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.

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 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

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

“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

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 $\sqrt{2}$, but you can name as many of them as you like, such as $\sqrt{17}$ or $\sqrt{99}$. (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, $\sqrt{2}$begins $1.41421356237…$ . 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

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

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

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

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

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

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

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

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 $y=e^x$. 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

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 $x^2+1=0$ (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 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

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

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

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

The span sheep

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

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

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

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

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

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.

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!