1.5 General Terminology: 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 and written email exchanges. You will read and analyse a dialogue between two students, and an email correspondence between a student and a lecturer.
The goal of this section is not just to understand the dialogue and the email, but to help you feel confident in having similar discussions and writing emails 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:
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notice useful phrases and sentence structures in the dialogues,
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memorise them,
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practise saying them,
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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 texts, 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, Sam and Elise, sitting together in a study space after a lecture, discussing mathematical terminology and notation.
Select the ‘Play’ button to watch the video. The activities are based on the dialogue below the video.
| Line | Speaker | Conversation |
|---|---|---|
| (1) | Sam: | Elise, I’m so confused about the language in this lecture. The lecturer keeps saying things like ‘this is sufficient for that’ and ‘necessary and sufficient’. What’s the difference? |
| (2) | Elise: | Oh, I struggled with this too, Sam! So, if one thing is necessary for another thing, it means you need the first thing. You can’t have the second without the first. For example, a number being divisible by two is necessary for being divisible by four. Meaning that if the number is not divisible by two, then certainly, it is not divisible by four. |
| (3) | Sam: | Okay, so necessary means required? |
| (4) | Elise: | Yeah, that’s right, but sufficient is different. Sufficient means enough. A number being divisible by twelve is sufficient for being divisible by four; in other words, it’s enough to ensure divisibility by four. Meaning that if a number is divisible by twelve, then it’s definitely also divisible by four. |
| (5) | Sam: | Is it also necessary then? |
| (6) | Elise: | Ehh … no, being divisible by twelve isn’t necessary for being divisible by four. Because a number can be divisible by four without being divisible by twelve, like the number sixteen. |
| (7) | Sam: | So, something can be sufficient but not necessary? |
| (8) | Elise: | Yes! At other times it can be necessary and not sufficient, and sometimes it can be both. For example, a number having zero as its last digit, is both necessary and sufficient for being divisible by ten. You need to have the last digit zero, so it’s necessary, and if you have the last digit zero, then you also know that the number is divisible by ten. |
| (9) | Sam: | That makes sense. But here’s what really gets me, the lecturer uses the phrase ‘if and only if’. Is that different from just saying ‘if’? |
| (10) | Elise: | Great question! This is really important, and it is also related to the necessary and sufficient conditions above. ‘If’ is one-directional, but ‘if and only if’ goes both ways. It’s like saying something is both necessary and sufficient. |
| (11) | Sam: | Can you explain this more? |
| (12) | Elise: | OK, if I say, ‘if you study hard, you’ll pass the exam’, that only goes one way. Studying hard will help you pass, but someone might also pass the exam without studying hard. But if I say, ‘you’ll pass the exam if and only if you score above forty percent,’ then both directions are logically true. Scoring above forty percent guarantees a pass, and a pass guarantees you scored above forty percent. |
| (13) | Sam: | Oh wow, that’s a really precise way of thinking! So ‘if and only if’ is like saying the two things are equivalent? |
| (14) | Elise: | Exactly! And mathematicians abbreviate it as ‘iff’ sometimes, which is why you’ll see that in textbooks. It’s very common in proofs. |
| (15) | Sam: | Thank you Elise, you’ve made this so much clearer. I feel like I can follow the lectures now without getting lost in the terminology! |
| (16) | Elise: | Happy to help! The truth is, once you understand the mathematical language, the actual mathematics becomes easier. The language and the mathematics really go hand in hand. Keep asking questions like you’re doing. |
Activity 1: Global Understanding
From the options below, decide which summary you think most accurately describes the overall structure of the dialogue. Next, select the ‘Answer’ button below the options.
Option 1: Sam and Elise help each other to understand a number of technical terms in mathematics.
Option 2: Sam and Elise use their understanding of a number of technical terms in mathematics to answer a set of questions.
Option 3: Elise helps Sam solve some equations by explaining the meanings of a number of technical terms to Sam.
Option 4: Sam is confused about and wants to understand a number of technical terms in mathematics and Elise is able to help Sam with that.
Answer
Option 4 is the correct answer: At the beginning of the dialogue, Sam is very unsure about the meaning of ‘sufficient’, ‘necessary’ and ‘if and only if’ in a mathematical context. Elise carefully explains these terms and Sam is clearer about them at the end. Option 1 is incorrect as there is no sense of the two students helping each other; Elise is helping Sam throughout. Options 2 and 3 are incorrect as there is no reference to a set of problems that either student is trying to solve.
Activity 2: The structure and function of the discussion
This conversation consists of two approximately equal parts, each with a similar function. In lines (1) to the beginning of (9), Sam seeks clarification from Elise about the meaning of the two terms sufficient and necessary. Then, from (9) to (16), Sam seeks a further clarification about the meaning of the term if and only if.
This is the overall structure of the dialogue; as regards its function, in each notional half, Sam seeks clarification and Elise assists him. Working with other students either to request assistance or to provide it (or both) are frequent, normal parts of student life. In this activity, therefore, we look at how the two students use language to solve Sam’s problem.
In the first section, Sam initiates the dialogue by saying that he is confused, giving some context and explicitly asking for help. On a sentence-by-sentence basis, here are the functions of each of Sam’s sentences.
| Line | Word, phrase or sentence | Function(s) |
|---|---|---|
| (1) | Elise, I’m so confused about the language in this lecture. | Sam states how he is feeling (confused) and makes clear what the problem is (the language in this lecture). |
| The lecturer keeps saying things like ‘this is sufficient for that’ and ‘necessary and sufficient’. | Sam goes on to give specific examples of what language is troubling him. | |
| What’s the difference? | Finally, Sam asks for assistance. |
The functions in column 3 are sometimes called rhetorical moves in linguistics. The term ‘rhetorical’ means something like ‘communication’ or ‘communicating intentionally with others’ and the term ‘move’ refers to a particular act in which a speaker or writer contributes to the wider communication. Each of the sentences that comprise (1) are rhetorical moves.
By separating (1) into three different sentences, we can analyse each sentence and assign each a function as a rhetorical move. We now do this for the remaining lines in the first half of the dialogue (2) to the beginning of (9).
| Line | Word, phrase or sentence | Function(s) |
|---|---|---|
| (2) | Oh, I struggled with this too, Sam! | Elise shows empathy with Sam which establishes rapport and signals that the question is reasonable. |
| So, if one thing is necessary for another thing, it means you need the first thing. | Elise gives a definition which is informal, or language-based definition of ‘necessary’. | |
| You can’t have the second without the first. | Elise paraphrases the definition above or restates in a slightly different way. | |
| For example, a number being divisible by two is necessary for being divisible by four. | Elise gives an example which is easy to understand. | |
| Meaning that if the number is not divisible by two, then certainly, it is not divisible by four. | Elise provides a commentary on her example. | |
| (3) | Okay, so necessary means required? | Sam asks a checking question. |
| (4) | Yeah, that’s right, but sufficient is different. | Elise confirms that Sam understands and then moves directly to the term sufficient. |
| Sufficient means enough. | Elise gives a single word, informal paraphrase. | |
| A number being divisible by twelve is sufficient for being divisible by four; in other words, it’s enough to ensure divisibility by four. | Elise gives an example followed by another longer paraphrase. | |
| Meaning that if a number is divisible by twelve, then it’s definitely also divisible by four. | Elise provides a commentary or gloss on the above. | |
| (5) | Is it also necessary then? | Sam asks a checking question. |
| (6) | Ehh … no, being divisible by twelve isn’t necessary for being divisible by four. | Elise directly indicates that Sam has misunderstood (we assume politely) and then provides full correction. |
| Because a number can be divisible by four without being divisible by twelve, like the number sixteen. | Elise gives an explanation to illustrate the correction she provides above. | |
| (7) | So, something can be sufficient but not necessary? | Sam asks another checking question. |
| (8) | Yes! At other times it can be necessary and not sufficient, and sometimes it can be both. | Elise directly indicates that Sam has understood (we assume somewhat enthusiastically or at least positively). |
| For example, a number having zero as its last digit, is both necessary and sufficient for being divisible by ten. | Elise provides an example. | |
| You need to have the last digit zero, so it’s necessary, and if you have the last digit zero, then you also know that the number is divisible by ten. | Elise provides a commentary on her example. | |
| (9) | That makes sense. | Sam indicates that he has understood the preceding comments. |
| But here’s what really gets me, … | Sam raises a second question for which he wants some help (see below). |
Analysing a dialogue (or any text) using this sentence-by-sentence functional analysis offers a range of benefits. These include the following:
- The technique allows us to break down the dialogue into its component parts and assign each part a function. In other words, it is at one level, simply a tool for analysing a stretch of language.
- As a result, the technique allows us to be able to identify recurrent or repeated patterns including patterns across sentences. For example, in the above dialogue, Elise uses the pattern of ‘paraphrase + example + commentary’ on two occasions.
- The technique allows us to match particular strings of words to particular functions. We are then able to focus on any particular strings of language that are not wholly familiar or automatic, study and learn them, and ideally use them in our own interactions.
Your Task: Using the above model, conduct a similar sentence-by-sentence functional analysis of (9) to (16) inclusive. Next, select the ‘Answer’ button below.
Answer
| Line | Word, phrase or sentence | Function(s) |
|---|---|---|
| (9) | But here’s what really gets me, the lecturer uses the phrase ‘if and only if’. | Sam notes that he is feeling confused about something else (what really gets me) and then states what it is, namely another piece of mathematical terminology. |
| Is that different from just saying ‘if’? | Sam asks a question about the terminology that is confusing him. | |
| (10) | Great question! | Elise affirms that Sam has raised a sensible question (which helps builds rapport). |
| This is really important, and it is also related to the necessary and sufficient conditions above. | Elise provides some context before giving her answer. | |
| ‘If’ is one-directional, but ‘if and only if’ goes both ways. | Elise provides an informal language-based definition. | |
| It’s like saying something is both necessary and sufficient. | Elise draws a parallel to the previous discussion. | |
| (11) | Can you explain this more? | Sam asks for further information. We might comment that this is a very open ended question from Sam. He might instead have asked something a little more specific for example I’m just not getting this distinction I’m afraid. Could you give another example or perhaps write something down?. |
| (12) | OK, if I say, ‘if you study hard, you’ll pass the exam’, that only goes one way. | Elise gives an example from the real world for the term if. |
| Studying hard will help you pass, but someone might also pass the exam without studying hard. | Elise provides some commentary on the example. | |
| But if I say, ‘you’ll pass the exam if and only if you score above forty percent,’ then both directions are logically true. | Elise offers an example of the other term, if and only if. | |
| Scoring above forty percent guarantees a pass, and a pass guarantees you scored above forty percent. | Elise provides some commentary on this second example. | |
| (13) | Oh wow, that’s a really precise way of thinking! | Sam appears to understand Elise’s explanations and expresses his surprise at the precision of mathematical language. |
| So ‘if and only if’ is like saying the two things are equivalent? | Sam checks his understanding; he usefully volunteers a statement that allows both him and Elise to judge to what extent he has understood. | |
| (14) | Exactly! And mathematicians abbreviate it as ‘iff’ sometimes, which is why you’ll see that in textbooks. It’s very common in proofs. | Elise indicates that Sam has indeed understood and, with this understanding in place, goes on to add further information, here some context about mathematical notation. |
| (15) | Thank you Elise, you’ve made this so much clearer. | Sam thanks Elise both with a thank you and an additional comment. |
| I feel like I can follow the lectures now without getting lost in the terminology! | Sam states how he is feeling (i.e. better!) as a result of this conversation. | |
| (16) | Happy to help! | Elise acknowledges Sam’s thanks. |
| The truth is, once you understand the mathematical language, the actual mathematics becomes easier. The language and the mathematics really go hand in hand. Keep asking questions like you’re doing. | Elise adds a couple of final comments which serve the purposes of encouraging Sam and bringing the conversation to a positive end. |
The final column of the tables above are, in effect, a subset of some of the common functional moves in spoken and written mathematics communication. Functions like offering a paraphrase, providing an example, commenting on an example, linking an example to a previous example are common ways of communicating. The functions above are not comprehensive: there are many, many more. For example, mathematicians often do things like defining a term of an equation, applying a formula to an example, explaining how a piece of code works. As a self-study task, you may wish to start developing a list of common mathematical functions and some of the language that goes with them. Of course, The Academic Language of Mathematics will return to this theme throughout the book.
A final comment on the sentence-by-sentence functional analysis in this activity: the ways in which people talk about mathematics varies from culture to culture. The dialogue here represents ways of speaking that are usual in many British English contexts. However, not all of these ways of speaking will necessarily translate to all other contexts. For example, in some cultures, it may be less usual to express how one is feeling (e.g. when Sam says I’m so confused). That said, in some cultures it may be usual to say more about how one is feeling. Similarly, in some cultures it may be less usual to say things like Well done!
Language Focus
In (2), Elise says For example, a number being divisible by two is necessary for being divisible by four.
The grammatical structure of this sentence might be analysed as follows:
- For example, …: This two-word linking phrase is at the head of the sentence, and links it to the previous material.
- … a number being divisible by two …: internally, this string has an -ing form verb (being divisible by) with its own subject (a number); externally, the whole string is itself the subject of the verb is in c) below.
- … is necessary for …: this string consists of a part of the verb be (is) plus an adjective predicate with a preposition (necessary for). As noted above in b), the subject of the verb is here is not a noun but a whole clause, specifically the string in b) above.
- … being divisible by four : this string completes the sense of what is necessary in c) above.
Grammatically speaking, the heart of this sentence is the fact that there is a clausal subject (b) to the main verb is in c). In other words, instead of having simply a noun or noun phrase as the subject (e.g., For example, a calculator is necessary for most mathematics exams), the example sentence above has the more complex structure, a clause, as its subject. This clause has its own verb in an -ing form.
Breaking the sentence down in this way allows us to generate additional sentences with similar structure. Some examples follow, where the clausal subject is underlined and the -ing form of the verb is in bold. Double slashes (‘//’) indicate breaks between the lettered divisions above.
- A recent study suggests that // lecturers publishing their office hours // gives students greater confidence in asking questions. This example differs from the one in the text in two ways. Firstly, lecturers is plural whereas in the example in the text we had the singular a number. However, both of these have general reference and can be expressed with any: any number that is divisible; any lecturer.
- As I was saying a moment ago, // students working on mathematics terminology with each other // is extremely helpful for // making the whole process more fun and in fact effective.
- In many people’s view, // preparing steadily for exams and even creating a study schedule // is much more effective than // simply rushing it all in the last few days.
Each of the sentences above, as well as the original sentence can be paraphrased using when as follows:
| Sentence | Original | Paraphrase |
|---|---|---|
| Sentence from text | For example, a number being divisible by two is necessary for being divisible by four. | For example, when a number is divisible by two, it is necessarily divisible by four. Or, For example, when a number is divisible by two, this means it is necessarily divisible by four. |
| i. | A recent study suggests that // lecturers publishing their office hours // gives students greater confidence in asking questions. | A recent study suggests that when lecturers publish their office hours, this can give students greater confidence in asking questions. |
| ii. | As I was saying a moment ago, // working on mathematics terminology with fellow students // is extremely helpful for // making the whole process more fun and in fact effective. | As I was saying a moment ago, when students work on mathematics with each other, this is extremely helpful for … |
| iii. | In many people’s view, // preparing steadily for exams and even creating a study schedule // is much more effective than // simply rushing it all in the last few days. | In many people’s view, when we prepare steadily for exams […], this is much more effective than simply rushing it all in the last few days. |
Activity 3: Your turn!
Goal
Practise having conversations which address confusion or misunderstandings about the meaning of key mathematical terms 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, a confusion or a misunderstanding about some key terminology in mathematics. 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 using examples, comments, illustrations.
Procedure
Step 1: Assign the roles – one student takes Speaker A and the other student takes Speaker B.
Step 2: Decide on some key terminology about which Speaker A has a misunderstanding or question.
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 Sam’s opening above (Elise, I’m so confused about the language in this lecture), or with something else, for example, Hi [Speaker B]. Do you have a quick moment to help me out with [Terms T1 and T2]. 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.
Email Correspondence: Student asking lecturer about missed class
An example email correspondence between a university student, Jin Park, asking their lecturer, Dan Foster, a question from a class that the student missed.
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Email Number |
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|---|---|
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(1) |
From: j.park@walsch-mcdonaldnu.edu.ac.uk Dear Dr Foster, I hope you’re well. I was unfortunately unable to attend yesterday’s lecture, and I’m hoping you could help me understand something that came up in the notes I’ve borrowed from a classmate. The notes mention the terms “converse” and “contrapositive” in relation to proofs. I understand these must be important ways of thinking about logical statements, but I don’t have a clear sense of what each one means or when we’d use them. Could you briefly explain these concepts? Also, the notes seem to suggest that a statement and its converse are not necessarily equivalent, but I’m not entirely sure what that means in practice. Thank you very much for your time. Best wishes, |
|
(2) |
From: d.foster@walsch-mcdonaldnu.edu.ac.uk Dear Jin, No problem at all, glad you reached out! These concepts are crucial for understanding mathematics, so it’s worth getting them clear. Let me explain with an example. Suppose we have a statement: “If a number is even, then it is divisible by 2”. This is our original statement. The converse of this statement reverses the direction: “If a number is divisible by 2, then it is even”. Now, in this particular example, the converse happens to be true as well. However, converse statements are not automatically true just because the original statement is true. For instance, consider the statement “If A is a subset of B and B is a subset of C, then A is a subset of C”. The converse would be “If A is a subset of C, then A is a subset of B and B is a subset of C”, which is clearly not always true – the set B could contain the whole of C! The contrapositive is different. The contrapositive reverses the statement and negates both parts. For our original statement “If a number is even, then it is divisible by 2”, the contrapositive is “If a number is not divisible by 2, then it is not even”. Here’s the key insight: a statement and its contrapositive are always logically equivalent. If one is true, the other must be true as well. In practice, sometimes it’s easier to prove the contrapositive of a statement than to prove the original statement directly. This is a powerful technique in mathematical proofs. I hope this clarifies things. Feel free to pop by my office hours if you’d like to discuss further. Best regards, |
|
(3) |
From: jin.park@walsch-mcdonaldnu.edu.ac.uk Dear Dr Foster, Thank you so much for that clear explanation! The examples really helped me understand the distinction. I can see now why proving the contrapositive might indeed be easier in some cases. I really appreciate you taking the time to explain this. Could you please remind me when you have your office hours? I would like to discuss a part of the proof from the lecture that I missed. Best wishes, |
|
(4) |
From: d.foster@walsch-mcdonaldnu.edu.ac.uk Dear Jin, I am glad that you found it helpful. My office hours are Mondays 14:00-15:00 and Wednesdays 11:00-12:00. Best regards, |
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(5) |
From: jin.park@walsch-mcdonaldnu.edu.ac.uk Dear Dr Foster, Thank you so much for letting me know. I will see you on Monday. Best wishes, |
Activity 1: Global understanding
From the options below, decide which summary you think most accurately describes the email correspondence. Next, select the ‘Answer’ button below the options.
Option 1: Jin’s initial question in (1) is partially resolved by the email exchange, but Dr Foster suggests that Jin attend office hours in order to go over the questions.
Option 2: The email exchange has three broad functions: a) clarifying the meaning of the two terms converse and contrapositive; b) ascertaining when Dr Foster’s office hours are; c) working through the proof from the lecture which Jin missed.
Option 3: At the end of the email exchange, Jin is clear both on the use of the two terms converse and contrapositive and on when Dr Foster’s office hours are.
Option 4: Jin will attend Dr Foster’s office hours with a fellow student from the course.
Answer
Option 3 is the correct answer: Dr Foster’s email resolves Jin’s confusion about the two terms and also deals with the practical matter of the timing of Dr Foster’s office hours. Option 1 is not correct, as Jin states that the email exchange has been sufficient to resolve his confusion over the two terms converse and contrapositive; Jin will attend office hours on a different matter (a proof from the lecture). Option 2 is wrong because the email exchange only deals with clarifying the two terms converse and contrapositive and with the practical matter of office hours. The third matter, the proof from the lecture, will be discussed at the office hours. Option 4 is not correct since Jin does not state he will attend office hours with a fellow student but alone.
Activity 2: Identifying useful language 1 – Rhetorical functions in written text
The central part of the email exchange is Dr Foster’s detailed reply in (2) to Jin Park’s question about mathematical terminology. This is given in the body of (2) in the exchange, reproduced below for convenience.
Let me explain with an example. Suppose we have a statement: “If a number is even, then it is divisible by 2”. This is our original statement.
The converse of this statement reverses the direction: “If a number is divisible by 2, then it is even”. Now, in this particular example, the converse happens to be true as well. However, converse statements are not automatically true just because the original statement is true. For instance, consider the statement “If A is a subset of B and B is a subset of C, then A is a subset of C”. The converse would be “If A is a subset of C, then A is a subset of B and B is a subset of C”, which is clearly not always true – the set B could contain the whole of C!
The contrapositive is different. The contrapositive reverses the statement and negates both parts. For our original statement “If a number is even, then it is divisible by 2”, the contrapositive is “If a number is not divisible by 2, then it is not even”. Here’s the key insight: a statement and its contrapositive are always logically equivalent. If one is true, the other must be true as well.
In practice, sometimes it’s easier to prove the contrapositive of a statement than to prove the original statement directly. This is a powerful technique in mathematical proofs.
In the dialogue above between Sam and Elise, we used the technique of sentence-by-sentence functional analysis to identify the different rhetorical moves that Sam and Elise made. We can use the same technique with the email exchange between Jin and Dr Foster. Consider the first paragraph of the reproduced text above (which is the second paragraph of the email in (2)).
| Sentence | Function |
|---|---|
| Let me explain with an example. |
Here, Dr Foster states what he will do in the rest of the paragraph. In other words, the purpose of this first sentence is simply to state what will come next. Dr Foster is writing about the text, not about mathematics. |
| Suppose we have a statement: “If a number is even, then it is divisible by 2”. | Dr Foster provides a mathematical proposition. You will be aware from reading the text that this proposition will be used in the following paragraphs. In other words, this proposition does not exist ‘on its own’ in the text but is a starting point for a subsequent mathematical argument. |
| This is our original statement. | Finally in this paragraph, Dr Foster labels the proposition so that he can refer to it efficiently and accurately in the subsequent text. |
Dr Foster’s opening paragraph in (2) is extremely clear, logical, explicit and makes no assumptions. Dr Foster is writing like a mathematician! He does not assume anything; he states what he is beginning his argument with, explains why, and annotates it.
Your task: Perform a sentence-by-sentence functional analysis for the second and third paragraphs of the extract above. Next, select the ‘Answer’ button below.
Answer
Analysis of paragraph two of the extract above:
| Sentence | Function |
|---|---|
|
The converse of this statement reverses the direction: “If a number is divisible by 2, then it is even”. |
Using the original statement as the starting point, Dr Foster gives an example of the term ‘converse’. |
| Now, in this particular example, the converse happens to be true as well. | Dr Foster offers a caveat to his example, signalled by the linker Now. |
| However, converse statements are not automatically true just because the original statement is true. | Dr Foster proceeds to a more general principle. Beginning with the linker However, …, Dr Foster gives a general, informal definition of the term converse. |
| For instance, consider the statement “If A is a subset of B and B is a subset of C, then A is a subset of C”. | Dr Foster sets up an example: the phrase for instance indicates that an example is to be introduced, the string consider the statement introduces the example and the material within inverted commas (“…”) is the example proposition. |
| The converse would be “If A is a subset of C, then A is a subset of B and B is a subset of C”, which is clearly not always true – the set B could contain the whole of C! | Dr Foster develops the example which he began in the previous sentence indicating how the converse of the statement above is not the same as the original statement. |
Analysis of paragraph three of the extract above:
| Sentence | Function |
|---|---|
|
The contrapositive is different. |
This first sentence of a new paragraph signals that the text is turning to the second term. |
| The contrapositive reverses the statement and negates both parts. | Dr Foster gives an informal definition. |
| For our original statement “If a number is even, then it is divisible by 2”, the contrapositive is “If a number is not divisible by 2, then it is not even”. | Dr Foster returns to the original statement to give an example of a contrapositive statement. |
| Here’s the key insight: a statement and its contrapositive are always logically equivalent. | Dr Foster signals that the key point is coming, then states it directly. |
| If one is true, the other must be true as well. | Dr Foster paraphrases the previous statement. |
Activity 3: Identifying useful language 2 – polite and appropriate language
The central purpose of the email exchange above is to communicate about mathematics. However, in addition to effectively and efficiently communicating about mathematics, the email exchange is also polite, respectful and appropriate. Consider, for example, Jin Park’s opening email, (1), reproduced below for convenience.
Dear Dr Foster,
I hope you’re well. I was unfortunately unable to attend yesterday’s lecture, and I’m hoping you could help me understand something that came up in the notes I’ve borrowed from a classmate.
The notes mention the terms “converse” and “contrapositive” in relation to proofs. I understand these must be important ways of thinking about logical statements, but I don’t have a clear sense of what each one means or when we’d use them. Could you briefly explain these concepts?
Also, the notes seem to suggest that a statement and its converse are not necessarily equivalent, but I’m not entirely sure what that means in practice.
Thank you very much for your time.
Best wishes,
Jin Park
Examples of polite, respectful and appropriate language include the following:
| Example | Comment |
|---|---|
| Dear Dr Foster, | Jin Park uses a standard address form. |
| I hope you’re well. | The email begins with a standard greeting. This is clearly nothing to do with the mathematical question which the email is about. |
| I was unfortunately unable … borrowed from a classmate. | This sentence may be seen as polite and respectful due to the adverb unfortunately, which signals that Jin regrets his absence, and the phrase borrowed from a classmate, which shows he has made his own effort to keep up. |
| Could you briefly explain these concepts? | Jin Park frames his request politely using Could you rather than a direct imperative. The word briefly shows consideration for Dr Foster’s time. |
|
Thank you very much for your time. |
The email includes an expression of thanks. |
| Best wishes, | The email ends in a standard way. |
Answer
Analysis of Dr Foster’s first reply in (2).
| Example | Comment |
|---|---|
|
No problem at all, glad you reached out! These concepts are crucial for understanding mathematics, so it’s worth getting them clear. |
In Sentence 1, Dr Foster warmly welcomes Jin Park’s email. In sentence 2, Dr Foster notes the importance of Jin Park’s question which indicates that he considers the question valid and worthwhile, which is encouraging for the student. |
| Let me explain with an example. | Arguably, this is more part of the mathematical presentation. However, the string Let me explain … is a polite way of signalling that an explanation is coming, rather than launching into it without preparation. |
|
I hope this clarifies things. |
This kind of language is a standard way to close a section of explanation and check that communication has been successful. |
| Feel free to pop by my office hours if you’d like to discuss further. | Dr Foster invites Jin Park to follow up with him if he would like. |
Analysis of Jin Park’s second email in (3). Note that nearly the whole of this email is dedicated to an extremely polite expression of thanks.
| Example | Comment |
|---|---|
| Dear Dr Foster, | An appropriate address form |
| Thank you so much for that clear explanation! The examples really helped me understand the distinction. I can see now why proving the contrapositive might indeed be easier in some cases. | Jin Park expresses his thanks, and provides further comments on how Dr Foster’s reply helped him. |
| I really appreciate you taking the time to explain this. | A polite expression of thanks. |
| Could you please remind me when you have your office hours? | A polite request for a reminder of timings. |
| Best wishes, | A standard sign-off. |
Activity 4: Your turn!
You have read and discussed the email exchange between Jin Park and Dr Foster. Now work with a partner to practise writing polite academic emails about mathematical terminology. This activity will help you practise writing formal academic emails, asking for clarification, and explaining mathematical terminology accurately and clearly.
Step 1: Individually, write an email to a lecturer asking for clarification of two mathematical terms. You may choose terms from this chapter or from your current mathematics course.
In the email, make sure you:
- use a polite and respectful tone
- include an appropriate greeting and closing
- clearly state the two terms you would like explained
- briefly explain what you find confusing or unclear
- ask the lecturer to provide clarification.
Use Jin Park’s email as a model for the structure, language, and level of formality.
Step 2: Exchange your email with your partner. Read your partner’s email carefully, and then write a reply as if you are the lecturer.
In your reply, make sure you:
- thank the student for their email
- explain both mathematical terms clearly
- include examples to support your explanations
- add brief commentary to help the student understand the difference or connection between the terms
- maintain a polite, helpful, and respectful tone.
Use Dr Foster’s reply as a model for the structure, language, and style of explanation.
In this section, you practised reading and performing a dialogue (spoken communication) about mathematics, and explored written communication through email correspondence. In the next section, you will focus on longer written mathematical texts.
