When doing algebra and solving equations, there is this move we often make which is usually called “doing the same thing to both sides”. Quite recently this phrase of “both sides” has begun to bother me.
Category: Reflections
Reflections on learning and teaching and research and life.
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The right order for the fundamental trig identity
If you google “fundamental trig identity” you will get many many images and handouts which all list the fundamental trig identity as:
sin2 t + cos2 t = 1
This is, of course in the wrong order and it should really have cos first then sin, like this:
(cos t)2 + (sin t)2 = 1
“But David,” you say, “it’s addition, so it doesn’t really matter what order it’s in does it?” Of course it does! Mathematically it’s the same, but psychologically it’s different. If it really wasn’t different then you would sometimes write cos first and sometimes write sin first, but I can bet you always write it in a particular order. And if you write it with sin first, then you’re making it harder for yourself.
Let me explain.
The reason we have the fundamental trig identity is because the angle t there is a piece of the circumference of a unit circle, and cos t and sin t are the coordinates of the points on that unit circle. If I asked you to write down an x-y equation for the unit circle, you would naturally write x2 + y2 = 1 with the x first. But the x-coordinate of a point on the unit circle is cos t, and the y-coordinate is sin t, so of course that means it’s (cos t)2 + (sin t)2 = 1. Writing your trig identity with the cos first makes it easier to make the connection with the equation of the unit circle. If you write it with sin first you’ll have to continually switch it round!
Also, the order does matter if you’re using hyperbolic trigonometry. Then the formula is (cosh t)2 – (sinh t)2 = 1 and having sin first would be definitely mathematically wrong. For years, I had great trouble remembering which way around this was supposed to go until I realised that the cos and sin were in alphabetical order. From that point forward I always wrote my ordinary trig identity in the same order as the hyperbolic trig identity (in alphabetical order) so that through force of habit I would never get the hyperbolic one wrong.
So, I recommend you start writing your fundamental trig identity in the right order. It might help you remember and make connections to other things!
PS: You may be wondering about the way I put brackets in my trig identities. Don’t even get me started.
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Do you get tired of the same topics?
In the Drop-In Centre, the majority of students visit to ask for help learning in a very small number of courses, mostly the first-year ones with “mathematics” in the title. Of course, any student from anywhere in the uni can visit to ask about maths relating to any course, and we do see them from everywhere, but the courses called Maths 1X have between them a couple of thousand students per semester and that’s a lot of people who might need help to learn how to learn.
Anyway, the upshot of this is that I help people with the same topics semester after semester, year after year. Sometimes people ask me this question:
Question:
“Do you get tired of the same topics?”
Short answer:
No.
Long answer:
I actually really love the topics in first year maths. Row operations and the fact that they help to solve equations and decide independence and find inverses are fascinating. Nutting out how to do an integral is a fun game. Eigenvalues are the Best Thing Ever. And don’t get me started on conics and quadrics. To me, seeing them every semester is like watching the Muppet Christmas Carol every December. I get to be reminded of a story I love, and notice something little I had never noticed before every time.
Also, it’s not just the topics I get to see each semester, it’s the students learning the topics. So many of them have a perfectly appropriate and successful way of understanding it that never occurred to me and these make the topics fresh again. Who ever thought of checking vectors are parallel by making sure that cos of the angle between them is 1 or -1? Not me until yesterday when a student did it.
And then finally, I get to be there at the moment everything clunks into place and see the light in their eyes as they feel the buzz of understanding it for the first time. And that never gets old!
Short answer:
Hell no.
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Showing how to be wrong
After writing the previous blog post (Finding errors by asking how your answer is wrong) and rereading one I wrote three years ago (Who tells you if you’re correct?), I got to thinking about how students are supposed to learn how to check if they are right.
It occurred to me that, at least at university, we almost always show students how to be right, but almost never show them how to be wrong. We give them highly polished examples in lectures that proceed smoothly from the original information to the final answer, and then we move on. We very very rarely check our answers to see if they are correct, and even if we do check them, they are correct.
So the students never see any examples of how to deal with the situation where they are wrong. Is it any wonder, then, that they don’t know to find and fix their own errors?
If we’re going to give them examples in front of the class, I think we could spend a bit more time showing them how to check their work, and at least sometimes we should actually find errors that need fixing and fix them. Then they might actually see some strategies they can learn, rather than simply being lost when they’re wrong.
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Finding errors by asking how your answer is wrong
One of the most common situations we face in the MLC is when a student says, “I’m wrong, but I don’t know why”. They’ve done a fairly long calculation and put their answer into MapleTA, only to get the dreaded red cross, and they have no idea why it’s wrong and how to fix it. One of the major problems is that many students can’t tell if it’s because they’ve entered the syntax wrong, or done something wrong in their algebra, or completely misinterpreted the question, or if MapleTA itself has a bug and isn’t accepting the correct answer.
The other day, I was helping an Engineering Maths IIA student in exactly this situation. He was solving a differential equation and his answer was wrong, but he didn’t know why. As usual in this situation, I encouraged him to think of a way they could check his answer for himself (I commented on this a few years ago, actually Who tells you if you’re correct?). In this case, subbing the solution back into the original equation is a useful approach.
When he subbed his solution into the left-hand part of the equation, he got a result of -3/16 cos(1/4 t). Unfortunately, the right-hand part of the equation was -3 cos(1/4 t). So yes, his solution really was wrong. This left us with the much more difficult question of how to fix the error.
In a sudden flash of inspiration, I realised that the way that his solution was wrong might tell us something about the kind of error he had made. How could he have gotten -3/16 cos(1/4 t) instead of -3 cos(1/4 t) when he subbed into the equation? Perhaps because his solution was 1/16 of what it should be. I went looking for a 1/16 but couldn’t find one. Ok then, how could you produce a 1/16 in a less direct way? Perhaps you could divide by 4 twice. So this time I went back through his working looking for 4s. Like a moth I was attracted to this line in his working: “A/4 + B/4 = -3 ⇒ A + B = -3/4”. Of course! Dividing by 4 instead of multiplying by 4 would have the same effect as dividing by 4 twice, which could totally have produced that 1/16.
I was floored by the amazing effectiveness of this approach, and I wondered that I had never thought to do it before. It seems like such an obvious way to come up with something specific to look for. Admittedly it might not always yield useful results, but the evidence from this episode suggests that it might, which is certainly better than no strategy at all!
The student himself was suitably impressed and you could see him consciously committing the idea to memory for future reference. So now at least two people have a new strategy to find errors: when you sub your answer in to the original and it doesn’t work, investigate the way that your answer is wrong – it might help you find something specific to look for to find your error.
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Pretending not to know
Yesterday the Maths 1M students handed in an assignment question that asked them to prove a property of triangles using a vector-based argument. It’s not my job to help students do their assignment questions per se, but it is my job to help them learn skills to solve any future problem. This kind of problem, I find, is really hard to learn generalisable skills from.
For most students you really need to be there every step of the way as you try to solve the problem together, so that at the end you can look over what happened and figure out the sorts of things that made it possible to come up with a proof today. Students need to be hear the sorts of general self-questions you ask to help progress your thinking, even if they don’t lead anywhere straight away, and they need to see the dead-end paths you went down only to come back and go a different way.
The big problem is that if you’ve already seen the proof of this 15 times this week, it’s very very easy to guide students down a particular path that they could never possibly think of by themselves first go. It’s very easy to ask specific leading questions rather than general questions that might not lead anywhere. It’s very easy to push them away from the dead-end paths towards something that will give a result more quickly. You want to avoid doing that as much as possible, and the only way I know to do that is to pretend you haven’t seen the solution.
You’re going to have to pretend that you really don’t know how to do it and you really are just figuring it out with them today, and pretend to be surprised that something turned out nicely, and pretend to be frustrated when things don’t. It’s a real art and it takes a lot of practice and a lot of energy to pull it off.
I was very pleased the other day when I did pull it off. I was helping some students with this proof, and I said and did all the right things, including the dead-ends and everything.
After these students were happy with what we’d achieved and had a nice moral about problem-solving to take away, I turned to my other side to help the student who had been sitting there patiently. He had a whole different kind of proof to work on (mathematical induction), and I started as I often do by looking up the definition and writing that down, then saying “Now I’m not sure if this is going to help yet”. He responded to this by saying, “I don’t think I’ll ever believe you again when you say that.”
You see, I had helped him with the geometry proof only a couple of days before, and he had patiently sat there listening to the deja vu of me go through all the same things I went through with him. I looked him in the eye at that point and he said, “That was very impressive.” And he meant it. It’s nice when someone appreciates your craft.
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My cat’s bottom
Did you know that cats have scent glands just inside their bottoms that are constantly being filled with liquid and are squeezed as their poos come out, and if their poos are too skinny the glands are not squeezed enough and get over-full making them very painful and inflamed? Neither did I, until my cat Tabitha Brown started bleeding out of her bottom.
The vet literally reached into Tabitha’s bottom and squeezed the glands, making scent-liquid squirt all over his face (I have very great respect for vets, especially this one!). Then he told us some very useful things about how to look after our cat properly, such as the fact that many cats need Metamucil (yes, human Metamucil) every so often.
The point of this story is that we would never have known about how to look after our cat properly if she had never been in this pain. Because of her bleeding bottom, we now know for her, and for any future cats we might have in our home, to keep an eye on the size of their poos and give them Metamucil if the poos are consistently a bit skinny. All because of her bottom.
As always, my life makes me think about my teaching…
Students often say that I am a better teacher than their own lecturer, and it has become clear to me that the main reason this might be true is because I spend so much time with students. I am there when they have problems, and so I am able to see the problems students might have, and thus I am better equipped to help future students. There are so many things about the way people learn or don’t learn that I would never know if not for me being there when the students were in pain. Not that I wish anyone pain of course! I’m just saying their pain helps me prevent unnecessary pain for future students.
So thank you to all the students who were in pain and let me be there when they were. You make me more able to help others.
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Let’s
One of my friends and a past MLC staffer graduated from her PhD yesterday (congratulations Jo!). One of my strongest memories of Jo is when she told me something about my teaching that I never knew I was doing, but that she saw as an essential part of what I was trying to achieve at the MLC. That’s what I want to share with you today.
Jo noticed that there was a particular word I used rather often when working with students in the MLC. It was only a small word, but it was able to make a difference to the way students felt about our interactions.
The word was “Let’s”.
Very often, we tell students to do things that will help them learn, but usually those instructions come out as commands: “Find the definition in your notes”, “Read the original assignment instructions”, “Have a go at this first part of the question”, “Look at this handout”.
Many of the students who come to the MLC are in a fairly heightened state of stress; some of them really feel embarrassed that they had to ask for help; at least a few are just waiting for an excuse to pack up and run if they feel threatened. These students need reassurance that we really are committed to helping them today – that we’re willing to stick with them until they get it.
It turns out you can let them know you want to stick with them by simply adding a “let’s” to the front of your commands: “Let’s find the definition in your notes”, “Let’s read the original assignment instructions”, “Let’s have a go at this first part of the question”, “Let’s look at this handout”.
It’s amazing how one simple word (and the attitude that goes with it) can make such a huge difference to the experience a student can have of learning today.
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When will I ever use this?
“When will I ever use this?” is possibly a maths teacher’s most feared student question. It conjures up all sorts of unpleasant feelings: anger that students don’t see the wonder of the maths itself, sadness that they’ve come to expect maths is only worthwhile if it’s usable for something, fear that if we don’t respond right the students will lose faith in us, shame that we don’t actually know any applications of the maths, but mostly just a rising anxiety that we have to come up with a response to it right now.
There’s an interesting discussion in this pdf article [1] of the various responses that are commonly given to this question and their various drawbacks. The author is mainly concerned that we often inadvertently confirm the uselessness of maths by our very attempts to make it seem useful. While this is a legitimate concern, I have another one: in our attempts to justify the mathematics, we forget to listen to what the student actually needs.
In my experience, when a student asks this question, it’s a sign that they are starting to lose faith. They are having trouble motivating themselves and are seeking a reason to keep working at it. Being able to use it someday is the first thing they think of to motivate themselves, so they ask the question. But really most students will be happy with any reason that encourages them to stick at it today.
I had been thinking about this for a couple of days, after following a Twitter conversation and the comments on a post on Dan Meyer’s blog . Then one one of the students in the MLC actually asked the question, so I was all ready with my response. I said, “Actually, I’m not going to answer that question, but instead I have my own question to ask: how are you feeling about this topic right now?”
It is a testament to the trust I’ve built up with the students that he answered my question honestly! He said that he couldn’t see how the bits fit together or how they related to other things in the course. So I talked about how this topic fit in with the big ideas in maths, and how it connected with what they did last semester and last week. Then I helped him to organise some of the information in the topic so it was clearer how it was structured.
And you know what? After this discussion it didn’t matter so much that he might never use it. He had what he needed to have the courage to keep going, because I took the time to find out what was really bothering him.
[1] Otten, S. (2011) Cornered by the Real World: A Defense of Mathematics, Mathematics Teacher, 105-1, 20-25
Alexandre Borovik 27 April 2016:
It is like learning to swim: how many people actually have to use swimming for *practical* purposes?
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Really working together
Yesterday, I had one of those experiences in the MLC that makes me love my job.
The Maths 1B students were working on a linear algebra proof today, and as I came up to one of the tables, Fred (name changed) was explaining the beginning of his proof to the rest of the table. When I arrived at the table, he was leaning over two of the other students to point at a section of the lecture notes. He noticed I was standing there and said, “But David can probably do this better than me.” I responded, “Not necessarily. You keep going,” and I sat down in his chair.
Fred continued to explain, and I think he did a very good job. I was very pleased that he kept flicking through the lecture notes to point out different theorems, though I thought it was interesting that at no point did he write anything down.
Then one of the other students said, “so is that the end?” And Fred said that no, this was just the beginning, there was still more after that, and I could see in his eyes he was having a sinking feeling as he tried to think of how to move on to the next bit.
So I asked him if he could pass me a whiteboard marker. I stood up to the wall and said, “I just want to write down where we’re up to.” I asked the students he had been talking with to tell me what they’d done so far, and I transcribed it to the wall, asking them to explain why each line worked. And then we got up to the end of what they had already done.
“So what now?” I asked. There was a short silence, and then Fred piped up with a comment about what we needed to know next. I asked why that was important to know, and this started a discussion of what goal we were heading for.
And here is where the really great stuff happened. The students at the table offered suggestions of things to try, looked up definitions and theorems in their notes, helped each other refine their maths language, asked each other questions when they weren’t sure of things, welcomed new students into the discussion when they wandered over to listen, discussed how to make the proof their own when they wrote it to hand in, and basically really worked together to construct the proof. It was a pleasure to be a part of it.
It’s this sort of thing that makes my job such a joy – seeing students learning and supporting each other to succeed. On a day containing many other parts of my job that are much less joyful, it was something I really needed to see.
This comment was left on the original blog post:
Steven 8 February 2016:
Indeed, I also did enjoy reading your post regarding on how crucial and effective to have a group discussion i.e. working together. I’ll be sitting UMAT this year and I hope i can find someone/group as well to discuss on some UMAT questions and produce interesting results like the above.