In case you are not familiar with it, the ‘Notice and Wonder’ prompt involves asking two questions: ‘What do you notice?’ and ‘What do you wonder?’. These are powerful questions to engage students. ‘Notice and Wonder’ helps lower the barrier to entry for all students and encourages sense making.

Notice and Wonder is definitely one of the top five actions that have transformed my teaching. Like Kate, I love how expansive and inclusive these two simple prompts are.

Me too. #NoticeWonder has been transformative in both my teaching and thinking.

— Amie Albrecht (@nomad_penguin) May 26, 2017

In previous posts, I’ve focused on using Notice and Wonder in problem-solving contexts like Conway’s Rational Tangles or a paper-folding investigation, intriguing prompts like the Prime Climb Hundreds Chart or in the world around us, and as a sense-making activity.

In this post I want to quickly share two recent experiences I’ve had that connect ‘Notice and Wonder’ with the types of concepts or questions that we encounter everyday in our classes, rather than more occasional problem-solving puzzlers.

In today’s lecture we were revising in preparation for the final exam. My focus was on helping students work out how to get started on questions when they don’t know what to do. On the spur of the moment, I started this question by asking what they noticed.

We noticed:

- The terms go negative, positive, negative, … .
- The denominators are multiplied by 3 each time.
- The numerators go up by 1 each time.
- There are five terms.
- The question has the sigma symbol in it.
- It says it’s a series, which we did in the context of sequences and series.

I asked if we could use these ‘noticings’ to write down one general term that could be used to describe any term in the series. And we were off! The alternating signs caused some consternation, and a wondering about how we could make that happen when the sum would ‘add everything up’. That was fun to tackle. There was another wondering about whether we were required to give a final numerical answer, which focused our attention on the word ‘express’.

What was particularly powerful was having a checklist of features from our ‘noticing’ work that we needed to be sure we incorporated in our final ‘sigma notation’ expression of the series.

A few weeks ago I was preparing for a lecture that introduced graphs of quadratic functions for the first time. At the last minute, I decided to show this graph and prompt for Notice and Wonder.

We noticed:

- The graph has two
*x*-intercepts, one positive and one negative. - The graph has one
*y*-intercept which is positive. - There is a maximum value at (2,9).
- The curve is in all four quadrants.
- The shape is ‘downwards’.
- (They probably noticed more features.)

We then went back through this list and expanded it into things we wondered:

- Could we have two positive or two negative
*x-*intercepts? What would that look like? (We sketched or talked about some possibilities.)

Instead of two*x*-intercepts, could we have one or even none? (We sketched some possibilities.) - We asked similar questions about the y-intercept.
- I introduced the term ‘vertex’ for the maximum. We wondered what other possibilities there are, and talked about the vertex being a minimum.
- We wondered whether the curve could be in exactly one quadrant? Or two quadrants? Or three quadrants? (I was not expecting this!)
- What other possibilities are there for the shape? I introduced the terms ‘concave up’ and ‘concave down’. We drew a concave up quadratic. We connected these to whether the vertex was a maximum or a minimum. We wondered about putting the shape ‘sideways’, but then discovered that it wouldn’t be a function.

This was a pretty strong start to the lecture as it previewed everything I planned to introduce in the next 90 minutes. I then took it one step further and showed three different forms of the equation of the quadratic. In the spirit of full disclosure, in my haste before class I made a mistake with two of the signs. When we discovered this, there was a nice ‘sense-making’ diversion as we expanded/factored the RHS of each equation. Establishing that these were alternative, equivalent forms of the same function turned out to be useful later too. Below I am showing the corrected versions.

Now we worked on connecting the different representations with the features that we had just noticed.

- From the standard form, we could ‘see’ the
*y-*intercept. - From the factored form, we could see that the factors related somehow to the
*x*-intercepts. - From the third form, we could see that the terms related somehow to the vertex. We then named it ‘vertex form’.
- We also discussed the negative sign in front of the
*x*^{2}and how that related to the shape. I don’t think we discussed it in class, but I can see now how we could develop that idea from the vertex form and the observation that, for any value of*x,*the*y*value will be less than or equal to 9.

This short introductory discussion motivated the rest of the lecture in such a way that some of the later material didn’t need to be discussed in the depth that is usually required. I’ve created more time within this topic!

I constantly marvel at how these two simple questions — asked together or independently — have such a positive impact on the learning that happens with my students and for me.

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One of the activities I chose for this year was Conway’s Rational Tangles. I’ve previously written detailed notes about running the activity with pre- and in-service teachers. For the Maths Experience, apart from the inherent fun of ‘playing’ with ropes, I wanted students to have a collaborative and authentic problem-solving experience. I introduced the activity as one that mirrors mathematical research — full of questions, puzzling moments, uncertainty, frustration and hopefully also joy. I emphasised that we might not solve the problem, but that the experience of working mathematically was our goal, which includes making wild conjectures and having out-of-the-box ideas!

In this post I want to highlight one addition I made to the activity described in my earlier blog post — the inclusion of the ‘Notice and Wonder’ prompt^{1}.

I started the session by showing students the short video below, edited from one I found on Youtube by Tom Hildebrand. Specifically, I turned off the sound, cut out the whiteboard, and sped it up significantly. Then I asked the two magic questions: ‘What do you notice? What do you wonder?’ Take 70 seconds to watch the video, and see what you think.

Here is what they **noticed**.

**Group A**

- They are trying to untangle the ropes.
- One person hangs on to one end of the rope for the whole time.
- They rotate 90 degrees clockwise.
- There is a plastic bag.
- Twist involves exactly two people and occurs in exactly the same position.
- They untangle using exactly the same types of moves they used to tangle.

**Group B**

- Four people holding two ropes.
- Same person holding the same end for the whole activity.
- When rotating, one person moves clockwise. (Later refined to each person moves one position clockwise.)
- The twist movement always involves the two people on the right. The same position goes under each time.
- There was some pattern they kept repeating.
- They did some moves to get a knot. Then they did some more moves and there was no knot.
- There was a bag.
- There were four rotations before the bag appeared and eight rotations after.
- Sometimes there is a different number of twists after a rotate.
- A twist after a rotate goes ‘perpendicular’. (Not sure what that means!)

And here is what they **wondered**.

**Group A**

- What’s the deal with the plastic bag?
- What’s the deal with the teacher?
- How did they decide when to stop tangling and start untangling?
- How tangled was the rope?
- What did the teacher and the student pass to each other? (Scissors.)
- How did they work out how to untangle? (I explicitly prompted this question — although I’m sure they were all thinking it.)

**Group B**

- How did they know how to untangle the ropes? Was it from memory?
- What is the point of rotate? It doesn’t seem to change the rope.
- Does the bag have something to do with the tangling?
- Is it a proper knot? Or just a tangle?
- What is the teacher doing?

There was more conversation that I didn’t manage to capture. (Next time I’ll record it!) Group A spent around 10 minutes on Noticing and Wondering. Group B spent 15-20 minutes. We then largely ran the session as I’ve detailed in the earlier blog post.

What effect do I think Notice and Wonder had? I noticed that students were keen to try the problem for themselves. They made sense of the situation, became intrigued and engaged, and then made the problem ‘theirs’. As a group, students saw that others had interesting ideas. They added on to each other’s thinking. I suspect that it also smoothed the way for working together more intensely once we broke into smaller groups where students didn’t necessarily know one another. It also became more natural for them to Notice and Wonder as the session progressed. All in all, it’s a great modification to a thoroughly engaging activity.

[1] I trialled this with teachers at the MASA conference in April.

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Well, it turns out that many of us want to read about lessons that are either uninspiring or failures. We want to know that there are others who have the same experiences that we do. That not every class is sunshine and lollipops.

Yesterday, Annie Perkins tweeted the following.

Hey #mtbos, had good convo w/ @PettisChristy which brought up good Q: are any bloggers writing about their lesson fails? We should be.

— Annie Perkins (@Anniekperkins) May 7, 2017

It led to the hashtag #lessonfail (my suggestion of #lessonmeh was far less popular ) and a huge influx of tweets of people sharing their own less-than-perfect teaching moments. Because we are all imperfect humans who make mistakes. Often. And we are constantly learning how to be better.

If you are like me, you view your students’ mistakes as learning opportunities, not learning failures. We need to permit ourselves the same freedom to fail, to learn, and to grow. And we need to talk about it with each other, from beginning teachers to those who have been in the game for a long time. I shared two of mine here. Thanks to those who has shared theirs in the last day or two: Annie F, Annie P, Bryan, Christine, David, Ilona, Madison, Tracy, and those that I missed. And if you aren’t ready to share, I understand that too. It’s hard to expose our vulnerabilities to others, for whatever the reason. Thank you to everyone for warmly responding to those who share. We are stronger together.

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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), 16 books in Part 2 (May – August) and 18 books in Part 3 (September – December). That makes 53 books in 2016.

*Black Rock White City, A.S. Patric.*It took me ~50 pages to warm to the story, but then it was unputdownable. A decidedly worthy winner of 2016 Miles Franklin Award.#read2016: A couple almost unknowingly clinging to each other through deep wordless grief, and yet a hopeful book.

*Commonwealth, Ann Patchett.*Wonderful writing. Skilled in capturing moments and innermost thoughts in few words. As a promising relationship between a famous author and a waitress begins: ‘He patted the top of her hand, which she had left close by on the bar in case he needed it.’ A glimpse of a left-behind, overworked mother of four young children: ‘The speed at which their mother ran from work to school to the grocery store to home had doubled. She was always arriving, always leaving, never there.’#read2016: Vignettes spanning 50 years woven together to tell the story of complex blended family relationships.

- Postcards from Surfers
*, Helen Garner.*Eleven short stories in true Garner style.#read2016: Stories that never use more words than they need. Expertly constructed.

*Dying: A Memoir*,*Cory Taylor.*I first learned of Cory Taylor on the fabulous ABC ‘Terminally Ill’ program of the ‘You Can’t Ask That‘ series. Cory was frank — the same as in her memoir.

#read2016: Clear-eyed. Unsentimental. A deeply reflective view of dying and of life. Moving.*Notes on An Exodus,**Richard Flanagan.*Richard Flanagan won the Man Booker Prize for his remarkable ‘The Narrow Road to the Deep North’. His ‘notes’, along with sketches by Ben Quilty (Archibald Prize winner), paint powerful portraits of Syrian refugees in Lebanon, Greece and Serbia.#read2016 (1): Devastatingly moving portraits of Syrian refugees from two of Australia’s most acclaimed in their crafts.

#read2016 (2): A slim volume but not at all light. Honours their dignity + courage. ‘Refugees are not like you and me. They are you and me.’*The Good, the Bad and the Unlikely: Australia’s Prime Ministers*,*Mungo MacCallum.*More than 29 biographies, this also brings together the story of Australian politics.

#read2016 (1): A lively and humanising view of each of Australia’s 29 PMs. Witty + concise writing that had me laughing (or snorting!) out loud.

#read2016 (2): I learned many things, but am still struck by the news that we had a PM with the middle name of ‘Christmas’.*The Curious Story of Malcolm Turnbull, the Incredible Shrinking Man in the Top Hat*,*Andrew Street.*I preferred the first book, ‘The Short and Excruciatingly Embarrassing Reign of Captain Abbott’, but only because Abbott was so laughably hopeless. The sequel certainly reveals some of the ineptitude of Turnbull.

#read2016: Grab the popcorn and dig into the spectacle. (If you don’t laugh, you’ll cry.) Street dishes out snark in spades.*Salt Creek*,*Lucy Treloar.*An interesting blend of fact and fiction.

#read2016 (1): Set on the South Australian Coorong in the 1850s as white settlers first encroach on the lands of the Ngarrindjeri people.

#read2016 (2): A beautifully-told, heartbreaking shameful story that matches historical truths. Starts painfully slow; gripping past pg 90.*The Long Green Shore*,*John Hepworth.*#read2016: Published 50yrs after writing of Australians in PNG during WWII. Banality alongside barbarity. Matter-of-fact yet almost poetic.

*The Hate Race*,*Maxine Beneba Clarke.*An Australian of Afro-Caribbean descent, Maxine Beneba Clarke tells what it is like to grow up as a person of colour in Australia.

#read2016: Packs a powerful punch, right to the stomach. A difficult, but important read. Particularly now.- *
*Which One Doesn’t Belong? Teacher’s Guide*,*Christopher Danielson.*The premise of ‘Which One Doesn’t Belong?’ is to consider four shapes, and ask the question. In the children’s picture book and its companion teacher’s guide, Danielson focuses on geometry and uses WODB to draw out rich mathematical ideas. The teacher’s guide provides convincing rationale and practical advice. There are plenty more WODB out there; try www.wodb.ca and the hashtag #wodb.#read2016: A delightful way to discuss + explore maths. The writing is crisp, purposeful, insightful + welcoming. A must-have for tchrs.

*Girt: The Unauthorised History of Australia,**David Hunt.*Australian history like it should have been taught at school. The ABC Radio podcast Rum, Rebels & Ratbags with Dom Knight is also worth a listen.

#read2016: Peppered with witticisms and dripping in places with sarcasm, this is a lively telling of Australian history like no other.- *
*Becoming the Math Teacher You Wish You’d Had: Ideas and Strategies from Vibrant Classrooms*,*Tracy Johnston Zager.*I have been madly awaiting this book for at least a year. It was so good I read it cover-to-cover almost as quickly as I could. It’s a beautiful, important book. Truly something special for all maths teachers. My full review is in this blog post.

- *
*Avoid Hard Work!*,*Maria Droujkova, James Tanton, Yelena McManaman.*There is a lot to like about this book, but ultimately I found it too light. I also have problems with the title.#read2016: A gentle approach to encourage mathematical problem solving with very young children.

*All That I Am*,*Anna Funder.*#read2016: A fictionalised biography of political activism against the Nazis in WWII. Crushing. Beautifully written. A page turner.

*Our Souls at Night*,*Kent Haruf.*The*‘Addie Moore and Louis Waters have been neighbours for years. Now they both live alone, their houses empty of family, their quiet nights solitary. Then one evening Addie pays Louis a visit.’*#read2016: A tender, quiet and impossibly beautiful tale of growing old together with grace.

*Victoria: The Queen*,*Julia Baird.*Loved the narrative-style approach, particularly once I realised it was built around impeccable research.#read2016: A hefty portrait of a formidable + intriguing queen. Flowing, engaging, well researched. Fascinating details of V as a woman.

*Monkey Grip**, Helen Garner.*Her acclaimed first novel. Reads like diary entries, with Garner’s perceptive view.

#read2016: Explores addiction, to hard drugs + to love. Written in the 70s; curious to see where Garner started. Still wondering if I liked.

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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.*(pg 49)**There’s nothing to try if everything is prescribed.**‘*.*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).*‘*How can I help students gain the skills to diagnose and learn from their mistakes, by themselves?**to teach students to make the most of the knowledge and experience they gained by figuring out their mistake**‘. - From Chapter 5, Mathematicians Are Precise: ‘
(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.**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.’ - 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.

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*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.

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 . 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. I’m also not going to tell them what ROTATE does; that’s for them to figure out.

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?

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, . 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:

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:

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.

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: . 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 . This works!

We test it on one TWIST: . This also works.

We predict what should happen with two TWISTS: . 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 . **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:

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.

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: . 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: and . We confirm that , so two RR leave the state unchanged. We talk about composition of functions, and how RTT is represented by

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.

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*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!

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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 :).

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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.

*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.

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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.

*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.*

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.)

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.

**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.

**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?

**Patterns**

- 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?

**Other**

- 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?

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