Making Your Own Sense

Reflections on maths, learning and maths learning support, by David K Butler

Tag: play

  • Maths Sheep Play Sheep

    It’s Children’s Book Week in Australia, and so it’s a good time to write a little something about a project I did in honour of Book Week in 2023: a picture book called “Maths Sheep Play Sheep” written and illustrated by me, inspired by “Where Is the Green Sheep?” by Mem Fox and Judy Horacek.

    The book

    You can download the book in PDF form here, and you can watch me reading the book aloud here:

    (Note the PDF version differs a little from the one in the video, because I have updated the book since I made the video but didn’t want to redo the video.)

    The making

    I’ve never written anywhere the story of how I created this book, and I thought this might be a good time to do so.

    It was Book Week 2023 and since my wife is an Early Childhood Educator, that meant a lot of discussion of children’s books in our house. During the discussion in the first few days of Book Week, an idea grew inside me to maybe create a maths version of the classic Australian picture book “Where is the Green Sheep?” by Mem Fox and Judy Horacek.

    Green Sheep begins like this: “Here is the blue sheep. And here is the red sheep. Here is the bath sheep. And here is the bed sheep. But where is the green sheep?” The rest of the book follows in a similar pattern: “Here is the A sheep. And here is the B sheep. Here is the C sheep. And here is the D sheep. But where is the green sheep?” Where A, B, C and D are all one-syllable words and B and D rhyme. The couplets are accompanied by extremely charming illustrations of sheep doing various things (for example, splashing in a bath or reading in bed).

    I knew that if I was going to make a maths version of this book, I would need a collection of one-syllable maths words with half of them in rhyming pairs. There are exactly six rhyming couplets in the original book before the one involving the final reveal, so I wanted to have six rhyming couplets in mine too. It was important to me that it sounded right. I brainstormed a fair number of words, but not enough rhyming pairs to do the job. It was Wednesday night of Book Week. So I thought I would enlist some help. I went to Twitter and asked my followers (who were mainly maths teachers and mathematicians) to help me brainstorm one-syllable maths words.

    Quite a few people joined in over the next half a day and gave me many lists of words you might encounter from early childhood maths to research maths. It was a delightful little community activity that I remember very fondly. Of course this all happened in between the meetings and teaching sessions of a regular work day.

    It was now Thursday afternoon of Book Week and I had a big list of words. Next I had to choose which ones to use in my book.

    There were several things I wanted to achieve: I wanted most of my words to be ones that the majority of people had heard before. I wanted them to come from a variety of mathematical domains. Half of them had to be in rhyming pairs, and the other half had to connect thematically to the rhyming words. They had to make grammatical sense in the phrase “X sheep”. And it had to be possible to illustrate them in a way that appealed to me. As you can imagine, it was quite a difficult task to satisfy all of these criteria, but I actually really enjoyed the process and managed to make a list that appealed to me.

    One aspect I didn’t enjoy was that there were many words left out, several of them that I had ideas for how to illustrate but I just couldn’t fit into the book. Then I remembered that the second-last illustration of Green Sheep is a wide shot of all sorts of different sheep doing different things, so I could include my favourite of the unused words on a page like that.

    There was one word that I really really wanted to include, which was “play”. For me, play is such a huge part of doing and learning maths, and I wanted it to feature in the book. But it never seemed right in any of the rhyming couplets. Then I realised that the problem itself was the answer: the play sheep must be the sheep we were looking for, like how you are looking for the green sheep in the original. This is what brought the whole thing together.

    I still had to firm up my initial thoughts of what the illustrations would actually look like. Here are my original hand-drawn sketches of the sheep for each of the words in the rhyming couplets, including edits to their shapes and notes about what order they should go in across the book.

    Three yellow A4 pages with pencil sketches of sheep. There is some crossing out and drawing over the top of previous drawings, and also some red loops and arrows indicating a reordering of the sheep.

    It was now Thursday afternoon of Book Week. It all had to be done by tomorrow if I wanted to get it done before the week ended.

    To Inkscape! I studied Judy Horacek’s sheep carefully and made several heads with big ears and dot eyes and big smiles. I also constructed a sheep body and some sheep legs. Then I set to work putting them together in various combinations and making the surrounding situations for my sheep that my words described. Some of them changed a bit from the original sketches when I came to make them, but only because I realised they would be even better that way. (I have saved telling you about the decisions I made while drawing the sheep for later in the post.) The sheep on the second-last illustrated page never had original hand-drawn sketches and I just did them straight in Inkscape. I worked past midnight and got up early, with only a few hours’ sleep in between. (Oops.)

    But I did finish it. It was Friday morning, the last day of Book Week.

    I recorded myself reading it aloud and shared the video and the PDF everywhere online. I went through the one-syllable maths words replies on Twitter and thanked everyone individually, telling them what I did with the words. I also did a half-day in the Maths Learning drop-in centre, though I did carry a printed copy of the book around with me.

    There were a couple of things that upon seeing the book printed, I wanted to change a little, so on the weekend I edited them and redid the video. (Even later, I tweaked a couple more things in the book, but I thought it was ok to keep the video as is. And even even later as I wrote this blog post, there was one more thing I wanted to change.)

    And that’s the story of the making of Maths Sheep Play Sheep.

    Some feelings and thoughts

    Most people love the book, and it really is a joy to watch something I made more-or-less for my own enjoyment bring other people enjoyment too. Many people have a specific favourite sheep, and it can say a lot about someone on a particular day which sheep it is today. This brings me joy too.

    But one response gave me mixed feelings.

    A few days after creating the book, someone came up to me to say they loved the book and asked me what program I used to make it. I replied, “I drew the pictures in Inkscape.”

    They looked confused and said, “No I meant which AI you used.” I told them I didn’t use any AI at all. I chose the words and drew all the pictures myself. They were surprised.

    And I was surprised at their surprise. Because why on Earth would I use an AI to do this and miss out on all the thinking and trying and tweaking and community-building and pride involved in the process of making it?! And I was also slightly offended that they would see me as someone who would outsource all of that. On reflection I was flattered that the thing I made was so good they believed it couldn’t possibly be made by a human mind and hands. But then I thought how sad it was that someone could believe that. It was a real rollercoaster of emotion.

    Honestly, I think using AI to make this is completely against the message of the book. The goal wasn’t for there to exist in the world a maths-themed Green Sheep; the goal was for me to make a maths-themed Green Sheep. The existence of the book wasn’t the real goal, the making of it was. And seems to me that that’s the real goal of maths too. Maths is not about the answers but about the finding of them. Maths is not about the facts but the understanding of them. And the punchline of this book is that the what-ifs and I-wonders that makes maths play also make maths maths.

    The sheep

    I want to do a comprehensive look at each sheep, describing the meaning behind the pictures and the decisions I made when I drew them or even just decided they should be there. Now if anyone ever wants to know, there’s a record for them to look to.

    The first sheep and the third sheep

    A line drawing of three sheep standing on a grey slope. The sheep have cloud-shaped bodies with small spirals indicating wool, and dots for eyes. The highest sheep has a smile, as does the lowest sheep. The middle one wears a frown and its head is slightly lowered.

    As soon as I saw the word “first” appear in the brainstorm I knew I wanted it to be the first sheep in the book. It still makes me laugh that the first sheep is the first sheep.

    But it’s not just the self-referential vibes that appeal to me. It’s also that I think that listing things is the most fundamental purpose of number, even before counting things. That’s why I define that the set of natural numbers doesn’t include zero, because the most fundamental use of the natural numbers is for listing things, not for counting things. You don’t say “here is the zeroth sheep”, you say “here is the first sheep”, which is why the natural numbers start at 1.

    Having both the third sheep and the first sheep on the first page really emphasises this point to me, even though the main reason the third sheep is there is because “third” rhymes with “surd”.

    In the original sketch, the sheep were standing on just their hind legs like people, but I remembered that in Green Sheep the first sheep in the book is very ordinary-looking (apart from being blue) and so I also wanted the first sheep to be ordinary. I had them standing on a slope to give the impression that the first sheep is first because it’s the first to climb the hill.

    The second sheep is frowning because it didn’t get a mention in the text even though both of the other sheep did.

    The mixed sheep

    A line drawing of a sheep standing on its hind legs and holding in one hoof a glass bowl while stirring with a wooden spoon held in the other hoof. Inside the bowl is the mixed number 1 and 2 thirds. The sheep has a big grin.

    “Mixed” is a description of a number, and since the word that rhymes with “third” is “surd”, which is also a kind of number, I thought it was appropriate to have them on the same page.

    In the original sketch, the sheep was holding the 1 and the 2/3 in opposite hooves, but then I realised it would be much cooler if it was actually literally mixing them, so I changed it to holding the bowl and spoon.

    I carefully made sure the two parts of the mixed number were at slight angles to show they had been dumped into the bowl separately. This is because the notation for mixed numbers really is a bit slapdash. Many mathematicians are very disparaging about mixed numbers, preferring vulgar fractions, but I don’t mind them all that much. They do at least tell me where my number is between the natural numbers on the number line. I do admit the notation is hard to represent in typed text. For example if you type “1 2/3” it’s easy to think it was supposed to be “12/3”. Often I’ll put a + as in “1+2/3” since it’s definitely correct, but if I want to emphasise the one-number nature of it as opposed to making people feel like it’s an unfinished problem, I’ll type “1&2/3”. Anyway the picture doesn’t technically say any of this, it’s just what was on my mind when I made it.

    The surd sheep

    A grey root sign has a line drawing of a sheep sitting cross-legged beneath it. The sheep has its eyes closed and no visible mouth. It's holding its hooves over its ears.

    I was so happy that someone mentioned “third” in their list of one-syllable maths words because it meant I could definitely include “surd” too, since they rhyme. A surd is a number that is a root of a natural number that isn’t itself a square number. The most famous surd is 2\sqrt{2}, but you can name as many of them as you like, such as 17\sqrt{17} or 99\sqrt{99}. (Technically, you’re allowed to put any rational number that’s not a square in there and it’s still called a surd, but when people talk about surds it’s the ones with natural numbers that are almost always on their mind.)

    A surd is an irrational number, meaning that you can’t write the answer as a fraction of natural numbers, and if you figured it out as a decimal, the decimal digits would go on forever without ever ending up being the same block of digits over and over. For example, 2\sqrt{2} begins 1.41421356237…1.41421356237… . What this all means is the only way to talk about a surd exactly is to say what it’s the square root of. Essentially, they don’t have names of their own. If you were being poetic, you could say their names are unspeakable.

    And that’s exactly what they were called in ancient times: unspeakable or unhearable numbers. This became “deaf numbers”, and the Latin word “surd” means “deaf”. This long description is the reason why the surd sheep is covering its ears.

    The reason it’s got a serene expression and sitting crossing its legs is because the image of the sheep hiding under the root sign reminded me of a scene in the Australian drama “A Country Practice” in the early 1980’s where Shirley was meditating under her pyramid.

    The eight sheep and the nine sheep

    Two line-drawing sheep sit on the floor facing each other with smiles on their faces. The left-hand one is just placing the final block on a 2 by 2 by 2 cube and is pointing at the right-hand sheep. The right-hand sheep is just placing the final block on a wall with four blocks on the bottom row, three on the next and two on the next, and is pointing at the left-hand sheep.

    One of the people who replied to my request for one-syllable maths words listed only one-syllable numbers and suggested they might get some hate for it, but honestly numbers have the coolest properties and I wanted to display some of that in my sheep. Even some things that you can do with young children’s blocks are deeply interesting.

    The eight sheep here is just finishing a 2 by 2 by 2 cube with its blocks, which is showing that 8 is a cubic number. But also 8 is the first number of blocks that can be put into a rectangular prism that’s wider than 1 in all directions. You need your number to have at least three (not necessarily distinct) prime factors, and 9 only has two, so it can’t be done with 9 blocks. This is why the nine sheep is pointing happily at the eight sheep, because the eight sheep has achieved something the nine sheep can’t do.

    The nine sheep is just finishing a wall that’s 4 blocks wide on the bottom layer, 3 blocks wide on the next layer and 2 blocks wide on the next. Not every number of blocks can be stacked like that. That is, in a wall of some height more than one layer, and with the widths of the layers descending by one each time. In fact, the only numbers that can’t do this are the powers of 2. (This is the sausage-stacking theorem.) And 8 is a power of 2, so it can’t be done with 8 blocks. This is why the eight sheep is pointing happily at the nine sheep, because the nine sheep has achieved something the eight sheep can’t do.

    I could have given them shocked faces to convey this message, but I liked the idea of the sheep each being happy for the other when the other did something they couldn’t do. Maths should be like that, where we celebrate others’ achievements.

    The plane sheep

    A line-drawing sheep stands sturdily on its hind legs with one arm in front and one behind and a smile on its face. It stands on a square like a magic carpet. The design is nine smaller squares with three decorated with O and three decorated with X.

    Planes and lines are a big part of the maths I teach every day, and finite planes in particular are what made me want to study pure maths all those years ago, so I definitely wanted both of them in the book somewhere.

    From the start it made sense for the plane sheep to be riding the plane like a magic carpet. It also appealed to me that it’s flying because then the plane is acting like a plane in the sense of an aeroplane. The actual stance is strongly influenced by the wave sheep from the original book, which is surfing.

    I added the pattern to the plane quite a bit later. In my own favoured area of finite geometry, a plane is a collection of objects called points and lines with specific rules for when you say a point is on a line. One of my favourite planes is the affine plane made by adding extra lines to the game noughts and crosses, specifically the diagonal lines that go off one side and come in the other. I thought that adding a grid pattern with two of these extra lines would signal my love of finite geometry but also make it look even more like a magic carpet. Also it would mean that the sheep is riding an entire plane, as opposed to just a square section of a plane in Euclidean space that is actually supposed to be infinite.

    The line sheep

    A black line slopes downwards from left to right, and a line-drawing sheep is sliding down it on its bottom with its legs splayed outwards and a big open-mouthed smile on its face.

    My first idea was to have the line sheep “walking the line” but I felt like it wasn’t dynamic enough compared to the plane sheep. So I decided to have it sliding down the line. When I came to make the page I did toy with having it hang off the line so that it was a zip line, but there really wasn’t enough room.

    It was important for me to make sure the line went off both sides of the page to give the impression that maybe it goes forever, which is the definition of a line in maths (well, in Euclidean geometry anyway — lines are finite in finite geometry of course). A line that has ends is called a line segment.

    It was also important that the line was on the left-hand side of the page, because then the line has negative slope. We often say to students that a line has negative slope when it goes downwards as you travel from left to right, so I definitely wanted the sheep to be sliding down as it travels from left to right.

    I’m very happy with the tilt of the sheep’s body and the placement of its arms and legs to give just the right impression of carefree excitement.

    The plot sheep and the chart sheep

    Line drawings of two sheep standing on their hind legs and looking down at their own bodies. The left-hand sheep has a square body, which has an x- and y-axis drawn with several little spirals making a scatterplot. Four of the spirals are darker than the others. Some of them are quite far from the cluster of the other ones. The right-hand sheep has a circular body divided into four unequal sections. The narrowest section is decorated with tiny spirals, the next one is blank, the next one has middle-sized spirals and the biggest section has big spirals.

    Almost all of the sheep in the rest of the book are number or geometry sheep, so I really really wanted to have at least one statistics sheep. “Chart” was a great statistics word on the big list, and it did rhyme with “part” so that was useful. Interestingly even though several graphical data representations are called charts, statisticians call all of them plots when talking in general, so it was natural to have a plot sheep too.

    The first graphical representation most people meet at school that is called a plot is the scatterplot, so I really wanted to have a scatterplot, just so that the word plot made sense to most readers.

    Most statisticians are very disparaging of the pie chart, preferring bar charts where it’s much easier to compare the size of the categories. But the pie chart persists because it does make sense to people and it is useful for the specific case of showing one category is much bigger than the others. To me, the pie chart actually points to the important lesson that you do have a choice of how to present data and your audience and your purpose are part of that choice. So I chose to use a pie chart for the chart sheep.

    My first thought was to have the sheep pointing at their respective chart like they were giving a presentation. But I did want to also make the charts themselves more sheepy. I thought maybe the decoration on the charts could be wool spirals, and then I thought if the decorations were wool, then they could actually be wool and be on the sheep themselves. Then the plot and chart sheep would be the plot and chart sheep not because they made plots and charts but because they are a plot and a chart!

    And so they became what they are. The final choice I made was to have them looking down at themselves, which was harder to get right in the design of their faces than I expected! In the end I am so happy with how they turned out and I love them so much!

    When I made the original plot and chart sheep, I just made some nice-looking charts with some features worth talking about (such as an outlier on the scatterplot). But writing this blog post I realised I wanted more. Maths should make sense and data should tell a story, so I wanted the graphs to actually represent something. So I found some data in the book itself to represent.

    The plot sheep is now showing a scatterplot of each of the sheep in the book with the x-axis being the width on the printed A4 page and the y-axis being the height on the printed A4 page. I did take a bit of license and moved the points that were very close together a little so their spirals weren’t overlapping, but it’s pretty close to accurate. There were four pairs of points with exactly the same measurements, and I didn’t move those but put them in bold instead, because that’s an interesting feature of the data.

    The chart sheep is now showing the proportions of sheep in the book with the four mouth shapes: open smile, closed smile, frown and no visible mouth. I leave it to you to figure out which region is which, though I have to tell you the no-mouth region is the blank one, because of course it had to be.

    The whole sheep and the part sheep

    A line drawing of approximately one and a half sheep. The whole sheep is standing on two legs with the others hanging down at its sides. The part sheep is the same except the right-hand half has been removed with a neat vertical cut. There is no edge drawn along the cut.

    Someone suggested “whole” and “part” and I very much wanted to include it because part-whole thinking is such an important thing in early maths. After starting to draw it I also realised that I could make a reference to one of my favourite books of all time: The Phantom Tollbooth by Norton Juster.

    In the Phantom Tollbooth, when Milo visits Digitopolis, the City of Numbers, he meets a boy who is only half a boy: the right-hand half divided top to bottom. He explains that the average family has 2.58 children and he is the .58. Milo says he’s never met half a boy before and the boy stresses that it’s a bit more than half. “A few years ago I was only .42 and it was terribly inconvenient.” I have carefully compared the areas of the whole sheep and part sheep and the part sheep is 0.42 of the area of the whole sheep. It wasn’t necessary to go to this level of detail but I like knowing that level of detail is there.

    The height sheep and the base sheep

    A big grey triangle has two sheep in front of it. One is standing, stretching its body upwards and its arms up high to reach the top vertex of the triangle. The other is lying on the ground, stretching outwards, with two legs at one vertex of the triangle and one arm at another vertex, and one arm crooked over its body. Both sheep have line-drawing smiles.

    I was very enamoured with the base-face rhyming pair, not least because the s-sound is written differently in each, so I had to figure out how to do it.

    Base has multiple uses as a maths terminology but I had no inspiration for how to draw the numerical ones, so a geometrical meaning it had to be. Which made me think of the height and base of a triangle for the purposes of calculating the area. A triangle doesn’t naturally have one base and height — you have to choose them for your own convenience. So it occurred to me that the sheep should be showing you where the base and height are.

    I deliberately stretched out their bodies including the wool spirals to make it clear that both sheep are stretching. (Because of this, It may interest you to know that the base and height sheep appear as outliers on the plot sheep.) And I immediately had a very clear idea of the vibe that the base sheep should be giving, right down to the crooked arm.

    The overall vibe is of a cringe family photo of just the teenagers, which is exactly what I hoped for.

    The edge sheep and the face sheep

    A pyramid with two of its faces visible. The front face has a huge drawing of a smiling sheep's face on it. The opposite edge has a sheep holding onto the edge with three feet, and the other hoof is pointed outwards into the sky, where it is looking with a big smile.

    The face sheep was immediate: it should be the face of a polyhedron and the face of the polyhedron should be showing a sheep’s face. Because obviously what else could I possibly do but display that double meaning? There was never any doubt about this.

    And if there was face, then there should be edge too, since they are the two one-syllable words that describe parts of a polyhedron. It was a little harder to think of what to do for the edge sheep. I couldn’t have it lying along or sliding down the edge because I’d already done both of those. Really that just left climbing the edge, except I couldn’t help making it a bit more dramatic than that. This drama is why the polyhedron is a pyramid rather than, say, a cube, because scaling a mountain is much more exciting.

    The e sheep and the π sheep

    At the bottom of the picture is a number line with regular marks. Three of the marks are labelled 2.5, 3 and 3.5 below the line. Above the line are two line drawings of dancing sheep. The left-hand sheep has a slender body and is balancing on one leg just past 2.7. Its arms are curved, the two of them sloping upwards left to right. Its smile has a similar curve. The right-hand sheep has a circular body and is balancing on one leg just past 3.1. It has a big open-mouthed smile.

    Even before I asked for help with the brainstorm I wanted to make sure I had e and π, the famous numbers with letter names. I’ve always liked how these two useful numbers both have 3 as their nearest natural number, so I thought I’d somehow put them on the number line.

    Having them standing was a bit boring and also it would be a bit ambiguous exactly where they were on the line, so I decided I would have them dancing so that they could stand on one foot and use that foot to point out the correct place on the number line more accurately.

    But I did even more than that. The arms of the e sheep are curved (the only sheep in the book to do so) and the curve matches the shape of the graph of y=exy=e^x. That exponential growth graph is one of the reasons why the number e is so useful. Its smile is also curved along that same shape.

    The π sheep’s body is circular to reference the number’s importance to circles. I even made the e sheep a little thinner just to emphasise even more the roundness of the π sheep.

    The 1 sheep and the i sheep

    Text and line drawings. On the left-hand half of the page at the top it says "Here is the 1 sheep." Below this is a line drawing of a smiling sheep standing on its hind legs. On the right hand half of the page, the text reads bottom to top "And this is the i sheep." The same drawing of the sheep has been rotated 90 degrees anticlockwise.

    The number i is the number we attach to the real numbers thus creating the complex numbers and allowing us to solve all the polynomial equations. In particular it’s one of the two solutions to x2+1=0x^2+1=0 (the other solution is -i).

    The complex numbers are usually drawn on a plane, with the real number line in the middle like an x-axis and the unreal numbers extending up and down. You can see in my original sketch I was going to have the 1 sheep standing at 1 on the real axis and waving up at the i sheep sitting at i on the imaginary axis like it’s sitting in a tree. But I felt there wouldn’t be room for all that, and also I suddenly had a much better idea.

    One thing the complex numbers allow us to do is represent rotations as multiplication. In particular, multiplying any complex number by i results in a 90° rotation anticlockwise. There was once a meme that had a picture of a mobile phone labelled “Phone” and then a picture of one rotated 90° clockwise labelled “iPhone”, and I am both proud and slightly embarrassed to say I laughed out loud when I saw it for the first time. I thought that a sheep version of this meme would be in just the right spirit for this book. The moment I had the inspiration to turn the words too was a wonderful moment. I still laugh at myself looking at it. (That’s why this picture is the only one in the blog that includes the words too.)

    I chose to write it “1 sheep” and not “one sheep”, like I did back with the eight and nine sheep, because I really wanted to emphasise the symbol for the number in this rhyming couplet since it involves three other numbers each with a one-letter symbol.

    One final choice I made on this page was to use the exact same illustration I did when I drew the whole sheep, since 1 is also one whole.

    The point sheep

    A line drawing of a sheep standing on its hind legs and pointing with one hoof at a round black point hanging in the air.

    A theme has been developing through all of these designs where I seem to enjoy a good double meaning pun. So of course the point sheep is pointing at a point. I carefully turned one half of its hoof to make the pointing even more pointy.

    The set sheep

    Two big curly brackets have a line drawing of a sheep between them. The sheep is bracing with its legs and holding the brackets outwards with its arms. It has a frown on its face.

    Sets are pretty much what the whole of maths is built on so I definitely wanted to include a set sheep. The notation for describing a set uses curly brackets (or braces) and the information describing the set is enclosed between. So my first thought was to put a sheep in there. As soon as I thought that, I couldn’t help but feel it was a bit claustrophobic in there and that the poor sheep might feel like the brackets are not just enclosing but closing in, which is why this sheep is one of only two with a frown.

    I did worry about that frown going against the play theme of the book, but I kept it because it pointed towards something else important about sets. If those two curly brackets did come all the way in, the sheep would have to escape and we’d be left with nothing between them at all, in which case it would be empty. In fact, it would be the empty set. And the empty set is such a special idea in maths. In fact, you can construct the natural numbers and the operations on them from the laws of set theory and the existence of the empty set. In that sense it’s almost like the natural state of a set is emptiness and we prise those brackets apart to squeeze things in and make other sets. So I kept my sheep there to hold the brackets open.

    The cube sheep and the net sheep

    In the top right corner of the picture is a sheep whose body is a 3D sketch of a cube. Its head is at one corner of the cube and it has one leg coming down from each of the four bottom edges. In the bottom left of the picture is a cross-shaped net of squares with wool spirals on them. In the middle of this stands a naked sheep with no wool, covering its body with its arms and with an embarrassed expression.

    My vision of the cube sheep was clear from the moment I read the word cube on someone’s list, though it still took some work to make it look right on paper.

    Not until I put the words cube and net together did the inspiration strike for the net sheep. A net is what you get when you unfold the faces of a polyhedron. And if the faces of the polyhedron are the wool of a sheep of course that means the sheep is left with no wool when they unfold.

    I am very proud of my design for the net sheep. The slight forward lean,the tilt of its head, the closed eyes, the demure straight legs, and the crossed arms all work together to give just the right impression, if I do say so myself.

    The net itself is the traditional cross-shaped net. There are ten other nets for a cube and my favourite is the one that’s a zigzag, but I wanted to use the traditional one so that the reader might be more likely to recognise it, and also it’s the one that feels most to me like it fell downwards and outwards from the top.

    As I said, I am very proud of the net sheep’s design, and I thought it was such a good joke that I deliberately saved it for last. It is such a thrill to see people laugh when they get to that page.

    The other sheep

    In the original Green Sheep book, when the narrator asks the final “But where is the green sheep?”, there is an illustration of a hillside with many sheep doing all sorts of interesting things. (My favourite is the sheep wearing a fruit headdress like Carmen Miranda.) But there is no text to describe what the sheep are. You get to decide for yourself what they get to be called.

    I decided that I would try to squeeze in on a similar page in my book as many sheep as possible for the words I collected but hadn’t used so far. I made these all pretty quickly, but still some thought went into each one. These are the sheep:

    The kite sheep

    A line drawing of a sheep whose body is shaped like a kite and whose legs and arms splay upwards. It has a big smile on its face. A black line extends downwards from the pointiest part of its body.

    There is an official quadrilateral called a kite. It has two pairs of adjacent equal sides. There’s a kite-flying sheep in the original Green Sheep, so to make it different, I made the actual sheep itself kite shaped and also being a kite.

    The arc sheep

    An arc of a circle is drawn in black and curves upwards. Along its curve, a sheep lies on its back with its eyes closed.

    An arc is a part of the circumference of a circle. It felt right to have the sheep using it like a hammock.

    The span sheep

    A line drawing of a sheep sitting on the ground holding its arms out in a V shape. Between them, a grey plane extends outwards.

    In linear algebra, the span of two vectors is the set of all points you can reach by doing linear combinations, which is a plane, if they’re not pointing in the same direction. When I describe span to students I use my arms to mark out the vectors and sweep out the span between them, so the sheep is doing the same.

    The plus sheep and times sheep, who are together the field sheep

    A line drawing of two sheep standing on their hind legs, holding hands and smiling at each other. One has a plus sign on its tummy and the other has a times sign on its tummy.

    These two operations have one-syllable names when you talk about doing them to actual numbers (as opposed to minus and divided by, which have more syllables). This is appropriate because they are in some sense the most fundamental of the operations. Indeed, there are ways to define these operations for other things that aren’t quite numbers, or define them in new ways for numbers. When you do that, you can call your set of things a field. That’s why they’re holding hands, because it’s only together that they define a field.

    The Venn sheep

    A line drawing of two sheep standing on their hind legs and looking down at their perfectly round bodies. Their bodies overlap in the middle. The left sheep has lots of small anticlockwise spirals indicating wool and the right sheep has big clockwise spirals. The place where their bodies overlap has big anticlockwise spirals.

    I was disappointed to not put the Venn sheep elsewhere in the book, so I definitely wasn’t going to leave them off this page! These two are sorting their spirals. The left-hand Venn sheep has anticlockwise spirals. The right-hand Venn sheep has big spirals. The place where they overlap has both. I made it this way because Venn diagrams are most useful for sorting things, over and above anything else they might do.

    The shear sheep

    A distorted picture of a shorn sheep crossing two arms across its naked body.

    The action of shearing in maths is what happens when you keep the bottom of something still but stretch the top sideways. Everything except the bottom moves sideways but things further up move further to the side. It’s the action that turns a rectangle into a parallelogram. I absolutely had to have a sheep without its wool because of the meaning of shear that normally goes with sheep. So this is the exact net sheep demonstrating the result of a shear.

    The dice sheep

    A line drawing of a sheep sitting on the ground rolling two big dice between its feet. It has a smile drawn on its face.

    The only probability word in the whole book, so I had to have it. I’ve made sure that all six faces of a regular die are shown across the two dice.

    The tree sheep

    A tree made of black dots joined by black lines starts with one branch at the bottom, then two, then four, then eight. A line drawing of a sheep sits in the fork of a branch on the right-hand side. It holds its legs out to the side and has a big open-mouthed smile.

    In graph theory, a graph is a diagram where nodes are connected by edges. So this sheep could also be called the graph sheep. However, a special kind of graph is a tree, where there is a base node and it’s connected to some number of notes, and then each of those is connected to some and so on. This one is a binary tree, where each node after the base is connected to two nodes further up. The sheep definitely had to be sitting up in the branches of the tree.

    I did mess with the order compared to the original Green Sheep. In the original, there is a line that isn’t part of the usual pattern, asking, “Where is that green sheep?” This line is after the wide shot of all the extra sheep. But I wanted mine to be before, so that you really felt like you were looking further. Also I wanted my line to be even more different to the others. I asked, “IS there a play sheep?”. The idea is that you’re losing faith and wondering if the play sheep is even there to find. This is supposed to mirror the experience of many people in maths class that it doesn’t feel playful at all. It felt more impactful to put the page with no words and lots to look at after this admission of hopelessness. You have to pause and consider. Plus, I think it sets you up to read it aloud better in the right rhythm.

    “The” play sheep

    After the reader has had a good look at the previous page with so many sheep, the text instructs, “Think very carefully. Don’t get it wrong…” I chose those words with care. These are the sorts of things people might hear in a maths class that they don’t think is playful. There is usually quite the fear of getting things wrong and fear of looking stupid is the antithesis of play.

    A line-drawing sheep stands on two legs, with one arm down and one arm up to its chin. Thought bubbles rise from its head with the words "I wonder..." in one and "What if..." in the other.

    We finally get to the last sheep. This one is thinking “I wonder…” and “What if…” with a smile on its face. And the text reads, “Maths was play all along.”

    It turns out that the thing you can get wrong isn’t your maths problem, but coming to the belief that maths isn’t playful. If you have ever thought anything beginning with “I wonder…” or “What if…”, then you have been playful, and all maths started with someone somewhere thinking those things.

    And if maths was play all along, then all those sheep we’ve seen so far have all been play sheep. Because a maths sheep is a play sheep.

    I deliberately chose this sheep to be as plain as possible, other than the thinking pose, because I wanted it to be clear that it wasn’t the activity the sheep was doing or anything in the way it looks that made it a maths or a play sheep; it’s the way it’s thinking. Playfulness and mathematicalness are mindsets.

    The end

    I hope you’ve enjoyed this, the most complete story I could tell of the creation of Maths Sheep Play Sheep. I am glad to have finally written it all down so people, including me, can come and look at the decisions I made and some of the maths behind the sheep. Thank you for reading!

  • Two-sided ruler constructions 2: Fundamentals

    This is the second in a series of blog posts about two-sided ruler constructions. Here are all the blog posts in the series:

    1. Introduction
    2. Fundamentals (you are here)
    3. Rhombuses
    4. Copying and cutting
    5. Perpendicular lines
    6. Parallel lines
    7. Circles without circles
    8. Equilateral triangle and regular pentagon

    This blog post is about the fundamentals of what a two-sided ruler can do, leading up to some basic constructions.

    So, what can a two-sided ruler do?

    It can draw lines.

    Draw a line anywhere
    1. Put the ruler down and draw along one side.
    Draw a line through a specific point
    0. Start with an existing point.
    1. Align the point on one side of the ruler and draw along that side.
    Draw a line through two specific points
    0. Start with two existing points.
    1. Align both points on one side of the ruler and draw along that side.

    A one-sided ruler can do all of this of course. What a two-sided ruler can do that a one-sided ruler can’t do is draw two parallel lines a ruler width apart.

    Draw two parallel lines anywhere (a ruler width apart)
    1. Put the ruler down and draw along both sides.
    Draw a line parallel to an existing line (and one ruler width away)
    0. Start with an existing line.
    A long straight line
    1. Align one side of the ruler along the existing line, and draw along the other side.
    A line on a piece of paper with a ruler aligned along it. One hand holds the ruler in place against the line, while another hand draws along the other side of the ruler with a blue pencil.
    Done! The second line is parallel and one ruler width away from the first line.
    Two lines on a piece of paper, one black and one blue.
    Video here

    You don’t have to actually draw the first line either. If you have two points, you can align them both on one side of the ruler and then draw along the opposite edge. This draws a line that is a ruler width away from the line that joins the two points without actually drawing the line that joins the two points – you don’t have to draw the line to know where it is.

    Draw a line a ruler width away from both of two points
    0. Start with two existing points.
    Two black points on a piece of paper.
    1. Align both points on the same side of the ruler, and draw along the other side.
    Two points on a piece of paper with a ruler aligned to both of them on the same side of the ruler. One hand holds the ruler in place against the line, while another hand draws along the other side of the ruler with a blue pencil.
    Done! The line just drawn is parallel to the line joining the two points.
    Two black points on a piece of paper with a blue line a short distance away from them.
    Video here

    There is a second way to use two existing points to draw lines using the two-sided ruler, and that’s to align the two points on opposite sides of the ruler.

    Two points drawn on paper, with a ruler aligned to both, but on opposite sides of the ruler. The ruler reaches from the top left left to the bottom right of the picture.

    (Of course, this can only be possible if the points are at least a ruler width apart or you won’t be able to fit the ruler between them. For points aligned on the same side of the ruler, it doesn’t matter how far apart they are.)

    This move of aligning one point on one side of the ruler and the other point on the other side of the ruler is so common in the later constructions, it needs its own name. I will call it cross-aligning, as in, I’ll write an instruction like, “Cross-align these two points.” I’ll also talk about cross-aligning a line segment, by which I mean cross-aligning the endpoints of the line segment.

    While I’m at it, I’ll call the action of aligning two points on the same side of the ruler, or aligning a line along one side of the ruler, side-aligning.

    I’m going to need that saving in word count because there are two ways to both side-align and cross-align and there will be times I’ll have to use up some words to specify which I want.

    Two points drawn on paper, with a ruler aligned to both on the top edge of the ruler.Two points drawn on paper, with a ruler aligned to both on the bottom edge of the ruler.
    Two points drawn on paper, with a ruler aligned to both, but on opposite sides of the ruler. The ruler reaches from the bottom left to the top right of the picture.Two points drawn on paper, with a ruler aligned to both, but on opposite sides of the ruler. The ruler reaches from the top left left to the bottom right of the picture.

    You might not believe there’s only two ways to cross-align two points, thinking surely you can fit the ruler in there in lots of ways. I urge you to give it a try and you will soon believe it, just like I did. On top of that, I can prove it.

    Proof:

    Let the width of the ruler be \(1\), and consider two points \(A\) and \(B\) a distance of \(d\) apart with \(d > 1\). Cross-align \(A\) and \(B\) in one direction and draw along both sides of the ruler. Imagine a line from \(B\) perpendicular to both sides of the ruler, meeting the opposite side in the point \(C\). Let the angle \(\angle CAB\) be \(\theta\).

    A diagram. Two points labelled A and B are joined by a blue line labelled d. There are two parallel lines, one through A and one through B. The angle between the line through A and the line from A to B is labelled theta. A dotted line starts at B and goes from one parallel line to the other, with right-angle markers where it meets each line. The endpoint of this dotted line on the line through A is labelled C. The line itself is labelled 1. Text on the right hand side says sine of theta = 1 over d and theta = arcsine of 1 over d

    The triangle \(\triangle ABC\) is a right-angled triangle, and the side opposite the angle marked \(\theta\) is \(1\) since it’s the ruler width, while its hypotenuse is \(d\). Therefore \(\sin(\theta)=\frac{1}{d}\).

    End proof!

    At the most basic level, this means that given a specific distance between two points, cross-aligning them will always produce a specific angle between the edge of the ruler and the line joining the points. And it works the other way too: given a specific angle between the edge of the ruler and another line, the ruler-edges will always cut off the same length segment of line between them.

    More precisely what this means is that the two-sided ruler can calculate sine and arcsine. (Technically it’s cosecant and arccosecant, but that’s much too hard to say.)

    Anyway, that’s my Lemma.

    The cross-align arcsine lemma.

    Let the width of the ruler be \(1\).

    When two points that are \(d > 1\) apart are cross-aligned, the acute angle that the ruler edges make with the line joining the points is \[\arcsin\left(\frac{1}{d}\right)\]

    When the sides of the ruler meet a line in an acute angle \(\theta\), then the distance between the points where the line meets the sides of the ruler is \[\frac{1}{\sin(\theta)}\].

    A diagram. Two points labelled A and B are joined by a blue line labelled d. There are two parallel lines, one through A and one through B. The angle between the line through A and the line from A to B is labelled theta. A dotted line starts at B and goes from one parallel line to the other, with right-angle markers where it meets each line. The endpoint of this dotted line on the line through A is labelled C. The line itself is labelled 1. Text on the right hand side says sine of theta = 1 over d and theta = arcsine of 1 over d

    There’s video explaining the lemma and its proof here.

    I’ll be using the full trigonometricality of it in the last two blog posts, but for now the most important thing to remember is that cross-aligning a specific length always produces a specific angle, and vice versa.

    Right now, I can do the first traditional construction, which is to double a length, at least when the length is longer than the ruler width.

    Double a line segment (longer than ruler width)
    0. Start with a line segment longer than the ruler width.
    A line drawn on a piece of paper.
    1. Cross-align the opposite ends of the line segment, and draw along one side of the ruler.
    A ru;er exactly covers a line. One hand holds the ruler still and the other draws along one side of the ruler with a pencil.
    A horizontal line on a piece of paper, with a diagonal line passing through its righthand endpoint.
    2. Side-align the line you just drew, and draw along the opposite side of the ruler to produce a parallel line.
    The previous diagram but now a ruler is aligned with the diagonal line and a hand draws along the other side of the ruler with a pencil.
    The previous picture with no hands now has an extra diagonal line parallel to the first.
    3. Finally, side-align the original line segment and draw along that side of the ruler to extend the segment to where it meets the parallel line you just drew.
    The ruler is aligned to the horizontal line and a hand draws along its edge with a pencil.
    Done! This extended segment is twice as long as the original.
    The previous picture without hands. Now there are two horizontal line segments one after the other, each with a diagonal line at the righthand end.
    Video here

    This procedure works because of the cross-align arcsine lemma, since I know a certain angle will always cut off a certain length. It’s in Wernick, but I thought of it before reading that paper, and he doesn’t use my lemma to prove it.

    You can repeat the procedure as many times as you like to multiply the length of a line segment by any natural number, though perhaps you’ll want to extend the line segment first rather than second. It will also produce as many points as you want on an existing line all an equal distance apart by just putting your ruler anywhere to make the first segment.

    I can also do another traditional construction, which is to double an (acute) angle.

    Double an angle
    0. Start with two lines meeting to make an acute angle.
    An angle drawn on a piece of paper.
    1. Side-align one arm of the angle and draw along the opposite side of the ruler. You only need enough of this line to see where it meets the other arm of the angle.
    A ruler is aligned with one arm of the angle, with fingers holding it down, while another hand draws along the other side of the ruler with a pencil.
    An angle with a small line marking a point on one side of the anfel.
    2. This point just made and the vertex of the angle have already been cross-aligned. Cross-align them in the other direction and draw along the side of the ruler through the vertex.
    The vertex and a point on an arm of the angle have been cross-aligned by the ruler. A hand draws along the side of the ruler through the vertex.
    Done! This line just drawn and the nearest arm make the same angle as the original.
    A diagram with two equal angles sharing an arm, and that arm has another small line crossing it.
    Video here

    This works because whenever you cross-align any line segment you always get the same angle, due to the cross-align arcsine lemma.

    I find it very interesting that neither Wernick nor Birrell contain this construction. Instead, Wernick has a much more complicated construction built from making right angles and doubling lengths, both of which take quite a few lines to draw. I think it’s because they were excited about rhombuses, which on reflection is not that surprising, because rhombuses are pretty cool.

    The next blog post is all about rhombuses.

  • Too many presents

    Once upon a time when my daughter was very young, she was given a lot of presents – it was a birthday or Christmas but I can’t remember which. What I do remember is that she played with just one present all day long, leaving all the others untouched.

    I think we sometimes do the same thing to our students that friends and family did to Charlotte: we give them too many presents and then get upset when they don’t play with them all.

    Let me explain.

    Playing with it is one of the main ways to get a deep understanding of concepts and to get fluency with procedures. You ask yourself, “What would happen if…?” and say to yourself, “I wonder…”, and you try things out in different combinations. It’s awesome when it happens and you feel all sorts of positive feelings like curiosity and joy and satisfaction. Even teachers who don’t consciously subscribe to a play-based approach are usually happy when they see this sort of thing happenning. Many of the people who become university lecturers had similar experiences when they were students and assume their students also play with the ideas in their courses.

    And the students actually do. It’s amazing how often even the struggling students are trying to explore. And the students who were engaged with the content long before they joined your course are sometimes aching for chances to explore that aren’t being given to them. But there’s a big problem: there’s just not enough time.

    A university course has multiple new concepts and procedures every week, and there’s just too many of them to play with all of them. Yet the assignment questions tend to assume a level of familiarity with every single thing in the course that only comes with a decent amount of playing with every one. There’s just too many things in the course to be able to give all of them the time they need. And if a student gets nerdsniped and goes on a deep dive on one thing, they are forced to sacrifice play time with the other things.

    I see it most clearly in two places.

    First, in a course like Nursing where students are expected to be fluent in all the various ways to do calculations with multiplication and division quickly without a calculator. This fluency comes to most people through years of play: trying new problems, seeing how others do them, noticing strategies worth trying, and noticing when they’re not worth trying, storing away relationships between numbers. But in a first-year Nursing course with students who have past traumatic experiences with maths, there is literally not enough time for this kind of play, even if a student realised that the play was the thing that helped them be better at getting the answer, because they also have to play with how to listen to patients and what all the drugs do and any number of other things that go into becoming a nursing professional.

    Second, in a pure maths course for students who chose a pure maths degree because they were interested in pure maths. These students deeply want to play, but there are so many concepts coming at them, there is just not enough time to play with them all, and they feel overwhelmed. Especially when their lecturer assumes unconsciously they have already done the play just because they’ve got previous experience with maths.

    My great hope is actually to give students less to play with, so they have time to play with each of them as they go. But failing that, we need to at least not get upset when they don’t play with everything, and definitely not assume that they are lazy or uninterested or ungrateful. They’re just toddlers with too many presents.

  • Quarter the Cross: Colouring

    Quarter the Cross is one of my favourite activities of all time, whether in maths or just life. I learned about it way back in 2015 and have been mildly or very obsessed with it ever since. This blog post is about one particular version of the Quarter the Cross problem you might like: the colouring version!

    You can read the rest of this blog post, and four other related posts, in PDF form here 

    The titles of the five posts in the series are:

    • Quarter the Cross (2016)
    • A Day of Maths: Quarter the Cross (2016)
    • David Butler and the Prisoner of Alhazen (2016)
    • Quarter the Cross: Colouring (2020)
    • Quarter the Cross: Connect the Dots (2020)

    Some resources linked from this post:

  • Struggling students are exploring too

    I firmly believe that all students deserve to play with mathematical ideas, and that extension is not just for the fast or “gifted” students. I also believe that you don’t necessarily need specially designed extension activities to do exploration – a simple “what if” question can easily launch a standard textbook exercise into an exploration.

    This is lovely, but one problem is those students who on the face of it don’t want to play. The majority of students I work with in the MLC are not studying maths for maths’s sake, but because it is a required part of their wider degree. That is, I am helping students who are studying maths for engineering, or calculations for nursing or statistics for psychology. A lot of these students just want to be told what to do and get the maths over and done and don’t like “wasting” time playing with the ideas.

    Or so I thought. I have realised recently that actually they do like playing with the ideas. I just couldn’t see that this was what they were asking for.

    One of the questions I like to use to play with maths is “what if?” To ask it requires the asker to notice a feature they think might be different, and wonder what would happen if it was, and so investigate how this feature interacts with the other features of the situation. This investigation of how ideas interact is exactly what mathematics is, to me. I am very very used to doing it at extracurricular activities like One Hundred Factorial, and it’s easy to do with students who are doing very well with their maths and have the breathing space to wonder about this stuff because they’ve finished their work. For students working on an assignment they are really struggling with that they wonder about the usefulness of for their degree and which is due in the next 24 hours, it’s not so easy.

    Only maybe it’s a little easier than I thought, because I started to notice that the students were already asking questions about the connections between things.

    A very common question students ask around exam time is “What would you do if the question was like this …?” as they suggest a change in a small detail that makes it more similar to the past exam question they have at hand. I used to get really annoyed at this sort of question, but then I realised that in order to ask this question they had already noticed that the two problems were similar in most respects and different in this one. Sure, they may be motivated by a belief that success in maths is about a big list of slightly different problems and remembering ways to deal with each, but on the other hand they have noticed a relationship and are trying to exploit it, This is a mathematical kind of thought, to look for similarities and differences and relationships, and we can hang on tight to it and actually learn something!

    Another kind of question students ask is “Why is this here in this course?”. I used to get annoyed at the whinging tone of voice here, but then I realised that a student is begging for a connection between this and the rest of what they are studying, and connections is precisely what understanding maths is all about. I can respond to it by saying that yes I am also confused about the curriculum writer’s logic for including it, but then we can search out connections together to see if we can find them.

    A third kind of question is the one where the student looks at the lecture notes or solutions and asks “How did they know to do this?”. Sure, it’s usually motivated by wanting to be able to successfully do it in their own exam in restricted time, but on the other hand it does recognise that there ought to be reasoning involved in that decision, as opposed to just guessing. This expectation of reasoning is the beginning of believing that they themselves could reason it out too.

    My second-last kind of question is “Why is this wrong?” as the student points to the big red cross on their MapleTA problem on the screen. Sure they just want to make it all better so they can submit the damn thing. But they are also recognising that there must be a reason why it’s wrong, and so are looking for meaning. These students are often ripe for the experience of looking at the information and how it’s related to what they’ve entered, thus doing exploration. They are also usually ready to try various different ways of entering the result, or different strategies of getting a right answer to see where the edges of the idea are.

    The final kind of question I want to mention isn’t even a question, it’s an exclamation of “It doesn’t make sense!” Even this is telling me that the student thinks it ought to make sense. They are crying out to do something to make sense of it. Often they describe being almost there and needing something to push them over into understanding. This student is ready to explore the edges of the understanding they do have to see where the nonsensical stuff can fit, or where their ideas need some tweaking to fit together better.

    It is very helpful to me as a one-on-one or small group teacher, or even a lecturer with a big room of students in question time, to be able to see the questions struggling students are asking as cries for sense-making exploration. It doesn’t matter that they are struggling or don’t understand things. Indeed, being in a situation of not knowing is exactly what we are doing for the fast students when we give them extension activities, so why is the everyday maths the slower students are not understanding any different? In my experience students like being treated as if they are behaving like mathematicians when they have these struggling sorts of questions. And it was so much easier to treat them that way after I realised they are behaving like mathematicians when they have these kinds of questions.

  • Fairy Bread

    Fairy bread, in case you don’t know, is an Australian children’s party food.

    A tray of fairy bread triangles. That is, bread spread with butter and with hundreds and thousands (tiny ball shaped coloured sprinkles) sprinkled on top.

    Here’s how to make fairy bread: take white bread, spread it with margarine, and sprinkle with hundreds and thousands. Now cut into triangles and serve.

    Notes:

    • It has to be white bread. If you try to make fairy bread with wholemeal bread, or multigrain bread, woe betide you!
    • It has to be margarine, not butter. Butter may just be acceptable only if it’s the kind that is spreadable directly from the fridge. It may be that “margarine” means something different in other places in the world, so just in case, what I’m thinking of the butter-like spread made of plant oils that is spreadable directly from the fridge and can spread very thinly.
    • Hundreds and thousands are a kind of brightly-coloured sprinkles that are shaped like very tiny balls. If you use chocolate sprinkles, or sprinkles shaped like little sticks, or coloured sugar, then it’s not fairy bread.
    • It has to be cut into triangles. Don’t ask me why. Triangles are more magical than rectangles I suppose.

    When I went to Twitter Math Camp in the USA in 2017, one of the lunchtimes I made fairy bread for everyone and passed it out. It was heaps of fun seeing people’s reaction to it, which was mostly good, though mixed with various levels of surprise and confusion.

    A Twitter post from Heather (Kohn) Russo @HeatherRusso99 on 30 Jul 2017 with text and a photo. The text says: Fairy bread from Australia is delicious! Thank you @DavidKButlerUoa #TMC17 The photo contains: Six people smiling at the camera. Behind us is a big room with chairs and tables and lots of other people. I am the second person from the left, holding a tray of fairy bread. The other people are all holding a triangular piece of fairy bread.
    https://twitter.com/heather_kohn/status/891339803056275456 

    For me, fairy bread is strongly linked to memories of my childhood, and every time I eat it I am surprised again at how good it is. I mean, it’s the stupidest thing: bread and margarine with sprinkles. Yet somehow all the more awesome for that.

    And here is where I am supposed to make a point about maths or teaching or maths teaching. But that might ruin the whole thing. Like those horrible people who try to make fairy bread “more healthy” by using wholemeal bread. Honestly people! It’s a party food – just own it!

    Actually this reminds me of people who are always trying to get me to make a mathematical moral to my play. Yes there are times when the mathematics people do is deeply meaningful or useful for solving real world problems, and there are other times when it’s just for fun and there is no other purpose to enjoy myself and spend time with good people. Sometimes I need to be left to simply enjoy it, thank you very much.

    Oh look, I did make a point. I hope it didn’t ruin the experience too much.

    This comment was left on the original blog post:

    David Roberts 10 July 2018

    As I’m sure you know, David, Dutch people love sprinkles of all kinds on bread, and for some reason especially for breakfast. When I was in the Netherlands a few years back, at a supermarket, there were (at least) two whole shelving units for different kinds of sprinkles. I do wonder if fairy bread was introduced via some widely-sold party-food cookbook a few decades back (edit: well, it’s at least 90 years old, according to https://en.wikipedia.org/wiki/Fairy_bread  !), where the author/compiler was inspired by this cultural phenomenon.

  • Childhood memories

    Two books I’ve read recently have encouraged me to investigate my memories from childhood. In Tracy Zager’s “Becoming the Math Teacher You Wish You’d Had“, she urged me to think about my maths autobiography to see what influenced my current feelings about maths. In Stuart Brown’s “Play“, he urged me to think about my play history to see what influenced my current feelings and tendencies about play. In the spirit of those two, here are some of my earliest memories about maths and play.

    In primary school, I have very few memories of actually being in a maths class, and all of them are negative. I’ve related two of them already in this blog. One was my memory of doing a maths assignment about one million dollars, where the financial aspect distressed me to tears. Another was my memory of my Year 6 teacher attempting to teach us averages using cricket.

    The only other maths class memory is of a test I did in Year 3. I had been sick with asthma for a couple of weeks and came back to school on the day of a test. I dutifully did the test and actually got almost full marks. The only thing I got wrong was the meaning of the word “net” in the phrase “net weight” as you might see listed on a packet of food. I distinctly remember it being a multiple choice question and ruling out two of the answers as ridiculous, but basically having to guess between the other two. I was angry because how could I possibly know that? Everything else was just logic and so I could figure it out for myself, but you can’t figure out the meaning of a word without more context. Eight-year-old me was an astute little person.

    Across my primary school career, I do remember a strong feeling of pleasure and fascination associated with construction toys. I remember absolutely loving the MAB blocks, in particular the moment when I replaced ten units with a long, and ten longs with a flat and ten flats with a block. Interestingly, my memory is only of the blocks themselves and I can’t pinpoint a year level or a teacher that goes with this. I also remember loving playing with polydrons and attribute tiles, but again the memory is just about the fascination of playing with them, and not about any particular maths class. In fact, thinking carefully about what is around me in these memories, I seem to be in a hall or a library, rather than in a classroom.

    Outside of school, I remember playing a game in each new playground, where I would try to do every part of the play equipment exactly once without crossing my path. Would I have to interpret the slide as both a sliding down and a climbing up in order to do it? Would I end up trapped on the top, or could I finish on the ground where I started?

    At home, we’d build elaborate maze-like cubby houses out of spare mattresses and sheets (we lived in a house where visitors often stayed over). I remember planning these out with my brother with explicit conversations of how we would fit more rooms and pathways into the space of our shared room. I also remember spending hours making designs with a ruler and compass. Or by folding paper several times and cutting out holes then unfolding and sticking on a contrasting colour.

    It seems that for me, geometrical play holds the strongest positive mathematical memories from my primary school years.

    Indeed, my very first memory of primary school is about geometrical play. It’s the moment I walked into my kindergarten classroom for the first time. We walked into a carpeted play area, and the desks and blackboard were some distance away at the other end of the classroom. Here in the play area was a bookcase filled with big thick brown blocks. Some of them were on the floor being made into a car track by some other children. I remember immediately wondering about how the various straight and curved pieces might fit together. I have some vague memories of tying various combinations on other days in kindergarten.

    Earlier than this, one of my only memories of Happy Days Pre-School was getting out the giant foam blocks from the store room under the building and playing with them on the grass.

    It’s funny that so many of my positive mathematical memories are geometrical when now I also have such a love of the structure and behaviour of numbers. Maybe that came later, though my mother says as a very young child I was always “playing number and letter games in my head”. I myself can’t remember doing that, but my mother is a very astute person and I am not about to doubt her observations.

    My earliest memory of any kind is of a cool hard flat greenness. My mother says this is probably a memory of the back verandah at the house we lived in before I was two years old – it had a green-painted concrete floor. I wonder if other people’s earliest memories are about feelings of space and colour. If so, maybe it means we’re all geometers from birth. Or maybe it’s just me.

    What is clear is that it’s hardly surprising that I ended up doing a PhD in finite geometry even though the original undergraduate degree I enrolled in was mathematical physics. I think the fundamental pull towards that geometrical play was calling me all along, considering how strongly I gravitated towards it in primary school despite the rest of maths not being so inspiring.

    If you’re reading this, I don’t know what you might learn from my story. But for myself I realise I am right where I belong.

    This comment was left on the original blog post:

    V Lakshmi 27 September 2017:

    Nice article! Infact, childhood memories have something to learn and plays an important role in future they are like the learning stages check this peace very interesting http://www.publicdebate.in/childhood-happiest-part-life-agree/ 

  • Book Reading: Play – How It Shapes the Brain, Opens the Imagination and Invigorates the Soul

    Looking back at my blog over the past few months, I’ve done a lot of these “book reading” posts. I really did mean to do some more on other ideas, but I felt I had to get these thoughts out of the way first. So here’s another book reading post, this time about the book “Play: How It Shapes the Brain, Opens the Imagination and Invigorates the Soul” by Stuart Brown (with Christopher Vaughan).

    (You can read this blog post and all other Book Reading posts in PDF form here. )

    Kassia Wedekind gave a talk about play in maths where she used some ideas from the book (you can see the video here ), but I didn’t realise what the book was until later when she responded to my request for how people define play. I immediately looked it up in the library and placed a hold on it so I could borrow it as soon as I could. I finished reading it pretty quickly, but I’ve only had the chance to stop and write about it now.

    The basic message of the book is that humans are actually wired to play and that play is essential for our survival and wellbeing, both individually and as a species. He doesn’t set out to apply this to education and certainly not maths education, but I have to say I agree with Kassia that it does indeed apply.

    The first few chapters describe what play is, and how it seems our brains are built with a need for play, even as adults. The last several chapters talk about applying play to various aspects of our lives including parenting, work and relationships. I will talk about Part One a lot, and then about Part Two a little.

    What play is

    One of the most interesting and useful bits of the whole book was Brown’s description in Chapter 2 of what play is. He is loath to give a definition, but he does describe a couple of lists of noticeable features that play tends to have. These were really useful both to broaden the scope of what can be called play, and to narrow my focus to the key features.

    His first list of features are things you can notice about play and people engaged in it. You can tell someone is playing when you notice most of these things (p17):

    • Play is apparently purposeless: that is, it seems to be done for its own sake and not because it has practical value.
    • Play is voluntary: it’s not required by duty or forced upon you.
    • Play has inherent attraction: it makes you feel good so you want to do it.
    • Play has improvisational potential: there is scope to put things together in new ways, to do things differently and try things out.
    • People who play experience freedom from time: they lose a sense of time passing.
    • People who play experience diminished consciousness of self: they stop thinking about their thoughts or how they look to others or whether they’re making mistakes.
    • People who play have continuation desire: they want to keep going and find ways to make it keep going.

    The last three are the aspects that Kassia mentioned in her ShadowCon talk, and at the time they really spoke to me. I really could imagine the times when I had experienced all three of these and they really were playful activities – those activities when you were so engrossed that you missed lunch or turned around to notice ten people watching you that you didn’t know were there. Interestingly, when I have noticed people watching, my play usually stops, or at least turns into more a performance, which isn’t really the same thing.

    So those last three just put names to how play already felt to me. What the full list added for me was a description of the sorts of things that might possibly encourage play. It would seem that an activity with potential to choose whether to do it, where there’s no particular performance goal in mind, and where you have scope to try different options, would be the sort of activity where play is possible. On the other hand, an activity that is tightly constrained with a specific goal in mind seems much less likely to produce play without the people involved being brave enough to break the constraints.

    The second list presented is a number of stages a player will go through as they play, taken from Scott Eberle. He says that all players may not go through all stages and not necessarily in this order. Still I agree with Brown that it’s still useful.

    • Anticipation: Curiosity and sometimes a bit of anxiety as you think about what will happen when you engage in the play.
    • Surprise: Something new happens or you see something a new way.
    • Pleasure: Usually caused by the surprise.
    • Understanding: Incorporating new ideas into what you know.
    • Strength: Being empowered because you have done something new and succeeded.
    • Poise: Feeling contented and composed.

    I particularly like the idea of anticipation. I can feel it when I do a puzzle and I’m investigating the ideas connected to it, getting a growing feeling that something cool is about to appear. And then it happens and it’s a surprise but not a surprise. When I dig into it to understand I feel ready to face something new in the future. That final idea of poise is also a good one. I see it as that quiet feeling of contentment, different from the intense spark of pleasure caused by the surprise, that allows you to feel ready to leap into anticipation once more.

    At this point I was in a dangerous place. I could recognise these aspects in myself and my children for activities that I knew already were play, but I hadn’t fully realised the extent of what Brown was trying to achieve here. The thing is, as he says on page 60: “play is a state of mind, rather than an activity.” Chapter 3 really opened my mind to understand that there are many experiences and activities that can be playful – even things that I might consider work. The key is in the state of mind that goes with it.

    In this chapter, Brown sets out a list of several “play personalities”. He describes eight types of people based on the sorts of things they prefer to play with, for example, the “creator”, the “storyteller”, the “joker” and the “director”. I’m not sure I agree totally that there is such a thing as a play personality, but I do agree that people would have certain preferences. The key thing I realised reading this is that there was a much wider scope of things that could be called play than I had realised. For example, the “competitor” plays by aiming to win. Their improvisation potential comes from figuring out how to get the furthest or the most within the confines of the rules, and how far they can push themselves. I had never considered this as play at all, since competition usually turns me off completely and makes things feel like work. But reading this I understood where the playfulness was and why a person competing against me may not actually have any grudge against me at all; instead I’m just a part of their game. Another example is a “collector”, who plays by collecting things. Their improvisation potential stems from the choices they make of what is in or out, or how to classify and order the collection of things they have. Again this was a completely new way to look at play for me.

    Why play is so good

    Brown argues that play is essential to help learn, to be creative and to be happy. He notes how in animals (in particular in bears), play helps them learn social cues that mean they can function successfully in their communities. Without play they can’t test the boundaries of what is acceptable and what is not acceptable. In Chapter 4: “parenthood is child’s play”, he outlines many interesting things that children learn through play, not least of which are the physical laws of nature and their own personalities.

    The really profound thing I got from the book was the importance of adult play. I have read some things recently saying that play is only appropriate for young children and shouldn’t be encouraged in high school because adults have to work. Brown argues that actually, adults are only capable of surviving work because they are able to infuse play into it. Work is soul-destroying when there is no scope for improvisation, no part of it gives you a choice, and when it focuses entirely on key outcomes you must be assessed against. These are the opposite of the lists of the features of play. On the other hand, given a choice to voluntarily do something, with freedom to try out things and fail, you are uplifted. So in short, making work more playful makes it more fulfilling. Even if you can’t change the constraints of your work, Brown says that allowing play elsewhere in your life will make work better too.

    Basically his argument is that as humans we need the opportunity to try new things in a safe environment, and only play allows for this. We need to be able to continue to develop, and play is the catalyst for development. As he says rather harshly on page 73:

    When we stop playing, we stop developing, and when that happens, the laws of entropy take over – things fall apart. Ultimately, we share the fate of the sea squirt and become vegetative, staying in one spot, not fully interacting with the world, more plant than animal. When we stop playing, we start dying.

    Helping myself and others play

    One question that niggles at me is this: if play is a state of mind, then how can I help anyone to play, or even myself? How can I change anyone’s state of mind? It’s a big question, and Brown goes some way to answer it in the final chapter, at least for helping yourself to play. He suggests a few strategies and I think they fall into three main categories what I want to synthesise here:

    • Move: Brown says “motion is perhaps the most basic form of play”. He says that basically most of your cognition and perception are actually encoded into your brain in the first place via movement. Therefore movement has a way of shortcutting all of your cognitive and emotional inhibitions to play. In short, if you want to relearn to play, then move.
    • Be near others who are playing: Brown says several times across the books that one of the best ways to learn to play is to be with a dog, or a toddler. When you see them playing, you often can’t help but join in. I’ve seen it myself at One Hundred Factorial — people relearn to play by being with others who are playing.
    • Find a safe environment: There is nothing more toxic to play (and indeed general wellbeing) than being in an environment where people judge you or where you are afraid to be yourself. In order to be free to play, you need to be free from fear, so you need to find a place where you don’t have to be afraid. Usually for us adults, it has to do with having the right people around us. As he says on page 216: “If people around you cannot learn to understand your need for play, find people who do.”

    As a teacher and a team leader, I think these three things tell me a lot about how I can help my students and my staff to play and so be happier and more productive people. I need to work super hard on creating that environment where it is safe. My classrooms need to be places where it’s ok to make mistakes, to try out new ideas, to suck when you do something for the first time. I myself might need to be the person nearby who is playing so that others can see it’s ok and learn how. And I can get my students to move, at the very least to move their hands, to forge that new connection to the fundamental state of play.

    All in all, I really enjoyed the book. At times it did slip into the style of a motivational speaker making grandiose claims with references to specific people’s stories. But I still think the points it made are valid points, and it has made me think differently about what play is and where it fits in mine and my students’ life.

    This comment was left on the original blog post:

    Kassia Wedekind 8 June 2017

    I love your observations adults and play.

    It particularly makes me think that teachers need opportunities for mathematical play too! Both because I think experiencing the feeing of play and math together is really important for Ts when they’re thinking about what they want for their students and because I think the play process really lends itself to thinking about reasoning and how we figure things out in math.

    Thanks for writing about this and thinking about it with me too.

  • Book Reading: Math on the Move

    Over the last week or so, I have been reading the book “Math on the Move” by Malke Rosenfeld (subtitled  “Engaging Students in Whole Body Learning”). Ever since connecting with Malke on Twitter back in June or July, I’ve wanted to read her book, and I finally just bought it and read it. Now that I’ve finished, it’s time to write about my thoughts.

    (You can read this blog post and all other Book Reading posts in PDF form here. )

    The book is all about whole-body learning as it relates to maths and dance, mostly focussing on pre-school to Year 6. Some of you may be wondering why I, a university lecturer with a doctorate in pure maths, would be so very interested in something to do with dance-and-maths at the primary level. My first response to that is that you clearly need to get to know me a bit better! Perhaps start by checking out the following past blog posts: Kindy is awesome and The Pied Mathematician of Hamelin.

    My second response is that seeing things from a new perspective is one of the best ways to understand them better and to understand how you understand them in the first place. I was fascinated by this new medium of a moving body for thinking about maths and I wanted to get the benefit of reading the thoughts of someone who has already considered it deeply. And Malke Rosenfeld is just that person, because reading the book you can tell immediately that she has thought very deeply about it.

    The book has two main parts. The first part is about the concept of movement-scale activities and the body as a thinking tool in mathematics. The second part is about the Math in Your Feet program, which is also about the body and its movement as a thinking tool, but even more than that, that dance itself is a mathematical thing worth thinking about.

    The first part had me thinking from the moment I started reading. Malke argues that scaling up a mathematical idea to the scale where your whole body can interact with it or be it can give insights and understandings not available in any other way. Malke gives examples of number lines and hundreds-charts of a scale you can walk on, and building polygons out of knotted rope that has to be held by multiple team-members. My head was whirring with the possibilities. Immediately I imagined what it would be like to stand on a surface defined by a two-variable function and questions about directional derivatives occurred to me that never had before. Imagine what would have happened if I could actually stand on the surface itself!

    Malke makes the very important point that meaningful moving-scale mathematics learning is not about using your body to memorise things, or to copy what is on the page. It is about using your body to make movements that are intrinsically related to the thing you are trying to understand. Stretching your arms to copy the drawn shape of a linear graph while saying its formula is not really meaningful. Perhaps more meaningful would be walking on a graph drawn on a basketball-court-sized coordinate grid and explicitly discussing how you move relative to the x and y axes. (And just now writing this, I suddenly have this cool idea to really understand discontinuities as places in the graph where the mover has to literally jump to get to the next point.) The discussing I mentioned is important too – meaning happens when the ideas are discussed and compared.

    The second part of the book, as I said earlier, described the Math in Your Feet program. Children are given a two-foot by two-foot square to dance in and a number of possible ways to move. They create steps within this framework and work with partners to make dance steps the same and different, to combine patterns of steps into longer patterns, and to transform dance movements through rotation and reflection. There’s detailed information about how the program moves forward, and the ways to facilitate work and play and thinking and discussion, as well as lots of linked videos to really see the action. You could be forgiven as a high school or university maths teacher for thinking this part of the book doesn’t really apply to you as much as the movement-scale exploration of existing maths ideas. I say you could be forgiven, but you’d still be wrong.

    Firstly, there is a whole heap of very deep discussion on what it means to give the students the power over their own learning. Malke discusses the importance of clear simple boundaries, of precise language, of encouraging language, of reflection, of getting students to share, and of ways to help children to focus. All of this is vividly displayed throughout the Math in Your Feet chapters of the book, and what you can learn here would translate to all sorts of other teaching situations. It is worth watching all the videos jut to revel in Malke’s skill of never praising product but always excitedly praising participation and practice.

    Secondly, it is this part of the book that is the most mathematical, from my perspective as a pure mathematician. The dance moves within the tiny square space are an abstract mathematical idea that is explored in a mathematical way. We ask how the steps are the same or different from each other, identifying various properties that distinguish them. We investigate how these new objects can be combined and ordered and transformed. We try out terminology and notation to make our investigations more precise and to communicate both current state and how we got there. These are all the things we pure mathematicians do with all our functions, graphs, groups, spaces, rings and categories. The similarity of this to pure mathematical investigation in striking.

    I have been changed by reading this in ways that I am not capable of processing completely at the moment. Not until I have more chances to try out movement-scale investigation of maths, and mathematical investigation of movement, will I feel I have a handle on it. But it’s a pleasant sort of feeling all the same.

    One final warning: If you read this book, don’t attempt to do it in an armchair, or on the train, or while walking. It won’t work. In order to read this book effectively, you need to sit with access to a computer to watch the video clips, and with access to a 2 foot by 2 foot square on the floor to try the dance steps in. Also if you’re like me, you’ll need somewhere to write down quotes which speak deeply to you. Quotes like this:

    Using the moving body in math class is about more than getting kids out of their seats to get the wiggles out or to memorize math facts. Instead, we need to treat the movement as a partner in the learning process, not a break from it.  pp 1

    Using tangible, moveable objects (including the moving body) can be useful in math learning as long as attention is paid to the math ideas as well as what you do with the object. pp 13

    Using language in context to label, describe, and analyze this work is one of the most powerful ways to help learners create meaning and understanding. pp 112

    Grading or judging a child on his or her ability compared with others’ is harmful in this creative environment. This is a place where the focus should be firmly on the ideas expressed, not on the facility or ease of that expression. pp 146

    We want math to make sense to our students, and the moving body is a wonderful partner toward that goal. pp xvii

    Thank you Malke.

    These comments were left on the original blog post:

    Joy 5 December 2016

    What about people with disabilities? How can teachers and students teach/learn if they are disabled?

    David Butler 6 December 2016

    Malke has a section of her book specifically about how to include children with special needs. She also encourages teachers to use the children to show examples of dance, so a teacher with a movement disability I think would be very successful with the Math in Your Feet program.

    Jeremy 15 January 2017

    Very interesting David !! I am very glad to have found this blog,

    I’ll admit that the life size maths visualisation technique seems to me to be much more helpful than the maths in your feet program. I does sound like an interesting book 🙂!

  • David Butler and the Prisoner of Alhazen

    Once upon a time, I did a PhD in projective geometry. It was all about objects called quadrals (a word I made up) – ovals, ovoids, conics, quadrics and their cones – and the lines associated with them – tangents, secants, external lines, generator lines. During the first two years, I did talks about my PhD research, which I could not resist calling “David Butler and the Philosopher’s Cone” and “David Butler and the Chamber of Secants”.

    At that time, my use of JK Rowling’s titles had to stop because there was no suitable mathematical thing to insert into the third title. It’s been ten long years since “David Butler and the Chamber of Secants”, and finally I have found something to use. Hence, welcome to…

    You can read the rest of this blog post in PDF form here.