Long Paper Assignment: Gestures as Bridges to Understanding: The Role of Embodied Learning in Children's Geometric Reasoning

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Summary

Children Learn When Their Teacher’s Gestures and Speech Differ Melissa A. Singer and Susan Goldin-Meadow

The authors of this article carried out this study on the effect of gestures by instructors on children's learning of mathematics, specifically on mathematical equivalence problem-solving. The researchers, in their study, asked whether gestures illustrating a different solution to the teacher's speech — gesture-speech mismatch — would facilitate the learning process or not. The researchers tested 160 third and fourth-grade students, who were divided into six groups. Different groups learned either one speech strategy or two. Students also learned under different conditions of gestures: no gesture, gestures that reflected the speech, or gestures that disclosed a different strategy from that disclosed in the verbal explanation.

It was discovered that children learned best when gestures were reflective of a different strategy from the one explained through speech. This mismatch between speech and gesture allowed students to learn two strategies without being overwhelmed with too much verbal information. Children who were taught two strategies only in speech did not perform as well, as expected, which means too much spoken information can mislead students. However, when the first strategy was explained through speech and the second strategy was presented only by gestures, students performed much better. This finding identifies that gestures have the potential to provide a valuable second channel of learning, complementing key concepts in a more palatable way.

The authors concluded that mismatching gestures can enhance learning by offering students an alternative way of learning the material. Gestures appear to contribute a little extra information in a soft, but effective, manner and make it easier for students to connect ideas. This study suggests that teachers intentionally supplement instruction with gestures — not as a rewording of what is being said, but as offering complementary methods for enhancing students' understanding of mathematics concepts.

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Even teachers routinely produce gestures as they instruct children in both individualized tutorials (Goldin-Meadow, Kim, & Singer, 1999) and the classroom (Crowder & Newman, 1993; Flevares & Perry, 2001; Neill, 1991; Roth & Welzel, 2001; Zukow-Goldring, Romo, & Duncan, 1994). And children pay attention to those gestures, often gleaning substantive information from gesture that cannot be found anywhere in the teacher’s speech (Goldin-Meadow et al., 1999).(p.g.85)

I stopped at this quote there because it reminded me of my own teaching experience. During my teaching career, I used gestures many times without ever consciously thinking about them — they simply seemed to be a natural addition to what I was saying. Whether I was demonstrating addition by spreading out my hands to show grouping or spreading out a finger to gesture towards different spots in an equation to show key points, gestures were an essential aspect of my classroom. I realize now that gestures played a significant role in helping students to understand certain concepts. After reading this quote, how effective are the spontaneous hand movements and being mindful of my gestures can make my explanations more clear and more effective for students.

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Gesture-speech mismatch occurs when gesture conveys information that is different from (although not necessarily contradictory to) the information conveyed in the speech it accompanies.(p.g.85)

I stopped at this quote because it helped me realize that gestures can be more effective than simply repeating what we say. Sometimes, a teacher's gestures express another solution to an issue or highlight an important idea that was not addressed verbally. This doesn’t mean the gesture is contradicting the verbal message; it is simply adding information that can help students think about the concept in a different way. As mentioned in the article, mismatched gestures can offer a second problem-solving strategy. However, in my opinion, this can’t happen all the time, especially when teaching younger students. If we mismatch gestures, it can confuse them and make it harder for them to understand the concepts clearly.

Do you have experience using gestures to enhance your teaching, and how do you balance them with verbal explanations to ensure clarity?

A Linguistic and Narrative View of Word Problems in Mathematics Education

By

SUSAN GEROFSKY

Susan Gerofsky's paper analyzes mathematical word problems from the perspective of linguistics and discourse analysis. Author highlights that while word problems appear to tell stories, they actually follow a pattern based on arithmetic and algebraic formulas rather than real-world stories. Susan first describes the three general components of a word problem which include:

1. A 'set up' component typically viewed as a narrative and generally non-essential

2. An 'information' component providing information to solve the problem

3. The question

Though some believe narratives are engaging, students tends to disregard them, focusing only on analysing them.

Susan takes into account the language features of word problems, their absence of empirical relevance in the world, and so they are indeterminate in the sense of locutionary force (i.e., what they literally mean). She outlines how there is a requirement for students to do some kind of "pretend" reading, reading fictionalized hypotheticals as if they were real, but completely fabricated. The study also discusses verb tense inconsistencies, showing how math word problems mix up grammatical tenses in ways not found in ordinary speech or storytelling.

Susan concludes the paper by explaining David Pimm's idea that word problems can be considered parables. Although they share some things in common, she wonders if this is an accurate comparison. Her general thesis is that we must critically examine the role of word problems within mathematics education and re-look at why they continue to be used so prevalently.

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"You can't go out and use them in daily life, or in electronics, or in nursing. But they teach you basic procedures which you will be able to use elsewhere."(p.38

From my childhood experience I learned word problems following specific steps, without showing how the steps worked to solve problems related to real life. It led me to question whether the students were simply being taught to memorize steps to solve problems rather than to think.

While learning processes is important, I felt like not relating them to real applications made math seem less applicable. Students can possibly do math problems, but maybe not understand how they can utilize those skills in their future occupations or daily life.

If mathematical problems were related to the real world, students would be able to recognize how mathematics functions in their world. For example, understanding mathematics in nursing or electronics would allow students to visualize the relevance of their studies in real-life situations. The realization would allow students to be critical and understand the application of mathematics in life.

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That delineating the boundaries of the word problem genre can allow us to play with those boundaries in interesting ways(p.g.43)

I stopped at this quote because in my opinion we can identify new ways to use the word problems by delineating the boundaries. We can use open-ended questions and by creating word problems more interesting to the students can help them to explore math beyond just applying formulas. Moreover, by connecting word problems to real life, we can encourage students to think critically rather than treating it as simple exercises. As a result, students recognize that math is relevant and useful.

One of the experiences that remain in my memory is when I witnessed how my students solved a division word problem. I knew that traditional word problems have the effect of making students think about calculations and not so much about thinking. But when I altered the problem a bit—by asking questions, allowing different answers, and making it real-life related—the students started thinking more deeply. This opened my eyes to the fact that as teachers, we can make word problems more than just exercises. We can use them to help students explore and understand math on a deeper level.

 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.

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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.

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'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.

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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.

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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.

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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?"

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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?