I am teaching two students (J and K) Higher maths. Today, I wanted reflect on what we have done so far, by working on a large set of exam questions. However, I wanted to find ways of working on the problems, not just working through them.
I have been thinking about the role of (inner) dialogue when solving problems, particularly with regards to naming/describing what is there in order to bring associations and actions to mind, and then articulating what might be a useful approach, and why.
What follows are dialogues about five of the problems. I am talking - probing, asking questions - much more than usual, in order to force the students to practice articulating what they are doing, or going to do, and why. I am doing this in the hope that they might internalise this type of dialogue.
I chose this question to model the process, as I suspected they would find it difficult (it is not clear what needs to be done, unless one knows what to do!). [See Dialogue #5 for more on this question.]
Me: What type of question is it?
Me: There’s a quadratic in it… you’ve named it… what does that bring to mind for you?
K: The roots, and the quadratic equation…
Me: What are you thinking about, J, what type of question would you call that?
J: Something to do with the parabola [making a U-shape with hands].
Me: So that suggests to you what?
J: Find the roots.
Me: And maybe that sketching it might be useful? … So when you [J] see a quadratic and the word roots, you’re thinking about sketching, and when you [K] see a quadratic and the word roots, you’re thinking about…?
K: The quadratic equation…
J: I thought you could just factorise it.
Me: So you have different actions coming to mind… can you now articulate in words exactly what you are going to do?
J: We’re trying to find the value of k.
Me: What is K, can you give it a name? what type of mathematical object is it? [pointing to equation]
K: The value that is multiplying x squared.
Me: Yes, it’s the coefficient of x squared that you’re looking for… OK… what is using the quadratic equation going to find, is it going to find what we want?
K: No, it’ll give you the roots
Me: But I don’t want the roots.. and J, will what you are going to do find what you want?
J: Probably not.
Me: Is it even possible to find the roots of this equation?
K: No, unless you know k.
Me: So all the methods that are coming to mind are methods for finding roots, perhaps because that is normally what you do with quadratics, but that is not what this question is asking us to do… so we need sometthing else to find this coefficient of x squared… have either of you got any other alternatives?
K: Just rearrange for k.
Me: Imagine doing that, what’s going to happen?
K: You’ll have two x values.
Me: Will this give you what you want, the value of k?
J: You must be able to separate k from the x’s at some point?
K: We could factorise it.
Me: That was your first thought, wasn’t it J? Can you imagine what is going to happen when you do that?
J: You’re going to get kx and x, but you’ll never be able to get the three x, because of that k there.
Me: Yes… all of these ideas are ways of finding roots, but roots are not the things you want. So when I ask you to ask yourself if some approach is a good idea, this is what I’m talking about. But by articulating we might get some idea of what to do and also what might not be useful. Can one of you try reading it aloud and see what happens?
J: The roots of the equation k x squared minus three x plus two equals zero are equal, what is the value of k?
Me: So what bits of that sentence felt important?
J: That the roots are equal.
Me: Ah! That the roots are equal. Have we even talked about the roots being equal yet?
Me: But by reading it… we haven’t even considered that fact yet, but it’s the only fact you’ve got. So what do we know about this? Does anything else come to mind now?
K: It’s just sitting on the x-axis, one point on the x-axis… and it crosses the y-axis at two.
Me: So that’s an image, isn’t it? Does anything else come to mind?
J: Finding ranges and stuff, where k is bigger than…
Me: Can you give that a name?
Me: OK, is it about inequalities? It might be…
K: It has to be a positive value.
Me: What does?
K: Because it’s not upside down, becuase it passes through the y axis at two.
Me: Oh yeah! True. So we know k is positive… So this is an image… is there something else we can say algebraically? What else comes to mind?
K: squared brackets… brackets squared… x minus something or x plus something.
Me: Yeah, something like this [draws (x+-a)^2]… so now you know two things that are true about this quadratic. Anything other links?
J: Would completing the square help, or not?
Me: Can you complete the square for this?
J: You could put k x… but… you’ll get k squared…
Me: So again, the methods we use for solving or finding roots are not useful… The question we have is: If the roots of this are equal, what else is true? Is there anything else we know about quadratics that might be useful?
K: I give up.
J: [long pause] There must be something we can do.
Me: In an exam, would you try just doing something anyway, or leave it blank?
K: I would just write down everything I know.
J: I would leave it and move on to the next question and come back if there is time at the end.
Me: Your brain might subconsciously work on it in the meantime, or something else in the paper might trigger some thoughts for you. K, what would you write down?
K: That there’s one root, it passes the y-axis at two..
Me: There’s one root! We haven’t said that yet. OK, write down what you think is useful.
[Both spend some time writing down what they know]
Me: OK, we’re stuck.
J: Could you think of it without the k for a minute?
Me: Just call it x squared minus three x plus two equals zero for a minute…? Go on then, do that. What would you do K?
K: Go and feed some cows. [laughter]
Me: It’s not the craziest idea, relax the mind… maybe not during an exam...
J: [works on solving x^2 - 3x + 2 = 0 and sketching it] it’s not what you want…
Me: So you’re looking for the k that just makes it touch.
J: It could be any number.
Me: You could try all the numbers in the world, but this suggests that there’s an algebraic way of solving this… the question is, what is it? OK, let’s come back to it.
K: It’s really frustrating when you don’t know the answer.
Me: Yes, I know.
I chose this question next as I felt they would have some success on it, as we have worked on these ideas more recently. It is also a nice example of the role of making associations from what is discerned.
Me: Let’s look at this question. Read it out loud.
[K reads it, and then immediately writes -2 on the diagram next to the radius, then 1/2 next to the tangent]
Me: Mmm, right. Hold on.
J: You need to find the radius.
Me: OK, when you say you want to find the radius, what do you mean?
J: I want to find the length of it.
Me: [To K] Pause there, because it’s not about getting the answers that I’m bothered about, it’s about how we go about solving problems. [To J] You want to find the length of the radius, can you say why that’s a good idea?
J: [pause] Normally, you would try and find the equation of the circle, and that would help you find the equation of that line.
Me: Mmm. How will it help you do that?
J: You would need to find the centre.
Me: So that you could find the equation of the circle?
Me: OK. [To K] What do you think about that?
K: I would say you don’t need to know anything about the circle at all.
Me: Why not?
K: Because you can see the gradient is a half… and you know that point [pointing at P]… so you don’t need to know anything about the circle, but I might be wrong.
Me: You’re articulating what you think and why you think it. So, J, I would ask you again: Why would finding the equation of the circle be useful for finding the equation of the tangent?
J: You could just use y equals m x plus c, you have m, and putting in the coordinate [P] would give you c…
Me: K’s right isn’t he, that you don’t need to know the equation of the circle. You sometimes want to know the equation of the circle, and then you would need to know the radius, and you would need to know the centre, but we don’t need to know that here… the question: why is it a good idea? is useful… you could spent quite a long time doing this here, but there is no reason for it. What you said, that you normally do that, is important. You have to ask yourself if what you are about to do is a good idea in this case… So, K, why is this one easier for you than the last one?
K: Because we’ve done it before?
Me: We’ve done both of these types of questions before. What do you mean by that?
K: We’ve done this process before.
Me: You remember doing this process before? What type of question would you call this?
K: Finding the equation of a tangent...
Me: ... to a circle. Yes, and all of the stuff you do is in the name for you. You don’t need anything else if you know what type of question it is.
From this point on, J and K decided to work on all the questions involving quadratics in the pack (taken from two years of exam papers).
Me: [I read the question out loud] I feel like I stressed the words values of x, graph and above the x axis. Has this type of question got a name?
J: Because those are the questions when you take different parts of the graph, and then see.
Me: What tells you that you’ve got to do that?
J: Above the x axis.
Me: [To K] Can you articulate what you are going to do?
K: Write less than minus three and greater than five [writes -3>x>5]
Me: You’re going to go straight to the answer? Is that a good idea?
K. I know it’s minus five and three, so it must be…
Me: What’s minus five and three?
K: The roots.
Me: That’s useful. So you think you can go straight to the answer from that? [To J] What do you think?
Me: Why do you want to sketch it?
J: Because then you know if you’re right or not.
Me: The fact that it is talking about a graph makes me think we should draw the graph. K, you can write that answer if you want, but you have no way of knowing it is right, you’ve got just what you think is the right answer. You still have to check it, so you probably need to sketch it anyway.
[Both sketch the graph]
Me: So what are we looking for?
J: [Pointing to the parts of the curve that is above the x-axis] That bit there, and that bit there. [Writes x>3, x<-5]
Me: You’ve got different answers, how do you know if your answers are right or wrong?
K: I know that x is definitely bigger than three, and [tracing his pen along the curve in the negative x direction] if it’s less than three then it’s under the x axis, then when it gets to minus 5 it’s on the x axis, and then it goes above it.
Me: Can you read to me what you’ve written there?
K: [Reading -5>x>3] the x is less than minus five and greater than 3.
Me: [To J] Is that what you’ve got? Is it saying what you want it to say?
J: x is less than minus five... and x is greater than three.
Me: Is that what you want?
Me: [To K] Is that what yours says?
K: No. [changes his to two inequality statements, x<-5 and x>3]
Me: Yes, you need that, why do you need that?
K: Because it’s two bits.
Me: Yes. Can you see that your original response was not what was needed. I feel you don't want to do it, but what I’m asking you to do is pause in some way.
The students are now choosing the questions. I am pleased as it will bring up the discriminant, which has not been mentioned yet, but is what is required to solve the first problem that we got stuck on.
Me: What type of question is it?
J: Inequalities again?
Me: Let’s read it out.
[K reads it out]
Me: Can you name anything there, what’s it about?
Me: OK, yes. What’s a?
J: The coefficient of x squared.
[Both students are selecting an answer. K points at B, J Points at A and B]
Me: OK… so what’s the question about?
J: About working out what these things do [pointing to the two inequalities], how they look?
Me: Yes, how the algebra - the coefficient of x squared, and the discriminant - is connected to the graphs. [To K] OK, you’re saying B, can you say how you came to that conclusion?
K: The discriminant has to be positive so it has to be A or B.
K: Because if the discriminant is negative, then it’s…
Me: Which one is the discriminant?
K: This one [points to b^2-4ac]. But then a is negative as well… if it was negative it would be upside down.
Me: It’s more about a being negative isn’t it?
K: Yeh… and then… erm… If it was that one [points at A] then it would be zero is less than that, or something… it’s in two bits so it must be under the x axis… [getting muddled]
Me: You’re thinking about inequalities, but is that what this is about? [To J] What do you think?
J: I think it’s B as well.
J: Because if b^2-4ac > 0 there are two real roots, but there is only one there [pointing to A].
Me: [To K] That’s a different reason to your reason. [To J] Talk to me about the mathematics that’s going on here.
J: I’m not sure if I’ve remembered this right, but if the discriminant’s greater than zero there are two roots, and if it is equal to zero there is only one root.
Me: And if it’s less than zero…?
Both: No roots.
Me: Right. [I remind them about the appearance of the discriminant in the quadratic formula] OK, so that would be a good rationale for the answer being B. Now, see how important it was naming that thing as the discriminant. By naming it, all the facts you know about the discriminant come to mind.
This dialogue is about the following question. I though it would be tricky, but was pleasantly surprised how straightforward they found it, and even more pleasantly surprised when J turned our attention to reconsidering the first question half way through...
[J reads the question out, K starts crossing out options, J writes b^2-4ac=23]
K: I would say it’s B.
Me: OK… Can you tell me what you are thinking about here?
K: If you think about this equation [starts writing the quadratic formula with sqrt(23) in] then that’s not going to give you a nice number, it’s going to be an irrational number, so the roots are not rational.
Me: Nice, a nice thought. So you’re imagining 23 in the equation… Lovely. Is that what you were going to say J?
J: No, I was going to look at statement 1 first… The roots must be real because the discriminant equals 23.
Me: Why does the discriminant being 23 mean that the roots are real?
J: Because, as K was saying, the square root of 23 [in the formula] is going to give you two roots.
Me: Yes. What would it have to be for it not be have real roots.
Me: Yes. OK. Now then, what question shall we do next?
[pause, then out of the blue...]
J: Would you use the discriminant in the one that we did before, that one we couldn’t do?
Me: You could try it, it’s something new isn’t it? Let’s go back to it. Can we use the discriminant to do that? Can you articulate what you are going to do?
J: The b squared minus four a c equals zero, because the roots are equal.
K: But what about the k?
Me: We’ve got a new idea to work on. The b squared minus four a c equals zero, because the roots are equal. Is this going to be a good idea?
J: By doing this, you get rid of your x, so you could rearrange to find your k.
Me: Got rid of your x, what do you mean by that?
[J doesn't reply and starts working on it, writes down b^2-4ac=0 and populates it with values. As he works on it, K is watching, contributing now and then with algebraic steps. They arrive at the correct solution.]
Me: Have you answered the question you’re trying to answer?
J: Well, we have a value for k.
Me: Well, you have a logical basis for believing it’s true, you did what you did for a reason, it feels as though it must be true somehow. Perhaps you could check by substituting these values back into b squared minus four a c, but if you’ve done your algebra right it must equal zero… Well, that felt important, didn’t it? Not solving that problem, but…
J: Doing another question helped us do this one.
J: Because it was one of the only things we hadn’t tried.
Me: Yes, the connection with the discriminant wasn’t close enough before. When looking at similar questions in the future, the association with roots being equal may lead us to consider the discriminant, we might look at this question and think ‘Ah, this is a discriminant question’. This association is something we would like to happen. Can you see how articulating and naming might help us figure out what is required, for example when you said ‘there is only one root’?