Week 8: Response to "Experience Meanings in Geometry" by Henderson & Taimina

Mathematics is actually an aesthetic subject almost entirely.

---John Conway, in Spencer, 2001, p. 165

Summary:

First of all, Henderson and Taimina point out the truth that intuitive understanding plays a major role in geometry. They agree with the tendency toward intuitive understanding fosters a more immediate grasp of the objects on studies, which stresses the concrete meaning of their relationships. In the field of geometry, concrete intuition is of great significance for anyone who wishes to study and appreciate the result of geometry. Amazingly, the artist and scientist both are contributors to constructing human mental and physical landscapes and are eager to show intuitions in their works. Henderson and Taimina's stories of their experience of meanings in geometry and art claim that aesthetics has always been a driving force in their experience of mathematics. 

My stops:

1. Without understanding, we will never be satisfied; with understanding, we want to expand the meanings and to communicate them to others.

I totally agree with this quote. I believe that the construction of a knowledge system is based on understanding. Every now and then, I think back to my undergraduate PDE (Partial Differential Equation) and real analysis courses, I can still see our teacher energetically writing symbols at the front of the room. The teacher would spend almost the whole three hours at the board working with equations or formulas. We copied down everything he wrote, being afraid of losing one point on the quiz. We learned math by copying instead of understanding. I think I encountered with mathematics learning block when I was in junior year, I no longer learned mathematics with understanding and lost interest in learning mathematics. I wished I understood more about what all the symbols represented and which I was meant to put where in my homework or examinations. 

Before I was in junior year, I had a different experience of learning mathematics. Whether or not I understand mathematics is responsible for the different experiences. When we understand, we can remember, transfer knowledge to new contexts, apply concepts to novel situations, look at problems from varied perspectives, and explain in ways that make sense to others.

2. Is creative thinking really different in its very essence?

Taimina shared her story with her students in her mathematics classes. Her students told me that the reason they took her class was to fulfill a distribution requirement. The students asserted that they were no good at mathematics, because they were artists, poets, musicians, actors, painters but not mathematicians, and their thinking was different. This made Taimina wonder: is creative thinking really different in its very essence? I also had the same assertion when I took two art courses last semester. The main reason for taking two art courses was to fulfill the credit requirement. I thought I was a math teacher and I would not be a qualified student in art courses. But  I would be a good listener in those courses and now and then gave my peers nods and applause. Thanks to these two art courses, I realize that arts present a way to tap into these ways of understanding and learning knowledge. 

When I look back to my math class, I found I have tried to integrate art into my Grade 2 mathematics class. “Making a Hat” was an activity to design a hat for the ping-pong ball with the shapes that students had learned such as rectangles, triangles, columns, or cones. Each student received a ping-pong ball and the first step for them to consider was how to keep the balls balanced. Some students stuck the balls on the desk with adhesive tape, and some students poked a hold on the top of the ball and put some plasticine inside. They cut the coloured paper into different shapes to design a unique hat for the ping-pong ball. 

With art integration in mathematics class, it can make my own teaching more engaging and relevant to the lives of the children, and the students have more opportunities to practice and express ideas through language and their senses. There is also the pleasure of learning mathematical concepts through play and getting creative with and through maths. 

Question:

Do you have the experience of making sense of mathematical concepts/context/knowledge through art or art activities?

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

Week 3: Response to "The 'Verbification' of Mathematics: Using the Grammatical Structures of Mi'kmaq to Support Student Learning" by Borden (2009)

Brief Summary:

Lisa Lunney Borden is a mathematics educator and researcher working in a Mi’kmaw Kinamatnewey (MK) school for ten years. She argues that teachers and students are supposed to acknowledge their identity and culture in order to stop the further disengagement of many students from marginalized groups. In addition, she worked on developing her teaching philosophy based on the concept of “verbification” of mathematics. Borden gave an example of conducting a mathematics lesson on prisms and pyramids in a grade three class by adopting the unique grammatical structures in a verb/action-based language to support student mathematics understanding. She believes that mathematics is not just about objects and facts, things that can only be described as nouns, but verbification is also an efficient way to describe mathematics. 

Three Stops: 

            a. “Everyone has something that they can learn.” (p. 9)

I love the attitude that Borden sought the advice of many community elders. The community leader suggested she use the word mawikinutimatimk which means “coming together to learn together”. How powerful a word is! As a teacher, we must acknowledge that each participant that joins in the class has something unique to contribute. This piece of discussion reminds me of the First People Principle of Learning and how they really benefit all learners in a classroom. It is worth trying to find an entry point for us to reach out to students of (Indigenous) culture.

b. When Borden articulates the model for examining the complexities of mathematics learning for Mi’kmaw students, she mentions the significance of making ethnomathematical connections for students.

I totally agree with Borden’s idea, we should not ignore the influence of one’s culture. Although ethnomathematics is widely known, as well as the frame of the curriculum approaches of employing ethnomathematics in classrooms gradually diverse, however, ethnomathematics in the Chinese mathematics classroom seems superficial and only is used for the introduction part in the mathematics concept. From a math teacher’s perspective, I think it might be time for our Chinese mathematics teachers to take our cultures into account and bring them to our mathematics classroom. It is believed that mathematics education can be more effective if examples are taken from culturally specific contexts (Barton, 1996). Teachers should look over and analyze the proper activities from different cultural backgrounds, then find the activities that are appropriate to be integrated into the class, and create a really rich and inspiring environment to help students develop their potentials” (Gerdes, 2001). All students, all people, have the capacity to do and study mathematics, to study and make sense of the patterns that are part of the land/place we experience.

c. I love the part that Borden had a communication with a Mi’kmaw speaker around the word “flat”. In Mi'kmaq, verbification can be seen everywhere, they would say the tire was losing air instead of a flat tire, they would say the prism could “sit still” instead of the prism was flat. I am wondering, if we more often use verbification in our math class, mathematics would be visible and vivid. Even the students’ descriptions of the prisms as “going like this” along with gestures indicated the motion embedded in their conceptual understandings (p. 12)

My Questions: 

Have you ever thought about how traditional/regular mathematics classes discourage students from constructing a personal understanding of mathematics? Do you have any ideas or resources to develop an intercultural class for mathematical education?

Reference: 

Barton, B. (1996). Making Sense of Ethnomathematics: Ethnomathematics is Making Sense.

Gerdes, P. (2001). Exploring the Game of Julirde: A Mathematical-Educational Game Played by Fulbe Children in Cameroon.