# Tangling and untangling

This is the seventh 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 hereWARNING: It’s a long post.

Edited to fix the confusion between × (multiply) and x (the letter).

I have been itching to try Conway’s Rational Tangles with a group of students. I first read about this problem a couple of years ago in Fawn Nguyen’s excellent post. It looked super interesting, but I was still somewhat confused with how it works (not to mention why it works). So I was excited to be in Fawn’s ‘Conway Tangles’ Math Micro-Session at the NCTM Annual Meeting in San Francisco this year, where it started to make some more sense.

This week I tried the activity with my #math1070 students. I waited until the last week of the course because: (1) now that we know each other better, I thought they’d tolerate me muddling through it, (2) their resilience and problem-solving skills for more challenging and ill-stated problems have increased. (Note that the ‘ill-stated’ part is my fault, not that of the problem!) I was upfront with them about how I was both excited and nervous about the session. It was a bit sketchy with the first group of students, but I was able to make some adjustments with the second group.

Below is a mash up of how I did it this week and how I would improve it in the future. This outline is based on Fawn’s write up, but I also pulled in ideas from Tom Davis’ thorough notes for a Maths Teacher Circle, along with the three-part outline from nrich maths: Twisting and Turning, More Twisting and TurningAll Tangled Up. We spent ~1.5 hours on the activity. Perhaps half of that was outside, with students doing it themselves.

#### Getting started: the rules

Have four volunteers come to the front. Each person holds the end of a rope so that the two ropes are horizontally parallel. This is the starting position. This state has a value of 0.

There are only two moves that can be made: TWIST (T) and ROTATE (R).

TWIST is when the person at the bottom left moves under the orange rope to the top left, as shown below. This new state has a value of 1. We notate this as $0 \xrightarrow{T} 1$. I tell the students that every time a TWIST operation is performed, the value of the ropes increases by 1. So, TWIST is +1.

A ROTATE is when every person moves clockwise to the next position, as shown below. (Note that this is from the starting position.) I say that I am not going to tell them the value of this new state. $0 \xrightarrow{R} ??$ I’m also not going to tell them what ROTATE does; that’s for them to figure out.

#### The ‘aim’

Our aim is to (in my words) ‘tangle the crap out of the ropes’ by performing any number of TWISTS and ROTATES and then work out how to untangle the ropes back to horizontally parallel (with value 0). But, remember that there are only two available moves: TWIST and ROTATE. ‘Untwist’ and ‘Unrotate’ are not possible moves.  (I wrote ‘aim’ because this isn’t the only goal, but it’s the one that students will initially want to work towards.)

At present, we have two questions:
(1) What does ROTATE do?
(2) How to get out of any tangle?

#### A first go at experimenting with ROTATE

Before I let them loose with some ropes, we try a few more systematic experiments.

We reset the ropes to 0, and try ROTATE followed by ROTATE (RR). We discover that the ropes end up horizontally parallel again. That is, $0 \xrightarrow{R} ?? \xrightarrow{R} 0$. We decide not to ever do RR unless we want to waste energy. We test this further by resetting the ropes to 0, and trying TWIST followed by RR. As expected, we end at a state with value of 1. We summarise: $0 \xrightarrow{T} 1 \xrightarrow{R} ?? \xrightarrow{R} 1.$

The other possibility after an initial rotate is to try a twist. So we reset the ropes to 0, and try ROTATE followed by TWIST (RT). This action is kind of strange; the ropes stay vertically parallel no matter how many twists we do: $0 \xrightarrow{R} a \xrightarrow{T} a \xrightarrow{T} \cdots \xrightarrow{T} a.$

There are already some interesting conclusions that we could come to as a group, but I decide (based on experience) that it might make more sense if everyone is participating instead of watching.

#### Trying it for themselves

Students get into groups of at least five: four on the ropes and at least one person recording the steps. I distribute the ropes (\$1 each at Kmart; bargain!). We go outside. Maths classes in university are never held outside, so this is novel for all of us.

I suggest that to help answer Question 1 (What does ROTATE do?) we might want to break Question 2 down into further sub-questions.

(1) What does ROTATE do?
(2a) Work out how to get out of one TWIST; two TWISTS; three TWISTS; four TWISTS; any number of TWISTS.
(2b) Work out how to get out of a mixed up sequence, like TTRTTTRT, shown below.

Everyone starts with (2a), and works it out fairly quickly (15 minutes?). Their strategy to untangle is to always start with a ROTATE (otherwise we would be further tangling the rope), then to look at the ropes and ‘see’ what to do next. Eventually they write down how to get out of these positive integer states (T, TT, TTT, TTTT, …) and see a pattern. Try it for yourself! (Or look on page 5 of Tom Davis’ notes.)

In general, my students find it hard to conjecture what ROTATE does. I talk to each individual group in turn. To get them started, I write down something they’ve just done: $0 \xrightarrow{T} 1 \xrightarrow{R} a \xrightarrow{T} 0$. We can work backwards from T to realise that a=-1.  We also realise that when we start with just twists, the value of the state keeps increasing from 0. OBSERVATION: To return back to 0, ROTATE must involve a negative somehow.

I suggest perhaps ROTATE is ×(-1). We look at $0 \xrightarrow{T} 1 \xrightarrow{R} -1 \xrightarrow{R} =1$. This works!
We test it on one TWIST: $0 \xrightarrow{T} 1 \xrightarrow{R} -1 \xrightarrow{T} 0$. This also works.

We predict what should happen with two TWISTS: $0 \xrightarrow{T} 1 \xrightarrow{T} 2 \xrightarrow{R} -2$. To get this untangled, we should be able to do TT. We try it with the ropes. Groan as it doesn’t work. (Note that some students have already forgotten that they know how to get out of two twists, from (2a).)

There is more conjecturing about ROTATE. For example, some students try ROTATE is -2. Later in the class discussion we realise that ROTATE can’t involve just adding or subtracting as RR would take us further away from 0 (positive or negative), and we know that $0 \xrightarrow{R} ?? \xrightarrow{R} 0$. OBSERVATION: ROTATE must involve multiplication or division (or perhaps some other operation).

Most are still stuck. I ask them if they’ve done (2b) and untangled TTRTTTRT. If so, I tell them that the tangled state has value 3/5. OBSERVATION: ROTATE must involve a negative and fractions somehow. Some more cautious conjecturing eventuates.

If they are still stuck, I tell them that TTRTTR has value -2/3.

After working on it for ~45 minutes, some of them give up and demand the answer. I know there is more problem-solving work to come so I tell those that haven’t worked it out that ROTATE is ×(-1/a), where a is the previous state value.  To summarise:

• $x \xrightarrow{T} x+1$
• $x \xrightarrow{R} -\left(\frac{1}{x}\right)$

#### Efficiently getting out of any tangle

I ask them to come up with a scheme to efficiently get out of any tangle. (Later we decide that we aren’t sure that it is the minimum number of moves, but it seems efficient.) It works a bit like this: Get as close to zero with a numerator of 1 and a positive denominator (like 1/m) then ROTATE. This leaves you with a negative integer, –m, and you can TWIST your way m times back to 0.

#### Wrapping up

Back in the classroom as one group, we summarise what we discovered, and make a few more observations.

• We go back and think about starting with a single ROTATE. Now that we know what ROTATE does, we see that the state becomes -1/0. This is like infinity. Another ROTATE brings it back to 0. When we start with a single ROTATE, TWIST leaves it exactly the same: $\infty + 1 = \infty$. So we can have a tangle value of infinity. This is all kind of cool.
• We wonder if every rational number can be reached through tangles, and then be untangled.
• We wonder about how to prove the minimum number of moves to get out of each tangle.
• We talk briefly about function notation: $T(x) = x+1$ and $R(x) = -\left(\frac{1}{x}\right)$. We confirm that $R(R(x)) = x$, so two RR leave the state unchanged. We talk about composition of functions, and how RTT is represented by $T(T(R(x))).$

We talk about how this activity is suitable for a range of students and different areas of focus:

• problem solving and team work just by trying to untangle a tangle (no investigation into TWIST and ROTATE)
• practicing fluency with fractions
• older students can work with function notation and tackle some of the more challenging questions.

I reflect later how there is so much more depth in this activity than I had realised. I also realise that because it has so many different dimensions—physical manipulation, symbolic notation, numerical calculations, pattern recognition, conjecturing, teamwork, leadership—it gave students opportunities to shine in different ways.

# How many triangles?

It’s been quiet on the blog, but a lot has been happening. University classes in Adelaide have just resumed after a two week mid-semester break. To warm up, I gave my MATH 1070 students the following problem. I found it via Tanya Khovanova who states that it was an entrance problem for the 2016 MIT PRIMES STEP Program. (Read more on Tanya’s blog.)

I drew several triangles on a piece of paper. First I showed the paper to Lev and asked him how many triangles there were. Lev said 5 and he was right. Then I showed the paper to Sasha and asked him how many triangles there were. Sasha said 3 and he was right. How many triangles are there on the paper? Explain.

Here are some solutions from my students, all considered to be correct. The ones in blue originally appeared in Tanya’s blog post. Additional ideas are shown in red below. The black rectangle shows the piece of paper. Two of the rectangles contain instructions instead of diagrams.

I loved this as an opener to encourage creative problem solving. Thanks Tanya!

# 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.)

#### ‘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?
• 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.

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?

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?

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.

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

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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# Notice and wonder: sense making

This is the third 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.

It’s been three weeks (how time flies!) since I last posted about this course. There are 1.5 workshops that I haven’t written about. We also missed three classes (1.5 weeks = 3 workshops) due to my travel and illness. We finally got back into the regular rhythm of class last week. In brief, I am not going to write about how we: dug into the importance of multiple representations; tackled our first substantial problem—’How many squares on a chessboard?’—and used it to discuss specialising and generalising; looked at ways to organise our work; talked about the importance of writing down our thinking; introduced the three phases of work (Entry, Attack, Review) and tackled a couple of problems along these lines; started maths talks with a dot talk and then a `duck’ talk. There is too much to write about!

I am continually inspired by the power of the ‘Notice and Wonder’ routine to help in all stages of problem solving, not just when starting to tackle a problem. We spent a few hours in class experiencing different dimensions of ‘Notice and Wonder’, and I hope to summarise our work over a few posts.

#### Introducing Notice and Wonder

In case you are not familiar with it, the ‘Notice and Wonder’ strategy involves asking two simple questions: ‘What do you notice?’ and ‘What do you wonder?’. These are powerful prompts to engage students. ‘Notice and Wonder’ helps lower the barrier to entry for all students. It encourages sense making. Students are more invested because they are connecting their own thinking to the scenario and are generating questions that they are interested in solving. ‘Notice and Wonder’ is enthusiastically promoted (with good reason!) by the team at The Math Forum. For more, read Max Ray-Riek’s post or watch one of Annie Fetter’s Ignite talks: NCTM 2011 and Asilomar 2015.

#### ‘How old is the shepherd?’ — Observing a lack of sense making in others

‘Notice and Wonder’ is most commonly implemented by displaying an image or a video and recording everything that students brainstorm. However, I decided instead to first have my pre-service teachers observe a lack of sense making in others. For this I used Robert Kaplinsky’s video showing eighth-grade student responses to the nonsensical question ‘How old is the shepherd?’. The question here is seemingly straightforward:

There are 125 sheep and 5 dogs in a flock. How old is the shepherd?

Before I showed the video, I asked students to estimate how many of the 32 students they thought would give a numerical — and thus nonsensical — answer. Try it for yourself.

Then I played the video. If you skipped the link, go watch it. It will only take 3:07 minutes. Students were incredulous; they thought the video was a joke. They just could not believe that 75% of the students that Robert interviewed gave a numerical response. Once that moment passed, we spent some time suggesting different reasons why this might happen, including:

• students wanting to do ‘something’ with the numbers because it’s a maths question
• students assuming that they’ve misunderstood something because the teacher has always previously asked sensible questions
• students not feeling comfortable challenging the questioner, who is the authority in the room.

For further reading, I suggest starting with blog posts by Robert Kaplinsky, Tracy Zager and Julie Wright.

Rather than simply laugh at the video (which did happen; sorry), I then focused on the role of teachers in all the reasons we had brainstormed. I emphasised the responsibility that we have as teachers to empower students to question and to make sense of problems. There was quiet in the room as this powerful moment resonated.

This led nicely into the idea of sense making — in all stages of problem solving. For example, the students in Robert’s interview who said ‘Hey, what? I can’t solve this!’ were making sense of the problem at the start of problem-solving: they had thought about the context and realised that it didn’t make sense. We talked about the students who jumped straight into a calculation, and how they hadn’t even made sense of which calculation they should do. (Hint: there is no sensible calculation one can do!)

We also noticed that most of the students who gave a numerical answer did not make sense of their answer at the end of problem-solving, even as Robert probed their thinking. We did note that no students used a multiplication strategy, so presumably there was some sense making in that the shepherd couldn’t possibly be 625 years old!

From here I wanted to draw attention to the value of units in making sense of problems. For example, if students had reflected that 125 – 5 meant subtracting dogs from sheep to get years, perhaps more sense making would have taken place.

#### ‘The flour problem’ — working with units

The best problem that I know to encourage making sense of problems via units is ‘the flour problem’. I discovered this via Fawn Nguyen, who has written a great blog post about how her sixth graders worked on the problem, including a ‘Notice and Wonder’ routine on the top half of the image.

Of course, it’s not just a problem for sixth graders. I have begun using it in the first class of my university mathematics course for laboratory medicine students. (If they can’t make sense of units, then we have a real problem!) I’ve watched pre-service and in-service teachers in my sessions dig deep into this task; it encourages sense making in so many directions. I also showed John Rowe’s makeover of a trigonometry question, which similarly gets to the heart of sense making.

We also briefly talked about estimation as a sense-making strategy: starting with an estimation, and then reviewing any calculations in light of the estimation that we made. This is a theme we’ll return to later in this course.

#### Where to next?

With such a persuasive start to the importance of sense making, the stage felt ready for students to experience ‘Notice and Wonder’ for themselves. I hope to write soon about how that unfolded.

# Activities for Day One: Fold-and-cut, Quarter the Cross

This is the second 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.

Last week I wrote about my three main goals for my Developing Mathematical Thinking course. You can read that post here. Today I want to talk about the activities I used in the first workshop: why I chose them, how they unfolded, and the discussion that resulted.

The two activities are Fold-and-Cut and Quarter the Cross. Along with ~35 minutes of other discussion, these activities took the entire 1hr 45 min workshop and another 30 minutes of the next. Regular readers will think by now that I’m somewhat obsessed with both of these activities (it is true; I’ve already written about them here and here), but they are such rich tasks. In this post I add more detail, particularly from the perspective of using them in the (pre-service teacher) classroom.

### 1. Fold-and-cut

Duration: ~50 minutes
I wanted the very first activity to: be non-threatening and hands-on, encourage students (who may not have met) to talk with one another, not look like a typical maths task, accommodate a variety of problem-solving strategies, and allow me to gently start eliciting explanations of their thinking. Sounds like fold-and-cut is ideal!

As before, I started by showing the first 2:30 min of Katie Steckles’ Numberphile video on the fold-and-cut theorem. (As a reminder, the task is to fold the shape so that it can be removed from the paper with one straight cut.) I gave out the first three pages of this revised handout with the yellow, green and blue shapes in the photos above. (Credits for the handouts due to JD Hamkins and Patrick Honner; see my earlier fold-and-cut post.)

After everyone had tackled most of the shapes, we stopped to reflect. I made reference back to the four lenses with which they could view each task (maths learner, maths thinker, student, teacher), and asked them to brainstorm possible maths content in the task, maths thinking they used, thoughts that I may have had as a teacher in presenting/selecting the task, any other ideas. An incomplete list is below:

• Maths content. Shapes terminology (for example, regular and irregular polygons, convex and concave polygons, equilateral, equiangular), symmetry properties, angles, angle bisectors, triangle incenter, optimisation/efficiency (minimum number of cuts). Later we added fractals and Koch curve.
• Maths thinking. Trial-and-error, guess-and-check, noticing (what we did, what others did) and wondering (for example, can I do it with fewer folds?), making a problem smaller, reducing to a problem we already know how to solve, reflecting on what went wrong/right, transferring skills to a new problem.
• Teacher lens. We thought that the task: gives students choice about where to start and how to do it; has an ‘easy’ entry; builds confidence; is ‘open’; is engaging, visual and hands-on; allows multiple approaches; is low risk and low threshold, high ceiling.

I shared that this was the first time that I had handed out all three pages without suggesting that students start on the yellow (easiest) page. We reflected about whether some people started with one that looked easiest (to give them confidence), the one that looked hardest (because they wanted a challenge), or some other shape for some other reason.

I then gave them the last two pages (pink and orange in the photo) with irregular shapes and the Koch curve. Discussion points included: we couldn’t necessarily rely on properties of symmetry anymore; they felt they could do these ones now that they’d build confidence on the first three pages; the Koch curve looked hard but turned out to be easier than expected (an experience that might hold them in good stead in the future!). I also directed them towards Chapter 3 of the free book ‘Art & Sculpture’ from Discovering the Art of Mathematics, which has a well thought-out sequence of prompts that might make a suitable classroom investigation.

In all, I was really pleased how this activity unfolded, and how we were immediately able to use the ‘four lenses’ idea as a framework for reflection.

### 2. Quarter the Cross

Duration: ~45 minutes (spread over two classes)
I am very thankful to David Butler for sharing here about the mathematical ideas he sees in this activity, and here about using this activity in his daughter’s Year 7 classroom; this was excellent guidance.

My reasons for using this task were: promote creativity, encourage students to find different ways of achieving the same goal, prompt students to give reasons for their work, lead into multiple representations.

I started by showing the slide. I then handed out six smaller copies of the cross. I encouraged students to use whatever method they wanted; some chose to cut them out (most likely because we had just been doing fold-and-cut), others used rulers, some drew free-hand.

After students had coloured in six, I picked two of the shapes that most people find in their first few: the ‘house’ and the ‘L’. (See below.)

• We started with the ‘house’. Some students shared that they found it using symmetry; others shared that they saw it as one full square plus a quarter of another one.
• We moved onto the ‘L’. Along with symmetry, another approach was to decompose the whole cross into a smaller number of equal parts. A key observation was that making the number of equal parts divisible by 4 made it easier to shade a quarter. (The orange cross on the bottom right is another example of this.)
• We then discussed the one with five blue triangles, and noted that it could be made from the house by moving bits around (which doesn’t change the area). (The red triangle-type shape in the bottom row is also like this).
• Finally we talked about how adding a bit and taking the same bit doesn’t change the area, which is how the green one with circles came about.

I also used different examples of student work to illustrate these strategies.

I then showed them David’s one hundred solutions (see his blog post). I zoomed in on some of the more complicated-looking ones as examples of the mathematical scope of the activity. Their homework task was to colour in another six based on their ideas from our discussion, and from examining their classmates’ work.

#### Multiple representations

In the next workshop, I started by noting that we had two ways of considering the cross: as one whole, or as composed of five smaller squares. I asked them to concentrate on the second method. If one square is a unit, then what does it mean to ‘quarter the cross’? We need to shade 1/4 of 5 units, so 5/4 units.

Now we started exploring different ways of expressing 5/4, and connecting them to corresponding visual representations. Once again, we used student work to illustrate these. For example:

• 5/4 = 5×(1/4). Colour in 1/4 of one unit. Do this five times. (We can move those bits around later if we want.)
• 5/4 = 1 + 1/4. Colour in one whole unit and a quarter of another. (An example is the house.)
• 5/4 = 2×(5/8). Colour in 5/8 of one unit. Do this two times.
• 5/4 = (1/2)×(5/2). This one was interesting, as it was the first one with an improper fraction. Start with shading 2.5 units, and then colour in 1/2 of the shaded area.
• 5/4 = 1/2 + 3/4. Colour in 1/2 of one unit and 3/4 of another unit.

We reflected on how much richer this task had become with the introduction of another representation, and how being able to translate between the two representations promoted deeper understanding about fractions. This was a great lead-in to the rest of Workshop 2; I’ll write about this soon.