4.6 Sequences and Series: Exploring Mathematics Through Text

The purpose of this activity is to help you think about three features of a text: its audience (who reads it), its purpose (why it was written), and its structure (how it is organised). Academic writing takes many different forms. Each form has its own conventions, purposes, and ways of organising information. Noticing these differences will help you become a stronger academic reader and writer.

Question 1:  Look over the whole text and select the single best option of the audiences listed below. Next, select the ‘Answer’ button below the options.

Audience 1: Teaching staff who are supervisors of final-year undergraduate projects who wish to engage in professional development.

Audience 2: Students considering applying for a PhD in mathematics.

Audience 3: Undergraduate students in mathematics from this university only.

Audience 4: Undergraduate students in mathematics from around the world.

Question 2: Read the whole text briefly and select which of the following best describes the central purpose of the text. Next, select the ‘Answer’ button below the options.

1. To invite undergraduate and postgraduate mathematics students from only Upper Northbridgerly University to submit a proposal for the conference.
2. To advertise the Undergraduate Mathematics Society to potential new undergraduate students.
3. To demonstrate to a government body that the School of Mathematics at Upper Northbridgerly University conducts world-leading research in statistics and data analysis.
4. To invite undergraduate students of mathematics across the world to consider submitting a proposal for the conference.

Follow-up task:

Before proceeding, you may wish to practise summarising what you have just noticed. Try to write one or two sentences in your own words stating:

• who the audience of this text is, and

• what the purpose of the text is.

You might begin with: The intended audience of this text is … / The purpose of the text is …

There is no single correct answer here. The aim is to express these ideas clearly and concisely in your own words.

A Call for Abstracts is a highly structured text. It is also a type of text we might skim-read fairly quickly to see if the topic, dates and format are relevant to us. It is therefore useful to have a mental structure of a Call for Papers. To practice familiarising yourself with this structure, this activity invites you to decide which of the 8 numbered sections would help answer different questions.

Your task: Read each of the following five questions below, that a student considering submitting to the event might have. For each question:

• Which of the 8 numbered sections would you refer the student to?
• What is the answer to the question?

If you are unsure, choose your best answer, then read the relevant section to check if you were correct or not.

The 8 numbered sections are:

1. Conference Information
2. About the Conference
3. Calls for Abstracts
4. Abstract Specifications
5. Further Opportunities
6. FAQs (which stands for Frequently Asked Questions)
7. Key Dates
8. Contact Information

Example: If the question was What form should my abstract take?, the answer is in Section 4, which states a 300-word limit (±10%) and provides a 6-part structure. Information about the 6-part structure is then repeated in Section 6, the FAQs. Note that from this example, you can see that any given question may be addressed in multiple sections of the text.

Question 1: I can’t use LaTeX! Can I submit in a .pptx format?

Question 2: Can my cousin, who is in her final year doing a PhD on sequence transformations at a university in Spain, submit an abstract?

Question 3: Can I write up my presentation after the conference?

Question 4: I’ve just done a really well received presentation on my explorations of the Fibonacci sequence? Can I repeat this at the conference?

Question 5: Can I present something about the learning and teaching of series and sequences?

Next, select the ‘Answer’ button below.

This Call for Abstracts contains many direct, unambiguous and non-negotiable instructions. However, it also offers choices and provides options for certain aspects of the conference.

For example, in the second sentence of paragraph 1, we read This year’s conference focuses on Sequences and Series, though papers related to adjacent topics are also welcome. The final words of this sentence offer a choice: presenters have the option to present on topics adjacent to sequences or series. However, what counts as an ‘adjacent topic’ is left to individual judgement.

Your task: Consider the extracts below taken from the text. For each, determine where there is a direct, unambiguous, non-negotiable instruction (which we label as Direct) and/or a choice, range or option (which we label as a Choice). Identify the language that indicates whether the sentence is Direct or a Choice. Next, select the ‘Answer’ button below the extracts.

1. Presentations may be based on original research, extended coursework, projects, historical investigations, pedagogical explorations, or novel applications of sequences and series in real-world contexts.
2. Your abstract should be 300 words ±10%. Use plain, accessible language and avoid excessive jargon.
3. Submissions may be individual or collaborative, with up to three authors per presentation.
4. Please use the provided LaTeX template and submit the generated PDF file.
5. Ensure that you write in English using UK spelling conventions.
6. Your abstract should follow a clear six-part structure: title, background and motivation, aims and scope, methodology, key findings, and significance.

Academic conferences have many functions. They certainly offer the opportunity to travel and socialise. However, their core purpose is about knowledge creation and knowledge building. This is often a highly collaborative effort where researchers share work, discuss and develop ideas together.

In contrast to this collaborative approach, however, a conference also offers individuals the chance to present and promote their own work so that others notice it. We sometimes call this as having an impact – having your work recognised and valued by other academics.

Two parts of the text focus on the potential impact that a participant may have. The first part is the final paragraph of Section 2, reproduced below with the relevant language of impact underlined.

If you are an undergraduate mathematics student, this academic conference offers you a unique opportunity to showcase your work in a welcoming setting and discover exciting research developments across a wide range of mathematical disciplines.

Your task: Look through the rest of the text and find the second part that uses language of impact. Identify particular uses of language that are to do with having an impact. Next, select the ‘Answer’ button below.

In the FAQs section, consider the paragraph which responds to the question How do I decide between a presentation and a poster? This paragraph is reproduced below for convenience.

[S1] Both formats are equally valued, but they suit different types of work. [S2] Presentations work best for work that is more developed, substantive, and requires sustained explanation or theoretical depth, for example, a complete investigation with methodology, findings, and implications. [S3] Poster presentations are ideal for work that is exploratory, preliminary, or demonstrates a specific concept or application. [S4] Consider the maturity and scope of your work: if it represents significant engagement with a topic, choose a presentation, whereas if it is more focused or exploratory, a poster may be more appropriate.

The framework below analyses each sentence by examining:

• Structure: the framework looks both at the internal structure of each sentence and any linking words or phrases such as but, although, on the other hand that connect the sentence to what precedes or follows it.
• Purpose: the framework relates each sentence to the sentences that precede and follow it in terms of the communicative purpose.

Here is a sentence-by-sentence commentary, with comments on both structure and purpose.

Sentence 1 (S1):

• Structure: This sentence has two clauses:
  1. Both formats are equally valued, …
  2. … but they suit different types of work.

Clause a) is the ‘simple’ answer to the question in the FAQs, namely that neither format is better. Clause b) provides further information that adds more detail. The but of clause b) is a fairly weak but: it does not contrast with clause a), it instead develops the ideas of clause a).

• Purpose: S1 is a controlling sentence for the rest of the paragraph. S2 and S3 give further detail about the two different types of format mentioned in S1, while S4 consists of advice on how to make the choice between them.

Sentence 2:

• Structure: This sentence contains two main ideas: it first characterises work that might work best in the presentation format, and then provides an example. This two-part structured might be bracketed in the following way:

[ Presentations work best for work that is more developed, substantive, and requires sustained explanation or theoretical depth ] , [ for example, a complete investigation with methodology, findings, and implications. ]

This bracketing shows that the material beginning with for example, … is an optional addition. In fact, this material could form a separate sentence and be rewritten as An example of this is a complete investigation … 

• Purpose: This sentence characterises work that may be best presented in presentation format, and to give an example of such work. That characterisation comes first, followed by the example. A reasonable expectation for the following sentence is either further material on what might work best for a presentation, or (as in the text), a similar sentence on what kind of work might best fit the poster format.

Sentence 3:

• Structure: This sentence is structurally parallel to S2, as it characterises what type of work is appropriate for the poster format and what it might look like. It does not include an example. Like S2, it characterises the type of work in three ways:

S2: work which is: (a) more developed, (b) substantive, and (c) requires sustained explanation or theoretical depth

S3: work which is: (a) exploratory, (b) preliminary, or (c) demonstrates a specific concept or application

• Purpose: As has already been stated, this sentence echoes the previous sentence in terms of purpose.

Sentence 4:

• Structure: This sentence has multiple clauses and it is divided by a colon. Its structure might be bracketed in the following way: [ Consider the maturity and scope of your work: ] { [ if it represents significant engagement with a topic, choose a presentation, ] [ whereas if it is more focused or exploratory, a poster may be more appropriate. ] } We note that there is a pre-colon controlling clause, followed by a pair of clauses after the colon which contrast each other (linked by whereas). Both parts after the colon use if-structures.
• Purpose: This sentence gives advice for potential contributors. It concludes the paragraph well because it offers practical advice based on the distinctions made in the previous sentences.

Your task: Now consider the answer to the fifth question of the FAQs which is I am studying mathematics in English as a second language. Will my abstract be disadvantaged? The answer to this question is reproduced below with divisions into sentences.

[S1] Submissions are evaluated primarily on mathematical content, rigour, and originality, not on English language proficiency. [S2] However, clarity of communication does matter. [S3] Your abstract should be as clear as possible so that the reviewers and the audience understand your mathematical ideas. [S4] We encourage you to use simple, accessible language. [S5] If your abstract is accepted, we can offer support in preparing your presentation and refining your written explanations. [S6] Please note in your submission if you would like language support assistance.

Using the framework above, analyse the structure and purpose of each sentence. Next, select the ‘Answer’ button below.

Locate specific words and phrases in the text with the following meanings. Next, select the ‘Answer’ button below.

1. Identify a noun in Section 3 which means something like the quality of being extremely thorough, logically precise and mathematically correct, without any gaps.
   
   Answer

   rigour (note that the adjective is rigorous with a spelling shift from –our to -or-)

2. Identify a verb in Section 4 which means something like make sure.
   
   Answer

   ensure

3. Identify an adjective in the question column of Section 6 which means something like to be treated unfavourably or to have a less chance of.
   
   Answer

   disadvantaged (disadvantage and advantage are nouns, however we can use both as verbs as this example shows).

In the FAQs section you have encountered the following sentence: Your abstract should be as clear as possible so that the reviewers and the audience understand your mathematical ideas. 

This can be broken down in the following way:

This basic structure can be used to generate many different sentences. We consider three variations on the structure.

Variation 1: variations on possible. The following alternatives to possible are available:

1. as clear as you can
2. as clear as the example
3. as clear as is required

Variation 2: verb + [as Adjective as + possible] → verb + [as Adverb as + possible (or variations)]:

1. to write + as clearly as you can
2. to respond + as soon as is required (soon is an adverb even though it does not end in -ly)
3. to present + as closely as possible to December 2^nd

Variation 3: the linker and all subsequent material can also be varied:

1. to write as clearly as you can so as to allow easily readability
2. to respond as soon as is required because of the approaching deadline
3. to follow the example as closely as possible as this will prevent unnecessary mistakes

Your task: Using both grammatical and meaning clues, match the parts of the following three sentences. Next, select the ‘Answer’ button below.

Make sure you are familiar with the grammar of this construction. Then try to create one or two more sentences using this structure.