It is no secret that Quarter the Cross is one of my favourite tasks. I’ve written about it twice before: as a Day 1 activity and in connection with Fraction Talks. The original source is apparently T. Dekker & N. Querelle, 2002,

Quarter the Cross promotes creative thinking, encourages students to find multiple ways of achieving the same goal, and compels students to justify their reasoning.

In the task, students verbally justify their visual representation of one quarter of the cross. This connects two representations together: verbal and visual. So far, students have been considering the cross as one whole.

We can incorporate a richer symbolic representation by considering the cross as composed of five smaller squares. (I was originally inspired by David Butler’s mention of this in a blog post.) In this second method, if one square is a unit, then what does it mean to ‘quarter the cross’? We can colour 1/4 of one unit and do this five times. This is represented symbolically as 5/4 = 5×(1/4). One way to represent this visually is shown below.

We can now explore different ways of expressing 5/4, and connect them to corresponding visual representations. For example, 5/4 = 1 + 1/4. Colour in one whole unit and a quarter of another. One way to represent this visually is shown below.

In the past, I’ve shown students a few different ways to symbolically express 5/4 and their corresponding visual representations. Occasionally I’ll pose one as a challenge. (‘Show me a visual representation of 5/4 = 2×(5/8)’.)

This year I wanted to try having students actively connect the representations. Inspired by Maureen Hegarty, I decided to have students try a card sort, with the following instructions:

- Cut out the cards.
- Match each visual representation to the symbolic representation. (You can write your own card, if needed.)
- When you are finished, try writing alternative expressions using different operations.

The pages are shown below. You can also download the entire PDF here.

What I found interesting is that because *all* representations show 5/4, students are focused on *which* symbolic representation best suits a particular cross. They also naturally debate whether more than one symbolic expression could be used. Next time I might include a few easier representations (students found some of these to be very challenging). In summary, I loved this way the discussion was mathematically rich, focused on connecting representations, and encouraged students to justify their reasoning to one another.

Responses from my #math1070 students:

- A koala because maths is interesting and unique.
- A spider because of the pattern organisation of their web.
- A cat because it has lots of layers.
- An ant because lots of small parts make up a big part.
- A zebra because it has many patterns and it is black and white with one solution. The patterns are often confusing but has many approaches: black with white stripes or white with black stripes.
- A chameleon cos once you think you’ve got it, it can disappear.
- A snail because it’s slow.
- An octopus because they’re calculated yet inspire intrigue.
- A starfish because there are many different aspects and ways to solve problems.
- A giraffe because it’s leagues above my level.
- A dog because while it has complex intelligence, its commands and responses are consistent.
- A leopard because it has a sophisticated pattern but also because it has an elegant mannerism.
- An eagle because it’s precise and adaptable.
- A turtle because you need to work through slowly and thoroughly.
- A tiger because it has a soft fur but could bite you.
- A monkey because its unpredictable.
- A zebra because there is an unknown amount
^{1}of stripes. - A bear because it looks scary but it’s actually soft and cuddly.
- A sloth because that’s how it makes you feel.
- A spider because we create webs of problems, solutions and challenges.
- A chameleon because just because you can’t see it doesn’t mean it’s not there.
- An ant. Maths underpins almost all concepts, it may be invisible at times but you can always find it if you look hard.
- A lion because a lot of people are scared of it until they know more about it and find the friendly side.
- A puppy because they are fun but then sometimes annoying.
- A penguin because they are fascinating.

[1] I’m transcribing student work. I know it’s ‘number’. Don’t @ me.

]]>As is customary, in the last few days there has been a flurry of reflective blog posts. (Eventually they’ll be linked here on the TMC website.) I’ve been reluctant to write one, and not just because it feels like I’ve been too busy since I got home to even breathe.^{2} Without wishing to sound overly sentimental, #TMC18 feels like a perfect memory that I’ll spoil if I try to capture it in words. I also don’t want to single out any moments for fear that I’ll forget to mention someone.

And, to be honest, I don’t go to conferences for the talks (as fabulous as they were). I go to conferences for the *people *and for the conversations that happen outside of the conference program. And my goodness, TMC has *so* many opportunities for those conversations, thanks to the careful design by the organisers. There is something special about meeting someone in real life after chatting online, sometime for years—the conversation naturally picks up from where it left off, irrespective of the medium. (It’s also fun to find out if someone sounds just like or nothing like you imagined!)

Perhaps more surprisingly, there is also something special about meeting someone at TMC who you *haven’t* yet interacted with online. Members of the #MTBoS community have a shared enthusiasm for maths in many forms, and so it is easy to strike up a conversation about teaching maths, playing with maths, making beautiful mathematical art, or any other variant. I knew the #TMC18 crowd would be my kind of people, I was just delighted at how quickly they felt like my kind of people.

During TMC18, there was some talk about what makes the MTBoS feel different to NCTM (the North American mathematics education organisation with more than 60,000 members), and TMC (limited to ~200 attendees) different to the NCTM Annual Conference (with thousands in attendance). There are, of course, some obvious differences (size, financial imperatives, charter, …) but I think it comes down to this. With a large organisation, you start by *paying*^{3 }to be a member and so it’s easy, subconsciously or otherwise, to automatically seek to get something in return. With the MTBoS, you start by *receiving—*good ideas, encouragement, friendship—and so you automatically seek to give back by contributing whatever you can. The generosity and warmth of the MTBoS community is remarkable.

In this kind of community it is easy to feel like an imposter. (This was a theme of Friday’s keynote.) To this I offer two thoughts. First, we tend to compare the inside of ourselves with the outside of others. Second, we all suck sometimes. With courage, sharing the failures also contributes to the community. Seeing that others are also imperfect helps to alleviate imposter syndrome. I wrote a little more about it here, and am reminded to use the hashtag #lessonfail more. (Because I know I’ve missed plenty of opportunities in the last year to do so!)

Social media is often derided as either frivolous or a hellscape. Mostly, it is what you make it^{4}. It’s fair to say that the #MTBoS has invigorated and enhanced my teaching in ways I’d not thought possible. So, if you are waiting for an invitation to join in, here it is: in the words of the TMC18 song, ‘Tweet at me now!’. I’d love to talk with you.

[1] I’m not going to get into the whole ‘teacher’ versus ‘educator’ debate except to say that not everyone in the #MTBoS is a teacher, and educator feels a little more inclusive. Or maybe we could just dispense with labels and say ‘people’.

[2] Our semester started while I was winging my way back to Adelaide. I didn’t even teach my first two classes. Shhh…

[3] Sometimes a lot. The NCTM conference? Well out of the price range of most everyday classroom teachers.

[4] Admittedly, it might require steady use of mute and block.

GASP!

I implore you to find a quiet moment away from the madness and ask yourself what it is that you want to achieve over the next semester. One of my ambitions — as it has been for many years — is to respect my students: to honour them as individuals, as learners, as contributors, and as mathematicians.

This is not a goal lightly undertaken. I fail at this regularly. There is always more I can learn and do, which is one reason I set the same goal every year. It is also a selfish goal; for me, building relationships with students is what turns bearable teaching experiences into enjoyable and deeply fulfilling ones.

I wrote about my five pillars for focusing on relationships a couple of years ago. They are by no means comprehensive but they’ve served me fairly well. You might also appreciate Art Duval’s recent post on the AMS Blog, ‘Kindness in the Mathematics Classroom‘ and Francis Su’s ageless blog post, ‘The Lesson of Grace in Teaching‘.

Some snippets:

*“GRACE: good things you didn’t earn or deserve, but you’re getting them anyway. … **Grace gives people dignity they don’t have to earn.” *(Su)

*“Grace amplifies the teacher-student relationship to one of greater trust in which a student* can thrive.” *(Su; * I’d also suggest ‘teacher’ here too.)

*“No single change I made was especially innovative or earthshaking, but the effect of each one was amplified by the others, and especially by my attitude.” *(Duval)

*“This is a good time to mention ‘kind’ is not the same as ‘nice’. Being kind does not mean just giving everyone A’s, or assigning less work, or never criticizing; it does mean listening to students, respecting their lives, and responding accordingly.” *(Duval)

*“I will continue to be kind to my students, though not because it will help them with mathematics, but because it is the right thing to do.” *(Duval; emphasis is mine.)

As Art Duval asks: What can you do today to show your students a little more kindness?

]]>This year, blogging about the books fell by the wayside. But, like 2016, I did tweet short character reviews with #read2017. I’ve copied them below, along with links to the books on Goodreads. The maths books are indicated with (*). Teaching-related books are indicated with a (^). But first, a few statistics and lists.

Out of the 62 books:

- 33 books with Australian authors
- 29 books by women
- 25 fiction books
- 13 books about political/current events
- 13 memoirs
- 12 books about teaching or maths (or teaching maths)
- 6 authors, multiple books (Georgia Blain: 4, Helen Garner: 4, Geraldine Brooks: 2, Peter H Johnston: 2, Graeme Simsion: 2, Elizabeth Strout: 2)
- 4 books aloud (We resumed this habit in June, with books 35, 38, 41, 60.)

(Note that there is some overlap with political and memoir.)

It’s not a ‘Best Of’ list, but twelve books that I either couldn’t put down or couldn’t stop thinking about afterwards. (There are probably more in this category on the long list.)

- Between a Wolf and a Dog, Georgia Blain
- Year of Wonders, Geraldine Brooks
- Light and Shadow, Mark Colvin
- Pushing the Limits, Kurt Fearnley
- Talking to My Country, Stan Grant
- Turtles All the Way Down, John Green
- The First Casualty: A Memoir from the Front Lines of the Global War on Journalism, Peter Greste
- Lab Girl: A Story of Trees, Science and Love, Hope Jahren
- Choice Words, Peter H Johnston
- Things to Make and Do in the Fourth Dimension, Matt Parker
- The Paper House, Anna Spargo-Ryan
- Anything is Possible, Elizabeth Strout

- A Pale View of Hills, Kazuo Ishiguro

One pronoun that I initially thought was a typo, and then it hit me. Wow. (Could have joined the previous list for this reason.)

*My Name is Lucy Barton, Elizabeth Strout*.

#read2017: A story of a mother who loves her daughter imperfectly. And of a daughter who accepts. It’s what isn’t said that lingers after.*Mrs Dalloway, Virginia Woolf.*#read2017: Success! To be read patiently and perhaps as unhurriedly as the story unfolds.

*Reaper Man, Terry Pratchett.*#read2017: Part of my slow journey to re-read the Discworld novels in their published order. Not a fav, but still some great writing.

** Can You Solve My Problems?, Alex Bellos.*#read2017: Brimming with an assortment of puzzles and their history. Filled with old and new favourites. A rich source for my classes.

*Take it Easy, Danny Allen, Phil Cummings.*

#read2017: Convincing portrayal of an Aussie country kid, their experiences + their fears of moving to the big smoke. Surprisingly touching.*The Phantom Tollbooth, Norton Juster.*

#read2017: A hero’s quest through a strange land. Ultimately a tale about learning. Delightful wordplay, wisdom and wit. Loved it.*The Paper House, Anna Spargo-Ryan.*#read2017: The shape of new and old grief, loss and mental illness. The support of loved ones. Exquisite. A stunning debut novel.

*True Stories: Selected Non-Fiction, Helen Garner.*#read2017: More Garner. (Nearly through her oeuvre!) Her ability to illuminate what at first glance appears simple is magnificent.

*Do Not Say We Have Nothing, Madeleine Thien.*#read2017: A intricate generational tale of the Cultural Revolution and its effect on creative expression, music, free thoughts and love.

** Yes, But Why? Teaching for Understanding in Mathematics, Ed Southall.*#read2017: A valuable resource for beginning maths teachers (and I learned a thing or two myself!).

*Honour & Other People’s Children: Two Stories, Helen Garner.*#read2017: One day I’ll run out of Garner to read. One day.

** Things to Make and Do in the Fourth Dimension, Matt Parker.*#read2017: A recreational maths book with a difference.

~~@~~standupmaths is an unashamed nerd and his delight+playfulness leap off every page.** Math on the Move: Engaging Students in Whole Body Learning, Malke Rosenfeld.*#read2017: Some great ideas for using the moving body as a tool to make sense of and connections between mathematical ideas.

*^ Choice Words: How our Language Affects Children’s Learning, Peter H Johnston.*#read2017: How our choice of words helps learners grow + develop agency. Set in a literacy context with many parallels to learning maths.

*Wrong about Japan, Peter Carey.*#read2017: A manga-anime pilgrimage to find ‘Real Japan’ in the enigmatic mix of old + new. Enjoyable read but felt Carey missed the point.

*The Good Girl Stripped Bare, Tracey Spicer.*#read2017: A ‘femoir’ from one of Australia’s best-known journalists. Spicer reveals gross inequities. Brutal. Insightful. Funny. Powerful.

*^ Play: How It Shapes the Brain, Opens the Imagination, and Invigorates the Soul, Stuart Brown, Christopher Vaughan.*#read2017: Play as an integral part of how we learn, love and live.

*^ Embedded Formative Assessment — practical strategies and tools for K-12 teachers, Dylan Wiliam.*#read2017: Provoked deep thinking on how I can improve my teaching. Full of practical ideas + concise integration of evidence-based research.

*The Subtle Art of Not Giving a F*ck: A Counterintuitive Approach to Living a Good Life, Mark Manson.*#read2017: After tolerating a blustery and vulgar start targeted at a specific audience, the message is strong and worth considering.

*Between a Wolf and a Dog, Georgia Blain.*#read2017: Slow to draw me in but I was eventually enveloped. So exquisitely beautiful. Sorrowful, but still joyful and hopeful. Affecting.

** A Mathematician’s Apology, GH Hardy.*#read2017: I really wanted to like this but, on the whole, I found Hardy to be smug with an elitist view of ‘real’ maths.

*The Secret Lives of Men, Georgia Blain.*#read2017: Short stories done right — dipping for a moment into the lives of others.

*Anything is Possible, Elizabeth Strout.*#read2017: Unexpectedly for me, interwoven short stories linked to earlier book. I couldn’t put this down. Skillful character development.

*Stoner, John Williams.*#read2017: Hard to pinpoint why I liked this. A simple story of an ordinary life filled with setbacks and sadness, and yet still uplifting.

*Astonishing the Gods, Ben Okri.*#read2017: Mystical, with many koan-like statements. I’m not sure I understood anything. Strangely compelling.

*Unpolished Gem, Alice Pung.*#read2017: A self-deprecating coming of age memoir. Laugh-out-loud start which became rambling + introspective. Strong, vivid storytelling.

*^ Opening Minds, Using Language to Change Lives, Peter H Johnston.*#read2017: A strong follow-on from the powerful ‘Choice Words’. Eye-opening examples of dialogic classrooms. Inspirational.

*The Feel of Steel, Helen Garner.*#read2017: The clarity of her writing is incredible. Another magnificent collection of essays.

*Lab Girl: A Story of Trees, Science and Love, Hope Jahren.*#read2017: An intimate and fierce memoir of science, plants and academia. Her writing is enlivening and beautiful.

*True Girt (The Unauthorised History of Australia #2), David Hunt.*#read2017: The way Australian history should be taught.

~~@~~DavidHuntGirt makes you laugh, squirm, and think. Can’t wait for Vol 3.*People of the Book, Geraldine Brooks.*#read2017: A good holiday read. I found Hanna implausible, but the interleaved stories of the People of the Book were enthralling.

*Random Life, Judy Horacek.*#read2017: The new collection from Judy Horacek. A perfect mix of irreverence, observation and sharp social commentary.

** The Joy of SET: The Many Mathematical Dimensions of a Seemingly Simple Card Game, Liz McMahon, Gary Gordon, Hannah Gordon, Rebecca Gordon.*#read2017: A comprehensive and conversational look at SET. From simple to deep maths with accompanying exercises and projects. Recommended.

*Why Men Are Necessary and More News From Nowhere, Richard Glover.*#read2017: A witty collection of columns on ordinary family life. Breezy read.

*Depends What You Mean by Extremist, John Safran.*#read2017: When extreme becomes mainstream. The far left + far right have more in common than you might think. Written in true Safran style.

*Booze Territory, Anna Krien.*#read2017: I’ve tried and failed to find another word to describe this. A truly sobering read. The statistics are shocking.

*Option B: Facing Adversity, Building Resilience and Finding Joy, Sheryl Sandberg and Adam Grant.*#read2017: A hopeful and practical book about the kind of growth that can come out of deep grief and loss, with lessons for all.

*One Halal of a Story, Sam Dastyari.*#read2017: Part memoir, part manifesto. A glimpse of the many sides of the ebullient politician. A little uneven but a good read.

*The Museum of Words: a Memoir of Language, Writing, and Mortality, Georgia Blain.*#read2017: Loosely chronicles the devastating final chapter of Blain’s life. An important voice in Aust writing taken far too soon. Vale.

*What Happened, Hillary Rodham Clinton.*#read2017: Plenty of hubris and anger, but also deep hurt and bewilderment (and some snark!). A sorrowful reminder of what could’ve been.

*Pushing the Limits, Kurt Fearnley.*#read2017: ‘Kurt Fearnley gives life a good name.’ Lots to learn from his infectious attitude and gritty determination. Great read.

*Moral Panic 101: Equality, Acceptance and the Safe Schools Scandal (Quarterly Essay #67), Benjamin Law.*#read2017: Measured and clear writing. A timely and important read. Strongly recommended.

*The Rosie Project, Graeme Simsion.*#read2017: Wasn’t sure whether to like or hate this. In the end I was quite fond of it. A somewhat more satisfying way to spend the arvo.

*What Alice Forgot, Liane Moriarty.*#read2017: Found the first half painful and repetitive with mostly vapid characters. Second half picked up with some brighter moments.

*Hateship, Friendship, Courtship, Loveship, Marriage, Alice Munro.*#read2017: Somewhat embarrassed that this is the first I’ve read of Munro (finally!). So much substance in her short stories.

*Turtles All The Way Down, John Green.*#read2017: Absorbing. Glad to see mental health tackled deftly and realistically. I love how it’s never quite perfect in Green’s books.

*Births Deaths Marriages, Georgia Blain.*#read2017: Saddened to have found Blain’s writing at the very end of her life. Slowly making my way through her work.

*The Rosie Effect, Graeme Simsion.*

#read2017: A fun follow-up to the first book.** How Not to Be Wrong: The Hidden Maths of Everyday Life, Jordan Ellenberg.*#read2017: mpeccably researched. Meanders in a way that struggled to hold my attention. A shame; I wanted to like this more than I did.

*Talking to my Country, Stan Grant.*#read2017: A powerfully lyrical reflection on the past and present injustices towards Aboriginal Australians. A must-read.

*Light and Shadow: Memoirs of a Spy’s Son, Mark Colvin.*Can’t believe I missed writing a #read2017 review of this. Exceptional book by one of Australia’s finest journalists.

*A Pale View of Hills, Kazuo Ishiguro.*#read2017: Foreboding. The last few pages make you want to flip to the start and immediately re-read.

*Year of Wonders, Geraldine Brooks.*#read2017: I love the occasional historical novel. This was well-paced and engaging.

*Rogue Nation, Royce Kurmelovs.*#read2017: Making sense of the disconnect between ’ordinary’ Australians and mainstream politics to help explain the rise of One Nation + co.

*Autumn, Ali Smith.*#read2017: Interweaves the story of an extraordinary trans-generational friendship with post-Brexit Britain, and more. Wish I understood more of the delicate construction, but that’s not a criticism of Smith.

*The Enchanted Wood, Enid Blyton.*#read2017: The comfort of a childhood favourite. Given to me on my 7th birthday. First published in 1939 and still a delight.

*Why Weren’t We Told?, Henry Reynolds.*#read2017: Filling in the gaps of what every Australian should know about white Australia’s black history. This is a confronting but important book.

*Black + White: Race, Politics and Changing Australia, Nyunggai Warren Mundine.*#read2017: An absorbing memoir from one of Australia’s most opinionated and, at times, controversial Aboriginal leaders. A good blend of personal and political.

** The Magic of Math: Solving for X and Figuring Out Why, Arthur Benjamin.*#read2017: A conversational look at some key ideas in maths: algebra, geometry, trigonometry, e, i, calculus, infinity, …. ‘Magic’ is overstated, but the book provides a useful introduction.

*The First Casualty: A Memoir from the Front Lines of the Global War on Journalism, Peter Greste.*#read2017: Combines Greste’s account of 400 days in an Egyptian prison on bogus charges with his insightful analysis of the attack on freedom of the press. An excellent book.

*Community: Salad Recipes from Arthur Street Kitchen, Hetty McKinnon.*#read2017: Read cover-to-cover yesterday — twice! — then started cooking. All hearty salads. Each showcases a core vegetable. First two dishes were amazing. Lots of hours of good eating here.

*The Last Days of Chez Nous and Two Friends**, Helen Garner.*#read2017: Screenplays are quite a different story-telling device, driven mainly by dialogue. Garner expertly draws out the personalities of her characters.

That feeling when you try something new for class and it only kind of works. https://t.co/t5u8Ch8sNh

—

Amie Albrecht (@nomad_penguin) October 03, 2017

We’ve had a two week break, culminating with a public holiday Monday. In our first class back, I wanted to add some supporting activities for the difficulties we were having in our last class. Let’s just say that I’m not sure that what we did today helped! Luckily, I have another opportunity with the other half of the class tomorrow. I thought I’d write a blog post to help organise my thoughts (and provide some kind of detailed post-class lesson plan), and then press send so that others can have a peek into my thought process (for what it’s worth!), and perhaps offer suggestions. If you want to focus on what didn’t work so well, skip right to the last section.

We’ve been developing mathematical reasoning using the key elements of *noticing patterns, conjecturing and generalising, crafting claims, *and* exploring why (proving). *(I’ve leant quite heavily in places on Chapter 10 of Tracy Zager’s *Becoming the Math Teacher You Wish You’d Had* for what follows.)

In our last class (before the break), we were working on claims. For each claim, I asked students to think about *whether* the statement is true, and, if so, *when* the statement is true. To help with this, I asked students to find counterexamples (examples where the claim doesn’t hold true). Then, they needed to revise the conditions of the claim to make it always true.

For example: **‘If you subtract a number from 5, your answer will be less than or equal to 5.’ **This is definitely not always true. A counterexample is -7; if we subtract -7 from 5, the answer is 12, which is *not* less than or equal to 5. With some more investigation we can see that this statement is always true if we revise the conditions on the claim to be ‘for all numbers greater than or equal to zero’.

Here are some of the others that students worked on:

- A rectangle is a square.
- The sum of four even numbers is divisible by 4.
- One-third is one of three pieces.
- An even number divided by an even number is an even number.
- Division always makes a number smaller.
- The square of a number is greater than or equal to the number.
- When you add two numbers you get the same result as when you multiply them.

These kinds of statements are used in the instructional routine *Always, Sometimes, Never. *For more on this, I thoroughly recommend Chapter 10 of Tracy’s book. If you don’t have access to the book, you can still check out the links at the companion website.

Of the wide variety of statements, students struggled (perhaps most) with the nature of statements 2 and 4. The discussion around ‘A rectangle is a square’ led me to think that our experience with classifying objects (in particular, shapes) and identifying properties they had in common needed to be richer. Which led me to the plans for today’s class …

If I’m honest, I kind of skirt around geometry, in particular the properties of shapes — too many definitions that differ depending on which part of the world you are from. But shapes offer lots of opportunities for classifying and characterising, and for nailing down some precise terminology. So, I started by sharing in a matter-of-fact way that I was a little uneasy with what we were about to do but that, as teachers, we can’t simply avoid a topic.

The progression that follows includes ideas from Pauline Carter and Kath Ireland, as well as from Van de Walle et al. (2018) *Teaching Student-Centred Mathematics* Vol III (Grades 6-8). Underlying all of this are the Van Hiele levels of geometric thought. I won’t elaborate, but a Google search should help — and Christopher Danielson’s excellent Teacher Guide for his *Which One Doesn’t Belong?* book gave me a useful way to describe the different levels to my pre-service teachers at the end of the session.

I gave this handout and asked students to name as many of the shapes as they could. (I chose not to pre-cut shapes for the students so that orientation was initially fixed.) The purpose of talking about the shapes was to draw out some terminology and some misconceptions. For example, we talked about ‘diamonds’ and the orientation of shapes. We talked about concave and convex. But, I refused to arbitrate on the precise names of shapes at this stage — that was for them to determine!

We kept this page up throughout the workshop, and added notation for parallel sides, congruent sides, right angles,and congruent angles.

Next we worked on dividing our quadrilaterals into two groups. The items in one group need to have something in common. That property must not present in the items in the other group. An example is below. Can you work it out?

(The items on the left do not have interior right angles. The items on the right do.) I asked students to make up puzzles for each other. This seemed to go well. In retrospect, I should have had a few more examples up my sleeve. I could have also asked them to write theirs down.

We moved into a few rounds of ‘Guess Who?’ with shapes. We reviewed the board game (in the photo above). Key points: each player selects and keeps hidden a mystery person. The other player asks ‘Yes/No’ questions to determine who the mystery person is. We talked about what makes good questions (for example, properties that divide the remaining people as evenly as possible into two groups).

When playing with quadrilaterals, we used sheet protectors (see the photo) to eliminate candidates. This ‘open play’ was also useful in checking each other’s deductions, and in making the game a little more collaborative. This felt successful, though I’d like to listen in a little more closely to some of the conversations.

To hone in on the nested properties of quadrilaterals, I gave them these cards and asked which properties each of their quadrilaterals had.

Students were able to sort them with relative ease, and so we moved onto the next task.

I handed out the following Euler diagram (A3 sized). The task was to put the property cards and the shapes in the relevant regions. We talked about the seven regions, and noted the overlap of the red and the purple. We discussed how an object in the very middle of the diagram has *all* of the properties and an object at the edge of the diagram (outside the blue ring) has *non**e *of the properties.

Some students unintentionally ignored the property cards and made up their own. Few realised that the property cards were numbered in a ‘useful’ way, but I’m not worried about that.

Each region now contains a class of shapes. What are the names of each class of shapes?

This seemed to come far more quickly. We also connected our nested diagram to a ‘quadrilateral family tree’ (credit to David Butler for his post on this).

Each class of shape inherits the properties of those above it. What I missed here (which is important for what comes next) was nailing down the description of the shapes. What differentiates (say) a kite from a rhombus? As David Butler helped me think about later, what distinguishing features do each class of shapes have?

And finally we come to the troubles … back to claims that are ‘always, sometimes, never’ true.

- A rectangle is a rhombus.
- A rhombus is a rectangle.
- A square is a rectangle.
- A rectangle is a square.
- A parallelogram is a quadrilateral.
- A trapezium is a parallelogram.
- An equilateral quadrilateral is a square.
- A parallelogram is a rectangle.
- An equilateral parallelogram is equiangular.
- A rhombus is a square.
- A square is a rhombus.
- A kite is a square.
- A square is a kite.

I found that it helped to say each statement three times, as follows:

- A rectangle is
**always**a rhombus. - A rectangle is
**sometimes**a rhombus. - A rectangle is
**never**a rhombus.

Only one of these can be true. Which is it? If it is *sometimes* true, then let’s think of examples (and later, properties) where it is true and examples where it is not.

Our problems were with the ‘sometimes’ statements. Consider the statement: ‘A trapezium is a parallelogram.’ As David Butler put it, the answer to a question like ‘When is a trapezium a parallelogram?’ is ‘When it’s a parallelogram’. Not very satisfying!

And ‘not very satisfied’ is a good description of how some students felt. ‘Magic’ is probably what others thought; I think they viewed it as redefining what it meant to be ‘a trapezium’ or ‘a parallelogram’.

Now I wonder if it is helpful instead to start considering ‘sometimes’ statements by first leaning heavily on their definitions and then looking for distinguishing features of **isn’t **and **is**.

For example, consider the statement ‘A rectangle is a rhombus’, which we agree is sometimes true. Let’s review our definitions:

- A rectangle is has two pairs of parallel sides and angles that are 90 degrees.
- A rhombus has two pairs of parallel sides and all sides are the same length.

So, when **isn’t** a rectangle a rhombus? When the pairs of opposite sides of the rectangle are of different lengths. When **is** a rectangle a rhombus? When all sides are of the same length. If we add this condition to our definition of a rectangle then, to be a rhombus, our rectangle must have two pairs of parallel sides, right angles, and all sides of equal length. (Note that I’ve just described a square, so a rectangle is a rhombus when the rectangle is a square.)

I’ve written about 1700 words to discover that most of what happened today seems okay, but could be tweaked. So in tomorrow’s repeat of this workshop, I’ll include definitions of these quadrilaterals and I’ll reframe the ‘sometimes’ statements as I outlined above. Also, when naming shapes, I’ll use the prompt ‘What is the most precise name that we can give this shape?’ Finally, I’ll try to articulate more clearly the role of these activities in ‘developing (our) mathematical thinking’ — the whole point of the course.

I’ll report back with an addendum to this post to see whether any of these ideas help it go smoother.

]]>This morning I woke up to a tweet from Bryn Humberstone.

Today’s date is a Pythagorean triple (written as 15/8/17 or 8/15/17). To ponder: How long until the next one? #iteachmath #MTBoS

— Bryn Humberstone (@brynhumberstone) August 14, 2017

I love using topical moments — like today’s date — as a lesson starter. While I understand the fatigue that days like Pi Day (14 March) can cause, I think it can be a great opportunity to briefly introduce some maths that might not be in the lesson plan. (Sara Van Der Werf has a recent and positive blog post on math(s) holidays that you should read.) Plus, I like my students to witness the joy I have in mathsy moments — even if they think I’m weird!

After the tweet, I opened my email to this image from brilliant.org and the 100 Day Summer Challenge.

To top it off, my plan for today’s #math1070 class was to review last week’s task: choose one of the four images below and tell me what you Notice and Wonder about it. You might be able to see how my lesson starter was going to unfold!

I started with today’s date: 15/8/17. I said it was an interesting mathematical date. Could anyone work out why? I then showed this image. We had a quick refresher of the Pythagorean theorem. This image already throws up some interesting questions: is it possible to always colour the squares on the hypotenuse with a complete square of one colour and a double thickness ‘half-border’ of the other?

Next, I showed the image from brilliant.org. I was careful to draw the squares on the sides of the 3-4-5 triangle, and connect Pythagoras’ theorem to the area of the squares that can be drawn on each side of the triangle. Then I asked them what they noticed and wondered. What do you notice and wonder? We had a little discussion based on this prompt, including generalising the side lengths.

Finally, I showed the image from their weekly task. Those who chose this image as their prompt generated a lot of noticings and wonderings; they are at the bottom of this post. For now, I directed their attention to the number of isosceles trapeziums making up the larger trapeziums on each side. The counts are 9, 16 and 25. What do you wonder now?

We explored whether this was a 3-4-5 triangle (it is) using the long base of the smaller trapezium as one unit. The overwhelming question in the room was why Pythagoras’ Theorem works for non-square shapes. I acknowledged that it works for similar shapes, and left it there. (By now, the lesson starter was threatening to take over the lesson!)

I finished with a quote from Chapter 7 of Tracy Zager’s book, from Peter Hilton: ** “Computation involves going from a question to an answer. Mathematics involves going from an answer to a question.” **I hope that these images and initial discussion prompted many more questions that they could go on to explore in their own time.

Incidentally, it was another serendipitous find this morning, shared on Twitter by John Golden, that sharply reminded me of this quote. John shared this blog post by Pat Ciula. It uses the same trapezium image to launch a complete different exploration. I love this post; do check it out for yourself.

This wasn’t really what I had planned for today’s class, but when these moments come along, you need to grab them!

These are from last week’s task, submitted in advance of today’s class. I’ve made some minor edits.

**I notice (that) …**

- Different colours are used: red and pink.
- A triangle has been made in the middle.
- The triangle formed is a right-angled triangle.
- All three shapes are the same.
- All three shapes are made up of smaller versions of the overall shape.
- The shapes are created by wooden blocks all put together.
- Each individual piece is a trapezium.
- There are three larger trapeziums made out of individual tessellating trapeziums.
- That the four-sided polygons go in an anti-clockwise movement ranging from smallest to largest.
- The blocks are arranged in different ways in each trapezium.
- The three trapezoids are similar trapezoids.
- There are nine sides forming the outer perimeter of the shape.
- In the centre of the three larger trapezium shapes, the points of the bases of these shapes make a right-angled triangle.
- If you were to move the largest trapezium shape so it was opposite where it is now, the shape in the middle made by the sides of the trapeziums would still be a triangle.
- Each tile is the same shape – trapezoids. In each tile, three sides (top and the two edges) are equal, and the base (the longest side) is twice the length as the other three sides.
- If the length of the base of the tile is one unit, then the length of the middle triangle’s three sides are 3 units, 4 units and 5 units. The layers of the tiles within the trapezoids are also 3, 4, and 5.
- Small trapezium is made out of 9 smaller trapeziums. Bottom trapezium is made out of 16 smaller trapeziums. Larger trapezium is made out of 25 smaller trapeziums.
- The number of tiles on each side of the triangle follows Pythagoras Theorem: 3
^{2}+ 4^{2}= 5^{2}, which is 9 + 16 = 25. - The image includes an aspect of scale.
- The small pink polygons shapes make a negative internal outline of a right-angled scalene triangle.
- That the fewer pink polygons on each section related directly to the length of the sides of the internal outline of the triangle, that is, fewer pink polygons, shorter sides.
- I noticed that the depth and width of the outer shapes was linked, that is, left side shape is 3 pieces deep and 6 ‘lengths’ wide and consists of 9 pieces, bottom shape is 4 pieces deep and 8 ‘lengths’ wide and consists of 16 pieces and finally upper right shape is 5 pieces deep and 10 ‘lengths’ wide and consists of 25 pieces.

**I wonder …**

- Is the angle exactly 90 degrees?
- How many trapeziums there are?
- How long did it take to make?
- Why are there two different colours?
- How many blocks would it take to fill the right-angled triangle?
- Are sides ‘a’ and ‘b’ equal to ‘c’ in length?
- If the bottom section would fill the triangle perfectly?
- Are the small trapeziums arranged in a particular way for a reason? Or is it random to fit the desired shape?
- Do the larger shapes go down in size proportionally and is there a specific number to make up each shape?
- Would the area of the trapeziums be bigger or smaller than the area of the triangle in the middle?
- What the two remaining internal angles would be (excluding the right-angle).
- Why using trapezoids to represent Pythagoras Theorem also works? Is it because the ratio of the area of the trapezoid to the square is fixed?
- I wonder if the sizes of the trapezium differed, what the effect would be on the right-angled triangle on the middle? Is it possible to make an isosceles or equilateral triangle in the centre?
- How many triangles can each Isosceles Trapezoid be split into?
- How many other shapes are present in the Isosceles Trapezoids? For example – I have already noticed that triangles can be found — what other shapes are there?
- What is the reasoning for this image? Why was it created?
- How I could start a conversation with the students regarding this shape and what they may come back with?

Skyscrapers are one of my favourite logic puzzles. They are a Japanese creation, introduced at the first World Puzzle Championship^{1} in 1992.

Skyscrapers are a type of Latin Square puzzle. A Latin Square in an *n × n *grid filled with *n *different symbols, each occurring exactly once in each row and exactly once in each column. (Sudoku is another type of Latin Square puzzle).

In a Skyscraper puzzle the objective is to place a skyscraper in each square, with heights between 1 and *n*, so that no two skyscrapers in a row or column have the same height. The numbers (clues) on the outside of the grid tell us how many skyscrapers are visible from that position. (I like to imagine that I’m standing on that number and along the street that is the row/column.) Shorter skyscrapers aren’t visible behind taller ones.

We use logical deductions to solve the puzzle. For example, in the puzzle below, the clue ‘4’ tells us that the skyscrapers must appear in ascending height order in that row/column. Similarly, the clue ‘1’ tells us that the tallest skyscraper must be adjacent to the clue. That leads us to the partially-filled grid below. If you want to solve it yourself, the solution is at the bottom of this post. You can also play them online at Brain Bashers.

It is fairly easy to turn a skyscraper puzzle into a hands-on activity — just choose objects of different heights. Teachers often use linking cubes. You can also be more creative; at David Butler‘s One Hundred Factorial gathering at the University of Adelaide in May, we experimented with video cassettes (remember them?) and cups of different sizes.

A while back, I wanted to make several hands-on sets for 5 × 5 grids to use with groups of school students. They needed to be cheap, lightweight, compact and portable. So, I made paper cylinders that nestle inside each other. You can download and print the skyscraper cylinders. The tabs are meant to show where to overlap and tape. You can use them with these puzzles (print A3 size): Puzzle 1, Puzzle 2, Puzzle 3, Puzzle 4, Puzzle 5.

My plan for MATH 1070 was curtailed by our short week (Week 3). I had planned the activity with these goals:

- Form visibly random groups with four students so that students could meet a few more classmates.
- Work collaboratively towards a common goal (and contrast this with the competitive nature of Prime Climb last week.)
- Practice claims and warrants as part of the focus on Maths Disputes: ‘I think <claim: this number goes here> because <warrant: my reason>.’

There are a variety of reasons to use skyscrapers in the classroom; you might like to read these posts by teachers: Mary Bourassa, Mark Chubb, Sarah Carter. Any activity introduced into the classroom should be intentional. You might like to think about these dot points. Mark has a fuller list in his blog post.

- If giving these puzzles to individual students is different than to groups of students.
- If a physical model is different than a pen-and-paper version.
- If you’ll use it as part of a lesson or as a ‘time filler’.
- What you’ll do if students give up easily.

If you give them a go, let me know what you think!

[1] Source: www.gmpuzzles.com/blog/skyscrapers-rules-and-info

]]>This year, I wanted to turn this into a student-driven rather than teacher-led activity. I also wanted to create opportunities for even deeper mathematical exploration. In this post I’ll briefly outline how this unfolded.

A goal for this year is to do better at helping students be good mathematical colleagues. I’ve been making heavy use of Chapter 12 of Tracy Zager’s book to guide this endeavour. I also wanted to begin using visibly random groups to build our mathematical community. However, I’m mindful it can be confronting to work with complete strangers, so I had students pair up first with someone they knew. (Pairing students up was also to serve another purpose that will become apparent later.)

I randomised the pairs by handing out cards from another of Dan’s games, the delightful Tiny Polka Dots. I deliberately used cards with different representations of 1, 2, 7 and 9 (with 8 as a back-up). Pairs with a different representation of the same number formed a group, one at each table. I gave them the rest of the cards for their number, and we did a quick ‘Notice and Wonder’ on the different representations.

Rather than show students the Prime Climb hundreds chart arranged in columns of 10 (the image earlier in this post), I wanted to foster noticing and wondering by having them construct the charts *themselves* with mini-cards of the numbers: physically handling, examining, ordering and organising. The (roughly) six students at each table arranged the cards into charts with the number of columns corresponding to their Tiny Polka Dots card (with 1 and 2 corresponding to 11 and 12, respectively). Click on the images below to make them bigger. What do *you* notice? What do *you *wonder?

To add impetus to the discussion, I relayed that part of the weekly task (contributing towards their course grade) was to individually write a forum post with at least five things that they noticed and wondered about their charts. It was heartening to see students collaboratively generating lists of their many observations.

The final part of this activity was to have students play Prime Climb in a way that compels them to articulate their thinking. One observation from playing Prime Climb at One Hundred Factorial with David Butler was that playing in pairs had an initially unexpected benefit of making thinking visible. (Side note: Bodyscale Prime Climb—where the numbers are A4 sized and the player is the pawn—is the most wonderful way to experience this game. Walking the board gives a different perspective to the relationship between numbers. You also need at times to shout your thinking across the board to your partner, which really makes thinking visible! Read more in David’s blog post.) We used a modification of the rules devised by David. This is the printable version that I wrote based on the rules that come with the game.

Collaboration and competition reached ear-splitting levels, so much so that we were asked to quieten down from those in the classrooms around us. (I sheepishly and profusely apologised later.) This was definitely a good start to building community.

While the in-class activities concluded with playing Prime Climb, there is so much more exploration provided by the cards. David, and later I, explored the different patterns that emerge when the cards are arranged in different ways. Others chimed in on Twitter with ideas; click the links above or look at the images below for more. The Prime Climb colouring is such a rich medium!

]]>I have always loved maths, but the reasons why have changed dramatically over time.

This is my Year 1 work. It reminds me about what I thought it meant to be good at maths: lots of ticks on neat work, especially if it was done quickly.

This attitude was reinforced by my report cards in primary school. A typical one looks like this. Note the focus on speed and accuracy. I loved maths because I was good at it.

Our Year 2 classroom had a corner filled with self-directed puzzle-type problems. If students finished their work early, they could go to the puzzle corner. I recall spending a lot of time there (my report says I was put in an extension group). Looking back, I’m sad that not every student had the same opportunities to engage with these richer, stimulating problems.

Outside of school, I loved doing and making up puzzles. I looked for patterns everywhere. I was always thinking about different ways to count, to organise, and to get things done more quickly. Growing up on a rural property, I had a lot of chores and time to think. For example, I’d think about how many buckets of oranges I could pick in an hour, how long it would take us to fill an orange picking bin, the different ways I could climb the rungs of the ladder, and so on. But I didn’t connect these ideas to maths.

Most of my school maths memories involve doing exercise after exercise from the textbook, but that was fine by me because I could put a self-satisfied tick next to each neatly done problem (after checking the answer in the back of the book!). I remember one high-school maths project to work out the most efficient way to wrap a Kit Kat in foil. It stands out in my memory because it was so different to the rest of maths class.

There were gaps in my knowledge along the way that I tried to cover up. I missed a month of Year 4 due to illness, and a substantial chunk of that time was devoted to fractions. When I got to algebraic fractions in later years, I would furtively use my calculator on simple examples to see if I could work out the right ‘rule’. Now I congratulate myself on having the sense to work it out for myself by generalising from specific examples. In Year 12 I felt embarrassed for using straws and Blu Tack to make visualisations of 3D coordinate geometry; everyone else could do it in their heads. Now I’m proud that I found a tool to help me make sense of the maths.

In Year 12 I hit a big obstacle. All my grades went downhill, including in maths. My maths report card says that I was ‘prone to panic attacks when working against a time constraint’. I don’t remember that, although I do remember crying (which I almost never do) in my maths teacher’s office and thinking that I didn’t know anything. I realise now that much of my maths schooling was about memorisation but not about understanding, and that it had caught up with me by Year 12.

Despite my mostly mediocre grades (I got a D in physics!), I did okay and was offered several university places. My love of the English language drew me to careers such as law, journalism and psychology. But I had also applied for and been offered a place in mathematics. Despite this, I chose to repeat Year 12. I took maths again because I still enjoyed it. The second time around it seemed to make a lot more sense; my scores were 19 and 19.5 for Maths 1 and 2. At the end of Year 13 I was awarded one of the first UniSA Hypatia Scholarships for Mathematically Talented Women. This boosted my confidence and made University study more affordable for a country kid. So, I decided to do mathematics. I also enrolled in a computer science major because I wasn’t sure what kind of job you could get with a maths degree.

Most of my undergraduate mathematics experience was the same as high school. I got Distinctions or High Distinctions for all my subjects (except Statistics 3B where I scraped a pass). I did most of my thinking in my head and then committed it to paper. I produced beautiful notes, and would rewrite a page if it had a single mistake on it. On reflection, I had a fairly superficial understanding of mathematics, but knew what to do to get good marks. I got disenchanted in the third year of my four-year degree and briefly considered quitting, but I had never quit something so important so I kept going.

At the end of third year, I had an experience that made me sure I wanted to be a mathematician. I attended a Mathematics-in-Industry Study Group. This is a five-day event that draws together around 100 mathematicians. On the first day, we listen to five or six different companies tell us about a problem they have that needs solving. For example, they might say ‘we want to stop washing machines from walking across the floor when they are unbalanced’ or ‘we want to know the best way to pack apples in cartons’. The mathematicians then decide which problem they want to work on, and smaller groups spend the next three and a half days feverishly trying to find a solution.

It was transformative as I witnessed, first-hand, mathematics put into action. I also saw how mathematicians creatively and collaboratively approach solving problems. I watched accomplished mathematicians initially not know how to start. I saw them making mistakes. They had intense (but friendly) discussions about whether something was the right approach. It was a defining moment, because it showed me how mathematics is *really* done, beyond learning mathematics that’s already known, or applying algorithms without a sense of why we would do so. I saw the true habits of mathematicians in action. I also discovered the important role that communication plays in mathematics, and that I could put my love of the English language to good use.

The transition from doing maths exercises with answers that were ‘perfect’ the first time to the more authentic and messy problem-solving required for mathematical research was not an easy one for me. I found it difficult in my PhD to accept that I was not perfect and that I had to constantly draft and refine both my mathematical ideas and my writing, especially because I had never been taught these skills. But I was helped in being surrounded by more experienced mathematicians who modelled, if not explicitly articulated, that this was how mathematicians really work.

It’s eight years since I was awarded my PhD, and I can now say that I am quite comfortable with this ‘messy’ approach to maths. I like to say that mathematicians are chronically lost and confused, and that is how it is *supposed to be*. It would be ridiculous for mathematicians to spend their days solving problems that they already know how to solve. So, being uncertain about whether something will work, or uncertain about what to do next, is a natural way for mathematicians to be.

I started teaching mathematics during my PhD. At first I taught exactly as I had been taught, with procedures and algorithms. But I also didn’t want to respond to a student with ‘Because that’s the rule’, so I started trying to really understand why maths concepts worked the way they did. I learned so much more about maths when I started to explain it to others. The way I taught expanded to include visual ways to think about maths, a variety of representations and approaches, and other flexible ways of thinking. It wasn’t natural to me at first (and at times I still solve arithmetical problems in my head by imagining a pen writing the algorithm) but it has immeasurably enriched my own understanding of mathematical concepts.

I also realised that the way I was taught was not the way I wanted to teach, but I wasn’t sure how to change that. I sought ideas from the internet, and eventually stumbled into the early days of the online community that is the MTBoS (the Math Twitter Blogosphere), although I didn’t realise that until much later. I lurked for a long time because I felt like an outsider: I wasn’t a school teacher (what did I know about education?!) and I wasn’t located in North America. Today I couldn’t imagine teaching without the support of my professional community on Twitter which extends all around the world.

Around five years ago I decided that I could help break the cycle of traditional procedural-based teaching by supporting students, particularly preservice teachers, in experiencing maths in the ways that I and other professional mathematicians do. So, I designed a course that gives students these problem-solving experiences alongside learning skills for thinking and working mathematically. I hold these word clouds from Tracy Zager in my head as a reminder and a motivation of what I am trying to accomplish. (You can read more about how I found them here.)

I still love ‘cracking puzzles’ in maths like I did in Year 2, but my love of maths has expanded to include learning how others think about mathematical ideas. In almost every class I see a student think about a problem in a way I’d never imagined, and I love it. Listening to student thinking is why I’ll never tire of teaching, and it helps me to be a better teacher. I can’t wait to learn from you.

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