Summary

Chapter 3 : Quantity: Trapping Numbers in Grammatical Nets

Chapter 6:A Never Ending Braid: The Development of Mathematics 

By Bill Barton

In Chapter 3, the author describes the intricate relationship between language and mathematics by focusing on how different languages structure numerical concepts. It examines the grammar of numbers in different languages, including Maori, Kankana-ey, and Dhiveli. In Maori, numbers are treated as verbs rather than nouns to make them more dynamic and in a lively way. But, in Kankana-ey, numbers are used like describing words (adjectives) to modify nouns. In Dhiveli, the language of the Maldives, numbers are used as both adjectives and nouns. The chapter ends by saying that while English makes it easier to express math, it's important to appreciate how other languages handle numbers. Understanding these differences can improve how we teach math and help make math education more inclusive for people from different cultures.

Chapter 6 explains how mathematics has grown through its connection with society, culture, and human creativity. The author compares mathematics to a braid, where different strands represent ideas from various cultures. Instead of seeing math as a single universal system, the chapter highlights its diverse roots, showing how cultures have contributed unique mathematical ideas. Examples like Pacific navigation and Indian Kolam art demonstrate how math borrows and adapts ideas, often without recognizing their original creators. This exchange, shaped by language and society, shows the importance of a more inclusive and flexible view of mathematical development.

Stop 1

" Mathematics absorbs good ideas, techniques, even symbol systems, and makes them part of the mainstream of the subject. The worth of the ideas is judged on mathematical grounds. But this is not a braid with independent strands woven together but retaining their individuality, this is a river with tributaries flowing in. "(p.103)

This quote stopped me because it made me think about mathematics-how it grows and evolves. Here mathematics is considered as a flowing river which absorbs all ideas, techniques and symbols from different cultural tributaries. Once an idea enters mathematics, it loses its original identity and it becomes part of mathematics mainstream where it is judged only based on the math grounds. As an example, Calculus originated from Newton and Leibniz, but today, it is no longer seen as their personal creation—it is simply part of mathematics. And different countries contribute ideas to mathematics but, it is no longer considered as their contribution. Is this a good thing? Should we recognize and appreciate the cultural roots of these ideas? The author says that if math were like a braid instead of a river, each contribution would remain distinct, woven together without losing its uniqueness.

 As teachers, when we teach a concept to students from diverse backgrounds we have to reflect on the contributions of people from different parts of the world. It helps the students see math as more than just numbers and formulas but as a story of human creativity across time.In this chapter,the author describes that "all mathematics is proceeding together in one large stream, a stream of different interests, but one stream nevertheless, with the happy family of mathematicians floating together along it. This may be what mathematicians feel, but below the surface, mathematics is made up of quite different ideas being developed, often interacting, and knowing of each other's existence, but conceptually different in important ways, Hence, the metaphor of a braid of many strands and fibres, is more appropriate than that of a river with tributaries."(p.106)

Stop 2

"Universalising and Isolating mechanisms not only occur as part of the colonial process when mathematical ideas from two cultures meet—as when Westem reference systems dominated Pacific ones— but also operate internally within mathematics."(p.115)

Mathematics is often seen as a universal language, but there are forces that make certain types of math more common and ignore others' contributions. Universalizing happens when one culture's way of thinking about math becomes dominant, pushing aside others. For example, Western coordinate systems replaced the navigation methods used by Pacific Islanders. Isolating happens when some important ideas are left out, like how algebra from the Islamic world and the concept of zero from ancient India were ignored by Europeans at first. 

These ideas affect how we teach math, often making it seem like math is separate from culture, which can leave out important perspectives. By recognizing these issues, we can make math more inclusive by connecting it to different cultures and stories .This way, math becomes more open to different ways of knowing and helps us better understand the world.

How can we ensure that our math lessons recognize and value the contributions of different cultures, while still teaching the core concepts students need to learn?

A Reflection on Discourse Analysis and Mathematics Education

 by

 David Pimm

David Pimm, a researcher from the University of Alberta, explores the connection between discourse analysis and mathematics education. Discourse analysis is a method of studying language that looks at connected speech or writing beyond individual sentences. His paper marks the 25th anniversary of a key review article, Language and Mathematical Education by Austin and Howson, published in 1979.

Some features of discourse and their mathematical and mathematical education

versions

• Voice and Agency – How mathematical ideas are presented, including pronoun use, references, and the role of the speaker or writer in shaping meaning.
• Meta-discourse – How people express opinions, politeness, and uncertainty in mathematical communication.
• Temporal Structure – Using verb tense and sequence markers to show the flow of mathematical reasoning.
• Style and Genre – The different ways math is written and spoken, including specific styles used in education and research.

Here, Pimm discusses the first 3 features.

1.Voice

Voice shows how the speaker or writer positions themselves in a conversation. It shows whether they are asserting authority, being informal, or addressing the audience directly. In math problems, phrases like "Let us consider" or "We can observe" suggest the speaker is guiding the reader.

2. Meta-Discourse

In mathematical communication, meta-discourse refers to elements of language that reflect the speaker or writer’s attitude toward their statements. A key aspect of meta-discourse is hedging—using language to soften assertions and manage uncertainty.

3. Temporal Structure

Mathematics is often described as “timeless,” meaning that its truths do not change over time. Temporal structure looks at how time is shown in a text. It focuses on how events are ordered—whether they happened in the past, are happening now, or will happen in the future. In math problems, the tense helps clarify when things are happening, making the problem easier to understand.

Stops

1)"Word problems have no truth value: the people and the events are fictional. Yet by using the names of real girls from the class in this problem, there may have been some interaction between the problem authors’ real and fictional worlds."(p.9)

As a math teacher, this quote reminds me that while word problems may be fictional, they become more meaningful when they incorporate students' names, familiar situations, or cultural traditions. This approach makes math more engaging and helps students connect with the problems on a personal level. At the same time, students bring real-world thinking into these problems such as figuring out how to create a perfect design for a flower mat by dividing a circle into six or eight equal parts. This highlights the importance of integrating diverse cultural experiences in math, such as Onam Pookkalam (flower mat designs), Indigenous drum making, or Mi'kmaq patterns, when we teach geometry or drawing circles, equal division of circles, ensuring that all students see themselves reflected in their learning. Moreover, the way problems describe students can shape their identity, so as educators, we must be mindful to present diverse and inclusive representations, avoiding stereotypes and fostering a sense of belonging.

Mi'kmaq Patterns

                                 

                                                            Pookkalam (flower Mat)

2) "This complex and problematic switch of tenses from present to past to present to past to present and the presumed, created context of the problem makes this hard to rationalize." (p. 8)

This quote made me think about how switching between different tenses in word problems can make them harder to understand. In math, clarity is important, and tense changes can confuse students about the order of events or the problem's meaning.

 I noticed this in my own teaching when a word problem about a student planting flowers used both past and future tense in the same sentence. Some students got confused about whether the action had already happened or was going to happen, which distracted them from solving the math when they are too concerned about the tense and grammar. This reminded me that using consistent verb tenses in math problems helps students focus on the math itself instead of struggling with the wording.

How often do we think about the language we use in math problems? Is language a barrier to learning math for students from diverse backgrounds?

Could making the wording simpler help more students focus on the math?

Language as social semiotic The social interpretation of language and meaning by M.A.K Halliday

  

Stop 1

"What matters most to be a child is how much talking goes on around him, and how much he is allowed and encouraged to join in. There is strong evidence that the more adults talk to a child and listen to him and answer his questions, the more quickly and effectively he is able to learn."(p.201)

It resonated with me because, just last week, when I phoned my husband to ask about our son, he mentioned that our son frequently asks questions and shares his thoughts. I encouraged him to motivate and nurture our son's curiosity by welcoming his questions and doubts. I believe that fostering curiosity is essential for learning. When children are encouraged to express their thoughts and ask questions, they develop critical thinking skills and a natural desire to explore the world around them. This creates a strong foundation for lifelong learning and builds their confidence to engage actively with their environment.


Reflecting on my own experience as a student, I remember how we were often afraid to ask questions. We grew up with the belief that teachers were the ultimate authority and sole source of knowledge. This traditional approach to teaching discouraged us from questioning and made us suppress our doubts, creating an environment where students were passive learners rather than active participants in their education. Now, as an educator, I recognize the importance of cultivating a classroom culture that values curiosity, encourages dialogue, and empowers students to voice their thoughts.

Stop 2

"Many languages have the same word for yesterday and tomorrow. This doesn't mean that their speakers cannot distinguish between what has already happened and what is still to come."(p.198)

This thought stopped me because it challenges the way we usually think about languages. The idea that some languages use the same word for "yesterday" and "tomorrow" may seem confusing at first, as it might appear that their speakers can't tell the difference between the past and the future.However, this is not true.

 The Indian language Hindi/Urdu does this, so kal means either "yesterday" or "tomorrow" (and parsõ means both "2 days ago" or "2 days from now", and tarsõ means both "3 days ago" and "3 days for now").  It shows that people can understand complex ideas like time in different ways, even if their language doesn't always make it obvious.

"Hello" in many languages

It made me think about how language and culture shape how we understand the world. This idea reminds me that language isn't just about words—it's also about context and shared understanding. As an educator, it encourages me to look beyond surface assumptions and appreciate how people from different cultures and languages may think and communicate differently.

Reference

M. A. K. Halliday. (1978). Language as social semiotic: The social interpretation of language and meaning. London: Edward Arnold.

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Hello World!!!!

 Hello All!!! I am Renu Maria Jose,a second-year M Ed Mathematics Student at UBC. In this blog, I explore math and language. I am looking forward to learning new things from you!!!