#read2016: Part 3

Tracy Zager’s new book

Tracy Zager’s new book ‘Becoming the Math Teacher You Wish You’d Had‘ is out, and it’s a treat. The central tenet of this important book is to ‘close the gap’ by making maths class more like mathematics, orienting our students towards the habits of mind of professional mathematicians. ‘Good teaching starts with us’ and Tracy companionably guides us through ten practices of mathematicians: taking risks, making mistakes, being precise, rising to a challenge, asking questions, connecting ideas, using intuition, reasoning, proving, working together and alone.

Tracy skillfully blends academic research, illuminating classroom dialogues, the thoughts of mathematicians and maths educators, and her own perceptive observations. This seamless mix is a real strength of the book; we not only see what habits are important and why, but how they can be enacted through specific teaching strategies, and the powerful effects they have on our students’ development as confident and capable mathematicians. The reader can’t help but be inspired by the teachers that Tracy holds up as exemplars of good practice. These teachers have so much respect for each of their students as serious mathematical thinkers. I was struck by the extent to which they would go to adapt instruction in response to student ideas and to support them in pursuing their own line of enquiry.

Tracy warns early on that the book is long—and it may be—but it is also captivating! The organisation is immensely practical; each chapter can be used as a self-contained guide for a particular mathematical habit. I can see myself repeatedly delving back into specific habits as the teaching year progresses. I read it cover-to-cover over a couple of days while curled up in a secluded cabin, pausing occasionally to stare out into the Australian bush and ponder what I can change in my own teaching. Some of my highlighted passages:

From Chapter 3, Mathematicians Take Risks: ‘When we assign problems that have a single, closed path from start to finish, we’ve eliminated the possibility that students will take mathematical risks. There’s nothing to try if everything is prescribed.‘ (pg 49). In my skills-based courses, I too infrequently give students opportunities to try and be successful with their own approaches. That’s something to work on.
From Chapter 4, Mathematicians Make Mistakes: ‘If we want students to learn from mistakes, we need to teach them how.’ (pg 57). Tracy outlines a three-part goal: to teach students to take mistakes in their stride, to keep going when they’ve made a mistake, and the one I need to focus on: ‘to teach students to make the most of the knowledge and experience they gained by figuring out their mistake‘. How can I help students gain the skills to diagnose and learn from their mistakes, by themselves?
From Chapter 5, Mathematicians Are Precise: ‘Math without inquiry is lifeless, but math without rigor is aimless. There is no tension between teaching students how to solve problems accurately and efficiently and teaching students how to formulate conjectures, critique reasoning, develop mathematical arguments, use multiple representations, think flexibly, and focus on conceptual understanding.’ (pg 80). In my problem-solving course, I deliberately swung the pendulum from the typical procedure-based courses my students had mostly experienced towards creative, collaborative problem-solving. But I also need to find the middleground, where I place as much emphasis on rigour as I do on inquiry.
From Chapter 12, Mathematicians Work Together and Alone: ‘If a major part of doing mathematics involves interacting with other mathematicians, then a major part of teaching students mathematics must be to teach students how, why, and whether to interact with one another mathematically. Students need to learn how to ask for what they need from each other and to be what they need for each other … we need to teach students how to be good colleagues … it’s important we honor individual thinking and working time. It’s not reasonable to expect students to collaborate at every moment, and that’s not how mathematicians work.’ (pg 312). This past semester, a few students in my problem-solving course commented that they needed more opportunities to work alone first, and more strategies to work effectively with group members. I’ll definitely be digging further into this chapter next year.

And, these phrases are going straight into my repertoire:

‘Do you have more questions after doing this? What are you wondering about now? (pg 149).
‘What does ______ have to do with _____?’ (Debbie Nicols, pg 191).
‘Remember that it’s hard to find mistakes when you assume that you’re right. So go back into it assuming something went wrong.’ (Jennifer Clerkin Muhammad, pg 284).
‘Would you recommend that strategy to someone you like?’ (pg 118). 😂

There is so much to love about this book. The writing is both encouraging and empowering. It’s labelled K-8 but Tracy offers important insights to help teachers across all year levels; I have been nodding furiously and making notes throughout. This particular passage had me shouting ‘yes!’:

‘We need to give ourselves permission to say, publicly, and with delight, “I never thought about it that way before!” whether it refers to addition, fractions, or place value. It is long past time for us to respect the beauty, power, and importance of elementary mathematics, instead of having contempt for “the basics.”’ (pg 208)

Listening carefully to student thinking, especially about ideas I thought I understood, always gives me new insight. It’s why I’ll never tire of teaching.

I can confidently say that, alongside ‘Thinking Mathematically‘ (Mason, Burton and Stacey, 1982; 2010), Tracy’s book will become a cornerstone for my teaching. It is a gift to all maths teachers. But don’t just take my word for it; you can preview the book in its entirety here. The companion website promises more, and I can’t wait to look around!

Update (22 December 2016): The companion website is now live, and it is packed full of goodies. Be sure to check out the free study guide under ‘Getting Started’, which works for either an individual or group book study.

Tangling and untangling

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This is the seventh in a series of posts about my course ‘Developing Mathematical Thinking’, a maths content elective for pre-service teachers training in primary and middle maths. All posts in the series are here. WARNING: It’s a long post.

Edited to fix the confusion between × (multiply) and x (the letter).

I have been itching to try Conway’s Rational Tangles with a group of students. I first read about this problem a couple of years ago in Fawn Nguyen’s excellent post. It looked super interesting, but I was still somewhat confused with how it works (not to mention why it works). So I was excited to be in Fawn’s ‘Conway Tangles’ Math Micro-Session at the NCTM Annual Meeting in San Francisco this year, where it started to make some more sense.

This week I tried the activity with my #math1070 students. I waited until the last week of the course because: (1) now that we know each other better, I thought they’d tolerate me muddling through it, (2) their resilience and problem-solving skills for more challenging and ill-stated problems have increased. (Note that the ‘ill-stated’ part is my fault, not that of the problem!) I was upfront with them about how I was both excited and nervous about the session. It was a bit sketchy with the first group of students, but I was able to make some adjustments with the second group.

Below is a mash up of how I did it this week and how I would improve it in the future. This outline is based on Fawn’s write up, but I also pulled in ideas from Tom Davis’ thorough notes for a Maths Teacher Circle, along with the three-part outline from nrich maths: Twisting and Turning, More Twisting and Turning, All Tangled Up. We spent ~1.5 hours on the activity. Perhaps half of that was outside, with students doing it themselves.

Getting started: the rules

Have four volunteers come to the front. Each person holds the end of a rope so that the two ropes are horizontally parallel. This is the starting position. This state has a value of 0.

There are only two moves that can be made: TWIST (T) and ROTATE (R).

A TWIST is when the person at the bottom left moves under the orange rope to the top left, as shown below. This new state has a value of 1. We notate this as $0 \xrightarrow{T} 1$ . I tell the students that every time a TWIST operation is performed, the value of the ropes increases by 1. So, TWIST is +1.

A ROTATE is when every person moves clockwise to the next position, as shown below. (Note that this is from the starting position.) I say that I am not going to tell them the value of this new state. $0 \xrightarrow{R} ??$ I’m also not going to tell them what ROTATE does; that’s for them to figure out.

The ‘aim’

Our aim is to (in my words) ‘tangle the crap out of the ropes’ by performing any number of TWISTS and ROTATES and then work out how to untangle the ropes back to horizontally parallel (with value 0). But, remember that there are only two available moves: TWIST and ROTATE. ‘Untwist’ and ‘Unrotate’ are not possible moves. (I wrote ‘aim’ because this isn’t the only goal, but it’s the one that students will initially want to work towards.)

At present, we have two questions:
(1) What does ROTATE do?
(2) How to get out of any tangle?

A first go at experimenting with ROTATE

Before I let them loose with some ropes, we try a few more systematic experiments.

We reset the ropes to 0, and try ROTATE followed by ROTATE (RR). We discover that the ropes end up horizontally parallel again. That is, $0 \xrightarrow{R} ?? \xrightarrow{R} 0$ . We decide not to ever do RR unless we want to waste energy. We test this further by resetting the ropes to 0, and trying TWIST followed by RR. As expected, we end at a state with value of 1. We summarise: $0 \xrightarrow{T} 1 \xrightarrow{R} ?? \xrightarrow{R} 1.$

The other possibility after an initial rotate is to try a twist. So we reset the ropes to 0, and try ROTATE followed by TWIST (RT). This action is kind of strange; the ropes stay vertically parallel no matter how many twists we do: $0 \xrightarrow{R} a \xrightarrow{T} a \xrightarrow{T} \cdots \xrightarrow{T} a.$

There are already some interesting conclusions that we could come to as a group, but I decide (based on experience) that it might make more sense if everyone is participating instead of watching.

Trying it for themselves

Students get into groups of at least five: four on the ropes and at least one person recording the steps. I distribute the ropes ($1 each at Kmart; bargain!). We go outside. Maths classes in university are never held outside, so this is novel for all of us.

I suggest that to help answer Question 1 (What does ROTATE do?) we might want to break Question 2 down into further sub-questions.

(1) What does ROTATE do?
(2a) Work out how to get out of one TWIST; two TWISTS; three TWISTS; four TWISTS; any number of TWISTS.
(2b) Work out how to get out of a mixed up sequence, like TTRTTTRT, shown below.

Everyone starts with (2a), and works it out fairly quickly (15 minutes?). Their strategy to untangle is to always start with a ROTATE (otherwise we would be further tangling the rope), then to look at the ropes and ‘see’ what to do next. Eventually they write down how to get out of these positive integer states (T, TT, TTT, TTTT, …) and see a pattern. Try it for yourself! (Or look on page 5 of Tom Davis’ notes.)

In general, my students find it hard to conjecture what ROTATE does. I talk to each individual group in turn. To get them started, I write down something they’ve just done: $0 \xrightarrow{T} 1 \xrightarrow{R} a \xrightarrow{T} 0$ . We can work backwards from T to realise that a=-1. We also realise that when we start with just twists, the value of the state keeps increasing from 0. OBSERVATION: To return back to 0, ROTATE must involve a negative somehow.

I suggest perhaps ROTATE is ×(-1). We look at $0 \xrightarrow{T} 1 \xrightarrow{R} -1 \xrightarrow{R} =1$ . This works!
We test it on one TWIST: $0 \xrightarrow{T} 1 \xrightarrow{R} -1 \xrightarrow{T} 0$ . This also works.

We predict what should happen with two TWISTS: $0 \xrightarrow{T} 1 \xrightarrow{T} 2 \xrightarrow{R} -2$ . To get this untangled, we should be able to do TT. We try it with the ropes. Groan as it doesn’t work. (Note that some students have already forgotten that they know how to get out of two twists, from (2a).)

There is more conjecturing about ROTATE. For example, some students try ROTATE is -2. Later in the class discussion we realise that ROTATE can’t involve just adding or subtracting as RR would take us further away from 0 (positive or negative), and we know that $0 \xrightarrow{R} ?? \xrightarrow{R} 0$ . OBSERVATION: ROTATE must involve multiplication or division (or perhaps some other operation).

Most are still stuck. I ask them if they’ve done (2b) and untangled TTRTTTRT. If so, I tell them that the tangled state has value 3/5. OBSERVATION: ROTATE must involve a negative and fractions somehow. Some more cautious conjecturing eventuates.

If they are still stuck, I tell them that TTRTTR has value -2/3.

After working on it for ~45 minutes, some of them give up and demand the answer. I know there is more problem-solving work to come so I tell those that haven’t worked it out that ROTATE is ×(-1/a), where a is the previous state value. To summarise:

$x \xrightarrow{T} x+1$
$x \xrightarrow{R} -\left(\frac{1}{x}\right)$

Efficiently getting out of any tangle

I ask them to come up with a scheme to efficiently get out of any tangle. (Later we decide that we aren’t sure that it is the minimum number of moves, but it seems efficient.) It works a bit like this: Get as close to zero with a numerator of 1 and a positive denominator (like 1/m) then ROTATE. This leaves you with a negative integer, –m, and you can TWIST your way m times back to 0.

Wrapping up

Back in the classroom as one group, we summarise what we discovered, and make a few more observations.

We go back and think about starting with a single ROTATE. Now that we know what ROTATE does, we see that the state becomes -1/0. This is like infinity. Another ROTATE brings it back to 0. When we start with a single ROTATE, TWIST leaves it exactly the same: $\infty + 1 = \infty$ . So we can have a tangle value of infinity. This is all kind of cool.
We wonder if every rational number can be reached through tangles, and then be untangled.
We wonder about how to prove the minimum number of moves to get out of each tangle.
We talk briefly about function notation: $T(x) = x+1$ and $R(x) = -\left(\frac{1}{x}\right)$ . We confirm that $R(R(x)) = x$ , so two RR leave the state unchanged. We talk about composition of functions, and how RTT is represented by $T(T(R(x))).$

We talk about how this activity is suitable for a range of students and different areas of focus:

problem solving and team work just by trying to untangle a tangle (no investigation into TWIST and ROTATE)
practicing fluency with fractions
older students can work with function notation and tackle some of the more challenging questions.

I reflect later how there is so much more depth in this activity than I had realised. I also realise that because it has so many different dimensions—physical manipulation, symbolic notation, numerical calculations, pattern recognition, conjecturing, teamwork, leadership—it gave students opportunities to shine in different ways.

How many triangles?

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It’s been quiet on the blog, but a lot has been happening. University classes in Adelaide have just resumed after a two week mid-semester break. To warm up, I gave my MATH 1070 students the following problem. I found it via Tanya Khovanova who states that it was an entrance problem for the 2016 MIT PRIMES STEP Program. (Read more on Tanya’s blog.)

I drew several triangles on a piece of paper. First I showed the paper to Lev and asked him how many triangles there were. Lev said 5 and he was right. Then I showed the paper to Sasha and asked him how many triangles there were. Sasha said 3 and he was right. How many triangles are there on the paper? Explain.

Here are some solutions from my students, all considered to be correct. The ones in blue originally appeared in Tanya’s blog post. Additional ideas are shown in red below. The black rectangle shows the piece of paper. Two of the rectangles contain instructions instead of diagrams.

I loved this as an opener to encourage creative problem solving. Thanks Tanya!

How Many Triangles.jpg

#SciPubMath

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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.

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.

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.
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.
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 :).

#read2016: Part 2

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I am tweeting 140 character reviews with #read2016, but I’ll also post the books here in three parts, one every four months. The maths ones (*) might be the subject of separate posts.

There were 19 books in Part 1 (January – April). Here is Part 2 (May – August) with 16 books.

After the Fire, a Still Small Voice, Evie Wyld. I raved (see Part 1) about her award-winning second book. This is a mind-blowingly good first novel. Can’t wait to see what’s next for Wyld.
#read2016: Incredibly moving in a matter-of-fact kind of way. Sadness seeps from every character.
Stop at Nothing: The Life and Adventures of Malcolm Turnbull, Annabel Crabb. This is an update to her 2009 Quarterly Essay, which I finally read in December 2015. I enjoyed re-reading the previous material and looking out for the additions.
#read2016: A snappy update to her 2009 Quarterly Essay. Witty and incisive, as always.
Teacher Man, Frank McCourt. From the bestselling author of Angela’s Ashes, which I read many years ago. Because Fawn Nguyen often mentions that Teacher Man is one of her favourite books, I’ve been looking out for it. I found three copies in a small secondhand book stall in the Penguin markets. (I resisted the urge to buy all three.) A fabulous read.
#read2016: The honest account of teaching. Master teller of stories. Words of wisdom on every page.
The Life-Changing Magic of Not Giving a F**k, Sarah Knight. A parody of Marie Kondo’s book The Life Changing Magic of Tidying Up (number 29 on this list). This was laugh-out-loud on every other page. I loved it. Great food for thought.
#read2016: After the day I had I read it in one sitting … while not giving a fuck about a whole lot of other things.
Weekend Language: Presenting with More Stories and Less PowerPoint, Andy Craig and Dave Yewman. There had been a lot of buzz about this book in the MTBoS, so I wanted to check it out. I liked it (see below) but for a deeper read I suggest two important books on the same theme: Made To Stick and Presentation Zen.
#read2016 (1): Snappy summary of elements of good presentations. Particularly liked chapter on mechanics of delivery.
#read2016 (2): Mechanics of delivery (my clumsy phrase): vocals, pausing, pacing, gestures and the like. Important to get right.
Faction Man: Bill Shorten’s Pursuit of Power, David Marr. In the lead-up to the Australian election (and wasn’t that a debacle!), Black Inc republished updated Quarterly Essay profiles by Annabel Crabb (see Item 21) and David Marr of the leaders of the two main Australian political parties. Some insight by Marr, but I felt it missed the mark.
#read2016: Adds detail to the sharp rise of the prime ministerial contender, but a rather disjointed piece of work.
The Little Coffee Shop of Kabul, Deborah Rodriguez. An impulse purchase at Cairns airport (with some encouragement from a companion) because I’d accidentally checked in my books and headphones. In places it was poorly written chick-lit, but the story stayed with me for many days — enough to buy the sequel.
#read2016: Fascinating, fictional account of Afghan culture. A little corny in places but surprisingly moving.
The Natural Way of Things, Charlotte Wood. I can see why this book has an ever-growing list of awards, including the 2016 Stella Prize. An imaginatively dark setting for an unlikely but captivating story.
#read2016: An engrossing dystopian tale of punishment and liberation. Almost unlike anything I’ve ever read.
Everywhere I Look, Helen Garner. This is a collection of Garner’s short stories, opinion pieces, diary extracts, essays and more. All but three have been published elsewhere, but the only one I’d previously read is still one of my favourites: The Insults of Age.
#read2016: Gulped when I should have savoured. Master of acute observation.
The Life-Changing Magic of Tidying Up: The Japanese Art of Decluttering and Organizing, Marie Kondo. I like to think that I’m an organised person who tries not to collect ‘stuff’, but I still like to read books on organising principles. After this I was inspired to get rid of half of my clothes. I can’t do the same with my books though!
#read2016: Slightly kooky but ultimately worth contemplating. Do your possessions spark joy?
The Spare Room, Helen Garner. After Everywhere I Look (Item 28), I had to read more of Garner’s work. Loved this book.
#read2016: An unflinching view of the complicated care of a dying friend. Deeply touching. Beautifully crafted.
The Virgin Suicides, Jeffrey Eugenides. I’ve not seen the movie, and I didn’t know the premise of the novel, but I’d been curious about it for a while. Dragged on, with the occasional great moment.
#read2016: Melancholic. Almost ethereal. Lyrical prose. Some intriguing moments but I never really connected.
* Thinking Mathematically (2nd edition), John Mason, Leone Burton, Kaye Stacey. This book changed the way I thought about mathematical thinking, so much so that I designed and developed a university course for pre-service maths teachers around it.
#read2016: Readying myself for another semester with 52 pre-service teachers by re-reading this foundational book.
Cosmo Cosmolino, Helen Garner. Based on what I’d read about this book I was expecting something a little different to her other works. I was not disappointed, but it was certainly unusual.
#read2016: Oddly engrossing with themes of new age, commune living and emerging from the shells of damaged lives.
* More Good Questions: Great Ways to Differentiate Secondary Mathematics Instruction, Marian Small, Amy Lin. Such an invigorating read.
#read2016: Immensely practical + deeply stimulating. With examples of open tasks to use, adapt or be inspired by.
High Sobriety, Jill Stark. ‘I’m the binge-drinking health reporter. During the week, I write about Australia’s booze-soaked culture. At the weekends, I write myself off.’ Jill makes an unsparing assessment of her relationship with alcohol — it’s worth doing the same.
#read2016: A frank look at Australia’s obsession with alcohol, along with a self-deprecating narration.

Notice and wonder: the Prime Climb hundreds chart

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This is the sixth in a series of posts about my course ‘Developing Mathematical Thinking’, a maths content elective for pre-service teachers training in primary and middle maths. All posts in the series are here.

This is the final post detailing how I introduced ‘Notice and Wonder’ to my pre-service teachers. We’ve used it for sense making. We’ve then looked at photos from the world around us and brainstormed what we noticed and wondered. The students later took their own photos and identified the mathematical ideas that they saw. (The photos and reflections were so much fun to look through!) Next, we transferred our ‘Notice and Wonder’ skills to more mathematical settings, including one of Dan Meyer’s Three-Act Maths Tasks, Toothpicks. I’ll now tell you about the consolidating task in which I had students tell me what they notice and wonder about an image bursting with mathematical ideas.

Update 6 August 2017: This post describes another way to incorporate ‘Notice and Wonder’ with the Prime Climb hundreds chart.

Note: although this looks like a long post, the first 200 words are an introduction; the last 1500 words are a summary of student ideas.

The Prime Climb hundreds chart

Prime Climb is a beautiful board game in which players deepen their understanding of arithmetic through gameplay. To be quite honest, I’ve never played! But that hasn’t stopped me appreciating the gorgeous hundreds chart that accompanies the game. A version is below; you can buy this image on a stunning poster here.

This hundreds chart compels us to notice and wonder. Take a moment and brainstorm for yourself. (Dan Finkel, creator of the game Prime Climb, talks about this image in his wonderful TED talk, ‘Five Principles of Extraordinary Math Teaching‘. It’s worth taking the ~15 minutes to watch.)

‘Noticing and Wonderings’ from my students

I asked my group of nearly fifty pre-service teachers to each tell me five things that they noticed, and one thing that they wondered. As a group, that’s potentially 250 different things that they notice, and 50 things that they wonder! Here is a collated list of about 100 of their ideas (with slight amendments to incorrect terminology), loosely grouped under my own section headings. I asked them to do this as individually. I’m sure that in a group discussion they would have built on and extended each other’s ideas. Next time!

Enjoy the read; I certainly did.

I notice that …

Colour and structure

Circles are numbered 1-100.
The chart is organised into a 10×10 system.
The numbers are ascending.
The numbers in each column increase by ten as you go down the list.
Colour has something to do with number, and vice versa.
There are different colours: blue, orange, yellow, red, green.
Some circles have only one colour.
With the exception of the whole red circles, each other colour appears as a whole circle only once.
Each circle is made up of one or more colours.
Colour is used to demonstrate relationships between numbers.
Every second number has orange in it (and similar statements about other colours).
All even numbers are yellow/orange.
Friendly numbers (5s and 10s) have blue in them.
Circles with blue end in 5 or 0.
There are a lot of red-only circles/numbers.
There are 21 solid red-only circles/numbers.
Red is the most prominent colour.
Purple is the least-used colour.
Completely green numbers are multiples of 3 (and similar statements about other colours).
The rings are broken into fractions that vary between a whole and 1/6.
Some of the red sections have little white numbers in them.
All the small white numbers that appear ‘randomly’ on the bottom of the circles are all odd numbers.
The red full circles only occur on odd numbers.
Numbers with orange in them (multiples of 2) are in a vertical pattern, as are numbers with blue in them (multiples of 5). But numbers with green in them (multiples of 3) are in a a diagonal pattern (right to left) when viewed from top to bottom.
If you place your finger on a number with purple, then move your finger up one row and then move it three columns to the right, you will end up on another number with purple (works with most purple numbers unless it is too close to the edge).
The greatest number of coloured sectors around a number is six.
The greatest number of different colours included in the sectors surrounding any number is three.
No number/circle has all the colours present.
There doesn’t seem to be a pattern in the colours.

The number 1

The number 1 has no colour, because it is neither a prime or a composite number.
The number 1 has its own colour and is not part of any particular pattern in the chart. Every whole number has a divisor of 1.
1 is not a prime number, which is why it is not coloured.

Prime numbers

The circles with full colours are prime numbers.
All prime numbers have a single unbroken circle.
97 is the largest prime number less than 100.
Prime numbers have their own specific colour up to the value of 7.
Red circle numbers are also prime numbers from 11 upwards.
Other than 2, all prime numbers between 1 and 100 are odd numbers.
There are 25 prime numbers between 1 and 100.
If there is a little number written at the bottom of a circle for a greater number then it means that greater number is divisible by a prime number. For example the number 92 has a small 23 written at the bottom of the circle, this indicates that 92 is divisible by the prime number 23.
There is only one prime number between 91 and 100. All other blocks of ten have at least two prime numbers.
The ‘3’s column has the most prime numbers between 1 and 100.

Composite numbers

Numbers that aren’t prime are a mix of colours. For example, 15 is 5×3 where 5 is blue and 3 is green, so 15 is half blue and half green.
All multiples of 6 have to have orange (2) and green (3) in them.
Any number ending in 4,6,8 or 0 isn’t a prime number.
Some non-prime numbers are made up of factors which are just (only) prime numbers.

Square numbers

All square numbers are comprised of one colour in several parts.
The sum of all the square numbers is 385.

Multiplication-oriented

We can use the colours around each number and multiply their ‘representing numbers’ together to make the number in the middle.
The circle fragments symbolise how many times multiplication has occurred. For example, the number 8 has three yellow circle fragments, indicating 2×2×2.
The colours of each circle represent the numbers in which the greater number can be divided by. For example number 95 is coloured blue and red. These colours represent 5 and the prime number 19. When multiplied their sum is 95.

Divisor and factor-oriented

There are only 2 numbers on this chart that are represented by a circle split into sixths. They are 64 and 96.
No more than six factors are required to make numbers up to 100.
Odd numbers more commonly have factors that are prime numbers.
The circles are divided into sections depending on how many divisors they have.
The factors of each number are obvious through the colouring.
Different coloured sections in the circle mean that the number is divisible by more than one number.
Odd numbers generally have fewer factors, even if they aren’t prime.

Prime factors

The colours that surround the number represent the prime factors of the number. For example, number 96 has five orange segments and one green segment, which suggests that the prime factors for the number 96 are 2×2×2×2×2×3.

Other

All numbers divisible by 11 have the number 11 in a subscript, and are in a diagonal line.
Consider numbers with the same digits (11, 22, …). The sum of the digits are all even numbers.
There are no explicit instructions or ‘key’ to explain what the chart is actually displaying.
The sum of the first nine prime numbers is 100.
If you squint your eyes, you start to see colour patterns rather than noticing numbers, which is how I noticed some of my previous points.

I wonder …