developing presentation skills in students? I’d love to hear about your strategies.

Another party puzzle

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More party puzzles! This one is from a thoroughly-recommended book, Puzzle Based Learning1.

Mr and Mrs Smith invited four other couples for a party. When everyone arrived, some of the people in the room shook hands with some of the others. Of course, nobody shook hands with their spouse or themselves, and nobody shook hands with the same person twice.

After that, Mr Smith asked everyone how many times they shook someone’s hand. He received different answers from everybody.

How many times did Mrs Smith shake someone’s hand?

At first glance it seems that there is not enough information to solve the puzzle—which is why I like it! Once we consider each piece of information, we can put the bits together to find a solution.

Warning: mathematical spoilers (but not the solution) ahead. I’ll post the solution in the comments.

Some prompts:

  • How many people are there at the party?
  • What is the minimum number of handshakes possible?
  • What is the maximum number of handshakes possible?
  • Can you draw a diagram to represent the handshakes made by the person who made the most handshakes?
  • What can you conclude from this?
  • What can you add to your diagram?
  • Can you now solve the puzzle?

Good luck!

[1] Michalewicz, Z., Michalewicz, M., Puzzle Based Learning, Hybrid Publishers, 2008. pp 99-102.

Ramsey’s party problem

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I love games that require no special equipment because they can be played at a moment’s notice. This is one of my favourite pen-and-paper games. It is played on the complete graph K6. In other words, a board with six dots where each dot is connected to every other dot by a line.

SimBlack

Although the game-board can be drawn up for each game, I like to have pre-printed boards which can be put inside plastic sleeves for use with erasable marker pens.

To play, two players each choose a different colour. They take turns colouring any uncoloured lines between two dots. The first player to complete a triangle made solely of their colour loses.

A natural question to ask when playing games is whether or not there will always be a winner (or loser), or if the game can end in a draw. (We are also often interested in whether there is an advantage in being the first (or second) player.) Play a few times. What do you think?

What I like about this game is that the reasoning required to determine if there will always be a winner is accessible by many children and adults, but it is also the entry point into a rich area of mathematics called Ramsey theory.

Warning: the rest of this post contains mathematical spoilers.

The game described earlier is called Sim, after the mathematician Gustavus Simmons who was the first to propose it. It is also equivalent to a puzzle posed in The American Mathematical Monthly in 1958. ‘Prove that at a gathering of any six people, some three of them are either mutual acquaintances or complete strangers to each other.’ That is, we can represent the six people by six dots. We can colour the lines between two acquaintances in red and the lines between two strangers in blue. The problem now is to prove that no matter how the lines are coloured, we can not avoid producing either a red triangle (between three acquaintances) or a blue triangle (between three strangers).

So, if we can prove that we cannot avoid producing a triangle of either colour, then we have proven that the game can never end in a draw. (In fact, computer search has verified that the second player can win with perfect play.)

Consider the diagram below. There are five edges leaving node A. In a completed game, they will be coloured red or blue. At least three of them must be of the same colour, say red. Now, in triangle ABC, edge BC must be blue (else we have a red triangle). Similarly, in triangle ABD, edge BD must be blue. And, in triangle ACD, edge CD must be blue. But, we have just formed a blue triangle: BCD!

SimProof

We could have posed this question in reverse, that is, what is the minimum number of guests that must be invited to a party so that at least three guests will be acquaintances or at least three guests will be strangers.

The game of Sim shows that it is possible to invite six people to a party and have either at least three guests as acquaintances or at at least three guests as strangers. We can show that this is the minimum number of guests by considering a party with five people and showing that it is possible to have no triangle of the same colour between any three people.

Sim is a demonstration of the Ramsey number R(3,3) = 6. In the language of parties, the Ramsey number R(m,n) = p says that p is the minimum number of guests required at a party so that at least m guests are acquaintances or at least n guests are strangers.

There is a lot written about Ramsey numbers; I’ve barely scratched the surface here. (I first read about them in Martin Gardner’s Colossal Book of Mathematics.) Finding Ramsey numbers is an active area of mathematical research, partly because they are fiendishly hard to calculate. At present, only a handful are known. The last word is for Paul Erdös, the great Hungarian mathematician. (The extract is from The Australian newspaper on Wednesday 21 April 1993 when Australian mathematician Brendan McKay and American colleague Stanislaw Radziszowski found R(4,5)=25).

If an evil demon threatened to destroy the Earth in two years unless we could tell him the value of R(5,5), our correct response, said Erdös, should be to devote all mankind’s resources to the problem — we could probably solve it in two years. But if the demon instead asked us to tell him the value of R(6,6) — then, said Erdös, we should devote all our resources to finding a way to kill the demon.

Tracy Zager’s word clouds

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Zager

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.

13177468_10154222357700712_1909259198696526835_n

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.

Friday Five: #4

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Turns out that Desmos already posts a ‘Friday Five‘. Wonder if I subconsciously stole the name from them?

I’m posting this early (although it is Friday here) because I have a hundred little things — and one big thing — to do today.

  1. Megan Schmidt (@veganmathbeagle) is wowing us on twitter with her number spiral investigations. See below. Megan’s own blog post is here. I love them so much that I’ve been storifying her tweets, including ones not in her post.
  2. I am a huge fan (like the rest of the #MTBoS) of Notice and Wonder. If you’re not sure what this is, go check out Annie Fetter’s 2011 Ignite Talk. I was planning on writing a post, but then Joe Schwartz wrote one about Notice and Wonder with second graders that is so much better than anything I could write. It’s important to allow students to notice in both mathematical and non-mathematical ways, but I like how Joe orients students towards the more mathematical wonderings.
  3. I love this paper-sharing activity for exploring infinite geometric series with students, thanks to Sam Shah and Bowen Kerin. It doesn’t need to be tied to a unit, either — a friend did it with a spare 10 minutes. The key for me is to ham it up; take the script that Sam suggests and really overact. I take different coloured paper, and make a big deal over each paper-master choosing their favourite colour. A recent improvement, at least for me, was to let the groups vary in size (within reason). That way, we explore several series in the same activity.
  4. Sara Van Der Werf writes how vocabulary can be the ultimate block to tackling a mathematical question — out of 80 students, 80 incorrect responses to a question involving ‘annual’ but 80 correct responses when changed to ‘in a year’. Sara offers a simple tweak, and some great reflections.
  5. This week I posted about focusing on relationships with students. Then I read Ilana Horn’s Who Belongs in our Math Classrooms. Powerful stuff. I was particularly interested in the linked article from PBS Newshour about the effect of teachers mispronouncing names. On this theme, I can think of no better way to end than with the transcript of Francis Su’s talk The Lesson of Grace in Teaching. David Butler called it a touchstone of what’s important in teaching. He’s absolutely right.

The tea towel of multiplication

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(If you want to understand the title, you can just skip straight to the end.)

When I think back on the inflexible way in which I remember being taught maths, I am often surprised that I became a mathematician. I don’t mean this to be unkind to my teachers — I enjoyed maths in school and I liked my teachers — but I don’t recall being taught to approach problems in multiple ways or to be flexible about which strategy I used.

Mental maths is a perfect example. Until relatively recently I have relied on one strategy — the standard algorithm. That is, I would find the answer by picturing writing the steps on a blank space1 in my head:

EE_Image_210516-1347

The only situation in which I would jump straight to a decomposition strategy was for percentage problems:

Percent

It wasn’t until I began teaching the distributive law by starting with numerical examples that I realised the power of the decomposition strategy:

Distributive

Seeing this as an area model for the first time almost made my head explode. How I had never explicitly made the connection between multiplication and area (or volume or …), I have no idea.Area

Discovering Fawn’s Math Talks and seeing how other people break these problems apart is captivating, as is looking at visual representations of strategies.

visual-image-fix-18-x-5

Source: Jo Boaler, Visual Math Improves Math Performance2

This really started to change the way I looked at maths; I now look—always—for visual representations of a mathematical concept. The two images at the bottom of the right figure are student responses to a prompt like: Explore a visual representation of (a-b)2.

Binomial01Binomial02

So, by now you are probably wondering why it took me so long to catch on to all this, and what it has to do with the tea towel of multiplication.

I’m now very interested in different methods of multiplication and collect them up as I find them. I like the ones that appear more obscure; you look at them and say ‘what?!’. After all, the ‘standard’ algorithm is only conventional to those who know it.

My collection includes various methods of finger multiplication, the line method, Russian peasant multiplication, grid methods, cross methods, paper strip multiplication, and copper plate multiplication. I offer these ‘puzzles’ to my pre-service teachers as a possible project investigation. This is another way to add to my collection!

Copper

Source: Thinking Mathematically (Mason, Burton, Stacey)

LineMultiplication

Finger

Source: Thinking Mathematically (Mason, Burton, Stacey)

And check out this method with Genaille-Lucas Rulers to determine 52749×4.

Last year, Twitter offered up this tea towel of multiplication. I have no idea (yet) how the circle method works. How cool is that?

 

[1] Yes. Ha ha.
[2] Visual Math Improves Math Performance

Focusing on the relationships

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I’m feeling a bit punchy tonight as I write this. You might find it preachy, confronting or self indulgent. Be warned. The most eloquent and uplifting writing about good teaching that I’ve read is by Francis Su: ‘Lesson of Grace in Teaching‘. You should probably read that instead.

Damn straight. For me, building relationships with students is what turns bearable teaching experiences into enjoyable and deeply fulfilling ones. Here are five of my teaching pillars.

Learn. Their. Names. So you teach a course with hundreds of students? You might not be able to learn them all, but you can learn some of them. Learn the names of all the students in your tutorial classes. Ask them when they come to your office hours. You can lead with ‘I’m sorry but I’ve forgotten your name.’ Don’t just learn the names of the high-achievers or the ‘trouble-makers’. Work hard to get the pronunciation right for all names. Ask students what they’d like to be called. ‘Ben’ is different than ‘Benjamin’; ‘Jenny’ is different than ‘Jennifer’. Ask it on a Getting To Know You form in the first week of class. I make learning names one of my highest priorities in the first two weeks of class. Students notice. It’s a good start to our relationship.

Show interest in their non-academic side. You don’t need to pretend to understand sport or care about the latest reality TV show, but you can say ‘that’s cool’ or ‘sounds like you enjoyed that’, or whatever response signals genuine interest. (Even if you aren’t really interested ;).) Plus, it helps with learning their names. My Getting To Know You form asks them to tell me something interesting about them, then I make sure to mention it when they bring their form to me. I connect it to my own experience, either in my head or in conversation, as a way to learn their name. That one item isn’t meant to define them, but it is a good starting point to, well, get to know them. I find out the most interesting things. I learn about pets, travel, hobbies, languages and weird body functions :shock:. The quietest girl in class loves rollercoaster rides, or horror movies; the coolest boy in class tells me how excited he is to be dating his best friend. They have talents well beyond my maths walls; they are singers, debaters, musicians, sporting champions — football, athletics, lawn bowls, stick-fighting (I didn’t even know that was a thing).

Care about their well-being. I’m not talking about knowing all the details of illnesses, relationship upheavals and deaths in the family. Respect their privacy and their distress. Gently inquire if they don’t look well or seem overly anxious. Accommodate as much as you can, even if it’s not ‘fair’ to other students. (What’s fair, anyway?)  Remind them that they are more than their academic transcripts. Refer them to support services for issues you aren’t equipped to handle. Make the appointment for them if needed. The best change I made to my Getting To Know You form was including ‘Is there anything you really want me to know about you at this stage?’ (I ‘stole’ this from Sam Shah’s ‘Getting To Know You‘ form.) I was taken aback, and then honoured, that they would share deeply personal information in the first class with someone they barely knew. I could offer support and direction to professional services from the beginning.

Show them respect. Ask before writing in their notebook or sharing their thinking with others. Deliver when you say you will. Apologise when you inevitably fuck that up. Keep private matters they share with you in confidence. Take care with feedback you write on quizzes or off-hand comments that you make — seemingly innocuous remarks can crush confidence. Recognise that university students are adults with jobs, families, bills, relationships, childcare issues, housing insecurity, cultural expectations on them that you might not understand, and other competing demands. They might walk into class late, or need to take a call in the middle. They might miss deadlines. If it’s not disrespectful or disruptive, you can extend them that courtesy. But call out bad behaviour. Don’t let them disrespect their classmates or you.

Treat them as equals. There are things that you can teach them; that’s why you’re the teacher. But there is plenty that you can learn from them. They know a whole lot more than you about a whole bunch of things. They’ll also teach you a lot about being a better teacher. Value their mathematical ideas, just as you would value those of a colleague. Show them you make mistakes. That you struggle. Model authentic behaviour of working mathematically. It will help them become better mathematicians.

This isn’t everything, I’m sure, but I’ve used 800 words when I could have just said ‘treat them how you’d want to be treated’. If you got to here, thanks for indulging me.