I think the most important and most common move in algebra is this:

Replace something with something else you know it’s equal to.

This rule isn’t even an algebra rule, it’s a rule you’ve used in plain arithmetic. Look at this working:

(3+4)×5-6

= 7×5-6

= 35-6

= 29

All of those steps were replacing something with something else I know it’s equal to. First, I replaced the (4+3) with 7, and then I replaced 7×5 with 35. Finally I replaced 35-6 with 29. The reason I was able to write the “=” signs there was because I knew each expression was produced by replacing something with another thing it’s equal to. Of course they are the same. If you have ever written your working in the way I did there, then you were using the replacement move.

Indeed, this move is the heart of how mathematicians write calculations. We always move from one step to another by replacing something with something it’s equal to. Knowing that this is what we do, we *read* other people’s working by comparing each expression in a chain to the next to see what it is that has been replaced.

It’s interesting that nobody has ever told me explicitly that this is how to read maths working that’s written with a chain of equal signs. I just somehow figured it out. I am sure quite a few of my students don’t actually know this strategy.

This idea that anything can be replaced with something it’s equal to is for me the major thrust of all the algebra laws and identities.

For example, the distributive law―a(b+c) = ab+ac―doesn’t tell me how to “expand brackets”. Instead it tells me any time I see a(b+c), I’m allowed to replace it with ab+ac, and every time I see ab+ac, I’m allowed to replace it with a(b+c). The same goes for all the algebra laws, even though some of them seem very complicated.

For me, completing the square illustrates this very well. For example:

3x² + 24x + 7

= 3(x² + 8x) + 7

= 3(x² + 8x + 16 – 16) + 7

= 3((x+4)² -16) + 7

= 3(x+4)²- 48 + 7

= 3(x+4)² – 41

The first move was to replace 3x² + 24x with 3(x² + 8x) by the distributive law.

Then I replaced 0 with +16-16.

Then I replaced x² + 8x + 16 with (x+4)².

Then I replaced 3((x+4)² -16) with 3(x+4)² – 48.

Then I replaced -48 + 7 with -41.

The second pair of moves are usually extremely surprising to students, and even I have to stop and look closely when I read already-written-down completing the square working. But they make sense when I compare the line of working to the one above and see which parts are the same, so that I can deduce that the parts that change must have been replaced because they are equal.

Incidentally, this is why I am such a stickler for keeping things lined up when you do maths working if you can. Changing the order makes it hard to deduce what’s been replaced from one line to the next. Of course, you can change orders, but I prefer to do that as its own move so you can see when it happened.

Finally, the replacing move is actually used when solving equations too, even though it’s usually hidden. Look at this working that I did in a chemistry lab with a student once.

ρ = m/V

ρ × V = m/V × V

ρ × V = m

The move from line 1 to line 2 was multiplying both parts of the equation by V, but the move from line 2 to line 3 was replacing m/V × V with m because you know they are equal.

The student who I did this with was ok with the multiplying both parts by the same thing, but they were *not* ok with the replacing. They complained that I hadn’t done the same thing to both sides. And this was when I realised that there was another move in algebra and it’s much more fundamental than doing the same thing to both parts of an equation. And we need to tell students explicitly about the existence of this move.

So there you have it. Algebra is not all about doing the same thing to both sides, it’s very very often about replacing something with something else it’s equal to. Keep an eye out for it next time you read or do any maths working, and maybe explicitly remind your students every so often when it happens.

Actually just a bit of an epilogue: the replacing move is really rather fun to do “in reverse” as it were. Usually we do it in arithmetic by replacing an expression with a single number, but there’s nothing stopping you replacing a single number with an expression and so finding some rather complicated expressions for familiar numbers. This first one is based on the fact that if 1 = 2-1, then any 1 can be replaced with 2-1 at any time. (If you click on the tweet you can see me replacing 1 to 9.)

Replacing 1. pic.twitter.com/ojsYcthQt9

— David Butler (@DavidKButlerUoA) August 19, 2019

May you enjoy your replacing too.

]]>- “Did you go to the lecture?”
- “Have you started yet?”
- “How many of the exercises have you done?”

These questions all have answers that are morally Right or Wrong. The answers a student gives make the student out to be a Good Student or a Bad Student. And if a student has the Wrong Answer, they will feel ashamed.

I know many people who believe it is very important to send students the message that they should go to lectures, start assignments straight away, and do all the exercises. While these are all things students could do to help themselves, they’re not the most important thing to focus on when they are here seeking support from me. They can’t change any of those things right now, so all a question like those does is make them feel ashamed. And, as Turnaround for Children CEO Pam Canto says in this blog post, “shame is toxic to positive outcomes”.

Shame is the feeling that you are a bad person, that there is something wrong with you. Guilt is a bad feeling about your actions, which is unpleasant, but may make you want to change those actions in the future. Shame is the next level, where you feel you have been exposed as the horrible person you really are. A person who feels shame won’t try to change their actions, they’ll just try to avoid situations that expose them, which will just make the problem worse. I don’t want this to happen to my students, and I certainly don’t want them to think that seeking support from me will expose them to shame, or they will decide not to seek help.

Once upon a time, I realised that I was causing a student shame, and I decided that I would give myself a new principle.

Never ask a question that has a morally wrong answer.

This is one of the rules I use to evaluate if my question is useful and choose a better alternative.

For example, I could ask “Did you go to the lecture?”, but there is definitely an answer to this question that is morally wrong and having to give that answer will cause shame. Do I really want to know if they went to the lecture? How will that help? Maybe what I really want to know is what the lecturer has to say about the topic, since that might be useful. In that case, I could ask “What did the lecturer have to say about this?” The student doesn’t have to reveal their attendance status to answer this question, thus avoiding the shame. Even better would be to avoid the awkward moment where they have to reveal they don’t know, and say, “It would be useful to know what the lecturer says about this. Can you tell me what they said, or tell me where we might go looking for that?”

For my second shame-inducing question of “Have you started yet?”, the first simple fix is to remove the “yet”. That implies they should have started already. The second fix is to think about why I want to know this? Maybe I want to know what they’ve done already so we can build on it. In that case I could just ask “What have you done so far?”, since that’s directly asking for the information I want. But there is still an implication that they should have done something, so causing shame if they have to reveal they’ve done nothing. So instead I could ask “What are you thinking about this problem?” or maybe “How do you feel about this problem?”. These let me get into their head and heart and I can help them move on from there. I might be able to ask them about what they’ve done so far later, or it might not even be important because they’ll tell me what they need to help themselves.

This second example highlights another principle, which is to ask open ended questions, preferably about student thoughts and feelings. This makes it much easier to ask questions without morally wrong answers, because there are no specific predetermined answers in particular! (Asking open-ended questions is actually one of the factors in SQWIGLES, the guide for action I give to myself and my staff at the MLC.)

So, I urge you, think about whether the questions you ask have a morally wrong answer, and if so, try a more open-ended question that is less likely to cause the shame that is so toxic to success.

]]>This Open Day I had a remarkable idea: instead of stating in the rules that the goal is to achieve the target, and trying to encourage people to take a different approach, what if I just *changed the stated goal! *I don’t know why I didn’t think of it before, to be honest!

So this is my new version of the Numbers game instructions:

NUMBERSHelp to make calculations that produce as many numbers near the target as possible, each calculation using some or all of these two big and four small numbers, and any combination of +, -, ×, ÷, and brackets. (Numbers can only be used in each calculation as many times as they are in the list.)

For posterity, I choose the numbers randomly from pop-sticks that I painted:

- the small numbers are chosen from a set with two each of of the whole numbers 1 to 10
- the large numbers are chosen from a set with one each of 25, 40, 50, 60, 75, 100, 120, 125.
- the target is chosen from a set with two 0’s and three each of the digits 1 to 9.

Since coming up with this on Open Day, I’ve put a daily Numbers game on the board in the MLC Drop-In Centre, and it’s been a delight to have students and staff join in and add their solutions.

After explaining to students that the goal is now to get as many numbers near the target as we can get, all of a sudden they start just saying things to write on the board. Mostly we don’t even get the target until quite late in the piece, because people are excited that they can modify what’s there to get other numbers. Even if the target *is *produced early, there is still a desire to fill in the numbers on either side, and then the whole set of 10 that contains the target.

That modification of existing answers is my favourite part. I used to work it in before when I would put a wrong answer up and fiddle with it to get closer to the target. But now it just happens naturally because the goal is just to make as many different answers as possible.

Here are four games from the last couple of weeks:

As a testament to how absorbing this new version is, I came back to work on Monday last week, after having worked from home on Friday and a student arrived early to see if there was a game yet, as well as to show me all the ones they came up with when I wasn’t there on Friday.

I am so very pleased that this new version of the game works so well, and a little ashamed that I didn’t think of it earlier. As much as my catch cry is “the goal is not the goal, the end is not the end”, really sometimes you have to explicitly change the goal.

PS: Every day, people have come to look at the board when I am starting the game, and they say “Ohh, this is like Countdown!” My response is always that yes it is like Countdown, but it’s BETTER because of three major differences:

- There is no clock, so it’s not stressful.
- We are working together, not competing, so it’s friendlier.
- The goal is to get all different answers instead of just one, so we get to keep playing.

]]>

**Players:**

- This game is for two players, or two teams.

**Setting up:**

- Each player/team choose six
*different*numbers between 0 and 10 (not including 0 and 10), each number with at most one decimal place.

*For example, you might choose 0.8, 3.2, 5.6, 5.9, 6.0, 8.7.* - Without the other player/team seeing, write the numbers in increasing order in the MY SET template.

*For example, the numbers above would end up looking like this:*

*(0.8, 3.2)U(5.6, 5.9)U(6.0,8.7).*

(I have a printable version of the game with the rules and templates to fill in, which can be turned into battleships-style stands.) - This notation represents a set on the number line made of three separate open intervals. Colour in the intervals on the MY SET number line.

*For example, your completed MY SET notation and coloured intervals might look like this:*

- Set up so that you can write and draw on the THEIR SET template and number line without the other player/team seeing the MY SET template and number line.

*One possibility is to use the printable template to create battleships-style stands. Another possibility is to play on either side of a freestanding whiteboard, though you would have to draw your own set and number line.*

**The goal:**

- The goal is to guess what the other player/team’s three intervals are. In other words, you must guess exactly their six numbers.

**Your turn:**

- On your turn, you suggest up to
**five**sets of numbers like “all the numbers less than D away from C”. - After each suggestion, the other player/team gives one of three responses:
- “All of those numbers are inside my set” — or “inside” for short.
- “All of those numbers are outside my set” — or “outside” for short.
- “Some of those numbers are inside my set and some are outside” — or “both” for short

- Note: the responses do not include the endpoints of any of the intervals. If the whole interval except the endpoints is inside their set then the answer is “inside”, and if the whole interval except the endpoints is outside their set, then the answer is “outside”.

*There are examples of suggestions and how to answer them in the next section.* - The suggestions made on the same turn must all have the same centre C, and the distance D must be smaller each time.

*For example, you could make the following suggestions on the one turn:**“All the numbers less than 2 away from 6.5”**“All the numbers less than 1 away from 6.5”**“All the numbers less than 0.5 away from 6.5”**“All the numbers less than 0.2 away from 6.5”**“All the numbers less than 0.1 away from 6.5”*

- You don’t have to use all five suggestions. If you want to ask about a different centre or a wider distance, you have to wait until your next turn.
- You can write any notes in any way you like at any time in the game. It is a good idea to keep a record of the suggestions you have made so far.

**Ending the game:**

- Once during the game, on your own turn and instead of making any suggestions, you can say you are ready to guess. Then you say the six endpoints of the intervals.
- If you are right for all six numbers, you win! If you are wrong for at least one number, you lose and the other player/team wins! Either way the game is over.

**Examples of answering suggestions:**

These examples show visually the situations where you would say “inside”, “outside” and “both” in response to the other player’s/team’s suggestion. Your set is (0.8, 3.2)U(5.6, 5.9)U(6.0,8.7), which is drawn in blue in the diagrams. The other player’s suggestion is drawn in yellow slightly higher on the diagram, with the centre marked as a red vertical line.

- They say “all the numbers less than 0.3 away from 0.3”.

Your response: “Outside”.

- They say “all the numbers less than 0.6 away from 5”.

Your response: “Outside”.

(Notice how the answer is still “Outside”, even though the other player’s suggestion shares an endpoint with your set.)

- They say “all the numbers less than 0.5 away from 2.4”.

Your response: “Inside”.

- They say “all the numbers less than 1 away from 7”.

Your response: “Inside”.

(Notice how the answer is still “Inside”, even though the other player’s suggestion shares an endpoint with your set.)

- They say “all the numbers less than 1.5 away from 3”.

Your response: “Both”.

- They say “all the numbers less than 0.3 away from 6.1”.

Your response: “Both”.

**Introductory play:**

When you are first learning to play, you may want to agree to each have only one interval in your set for your first game, then increase the number of intervals to two and then three in later games.

In this section I’ll describe some thoughts about the game I’ve had through playing it. Note there are some spoilers, so I recommend playing the game yourself before reading this section!

I’ve played several games of Number Neighbourhoods across the months since February, and I’ve thoroughly enjoyed them all. Even now, after getting quite a bit of experience, the game feels much harder than Digit Disguises, but I don’t think that’s a bad thing. I think one of the things that makes it feel harder is that the information you collect doesn’t join together as easily as it does in Digit Disguises, which tends to produce a cascade of figuring stuff out towards the end of the game. But there are still some really cool moments of logic that happen in most games.

Here are the whiteboard of notes from a recent game, so you can see the logic that goes into it. The left-hand set is me and a student Kristin. The right-hand set is another pair of students.

Interestingly, though Kristin and I were certain of our set, we got it “wrong” because our opponents thought that the answer should be “both” when an interval was inside the set but included an end-point. I’ve actually updated the rules here in the blog to explicitly point out that case, so it’s less likely to happen in the future!

One thing that’s really interesting about the game is the amount of concentration it takes to even answer the question by another player! The act of seeing whether an interval specified as centre and radius is inside an interval specified with two endpoints is surprisingly difficult, no matter how many times you do it. But that’s really cool, because it’s definitely a skill that Real Analysis students need to think about!

My absolute favourite moment is when someone picks a boundary point as their centre, in which case the answer is going to be “both” all the way down. If someone playing the game with me watched closely, they could probably guess they had a boundary point by my excitement and the sudden increase in the speed at which I answer.

I also like how every new player suddenly realises in the first two turns that once they get an answer of “inside” or “outside” they can stop that turn, because going further in won’t change the answer. I did actually toy with putting that stopping condition in the rules, but decided against it because it is such a nice little moment of “I figured out something cool” I can give to new players. (This is one of the spoilers I warned you about earlier.)

I haven’t really come up with any decent strategy for winning the game yet, or for choosing a “most hard to guess” original set. I have thought about it a bit, but I don’t know if I really *want *a perfect strategy, so that I don’t feel tempted to use it and ruin the game for new players!

For now I am enjoying the experience of playing, and especially playing in teams and discussing the logic and strategies with my team-mates, which is my favourite part of all my battleships-style games.

If you would like to see an entire game, then check out this game I played very slowly over Twitter with my good friend and colleague Lyron.

Oh Fun! Ok I'll play! I've randomly generated my 6 numbers in R using:

set.seed(…)

sort(floor(101*runif(6))/10)I'll tell you the seed when you guess my numbers correctly! Do you want to start?

— Lyron Winderbaum (@LyronW) July 23, 2020

Along the way in the game above, since it was sometimes days between turns and I had the time, I made a Desmos thing for recording guesses and answers, and you might be interested to see that too. This graph is Lyron’s guesses of my set.

Finally, as a bit of an epilogue, I thought you might like to hear how I went about designing the game and the choices that I made along the way.

The game Digit Disguises, which I invented last year, was inspired by a puzzle which I created specially to give a similar feel to the maths in a course like Abstract Algebra. In Algebra you investigate the properties of numbers (and other things) by the way they interact with each other, and so a game where you find numbers based on their interactions with each other matches the vibe of such a course. But there are two broad flavours of pure maths that tend to feel different. A course like Algebra is of one flavour, but a course like Real Analysis is a different flavour. So ever since I created Digit Disguises with Algebra in mind, I thought it would be cool to make a similar game that had Real Analysis in mind.

In Real Analysis, you define a set as open if at every point inside it, you can find an interval centred at that point that fits completely inside the set; and you define a set as closed if at every point *outside *it, you can find an interval centred at that point that fits completely outside the set; and a point is a boundary point if no matter how small you make an interval centred there, it always has some of it inside and some of it outside the set. So given any set, there are three kinds of points and you can decide between them by looking at intervals centred there. And as long as you know in advance if boundary points are in or out, you should be able to find the entire set in this way. This seemed like an idea you could build a game around!

Riffing off Digit Disguises, which uses the digits 0 to 9, I decided the set you were looking for should be between 0 and 10. I also wanted it to feel *small *because Real Analysis is about things being *close *to each other, so I went for decimal places. I did briefly toy with the idea of making it numbers from 1 to 99, since that would be equivalent and not require operations with decimals. But that felt so *huge *and also put a focus on integers, whereas Real Analysis is about the continuously full number line in all its detail and the tiny distances between things. Plus I decided that actually, I *wanted *a chance to practice my decimal calculations. So I stuck with 0.1 to 9.9. Plus as I said before, it did feel like it was a proper partner to Digit Disguises then.

And so the idea for the game was born. I wanted the idea of boundary points to happen several times during a game, so I decided that the goal set should be three intervals, in order to have six boundary points to consider. And I also liked the idea of something being outside the set even though it was between pieces of the set.

I thought about the concept in Real Analysis of finding an interval small enough to fit inside (or outside) the set, and decided I wanted that idea to be in the game too, so I came up with being allowed to make several guesses on the same turn as long as you make your intervals go inwards. After playing it against myself a couple of times, I decided I should restrict the number of guesses to five so each turn didn’t go too long.

I briefly considered changing it so that you specified the endpoints of the interval you guess on each turn, but it lost the cool |x – b| < epsilon feel, so I abandoned that idea.

The game was almost complete, and just needed a name. Neighbourhood is a terminology they use in Real Analysis to describe an interval centred at a point, and we were using numbers, and I used alliteration already for the titles of the other games, so Number Neighbourhoods seemed like the most natural choice.

I played the game against myself a couple of times, and conscripted a student in the MLC to play with me too. Then I played with some students at a games night. These helped me to define what examples I needed to include in the rules. And that’s how I created the game of Number Neighbourhoods.

]]>In 2016 I created the iplane idea, which allows you to locate the complex points on a real graph.

In case you haven’t heard of it or you need reminding. The idea is that at every real point (p,q) of the real plane, there is a planes-worth of complex points attached, all of which have coordinates (p+si, q+ti). The collection of all the points with real part (p,q) I called the iplane at (p,q), and I imagined the iplane as a transparent sheet attached at the point (p,q) which I could flatten out to see its points when I needed to.

Ever since I had this idea, I have wondered on and off about the complex points on a circle. It’s time to write about what I’ve found.

In this blog post, I will investigate the complex points on a real circle centred at the the origin. (In later blog posts, I’ll think about “unreal” circles, but you may need to wait a while for those.)

I’ll consider the circle with equation x²+y² = r², where r is a real number with r>0, and I’ll try to find the complex points on this circle in the iplane attached at the real point (p,q).

The points in this iplane have coordinates (p+si,q+ti) for real numbers s and t. If such a point is on the circle I’m considering, then it would have to satisfy the equation. So I’ll sub it in:

(p+si)² + (q+ti)² = r²

p² + 2psi + s²i²+q² + 2qti + t²i² = r²

p² + 2psi – s²+q² + 2qti – t² = r²

The number on the left is the same number as the number on the right, so their real and imaginary parts will be the same.

Real parts:

p² – s² + q² – t² = r²

p² + q² – r² = s² + t²

s² + t² = p² + q² – r² (Equation 1)

Imaginary parts:

2ps + 2qt = 0

ps + qt = 0 (Equation 2)

These equations together describe the (s,t) coordinates of the points in the iplane at (p,q) that are on the circle. Let me investigate them separately.

Equation 1 has real solutions for s and t — and they have to be real because s and t are real numbers — when

p² + q² – r² ≥ 0

p² + q² ≥ r²

√(p² + q²) ≥ r

Now √(p² + q²) is the distance of the point (p,q) from the origin, so there are solutions for s and t in Equation 1 when the point (p,q) is at least r from the origin. That is, when (p,q) is on our OUTSIDE the circle!

Just to be absolutely sure, let me just check.

When (p,q) is inside the real circle, then

p² + q² < r².

So p² + q² – r² is negative, which means that

s² + t² = p² + q² – r² has no solutions.

So there are no points of the circle in the iplane at (p,q) if (p,q) is inside the real circle.

When (p,q) is on the real circle, then

p² + q² = r².

So p² + q² – r² =0, which means that

s² + t² =0,

And that means s=0 and t=0.

This also satisfies Equation 2 since p*0 + q*0 =0.

So the only pair (s,t) that satisfies both equations is (0,0), which corresponds to the centre of the iplane at (p,q).

That is, when (p,q) is on the real circle, then the only point of the circle in the iplane at (p,q) is (p,q) itself.

Ok, so I can now get back to the iplanes at the points outside the circle.

Let (p,q) be a point outside the circle, which means p² + q² > r² and so p² + q² – r² is positive.

Therefore s² + t² = p² + q² – r² is the equation of a circle (inside the iplane at (p,q)), centred at (p,q) and with radius √(p² + q² – r²).

That radius √(p² + q² – r²) ought to be a short side in a right-angled triangle with hypotenuse √(p² + q²) and other short side r.

But I know those distances! The distance √(p² + q²) is how far (p,q) is from the origin, and the distance r is the radius of the real circle. If the distance √(p² + q² – r²) meets a radius in a right angle, then that means it’s a tangent to the circle! So that means I am looking for the distance along a tangent drawn from the point (p,q) to the real circle! Of course, this construction is happening inside the iplane, as it is lying flat on top of the real plane so I can see the real circle through it.

The points I am looking for are on a circle of radius √(p² + q² – r²) drawn inside the iplane centred at (p,q). But this is only based on Equation 1. The complex points on the original circle in the iplane also satisfy Equation 2:

ps + qt = 0

(p,q) · (s,t) = 0

So the vector (s,t) is perpendicular to the vector (p,q). The vector (s,t) is the vector from the centre of the iplane to the point (s,t) inside the iplane, while the vector (p,q) is the vector from the origin to (p,q). So the line ps+qt=0 inside the iplane passes through the centre of the iplane at (p,q) and is perpendicular to the line from the origin to (p,q). Which means the points I am looking for can be constructed like this.

To find the points of the circle with equation x² + y² = r² in the iplane at a point (p,q) outside the circle:

- Lay the iplane at (p,q) flat over the real plane.
- Draw a tangent from (p,q) to the real circle.
- Find where this tangent meets the circle, then draw a circle inside the iplane through this point, centred at (p,q).
- Draw a line perpendicular to the vector (p,q) inside the iplane at (p,q).
- Find where this line inside the iplane meets the circle inside the iplane.
- These two points are the points of the original circle inside the iplane.

This GeoGebra graph allows you to see the complex points on a real circle in the iplane at a moving point (p,q). You can turn on and off the construction lines used to find the points.

I feel like I need to do one final thing: to find the actual coordinates of those points, which means formally solving those equations.

s² + t² = p² + q² – r² (Equation 1)

ps + qt = 0 (Equation 2)

From Equation 2:

ps = -qt

p² s² = q² t²

Multiply both parts of Equation 1 by p²:

p²s² + p²t² = p²(p² + q² – r²)

Substituting the result of Equation 2 into this:

q²t² + p²t² = p²(p² + q² – r²)

(q² + p²)t² = p²(p² + q² – r²)

t² = p²(1 – r²/(p² + q²))

t = ±p√(1 – r²/((p² + q²))

From Equation 2:

ps = -qt

s = q/p t

So if t = p√(1 – r²/((p² + q²)), then s = -q√(1 – r²/((p² + q²))

and if t = -p√(1 – r²/((p² + q²)), then s = q√(1 – r²/((p² + q²)).

Technically, this argument doesn’t work if p=0, but then a similar argument using q instead will produce the same result. (And it doesn’t matter if they’re both zero because I already know there are no solutions then anyway.)

So, the points of the circle with equation x²+y²=r² in the iplane at (p,q) are

(p-q√(1 – r²/((p² + q²)),q+p√(1 – r²/((p² + q²)) )

and

(p+q√(1 – r²/((p² + q²)),q-p√(1 – r²/((p² + q²)) ).

(Note how these are the same point when √(p² + q²) = r and they are undefined when √(p² + q²) < r.)

The fact that the two points on the circle in the iplane at (p,q) are both on a circle centred at (p,q) tells me they are the same distance inside the iplane from the centre.

Thinking all the way back to my physical model, the iplanes can be folded up like umbrellas, like this picture but in reverse:

If I do that, then every point inside them would end up on a vertical line, at a height equal to the distance away from the centre point. That seems to me like a very nice representation! In this case it’s particuarly convenient because both points are the same distance from the centre of the iplane, and so the same height.

For our circle x²+y²=r², the points on the circle in the iplane at (p,q) are a distance √(p² + q² – r²) from the centre of the iplane, so my “folded umbrella” representation will have a point at height z =√(p² + q² – r²) at each point (p,q) and it will come out like this:

You can see an interactive version of this graph which you can drag around to look at from different perspectives here: interactive folded umbrella iplane graph.

One thing I love about this graph is the sense that some complex points are closer to being real than others. In the graph, you can see there are no points of the circle (real or otherwise) inside the circle, and as you move further away from the real circle, the complex points get further and further from being real. The points on the circle far away from the real circle are very far from being real, which appeals to me.

(Of course, if I had two such graphs, there would be no guarantee that the places where they met would actually indicate shared complex points, because every height can be created by anything in the iplane on a circle of that radius. But still useful for thinking about one graph.)

I am going to have to investigate the folded umbrella representation of the complex points on an object more at some point. I really want to see how it looks for the complex lines and parabolas I already investigated four years ago. But first I would like to investigate what happens for an *unreal *circle. What does the circle look like when it has x²+y²= R for some *negative *number R. In a sense, that would be a circle with negative area. And what if the area was any *complex *number? But those are other stories and shall be told at another time.

The story of how that happened was pretty cool from my perspective, but every so often I wonder about it from his perspective. The Duke is the patron of the Royal Institution of Australia, and was in Australia just as they were displaying their installation of the Institute for Figuring’s Hyperbolic Crochet Coral Reef. So they organised a special viewing for him, to which a few key people were invited. One of those people was me, since I was the mathematician involved in the project.

I had a short but pleasant conversation with an old man about geometry and what it was like to be a man crocheting in public. We’re laughing in the photo, so somebody must have said something funny, though I don’t remember what it was. And that was it.

I suppose His Royal Highness meets a lot of people, so I might not make much of a difference to him. But every so often I think about how when he thinks about his time in Adelaide, there I am, if even for a moment (perhaps made just a little more memorable due to the rarity of meeting people standing in a roomful of coral made of yarn). I am part of the life memory of the Queen’s cousin.

Which gets me thinking…

Many of the students who I meet through my work in the MLC, I only meet once. Yet what happens with me will be a part of their memory of their time at university, however small. My short time with them is entangled forever with the events of their lives during their study. Will that moment with me make their memories more or less pleasant? Will it encourage them or discourage them? Will I just confirm their worst fears about themselves, or will it be at all like this student, who one visit to a lab in one class made all the difference?

did wonders for my confidence! So whilst I did have experiences early on that made me fear/hate maths, take heart that they can be turned around by positive and empowering experiences with the right teacher. Love your work!

— Ashleigh Geiger (@ashleigh_bryar) August 22, 2020

Every one of these students is important to someone, much more than a Royal who lives on the other side of the world is important to me. And I am there in their memory for better or for worse. This inspires me regularly to be the best I can be with each and every one of the students, especially that very first time I meet them.

Because once upon a time, His Royal Highness the Duke of Kent met me.

]]>*Twelve matchsticks can be laid on the table to produce a variety of shapes. If the length of a matchstick is 1 unit, then the area of each shape can be found in square units. For example, these shapes have areas 6, 9 and 5 square units. *

* Arrange twelve matchsticks into a single closed shape with area exactly 4 square units.*

I will tell you soon why this is one of my favourite puzzles, but first I want to tell you where I first learned this puzzle.

Once upon a time, about ten years ago, I met a lecturer at my university who was designing a course called *Puzzle-Based Learning . *It was an interesting course, whose premise was that puzzles ought to provide a useful tool for teaching general problem-solving, since they are (ostensibly) context free, so that problem-solving skills are not confounded with simultaneously learning content. Anyway, when he was first telling me about the course, he shared with me the Twelve Matchsticks puzzle, because it was one of his favourite puzzles. The reason it was his favourite appeared to be because the solution used a particular piece of maths trivia.

And the puzzle has become one of my favourite too, but for very different reasons! Indeed, I strongly disagree with my colleague’s reasons for liking the puzzle.

The first thing I disagree with is the use of the word “the” in “the solution”, because this puzzle has *many *solutions! Yet my colleague presented it to me as if it had just one.

The second thing I disagree with is that using a specific piece of trivia makes it likeable. I do recognise that knowing more stuff makes you a better problem-solver in general for real-world problems, but even so I always feel cheated when solving a puzzle requires one very specific piece of information and you either know it or you don’t. It seems to me it sends a message that you can’t succeed without knowing all the random trivia in the world, when of course it is perfectly possible to come up with acceptable solutions to all sorts of problems with a bit of thinking and investigating.

The main reason I love the Twelve Matchsticks puzzle is because it *does *have many solutions, and because this fact means that different people can have success coming up with their own way of doing it, and feel successful. Not only that, if I have several different people solving it in different ways, I can ask people to explain their different solutions and there is a wonderful opportunity for mathematical reasoning.

Another reason I like it is that people almost always come up with “cheat” solutions where there are two separate shapes or shapes with matchsticks on the inside. I always commend people on their creative thinking, but I also ask if they reckon it’s possible to do it with just one shape, where all the matchsticks are on the edge, which doesn’t cross itself. I get to talk about which sorts of solutions are more or less satisfying.

The final reason is that most of the approaches use a lovely general problem-solving technique, which is to create a wrong answer that only satisfies part of the constraints, and try to modify it until it’s a proper solution. This is a very useful idea that can be helpful in many situations. As opposed to “find a random piece of trivia in the puzzle-writer’s head”.

Which brings me to the second part of the title of this blog post. Let me take a moment to describe two different experiences of being helped to solve this puzzle.

When I was first shown this puzzle, my colleague thought there was only one solution, and his help was to push me towards it. He asked “Is the number 12 familiar to you? What numbers add up to 12? Is there a specific shape that you have seen before with these numbers as its edges?” He was trying to get me closer and closer to the picture that was in his head and was asking me questions that filled in what he saw as the blanks in my head. And there were also some moments where he just told me things because he couldn’t wait long enough for me to get there.

I did get to his solution in the end, and I felt rather flat. I was very polite and listened to his excitement about how it used the piece of trivia, but inside I was thinking it was just another random thing to never come back to again. It wasn’t until much later that I decided to try again, wondering if I could solve the problem *without *that specific piece of trivia.

So I just played around with different shapes with twelve matchsticks and found they never had the right area. And I wondered how to modify them to have the right area. And I played around with maybe using the wrong number of matchsticks but having the right area, and wondered how to change the number of matchsticks but keep the same area. This experience where I used a general problem-solving strategy of “do it wrong and modify” was much more fun and much more rewarding.

When I help people with the Twelve Matchsticks puzzle, this second one is the approach I take. I ask them to try making shapes with the matchsticks to see what areas they can make. I ask what all their shapes have in common other than having 12 matchsticks. I notice when they’ve made a square with lines inside it and wonder how they came up with that. Can they do something similar but not put the extra ones in the middle? In short, I use what they are already doing to give them something new to think about. I encourage them to do what they are doing but more so, rather than what I would do.

The approach where you have an idea in your head of how it should be done and you try to get the student to fill in the blanks is called funnelling. When you are funnelling, your questions are directing them in the direction you are thinking, and you will get them there whether they understand or not. Often your questions are asking for yes-or-no or one-word answers to a structure in your head which you refuse to reveal to the student in advance, so from their perspective there is no rhyme or reason to the questions you are asking.

The approach where you riff off what the student is thinking and help them notice things that are already there is called focussing. When you are focussing, you are helping the student focus on what’s relevant, and focus on what information they might need to find out, and focus on their own progress, but you are willing to see where it might go.

Interestingly, when you are in a focussing mindset as a teacher, you often don’t mind just telling students relevant information every so often (such as a bit of maths trivia), because there are times it seems perfectly natural to the student that you need such a thing based on what is happening at that moment.

(There are several places you can read more about focussing versus funnelling questions. This blog post from Mark Chubb is a good start: https://buildingmathematicians.wordpress.com/2016/11/03/questioning-the-pattern-of-our-questions/ )

From my experience with the Twelve Matchsticks, it’s actually a rather unpleasant experience as a student to be funnelled by a teacher. You don’t know what the teacher is getting at, and often you feel like there is a key piece of information they are withholding from you, and when it comes, he punchline often feels rather flat. Being focussed by a teacher feels different. The things the teacher says are more obviously relevant because they are related to something you yourself did or said, or something that is already right there in front of you. You don’t have to try to imagine what’s in the teacher’s head.

So the last reason the Twelve Matchsticks is one of my favourite puzzles is that it reminds me to use focussing questions rather than funnelling questions with my own students, and I think my students and I have a better experience of problem-solving because of it.

]]>This blog post is about one particular version of the Quarter the Cross problem you might like: the colouring version!

Back in 2017 I made a version of the cross with a lot of straight and circular lines on it so that instead of drawing your own lines, you could simply colour in the existing ones. I say “simply”, but… well look for yourself:

There are a LOT of lines there. If you look closely you can see lines cutting each smaller square into quarters in two directions, the diagonals of each square, lines connecting the centres of each side of the square to the corners, and circular arcs of two different sizes in various different locations. Plenty of scope to create your own fancy designs, or just make designs you think are less fancy look more like stained-glass windows!

You can download the Quarter the Cross colouring template in PDF, SVG and high resolution PNG formats here:

Here are some designs made by me and Simon Gregg using the colouring template.

Some #QuarterTheCross colouring by @Simon_Gregg. pic.twitter.com/TzlH8D5XXJ

— David Butler (@DavidKButlerUoA) August 24, 2020

So there it is. I hope you have fun with it, and if you do make something I’d love you to share it with me!

]]>It’s happened to me a lot before. Many students in various disciplines are extremely good at coming up with worries about experimental design or validity of measurement processes, and so they never get to the part where they deal with the statistics itself. They seem to treat every problem like the classic “rooster on the barn roof” problem, essentially declaring that “roosters don’t lay p-values” and choosing not to answer the question at all.

Don’t get me wrong, I really do want the students to be good worriers: they *should* be able to think about experimental design and validity and bias and all those things that impact on whether the statistics answers the question you think it does. But what they can’t do is use it to avoid talking about statistics at all! There are quite a few students who seem to be using those worries to discount *all* statistical calculations, and to sidestep the need to understand the calculation processes involved. “Your question is stupid, and I refuse to learn until it’s less stupid,” they implicitly say.

The weirdest part is that the assignment or discussion questions don’t usually discuss enough details for the students to actually conclude there is a problem. They say “the groupss were not kept in identical conditions”, but nowhere does it say they weren’t. I realise that in a published article if it doesn’t say they were then you might worry, but this is just an assignment question whose goal is to try to make sense of what a p-value means! Why not give the fictitious researchers the benefit of the doubt? And also, take some time to learn what a p-value means!

I do realise it’s a bit of a paradox. In one part of the intro stats course, we spend time getting them to think about bias and representativeness and control, and in another part, we get grumpy when they think about that at the expense of the detail we want them to focus on today. It must be a confusing message for quite a few students. But on the other hand, even when reading a real paper, you still do need to suspend all of that stuff temporarily to assess what claim the writer is at least trying to make. It’s a good skill to be able to do this, even if you plan to tear down that claim afterwards!

I am thinking one way to deal with this is to start asking questions the other way around. Instead of asking only for “what ways could this be wrong?”, ask “how would you set this up to be right?” And when I ask about interpreting a p-value, maybe I need to say “What things should the researcher have considered when they collected this data? Good. Now, suppose they *did* consider all those things, how would you interpret this p-value?” Then maybe I could honour their worries, but also get them to consider the things I need them to learn.

So, here are some ways I don’t want to treat my students, based on ways I have been treated on Twitter. To my shame, I have done most of these to others on Twitter too, and I am trying hard not to. I know most of the people who have done this to me will be mortified to know they have, so I am not going to call anyone out here. I just want to share what I have learned.

So, here is a list of things I don’t want to do to the students, because I don’t like it when they happen to me:

- Offer solutions when they haven’t asked for any.
- Interrupt their problem-solving process.
- Ignore their feelings when they express them.
- Tell them their feelings are wrong.
- Respond with a story about me rather than seek more from them.
- Respond to them telling me something they like by giving recommendations for new things.
- Respond with a fire hose of even more technical terminology.
- Tell them they are wrong to be confused.
- Tell them the thing they like is wrong because I like something else.
- Discount their success by pushing to the extension straight away.
- Respond to everything with sarcasm.
- Focus only on the bit I think is wrong.

My original plan was to elaborate on each of these, but I have kept coming back to this post for months and feeling overwhelmed with that task, so I think it’s time to just push send. And maybe it’s a good thing each time I read this to have to imagine what each of these things looks like. I’m hoping it’s useful to you to have to think about what these might mean too. But of course if you want me to explain a particular one of them more, do ask and I will do my best.

]]>