Here in Australia, we are at the tail end of a reality cooking competition called “Zumbo’s Just Desserts “. In the show, a group of hopefuls compete in challenges where they produce desserts, hosted by patissier Adriano Zumbo. There are two types of challenges. In the “Sweet Sensations” challenge, they have to create a dessert from scratch that matches a criterion such as “gravity-defying”, “showcasing one colour” or “based on an Arnott’s biscuit”. The two lowest-scoring desserts from the Sweet Sensations challenge have to complete the second challenge, called the “Zumbo Test”. In this test, Zumbo reveals a dessert he has designed and the two contestants try to recreate it. Whoever does the worst job is eliminated.
I find it very interesting that the Zumbo test is the harder of the two tests. In the Sweet Sensations challenge, the contestants can choose to use whatever skills they are already good at, and design their dessert in a way that they can personally achieve. In the Zumbo Test, the contestants have no control over the techniques that are required, and must try to do things they are not familiar with in ways they may not have seen before.
And why am I talking about this? Because my medical students find themselves in similar situations. Our medical students have two projects to do as part of their research curriculum during their third year. One project is a research proposal: they work in a group with a supervisor to plan a hypothetical research project, including ethics, literature review and (this is where I come in) statistics. The other project is a critical appraisal: they work in pairs to analyse a published article, including where it fits in the research, the writing, the importance and (again where I come in) whether the statistics is appropriate.
I have noticed over the years that in terms of statistics, the critical appraisal is harder than the research proposal. A meeting with students about the critical appraisal usually takes twice as long as one for the research proposal, and twice as much preparation for me. Many more students come to me to talk about the critical appraisal, and the ones who do come are more worried about the statistics they find in the critical appraisal than the statistics they need in the research proposal. Why is this?
When watching Zumbo’s Just Desserts, it occurred to me that the reason why is the same as the reason the Zumbo Test is harder than the Sweet Sensations challenge.
When doing your own research you can choose to only investigate questions in such a way to use the statistical methods that you understand. Even if you need a new statistical method, you just need to learn that one. Either way, you have complete control over your own decisions and know the things you are measuring and what they mean. It’s just like in the Sweet Sensations challenge the contestants get to make all the choices and use methods they are familiar with.
On the other hand, when reading someone else’s research, you have no control over the wacky statistical methods they choose to use. Even if they are the appropriate ones (they often are in medicine, actually), the paper almost never describes how the researchers decided to use them – it just says what they used. And they often measure new things in new ways that you don’t deeply understand. It’s just like in the Zumbo Test the contestants have to do things that are new to them in ways that are new to them.
It’s much much harder to understand the statistics in someone else’s research than it is to make your own.
Let’s just hope we don’t eliminate all the students by asking them to do it with less support.
In Maths 1A here at the University of Adelaide, they learn the following theorem (this is taken from the lecture notes written by the School of Maths here):
Theorem 2.3 (Stewart p.312)If \(f\) is continuous on \([a,b]\), the function \(F(x) = \int_a^x f(t)dt\) satisfies \(F'(x)=f(x)\) for all \(x \in (a,b)\).
It says that, given a function of x defined as the integral of an original function from a constant to x, when you differentiate it you get the original function back again. In short, differentiation undoes integration. In many places in the world, including in the textbook for this course, this theorem is called “Part I of the Fundamental Theorem of Calculus”. (Part II says that integration can be done by doing differentiation backwards. Here at Uni of Adelaide, only Part II is called the Fundamental Theorem of Calculus. Go figure.)
And then they get questions something like this on their assignments and they don’t know what to do. The students always say something like “I would know what to do if that was an x, but it’s not just an x, so I don’t know what to do”.
If \(f(x) = \int_3^{x^2} t\sqrt{1+t^3} dt\), find \(f'(x)\).
I have a way of thinking about this that makes sense to me and it has to do with blocking what I see into blobs and a mantra I say to myself when I am faced with certain derivatives (I will probably write about that soon). However, I know that this doesn’t always make sense to students when I tell it to them for the first time. When I was with a student grappling with this last week, I suddenly realised that I had come to my own way of thinking because I had noticed a pattern in the way derivatives worked and noticed that I could apply my pattern to this situation. So I decided to take a different approach and help the student notice this pattern for themselves.
And what do you do to make a pattern more obvious? You line things up!
I made a grid and said to the student I would write problems in the grid for him to do, and to bear with me because I did have a point I promise. So for the next several minutes I wrote a new problem in the grid and passed it over for him to do it, asked him what his thinking was in that problem, and then wrote another one for him to do, and so on. I was very proud of myself for relinquishing the pencil and letting him do all the writing – this is the part of SQWIGLES I often find the hardest.
These were the problems we did, approximately (it was almost a whole week ago now). I was particularly proud I thought of f(x) = sec(x), because its derivative has two x’s in it:
At the end I asked him what he noticed about the problems and if anything he noticed would help him with the one at hand. And he figured it out! He noticed that all of them had an x in one version and an x2 in the second version, and he noticed that in every one, you did exactly the same move on the bottom row as the top, but multiplied by 2x each time. And he straight away wrote down the correct result for his problem with the derivative of an integral and articulated a general rule he could use in the future.
I couldn’t have been prouder of him. Internally, I couldn’t have been prouder of myself for using this approach. I am led to believe that this sort of thing is called a “problem string”, and I want to notice more opportunities to use them because I have a feeling it will be way more empowering for students. I’ll certainly use problems strings next time I am called upon to help a student understand using the chain rule when a function is defined as an integral!
I was talking to a student about his calculus last week. He was trying to differentiate \(x^x\). (Actually he was trying to differentiate \(x \ln(x)\) and had decided the best place to start was to raise e to the power of it, thus producing \(x^x\).) At first he tried this:
I asked him what he thought about that and he said he wasn’t sure it was right because of the x in the power. I asked him how you normally dealt with an x in the power. He thought about it and then produced this:
I confirmed that this is indeed what you would do if there was an x in the power but the base was constant. But I also confirmed that because there was an x in both the power and the base, it was going to be a bit trickier. He asked how you did deal with that, and I started out by writing \(x^x = \left(e^{\ln(x)}\right)^x\). (On reflection, if I was doing SQWIGLES properly, I should have helped him find this definition in his notes.) Together we continued this working as follows. Interestingly, we ended up having to differentiate \(x \ln(x)\) as part of this, but by this time he realised it would be much simpler to use the product rule!
Do you notice anything interesting about that result? Well, I did: it’s the sum of the other two wrong answers! If we treat the base as a constant we get a (wrong) result, and if we treat the power as a constant we get a (wrong) result, but if we add those two results together we get the correct result! This is WILD!
After we did the summing up what we’ve learned bit, I absolutely had to try this out generally to see what would happen:
Let \(h(x) = [f(x)]^{g(x)}\).
If \(g(x)\) is constant, say \(g(x)=b\), then \[\begin{aligned} h(x) &= [f(x)]^b\\ h'(x) &= b[f(x)]^{b-1}\cdot f'(x)\\ &= g(x)[f(x)]^{g(x)-1}\cdot f'(x)\\ &=g(x)f'(x)[f(x)]^{g(x)-1}\end{aligned}\]
If \(f(x)\) is constant, say \(f(x)=a\), then \[\begin{aligned} h(x) &= a^{g(x)}\\ h'(x) &= \ln(a)\cdot a^{g(x)}\cdot g'(x)\\ &= g'(x)\ln(a)a^{g(x)}\cdot f'(x)\\ &=g'(x)\ln(f(x))[f(x)]^{g(x)}\end{aligned}\]
Oh my goodness! It worked again! I am reeling with the wildness of this! Currently I don’t know if there’s a more conceptual/visual approach to this that will help me feel like I understand why it is true, but for now I’m just going to relish the wildness of these two wrong results adding up to the correct one.
PS: Isn’t it cool the stuff you learn when you listen to students? If I’d just told him what to do I’d never have noticed this.
UPDATE: There’s more to learn in a blog post the following day.
These comments were left on the original blog post:
David 21 September 2016
This can be explained using the multivariable chain rule. Let f(u,v) = u^v, u=x, v=x. Then df/dx = ∂f/∂u du/dx + ∂f/∂v dv/dx = ∂f/∂u + ∂f/∂v. The product rule is also a special case of the multivariable chain rule.
David Butler 21 September 2016
Ah you beat me to it! I was just going to do an update to say exactly this! In my head it was something like f=u^v with u=f(x) and v=g(x). So df/dx = df/du*du/dx + df/dv*dv/dx = (derivative of f treating v as constant) + (derivative of f treating u as constant)
Of course, this student won’t see multivariable calculus in their course for another several months yet…
It’s university holidays again (aka “non-lecture time”), which means I’m back on the blog trying to process everything that’s happened this term. Mostly this has been me spending time with students in the Drop-In Centre, since I made a commitment to do more of what I love, which is spending time with students in the Drop-In Centre. When trying to decide what to write about first, I realised that a lot of what I wanted to say wouldn’t make much sense without talking about SQWIGLES first. So that’s what this post is about.
SQWIGLES is an acronym that we use to help our staff (and ourselves) when teaching in the MLC Drop-In Centre. It is a list of eight actions we can do to help make sure our interaction has a better outcome and make it more likely students will learn to be more independent.
SQWIGLES
The story
It was originally Nicholas’ idea to have something like this. He wanted something to help the staff choose what to do in the moment, and also to help them reflect on their actions and choose ways to improve. We noticed that our staff (and ourselves) needed something focused on actions rather than philosophies, because then it could be used on the fly to choose what to do. Telling staff they need to be “encouraging” or “socratic” is not all that helpful when they don’t know how to put it into action. Yet this is what many documents giving advice to tutors do. So we decided to focus on the actions instead.
Back in 2013, I was away for the weekend and thought hard about what actions we could focus on and scribbled a lot in my ideas book until I came up with an acronym I could work with, and SQWIGLES was born. Over the past three years it has been remarkably good at helping our staff make improvements in their one-on-one teaching quickly, because it does focus on actions.
(Do note that we also have regular discussions about philosophy too, and one overarching goal to draw everything together: “In every interaction the student learns something they can use independently in the future”. But these are not as useful without a guide for action.)
Here is what the acronym means, with further details below:
(UPDATE 2018: You can see video of me talking about SQWIGLES at Twitter Math Camp 2017 here: https://youtu.be/u3v_7G5D5xI )
S: SPEAK your thinking
Every time we read or write in front of the student, we must speak our thinking. In order to learn how to do this stuff themselves, they need to learn about how to think, and they can’t do that without us speaking your thinking. I have yet to meet a telepath!
So, when reading student working, we can tell them what we are looking for and what we see. When reading a question to figure out how to do it, we can read it aloud and then say what our first thoughts are, including the ideas for attacking it that we are rejecting. When writing a solution to a maths problem, we can speak the decisions we are making.
This may seem such a basic thing it hardly seems mentioning, but believe me it does need mentioning. Our very new staff often need reminding of this – they will do a maths problem in front of a student without speaking at all. Or they’ll sit silently staring at a new problem, clearly thinking deeply, but not letting the student know what they are thinking. Or they’ll just present an existing solution without explaining the thinking that created it. We all need reminding to speak our thinking.
Q: Ask open-ended QUESTIONS
This action has two parts. First, we should ask the students questions, rather than just tell them stuff. Them responding means them thinking, which is much better than just watching. Second, we should ask the students open-ended questions if we can. There’s a much better chance of thinking then.
For example, instead of saying, “The definition of continuity is…”, we could ask a question like “What is the definition of continuity?”, or even better ask an open-ended question like, “What do you know about continuity?” Instead of saying, “Here’s how to start this problem…”, we could ask a question like, “How could you start this problem?”, or even better ask a more open-ended question like, “What do you think about this problem?”
Open-ended questions allow the students to tell us what they are thinking, whether it is what we were expecting or not, and so we can change our teaching accordingly. They may even surprise us by solving their problems by themselves if we give them the freedom to respond that way!
Some great all-purpose open-ended questions are “What do you think about that?”, “How do you feel about that?” and “Can you tell me more?”
W: Encourage them to WRITE
Writing is a great way to record thinking, and a great way to help structure thinking. So it’s a great thing to encourage the students to write. Us writing is not going to have nearly as much of an impact on their learning as them writing!
Moreover, if the goal is to learn how to solve problems, then the students need to solve problems. Us writing things for them is not going to give them this experience. Plus, everything looks easy when someone else is doing it!
So encourage them to write. This is the one I have the greatest struggle with. I have to remind myself of this one constantly and use every opportunity to hand them the pencil.
I: Help them to find INFORMATION
Students need to learn how to find the information they need for themselves. One of the most important skills they can learn at university is how to find information, rather than the information itself per se. So, we can help them learn how to find information. Moreover, if we just tell them, all they might learn is that the quickest way to get information is to ask us!
For example, instead of telling students the definition, we can go and find the definition with them. If they don’t have lecture notes to look it up in, there’s always the textbook or the internet – learning how to search the internet effectively is an important skill to learn! If they want examples to inspire them, we can help them find a source of these examples. If they need instructions on the expectations for their assignment, we can help them find these instructions.
Another worthwhile thing to note is that even if the student has a problem that it’s not our job to solve, we can always help them find information. At the very least we can help them find out who is the person whose job it is to help!
G: GUIDE their problem-solving
This one is the most vague of the actions, but I really wanted to mention the problem-solving thing here! There are two ideas here: guiding is different from telling, and the problem-solving is theirs and not ours.
We can tell students what to do, and we can do problem-solving for them, but it won’t achieve them learning something they can do on their own in the future. The best we can offer is guidance. We can give them signposts to point the way, like “What do all these words mean?” and “How is this thing related to that thing?” We can give encouragement. We can help them delineate smaller steps they need to follow.
But we can’t do it for them. It’s their problem-solving and we can’t do it for them.
L: LISTEN to their thoughts and feelings
This is one of the most important things on the list. We need to listen to the students. We can’t figure out what they need without listening to them. We can’t figure out where they are so we can guide them further without listening to them. We figure out what information they need to find without listening to them. We can’t choose an alternative explanation without listening to them. And we can’t convince them we care without listening to them.
It’s also important to note that we’re not just listening to hear if they are correct or not – we’re listening to their thoughts and feelings. We really want to and need to know what they think and how they feel. These are so very important to their experience of learning and our decisions of how to help them learn, especially the feelings. Never underestimate the impact how they feel has on their learning, whether positive or negative feelings!
This is one of the reasons why open-ended questions are so very important – it gives us something to listen to!
E: EXPLAIN another way
At some point in our interactions with students, it is likely we will need to explain something. This letter is about not sticking to just one explanation or type of explanation. It is likely the student has heard at least one explanation and it hasn’t worked so far, so we need to be ready to try other ways to explain, especially if it was us who gave them the first explanation they didn’t understand!
We can use a proof, or just an example. We can use an analogy, or a drawing, or play-dough, or movement. We can colour-code an existing example, or line several examples up next to each other. We can ask another student to have a go explaining it to them. There are so many options to try and the key is to keep trying, and build our repertoire of new ways to explain, and be brave in trying something new.
There is also the issue of what exactly they need explaining, which might be different to what we are used to explaining. Sometimes they want the concept explained in all its mathematical detail, and sometimes they just want the general idea. Sometimes they need to feel philosophically comfortable with it, and sometimes they need to understand the mechanism of the procedure. Sometimes they need to know the acceptable ways to write it on paper. We can only know which thing they need an explanation of by asking questions and listening to the response.
S: Help them to SUM UP the learning
Most of the time, students come to us with assignment problems, and the only thing they will think about the interaction with us is that they got their problem solved. But this does not service the goal of them learning something they can do on their own. They probably did learn something, but if they don’t know they did, then how can they use it when they need to? So we can help them sum up what they learned today. At a conference once, they called this cognitive closure – a bit like grief closure, but for learning.
Summing up what they’ve learned is good, but helping them to sum up what they learned is even better. Asking them what they’ve learned here, or what they can use in the future is an excellent open-ended question that will often give pleasantly surprising results.
Conclusion
So there is SQWIGLES: a list of actions we use when we work with students one-on-one. A list of things to reflect on when we think about our one-on-one teaching. Try using them in your own teaching and reflection.
Last week, I helped quite a few students from International Financial Institutions and Markets with their annuity calculations, which involve quite detailed stuff like this:
There were several small issues a lot of them had, which combined to stall their calculations. One of the more important problems was about how the calculator interprets what they type into it, which is really in essence about the order of operations.
When I look at the formula above, I see many pieces that are put together in various ways to make a number called “PV”. These pieces are combined using various operations, and are also held together in order to go into these operations with various grouping symbols. Some of these things have a stronger “hold” than others and will enforce certain things to be done before other things are done if I were to actually do this calculation. This is the “order of operations”. I’ve written about it before , but I’ve moved on in my thinking quite a bit since then!
While many people use some sort of acronym to remember the order of operations, that has never worked for me. Instead I have a picture of a hierarchy of operations like this:
When I draw this on paper, I start at the bottom, drawing the + and – first. Then I draw the × and ÷ above that. Then I draw the ^ and √ above that. Finally I draw the box on the side with the (), [], ___. The idea is that + and – are the most basic and hence least important to do before things nearby followed by × and ÷ and then by ^ and √ . The symbols on the side are for holding things together in order to override this hierarchy. It’s worth noting that I use this picture to remind myself which things expand out over which things when there’s brackets in the way. (If I was particularly clear-minded, I might draw the × and + even further to the left than the others because they work fine in both directions whereas everything else is different in different directions.)
Anyway, this order of operations allows me to analyse the expression above and tells me that the r/n on the bottom there is held together before it becomes part of the larger division. It tells me that the r/n inside the brackets is more important to calculate before the “1+”. I also know, though it’s not written in my order of operatons picture, that the little -nt is all inside the top of that power, and needs to be done before I can resolve the power itself.
Unfortunately, the scientific calculator (or Excel, or Wolfram Alpha) is less clever than this. When I type things into the calculator, it can’t see the physical arrangement of things on the page. It can’t tell that some things are smaller than others and arranged at the top. It can’t see that there’s a big horizontal line holding the r/n together. The only way the calculator knows to hold things together is with some brackets. So I’m going to have to turn all the things that visually hold things together into brackets. So suppose I wanted to calculate the following:
If I only typed the symbols I see like this:
1000÷0.06÷12×(1-(1+0.06÷12)^-12×30)
Then I would not get the result I hoped for. The calculator would certainly do the 0.06÷12 inside the brackets before the “1+”, but it will do the 1000÷0.06 before the “÷12” at the front, and it will do the “^-12” before the “×30” too. I have to tell the calculator explicitly what to hold together using brackets. Like this:
1000÷(0.06÷12)×(1-(1+0.06÷12)^(-12×30))
I find this sort of discussion very important and very helpful for the student. It’s so important to help them see what is held together by the brackets and by the horizontal lines and the powers. And it’s also so important to help them see that the calculator is like they were a few minutes ago — it can’t tell that those things hold other things together and just needs to be told every time.
These comments were left on the original blog post:
Kevin Hall 11 November 2016: I teach this skill, too, in my Algebra 1 classes. I sometimes wonder whether better computer/calculator interfaces will eliminate the need for this skill in a couple years. For example, I make a big deal out of the fact that, to square 3/5, you need to type (3/5) with parentheses. But that’s actually not true in the newer graphing calculators.
David Butler 14 November 2016: I don’t think we’ll eliminate this skill in its entirety for quite a long time! Firstly, there will be for quite some time yet, interfaces that are more computer-programmy (like Excel and the fancy statistics manipulation software). Secondly, this discussion often highlights they don’t actually know what the powers or fraction bars are doing anyway. Something that is able to copy their writing and interpret it might mask that misunderstanding!
NOTE: My thinking about this evolved over the years. Earlier blog posts on related ideas are here:
I have always loved the naming of quadrilaterals, right from when I first heard about it in high school. I’m not entirely sure why, but some of it has to do with the nested nature of the definitions – I like that a square is a kind of rectangle and a rectangle is a kind of parallelogram.
Most of the time you’ll see this classification organised in a sort of “family tree”. Like this:
(Warning: if you search for quadrilateral family tree in Google Images, you’ll get all sorts of things, many of which use American terminology, which is different to that here in Australia, and many others are just plain wrong.)
Of course, almost all people I meet are not aware of these nested definitions. Most students I work with are bamboozled when the answer to the problem of the largest area rectangular garden of fixed perimeter gives them a square. To them, a square is not a rectangle. Then again, there are others who aren’t aware that orientation is not one of the factors mathematicians use to name shapes. Just the other day someone asked me why a shape was called a diamond if it’s this way and a parallelogram if it’s this way. (That was a most interesting conversation about language and maths and maths language.)
Partly I think this is because the only examples we give of rectangles are non-squares — we never include squares in a collection of rectangles. Also, almost all examples we give of these things have their edges parallel to the top and bottom of the page. The casual viewer comes to believe that the orientation is part of the defining properties. A serious case of Beware of the Toast.
For a long time, I’ve wanted to make some posters showing collections of quadrilaterals including ones not parallel to the page edges and including the more special ones further down the tree. And yesterday I just decided that I would finally do it.
Here’s a picture of the majority of them. (You can find them in pdf form here .)
And finally, an upgraded family tree, showing heaps of examples of each type of quadrilateral. (You can find this in pdf form here .)
PS: If you want the original svg files in case you want to edit them (for example, to make them into local terminology), just ask.
This comment was left on the original blog post.
Anne Kelly 30 August 2016:
Hi David Many thanks for putting together and sharing the Quadrilateral Family Tree. I plan to share it with the classes I teach. Anne Kelly Geographe Primary WA
Every so often, someone brings up the thing with tau (τ) versus pi (π) as the fundamental circle constant. In general I find the discussion wearisome because it usually focuses on telling people they are stupid or wrong for choosing to use one constant or the other. There are more productive uses of your time, I think.
But for a while I have wanted to add just this one thought to the conversation and now is as good a time as any.
Some people advocate tau for pedagogical reasons, claiming it’s better for teaching radian measure of angles and also trigonometry. Meh. A good dose of cutting slices of circles is really a much better innovation than rewriting the books to use a different constant. Plus I have this one important point to make:
The values of the trig functions are calculated based on how far away the angle is from the nearest multiple of pi.
Look: Say I want to calculate the value of sin(3π/4). I draw the angle 3π/4 as a length around the unit circle and I remember that sin(3pi/4) is the y-coordinate of the point I have come to. Then I notice that this y-coordinate is the vertical side of a little triangle drawn down to the x-axis, and that therefore sin(3π/4) is the same as sin(π/4).
Did you see what happened there? sin(3π/4) was calculated by seeing how far 3π/4 is from π and using that distance instead! This is exactly how you do actually do it! The values of sin(t) and cos(t) are always calculated based on how far t is from the nearest whole multiple of π! You want cos(10π/3)? Well the nearest whole multiple of π is 3π, and we’re π/3 beyond that into the third quadrant, so cos(10π/3) = -cos(π/3).
This animated GIF shows how the angle of interest is always how far the actual angle is from the nearest whole number multiple of pi.
I shudder thinking how much harder this would be if we were using tau and had to find out how far our angle was from a whole multiple of a fraction of τ! My university students have enough trouble with fractions as it is!
So there you go: I’ll keep using pi because you calculate the trig functions based on how far the angles are from the nearest whole multiple of pi. That’s all I have to say on the matter for today.
This is the last (for now) in a series of posts about Where the Complex Points Are. To catch you up, I discovered a way of visualising where the complex points are in relation to the points of the real plane. All the complex points \((p+ri,q+si)\) are in a plane attached to the real plane at the point \((p,q)\). I call these planes the “iplanes”, and I imagine it as a transparent sheet that can be unfolded to lie flat on the real plane in order to see its complex points. Using this model, I’ve investigated where the complex points are on a line, on a parabola, and on the graph of a complex function.
Here are links to all the other posts in the series:
Where the complex points are: where idea came from (YOU ARE HERE)
There is one more thing I want to talk about and that’s where the idea came from, because I think it’s an interesting story and I want it recorded somewhere.
Between not beyond
Ever since I first learned about complex solutions to quadratic equations, I have wondered where the complex points are. I never really thought about it deeply, but the general picture in my head was that the complex points were somehow “beyond” the real plane. I imagined them as over there beyond the edge of the real plane, surrounding it so that the real plane was a small plane inside some much bigger plane.
But something doesn’t quite work about this picture. When I studied conics in depth much much later I learned that a “degenerate circle” with equation \(x^2+y^2 = 0\) wasn’t just the origin \((0,0)\) as I had originally thought. In fact it was a pair of complex lines that each happen to have one real point. (These lines have equations \(x+iy=0\) and \(x-iy=0\).)
Only recently did this start to bother me. Suddenly I asked myself, if the complex points are beyond the real plane, how can a line made almost entirely of complex points meet the real plane in the origin, right there in the centre?
It occurred to me that the line must somehow travel over or between the real points to arrive at the origin in the middle.
And the seed of an idea began to grow. If the lines have to travel between the other points to reach the origin, then the complex points must somehow be bewteen the real points, not beyond them.
Finite planes — the smallest plane
Once I realised that the complex points had to be between the other points, I started thinking about the problem in terms of finite planes. In a finite geometry, the points are discrete points, and there is plenty of space between them to put extra points, if indeed that’s where they are.
I started out with the smallest possible affine plane (our ordinary geometry is a kind of affine plane). It has four points with coordinates taken from the set \(\{0,1\}\) (also known as \(\mathbb{Z}_2\)) and six lines. The lines have equations \(x=0\), \(x=1\), \(y=0\), \(y=1\), \(x+y=0\), \(x+y=1\). One name for this plane is \(AG(2,2)\) – the Affine Geometry of dimension 2 and order 2.
What I needed was to find where the “complex points” were in relation to this plane. Unfortunately, if you’re using \(\mathbb{Z}_2\), \(1+1=0\) so -1 is the same as 1. This means the equation \(x^2 = -1\) already has a solution, so I can’t add the number \(i\). You actually have to add the solution to \(x^2 + x = 1\). I called this number \(\alpha\) so that I ended up with a set of four numbers to use for coordinates: \(\{0,1,\alpha, 1+\alpha\}\).
From this I can build \(AG(2,4)\), which has 16 total points, and 20 total lines. I wrote down the equations of all the new lines and tried to figure out how to add the points to my diagram so that the new lines looked like lines.
I knew the lines in my new plane would have to have four points — one for each “number” in the coordinates — and this meant each line already existing needed two extra points. It seemed reasonable to exend them outwards. It also seemed reasonable to try and make the new lines \(x=\alpha\) and \(x=1+\alpha\) vertical and the new lines \(y=\alpha\) and \(y=1+\alpha\) horizontal.
Fiddling around with this gave me a nice square arrangement for the points. Then I investigated what all the new lines looked like and I got something like this. (Note I was drawing in crayon on the paper tablecloths at my sister’s restaurant at the time. I didn’t take a photo on the day, but this is a reasonable facsimile.) You can see the lines a little more clearly in this GeoGebra applet.
Unfortunately the extra points were outside the original ones, and not between them as I had hoped. I did notice that each new line that wasn’t part of the original plane met the original plane in exactly one point. I knew this would have to happen because it’s a theorem from finite geometry, but I was glad that everything was working the way it should.
Finite planes — noughts and crosses saves the day!
After the half-success of the smallest affine plane, it was natural to move on to the second-smallest affine plane \(AG(2,3)\). This is the plane you get when you add in all the wraparound diagonals to a Noughts and Crosses board. When drawing it on a noughts-and-crosses board, the twelve lines look like this:
If you draw the points as dots and the lines as (curved) lines, you can make it look like this. You can see the individual lines more clearly in this GeoGebra applet.
As you can see, it’s already quite invovled, and I didn’t really like to imagine what would happen when I went up to the “complex” plane — that would mean going from 9 points to 81 points and from 12 lines to 90 lines. The 20 lines of the affine plane of order 4 were enough of a mess!
But I did start to investigate. \(AG(2,3)\) has its coordinates taken from \(\mathbb{Z}_3\), which is \(\{-1,0,1\}\) with the rule that \(1+1=-1\). This field doesn’t have any solutions to \(x^2=-1\), so it was safe to attach the number \(i\) this time. I drew my extra points clustered around the existing points and started to investigate the lines I would need. In order to get the lines with equations like \(x=i\) to be vertical, I needed to make sure that the points with coordinates \((i,something)\) were lined up above each other. This required me to make sure that within each cluster of points the same imaginary parts were added to the coordinates.
Here’s a picture of my ideas book around this time:
After drawing it on paper, it was time to investigate it more fully, so I worked on creating a GeoGebra applet that would show the points on a specific complex line. It took some fancy footwork with the spreadsheet to make it do what I wanted.
I noticed now that the extra points on every real line lined up neatly with the real line. This was totally awesome, and I was inspired to move on to the real plane to see what was happening there.
Finally iplanes are found
I was all satisfied with how my complex points worked in finite planes. The final step was to think about how this applied to the infinite real plane.
There was a big problem, of course. In the finite planes, there was plenty of space between the points to fit the extra points. In the contiuous real plane we’re already full. Where do I fit those extra points?
My first idea was that they were all squished into each point of the plane — “fat” points, as it were. But this was somehow unsatisfying because I couldn’t actually see where the points were. Then it occurred to me that if I couldn’t squish them into each point, they’d have to be above the plane but attached at the real point. And the iplane was born.
At the start, I was sketching and calculating on scrap paper, drawing a little square around the real point to visualise the iplane. This was giving me enough insight to get a feel for what was going on with lines. The GeoGebra applets in the previous posts are based on this visualisation. It wasn’t until I got out the cellophane and the tap-tap board that I really felt like I had a proper feel for what was going on. This pinkness has transferred to all of my diagrams and all of my thinking about iplanes, but you’ve seen the results of all that thinking already…
Conclusion
So what is the point of this story? I don’t know. I just thought you (and future me) might like to see my train of thought in creating this idea.
I suppose for me the main moral is that finite geometry helps give me insights into real geometry, and that my habit of making physical models of things is still a worthwhile thing to foster.
In maths, or at least university maths, there are a lot of statements that go like this: “If …., then …” or “Every …, has ….” or “Every …, is …”. For example, “Every rectangle has opposite sides parallel”, “If two numbers are even, then their sum is even”, “Every subspace contains the zero vector”, “If a matrix has all distinct eigenvalues, then it is diagonalisable”.
It’s possible to flip these around so that the implication goes the other way: “Every shape with opposite sides parallel is a rectangle”, “If the sum of two numbers is even, then they are both even”, “Every set containing the zero vector is a subspace”, “If a matrix is diagonalisable, then it has all distinct eigenvalues”. This is called the converse of the original statement. The big problem is that while it’s very easy to form the converse, it’s not usually true! In every one of my examples, the original statement is true but the new converse statement is false. It is possible for the converse to also be true, but it’s not the most common thing to happen!
Many students when faced with statements like those in the first paragraph automatically and unconsciously assume that it works both ways, especially when the subject matter is new to them. The number of students who I have seen show that the zero vector is in a set and declare that therefore it’s a subspace doesn’t bear thinking about. I have some theories about why this happens: one is that they assume the statement is the definition of the concept rather than something that happens to be true about it, the other is that they see us use this one rule to show that something isn’t a subspace and assume you can use this one rule to show that something is. But this post is not about the why. It’s about a way of helping students see the problem.
This is a Ball
There’s this awesome picture book called “This is a Ball” by Beck and Matt Stanton. In it, the adult is supposed to read the words, but the words don’t match the pictures that are shown. The idea is to let kids have a chance at being right for a change. This book is so very awesome that when we saw it in Aldi recently as a big book, we just had to buy it, even though there was nowhere in our house to store it!
Anyway, there’s a page in the book which describes a monster scaring a princess and her pet dog, only the picture shows a princess scaring a monster and her pet elephant.
When the child tells you that you’re wrong, you’re supposed to read the next part of the text, which says, “We talked about this before. Don’t you remember? Look at its tail! All dogs have tails.”
When I read this for the first time I said to myself, “Oh my goodness it’s a false converse!” I realised what power this would have for helping students understand the problem with converses not always being true. It’s such a simple example, and is pitched in such a way that I can use it as an experience rather than just an explanation. It only took a few months for me to get an opportunity to use it…
All dogs have tails
I was with a student last semester who was doing a problem about eigenvalues of matrices. He looked up a theorem in his notes and its corollary which said “If a matrix has all distinct eigenvalues, then it is diagonalisable”. Then he said to me, “So if this matrix is diagonalisable, it must be that all the eigenvalues are distinct.”
“This is it!” I said to myself and quietly drew a picture something like this:
I said, “Look. It’s a dog!”. He looked at me funny, and I said, “You don’t believe me? But look at the tail – all dogs have tails!” I let it sink in for a few seconds. “Oh,” he said. “I see what you did there.”
We were then able to talk about how things don’t always work both ways and to be careful to read things closely to make sure. I am so so very happy I now have a way to bring up this conversation. Thanks Beck and Matt Stanton!
This comment was left on the original blog post:
David Roberts 29 August 2016
FWIW, here’s a teacher reading the book to the class YouTube link they are loving it🙂
This one wasn’t part of the original plan, but some interesting thoughts have come up while discussing the previous posts on Twitter, and I needed to process them. So here we are doing an extra blog post before my final epilogue.
A problem of viewpoints
In the introduction, I talked about how I’ve always wondered where the complex points are when you talk about a parabola meeting the x-axis in complex points. I introduced the concept of an iplane: at each point \((a,b)\) in the real plane, there is an iplane attached, which contains all the points \((a+ci,b+di)\). These points are arranged with the x-axis showing the imaginary part added to the x-coordinate and the y-axis showing the imaginary part added to the y-coordinate. I imagine them as being a transparent sheet attached to the point itself and being able to be flattened out to sit on top of the real plane.
This representation of the complex points allowed me to investigate the complex points that lie on lines (both complex lines and real lines) and to investigate the complex points that lie on a vertical parabola, and also how lines meet such a parabola. I shared these thoughts on Twitter, and waxed lyrical about them in person to basically every person I met. People were impressed, but there was something in the way they spoke that niggled at me and I couldn’t figure out what it was.
Yesterday I figured out the problem: I was thinking about the lines an parabolas in totally different ways to other people! In my head, the lines and parabolas were sets of points whose coordinates satisfy a condition. To decide if a point is part of the set, you sub its coordinates into the equation and see if it’s true: if it’s true, then the point is on the object, if it’s not true then the point isn’t on the object. I was looking to see which points were on the parabola by simply subbing the coordinates in to see which ones worked. I did this one iplane at a time because it seemed the most reasonable way of organising my search.
But what was reasonable to me was not reasonable to others! Taking each iplane one at a time and looking to see which points in the iplane satisfy the equation was totally alien to quite a few people I talked to. These people totally understood the iplane concept and they thought it was a great idea, but they had no idea what I was getting at by looking in each iplane separately. Yesterday I realised why: they were thinking of the lines and parabolas as an organisation of the input/output relationship for a function. They were looking for an x-axis and a y-axis to “line up” the input and the output of the function formula.
This is of course a perfectly valid way to think of many graphs, especially for the parabolas I was looking at, whose equations all look like “y = something in terms of x”. It’s just that I wasn’t thinking about them that way so my iplanes weren’t designed to match with this viewpoint. So I started thinking about how to make it match.
The complex line
There is one key issue with making the graph relate more closely to the idea of lining up the input on the x-axis with the output on the y-axis: the input is a complex number and it doesn’t really live on the x-axis. But, I dealt with that problem already in the plane by adding an iplane to each point of the plane. Surely I could do the same thing to the line, right? Why can’t I add an iline to each point on the real line? The iline attached at the number a would contain all the numbers of the form \(a+bi\).
With the iplanes, I flattened them out to sit on top of the real plane so I could see the complex points, but I only flattened out enough to see the point I was looking for. I wanted to do the same thing with the ilines, but it’s not really the same feel. Somehow I can’t tell so easily what’s going on with the complex number. Then I realised what I could do is just draw an arrow from the point representing the real part to the complex number in its iline. That is, I could draw the number a+bi as the point a on the number line and an arrow of length and direction b starting from that point. So the following representation was born:
You can investigate what complex-number addition looks like in this representation with this GeoGebra applet, and what complex-number multiplication looks like with this GeoGebra applet. (This representation is not so good for complex multiplication, but that’s ok — different representations are useful for different things.)
A complex x-axis and y-axis
So I was now able to locate a point on the x-axis as an input for a function, and locate its output on the y-axis. Then something rather wonderful happened: I realised that I could combine this point-and-arrow x-coordinate and this point-and-arrow y-coordinate into one point-and-arrow in the plane. The point (p+ri,q+si) became the point (p,q) and the arrow (r,s) starting at that point.
But the point \((p,q)\) is the centre of the iplane in which the complex point \((p+ri,q+si)\) lives! And the arrow \((r,s)\) is the vector-version of the coordinates \((r,s)\) within that iplane! So it turns out that my point-and-arrow x-coordinates and y-coordinates matched up perfectly with my iplanes! I couldn’t be happier!
The complex graph of a real function
So now I was able to investigate graphs of complex functions from the perspective of input-output relationships. It’s a whole different way of thinking about things, and I haven’t investigated it to any great depth yet, but so far it is most interesting. One particular thing I’ve noticed is that quadratic functions are rather special when it comes to how the real images of the complex points are arranged — the tangents seem to play a much bigger role than they do for higher polynomials.
I’ve made a GeoGebra applet that will allow you to enter a real function and move the input around on the x-axis to see what happens to the output and the matching complex point on the graph. You can turn on the trace to see what shape is made when you do move the input around. I now have a host of new questions rising in my mind which I hadn’t even considered before, and it will probably take years to sort them out.
But for now I’ll just be content that I have a picture that works for both views of a graph: as a collection of points that satisfy a condition and as an encoding of the input/output relationship of a function.
