A lesson plan (of sorts) for quadrilaterals

This tweet sums up today’s class.

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.

Previously …

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:

1. $a \times \frac{1}{a} = 1$
2. A rectangle is a square.
3. The sum of four even numbers is divisible by 4.
4. One-third is one of three pieces.
5. $2-x = x-2$
6. An even number divided by an even number is an even number.
7. $p+12 = s+12$
8. Division always makes a number smaller.
9. The square of a number is greater than or equal to the number.
10. 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 …

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.

1. Names of shapes

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.

2. Sorting and mystery definitions

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.

3. Guess Who?

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.

4. Card/shape sort

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.

5. Nested diagrams and family trees

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 none 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?

6. Always, Sometimes, Never True

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

1. A rectangle is a rhombus.
2. A rhombus is a rectangle.
3. A square is a rectangle.
4. A rectangle is a square.
5. A parallelogram is a quadrilateral.
6. A trapezium is a parallelogram.
7. An equilateral quadrilateral is a square.
8. A parallelogram is a rectangle.
9. An equilateral parallelogram is equiangular.
10. A rhombus is a square.
11. A square is a rhombus.
12. A kite is a square.
13. 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.)

And now what?

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.

A serendipitous Pythagorean #LessonStarter

#LessonStarter is a Twitter hashtag, used particularly by Matt Skoss, to collect together ideas that might start (or take over!) a lesson. A #LessonStarter is usually a provocative image, but could also be an intriguing mathematical prompt. For me, lesson starters are often spontaneous. Today, a few serendipitous moments meant that I had a lesson starter just before walking into class.

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

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!

‘Noticing and Wondering’ from my students

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: 32 + 42 = 52, 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

This is a quick post mainly for the benefit of my ‘Developing Mathematical Thinking’ (#math1070) students.

Introducing the puzzle

Skyscrapers are one of my favourite logic puzzles. They are a Japanese creation, introduced at the first World Puzzle Championship1 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.

Hands-on skyscrapers

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.

Skyscrapers in the classroom

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 BourassaMark 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!

Redux: #NoticeWonder and #PrimeClimb

Last year I wrote a post about using the two simple questions ‘What do you notice?’ and ‘What do you wonder?’ with my maths pre-service teachers to dig into the mathematically-rich image that accompanies Dan Finkel‘s game, Prime Climb.

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.

Using Tiny Polka Dots for visible random groupings

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.

Different-sized grids for Prime Climb

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.

Playing Prime Climb to ‘make thinking visible’

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.

Further exploration

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!

My maths autobiography

School maths

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.

University days

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.

Teaching maths

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.

#NoticeWonder with everyday concepts

I often joke that my blog should be called ‘Notice and Wonder in Mathematics’ because I blog about the ‘Notice and Wonder’ prompt often enough!

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.

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.

Example 1: Revising series notation

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.

Example 2: Launching a lecture on graphs of quadratic functions

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:

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

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

1. 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.)
3. I introduced the term ‘vertex’ for the maximum. We wondered what other possibilities there are, and talked about the vertex being a minimum.
4. We wondered whether the curve could be in exactly one quadrant? Or two quadrants? Or three quadrants? (I was not expecting this!)
5. 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 x2 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.

#NoticeWonder and Rational Tangles

Yesterday we held the first of this year’s Maths Experience days. We invite students in Years 10 and 11 from different schools onto campus for an intensive one-day program. Students find out about mathematical research, talk to professionals who use mathematics in their careers in some way, and participate in hands-on mathematics workshops. Importantly, they also meet and connect with other students who enjoy mathematics.

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’ prompt1.

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.