Short Paper

Click here EDCP 553 A1 Rosmy Mathew.docx  to view my review of Embodied Learning in Early Mathematics Education.

Summary

LEARNING MATHEMATICS IN A SECOND LANGUAGE: A PROBLEM WITH MORE AND LESS

By

P E T E R L. J O N E S

The present study is an attempt to explore this area by tracing the development of understanding of the meaning of 'more' and 'less' when used in different mathematical contexts for two groups of learners. Comparing expatriate students who are native English speakers with Papua New Guinean students for whom English is a second language. Performed on 376 second-language learners and 225 first-language learners, the study discovered that though both groups made the same types of errors, second-language learners consistently fell 2 to 4 years behind in acquiring these concepts. The research classifies mathematically "more" and "less" uses into three contexts: comparative (for example, "Which is more, 10 or 13?"), direct (for example, "What number is 1 more than 5?"), and indirect (for example, "The number 8 is 2 more than what number?"). The results were that both groups learned the comparative context first before the direct and indirect context, with "more" typically preceding "less." Second language learners did have a harder time, however, and were more prone to be confused by relational terminology and learn basic misconceptions of math processes like addition and subtraction. The research concludes that the learning of mathematics demands the command of language and that delayed learning of fundamental relational terms can hinder overall mathematical learning. The research illustrates the necessity for specific educational interventions to assist second-language mathematics learners.

Stop 1

An important step in solving word problems in arithmetic is the translation from the verbal formulation to the underlying mathematical relationship.(p.g.270)

I stopped here because this quote reflects a basic difficulty in math learning—comprehending how to translate a word problem into an equation. Most students, particularly those studying in their second language, struggle not because they don't grasp the mathematics but because they misinterpret the words that are used in the problem.

For example, a farmer has 320 kg of wheat. He sells 75 kg and buys an additional 40 kg. How many kg of wheat does he have now?

Here, the student must recognize that first, they have to do subtraction (320 - 75) and then addition (+ 40), to get the right answer:

320 - 75 + 40 = 285 kg. When a student reads the language of math incorrectly, they may do the incorrect operation, like add when they should subtract. This quote reminded me how important it is to assist students in acquiring good problem-translation math skills.

Stop2

'Five is more than three' describes a relationship between the numbers 5 and 3 and can be expressed symbolically as 5 > 3, the phrase 'five more than three' describes an operation involving the numbers 5 and 3 which can be written symbolically as 5 + 3. Similar statements can be made for 'less'. Thus a subtle change in context can dramatically change the mathematical meaning and lead to gross errors of understanding if not recognised by the learner.(pg.270)

I stopped at this quote as it highlights how a small change in wording can completely change the mathematical operation needed. It made, me to reflect on my teaching experience with students learning in a second language, can confuse like the words ‘more’ and ‘less’. For example, "A bookstore contains 250 books. That is 75 more than the school library contains. How many books does the school library contain?" Even though the word "more" is present, the operation required is subtraction: 250 - 75 = 175. A student who believes "more" always signifies addition might incorrectly calculate 250 + 75, finding an incorrect answer. This sample highlights the importance of having a good understanding of math vocabulary in order to work out problems accurately.

This article made me think: What specific strategies can we use to help students build both their mathematical and language skills simultaneously? How to bridge the language gap in math learning.

Summary

Algebra Transforming by David Pimm

This article examines the effective view of algebra, arguing that algebra is all about transformation. It is highlighted in this chapter that algebraic operations take place between successive written statements, representing not only the results but also the process of change. In algebra, there are counterpart objects-symbols that can be manipulated well as descriptive elements that relate to real-world scenarios. For beginners, however, unifying the two perceptions may prove challenging. In the article, it is argued that as algebraic fluency increases, meaning matters less, and one is more efficient when manipulating symbols as objects instead of always connecting them to real meanings.

To reveal how algebra develops from arithmetic, the chapter gives two lessons from Dave Hewitt. Through these lessons, attention is redirected from determining the value of variables to manipulating equations. The "Think-of-a-Number" lesson introduces algebraic reasoning via verbal and written equation manipulation and complementary inverse operations. The "Rulers" lesson builds algebraic expressions through number line movement. In both lessons, structural knowledge takes over memorization, and students are encouraged to experiment with algebra in action without restricting themselves to symbols.

The chapter continues with a discussion on the role of technology in the teaching of algebra. Computer developments, specifically symbolic manipulation software, may reduce the need for conventional algebra instruction, according to the article. The Logo computer language is cited as a means of introducing students to algebraic notation, enabling them to interpret symbols. Symbolic descriptor software, including computer algebra systems (CAS), is also referred to in the context of its capacity to carry out intricate algebraic manipulations, including graph plotting, equation solving, and differentiation.

Moreover, the article focuses on symbols, going back to concepts of incompleteness or complexity. Symbols are representatives of mathematical objects, to be operated on, but not the objects themselves. This is taken beyond algebra, noting that symbols are important in all branches of mathematics. The chapter ends with a consideration of some of the most significant cultural symbols, including The Vietnam Veterans Memorial and the Holocaust Memorial, to draw an analogy between symbols in society and mathematics as representations of deeper, usually intangible meanings.

Stop 1

What is written in an algebraic demonstration are not the actions but the results of actions: sequences of equations show the results of transformations, not the transformations themselves.(p.g.89)

I paused on this quote because it points our attention to something significant but generally neglected in algebra—the distinction between what we lay out on paper and the process of thought that lies behind it. We document steps and conversions when we're solving equations, but what we document is a representation of what occurs, not the reasoning and choices that led us there. This is connected to the idea in the paper that algebra is a dynamic process and that most of the real work is done in between the steps. It also made me think about how students fixate on algebra sometimes because they're only interested in the printed symbols and not the transformations that bring them there. Like in the case of a science experiment when the results are put into writing but the process is more complex, algebraic proofs write down the results, but the mathematical mind working beneath is invisible.

Stop2

When we attempt to realize a piece of music composed by another person, we do so by illustrating, to ourselves, with a musical instrument of some kind, the composer’s commands. Similarly, if we are to realize a piece of mathematics, we must find a way of illustrating, to ourselves, the commands of the mathematician.(p.g.100)

I paused reading at this quote because it expresses the nature of algebra as demonstrated throughout the paper—algebra is not merely operating with symbols but is about understanding the transformation between steps. Similar to music, algebra is more than what it appears on paper. In the same way that the musician has to decipher and play the piece, the mathematician or student has to work with algebra beyond rule-based behaviour. This quotation also relates to the notion in the paper that algebraic expressions are not inert symbols but are engaged in an active interplay of structure and meaning. It called to mind the manner in which, when teaching algebra, the students should not just be so concerned with solving for x but with being made aware of the change that is occurring, in the same way a musician is more interested in how music progresses rather than the individual notes.

How can we encourage students to view algebra as a dynamic, evolving process rather than a collection of static equations?

Summary

USING TWO LANGUAGES WHEN LEARNING MATHEMATICS

JUDIT MOSCHKOVICH

In this article, Judit Moschkovich examines how bilingual mathematics learners use two languages. The author has studied two sets of studies

1) Psycholinguistics experiments- This study compares monolinguals and bilinguals using two languages during arithmetic computation. (language switching)

2) Sociolinguistic research – These studies explore young bilinguals using two languages during conversations(code-switching)

Research in psycholinguistics shows that adult bilinguals show arithmetic operations more rapidly in their preferred language than in their non-preferred language. The studies also found that language switching does not affect the quality and integrity of thinking at the conceptual level in second language production.

Sociolinguistic research demonstrates that code-switching is governed by specific principles and does not occur randomly. In certain communities, the capacity to switch between languages with ease is a gauge of successful bilingualism. Code-switching also allows students to use mathematical terminology in both languages, thus leading to greater student engagement in class discussions.

Based on these findings, teachers should allow bilingual students to choose which language they use for arithmetic calculations, whether speaking or writing. Assessments should also consider how bilingual students perform in timed math tasks.

Sociolinguistics suggests that understanding bilingual students’ communication requires looking at their background, including their language history, schooling, and attitudes toward using both languages.

The author also suggests that more research is needed to understand how bilingual learners use mathematical discourse and how their dual-language abilities support their learning.

Stop 1

One common misunderstanding of bilingualism is the assumption that bilinguals are equally fluent in their two languages. If they are not, then they have been described as not true, real, or balanced bilinguals and sometimes labeled as ‘semilingual’ or ‘limited bilingual.’(p.g.123)

I paused at this quote because it resonates with my personal experience. Malayalam is my native language and English has been a language which I used for my education, formal communication and professional growth. I am fluent in both Malayalam and English, but sometimes I struggle to recall mathematical terms in my native language, so I use the English terms instead. Similarly, there are times when I have to think in Malayalam because I am unsure of the English terms.

The idea that bilinguals must be equally strong in both languages to be considered “real” bilinguals is limiting and unfair. It disregards the dynamic nature of language learning and use. The fact that one may have difficulties with academic writing in one language or feel more comfortable expressing emotions in the other does not make one less bilingual.

This quote led me to think about the ways in which bilingualism is misconceived. We should recognize that bilingualism exists on a spectrum influenced by individual experience, culture and depends on the situations.

"How has your personal experience with bilingualism shaped your understanding of what it means to be 'truly' bilingual? Do you think fluency in both languages should be measured equally, or should it depend on context and usage?"

Stop 2

Language switching does not affect the quality and integrity of thinking at the conceptual level in second language production. (p.g.127)

I totally agree with this quote. I remember this quote with Renu’s experience as she used her first language to make arithmetic calculations more easy. This is true in my case too, when I need to quickly add numbers in my mind, I naturally count in Malayalam rather than in English. I have noticed the same pattern in my students too. At the same time, when I explain mathematical concepts to others on algebraic expressions, I naturally use English, as that is the language in which I studied these topics in school.

This shows that language switching is a natural process and does not affect my ability to understand or apply mathematical concepts. Instead, it helps me navigate my bilingual experience in a way that supports my learning and problem-solving skills.

Have you ever noticed yourself switching languages depending on the task you are performing? In what contexts do you find one language more natural or effective than the other?

Summary

Teacher Code Switching Consistency and Precision in a Multilingual Mathematics Classroom

By Clemence Chikiwa & Marc Schäfer

This document is a study that investigated teacher code switching consistency and precision in multilingual secondary school mathematics classrooms in South Africa. Methods used for data collection were interviewing and observing five lessons of each of three mathematics teachers purposively selected from three township schools in the Eastern Cape Province.

This study drew from the work of Dowling(1998) to understand and analyse teachers’ language(isiXhosa) use in the classroom and the mathematical contents of the meanings communicated in the classroom activities. This study has used the four domains proposed by Dowling.

· * Esoteric - use of highly specialised, formal and abstract mathematical language and

content.

· *Public - referring to forms of expressions and content expressed in entirely everyday

terms.

· * Descriptive - uses specialised mathematical language imposed on non-mathematical

content.

· *Expressive - non-mathematical language to refer to mathematical content.

In the data analysis two dominant teacher code switching practises emerged. These were referred to as borrowing code switching (BCS) and transparent code switching (TCS).

· Borrowing Code-Switching (BCS): Teachers borrowed English words and added isiXhosa prefixes (e.g., "ku-Cosine," "ngu-ABC").

· Transparent Code-Switching (TCS): Teachers used clear isiXhosa equivalents (e.g., "krwela umgca" for "draw a line").

Teachers used BCS in more common than TCS and they used same word in different concept. For example, “Bala” was sometimes used for ‘calculate’, other times for ‘write’ and ‘count’.

This study recommend that mathematics teachers in multilingual classrooms need to be made aware of and encouraged to use available multilingual mathematics resources to aid their teaching and learning of mathematics. In addition to this, study concludes that best practices for code switching need to be established to promote transparent code switching that is precise, consistent and beneficial to increasing access to mathematical understanding in multilingual secondary school classes.

Stop 1

Consistency and precision in teacher language during the teaching of mathematics multilingual classes is crucial for enhancing access to mathematical concepts.(p.g.246)

This quote highlights the importance of code-switching in class, especially for a multilingual class. I can use this quote on our recent class discussion on how to define odd and even in different languages. We observed that, in the math language, odd and even have clear-cut definitions but in other languages they can have different meanings.

Similarly, in this study we see how the word "Bala" is used by teachers in different contexts with different meanings which confuse children. I can compare it to the word "Gunam" in Malayalam; when used in mathematics, it is multiplication, but generally refers to quality or virtue.

These inconsistencies in math vocabulary underscore the need for instructors to utilize concise and precise language to facilitate higher understanding for multilingual students.

 

Stop 2

Promoting conceptual understanding of mathematics intertwined with good language practices that support thinking are key to meaningful teaching of the subject.(p.g.252)

I stopped on this quote because it highlights something so important in teaching math—both good explanations and good language support are required for students to have full access to mathematical ideas. As a math teacher, I have seen students struggle when they are unable to grasp the meaning of a word or concept because of language limitations. For instance, when I am teaching fractions, there are some students who do not comprehend the terms "denominator" and "equivalent," and it makes problem-solving harder for them. But when I define these terms in a manner that relates to their daily vocabulary and lives, they grasp the concepts so much better. Putting it in simple terms, using examples from everyday life, and even alternating between languages when required assist students in thinking critically about math and not memorizing procedures. This is the reason why it is so critical to combine strong language practices with clear math instruction—by doing this, learning with meaning and building students' confidence in problem-solving becomes possible.

REFERENCE

Chikiwa, C., & Schäfer, M. (2016). Teacher Code Switching Consistency and Precision in Multilingual Mathematics Classroom. African Journal of Research in Mathematics, Science and Technology Education, 20(3), 244–255. https://doi.org/10.1080/18117295.2016.1228823