Monthly Archives: August 2016

#read2016: Part 2

As I’ve mentioned, I love books. Real books, with paper and ink. None of those fancy ebooks. I spend enough time each day staring into screens. Plus, I like to read in the bath and the idea of accidentally dropping a $1000 device doesn’t appeal. (I’ve only ever once dropped a book in the bath. It was a library book. Go figure.)

The busier I get, the less I seem to read for pleasure. To redress this, my plan is to read 50 books in 2016. Fiction, mathematics, Australian politics, biographies, non-fiction, anything. Some books are short novellas which you might think of as ‘cheating’. Whatever. Despite the fact that I am counting, the number doesn’t count. It’s just a target to get me to read more.

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.

  1. 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.
  2. 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.
  3. 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.
  4.  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.
  5. 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.
  6. 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.
  7. 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.
  8. 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.
  9. 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.
  10. 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?
  11. 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.
  12. 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.
  13. 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.
  14. 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.
  15. 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.
  16. High SobrietyJill 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

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.


  • 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.


  • 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 …

Colour and structure

  • Why 1 is the only number that is grey?
  • Why some circles have extra numbers in white?
  • What do the sections of the circles mean?
  • Why are different numbers cut into different ‘fractions’? Is there an underlying reason for this?
  • Why do some numbers have parts in their colour, even if those parts are the same colour? For example, number 64 has six parts of orange, and orange is associated only with 2.
  • How did they work out to segment the outside circle of 24 into four segments? And why are three of them orange and one green?
  • What colour is used the most?
  • Would the chart be easier to read if all prime number had their own colour rather than the first 10?
  • Why do 96 and 64 have the most divisions?
  • Are there multiple ‘solutions’ to this problem?


  • If there is a pattern? And if I could figure it out?
  • Is there are pattern between the numbers and the number of parts in its coloured circle that can be used to work it out for any number?
  • Why didn’t they write the number of times that a particular number goes into the large number inside the appropriate colour section?
  • Why are the numbers coloured in randomly (no specific pattern)?
  • Can you use this number chart and extend it to find every single prime number without manual and tedious calculations?
  • Is there a systematic way of determining the greatest number of sectors or different colours that can surround any number in a set (1 to 1,000,000 for example) without having to sit down and multiply prime numbers?

Extending the chart

  • If this went to 1000, what number would have the most number of different colours?
  • If this went to 1000, would we start to see more and more red compared to other colours?
  • I wonder what the next 100 numbers would look like prime factorised in this way. I would imagine that the amount of red visible would decrease.
  • What would this look like if extended to 200?
  • If it went to 200, would the numbers have more than four or five colours?
  • How many prime numbers would there be in the next set of 100 numbers, as in from 101 to 200?
  • When is the first row of 10 with no prime numbers?


  • What maths learning this could be used for?
  • What are hundreds charts used for?
  • Could a chart like this be used to help introduce maths to young children before they use rote memorisation?
  • If knowing primes and composite numbers can help in everyday life?
  • What would this look like if we created an image like this based on addition?
  • If this chart would be as easily translated if squares or triangles or some other shape was used in place of circles?
  • What does this diagram represent? Who was it made for?
  • Why did someone choose this representation?
  • Why was this created?
  • How long did it take to create?
  • Who came up with this representation? It’s really cool!

If you read this far, well done! But to quote the last student, it is really cool, isn’t it?

‘Notice and wonder’ and ‘slow maths’: reviving an activity that fizzled

This is the fifth 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.

In my last two posts I’ve been explaining how I’ve introduced the ‘Notice and Wonder’ routine to my pre-service teachers. We started with the value of ‘Notice and Wonder’ for sense making. We then looked at photos from the world around us and brainstormed what we noticed and wondered. My intention was that students would gain experience with everyday situations before transferring their ‘Notice and Wonder’ skills to mathematical settings. In this post I’ll write about the next stage of this journey. But to do this, I want to first tell you about a great task and how I’ve never managed to do it full justice.

An activity that fizzled — because of me

Imagine a long thin strip of paper stretched out in front of you, left to right. Imagine taking the ends in your hands and placing the right hand end on top of the left. Now press the strip flat so that it is folded in half and has a crease. Repeat the whole operation on the new strip two more times. How many creases are there? How many creases will there be if the operation is repeated 10 times in total?

I originally saw this problem in Thinking Mathematically (Mason, Burton, Stacey). Looks like a great problem, right? Try it for yourself, either visually or physically. You might notice relationships between the number of folds, the number of creases, the number of sections, and more.

Paper Strips is an activity rich in opportunities to make conjectures and test them out. For the past two years I’ve positioned it in Week 7 of a 12 week program, when we are deep in conjecturing, justifying and proving. In this context, I’ve given students the description above, a few strips of paper, and asked them to record the number of sections and creases for a given number of folds, to make conjectures, and to try and justify their conjectures.

And it has bombed. Both times. A charitable student in either of those classes would say that it was ‘fine’ — hardly a ringing endorsement. This year I was planning on dropping the problem. I could see how rich it was mathematically, but I just couldn’t see how to make it shine.

And then it dawned on me.

It’s all in the presentation

I’ve mentioned before that I was fortunate to attend a micro-conference in June led by Anthony Harradine. This was a master-class in having people think and work mathematically. Anthony emphasised three key ingredients for a successful problem-solving experience for students.

  1. Pick a problem where students are likely to already have the required ideas and skills. My interpretation is that the problem-solving process is already cognitively demanding and so students shouldn’t also be grappling with calculations that they find difficult.
  2. How the problem is presented matters a great deal. Let students have ideas about the problem. (And, if needed, find a way to make them have the ideas that you need them to have!) Acknowledge and value all their ideas. If their ideas don’t suit your purpose, put them on an imaginary shelf to be pulled down and tackled later. (This is similar to Dan Meyer acknowledging all questions that students have in the initial stages of a Three Act task, and returning to them at the end to see if they can now be answered.)
  3. The way that students work on the problem is important. How much structure will you provide? Will students work individually or collaboratively? How will students share their resolutions? Will you provide a full resolution? What will you leave them to think about?

Back to Paper Strips. While planning Week 3 — and looking for a more mathematical setting for students to develop their ‘notice and wondering’ skills — it occurred to me that in the past, I’d had two out of these three elements for Paper Strips. But I was missing a vital ingredient: the right presentation that let students have more ideas for themselves.

The revival: notice and wonder to the rescue

This year I told students that we were going to do a visualisation activity, and that I would walk them through a set of instructions.

  • Imagine a long thin strip of paper stretched out on the table in front of you.
  • Hold each end.
  • Now fold the paper by moving your left hand over to your right.
  • Make a crease along the folded edge with your left hand.
  • Now hold the creased end with your left hand.
  • Fold it again by moving your left hand towards your right.
  • Make another crease.
  • Now slowly imagine the paper unfolding.
  • What does it look like?

This presentation is exactly as outlined in the Shell Centre’s Problems with Patterns and Numbers, of which John Mason is a co-author. And compare it to the earlier description. Similar, right? But not the same.

Rather than continue on and ask students to investigate the number of creases and folds, I lingered onWhat does it look like?’ I asked students to draw what they thought it looked like. Sketches ranged from the simple to the complicated. A typical sketch looked like this, although a few others were 3D.


I then handed out strips of paper and repeated the instructions. They compared their record of their mental image with the physical model. I asked them if there was anything else that they noticed about their physical model that was missing in their drawing.

Students then brainstormed other features they noticed about their strip of paper. As a group we noticed creases, folds, sections, sections of equal size, up creases and down creases, the pattern of the creases (two folds gives down, up, up), the dimensions of the strip of paper. We agreed on definitions for many of these terms. We thought about whether our diagrams could be more accurate. For example, were our sections of equal size? Have we distinguished in our diagram between up and down creases? Is accuracy even important here?

Next I suggested that students make more folds, and brainstorm anything new that they noticed. I also asked them to record what they wondered. How did the paper-folding process affect the features of the paper strip that we identified earlier?

The very first idea volunteered is shown below. Look at the profile that forms when the sections between two ‘up creases’ (∨ shape) are placed flat. How does this pattern change with more folds?


Other students added on to this idea. They noticed what would happen to the profile when we unfolded the paper. What do you think will happen?


Another group had wondered how the original dimensions of the paper strip affected the size of sections after each fold. They noticed that sometimes it resulted in a square, and sometimes in rectangles of particular sizes.

These were the first three observations in one group and I loved them. I hadn’t even noticed them as I’d only been considering the strip as a 2D object. Other ‘notice and wonderings’:

  • Is there a formula for the number of creases for a particular number of folds?
  • The number of sections seems to double with the number of folds.
  • The number of sections seems to be one more than the number of creases.
  • It looks like there is always one more down crease than up crease.
  • The more folds you do, the shorter the ‘bottom layer’ and so that affects things. (The physical process differs from the theoretical process.)
  • There seems to be a pattern between the number of consecutive down folds (∧) and the number of folds.
  • Is it possible that the maximum number of down folds in a row is three?
  • The number of creases appears to be one less the number of folds.
  • Could we predict the crease pattern after another fold?

Many of the ideas I wanted them to notice came out of the brainstorming exercise — and so many more interesting questions that I hadn’t even considered. Fifty brains are definitely better than one!

This was the endpoint for this activity. Notice how we didn’t resolve any of these questions as a group. Some students worked out formulae or explanations, but I asked them to keep those private for now. There will be time later when we dig into justifications to revisit this problem.

And now for the whole truth

When I started writing this post, I had thought that ‘Notice and Wonder’ was the key to making this activity shine. It’s the truth, but it’s not the whole truth. Looking back through last year’s photos, I had written a big ‘Notice and Wonder’ next to my instructions for students on the whiteboard. And yet it still kind of flopped.

As The Classroom Chefs say, how you plate a meal is important. And as with meals, how a maths problem is presented is everything! Jennifer Wilson’s latest blog post also reminds me that, throughout their book, John and Matt constantly encourage us to savour our meals, that is, to slow down.

Go back through the Entrée stories you just read, and look specifically at the questions each teacher asked the students. Notice how no teacher was in a hurry; they let students discuss a topic or an idea until they were satisfied that the students fully understood it.

Slow maths. Let students notice and wonder for themselves. Don’t rush them towards what you want them to focus on. As Anthony Harradine said: ‘Let students have ideas about the problem. And, if needed, find a way to make them have the ideas that you need them to have.’. For Paper Strips it was the right presentation, combined with slowing down, that meant I didn’t need to find a way for students to notice what I wanted them to see. They saw that — and so much more.

Notice and wonder: the world around us

This is the fourth in a planned 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.

In my previous post, I talked about how I used sense making as a powerful motivator for the ‘Notice and Wonder’ routine. My next step was to have my pre-service teachers experience ‘Notice and Wonder’ for themselves.

Entry task

I deliberately chose to start with an everyday, seemingly non-mathematical image. Study the image below. What do you notice? What do you wonder?


I displayed the image and asked the two magic questions. There was silence. Inwardly I was thinking ‘Oh, crap — this is going to be disastrous.’. I think it was at this stage that I reminded them that non-mathematical ‘notice and wonderings’ are as important as mathematical ones. After another quiet moment the buzz started. I quickly walked to the back of the room and made myself invisible. Once the intensity of the discussion subsided, each group shared some of their ideas. I wish I could remember them all, as there were as many non-mathematical ideas as mathematical ones, but their wonderings were questions like:

  • Why does a shot cost the same as a small?
  • Do you need to buy a coffee to get a free babycino for kids?
  • Where was the photo taken? (If you are Australian, you might recognise some clues. But there are also red herrings such as the units, which are in ounces.)
  • How can you get a shot of tea?
  • Which size is best value for money?
  • Do they really mean that skim milk is free? Or just doesn’t attract an extra charge?
  • How many people buy the large size when they really want a medium because it’s ‘just an extra 50 cents’?
  • Why do we measure coffee in ounces in Australia?
  • How did they decide the pricing structure?
  • Are the diagrams to scale?
  • Why does anyone buy coffee at a petrol station? (Yes — this is where I took this photo, while filling up my car.)

We then reflected on what had happened. I supplemented their ideas with those from Max Ray-Riek’s fabulous book, Powerful Problem Solving. Chapter 4 is dedicated to ‘Noticing and Wondering’ and can downloaded as a sample chapter from Heinemann here. Here is an extract:

These activities are designed to support students to:

  • connect their own thinking to the math they are about to do
  • attend to details within math problems
  • feel safe (there are no right answers or more important things to notice)
  • slow down and think about the problem before starting to calculate
  • record information that may be useful later
  • generate engaging math questions that they are interested in solving
  • identify what is confusing or unclear in the problem
  • conjecture about possible paths for solving the problem
  • find as much math as they can in a scenario, not just the path to the answer.

The #math1070 photo challenge

Next I shared how people were sharing images on Twitter. Some of my favourites are at the end of this post. Then I shared examples from the 2016 Maths Photo Challenge. Part of their weekly task is to take two photos from the world around them, and describe any mathematical ideas that they see. I look forward to seeing their ideas, and perhaps sharing them with you soon!

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Notice and wonder: sense making

This is the third in a planned 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.

It’s been three weeks (how time flies!) since I last posted about this course. There are 1.5 workshops that I haven’t written about. We also missed three classes (1.5 weeks = 3 workshops) due to my travel and illness. We finally got back into the regular rhythm of class last week. In brief, I am not going to write about how we: dug into the importance of multiple representations; tackled our first substantial problem—’How many squares on a chessboard?’—and used it to discuss specialising and generalising; looked at ways to organise our work; talked about the importance of writing down our thinking; introduced the three phases of work (Entry, Attack, Review) and tackled a couple of problems along these lines; started maths talks with a dot talk and then a `duck’ talk. There is too much to write about!

I am continually inspired by the power of the ‘Notice and Wonder’ routine to help in all stages of problem solving, not just when starting to tackle a problem. We spent a few hours in class experiencing different dimensions of ‘Notice and Wonder’, and I hope to summarise our work over a few posts.

Introducing Notice and Wonder

In case you are not familiar with it, the ‘Notice and Wonder’ strategy involves asking two simple questions: ‘What do you notice?’ and ‘What do you wonder?’. These are powerful prompts to engage students. ‘Notice and Wonder’ helps lower the barrier to entry for all students. It encourages sense making. Students are more invested because they are connecting their own thinking to the scenario and are generating questions that they are interested in solving. ‘Notice and Wonder’ is enthusiastically promoted (with good reason!) by the team at The Math Forum. For more, read Max Ray-Riek’s post or watch one of Annie Fetter’s Ignite talks: NCTM 2011 and Asilomar 2015.

‘How old is the shepherd?’ — Observing a lack of sense making in others

‘Notice and Wonder’ is most commonly implemented by displaying an image or a video and recording everything that students brainstorm. However, I decided instead to first have my pre-service teachers observe a lack of sense making in others. For this I used Robert Kaplinsky’s video showing eighth-grade student responses to the nonsensical question ‘How old is the shepherd?’. The question here is seemingly straightforward:

There are 125 sheep and 5 dogs in a flock. How old is the shepherd?

Before I showed the video, I asked students to estimate how many of the 32 students they thought would give a numerical — and thus nonsensical — answer. Try it for yourself.

Then I played the video. If you skipped the link, go watch it. It will only take 3:07 minutes. Students were incredulous; they thought the video was a joke. They just could not believe that 75% of the students that Robert interviewed gave a numerical response. Once that moment passed, we spent some time suggesting different reasons why this might happen, including:

  • students wanting to do ‘something’ with the numbers because it’s a maths question
  • students assuming that they’ve misunderstood something because the teacher has always previously asked sensible questions
  • students not feeling comfortable challenging the questioner, who is the authority in the room.

For further reading, I suggest starting with blog posts by Robert Kaplinsky, Tracy Zager and Julie Wright.

Rather than simply laugh at the video (which did happen; sorry), I then focused on the role of teachers in all the reasons we had brainstormed. I emphasised the responsibility that we have as teachers to empower students to question and to make sense of problems. There was quiet in the room as this powerful moment resonated.

This led nicely into the idea of sense making — in all stages of problem solving. For example, the students in Robert’s interview who said ‘Hey, what? I can’t solve this!’ were making sense of the problem at the start of problem-solving: they had thought about the context and realised that it didn’t make sense. We talked about the students who jumped straight into a calculation, and how they hadn’t even made sense of which calculation they should do. (Hint: there is no sensible calculation one can do!)

We also noticed that most of the students who gave a numerical answer did not make sense of their answer at the end of problem-solving, even as Robert probed their thinking. We did note that no students used a multiplication strategy, so presumably there was some sense making in that the shepherd couldn’t possibly be 625 years old!

From here I wanted to draw attention to the value of units in making sense of problems. For example, if students had reflected that 125 – 5 meant subtracting dogs from sheep to get years, perhaps more sense making would have taken place.

‘The flour problem’ — working with units

The best problem that I know to encourage making sense of problems via units is ‘the flour problem’. I discovered this via Fawn Nguyen, who has written a great blog post about how her sixth graders worked on the problem, including a ‘Notice and Wonder’ routine on the top half of the image.


Of course, it’s not just a problem for sixth graders. I have begun using it in the first class of my university mathematics course for laboratory medicine students. (If they can’t make sense of units, then we have a real problem!) I’ve watched pre-service and in-service teachers in my sessions dig deep into this task; it encourages sense making in so many directions. I also showed John Rowe’s makeover of a trigonometry question, which similarly gets to the heart of sense making.

We also briefly talked about estimation as a sense-making strategy: starting with an estimation, and then reviewing any calculations in light of the estimation that we made. This is a theme we’ll return to later in this course.

Where to next?

With such a persuasive start to the importance of sense making, the stage felt ready for students to experience ‘Notice and Wonder’ for themselves. I hope to write soon about how that unfolded.

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!


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?


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?


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?


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.


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.


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.


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.


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.

Activities for Day One: Fold-and-cut, Quarter the Cross

This is the second in a planned 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.

Last week I wrote about my three main goals for my Developing Mathematical Thinking course. You can read that post here. Today I want to talk about the activities I used in the first workshop: why I chose them, how they unfolded, and the discussion that resulted.

The two activities are Fold-and-Cut and Quarter the Cross. Along with ~35 minutes of other discussion, these activities took the entire 1hr 45 min workshop and another 30 minutes of the next. Regular readers will think by now that I’m somewhat obsessed with both of these activities (it is true; I’ve already written about them here and here), but they are such rich tasks. In this post I add more detail, particularly from the perspective of using them in the (pre-service teacher) classroom.

1. Fold-and-cut

Duration: ~50 minutes
I wanted the very first activity to: be non-threatening and hands-on, encourage students (who may not have met) to talk with one another, not look like a typical maths task, accommodate a variety of problem-solving strategies, and allow me to gently start eliciting explanations of their thinking. Sounds like fold-and-cut is ideal!

As before, I started by showing the first 2:30 min of Katie Steckles’ Numberphile video on the fold-and-cut theorem. (As a reminder, the task is to fold the shape so that it can be removed from the paper with one straight cut.) I gave out the first three pages of this revised handout with the yellow, green and blue shapes in the photos above. (Credits for the handouts due to JD Hamkins and Patrick Honner; see my earlier fold-and-cut post.)

After everyone had tackled most of the shapes, we stopped to reflect. I made reference back to the four lenses with which they could view each task (maths learner, maths thinker, student, teacher), and asked them to brainstorm possible maths content in the task, maths thinking they used, thoughts that I may have had as a teacher in presenting/selecting the task, any other ideas. An incomplete list is below:

  • Maths content. Shapes terminology (for example, regular and irregular polygons, convex and concave polygons, equilateral, equiangular), symmetry properties, angles, angle bisectors, triangle incenter, optimisation/efficiency (minimum number of cuts). Later we added fractals and Koch curve.
  • Maths thinking. Trial-and-error, guess-and-check, noticing (what we did, what others did) and wondering (for example, can I do it with fewer folds?), making a problem smaller, reducing to a problem we already know how to solve, reflecting on what went wrong/right, transferring skills to a new problem.
  • Teacher lens. We thought that the task: gives students choice about where to start and how to do it; has an ‘easy’ entry; builds confidence; is ‘open’; is engaging, visual and hands-on; allows multiple approaches; is low risk and low threshold, high ceiling.

I shared that this was the first time that I had handed out all three pages without suggesting that students start on the yellow (easiest) page. We reflected about whether some people started with one that looked easiest (to give them confidence), the one that looked hardest (because they wanted a challenge), or some other shape for some other reason.

I then gave them the last two pages (pink and orange in the photo) with irregular shapes and the Koch curve. Discussion points included: we couldn’t necessarily rely on properties of symmetry anymore; they felt they could do these ones now that they’d build confidence on the first three pages; the Koch curve looked hard but turned out to be easier than expected (an experience that might hold them in good stead in the future!). I also directed them towards Chapter 3 of the free book ‘Art & Sculpture’ from Discovering the Art of Mathematics, which has a well thought-out sequence of prompts that might make a suitable classroom investigation.

In all, I was really pleased how this activity unfolded, and how we were immediately able to use the ‘four lenses’ idea as a framework for reflection.

2. Quarter the Cross

Duration: ~45 minutes (spread over two classes)
I am very thankful to David Butler for sharing here about the mathematical ideas he sees in this activity, and here about using this activity in his daughter’s Year 7 classroom; this was excellent guidance.

My reasons for using this task were: promote creativity, encourage students to find different ways of achieving the same goal, prompt students to give reasons for their work, lead into multiple representations.

QTCI started by showing the slide. I then handed out six smaller copies of the cross. I encouraged students to use whatever method they wanted; some chose to cut them out (most likely because we had just been doing fold-and-cut), others used rulers, some drew free-hand.

After students had coloured in six, I picked two of the shapes that most people find in their first few: the ‘house’ and the ‘L’. (See below.)

  • We started with the ‘house’. Some students shared that they found it using symmetry; others shared that they saw it as one full square plus a quarter of another one.
  • We moved onto the ‘L’. Along with symmetry, another approach was to decompose the whole cross into a smaller number of equal parts. A key observation was that making the number of equal parts divisible by 4 made it easier to shade a quarter. (The orange cross on the bottom right is another example of this.)
  • We then discussed the one with five blue triangles, and noted that it could be made from the house by moving bits around (which doesn’t change the area). (The red triangle-type shape in the bottom row is also like this).
  • Finally we talked about how adding a bit and taking the same bit doesn’t change the area, which is how the green one with circles came about.


I also used different examples of student work to illustrate these strategies.

I then showed them David’s one hundred solutions (see his blog post). I zoomed in on some of the more complicated-looking ones as examples of the mathematical scope of the activity. Their homework task was to colour in another six based on their ideas from our discussion, and from examining their classmates’ work.

Multiple representations

In the next workshop, I started by noting that we had two ways of considering the cross: as one whole, or as composed of five smaller squares. I asked them to concentrate on the second method. If one square is a unit, then what does it mean to ‘quarter the cross’? We need to shade 1/4 of 5 units, so 5/4 units.

Now we started exploring different ways of expressing 5/4, and connecting them to corresponding visual representations. Once again, we used student work to illustrate these. For example:

  • 5/4 = 5×(1/4). Colour in 1/4 of one unit. Do this five times. (We can move those bits around later if we want.)
  • 5/4 = 1 + 1/4. Colour in one whole unit and a quarter of another. (An example is the house.)
  • 5/4 = 2×(5/8). Colour in 5/8 of one unit. Do this two times.
  • 5/4 = (1/2)×(5/2). This one was interesting, as it was the first one with an improper fraction. Start with shading 2.5 units, and then colour in 1/2 of the shaded area.
  • 5/4 = 1/2 + 3/4. Colour in 1/2 of one unit and 3/4 of another unit.

We reflected on how much richer this task had become with the introduction of another representation, and how being able to translate between the two representations promoted deeper understanding about fractions. This was a great lead-in to the rest of Workshop 2; I’ll write about this soon.