This post is about my reaction to the book “Choice Words: How Our Language Affects Children’s Learning” by Peter H. Jonston. I was lent the book by Amie and I am very grateful to her because it really is a good book (though it was tough to read with the forest of sticky notes marking her favourite pages ūüėČ ).

This thin little book is about how words have power to help children learn about reading, writing, learning, themselves and their place in the world. The majority of the book is a list of sentences spoken by teachers followed by an analysis of what those words mean for children’s learning. The focus is mostly on helping children learn to read and write successfully, but don’t let the “children” or the “read and write” fool you — I have so many thoughts swirling in my head about how this might possibly apply to my own teaching, and indeed my life.

Unfortunately, “swirling” is the appropriate word for my thoughts right now. The fact that the book is structured around analysing specific utterances by teachers made it all very concrete, but on the other hand it is making really hard for me to process the information coherently. At the moment it’s just a big cloud of things to think more about, a lot of which overlaps. I’m finding it hard to tease things apart to find something I can apply first, or a way for me to consistently apply it so it’s useful for my students. I’ve decided the best thing to do is to write this post so I can attempt to process it all.

The chapter titles might be a good place to start. Here they are:

  1. The Language of Influence in Teaching
  2. Noticing and Naming
  3. Identity
  4. Agency and Becoming Strategic
  5. Flexibility and Transfer (or Generalizing)
  6. Knowing
  7. An Evolutionary, Democratic Learning Community
  8. Who Do You Think You’re Talking To?

Even just listing those titles is helping me focus a bit more. While I was reading it, it might have helped me to keep a bookmark in the chapter heading so I could look back and remind myself what the big idea of the chapter was. Instead I found that I got a bit bogged down in some of the details as I went along and lost the focus. Now that I can look back from a higher vantage point, I reckon I might be able to pull out some bigger ideas…

Chapter 1 is about how much our language has power to¬†create¬†reality, in particular the reality of the listener’s identity. If I were to hold on to just one thing from the whole book then maybe this message would be it: I can make the world different for another person by choosing the words I use.

Chapter 2 is about how in order to learn and know what you have learned, you need to notice things. You need to notice how things are similar or different, how they are related or not. And then, things need to be named, so that it is possible to talk about them. This is remarkably similar to the Notice and Wonder idea from the Math Forum people, and to Chris Danielson’s way of getting to geometry ideas via Which One Doesn’t Belong. But here,¬†Peter goes deeper than this. He suggests that you can notice and name not just content, but also your processes as you work as a group, your thoughts about yourself as a learner, the things you have learned so far, and your behaviour. It is a fascinating idea to me that you can apply the same noticing and naming to mental and social processes as you can to the properties of quadrilaterals.¬†Something to hold onto from this chapter is that my words can draw attention to features worth noticing, and the act of noticing itself.

Chapter 3 is specifically about identity. Peter talks about how we construct a narrative with ourselves as one of the characters and the words we use to tell this story shape the sort of person we see ourselves as. We as teachers can make a difference to identity by the words we choose. Something that struck me most strongly was using words that don’t give people a choice to opt out of the identity. For example, the question “What problems did you have?” assumes that there must have been problems, and asking someone what choices they made assumes they made a choice. This is what I want to hold onto from this chapter, that I can give someone courage to be a writer or mathematician by using words that put them into that character.

Chapter 4 is about agency, and in a way is an extension of the previous chapter on identity. The identity in question here is that of a person who has power over their own choices. This chapter spoke to me most strongly as a maths teacher, since maths is a subject where so many students feel they have no choice and that choice isn’t even a thing that people ought to have (as evidenced by the constant request to tell them what to do). Peter advocates talking to students as if they did make a choice, and analysing the choices they could have made. This is one of the biggest ideas in the whole book to me, and I want most to hold onto this one as I go forwards.

Chapter 5 is about transfer, that holy grail of teaching where students are able to apply what they learn in one area to another. Peter pulls together the agency and the noticing/naming from the earlier chapters as the main mechanism for this. More explicitly, the questions listed here focus on noticing¬†explicit connections between things and also exploring the “what if” questions. He ends with a comment about the importance of play, which of course resonates strongly with me. The thing I want to hold from this chapter is the focus on connections, over and above answers.

Chapter 6 is about knowing, and in particular about who holds knowledge and who decides when we know something. In many teacher-student interactions, the assumption is that it’s the teacher who knows and the teacher who decides what is true and when we are correct. Yet really one day when they leave our care, our learners will need to know how to be sure of things for themselves. The thing I want to hold onto here is that I can give my students the power over knowledge. This is especially important in maths, which is set up so that you actually can be sure of things through your own arguments, rather than having to rely on the authority of others.

Chapter 7, while it has a very long title, is really about how our words can help people learn to work together. Peter has a lot of examples where teacher words encourage learners to consider the feelings and ideas of others, and to choose shared goals. He reuses the noticing and naming power of words to help learners notice their own group processes, and the identity-forming power of words to help learners put on the mantle of people who care about others. The thing I can hold onto from this chapter is that words can make group social and cognitive processes explicit in a way that makes them learnable.

Chapter 8 is about the interplay between your beliefs and your words. As a teacher, if you believe your students are not capable of learning something, your words (and your silences) will reflect this. However — and this is the big thing I want to hold on to here — if you choose to change your words, then some of your beliefs might follow. I see this in using SQWIGLES with myself and my staff where choosing to ask open-ended questions changes the ways that students respond to you and therefore ways that you respond to them. Your beliefs about what students have to say can change through this change in your words.

I think I’ve achieved my goal in writing about this book, in that I have a much clearer idea about how I want to respond to it in my work. I have clarified how much of an impact my words can have on learners’ realities, which I knew, but not to the level of specific detail I did before. In particular I think I want to hold on most strongly to the idea that I can help learners to see themselves as having choice and capable of making that choice, changing both their view of mathematics and of their place in it.

Posted in Education reading

Well I did it. I went to Twitter Math Camp 2017 (TMC17) in Atlanta, Georgia, USA.

I found out about TMC last year, when Tracy Zager mentioned me in her keynote at TMC16, effectively yanking me right into the thick of it. I could see that this was one of the things that cemented together the people of the MTBoS and I really wanted to feel first hand what it was like.

And somehow I managed to do it. I applied for a Learning and Teaching Development Grant from my University, available to me because I am part of the Adelaide Education Academy, and submitted a couple of proposals for sessions at TMC. Everything was accepted and I was able to go. Some of the process of organising the money and the travel was a bit arduous, with more red tape than I was expecting in order to make it happen, not to mention the close to 70 hours of travel time involved and the longest time I have ever spent away from my wife since we were married. But it really was totally worth it to have this (possibly once in a lifetime) experience.

I really do want to reflect on why I think it was worthwhile, but I’m having trouble doing that right now. So at the moment, I will simply describe what I actually did at TMC, diary style.


I left home early Tuesday morning and spent about 35 hours in taxis and airports and aeroplanes to arrive in Atlanta airport on Tuesday night (there’s some mind-boggling timezone maths in there if you want to figure it out). I consider this journey to be part of TMC because I spent a large amount of the time crocheting corals to be used in the triplet of morning sessions I would be running with with Megan Schmidt, called Mathematical Yarns. I also made a timetable and chose what sessions and activities I would go to across the five days I would be in Atlanta.

Tuesday night I was launched right into TMC when I shared a taxi to the hotel with Annie and Greta, with some pretty intense discussion of finite geometry along the way (not sure what the taxi driver thought of us!). When I checked into the hotel I didn’t even make it to my room for several hours because there was a crowd of people hanging out in the lobby/bar and I stayed to meet and talk with them (and to give out Tim Tams and Mint Slices).


On Wednesday, I went out on a tour of some of the sights of Atlanta with a small group of other TMC attendees: Megan, Henri and Andria. Megan, Henri and I went to the Civil Rights Museum and learned about the history of the struggle to gain rights for black people and immigrants, including the work of Dr Martin Luther King. This was extremely powerful. We wandered around the Olympic Park and ate in the CNN building food court. Megan and I went to the World of Coke, which was way more fascinating than I had imagined it would be. The innovations in marketing were quite amazing and mildly scary. Also worth noting this pro tip: thongs are not appropriate footwear for the sticky floor of the tasting room! Lastly Megan, Henri, Andria and I went to the Atlanta Aquarium, which was also way more fascinating than I imagined it would be, which is *really* saying something because I had pretty high standard of imagined fascination to meet! Whale sharks! Belugas! Sea turtles! Sea otters! Dolphins! Sea lions! Between and among all this, we discussed maths and teaching and how our lives intersect with that. Oh, and I also saw a bumblebee for the first time.

On Wednesday night, I went to the registration evening in one of the hotel ballrooms, and spent the night talking and playing games with even more people, and giving out Tim Tams to the crowd. Also my roommate Andrew arrived, though I didn’t see much of him until the next day!


On Thursday, the program of sessions began at the School which was hosting TMC. We started bright and early with the newbies session. I was late to this (because I had met Megan to do some last planning and smoothie breakfast along with Stephen), so all I heard about was how to use Twitter. I have a suspicion that before this there was advice about surviving your first TMC, which would have been nice, but it’s my own fault for being late. Then we had an opening session with the official welcome and those of us giving morning sessions each doing a one minute pitch to the crowd about what we were going to do. The one Megan and I were doing was called “Mathematical Yarns” and our pitch was literal because I threw corals into the crowd (though not far because corals are not known for their aerodynamic properties). Then off we went to actually give the first of our three sessions. How our morning sessions went deserves a whole blog post of its own, which I’ll do later. Spoiler: it was wonderful!

At lunch time, food trucks came to the School and we lined up in the sun to get something to eat together in the Dining Hall. Of course I also pulled out a game to play with people. After lunch we went into “My Favourites”. This is where anyone can go on the list to present for 5 to 10 mintues on something they like about a resource, a teaching strategy or anything at all really. Today I was particularly impressed by Sam‘s description of the “math joy bell” to make mathematical joy audible in his classroom.

Then we had the first keynote, which was by Grace Chen. It was about how teaching is political because it intersects with stories about who gets to do things or have things or be things. She was immensely brave and honest telling us about her own story and the stories of her parents and grandparents. I was partiularly struck by her comments on the power of listening to someone’s story how they tell it, rather than how others tell it to us.

The keynote was followed by two afternoon sessions. I went to Max and Malke‘s one about bodyscale maths learning and made some cool things out of rolled-up newspaper and sticky tape. I got talking to people after this session and so was a little late to Megan’s one on the patterns you can find in number spirals, though it was still great to be part of some investigations with the people there, like Christopher, who co-opted several of us into a collaborative effort to investigate quadratics.

At the end of the day there was supposed to be speed dating, but I just wasn’t sure I could cope with meeting any more people that day. In hindsight maybe I should have done it after all, based on the glowing reports I got from others about it. Oh well. It allowed me to have some more quiet conversations. Plus Christopher gave me a signed copy of Which One Doesn’t Belong!

On Thursday night, we all went to a restaurant for the newcomer’s dinner. I had lovely conversations with some great people. The hashtag controversy hikacked the discussion towards the end, but still it turned out ok. After dinner I went back to the hotel and spent most of the night playing games and singing. Later, I got back to my room and prepared for my My Favourites session tomorrow and had a longer chat with Andrew than I’m sure he really wanted before finally falling asleep much earlier in the morning than I had planned!


Early this morning, Andrew and I made a failed attempt to go to get bread (and breakfast) which made us late for My Favourites. One consolation was that I got to see two live squirrels as we drove along, which I have never seen before. We weren’t too late for me to do my My Favourites, which was all about SQWIGLES. It was nice to share something to the whole of TMC that was important to me, even though it was a lot less practiced than I had hoped. I was mildly surprised by how much people were interested in it. Of course I very much tried to listen to the rest of the My Favourites, but I was coming down off the thrill/terror of doing my own.

After this it was the second of our three Mathematical Yarns sessions. It was an island of calm and interesting discussion in the wild tumultuous sea that is the rest of TMC.

Lunch today was brought in by the school and so we all sat in the Dining Hall together to eat. There was a true community atmosphere to this that that I really loved. I used my time well in the lunch line by teaching Taylor how to crochet as we walked along, with a couple of other listeners on either side.

After lunch it was another My Favourites. I was particularly impressed with Pam‘s ideas for encouraging students and teachers using each of the five fingers with a meaning, especially the pinkie promise of I will be here with you. Graham Fletcher gave a keynote session about various ideas he learned from the MTBoS, including some interesting estimation challenges. The strongest idea I took from it was that everyone sees things differently – both different students and different teachers – and we need to realise that different is still smart and work together to learn.

After the keynote I did my afternoon session on One Hundred Factorial. I was deliberately late this time because I set up a One Hundred Factorial play space in the Dining Hall in order to bring my participants back to it when I’d finished presenting. I actually left this play space set up for the rest of the weekend and it was very gratifying to see it being used on and off throughout the next several days, even by people who hadn’t been at my session. But I’ll talk about that more in its own blog post.

And then there was one more afternoon session to go. I wandered around aimlessly and found my way into Bob and Scott‘s session about engaging with reluctant colleagues. I think this was something I really needed to be in, even though I had originally planned to go to a different session. (Still I was bummed to find I’d missed Kent’s session on base 8.) The biggest message I took from it was to start with your colleagues’ strengths and ask them to help you as a way of opening discussion.

I had the unexpected pleasure of having a group of people invite me out for dinner because they wanted to talk to me more about One Hundred Factorial. I don’t remember if we actually did talk about it in the end, but it was a wonderful time of fellowship and maths discussion nonetheless. Thanks to Jill, Kent, Jasmine, Ethan and Taylor I will always laugh when the mode is 1.

After dinner it was the second annual Trivia Night. I didn’t have a group pre-organised, so I just wandered in and a table of ladies flagged me down and said I had to join their group. It was loud loud fun. I particularly liked the round of books with numbers in the title. (Though again, the mode was 1.) When trivia was over, I wandered out into the bar/lobby to discover Malke and Max doing some Sierpinski Sponge and some pattern-machine/accordion music, which was fun to be part of. Later I pulled out Home in One Piece and Justin was unreasonably amazed by it. So much so that if I don’t blog about it eventually, I’m sure he’ll kill me. We had some pretty deep discussion along with Taylor until I finally had to call it and stagger back to bed.


Andrew and I were up bright and early again and this time we really did manage to get to Whole Foods, though not without missing a little of My Favourites. I did get to see Bob show us, which will attempt to guess your age from a photo and therefore provides a nice set of data to analyse.

Then we had our final Mathematical Yarns session and set up a gallery for everyone else at TMC to peruse our work. I was so grateful to Megan and all our participants for a really lovely time. (But as I said I’ll blog about that separately.)

At lunch time, I picked a table and slowly made tray after tray of Fairy Bread, which I gave to everyone in the whole room. People were sufficiently impressed with Australia’s favourite children’s party food. I did manage to squeeze in a moment for a quick sandwich, which I have to say was a welcome change to all the aeroplane and takeaway food I’d eaten so far!

After lunch it was My Favourites again. Joey began with a description of his Play With Your Math project, which reminded me a lot of One Hundred Factorial. I particularly liked the focus on making the puzzles very accessible with clear design and few words. I may have gotten nerdsniped by one of his puzzles and not paid too much attention to the other My Favourites sessions after Joey.

The final keynote came next, which was given by Carl Oliver and basically asked us to be brave and vulnerable and to just push send on our blog posts and tweets. There was also some fascinating analysis of when various people first used the hashtag #MTBoS. This made me reflect on what it was that brought me into the #MTBoS and I went trawling my old twitter to see how I had thought about it at the time. It was most interesting to reflect on my persistence with the #MTBoS despite varying levels of engagement at the beginning.

And then, there was one last afternoon session to go to, and I decided to go to Jonathan‘s one on Calculus for Algebra teachers. While I know a lot about calculus, I was interested in the sorts of things the others might like to know and how they responded to the explanations. I was so impressed with Jonthan’s gentle and respectful approach where he listened to the participants’ needs and responded to them gently and respectfully. I got into some quite deep discussions with Nik, which distracted the other participants, which I am very sorry for, everyone!

In the final afternoon timeslot, there were “Flex” sessions, which allowed people to do impromptu sessions on things that came up across the week. Malke and I decided to do one on Bodyscale Prime Climb because we had been itching to play it all week and hadn’t had the chance. To our surprise, eight other people arrived to join in too. We all had a great time walking on the numbers and noticing all sorts of patterns and relationships between the numbers. At the end of the session, I gave Malke my set of giant Prime Climb cards, which was completely worth it just for the expression on her face!

After the end of this session, I was invited out to tea again with a bigger crowd. After they waited for me to talk with my family because I was missing them very very much, we had a lovely conversation about all sorts of things, not least of which was the dangerousness of Australian spiders.

Finally, I made it back to the hotel for Games Night and played Home in One Piece and Flamingoes and Hedgehogs for many hours, plus had some wonderful talk with Max about the process of designing games, until the hotel staff chucked us out of the ballroom. Somewhere in the middle there, John asked me and a few other people (Jasmine, Edmund, Jim) to help make a video for his online Calculus students. We went up to Jim’s room and had a very pleasant conversation about derivatives and integration and problem-solving. It was an honour and a blast to be a part of this.


I got up bright and early and packed up all my stuff so that we could check out of the hotel before the final sessions. I got there nice and early to pack up the coral display and the One Hundred Factorial stuff only to discover people using them, so I just left them there until the last minute! Anyway, I had to go practice for the traditional TMC song, which I was roped into by Julie when she discovered I could sing on Wednesday night.

The final My Favourites was even more wonderful than the rest because so many people did one. I was particularly impressed with Glenn‘s about the power of small changes in the words you use, especially the words you use to name people, like “learner” versus “student”, and something other than “guys”.

After this, we did the TMC song, which was a bit of sustained ridiculousness celebrating the things that happened across the week, written by Sean, Julie, David and several others I shamefully never learned the name of! I really didn’t realise just how much of a public figure I had been at TMC until I saw just how many of the photos I or one of my games was in.

And suddenly the offical stuff was over! Lisa announced the date and location of TMC18 (Cleveland, July 21 2018) with the flair of a reality singing competition presenter, and that was it. There were a lot of hugs and “will you be coming next year”s. I got some lovely thank you cards from people, which made me cry completely unexpectedly. I gave away a lot of my stuff to people as souvenirs, including the giant Galaxy to Taylor, the giant Jigoku to Anna¬†and the Dragonistics cards to Bill (which I originally got from Nic Petty) . By this stage I had arranged¬†to wear all 12 of my home-made maths t-shirts, plus the TMC17 one Glenn brought me.

Yet it wasn’t quite over yet. Taylor and Emily invited me out to lunch and we had a most interesting discussion about the nature of MTBoS and TMC and welcomingness and joining in.

And then it really was over. I did try to connect to people who were still in the hotel or airport, but always contrived¬†to miss them. Finally, I was on the first of three planes home and 33 hours later was relieved¬†to run into my beautiful and longsuffering wife’s arms.


So that was what I did at TMC17. Looking back I crammed weeks of stuff into a few short days, and as I said to Megan, it was unreasonably awesome. It is going to take me months to process all of it, so keep an eye out for the various reflections to come. For now, thank you to everyone for welcoming me and making me part of your family for this week and beyond.

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I have a whole suite of maths t-shirts that I made myself. One of them simply has the number 65536 on it. It’s been getting a bit of attention over the past couple of weeks, so I thought I might write about it.

65536 is my favourite power of 2. More specifically, it’s 216, which means you can make it by starting at 1 and multiplying by 2 sixteen times. Even better…


But this cool stack of powers is not why it’s my favourite powet of 2. It’s my favourite power of two because of its connection to two very cool ideas in maths.

Firstly, 65536 is the last known power of 2 for which the next number is prime. It’s known that if a number one more than a power of 2 is a prime then it must be 2^(2^n)+1 for some n. The first five are all prime

  • 2^(2^0)) +1 = 3
  • 2^(2^1)) +1 = 5
  • 2^(2^2)) +1 = 17
  • 2^(2^3)) +1 = 257
  • 2^(2^4)) +1 = 65537

Fermat apparently conjectured in 1650 that they were all prime, which is why numbers of the form 2^(2^n)+1 are called “Fermat numbers” and if they’re prime they’re called “Fermat primes”. But so far, no more Fermat primes have been found. That is, every bigger number of the form 2^(2^n)+1 that we can calculate has been found to not be prime after all. Yet we haven’t been able to prove that there are definitely no more of them.

Isn’t that amazing? In close to 400 years we haven’t been able to find any more Fermat primes, but neither have we been convinced beyond a doubt that there aren’t any. I think it’s awesome that in maths there are things so simple that at the moment are unknown.

Secondly, 65536 is the only known power of 2 with no powers of two in order among its digits. Every other power of two where we have a list of most its digits, you can cross out some of the digits and have a power of two left behind. But we just don’t know if somewhere out there there’s a really big power of 2 that again lacks any smaller powers of two among its digits. Even stronger than this, every other power of two we’ve calculated has a 1, 2, 4, or 8 among its digits, but again we don’t know if somewhere out there in the distance there might be one that lacks these four digits.

What makes this surprising is that there is a perfectly good pattern to the final digits of the powers of 2. The last digit goes in the pattern 2, 4, 8, 6, 2, 4, 8, 6, … and the last two digits go in the pattern 04, 08, 16, 32, 64, 28, 56, 12, 24, 48, 96, 92, 84, 68, 36, 72, 44, 88, 76, 52, 04, … You’d think that the repeating pattern of the final digits might make it easy to tell what digits were in a power of 2, but it’s not nearly so easy.

What’s even more surprising is that the same concept for prime numbers is completely solved. If you cross out some of the digits of a prime number you might have a prime number left behind. For example, the prime number 16649 leaves behind the prime number 19 when you cross out the 664. So which prime numbers have no prime numbers among their digits? Well there’s exactly 26 of them and they are 2, 3, 5, 7, 11, 19, 41, 61, 89, 409, 449, 499, 881, 991, 6469, 6949, 9001, 9049, 9649, 9949, 60649, 666649, 946669, 60000049, 66000049, 66600049. That’s it. All of them. Every other prime number has at least one of these in order among its digits.

This is from a paper in 2000 from Jeffrey Shallit: “Minimal primes,” J. Recreational Math., 30:2 (1999–2000) 113–117. He talks about it here. Within a set of numbers, he calls the “minimial set” those ones with none of the others in the set in order among their digits. He references another author’s theorem which says that given any set of numbers, the minimal set within it must be finite.

Isn’t it amazing that the prime numbers with all their apparent randomness have allowed us to find their minimal set, but the powers of two with their obvious regularity haven’t?

So that’s why 65536 is my favourite power of 2. It represents to me some cool ideas, and more than that, it reminds me that maths is far from all done in the distant past, it’s got unanswered questions alive right now.

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The game of Prime Climb
Prime Climb is a wonderful game by Dan Finkel (aka @MathforLove), which you can find out more about here. The board is a path made of the numbers from 0 to 101, coloured by an ingenous and beautiful system. Each player has two pawns which they move around the board by […]

Posted in One Hundred Factorial
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Yesterday I talked about one of the common responses to people finding out I am a mathematician/maths teacher, that of saying, “I’m not a maths person.” The other common response I get is, “I don’t have a maths brain.” (John Rowe mentioned this in his comment on the previous post.)
This is how I reacted last […]

Posted in Being a good teacher, Thoughts about maths thinking | Tagged
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I am a mathematician and a maths teacher. Therefore it is an occupational hazard that any random person who finds out what my job is will respond with “I’m not a maths person.” The most frustrating people are my own students who I am trying to tell that my actual job is to help them […]

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Two books I’ve read recently have encouraged me to investigate my memories from childhood. In Tracy Zager’s “Becoming the Math Teacher You Wish You’d Had“, she urged me to think about my maths autobiography to see what influenced my current feelings about maths. In Stuart Brown’s “Play“, he urged me to think about my play […]

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Looking back at my blog over the past few months, I’ve done a lot of these “book reading” posts. I really did mean to do some more on other ideas, but I felt I had to get these thoughts out of the way first. So here’s another book reading post, this time about the book […]

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Here is another post about a book I’ve read recently. This time, I’m writing about the book “Making Number Talks Matter” by Cathy Humphreys and Ruth Parker.
In Cathy and Ruth’s words, number talks are “a brief daily practice where students mentally solve computation problems and talk about their strategies”. I had heard people talk about […]

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This post is about Tracy Zager’s most excellent book, Becoming the Math Teacher You Wish You’d Had. I actually finished reading it back in January, and I live-tweeted my reading as I went. The process culminated with this tweet:
I've just finished reading your #becomingmath book @TracyZager. This is the bit I liked:
— David Butler […]

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