6.6 Vectors and Coordinate Systems: Exploring Mathematics Through Text
In the previous section, you practised reading and performing mathematical dialogues. This is spoken communication. In this section, you will focus on written mathematical communication. This book uses several different types of text across the chapters. This gives you exposure to a variety of mathematical writing.
The text below is a fictional announcement for an inaugural lecture at the fictional Southbury University. It consists of a formal event invitation followed by detailed sections describing the lecture topic, the speaker’s biography, and a related book launch. Imagine that this announcement appears on the university’s events website or in a departmental newsletter.
Your task is to read the text and work through the activities. The activities have a logical structure: they begin with more global or whole-text activities, that examine the purpose and audience of the text, and they gradually narrow to a focus on individual language items. In mathematical writing, there is an inherent connection between ‘macro’ text-external elements (such as audience and purpose) and ‘micro’ text-internal aspects (such as individual words and phrases).
By working through the various activities below, you will be able to:
-
Work with extended mathematical prose texts to enhance your understanding of both mathematical concepts and the structure of mathematical writing.
-
Identify and develop your command of useful academic phrases and sentence structures within text.
-
Apply this language in your own writing and mathematical discussions.
Activity 1: The overall audience and purpose of the text
The purpose of this exercise 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: The text below is an announcement of an inaugural lecture from a fictional academic working in Mathematics and in Mathematics Education. In many academic cultures, an inaugural lecture is usually delivered by an academic who has recently been promoted to the role of Professor.
Your task: For each of the ten text items below, decide whether it is likely to be Present (likely to occur) or Absent (unlikely to occur) in an announcement of this kind. For the items you marked as Present, categorise when they are likely to appear in the text: Earlier (near the beginning of the announcement), Later (near the end of the announcement), Variable (could appear at different points; no fixed position. Next, select the ‘Answer’ button below.
- The speaker’s date of birth.
- Where and when the lecture will take place.
- Any awards or recognition that the speaker has received.
- The speaker’s own education development at UG and PG levels.
- A bio of the speaker.
- The speaker’s favourite colour.
- What the speaker enjoys doing in their own time.
- Some relevant intellectual context or background for the lecture.
- A comprehensive list of publications.
- The title of the lecture.
Answer
Present: 2) Where and when the lecture will take place; 3) Any awards or recognition that the speaker has received; 5) A bio of the speaker; 8) Some relevant intellectual context or background for the lecture; 10) The title of the lecture.
Absent: 1) The speaker’s date of birth; 4) The speaker’s own educational development at UG and PG levels; 6) The speaker’s favourite colour; 7) What the speaker enjoys doing in their free time; 9) A comprehensive list of publications.
There are various answers regarding the order. The following is one possibility:
- Earlier/towards the beginning: Title of the lecture; Some relevant intellectual context or background for the lecture.
- Later/towards the end: A bio of the speaker; Any awards or recognition that the speaker has received.
- Variable: Where and when the lecture will take place. This practical information may appear at the beginning of the text, as in this example, or it may be placed in a later section.
Question 2: In Question 1, you identified five text features that are Present in inaugural lecture announcements. Each feature is expressed using specific language patterns and formatting choices.
Your task: Read through the text and:
- identify where each of the five features appears. A given feature may be located in more than one place. Note anything about the format, i.e. whether the text is centred, in bold, in italics, or is a list of bullet points.
- note down any relevant language patterns used, e.g. key verbs, time and place expressions, linking phrases.
- consider the purpose of this part of the text.
For example, the first Present item is the title. It appears near the start of the text as a centred line. In terms of any relevant language patterns, it consists of two adjective + noun phrases, plus a colon (:), plus another adjective + noun phrase. The purpose of the title, is to provide a succinct summary of the event and catch the eye of people with an interest.
Next, select the ‘Answer’ button below.
Answer
| Function | Where; any relevant language patterns; purpose |
|---|---|
| Title | Location and format: At the head of the text as the second line; centred, as is conventional for titles.
Language: it consists of two adjective + noun phrases, plus a colon, plus another adjective + noun phrase. The function of the colon is to specify more information. Purpose: to provide a succinct summary of the event. |
| Some relevant intellectual context or background for the lecture | Location and format: This information occurs across the text including:
Language:
Purpose: After the brief title, the text provides detail, context, and background. This lecture is particularly rich in context because it links an area of mathematics, the teaching and learning of mathematics, and a mathematician’s professional journey, whilst also marking a book launch. |
| Where and when the lecture will take place | Location and format: This occurs after the opening sections. It takes the form of a list of phrases containing relevant, practical details.
Language: The list of headings are typical. The term Venue is a more formal equivalent of the term Location. We might also see this material in single-word or short phrase question form e.g. What?, Where?, How to register?, Whom to contact? Purpose: To cover practical arrangements and logistics. Similar details appear in the final paragraph of About the lecture and in the contact details at the end. |
| A bio of the speaker | Location and format: Mainly in the section About Professor Sandra Hernandez, but also scattered across the text.
Language: The biography mentions work, achievements, and awards in broadly chronological order, tracing her shift from mathematician to researcher in mathematical education. Purpose: This section serves two purposes:
|
| Any awards or recognition that the speaker has received | Location and format: This information occurs in line 1 of the main body of the text: … newly appointed to the Chair of Mathematics Education at Southbury University. It occurs in the final paragraph of the section About Professor Sandra Hernandez.
Language: The first occurrence above is simply Sandra Hernandez’ title. However, the phrase newly appointed conveys immediacy. The second occurrence lists her Fellowships and Awards; this conveys prestige of the speaker but might also allow the audience a way in to informal conversation with the speaker. Purpose: To market the event by emphasising the speaker’s achievements. |
Inaugural Lecture Professor Sandra Hernandez
Spatial Language and Mathematical Understanding: A Scholarly Evolution
You are cordially invited to the Inaugural Lecture of Professor Sandra Hernandez, newly appointed to the Chair of Mathematics Education at Southbury University.
This event also marks the official UK launch of Professor Hernandez’s major new monograph, Spatial Language and Mathematical Understanding: Teaching Vectors in Multilingual Contexts, which will be available as an open-access resource.
________________________________________
Date: Friday, 14 February 2026
Time: 17:30–18:45
Venue: Stephenson Lecture Hall, Southbury University
Refreshments: Tea, coffee and biscuits will be served from 16:45. A drinks reception will follow the lecture.
Registration: Please register via the form. Places are limited.
________________________________________
About the Lecture
How does a pure mathematician come to focus on the teaching and learning of mathematics? In this inaugural lecture, Professor Hernandez will trace her career path from research in Riemannian geometry to questions of mathematics pedagogy and mathematical communication, reflecting on the experiences and collaborations that shaped this scholarly evolution.
She will address three central themes:
- From Riemannian Manifolds to Pedagogical Inquiry. What prompted a shift in focus from abstract geometry to mathematics education? Professor Hernandez will discuss how her engagement with international student cohorts, and the linguistic and conceptual challenges they faced, gradually led her to formalise questions about how mathematical ideas are understood and expressed.
- The Linguistic Determinants of Mathematical Thought. Drawing on interdisciplinary collaborations with applied linguists, Professor Hernandez argues that language acts as a cognitive scaffold rather than a mere communicative vehicle. She will demonstrate how lexical imprecision generates conceptual fragility, particularly in the acquisition of abstract spatial concepts such as vectors and coordinate systems.
- Translating Evidence into Educational Practice. Bridging the divide between theoretical research and classroom reality remains a persistent challenge. Professor Hernandez will discuss the methodological complexities of translational research, sharing insights on how empirical findings can be effectively operationalised in university mathematics teaching.
The lecture is open to all members of the academic community and the public. The lecture can be viewed online (please contact the organiser, Joshua Wong-Benington, via the email address below for access details) and a recording will be made available.
________________________________________
About Professor Sandra Hernandez
Sandra Hernandez holds a PhD in Differential Geometry from the University of West Columbia. Her early career focused on Riemannian manifolds and vector bundles at the Institute for Advanced Geometries, with publications on coordinate-free methods and geometric invariants
Around 2010, her research began to expand in scope. While continuing her work in differential geometry, she undertook a visiting fellowship at the Westbury Centre for International Mathematics Education, where she observed mathematics instruction in international and multilingual contexts. The experience of explaining her own research to students with varying linguistic backgrounds prompted systematic reflection: what made certain geometrical concepts difficult to communicate? Were these difficulties primarily mathematical, or did they reflect gaps in precise mathematical language? What kinds of interventions might be effective in bridging any gaps?
These questions evolved gradually into a second research programme. Over the following decade, she established collaborations with applied linguists and conducted systematic studies of student understanding, eventually publishing her first papers on mathematics education in 2016. Her work now integrates both trajectories, maintaining engagement with pure mathematics whilst developing a substantial body of empirical education research.
Professor Hernandez has held visiting fellowships at the Sterling Institute of Technology, the Easterwether Combined Institute for STEM Pedagogical Research and the University of Port Phillip. Her research has been recognised with the International Society for Mathematical Communication Research Award, an honorary doctorate from the Prifysgol Frenhinol Cae-y-Bryn and a fellowship from the Foundation for Educational Inquiry.
________________________________________
About the Book
The open-access book Spatial Language and Mathematical Understanding synthesises fifteen years of empirical research. It argues that the “vector gap” (the ability of students to perform calculation without conceptual grasp) is frequently a linguistic phenomenon. The text offers a theoretical framework and practical pedagogical strategies for addressing this challenge in multilingual cohorts, as well as a third section on case studies from particular pedagogical contexts around the world. The open-access book will be available immediately after the lecture.
________________________________________
Further Information
For enquiries or accessibility requirements, please contact:
Joshua Wong-Benington
Faculty Events and Communications Officer
Southbury University
j.wong-benington@southbury.edu.ac.uk
Activity 2: Summaries of the central sections of the text
After the introduction, the text consists of three main sections which are titled as follows:
- About the lecture
- About Professor Sandra Hernandez
- About the Book
In this activity, you will select the best summary for each section. The purpose is to help you explore your broad understanding of the text. Each question is followed by a prompt for further reflection or discussion.
Question 1: Read the first major section titled About the lecture which ends with the words … and a recording will be made available. Select the best summary of this section from the three below. Next, select the ‘Answer’ button below.
- Summary A: The primary motivation for Professor Hernandez’ move to pedagogical mathematics was her experience as a PhD supervisor of international students and the questions she asked herself about the challenges this particular cohort faces in terms of researching mathematics.
- Summary B: Professor Hernandez will argue that mathematics as a discipline is essentially linguistic. In this novel approach to understanding the discipline of mathematics, Professor Hernandez will claim that language lies at the heart of any mathematical representation.
- Summary C: The central section of Professor Hernandez’ lecture will address the powerful role of language in building mathematical understanding including how vague or unspecific uses of language in mathematics communication may cause misunderstandings to arise for some students.
Question for reflection and discussion: What role does language play in mathematics for you? Is language simply a communicative vehicle, does language serve a wider role, shaping, framing and constituting your understanding of mathematics?
Answer
The best summary is C. The key extracts that point to this are:
- … language acts as a cognitive scaffold rather than a mere communicative vehicle. The meaning here is that language plays an active and constitutive role in mathematical thought and is not simply, only or merely a vehicle for transmission.
- She will demonstrate how lexical imprecision generates conceptual fragility, … This extract suggest that being unclear or vague in using language to teach mathematics may contribute to poor or less than robust understanding in students’ minds.
Summary A is inaccurate as the text does not say that motivation for Professor Hernandez to do this research was primarily (or indeed at all) from supervising postgraduate students.
Summary B is inaccurate for two reasons: Firstly, it is inaccurate in any case in that mathematics is not considered ‘essentially linguistic’; secondly, it is inaccurate as a summary of this text. Summary B radically overstates the role that Professor Hernandez attributes to language. She does not claim that ‘mathematics is essentially linguistic’; rather she is highlighting a dynamic and fundamental role that language may play in the teaching, learning and mental representation of mathematics at least for some students.
Question 2: Read the second section titled About Professor Sandra Hernandez and select the best chronological representation from the two below. The symbol -> denotes ‘occurs after’; in other words, X -> Y means that event Y occurs after event X (or alternatively, X occurs before Y). The incorrect representation may be wrong in terms of the chronology itself or in terms of the accuracy of the event summaries. Next, select the ‘Answer’ button below.
- Chronological representation A: Begins studying at the University of West Columbia -> obtains a PhD -> publishes in geometry -> research into geometry expands in scope -> teaches a range of classes at the Westbury Centre for International Mathematics Education -> develops a second research programme
- Chronological representation B: Obtains PhD -> focusses on Riemannian manifolds -> observes the teaching of mathematics in various contexts -> begins to ponder questions about the role of language in teaching and learning mathematics -> develops a research programme to address these questions.
Question for reflection and discussion: In chronological terms, what has been your own development as a mathematics student up to this point? What do you expect or hope your ongoing development will look like?
Answer
Chronological representation B is the correct answer.
Chronological representation A has several inaccuracies:
- … research into geometry expands in scope -> teaches a range of classes at the Westbury Centre for International Mathematics Education … Specifically, the phrase expands in scope in the context of this text means that Professor Hernandez’s research expanded away from differential geometry into pedagogical concerns.
- Moreover, the text does not state that Professor Hernandez taught a range of classes. Instead, it was her observation of various classes and explaining her own research to students that caused Professor Hernandez to become more interested in mathematics pedagogy.
- Suppose we correct the two events discussed above as follows: … research began to expand in to mathematics pedagogy -> observed mathematics instruction … Even with this correction, the -> symbol remains inaccurate as it was the second event (observing mathematics) that led to the expansion of the research agenda.
Question 3: Read the third section entitled About the Book and choose the best single line summary of the content of the book. Next, select the ‘Answer’ button below.
- Summary A: Based on a decade-and-a-half of empirical investigation, the book argues that language can play a constitutive role in students’ understanding of mathematics, not merely a representational one. Theory, application and case studies are offered.
- Summary B: Written over a period of 15 years, this book argues that the usual theoretical frameworks that mathematics uses to understand how students use language in their studies are not wholly correct. The author produces case study evidence from around the world to argue against these theoretical frameworks.
Question for reflection and discussion: If you were going to write a book, would you prefer it to be about mathematics itself, or the teaching and learning of mathematics? Would you be happy to contribute as a participant to someone’s else research on mathematics learning and teaching?
Answer
Summary A is the correct answer.
Summary B is incorrect for two reasons:
- the book was not written over 15 years but instead summarises 15 years of research
- the book does not specifically argue against established theoretical frameworks but instead offers a novel, single framework.
Language Focus 1: Titles for roles in academic contexts
In academic contexts, we use different terms to describe people’s roles. In this text, even though Professor Hernandez is delivering a lecture, she is referred to as a speaker, not a lecturer or presenter. This is the standard term for someone delivering an inaugural lecture or giving a talk at a conference. An inaugural lecture is given to an audience of colleagues and peers, not to students in a taught module. The term speaker reflects this peer-to-peer context.
In this context of speaking to peers, we also have the terms guest/visiting speaker (a colleague from another institution who is invited to give a talk), as well as keynote speaker (the main or opening speaker at a conference). Similarly, Joshua Wong-Benington is referred to as the organiser. To this term, we might also add the term chair which refers to an individual who oversees a series of presentations.
Your task: Consider the table below which lists ten titles that people may have or use in various academic contexts, including all of those mentioned above. Match the titles to the correct scenario. Next, select the ‘Answer’ button below the table.
| Role | Scenario | |
|---|---|---|
| speaker | Sam is a PhD student running a small conference for other PGR students in her department. She has sent out an announcement to various PGR mailing lists and is busy booking rooms and looking over the abstracts that have been submitted. | |
| presenter | Dr Kim has been asked to help at a conference. She has been asked to introduce the speakers in one of the morning presentation sessions. Her role is to introduce each speaker, manage the Q&A session, and support speakers with the technology. | |
| lecturer | Satoshi is a research student presenting a poster at a postgraduate research (PGR) conference within the Department of Mathematics where he is studying. Other attendees are PGR students and a few staff members. At this conference, the only thing that Satoshi is doing is presenting his poster. | |
| guest speaker / visiting speaker | Dr Liu has just obtained a PhD and now has a one-year postdoctoral contract to continue research initiated during doctoral study. The contract does not include any teaching. | |
| keynote speaker | Professor van Dijk is delivering a lecture to postgraduate taught (PGT) students as part of a teaching module. | |
| researcher | Dr Tywyn is one of five academics at a university in Finland who together are organising an annual mathematics conference. They have several tasks to complete as a group, including contacting speakers, organising the itinerary, and booking hotel accommodation for speakers as well as rooms on campus for the conference itself. They are working well together. | |
| organiser | Professor Tyler is an international authority on mathematical logic. She has been asked to present a lecture on her recent work as the first plenary lecture of a three-day mathematics conference in Argentina. | |
| member of the organising committee | Dr O’Grady is an expert on Riemannian geometry has been invited to share and discuss her insights and perspectives on the teaching of Riemannian geometry as part of a group of experts at a conference. This is all Dr O’Grady is doing at the conference: she is not presenting or giving a poster. | |
| chair | Associate Professor Jahnke works at a university in Spain. Today, she is travelling to another university in Spain to present a talk about her recent research to colleagues. There will be lunch afterwards and a chance to chat, then she will return home. | |
| panellist | Dr Wu is delivering a lecture to colleagues as part of a series of lectures. |
Answer
| Role | Scenario |
|---|---|
| speaker | Dr Wu is delivering a lecture to colleagues as part of a series of lectures. |
| presenter | Satoshi is a research student presenting a poster at a postgraduate research (PGR) conference within the Department of Mathematics where he is studying. Other attendees are PGR students and a few staff members. At this conference, the only thing that Satoshi is doing is presenting his poster. |
| lecturer | Professor van Dijk is delivering a lecture to postgraduate taught (PGT) students as part of a teaching module. |
| guest/visiting speaker | Associate Professor Jahnke works at a university in Spain. Today, she is travelling to another university in Spain to present a talk about her recent research to colleagues. There will be lunch afterwards and a chance to chat, then she will return home. |
| keynote speaker | Professor Tyler is an international authority on mathematical logic. She has been asked to present a lecture on her recent work as the first plenary lecture of a three-day mathematics conference in Argentina. |
| researcher | Dr Liu has just obtained a PhD and now has a one-year postdoctoral contract to continue research initiated during doctoral study. The contract does not include any teaching. |
| organiser | Sam is a PhD student running a small conference for other PGR students in her department. She has sent out an announcement to various PGR mailing lists and is busy booking rooms and looking over the abstracts that have been submitted. |
| member of the organising committee | Dr Tywyn is one of five academics at a university in Finland who together are organising an annual mathematics conference. They have several tasks to complete as a group, including contacting speakers, organising the itinerary, and booking hotel accommodation for speakers as well as rooms on campus for the conference itself. They are working well together. |
| chair | Dr Kim has been asked to help at a conference. She has been asked to introduce the speakers in one of the morning presentation sessions. Her role is to introduce each speaker, manage the Q&A session, and support speakers with the technology. |
| panellist | Dr O’Grady is an expert on Riemannian geometry has been invited to share and discuss her insights and perspectives on the teaching of Riemannian geometry as part of a group of experts at a conference. This is all Dr O’Grady is doing at the conference: she is not presenting or giving a poster. |
Activity 3: Triads (or trios or groups of three) in the text
The title of the text is repeated here:
Spatial Language and Mathematical Understanding: A Scholarly Evolution.
As noted above in Activity 1 Question 2, this title follows a standard academic form. Grammatically, it consists of two adjective + noun phrases separated by a colon (:), plus another adjective + noun phrase. The part after the colon gives more detail about the topic introduced before the colon.
In terms of the linguistic structure of the title, it contains three adjective + noun strings:
- spatial language (spatial is the adjective from the noun space)
- mathematical understanding (mathematical is the adjective from the noun mathematics)
- scholarly evolution (although ending in -ly, scholarly is an adjective from the noun scholar.
Academic texts – as well as other texts – can present information in groups of three. These are called triads or trios. This pattern creates a sense of completeness and rhythm.
Your task: Read through the whole text and identify six more examples of triads (groups of three). The first one (after the title) is found in the refreshments: ‘Tea, coffee and biscuits’. Next, select the ‘Answer’ button below.
Answer
- The overall shape of the lecture is expressed in three-part form: In this inaugural lecture, Professor Hernandez will trace her career path [1] from research in Riemannian geometry [2] to questions of mathematics pedagogy and mathematical communication, [3] reflecting on the experiences and collaborations that shaped this scholarly evolution.
- Professor Hernandez address three central themes: [1] From Riemannian Manifolds to Pedagogical Inquiry. … [2] The Linguistic Determinants of Mathematical Thought. … [3] Translating Evidence into Educational Practice.
- Professor Hernandez’s systematic reflection is expressed in a three-part form: The experience of explaining her own research to students with varying linguistic backgrounds prompted systematic reflection: [1] what made certain geometrical concepts difficult to communicate? [2] Were these difficulties primarily mathematical, or did they reflect gaps in precise mathematical language? [3] What kinds of interventions might be effective in bridging any gaps?
- Professor Hernandez has had three visiting fellowships:[1] Professor Hernandez has held visiting fellowships at the Sterling Institute of Technology, [2] the Easterwether Combined Institute for STEM Pedagogical Research and the [3] University of Port Phillip.
- Professor Hernandez has been recognised by three institutions: Her research has been recognised with [1] the International Society for Mathematical Communication Research Award, [2] an honorary doctorate from the Prifysgol Frenhinol Cae-y-Bryn and [3] a fellowship from the Foundation for Educational Inquiry.
- The book has a three-part structure: The text offers a [1] theoretical framework and [2] practical pedagogical strategies for addressing this challenge in multilingual cohorts, as well as [3] a third section on case studies from particular pedagogical contexts around the world. The open-access book will be available immediately after the lecture.
Language Focus 2: The language of cohesion
In this Language Focus we will look at how elements of language work to connect sentences and text together. This idea is sometimes called cohesion; the aspects of language that are in play here are sometimes called cohesive devices. Understanding cohesive devices can assist us both in reading ore fluently and in writing more clearly.
Consider the section of the text in which Professor Hernandez addresses three central themes, reproduced below for convenience.
She will address three central themes:
- From Riemannian Manifolds to Pedagogical Inquiry. [1] What prompted a shift in focus from abstract geometry to mathematics education? [2] Professor Hernandez will discuss how her engagement with international student cohorts, and the linguistic and conceptual challenges they faced, gradually led her to formalise questions about how mathematical ideas are understood and expressed. [3]
- The Linguistic Determinants of Mathematical Thought. [1] Drawing on interdisciplinary collaborations with applied linguists, Professor Hernandez argues that language acts as a cognitive scaffold rather than a mere communicative vehicle. [2] She will demonstrate how lexical imprecision generates conceptual fragility, particularly in the acquisition of abstract spatial concepts such as vectors and coordinate systems. [3]
- Translating Evidence into Educational Practice. [1] Bridging the divide between theoretical research and classroom reality remains a persistent challenge. [2] Professor Hernandez will discuss the methodological complexities of translational research, sharing insights on how empirical findings can be effectively operationalised in university mathematics teaching. [3]
In terms of the number and type of phrases and sentences, each of the three central sections has the following form:
- An opening phrase which labels the theme (end-annotated with [1] above).
- A first sentence which explains that theme (end-annotated with [2] above).
- A second sentence which responds to the first (end-annotated with [3] above).
We will now work through the first central theme:
- From Riemannian Manifolds to Pedagogical Inquiry. [1] This is the phrasal title. The use of From … to … in the title indicates clearly that a shift or change of some kind has occurred. The adjective Pedagogical means to do with teaching and learning. We therefore know that shift is in some sense from mathematics as a discipline to the teaching and learning of mathematics.
- What prompted a shift in focus from abstract geometry to mathematics education? [2] This is sentence 1. The expectations of the title are fulfilled: indeed, we see the noun shift in the string shift in focus from X to Y. In this sense, Sentence 1 is simply an explanation of the phrasal title. However, sentence 1 takes the form of a question (so technically it is not a sentence, but we will retain this term here). This sets up a strong expectation that the following material will answer the question.
- Professor Hernandez will discuss how her engagement with international student cohorts, and the linguistic and conceptual challenges they faced, gradually led her to formalise questions about how mathematical ideas are understood and expressed. [3] This is sentence 2. The overall shape of this sentence may be captured as … will discuss how X gradually led to her Y. Here, once again, the theme of the shift is again found. Moreover, a number of words and phrases pertain to pedagogy: engagement with international student cohorts; linguistic and conceptual changes they faced; how mathematical ideas are understood.
Your task: Now look at the second and third central themes and follow the pattern in the example. In other words, identify words, phrases, strings and forms of grammar that connect the phrasal title and the two sentences to each other. Next, select the ‘Answer’ button below.
Answer
Second central theme:
- The Linguistic Determinants of Mathematical Thought. [1]. The phrasal title has the form noun phrase + of + noun phrase, which is a common structure. It introduces the central contrast between language (as a tool for thinking) and mathematical thought. This phrasal title clearly echoes this pairing, which runs throughout Professor Hernandez’s presentation.
- Drawing on interdisciplinary collaborations with applied linguists, Professor Hernandez argues that language acts as a cognitive scaffold rather than a mere communicative vehicle. [2] This sentence adds new information (drawing on interdisciplinary collaborations with applied linguists), but in doing so it again echoes the linguistic theme of Professor Hernandez work. In this sense, this ‘new’ information remains very much within expectations. The two-word linker rather than alongside the adjective mere (which suggests something like only or simply) clearly contrasts two views of language, and indicates which position Professor Hernandez adopts (that language is a cognitive scaffold, not merely a vehicle for communication).
- She will demonstrate how lexical imprecision generates conceptual fragility, particularly in the acquisition of abstract spatial concepts such as vectors and coordinate systems. [3] This sentence provides greater detail as to Professor Hernandez’ argument. The theme of language in relation to thought is seen in words and phrases such as lexical imprecision, conceptual fragility, acquisition of abstract spatial concepts.
Third central theme:
- Translating Evidence into Educational Practice. [1] This phrasal title is structured around translate X into Y, although the verb translate appears in its so-called -ing form (sometimes called a gerund), giving it a more noun-like quality. (It is not, however, a nominalisation, strictly speaking; this would be Translation of Evidence into ….). The -ing form of translate with the preposition into indicates that the theme in Professor Hernandez’ talk will be about the application of her findings into concrete practice in educational settings.
- Bridging the divide between theoretical research and classroom reality remains a persistent challenge. [2] Sentence 1 picks up cleanly on the ideas in the phrasal title. It restates the two poles of the translation (theoretical research and classroom reality) and notes that they are a challenge. The challenge aspect fulfils expectations of the phrasal title in that the translation is likely only to have value if there is a challenge in play.
- Professor Hernandez will discuss the methodological complexities of translational research, sharing insights on how empirical findings can be effectively operationalised in university mathematics teaching. [3] Sentence 2 offers greater detail of what the translation or application will consist of. The string effectively operationalised in echoes previous words and phrases such as translate.
Activity 4: The structure of a particular paragraph
Consider again the third section of the main part of the text, namely the section titled About the Book. The section is reproduced below with some annotations indicating the start of each sentence. This section is a single paragraph that consists of four sentences. In this activity, we consider the function of each of the sentences, how it contributes to the whole section and to the whole text, and its relationship to other sentences in the text.
This activity will help you interpret texts more fluently and write more coherently in your own work.
[S1] The open-access book Spatial Language and Mathematical Understanding synthesises fifteen years of empirical research. [S2] It argues that the “vector gap” (the ability of students to perform calculation without conceptual grasp) is frequently a linguistic phenomenon. [S3] The text offers a theoretical framework and practical pedagogical strategies for addressing this challenge in multilingual cohorts, as well as a third section on case studies from particular pedagogical contexts around the world. [S4] The open-access book will be available immediately after the lecture.
Your task: Match each description (A–D) to the correct sentence (S1–S4). Next, select the ‘Answer’ button below.
| Letter | Description |
|---|---|
| A | This sentence offers a succinct summary of the argument of the text including providing a foregrounded two-word term which is intended to capture the problem or challenge which the text addresses. |
| B | This sentence is a very practical sentence which may be read as marketing the book. It reminds those who attend the lecture of one of the motivations for this event, i.e. the launch of a new and important book. |
| C | This sentence provides context in two ways: it restates the name of the book and gives a historical or time perspective on the amount of work that has gone into the book. As such, this sentence picks up on earlier portions of the text where the author’s empirical research has been foregrounded. |
| D | This sentence picks up on the challenge identified in the previous sentence and states three ways in which the book will respond to this challenge. |
Answer
- Sentence 1 -> C: The phrase fifteen years of empirical research provides the historical perspective. This phrase also serves to suggest that the book is a rigorous and data-driven piece of work that can be trusted to provide meaningful insights.
- Sentence 2 -> A: The two-word term is vector gap; the problem or challenge is the role that language plays in constructing understandings of mathematics.
- Sentence 3 -> D: The three ways are: a theoretical framework, practical pedagogical strategies, and case studies.
- Sentence 4 -> B: This final sentence gently promotes the book by reminding those attending the lecture that it is available.
Extension Task: Apply the same analysis to the four paragraphs of the second main section, About Sandra Hernandez. Try this yourself and then select the ‘Answer’ button below for the suggested answer.
Answer
The four paragraphs take a broadly chronological perspective on Professor Hernandez professional life, as noted in Activity 2, Question 2. A summary of each paragraph’s function and connections to other paragraphs now follows:
- Paragraph 1: The start of Professor Hernandez’s career and her initial focus on an area of mathematics without concern for pedagogy. As such, this paragraph provides a point of contrast for later material.
- Paragraph 2: A key event in Professor Hernandez’s professional life and how it triggered a series of questions for her.
- Paragraph 3: The intellectual result of the questions triggered by the events described in paragraph 2, including details of outputs in the Professor Hernandez’ second field of research.
- Paragraph 4: Similar to the final sentence in the example above, this is a slightly separate part of the section, simply providing a succinct academic biography of Professor Hernandez.
Language Focus 3: adjective + noun and noun + noun combinations
In the exercise, we look at adjective + noun and noun + noun combinations. These patterns are very common in academic texts. Language Focus 3 in Section 5.6 also discusses adjective + noun combinations as does Language Focus 3 in Section 5.3.
-
Adjective + noun: The adjective modifies (describes) the noun. Example: mathematical understanding.
-
Noun + noun: The first noun modifies the second noun. Example: vector gap.
Consider the opening section, reproduced below.
Inaugural Lecture Professor Sandra Hernandez
Spatial Language and Mathematical Understanding: A Scholarly EvolutionYou are cordially invited to the Inaugural Lecture of Professor Sandra Hernandez, newly appointed to the Chair of Mathematics Education at Southbury University.
This event also marks the official UK launch of Professor Hernandez’s major new monograph, Spatial Language and Mathematical Understanding: Teaching Vectors in Multilingual Contexts, which will be available as an open-access resource.
The table below shows examples of these combinations from the text above.
| Adjective + noun | Inaugural lecture; spatial language; mathematical understanding; scholarly evolution; official launch, major new monograph, multilingual contexts |
|---|---|
| Noun + noun | Mathematics Education, UK launch, open-access resource |
In the table above, the endings have been underlined: most of the adjective endings happen to be -al in this text with one -ly which looks like an adverb ending (e.g. compute something efficiently or solve the equation completely).
Your task: Now consider the paragraph of the text titled About the Book, reproduced below. Identify any adjective + noun combinations as well as any noun + noun combinations. Next, select the ‘Answer’ button below.
The open-access book Spatial Language and Mathematical Understanding synthesises fifteen years of empirical research. It argues that the “vector gap” (the ability of students to perform calculations without conceptual grasp) is frequently a linguistic phenomenon. The text offers a theoretical framework and practical pedagogical strategies for addressing this challenge in multilingual cohorts, as well as a third section on case studies from particular pedagogical contexts around the world. The open-access book will be available immediately after the lecture.
Answer
Adjective + noun: Spatial Language; Mathematical Understanding; empirical research; conceptual grasp; linguistic phenomenon; theoretical framework; practical pedagogical strategies [2 adjectives + one noun]; multilingual cohorts; third section; particular pedagogical contexts [2 adjectives + one noun].
Noun + noun: open-access book; vector gap; case studies
In this section, you have focused on longer written mathematical communication. To conclude the chapter, the final section provides a glossary of important mathematical terms from the chapter.
