Category Archives: teaching maths

#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.

Series.jpg

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. Quadratics.jpg

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.)
  2. We asked similar questions about the y-intercept.
  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.Quad.jpg

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.

Thank you for the #lessonfail

This blog has been accumulating a layer of dust. I could name a lot of reasons (I only teach four hours a week, I’m busy with other projects, and so on) but the truth of it is that I usually blog when I feel motivated by my teaching. And lately, well, it’s felt a bit ‘meh’, and who wants to read ‘meh’?

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

Yesterday, Annie Perkins tweeted the following.

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

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

Tracy Zager’s new book

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Tracy Zager’s new book ‘Becoming the Math Teacher You Wish You’d Had‘ is out, and it’s a treat. The central tenet of this important book is to ‘close the gap’ by making maths class more like mathematics, orienting our students towards the habits of mind of professional mathematicians. ‘Good teaching starts with us’ and Tracy companionably guides us through ten practices of mathematicians: taking risks, making mistakes, being precise, rising to a challenge, asking questions, connecting ideas, using intuition, reasoning, proving, working together and alone.

Tracy skillfully blends academic research, illuminating classroom dialogues, the thoughts of mathematicians and maths educators, and her own perceptive observations. This seamless mix is a real strength of the book; we not only see what habits are important and why, but how they can be enacted through specific teaching strategies, and the powerful effects they have on our students’ development as confident and capable mathematicians. The reader can’t help but be inspired by the teachers that Tracy holds up as exemplars of good practice. These teachers have so much respect for each of their students as serious mathematical thinkers. I was struck by the extent to which they would go to adapt instruction in response to student ideas and to support them in pursuing their own line of enquiry.

Tracy warns early on that the book is long—and it may be—but it is also captivating! The organisation is immensely practical; each chapter can be used as a self-contained guide for a particular mathematical habit. I can see myself repeatedly delving back into specific habits as the teaching year progresses. I read it cover-to-cover over a couple of days while curled up in a secluded cabin, pausing occasionally to stare out into the Australian bush and ponder what I can change in my own teaching. Some of my highlighted passages:

  • From Chapter 3, Mathematicians Take Risks: ‘When we assign problems that have a single, closed path from start to finish, we’ve eliminated the possibility that students will take mathematical risks. There’s nothing to try if everything is prescribed.(pg 49)In my skills-based courses, I too infrequently give students opportunities to try and be successful with their own approaches. That’s something to work on.
  • From Chapter 4, Mathematicians Make Mistakes: ‘If we want students to learn from mistakes, we need to teach them how.’ (pg 57).  Tracy outlines a three-part goal: to teach students to take mistakes in their stride, to keep going when they’ve made a mistake, and the one I need to focus on: to teach students to make the most of the knowledge and experience they gained by figuring out their mistake‘. How can I help students gain the skills to diagnose and learn from their mistakes, by themselves?
  • From Chapter 5, Mathematicians Are Precise: ‘Math without inquiry is lifeless, but math without rigor is aimless. There is no tension between teaching students how to solve problems accurately and efficiently and teaching students how to formulate conjectures, critique reasoning, develop mathematical arguments, use multiple representations, think flexibly, and focus on conceptual understanding.’ (pg 80). In my problem-solving course, I deliberately swung the pendulum from the typical procedure-based courses my students had mostly experienced towards creative, collaborative problem-solving. But I also need to find the middleground, where I place as much emphasis on rigour as I do on inquiry.
  • From Chapter 12, Mathematicians Work Together and Alone: ‘If a major part of doing mathematics involves interacting with other mathematicians, then a major part of teaching students mathematics must be to teach students how, why, and whether to interact with one another mathematically. Students need to learn how to ask for what they need from each other and to be what they need for each other … we need to teach students how to be good colleagues  it’s important we honor individual thinking and working time. It’s not reasonable to expect students to collaborate at every moment, and that’s not how mathematicians work.’ (pg 312). This past semester, a few students in my problem-solving course commented that they needed more opportunities to work alone first, and more strategies to work effectively with group members. I’ll definitely be digging further into this chapter next year.

And, these phrases are going straight into my repertoire:

  • ‘Do you have more questions after doing this? What are you wondering about now? (pg 149).
  • ‘What does ______ have to do with _____?’ (Debbie Nicols, pg 191).
  • ‘Remember that it’s hard to find mistakes when you assume that you’re right. So go back into it assuming something went wrong.’ (Jennifer Clerkin Muhammad, pg 284).
  • ‘Would you recommend that strategy to someone you like?’ (pg 118). 😂

There is so much to love about this book. The writing is both encouraging and empowering. It’s labelled K-8 but Tracy offers important insights to help teachers across all year levels; I have been nodding furiously and making notes throughout. This particular passage had me shouting ‘yes!’:

‘We need to give ourselves permission to say, publicly, and with delight, “I never thought about it that way before!” whether it refers to addition, fractions, or place value. It is long past time for us to respect the beauty, power, and importance of elementary mathematics, instead of having contempt for “the basics.”’ (pg 208)

Listening carefully to student thinking, especially about ideas I thought I understood, always gives me new insight. It’s why I’ll never tire of teaching.

I can confidently say that, alongside ‘Thinking Mathematically‘ (Mason, Burton and Stacey, 1982; 2010), Tracy’s book will become a cornerstone for my teaching. It is a gift to all maths teachers. But don’t just take my word for it; you can preview the book in its entirety here. The companion website promises more, and I can’t wait to look around!

Update (22 December 2016): The companion website is now live, and it is packed full of goodies. Be sure to check out the free study guide under ‘Getting Started’, which works for either an individual or group book study.

Notice and wonder: the Prime Climb hundreds chart

This is the sixth in a series of posts about my course ‘Developing Mathematical Thinking’, a maths content elective for pre-service teachers training in primary and middle maths. All posts in the series are here.


This is the final post detailing how I introduced ‘Notice and Wonder’ to my pre-service teachers. We’ve used it for sense making. We’ve then looked at photos from the world around us and brainstormed what we noticed and wondered. The students later took their own photos and identified the mathematical ideas that they saw. (The photos and reflections were so much fun to look through!) Next, we transferred our ‘Notice and Wonder’ skills to more mathematical settings, including one of Dan Meyer’s Three-Act Maths Tasks, Toothpicks. I’ll now tell you about the consolidating task in which I had students tell me what they notice and wonder about an image bursting with mathematical ideas.

Note: although this looks like a long post, the first 200 words are an introduction; the last 1500 words are a summary of student ideas.

The Prime Climb hundreds chart

Prime Climb is a beautiful board game in which players deepen their understanding of arithmetic through gameplay. To be quite honest, I’ve never played!  But that hasn’t stopped me appreciating the gorgeous hundreds chart that accompanies the game. A version is below; you can buy this image on a stunning poster here.

This hundreds chart compels us to notice and wonder. Take a moment and brainstorm for yourself. (Dan Finkel, creator of the game Prime Climb, talks about this image in his wonderful TED talk, ‘Five Principles of Extraordinary Math Teaching‘. It’s worth taking the ~15 minutes to watch.)

HundredsChart.png

‘Noticing and Wonderings’ from my students

I asked my group of nearly fifty pre-service teachers to each tell me five things that they noticed, and one thing that they wondered. As a group, that’s potentially 250 different things that they notice, and 50 things that they wonder! Here is a collated list of about 100 of their ideas (with slight amendments to incorrect terminology), loosely grouped under my own section headings. I asked them to do this as individually. I’m sure that in a group discussion they would have built on and extended each other’s ideas. Next time!

Enjoy the read; I certainly did.

I notice that …

Colour and structure

  • Circles are numbered 1-100.
  • The chart is organised into a 10×10 system.
  • The numbers are ascending.
  • The numbers in each column increase by ten as you go down the list.
  • Colour has something to do with number, and vice versa.
  • There are different colours: blue, orange, yellow, red, green.
  • Some circles have only one colour.
  • With the exception of the whole red circles, each other colour appears as a whole circle only once.
  • Each circle is made up of one or more colours.
  • Colour is used to demonstrate relationships between numbers.
  • Every second number has orange in it (and similar statements about other colours).
  • All even numbers are yellow/orange.
  • Friendly numbers (5s and 10s) have blue in them.
  • Circles with blue end in 5 or 0.
  • There are a lot of red-only circles/numbers.
  • There are 21 solid red-only circles/numbers.
  • Red is the most prominent colour.
  • Purple is the least-used colour.
  • Completely green numbers are multiples of 3 (and similar statements about other colours).
  • The rings are broken into fractions that vary between a whole and 1/6.
  • Some of the red sections have little white numbers in them.
  • All the small white numbers that appear ‘randomly’ on the bottom of the circles are all odd numbers.
  • The red full circles only occur on odd numbers.
  • Numbers with orange in them (multiples of 2) are in a vertical pattern, as are numbers with blue in them (multiples of 5). But numbers with green in them (multiples of 3) are in a a diagonal pattern (right to left) when viewed from top to bottom.
  • If you place your finger on a number with purple, then move your finger up one row and then move it three columns to the right, you will end up on another number with purple (works with most purple numbers unless it is too close to the edge).
  • The greatest number of coloured sectors around a number is six.
  • The greatest number of different colours included in the sectors surrounding any number is three.
  • No number/circle has all the colours present.
  • There doesn’t seem to be a pattern in the colours.

The number 1

  • The number 1 has no colour, because it is neither a prime or a composite number.
  • The number 1 has its own colour and is not part of any particular pattern in the chart. Every whole number has a divisor of 1.
  • 1 is not a prime number, which is why it is not coloured.

Prime numbers

  • The circles with full colours are prime numbers.
  • All prime numbers have a single unbroken circle.
  • 97 is the largest prime number less than 100.
  • Prime numbers have their own specific colour up to the value of 7.
  • Red circle numbers are also prime numbers from 11 upwards.
  • Other than 2, all prime numbers between 1 and 100 are odd numbers.
  • There are 25 prime numbers between 1 and 100.
  • If there is a little number written at the bottom of a circle for a greater number then it means that greater number is divisible by a prime number. For example the number 92 has a small 23 written at the bottom of the circle, this indicates that 92 is divisible by the prime number 23.
  • There is only one prime number between 91 and 100. All other blocks of ten have at least two prime numbers.
  • The ‘3’s column has the most prime numbers between 1 and 100.

Composite numbers

  • Numbers that aren’t prime are a mix of colours. For example, 15 is 5×3 where 5 is blue and 3 is green, so 15 is half blue and half green.
  • All multiples of 6 have to have orange (2) and green (3) in them.
  • Any number ending in 4,6,8 or 0 isn’t a prime number.
  • Some non-prime numbers are made up of factors which are just (only) prime numbers.

Square numbers

  • All square numbers are comprised of one colour in several parts.
  • The sum of all the square numbers is 385.

Multiplication-oriented

  • We can use the colours around each number and multiply their ‘representing numbers’ together to make the number in the middle.
  • The circle fragments symbolise how many times multiplication has occurred. For example, the number 8 has three yellow circle fragments, indicating 2×2×2.
  • The colours of each circle represent the numbers in which the greater number can be divided by. For example number 95 is coloured blue and red. These colours represent 5 and the prime number 19. When multiplied their sum is 95.

Divisor and factor-oriented

  • There are only 2 numbers on this chart that are represented by a circle split into sixths. They are 64 and 96.
  • No more than six factors are required to make numbers up to 100.
  • Odd numbers more commonly have factors that are prime numbers.
  • The circles are divided into sections depending on how many divisors they have.
  • The factors of each number are obvious through the colouring.
  • Different coloured sections in the circle mean that the number is divisible by more than one number.
  • Odd numbers generally have fewer factors, even if they aren’t prime.

Prime factors

  • The colours that surround the number represent the prime factors of the number. For example, number 96 has five orange segments and one green segment, which suggests that the prime factors for the number 96 are 2×2×2×2×2×3.

Other

  • All numbers divisible by 11 have the number 11 in a subscript, and are in a diagonal line.
  • Consider numbers with the same digits (11, 22, …). The sum of the digits are all even numbers.
  • There are no explicit instructions or ‘key’ to explain what the chart is actually displaying.
  • The sum of the first nine prime numbers is 100.
  • If you squint your eyes, you start to see colour patterns rather than noticing numbers, which is how I noticed some of my previous points.

I wonder …

Colour and structure

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

Patterns

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

Extending the chart

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

Other

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

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

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

This is the fifth in a series of posts about my course ‘Developing Mathematical Thinking’, a maths content elective for pre-service teachers training in primary and middle maths. All posts in the series are here.


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

An activity that fizzled — because of me

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

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

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

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

And then it dawned on me.

It’s all in the presentation

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

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

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

The revival: notice and wonder to the rescue

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

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

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

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

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

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

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

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

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

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

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

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

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

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

And now for the whole truth

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

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

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

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

Notice and wonder: the world around us

This is the fourth in a planned series of posts about my course ‘Developing Mathematical Thinking’, a maths content elective for pre-service teachers training in primary and middle maths. All posts in the series are here.


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

Entry task

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

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

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

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

These activities are designed to support students to:

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

The #math1070 photo challenge

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

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