3.2 Integration: Terminology in Context

The first paragraph of the activity above contains five sentences. As in any well-written paragraph, each sentence should show clear links to the other sentences in the paragraph and should help to create the meaning of the whole paragraph. In this language focus box, we analyse each sentence, we give it a name to capture its function (purpose) and we note some language features that help make that sentence do its work.

Sentence 1: We have now completed our study of differential calculus.

Function: This looks back on and recaps the previous paragraph. By doing this at the beginning of a new paragraph, the sentence strongly suggests that a new idea – or at least some fresh development of the previous idea – will be the focus of this new section. This is a good example of clear, logical, ordered mathematical writing.

Noteworthy language features: 

We have now completed … An alternative way of saying this includes The previous paragraph was concerned with differential calculus. We could also combine this sentence with the following one like this: Having completed our study of differential calculus, we now turn to …

Sentence 2: We turn next to integral calculus, which is based on a very different geometrical concept, but will nevertheless turn out to be closely related to differential calculus.

Function: This sentence does a number of things:

We turn next to … This makes it explicit to the reader that a new topic will now be considered;

… integral calculus, which is based on a very different geometrical concept, … This names that topic and establishes that is distinct;

… but will nevertheless … This provides a connection between the new and the former topic (hence nevertheless) and as such acts as a rationale for why this new topic is being considered now.

Noteworthy language features: 

We turn next to – This string makes explicit that a new topic is coming.

… which is based on a very different … but will nevertheless turn out to be closely related to … This balanced structure simultaneously establishes a distinction while, at the same time noting a clear connection between the two topics.

Sentence 3: Suppose we are given the graph of some function f of x, and suppose we want to evaluate the area under the graph, between x is equal to a and x is equal to b.

Function: This sentence is an example with two steps: first it states what the graph is and then immediately states the problem to be solved. The repetition of suppose marks the two-step nature of the process which the sentence is talking about.

Noteworthy language features:

Suppose … – This verb is fairly similar in meaning to assume. Here it introduces the example. We might also write Take the graph of … 

Sentence 4: Geometrically, we certainly have a very clear idea of what we mean by this area, but so far we have no idea of how to actually compute it.

Function: This is a comment on the problem in the previous sentence. Like sentence 2, it has two parts, the first of which states something which the reader knows, the second of which introduces what the reader doesn’t know. This is why we have but between the two parts.

Noteworthy language features:

The connections here are more implicit. The explicit connection to the previous sentence comes towards the end with … this area, … and … how to actually compute it. These strings refer the reader back to the problem in the previous sentence.

Sentence 5: One way might be to split the closed interval a b into lots of little subintervals, and approximate the area we really want by the sum of the areas of many little rectangles.

Function: An initial, tentative approach to the problem. The final sentence of the paragraph suggests a possible approach of calculating the area.

Noteworthy language features: 

One way might be to … This tentative/hedging language suggests that there may be other approaches to take.

Clearly, a sentence-by-sentence analysis of a paragraph is a rich and rewarding language analysis task to undertake. Using the three-fold framework used here of Sentence, Function, Noteworthy language features, analyse the second paragraph of the text.