Week 7: Response to "Children Learn When Their Teacher's Gesture and Speech Differ" by Singer and Goldin-Meadow

"People gesture when they talk."

"Teachers gesture when they teach, and those gestures do not always convey the same information as to their speech."

Summary:

The previous researches show that when gestures are presented simultaneously and align with a spoken message, the listeners do better at receiving and understanding the message. In this article, however, Singer and Goldin-Meadow point out that gesture per se did not promote learning—only gestures that conveyed mismatching information led to improved performance. Mismatching gestures refer to the gestures that do not match the information conveyed in the speech it accompanies. In addition, they also discover that teachers spontaneously increase the number of gesture-speech mismatches in their instruction when teaching children who are on the cusp of learning the task.

My stops:

1. The example of mismatching gesture in the article

The example that Singer and Goldin-Meadow gave in the article makes me confused at first reading, the example as follows:

"When giving a child instruction in how to solve the problem 7+6+5=        +5,  a teacher articulated the equalizer problem-solving strategy in speech:' We need to make this side equal to this side.' At the same time, she conveyed a grouping strategy in gesture: She pointed at the 7 and the 6 on the left side of the equation and then at the blank on the right side."

I am thinking about why it is a mismatching gesture here (I thought it is a matched gesture when I was first introduced to this example.) Now, I might get the point of what the author tried to deliver. The speech of the teacher in the example was talking about two sides of the equation, but her gesture just highlighted two parts of the equation (7 and 6 on the left side and the blank on the right side, rather than the whole equation.) I hope I have figured out the meaning of mismatching gestures now.

It is a surprising result that mismatching gestures enhance students' performance in mathematics class. I hold the same opinion as the previous research that the children are likely to profit from instruction with matching gestures. But here in the example, I don't think the mismatching gesture help student to think of this question, but bring their attention from the principle of the equation to focusing on arithmetic (the blank is equal to 7+6). 

2. Gestures in the mathematics classroom

I regularly use pointing gestures in mathematics lessons. When I point to objects or text on the chalkboard, those pointing gestures link my speech to its referents. In addition, I think students also frequently use pointing gestures when they speak. I don't think I can finish a mathematics lesson without a gesture which is a powerful tool for children’s math learning.

Question:

Could you recall an experience using a mismatching gesture in your mathematics class? (Please tell me more about your opinion of mismatching gestures because I am not sure if I understand it correctly.)

Draft Short Paper

Happy reading! 

https://docs.google.com/document/d/1TGPy7LvJF9Xpo-ZaOgnkIA90vJTOMuNEEhbqeCXncBI/edit?usp=sharing

Week 6: Response to The Writing on the Board by Artemeva and Fox

Two Research Questions:

Question 1: What are the genres of teaching university mathematics in lecture classes? Or, what repeats across global and local contexts?

Question 2: What differs in the enactment of these genres in local contexts?

Summary:

In this article, Artemeva and Fox are focusing on undergraduate mathematics lectures in 33 classrooms, in 10 universities and 7 countries in order to explore a pedagogical genre at play in university mathematics lecture classrooms. Across all the observed contexts, the study suggests that chalk talk, namely, writing out a mathematical narrative on the board while talking aloud, is the central pedagogical genre. Despite the participants from different backgrounds, using different languages in the lectures, all of them used the genre of chalk talk in response to similar situations in differing linguistic contexts.

In addition, this study also finds evidence of the same genre occurring across global and local contexts and across participants' differing personal dimensions, for example, gender, teaching experiences, educational and linguistic background, and so on.

The types/functions of chalk talk:

I think they are important to mention, help us to identify or reflect on the text, symbolism, graphs and diagrams we write on the board.

• verbalize everything they write on the board (running commentary)
  
• talk about what they write on the board (metacommentary)
  
• move in space and use pointing gestures to indicate relationships, signal references, highlight key issues, and so on
• refer to problem sets and textbook chapters
• refer to their notes (often handwritten)
• discursively signal shifts in the action
• use rhetorical questions to signal transitions, pause the action for reflection, or check student understanding
  
• turn to students and ask questions
• talk to and/or with students. 

My Stops:

1. About the "Chalk Talk"

I have my first introduction to "Chalk Talk" through this article. As the authors said "as practitioners, our participants had no need to name the genre, but they were well aware of the centrality of talking while writing in chalk on the board in teaching mathematics." (p. 14) Even though I don't have "chalk talk" in my mathematical register, I am aware of how important the act of writing on the board is as a means of introducing students to disciplinary thinking, practices, and procedures of "doing mathematics. I try to find the closest description to "chalk talk" in Chinese, what comes to my mind is "板书" means "board & writing" if translate it to English. The difference between "chalk talk" and "板书" is the former stresses the interactions between the texts on the board and teachers. 

I like the word how Artemeva and Fox named the chalkboard management - choreography. "Choreography" seems a fancy word to me, carrying aesthetics and thinking. As a teacher, a lot of thinking goes into writing on the chalkboard, thinking of what to write, where to write, what to keep... It's definitely a process of choreography.

2. Novice v.s Experienced teachers

I had the same feelings as the postdoctoral fellows (novice) mentioned in the article in the first year of teaching. I neglected the importance of chalkboards and did not think about managing the chalkboard prior to coming to class. In my last two weeks' blog, I have mentioned that I invited experienced teachers to come to my class for some advice. They suggested I recall my previous mathematics classes, how did my math teacher give lectures. I realized that every math teacher relied on the chalkboard and much more cared about the management of the board writing. Writing, talking involve thinking, that's what made mathematics class.

My Question:

Have you ever considered what role the chalk talk plays in your mathematics class? What dimensions do you take into account to design your chalk talk?

Week 5: Response to Using Two Languages When Learning Mathematics by Judit Moschkovich

Summary:

The paper reviews psycholinguistics research & sociolinguistic research to consider how these two sets of research may be relevant to the study of bilingual mathematics learners using two languages. The author argues that psycholinguistics views language as an individual cognitive phenomenon and uses the term "language switching" to refer to the use of two languages during solitary and/or mental arithmetic computation. On the other hand, she also gives the definition of "code switching" from a sociolinguistic perspective, reserving "code switching" to refer to using two languages during conversations. 

By studying psycholinguistic research, Moschkovich suggests that classroom instruction should allow bilingual students to choose the language they prefer for carrying out an arithmetic computation, either orally or in writing. Moschkovich analyzes a mathematical conversation between two bilingual students from a sociolinguistics perspective, finding that code switching can provide resources not only vocabulary but also phrases from the mathematics register in two languages and multiple ways to participate in mathematical discourse practices.

Three Stops:

1. I am not sure I should consider myself bilingual (English-Chinese) or trilingual  (English-Cantonese-Mandarin). Although Cantonese and Mandarin share the same Chinese characters, their pronunciations are totally different. My grandparents are Cantonese speakers, but they have no idea of Mandarin. In addition, my friends whose first language is Mandarin can't understand Cantonese at all. The one that used to get me was that among the Chinese community especially the overseas Chinese, knowing Cantonese and Mandarin would be considered as an ability to use two different languages. However, in the western world, they don't really care if you know how many different Chinese dialects. Cantonese and Mandarin both are Chinese, one language. Sometimes I get my wires crossed when I speak Mandarin as my first language is Cantonese. 

I can recall one funny experience, that was a time I need to give my ID number (consisting of 18 digits) to the school administrator. I failed to read out my ID number in Mandarin, then I had to write down my ID number on paper and hand it over to the administrator. I am pretty sure I would have difficulty reading out my ID number in English nowadays, it's a hard job!

This quote resonates with me "It may seem reasonable to conclude that a word in language A in the middle of an utterance in language B means that the speaker does not know or cannot retrieve that word in language B". I had English-based mathematics courses when I went to college, there were so many terminologies that I was strange. At that time, I just ascribed delays in responding to "not knowing the math facts". In fact, I knew that math fact, but not in English!

2. Spanish shows a pattern of loss from one generation to another (Tse, 2001).

Cantonese faces the same situation as Spanish nowadays. When I was a child, my teachers were accustomed to speaking Cantonese inside or outside the classroom. My grandmother told me they even had a national anthem in Cantonese version in their time. At some point, Mandarin became the "standard" language. All teachers and students were required to speak Mandarin on campus otherwise they would get warnings. How ridiculous! I agree with Moschkovich's view that classroom instruction should allow bilingual students to choose the language they prefer for carrying out communications.

3. Bilingual students' use of gestures to convey mathematics meaning has been documented in several studies (for example, Moschkovich, 1999, 2002).

Although the research of the gestures is not the key point in this article, I think it's worthy to explore this phenomenon. I notice myself using a lot of gestures while speaking English. I don't have that many gestures when I speak Chinese. Gestures are often used while speaking to aid in the speaker's packaging of the verbal message and/or to aid the listener in decoding the message (Nicoladis, 2007). Maybe, I am not confident enough when I speak English, using gestures as a complement for the listeners to understand my words.

Questions:

Do you think retrieval, response, or solution times may be slower or faster when bilinguals are not using their preferred language or are asked to switch from one language to another in your classroom? How do you help bilinguals out when they have difficulties understanding or phrasing mathematical discourses?

Week 4: Response to "Just don't": The suppression and invitation of dialogue in the mathematics classroom

"Tools for the monoglossic are especially powerful in environments structured with significant positioning distinctions. Mathematics classrooms are just such places."

"The power is in the subtlety."

Summary:

By examining a broad set of mathematics classroom transcripts from multiple teachers, Wagner and Herbel-Eisenmann find that the word just played a big part in the mathematics classroom and could be used to suppress and invite dialogue. They focus attention on 'just problem'  as central to mathematics classroom conversation. Students had the feeling that teachers should not use just in the class and thought just was kind of an aggressive word. In addition, the word just was the 27th most common word used in mathematics classroom discourse, more common than multiply, why and because. 

Wagner and Herbel-Eisenmann point out that just serves as an adverb that seems to be synonymous with simply in most cases (28%) where the teacher is positioned as one who authorizes processes or procedures. The second most common usage is relatively synonymous with only (21%), and this usage can be accepted by students. The third but most powerful usage of just includes situations that represented varying degrees of frustration (22%). The authors invite mathematics teachers and educators to pay attention to subtle words as this kind of subtlety is powerful.

Three Stops:

1. 'Exclusive" verbs v.s. 'Inclusive' verbs

I am interested in the classifications of 'exclusive' verbs and 'inclusive verbs'. Rotman (1988) distinguishes between 'exclusive' verbs, which describe an action that can be done independently from others (e.g., write, calculate, copy), and 'inclusive' ones, which include an action that requires dialogue (e.g., describe, explain, prove). I recall my mathematics class and can think of some exclusive and inclusive verbs that I apply in my teaching. I use a lot of describing, explaining, and gestures in my teaching process. For example, I have been using “Can you tell me more?” and “What do you notice?” in our dialogues. I tried to focus on students’ understanding by encouraging them to share their ideas and thoughts. I found that ‘inclusive’ is a perfect word to describe the role of teachers and students during the interaction. We must allow each other to play a role in the mathematics dialogue. Sometimes, when I was introducing a new concept or theory to students, however, I may use ‘exclusive’ verbs such as repeat, copy or do without knowing it.

2. Suppressing dialogue, like any suppression, is an act of power (p. 10).

Wagner and Herbel-Eisenmann find that the word just was one of the most common words to appear in the mathematics classroom, and also find that the word just acted as a monoglossic tool closing down dialogue. They gave us several examples to show how the word just closed down and gave pressure to students (e.g., “Just solve the equation”, “It’s multiplication just progress straight across”.) This suppressing dialogue expanded the levels of frustration among students because teachers didn’t respect students’ thinking and positioned students as incapable listeners. And I love what Aijmer (2002) said “The task of the emphatic just is to stop further discussion” and I will keep it in mind.

3. "What is my discourse like?"

The authors encourage us to note what our discourse is like among the mathematics class and to prompt reflective awareness. When I was in my first year of teaching, I invited some experienced teachers to come to my math class. Because when I was giving a class, I can hardly notice my problems in the dialogue. I learned a lot through their feedback. I realized that I often repeated students' questions or answers that made our conversations less succinct. In the beginning, from my point of view, why I did that was to make sure students get my point or gave them responses. But in most cases, repeating what students already said is not an efficient or positive response. If I didn't invite my colleagues to my class, I would have never known it was not good behaviour. As mathematics educators, we are supposed to keep asking ourselves, "What is my discourse like?" and "How might I change it to reflect my intentions?" (p. 13).

Question:

What do you think of the application of 'exclusive" verbs and 'inclusive' verbs in your mathematics class?

References:

Aijmer, K. (2002). English discourse particles: Evidence from a corpus. Philadelphia: John Benjamins.

Rotman, B. (1988). Towards a semiotics of mathematics. Semiotica, 72(1/2), 1–35.

Wagner, D., & Herbel-Eisenmann, B. (2008). “Just don’t”: The suppression and invitation of dialogue in the mathematics classroom. Educational Studies in Mathematics