2.5 Differentiation: Mathematics Communication
So far, you have focused on written mathematical language. In this section, you will practise understanding and taking part in spoken mathematical discussions. You will read and analyse two different dialogues: one between two students, and one between a student and a lecturer.
The goal of this section is not just to understand these dialogues, but to help you feel confident in having similar discussions yourself. You will learn how to have a discussion about mathematics, ask and answer questions, and explain your ideas clearly to someone else. This will help you use English for practical conversations about mathematics, both in class and out of class. To do this, you will need to:
-
notice useful phrases and sentence structures in the dialogues,
-
memorise them,
-
practise saying them,
-
and finally, use them in real conversations.
Each line in the dialogues is numbered to help you talk about specific parts easily. For example, if you have a question, you can say: What does the second word from the end of line 8 mean?
By working with these dialogues, you will improve your ability to discuss mathematics in English, ask and answer questions, and communicate your ideas clearly with others.
Dialogue 1: Two students
An example dialogue between two university students, Oluwaseun and Jamie, discussing differentiation. This dialogue occurs in a study space with the two students sitting close to each other. It is a relaxed study session between peers tackling homework.
Select the ‘Play’ button to watch the video. The activities are based on the dialogue below the video.
| Line | Speaker | Conversation |
|---|---|---|
| (1) | Oluwaseun: | Hey Jamie, I was going through our mathematics homework, and I’m stuck on this differentiation problem. Do you have a minute to help me out? |
| (2) | Jamie: | Absolutely, Oluwaseun! Let me come round to your side of the table. What’s the problem? |
| (3) | Oluwaseun: | I’m trying to differentiate this function. I know the basic rules, but I’m not sure if I’m applying them correctly. |
| (4) | Jamie: | No worries, let’s go through it together. The first two terms are polynomial terms, so we need to use the power rule. This states that the derivative of x to the power n is equal to n times x to the power, n minus one. So, let’s apply that to each polynomial term in your function. |
| (5) | Oluwaseun: | Okay, so for the first term, three times x to the power four, that would come to three times four x cubed, right? |
| (6) | Jamie: | Exactly, which is twelve x cubed! Now, what about the second term, minus five x cubed? |
| (7) | Oluwaseun: | For that one, would the derivative be minus five times three x squared, which is minus fifteen x squared? |
| (8) | Jamie: | Indeed, it would! And, here of course we applied the same rule that we used just a moment ago. Now, how about the third term, cos of x? |
| (9) | Oluwaseun: | That is not a polynomial function, right? |
| (10) | Jamie: | That's right. Remember that a polynomial function involves sums of powers of x with constant coefficients, whereas cos is one of the trigonometric functions, along with sine and tangent. |
| (11) | Oluwaseun: | But cos is also one of the standard derivatives, right? |
| (12) | Jamie: | That’s right, the derivative of cos is minus sine. Remember, we proved this in our lectures last week, using the limit definition of the derivative. And how about the last term, minus seven, which is a constant? |
| (13) | Oluwaseun: | That just differentiates to zero. So, bringing everything together, the derivative is twelve x cubed, minus fifteen x squared, minus sine x. |
| (14) | Jamie: | Exactly! You’ve got it. Differentiation is all about applying the rules step by step. Do you have any other questions? |
| (15) | Oluwaseun: | Not right now, but thanks a lot, Jamie! This really helped. |
| (16) | Jamie: | Anytime, Oluwaseun. Happy to help! |
Activity 1: Global understanding
From the options below, make a note of the summary that you think most accurately describes the beginning of the dialogue from (1) to (8). Next, select the ‘Answer’ button below the options.
Option 1: Jamie firstly asks Oluwaseun what he is unsure about before referring him to a certain page in the course notes.
Option 2: Jamie firstly revises a mathematical operation with Oluwaseun before working through two parts of an example with Oluwaseun.
Option 3: The first thing Jamie does is give his own example of how to apply a rule before turning to Oluwaseun's questions.
Answer
Option 2 is the correct answer: Jamie invites Oluwaseun to recall the power rule and then applies this to the first two (polynomial) terms of the function that Oluwaseun is trying to differentiate. Option 1 is incorrect as it is Oluwaseun who initiates the conversation and there is no reference to a page number. Option 3 is incorrect as the first thing that Jamie does is revise the power rule; only then does Jamie proceed to the examples.
Activity 2: Identifying useful language
Question 1: Identify useful phrases in the dialogue using the following prompts.
- Between (5) and (9), where does Jamie state that the two students have just used a procedure identical to that which they used just now.
Answer
And, here of course we applied the same rule that we used just a moment ago.
- After (11), where does Oluwaseun say something that means 'Now we are able to draw a conclusion for the calculation'?
Answer
So, bringing everything together, … .
Question 2: In (10) and (12), Jamie uses the term remember. Which of the following options is the better description of what Jamie means here? Make a note of your answer and then, select the ‘Answer’ button below the options.
Option 1: Jamie is saying that he (Jamie) can clearly recall studying this and that therefore he (Jamie) feels confident in explaining it to Oluwaseun. In other words, Jamie is saying 'I can remember this!'.
Option 2: Jamie is reminding Oluwaseun that Oluwaseun probably, or certainly, will recall the background to the information. In other words, Jamie is saying 'You probably remember this!'.
Answer
Option 2 is the correct answer. If we say I remember the lecture yesterday, we are likely to mean remember in its literal or usual sense, similar to I recall. However, when we use the instruction form e.g. Remember that Pythagoras' theorem only applies to right-angled triangles, we usually mean You need to know this to follow the next things I'm saying.
Activity 3: Your turn!
Goal
Practise having conversations about differentiation problems in English. Focus on communication, not necessarily on solving new math problems.
Set-up
In pairs, take the following roles:
Speaker A is the student that has a question about a problem in differentiation. Speaker A can either choose something that is an actual problem, or something that they already understand and pretend they don't.
Speaker B is the student that has the answer to the question and needs to explain it to Speaker A.
Procedure
Step 1: Assign the roles - one student takes Speaker A and the other student takes Speaker B.
Step 2: Decide a problem from the topic differentiation to discuss.
Step 3: Take 3 to 5 minutes to look over the dialogue above and note down any useful phrases.
Step 4: Run the dialogue. Speaker A should start, either with a variation on Oluwaseun's opening above, or with something else, for example, Hi [Speaker B]. Do you have a quick moment to help me out with [Problem P]. It shouldn't take more than a couple of minutes.
Step 5: Switch roles and practise again to experience both asking and answering.
Step 6: Together reflect on what went well. You might reflect on how accurately and fluently the dialogue was produced, how confident you felt, and if there are alternative ways of going through the dialogue.
Dialogue 2: A student and a lecturer
An example dialogue between a university student, Jamie, and his lecturer, Dr Smith, discussing differentiation. Jamie approaches Dr Smith just after a lecture, to ask some questions from the lecture.
Select the ‘Play’ button to watch the video. The activities are based on the dialogue below the video.
| Line | Speaker | Conversation |
|---|---|---|
| (1) | Jamie: | Thank you for today's lecture, Dr Smith. Do you have a moment now at all? I was wondering if you could please help me with a couple of differentiation questions I’m struggling with, from today’s lecture. |
| (2) | Dr Smith: | Hello, Jamie. Glad you found the lecture of value. Happy to help, of course. What are your questions? |
| (3) | Jamie: | OK, so firstly, when we were differentiating the function f of x is equal to e to the power two x, times sine of x, I know that we used the product rule, but I’m not sure how to apply it correctly. |
| (4) | Dr Smith: | OK, so what does the product rule state? |
| (5) | Jamie: | Just let me check my notes. The product rule states that if you have a function f of x that is equal to u of x times v of x, then the derivative, f prime of x, is u prime of x times v of x, plus u of x times v prime of x. |
| (6) | Dr Smith: | Perfect. So, let’s identify u of x and v of x in your function. |
| (7) | Jamie: | Okay, so I can let u be the exponential and v be the sine. |
| (8) | Dr Smith: | Exactly. Note that it doesn’t really matter which one is which. We can swap them around, due to the symmetry of the product rule. Now, let’s find the derivatives of u and v. What is u prime of x? |
| (9) | Jamie: | The derivative of e to the power two x is two e to the power two x. |
| (10) | Dr Smith: | Correct. And which rule did you use here? |
| (11) | Jamie: | This is due to the chain rule. |
| (12) | Dr Smith: | Very good, and what about v prime of x? |
| (13) | Jamie: | The derivative of sine is cos. |
| (14) | Dr Smith: | Right. Now, all that we need to do is apply the product rule. |
| (15) | Jamie: | So, if I insert everything in the product rule, I get that f prime of x is equal to two e to the power two x, times sine x, plus e to the two x, times cos x. |
| (16) | Dr Smith: | Perfect! You’ve got it. Differentiation it can be tricky but breaking it down step by step makes it more manageable. Do you have any other questions? |
| (17) | Jamie: | Not on this. This makes perfect sense. But I do have a question on notation. When we were taking the derivative of tan to the minus one of x, were we differentiating the inverse of tan, or tan to the power minus one? |
| (18) | Dr Smith: | That’s a good question. Notation can mean anything we want it to be, as long as we follow the general conventions. If we write tan to the minus one of x, we mean the inverse function of the tangent, which is also known as arctangent. If we write tan of x, all of it to the power minus one, we mean the reciprocal of the tangent, which is one over tangent, which is also known as cotangent. That’s one of the reasons why the notations arctan and cot are useful, because they allow us to easily distinguish between the two cases. |
| (19) | Jamie: | So, in our case, we were differentiating the inverse of tan. |
| (20) | Dr Smith: |
That’s right. Do you have any other questions? |
| (21) | Jamie: | Not at the moment, but this really helped. Thank you, Dr Smith! |
| (22) | Dr Smith: | You’re welcome, Jamie. I’m glad I could help. Feel free to contact me if you have any more questions. |
Activity 1: Global understanding
From the options below, make a note of the summary that you think most accurately describes the beginning of the dialogue, from (1) to (12). Next, select the ‘Answer’ button below the options.
Option 1: In the discussion with Jamie about the product rule, Dr Smith gets Jamie to identify another rule that is used.
Option 2: The central piece of advice that Dr Smith offers is that in certain circumstances, variables can be swapped around.
Option 3: Jamie's main concern is about how to use the notation correctly.
Answer
Option 1 is the correct answer: after a discussion about the product rule, Dr Smith gets Jamie to identify the chain rule as the second rule which is relevant to the problem. Option 2 is incorrect because, although Dr Smith does mention that certain variables can be swapped around in certain circumstances, this is a tangential or sidebar comment; this is not the central point. Option 3 is incorrect because although Jamie does have a question about notation later in the dialogue, this is not the focus of the opening section of the conversation.
Activity 2: Identifying useful language
Question 1: Note 6 occasions where Dr Smith makes an encouraging comment that means something like 'You understood that correctly, Jamie'. The first one is 'Perfect' in (6).
Answer
Perfect, Exactly, Correct, Very good, Perfect (2), That's right.
Question 2: Identify something Dr Smith says which means 'If we take apart a mathematical procedure into its component parts and work through each component part one-by-one, this can make the mathematics easier'.
Answer
breaking it down step by step makes it more manageable.
Question 3: In (18), locate a three-word string that means something similar to 'provided that'.
Answer
as long as. This phrase has two distinct meanings in English. First, it can express a comparison of length or degree, similar to as tall as or as large as. For example: In any rectangle that is not a square, the sides within each pair of equal sides are as long as each other. Second, and more commonly in mathematical writing, as long as functions as a conditional linking phrase meaning provided that. For instance: The formula holds as long as [latex]n>0[/latex]. Because the phrase is ambiguous out of context, readers should use the surrounding sentence to determine which meaning is intended.
Activity 3: Your turn!
Goal
Practise having conversations about differentiation problems in English. Focus on communication, not necessarily on solving new math problems.
Set-up
In pairs, take the following roles:
Speaker A is the student with a question in differentiation. Speaker A can either choose something that is an actual problem, or something that they already understand and pretend they don’t.
Speaker B is the lecturer, professor or seminar leader with the answer to the question and needs to explain it to Speaker A.
Procedure
Step 1: Assign the roles - one student takes Speaker A and the other student takes Speaker B. Think about how this student-lecturer dialogue might be different to the student-student dialogue in the beginning of the section.
Step 2: Decide a problem from the topic of differentiation to discuss.
Step 3: Take 3 to 5 minutes to look over the dialogue above and note down any phrases that you feel may be useful.
Step 4: Run the dialogue. Speaker A should start, either with a variation on Jamie's opening utterance above, or with something else, for example, Excuse me Professor Xi, would you happen have some time to help me out with [Problem P]. It shouldn't take too long.
Step 5: Run the dialogue a second time with the roles reversed.
Step 6: Together reflect on what went well. You might reflect on how accurately and fluently the dialogue was produced, how confident you felt, and if there are alternative ways of going through the dialogue.
In this section, you practised reading and performing dialogues (spoken communication) about mathematics. In the next section, you will focus on written mathematical communication.
