free writing

How can an examination of language and the body help us learn about math education?

Math itself is a language, involving a lot of terms (the meaning of words) different from ordinary English, such as product, difference, diagonal, multiplication, etc. Students need to develop a language register based on which they can express mathematics. Word problem solving is a very significant part in mathematics education. The high quality of interpreting and solving word problems is closely related to language, involving semantic and syntactic interpretation. The failure of understanding the semantic and syntactic aspects of word problems accurately will definitely baffle students to go through the whole interpretation and solution. Meanwhile, math can be embodied by many different ways, such as gestures, genres, signs,body-movement.

For teachers, the examination of language and the body can help us improve our math teaching and make math teaching more dynamic and meaningful. For students,we need to let them know that math is not a mechanical and boring subject. Let me still take word problem solving as an example. The examination of the use of language in word problems allows me to reflect: "What words and sentence structures are difficult to students and why?";"If the words used in word problems are clear for students to understand and interpret without any linguistic ambiguity and confusion?" , and etc.Therefore, I can focus more on these points when I design the word problems myself later.Moreover, the examination of language in math education can help me set up a sound mathematical register particulary for my teaching by which I can convey the mathematics explicitly. The investigation of the embodied mathematics offers me with new insights into teaching methodology,focusing more on using the visual tools, class activities involving body-movement and gestures, and other methods that include sensory perception, to not only enrich math teaching and learning but also enhance students' understanding of mathematics.

10. Some thoughts about Radford’s article

On the relevance of semiotics in mathematics education---Luis Radford (2001)
First, I’m in agreement with the fact that mathematics relies on an intensive use of different kinds of signs (letters, signs for numbers, diagrams, graphs, formulas, etc). Based on my own learning and teaching experience, I totally agree to the idea that signs, artefacts, and tools are beneficial for facilitating students thinking and thereby helping students’ cognitive development in mathematics learning. Also, I agree that signs are psychological tools, or as prostheses of the mind, or even as the external locus where the individual’s mind works. There are a lot of instances that I can illustrate to demonstrate the contribution of signs and tools in enhancing students’ math learning. I still remember I have ever illustrated an example regarding how to use some simple tools, including a string, two thumbtacks, a white board and a pen to conceptualize the understanding of the definition of ellipses in the response to Nemirovsky’s (et al) article: Motion experience, embodiment, math. The concrete tools helped me not only visualize the concept of ellipses but also discover explicitly and quickly the quantitative relationship between the distance of two fixed points and the length of the string, the relationship which is also the base to deduce the equation of ellipses. Here, I would like to give more examples to show the advantages of using signs and tools in math learning. For example, ‘To find the sum of odd numbers’, if students do not have the knowledge concerning ‘arithmetic sequences’, they probably view the problem as finding a pattern. As we know, 1=1, 1+3=4, 1+3+5=9, 1+3+5+7=16, 1+3+5+…+(2n-1)=n2, but it’s hard for younger students to relate 1, 4, 9, 16 to perfect squares just based on numbers, thus they may have difficulty in exploring the most general case 1+3+5+…+(2n-1)=?. So, if we change a way using the figures (signs), it may be easier for students to discover the pattern. See, we first use a unit square to represent 1 (1 is ÿ), and then based on the first case, we add 3 to 1, which means that we put 3 squares more around the first square (figure shown as below):
Students can see clearly that this is a 2 by 2 square. If we continue to add 5 to 1+3, it means that if we put 5 squares more around the two adjacent sides of the 2 by 2 square, students can see that they have got a 3 by 3 square, and so on. By using the diagrams, students will find it easier to relate each sum to a perfect square since each diagram is a square. As well, students may generalize the pattern more easily since 1 is a 1´1 square, the sum of 1+3 is a 2´2 square, the sum of 1+3+5 is a 3´3 square, and so on. Of course, in this problem, I admit that students may have difficulty in drawing such a n´n square (just as the same situation that the author presents in the part of ‘A classroom episode’), but the n´n square truly exists and we still can draw it with small squares and words. One more example is that for high school students, function is a very important algebraic topic for them to learn and grasp. In my teaching, I always encouraged students to solve function problems or function-related problems through drawing the corresponding functions. Based on the visualized graphs, they can see the properties of functions, the relationships between functions, which will definitely help students develop intuitive thinking rather than abstract thinking.

Second, I agree to the author’s idea that he sees signs playing a dual role in cognition from a cultural-semiotic perspective. Signs allow individuals to move along in two interrelated directions: (1) the ‘technique’ one, as a means to deal with the object of knowledge; (2) the other one which he calls the ‘social’ direction-in which individuals communicate with each other (Radford, 2001). In my view, I would like to add one role that signs can play in cognition, namely, signs also provide a kind of communication arising between individuals and the object of knowledge.

To sum up, signs and tools are a kind of embodiment in mathematics education providing students with sensory perception, such as visualization, touch, movement, and etc. In so doing, they can alter the entire flow and structure of mental function. Students may initially think about mathematics problems mentally, but with signs and tools intervening, I do believe that their thinking can be modified.

9. Some thoughts about Tall’s article

A theory of mathematical growth through embodiment, symbolism and proof (2006)
This is a nice article. It explicitly presents two parts: the first one is regarding the theoretical framework of long-term cognitive growth; and the second one explains some important issues related to long-term learning based on the previous framework.
The interesting points in the first part:
1. Mathematical learning from early childhood to adulthood is a super complicated process of accumulation. Tall views the coherence and structure of mathematics and the biological development of the human mind as two different frameworks. That’s exactly true, but, in my view, these two synthetically influence an individual’s math learning. Mathematical cognitive development is closely related to biological development. An individual’s development or maturity determines to what extent he can develop his mathematical cognition. So, this is one of the reasons resulting in people’s different learning abilities at a certain stage. For example, two children at the same age may have different acceptability to the same math knowledge. However, an individual’s mathematical cognition can prompt his biological further development. They are reciprocal to each other.
2. Based on my own learning experience and teaching reflection, I agree to Tall’s idea of using three mental worlds to explain the long-term complex construction of mathematical knowledge. These three mental worlds also form a chain representing the learning process from the basic stage to the most advanced and formal stage. Students initially learn math conceptually from seeing, hearing, and then doing based on the concrete manipulatives and embodiment. Then they gradually evolve the learning from conceptual embodiment to proceptual symbolism in which students can handle symbolic operations and manipulations more flexibly. The best example to show the transition from the conceptual embodiment to proceptual symbolism is the transition from the learning of arithmetic in early elementary grades to the learning of algebra in intermediate grades and high grades. Finally, after students become more mathematically cognitive, they can learn axiomatic systems and fulfill theoretical proof. These three mental worlds also reflect the setting of the learning expectations of the current school math curriculum, which starts from the operations of numbers in arithmetic, and next to the symbolic transformations and manipulations in algebra, and finally to the rigorous proof based on properties, definitions, theorems, axioms either in algebra or in geometry.
3. I agree to the notions of set-befores and met-befores. I always believe that people innately bring some different talents of math learning with their natural genes. This is also one of the reasons why not all people can learn math well (at least, I think so). Meanwhile, it’s quite normal for us to see students construct new knowledge based on their prior existing knowledge and use something familiar to assimilate something unfamiliar. I’m in agreement with the opinion that inappropriate met-befores will place negative impact on students’ later learning in terms of my own learning and teaching experience.
The interesting points in the second part:
1. Just as Tall, I also do believe that elementary school is the most significant stage for children when they should lay a sound foundation (including constructing the correct understanding of met-befores ) and develop a good learning habit for later learning. In most cases, I found that children’s difficulties in math learning developed at elementary school would carry over into secondary school math learning if children can not conquer them at the right time and thereby influence their life-long math learning. Also, Tall believes that there is a need to analyze the cognitive growth of ideas to help teachers and students to address inappropriate met-befores when they are likely to occur (Tall, 2006,P205)…His major concern in the UK is that students are learning necessary procedures to pass national examinations, yet seem to lack the flexibility to solve multi-step problems at university. What he says is just like a mirror reflecting the same situation of math education in China.
2. Tall mentions that focusing on essential connections should become the base on which math curriculum designers and teachers can organize mathematical matters to induce a kind of natural learning instead of rote-learning. This is exactly the same idea I have been thinking of for quite a long time. Many math concepts can be connected to each other. As math teachers, we need to help students construct a complete math image by relating relevant topics and concepts rather than pieces of image showing isolated concepts. To construct connections between ideas provides students with an opportunity not only for reviewing the met-befores and correcting the misunderstanding of them but also for learning some new knowledge based on their prior familiar experience.

8. Reflective thinking to Nemirovsky’s (et al) article: Motion experience, embodiment, math

PME Special Issue: Bodily Activity and Imagination in Mathematics Learning
Based on my own learning experience and teaching experience, I agree to the authors’ opinion that students can more easily get engaged with concrete materials that they manipulate with their hands and various activities in which they can move their bodies, hands, feet, and etc.(including different parts of their bodies) around. The advantage of the use of concrete materials and devices can facilitate the embodiment of math through the sensory perceptions, such as touch, movement, vision, kinesthesia. This article let me recall my own learning experience. I still remember clearly how my classmates and I learned the concepts of a circle, an ellipse, and a hyperbola in the high school. At that time, computer was not so prevalent as it is today in the classroom due to its expensiveness, but my math teacher just used very simple materials to demonstrate the formations of a circle, an ellipse, and a hyperbola and helped us conceptualize the understanding of these concepts quickly and explicitly. For example, what is the definition/concept of an ellipse? Every student was given a string, two thumbtacks, and a white card board. First, I fixed the two ending points of the string with the thumbtacks on the white card board and kept the distance between the two fixed points less than the length of the string. The string can be seen to be comprised a set of points. Then I picked up the string at any point to form an angle with my pen and moved my pen while keeping the string which has already formed an angle tight. The trace obtained with the pen moving is an ellipse. In my view, this is a very good hand-based activity with simple materials, but it embodies the concept of an ellipse efficiently. I still remember I deduced independently the definition of an ellipse based on this bodily activity quickly without referring to the one in the textbook which seemed more abstract and built up a good conceptual understanding of the concept. More importantly, the bodily activities usually can impress students with a longer term memory. I inherited this kind of activities from my math teacher and have kept them in my teaching over the past ten years even after the integration of technology permeated into each subject.

The second point I’m interested in is that the authors discussed the notion of a humans-with-media system. In the authors’ view, all technological means, including calculators, graphing calculators, computers, printers, videos, etc, are interconnected. Computer is not an isolated unit. So, students’ math learning can be a product made by collectives of humans-with-media. All of the technological means can create links between body activity and math representation. For example, the simulation function offered by computers and graphing calculators can help students visualize the concepts, representations of math. Vision is a part of body sense and action with the eyes moving. Through the simulation, students can visualize the math first, and then convey the visual information to brain cells through eyes moving, and then bring about thinking, prediction, decision making, verbal discussions and arguments through language and gestures. This is a learning process involving a series of bodily activities and actions. Also, I recognize that there is a major overlap between perception and imagination. To some extent, they are reciprocal to each other. Let me go back to the example above, simulation can visualize the imagination and then imagination can prompt further needs of different types of perception.

The third point I’m interested in is that teachers’ belief about the use of bodily experience in the math classroom, just as the authors stated: “We caution that widespread use of bodily experience in classrooms will depend on teachers being able to articulate how such activity is mathematical activity that is legitimate for the mathematics classroom.” This is the exact same thing as teachers’ belief about using technology in teaching. If a teacher is able to justify the intentions/purposes of the use of bodily activities and how to use them in his teaching, he can embed them in the math teaching purposefully, appropriately rather than randomly. In my view, only will these purposefully, appropriately designed bodily activities benefit students’ math learning. Meanwhile, the appropriate use of bodily activities should be an organic component which can not be isolated from the whole instruction and provide students with the insights and feelings that are hard or difficult to fully sense in other ways.

7. Rafael Nunez’s article

Mathematical Idea Analysis: What embodied cognitive science can say about the human nature of mathematics---Rafael E. Nunez

Assignment requirement: Write a short (imaginary) letter from Nunez to a skeptical math teacher who believes that math has nothing to do with the body or the physical world. In the letter, Nunez should be trying to convince the skeptical teacher of his ideas.

Dear Mr. Wang:

I still remember the issues about mathematics education that we discussed last time when we first met each other. Your comments on what the nature of mathematics is got me think further. From discussing with you, I know that you, as a secondary school math teacher, have been suspecting for quite a long time about a problem that is whether or not math has anything to do with the body or the physical world. In your view, not all of the math concepts / ideas can be explained and understood explicitly, for instance, you illustrated an example “Why the empty set is a subset of all sets?”, the question posed by your students. Your reply was that “The empty set is a subset of all sets” is merely a definition and asked your students to memorize it rather than understand its true meaning. As to this point, I totally disagree with you. So, this is the main reason why I’m writing to you now.

From my perspective, mathematics is neither transcendentally objective nor arbitrary. The nature of mathematics is about human ideas, not just about formal proofs, axioms, and definitions. These human ideas are grounded in species-specific everyday cognitive and bodily mechanisms. In the last century, many influential mathematicians viewed human intuitions (which are not theorems, axioms, and definitions) as basis for helping explain and understand mathematics. These intuitions, although they were not proved scientifically, demonstrated implicitly that mathematics is based on aspects of the human mind. In recent years, there are some empirical findings about the nature of mind discovered based on the contemporary embodied cognitive science. These empirical findings show that we are able to understand the real meanings of mathematical ideas through bodily mechanisms or people’s physical world.

In this letter, I will draw on Mathematical Idea Analysis which comes out of the embodied cognitive science as a theoretical tool to further address why we can understand mathematical ideas through bodily, physical mechanisms and people’s everyday experience embodied by image schemas, conceptual metaphors (this one is very important, constituting the very fabric of mathematics), and etc in the following:

First, by using Mathematical Idea Analysis, I have found that mathematical ideas are grounded in bodily-based mechanisms and everyday experience. For example, we can use the everyday concept of a collection of objects in a bounded region of space to conceptualize the math concept of a set; use the everyday concept of a repeated action to conceptualize the math concept of recursion; use the everyday concept of rotation to conceptualize the math concept of complex arithmetic; the everyday concepts as motion, approaching a boundary, etc to conceptualize derivatives in calculus…

Second, conceptual metaphors are fundamental cognitive mechanisms which project the inferential structure of a source domain onto a target domain, allowing the use of effortless species-specific body-based inference to structure abstract inference. They are not arbitrary, because they are structured by species-specific constrains underlying our everyday experience- especially bodily experience. For example, Affection Is Warmth, here, Affection is conceptualized in terms of thermic experience. The same happens in mathematics. For instance, Classes (Sets) Are Container Schemas—Grounding metaphors; use ‘more than’ and ‘as many as’ for infinite sets—Redefinitional metaphors; Functions Are Sets of Points, the Sets Are Graphs—Linking metaphors (see the details of three types of conceptual metaphors in Lakoff & Nunez (2000)).

Third, image schemas are basic dynamic topological and orientation structures that characterize spatial inferences and link language to visual-motor experience. Their inferential structure is preserved under metaphorical mappings. So, if we go back to the instance ‘Sets Are Container Schemas’, we can see that the Container Schema(a common image schema in mathematics) presents the link between language and spatial perception, involving three parts: an Interior, a Boundary, and an Exterior. Image schemas can fit visual perception, for example, Venn Diagrams are visual instantiations of Container Schemas.

Of course, conceptual metaphors are closely related to image schemas. Now, please allow me to still focus on the example of “Sets Are Container Schemas”, here, sets initially metaphorically conceptualized as containers, and then Venn Diagrams work as symbolizations of sets to show their bounded regions in space. Therefore, for this mathematical idea ‘Sets’, we can realize the understanding through metaphorically relating them to containers and then using Venn Diagram as the image schema for containment. It means that we can conceptualize ‘Sets’ through our everyday life experience and body-based perception.

Furthermore, the ‘Laws of Container Schemas’ called by Lakoff and I are conceptual in nature and are reflections at the cognitive level based on the analysis of neural structures. So, I sincerely invite you to read my paper which includes more regarding this issue.

Yours sincerely,

6. Reflective thinking to Susan Gerofsky’s article

Genre analysis as a way of understanding pedagogy in mathematics education
This is a paper regarding the application of genre analysis in mathematics education, focusing the lens on two topics: one is the genre features existing in word problems and the relationship between word problems, riddles and parables; the other one is the lecture genre of initial calculus at a university. After reading the whole paper, I want to propose some questions as below:
1. On pp38, you proposed a question for readers to think about: “What are word problems?” To be honest, when I arrived in Canada initially, I was confused about the concept of ‘word problems’. If we just talk about this concept from the view of mathematics, I have to admit that we have different notions between China and Canada. If I translate the words of ‘word problems’ into Chinese literally, ‘word problems’ means ‘wen zi ti’, which means that the problems only involve mathematical data and question without being embedded into a real or fictional story, for instance: 1 more than a number divided by 7 is 10. What is the number? . But with getting myself into the Canadian mathematics education, I realized that ‘word problems’ here is the same meaning as the problems that we call ‘ying yong ti’ in China (if I translate the Chinese words ‘ying yong ti’ into English literally, ‘ying yong ti’ refers to problems of application (applying mathematics to problems embedded into either life contexts or stories) So, here, I still need to clarify the question of ‘what are word problems?’. Are they the same as ‘ying yong ti’ or the combination of both ‘wen zi ti’ and ‘ying yong ti’? In terms of the features of word problems that you typified in the paper, a word problem is a three-component, sequenced rhetorical structure (a story element which can be disposable, data and question), I think that ‘word problems’ is the same as ‘ying yong ti’ in China. Then I’m not sure how to term the problem just like the one I illustrated above?
2. The use of verb tense in word problems is ‘tenseless and non-deictic’. Although I understand that the strange mixing of verb tenses in word problems may not disturb the understanding of English speaking students about the problems, I’m wondering if the mixing of verb tenses will confuse ESL students and to what extent it affects ESL students’ math thinking. From my perspective, I think that ESL students will generally have a long term obstacle of developing the English language register and genre. The English genre they have, to some extent, is more consistent with the genre of their mother tongue, different from the one that English speaking students hold. ESL students usually learn English in a very formal environment in which the use of verb tense in English is governed by norms of grammar, so if the word problems would be counterproductive to ESL students’ English learning?
3. The painting example (pp39) you illustrated in the paper is impressive. I have ever met the same situations in my previous teaching. What my students were more curious about was to explore the truth of the stories or the life contexts in which I embedded the problems rather than the math problems per se. So, if we have a number of stories that we can use to integrate the math problems, do we have any criteria that we can refer to when assigning an appropriate story to the problem and deciding the language we need to use in order to avoid distracting students’ attention?
4. In the second case study-initial calculus lectures as a genre, the features of the lecture genre evoke my resonance. You said: “The lecture genre, is already a mode of persuasive talk that tries to “sell” its audience on both of truth of the ideas presented, and the authority and status of both the lecturer and sponsoring institution as purveyors of truth and knowledge….In the lecture genre, tag questions, rhetorical questions (fake dialogue), non-standard use of ‘we’, hard sell persuasion techniques and ‘making encouraging noises’ can co-occur.” In my view, the features of the lecture genre can be shared by the genre of all teacher-centered environments in which there are no sufficient interactions, discussions, argumentations between the teacher and the students. I can see these features in my past teaching. However, in most cases, I feel that these features emerged spontaneously and unconsciously in my teaching. Even when I was explaining mathematics knowledge to one student rather than to all students in my big-size class, I realize that I would still use these tag questions, rhetorical questions, etc, such features belonging to the lecture genre naturally and unintentionally. So, I think that even in the student-centred learning environment, the same genre will still occur naturally. For example, when I invite students to enter into the communication, regardless of as a teacher or just as a facilitator (in student-centred environment), I will still say “Let’s look at…, Let’s do….” rather than “Let you look at…, or Let you do…”. If we always make explicit use of “we, us, our”, the communication between my students and me must sound weird. In this sense, I’m asking if the use of tag questions, rhetorical questions, the extensive use of the first personal plural pronouns sometimes are unconscious behaviors in a person’s speaking(including monologue and dialogue), and sometimes, if even the hard sell is also unconscious and spontaneous?

5. Reflective thinking to Tim Rowland’s article

Hedges in mathematics talk: Linguistic pointers to uncertainty (Tim Rowland,1995)
After reading Rowland’s article, I found that it is pretty interesting reminding me of something that I have never paid attention to before. This article also let me recall the project that I did in Ann’s course last term. In the project, I had 4 interviews which aimed to examining the mathematical thinking in different grades of students. What impressed me the most among these interviews was that my students always replied me “oh, there are many” if they found the math problem has more than one answer/possibility. For example, “Three way of sharing” was one of the problems I posed to my students. It has 21 possible ways to share 8 cookies among 3 kids. I still remember that my interviewee presented her answer with “there are many ways” after she thought about the problem for a while. What’s the meaning of ‘many’? It’s a vague math word, which explicitly shows that my student is not sure about the answer. In looking back the word transcripts that I recorded for the conversations of the project between my students and me, I found that either my interviewees or I frequently used the words, such as about, around, maybe, probably, think, basically, normally, suppose, (not) sure, (not) really/exactly, all the words that are termed as hedges by Rowland. But I have never thought that these words can be advanced to such a theoretically linguistic view, I thought that they are very common words that can be heard quite often either in our daily speaking or the talking in subject learning. Honestly speaking, I use these hedges frequently and unconsciously sometimes when I talk about something that I’m not so sure or I feel less confident.

In my view, these mathematical hedges still reside in the mathematical register. They just convey the mathematical vagueness or uncertainty. Rowland has made two major contributions in this article concerning the investigation of vague language use in math education. The first one is that he has outlined and exemplified a classification (which he cited from Pince et al. 1992) of hedges into more detailed functional categories. The other one is that he offered us with an interpretive framework which can be applied to explain why children use some ambiguous and vague language when they are predicting and generalizing something in the math learning.

I don’t care about so much how to call these vague words in terms of the taxonomy of hedges, for instance, which type should the word ‘maybe’ go into?, but what I’m more concerned about is that what inferences about students’ math learning they can convey to us (teachers), therefore, we can know more clearly about what we should do next, for example, adjusting our teaching strategies or the ways of inquiry or providing further explanations, etc. I totally agree to the thoughts that Rowland either came up with individually (e.g. the effect of the announcement ‘I shall have to think about that’, pp345) or cited from other literatures as theoretical supports (e.g. he cited Channell’s thoughts about a number of goals of why people use vague expressions, pp350) regarding why students use the uncertainty and vague expressions in math learning.

Furthermore, as far as I’m concerned, hedges are important in students’ math learning, at least, providing students with an opportunity to conceptualize math knowledge through discussing with their peers or teachers. These conversations with peers or teachers should involve questioning, self-reflection and argumentation. What we should do is how to help students transform the initial uncertainty to the assurance step by step.

4. Reflective thinking to Yackel and Cobb’s article

Sociomathematical Norms, Argumentation, and Autonomy in Mathematics (Erna Yackel & Paul Cobb)
When I first saw the title of the article, I thought that it would be very boring with many theoretical ideas related to social issues. However, after I read it, I really felt that this is a very enlightening article, reminding me of a fact that we need to place more emphasis on the discussion and interaction between the teacher and students in the classroom disregarding the subject areas. I have to admit that in my previous teaching, sometimes, I didn’t take my students’ mathematics voice seriously. Because of the time limitation and the main purpose which aims at how to let my students achieve high scores in all types of examinations, I took more teacher-centered way to treat my teaching, especially when I was teaching grade 12 students who have an urgent need to get good scores in the National Entrance Examination, at that period of time, I had less communication and interaction with my students in the classroom. What I usually did during that period was to convey the knowledge throughout the whole class and asked my students to copy down the important notes and solutions step by step. I always thought that the solutions I provided to my students were more elegant than what they got and the teacher-centered teaching could save their time. Also, I was also very satisfied with seeing that my students could follow the skills or solving techniques that I taught them to solve some similar problems. What I did was to view my students as passive recipients or empty vessels waiting to be filled and kill their interests and motivation to construct the knowledge themselves. I admit that if students can construct the knowledge themselves, it will be very beneficial for developing their conceptual understanding of mathematics. But, what I talked above doesn’t necessarily mean that I always used the teacher-centered way in my teaching.

The authors define the term of ‘sociomathematical norms’ and clarify the subtle difference between general social norms and sociamathematical norms in the paper. They stated: “normative understandings of what counts as mathematically different, mathematically sophisticated, mathematically efficient, and mathematically elegant in a classroom are sociomathematical norms. Similarly, what counts as an acceptable mathematical explanation and justification is a sociomathematical norm.” It means that mathematical difference, mathematical sophistication or efficiency, mathematical elegance, and an acceptable mathematical explanation and justification form the sociamathematical norms in the math classroom. Also, in my view, they also form an ascending chain in which each norm is developed one by one. These norms remind me of the fact that if students can make mathematic difference in their response or solutions, they are able to develop a sophisticated or efficient solution among different solutions themselves. If they can see or compare the similarities and difference in solutions through argumentation, it’s also nature to see that they are able to develop conceptual understanding of the problem and present an acceptable explanation and justification accordingly. I recognize that these norms are so essential and important that they can be seen often in the math classroom in which there is sufficient interaction between the teacher and students. In my former teaching, if I had enough instruction time when I taught in grade 10 and 11, I also encouraged my students to engage in math learning with group discussions, whole-class discussions, and the communication with me. So, I can say that I’m quite familiar with the episodes that the authors described in the examples although I’m devoid of the academic ability to advance these episodes to a theoretical level. In my math classes, as long as there is no conflict between the teaching pace and my teaching strategies, I preferred to use the way with many inquiries. I totally agree to the author’s opinion that mathematical difference is interactively constituted by the teacher and the students. To give the students enough time to present their solutions fully should be greatly emphasized by teachers. Students are different individuals, having different mathematical thinking. So, they are the source of generating different solutions although not of them are correct. Just like what the authors said that the accuracy is not an important issue in developing math understanding, what more important is whether students can reflect their understanding through identifying similarities and differences among various solutions. Such reflective activity has the potential to contribute significantly to children’s mathematical learning. (Yackel & Cobb, pp464) Students can truly establish the conceptual understanding when they protect their own solutions or learn from others’ argumentation. In addition, students’ different thinking or solutions also help the teacher broaden his/her thinking. For example, in my teaching, if I got quite stuck when solving a complicated problem, I would turn to ask help from my students rather than from my colleagues. I usually posed the question first, and then asked students to raise their opinions of how they thought about it. If students raised some ideas that I didn’t know, I definitely would ask the students to give me an explanation, and then I saw students discussing, arguing between the similarities and differences among various ideas. Students are always thrilled to see that they can solve a problem that the teacher is not able to, at that time, they are very willing to present their explanations and justify their explanations. It’s true that their explanation and justification also provide me with an opportunity to develop my better conceptual understanding.

The teacher’s reactions to a child’s solution can be interpreted as an implicit indicator of how a solution is valued mathematically, such as “Yeah”, “Perfect”, “Well done!”, “really?”, etc. I’m really impressed with reading the episode in example 4 because I always did the same way to see if my students could persist in their solutions or explanations in my teaching. The questioning, like “Is it right?”, “Really?” can help me identify whether my students have got a strong conceptual understanding. If they haven’t, they would swing in front of such questionings. I have ever used the same example when I saw my students protesting my questioning and told them what they needed is to justify their reasoning. “If you can justify the accuracy mathematically, why do you swing?” This was a question I would ask when seeing my students’ protest. I admit that these soicomathematical norms are helpful for students to foster the development of intellectual autonomy which is a major of mathematical education.

To sum up, this is a good paper, setting forth a way of analyzing and taking about mathematical interaction in the classroom from the perspective of sociomathematical norms.

3. Reflective thinking to Barwell’s article

Integrating language and content: Issues from the mathematics classroom

I agree to Barwell’s opinion that language and content can be seen as reflexively related, for example, through solving word problems. The various aspects of thinking mathematics and learning language are all closely interwoven, with attention to written form, to problem genre, and to mathematical structure. Barwell uses his analysis for two ESL students’ dialogue during word problem solving to demonstrate that there is a link between attention to mathematical structure (i.e. content) and attention to written form (i.e. language), including the correct use of words, word spelling, verb tense (grammar), which comprise important parts in language learning. His article enlightens me with a way of how to embed language learning into math learning and teaching, the way which will be beneficial for my future bilingual teaching in Shanghai.

Over the past ten years, Chinese Education Ministry has been engaged in making efforts to reform Chinese English instruction due to the fact that Chinese English instruction prior to 1998 showed a big failure in educating students to be English proficient. So, after investigating and observing the bilingual education in Singapore, Hongkong, Canada, China realized that to integrate English learning into other subjects involving math, physics, chemistry, PE, art, etc in formal school education, is an essential method to improve Chinese English teaching whose main goal aims to make Chinese students more proficient at English use and communication in the world because of the globalization. Shanghai municipal government encourages all schools to carry out bilingual teaching, which has already become an urgent task in accordance with the economic rapid development since 2000. Here, we need to note that in Shanghai (China), we say ‘bilingual teaching’ instead of ‘bilingual education’. The term ‘bilingual’ in Longman Applied Linguistic Dictionary is defined as:

(A person who) knows and uses two languages.

In everyday use, the word bilingual usually means a person who speaks, reads, or understands two languages equally well (a balanced bilingual), but a bilingual person usually has a better knowledge of one language than of the other.

　　For example, he / she may:

　　a) Be able to read and write in one language but speak and listen in another language.

　　b) Use each language in different types of situation, eg. one language at home and the other at work.

c) Use each language for different communicative purposes, eg. one language for talking about school life and the other for talking about personal feeling.

‘Bilingual Education’ means:

The use of a second or foreign language in school for the teaching of content subjects.

　　Bilingual education includes three models as below:

　　a) The use of a single school language which is not the child’s home language. This is sometimes called an immersion bilingual model.

　　b) The use of the child’s home language when the child enters school but later a gradual change to the use of the school language for teaching some subjects and the home language for teaching others or alternately use both home language and a second language in a subject teaching. This is sometimes called maintenance bilingual model.

c) The partial or total use of the child’s home language when the child enters a school and a later change to the use of the school language only. This is sometimes called transitional bilingual model.

(Wang, 2004)

Several influential educations, who have been long-term engaged in Chinese bilingual teaching field, including Wang, differentiated the concept of ‘education’ and the concept of ‘teaching’. Based on their discussion, ‘Education’ involves a series of planned activities influencing learners’ moral education, intellectual education, and physical education according to certain educational expectations. However, ‘teaching’ means the instructional interaction between teachers and students. From the above definitions, we can see that the concept of ‘teaching’ is narrower than the concept of ‘education’. Meanwhile, terming ‘bilingual teaching’ is more consistent with the current situation of Chinese language education. Our current situation discovers a fact that we lack sufficient bilingual teachers with good English proficiency whilst our students are generally identified as quite limited English proficiency. In so doing, we need to ground our bilingual teaching on the maintenance bilingual model as meaning to that we still have to depend on our home language to support students’ mathematical content understanding. All Chinese students are considered as ESL learners but different from the ESL learners in English spoken countries, because Chinese students only learn English at schools. If we integrate English learning and math content learning, we must note that both the understanding of math terminologies, concepts, problem structures and meanings in English and the ability to translate from English to mathematical symbols are hurdles to Chinese students. In the meantime, as far as I’m concerned, Chinese students as ESL learners need a math learning context in which they can practice and improve reading, listening, writing, speaking skills of English simultaneously rather than merely focus on one skill.

If I go back to Shanghai in the near future, I think that I will incorporate bilingual teaching into problem-based learning context where students can start language and math learning with carefully selected real-life problems. From my perspective, these problems arising from real life involve more than word problems. Over the past 40 years, many research findings demonstrated that students in PBL can become more active learners instead of passive recipients in the traditional environment. These problems are beneficial for helping students not only improve their reading skill but also enlarge their vocabulary reservoir subsuming both general English and mathematics English words and enrich their knowledge, including history, social development, etc about western countries. . They embrace the mathematics structures featured by word problems, so students also need to analyze math structures expressed in English and learn how to translate English words and sentences to mathematical symbols and expressions. They are ill-structured, so students need to seek for more information via different methods which can help students develop various learning skills, including problem- solving skill, critical thinking skill, cognitive skill, and etc. In PBL, students are usually divided into small groups of 5-6 in each. In the small groups, students can have more opportunities to participate in both verbal and written practices of English through searching the meaning of math technique words, comparing the meanings between general English words and math words, analyzing the problem sentence structures, explaining solution processes, describing conjectures, proving conclusions and presenting arguments. Students in the small groups can have less shyness and fear to speak English and share mathematical ideas more fully than in the big-size class.. If students have questions about what they are thinking and learning, the teacher, in PBL, as a facilitator, can help them dispel the puzzles and explain ‘why’ and ‘how’ questions either in English or home language to facilitate their understanding of math content and English language knowledge. At the final step of each lesson, each student will be encouraged to present their assimilation and applications of new knowledge (i.e. solving new similar problems) both verbally and in written form which purport to help them not only consolidate their math learning but also pratice their speaking and writing skills of English. In addition, I need to point out that, from my perspective, it’s not easy to be a blingual teacher, who should not only have sufficient math knowledge background but also be very competent at English language.

The above discussed is some of my immature thoughts for bilingual teaching in Shanghai. I’m still thinking about a better approach to make my future biligual teaching more efficient in accordance with the current language abilities and situations of Shanghai students.

Wang, B. H. (2004). Bilingual Education and Teaching in China. Journal of Education, Shanghai Education Press.

2. Reflective thinking to Pimm's article

Speaking mathematically---Communication in mathematics classrooms (David Pimm)

When I was reading Pimm’s paper, I went back to Halliday’s paper from time to time to review the meaning of a register of mathematics. I wanted to avail myself of reading these two articles to help me understand the meaning of register in a language context. In Halliday’s definition, a register is a set of meanings that is appropriate to a particular function of language, together with the words and structures which express these meanings. (Halliday, 1978,pp195). Pimm summarized that registers have to do with the social usage of particular words and expressions, ways of talking but also ways of meaning. (Pimm, 1987, pp108) Pimm’s summary of a register of mathematics helps me understand the term “register” further and better. Language itself is developing with new technology, ideas, products created. Mathematics itself is also a language, in which there are many words whose meanings are different from daily ordinary language. So, a register is also a place in which we can store and develop words with their meanings, expressions, structures, disregarding whether they have already been developed or are being developed or undeveloped. Halliday also illustrates some ways of developing a register of mathematics, such as reinterpreting existing words; creating new words and borrowing words from other languages, the ways that English favors; calquing words from other languages, the way that Chinese favors, etc. Although I was quite interested in knowing the various ways of how to develop a register of language, especially knowing about the way how Chinese create new words, I have to admit that I misunderstood the meaning of calquing initially. Your comment helped me realize my misunderstanding. I have never thought that “jing ji” comes from the Greek root word of economy meaning “household management”, democracy (min zhu) literally originates from the Greek root “demos” and “cracy”. I realize that I’m devoid of systematic knowledge concerning the roots of English words. I even have never thought that it is necessary for a Chinese to study how Chinese language developed, but your comment invokes my interest to study the relationship between Chinese and other languages, because I realize that the accurate and rich use of language is very important in our teaching.

In my view, English has the difference between ‘ordinary English’ and ‘mathematics English’. So does Chinese. Pimm also mentioned: there are some special terms, such as quadrilateral, parallelogram, hypotenuse, which are unlikely to encounter often in our daily language. As to those words, such as product, factor, function, reciprocal, face, degree, power, radical, legs, rational, etc, they are borrowed from everyday English but with different meanings in mathematics. Although students can refer to general meanings in ordinary English when they touch these words in mathematics classes, they still will get confused the special meanings of mathematics with the general meaning. One of my tutoring students has made several attempts to understand the meaning of a power in mathematics. She always confused the power with the exponent or was not able to recall the word power when she was asked to express the form of qk, because what she could recall naturally is the general meaning of power in everyday English. For ‘product’, what students can suddenly think of is the item either sold in the market or produced in the factory rather than the result of multiplying two or several numbers or objects. So, I totally agree to Pimm’s opinion: “the failure to distinguish between ordinary English and mathematics English can result in incongruous errors and breakdowns in communication (Pimm, 1987, pp88).”

I was so impressed with reading some examples with regards to the word meanings in mathematics that Pimm has illustrated. These examples evoke my resonance through reflecting my own teaching. The first one is the term ‘clockwise’. This is a very good example demonstrating that the everyday meaning of a word is helpful for students to understand its meaning in mathematics. When I taught trigonometric ratios in grade 10, I found that some of my students had difficulty in understanding ‘clcokwise’, ‘counter-clockwise’ or were not able to turn an angle clockwise or counter-clockwise. When I asked them to refer to their watches to see how the watch works, they told me that they couldn’t see clockwise on their watches since their watches are digital. In this sense, if a mathematics term loses its everyday referent, it will become a difficult knowledge point in students’ learning. The second one is the ‘if…then…’ format which almost all mathematics propositions fall into. I have never related it to the mathematics register with which it carries the force of logical implication. I always thought the structure ‘if…then…’ has become a rule when we compose a mathematics proposition before. Right now, I’m wondering if the structure (or these two words) can show a real logic relationship between the conditions and results of a mathematics proposition. The third example is the problem concerning diagonals in polygons. To be honest, in my previous teaching in Shanghai, I have never realized that students might have difficulty in understanding the meaning of ‘diagonal’ in mathematics. I found that the understanding of the term ‘diagonal’ is a problem to grade 7-8 students in Canada when I was tutoring some local CBC students who can not speak Chinese. When I asked them to explore the respective number of diagonals in different types of polygons, such as triangles, quadrilaterals, pentagons, hexagons, etc, they initially counted the number of diagonals by joining two vertices disregarding whether these two vertices are adjacent or not adjacent. Although they know the diagonals are the line segments connecting two vertices, their understanding is not precise. I have to admit that for the term ‘diagonal’, English is not a good language which can help students understand the exact meaning through referring to the word itself. In contrast, ‘diagonal’ in Chinese is much easier for students to understand its definition either in polygons or in polyhedrons. ‘Diagonal’ is translated into ‘对角线(dui jiao xian)’. Based on the two Chinese characters ‘对角(dui jiao)’ which means opposite angles in English, students can readily understand that this line segment needs to join two opposite angles (two opposite vertices) rather than two neighboring vertices. So, I think this is why Chinese students seldom have trouble in understanding the concept of ‘diagonal’. This is the language difference between Chinese and English. Of course, I admit that in some mathematics cases, English is much easier for students to learn math. For example, Chinese students have to memorize most mathematics notations imported from western countries through understanding English. Otherwise, they have to memorize them mechanically. Most Chinese students have such questions: why do we need to use f (the initial letter of ‘function’) to express a relation between x and y since you will never see f in Chinese words or pinyin: ‘han shu’, expressing function; why do we need to use log to express a logarithm or a logarithmic function since we can never see log in the Chinese translation ‘dui shu’, why do we need to use sin, cos, tan, cot, sec, csc to express 6 trigonometric ratios, etc.

In Pimm’s paper, he pointed out a fact that I’ve heard again and again after I came to Canada. When examples of figures are drawn to illustrate or merely invoke a concept, the orientation is seldom randomized, and many pupils seem to include the particular orientation in their concept (Pimm, 1987, pp85). Even in my tutoring job, I found my students only regarded an isosceles or equilateral triangle as a triangle. If the polygon has three sides which are not equal to each other, it’s not a triangle. They didn't realize that a scalene triangle is also a triangle. As a matter of fact, I seldom heard the same story in Shanghai. If the teacher only demonstrates some specific orientations of a polygon when they explain the concept verbally, in my view, it’s not weird for us to see that students are not able to understand it accurately. I think the problem not only involves the language interpretation of the concept of a triangle but also includes the teaching methodology. (Maybe I’m wrong.)

The last point that I want to talk about is the role of metaphor in mathematics language. Pimm mentioned that analogy and metaphor come to mind as powerful linguistic techniques for creating new meanings (Pimm, 1987, pp93). Both of them offer means by which the less familiar may be assimilated to the more familiar, by viewing the former in terms of the latter. It’s exactly true. I also often tried my best to use analogy and metaphor to explain math concepts in my teaching. Or simply saying, the means of relating something new or unfamiliar to something that students have already known is really helpful for students to learn math., especially when students learn some abstract new concepts and solve some new difficult problems. To change something unfamiliar to familiar and to transform from the abstract to the concrete need analogy or metaphor. For example, we can teach students how to solve the equation sin2x+3cosx-3=0 (cos2x-3cosx+2=0) based on the easy quadratic equation x2-3x+2=0. This is the means of analogy. We can use a room corner which connects three walls (one is the ground, the other two can be two vertical adjacent walls) who are perpendicular to each other to help students understand the 3-dimensional coordinate system, use fishing to express the process of expanding the product of two binomials, use SOH, CAH, TOA to help students memorize the trigonometric ratios, and relate ‘take away’ to subtraction, ‘have more’ to addition. I think that they are the means of metaphor, but they are the extra-mathematical metaphors as what Pimm defined. I think we may use extra-mathematical metaphors quite often in our teaching. There is not only one way of metaphor to express one certain thing. Different people may create various personal metaphors as long as they find them beneficial for them to understand mathematics. As to the structural metaphor, this is the first time I heard it, although I’m quite familiar with the example—a complex number is a vector, cited by Pimm. A complex number as a vector is a very important knowledge part when we teach complex numbers in grade 10, but I always regarded it as the geometric meaning of a complex number instead of a structural metaphor. Meanwhile, from this example, we can see the importance of using structural metaphor in math teaching, which can help students relate one knowledge fragment to another relevant fragment. In so doing, students can transfer the existing knowledge to new knowledge or contexts and also view math as a complete learning subject in which knowledge points relate to each other rather than separate from each other. However, it’s challenging for a teacher to use metaphors including extra-mathematical and structural metaphors appropriately and with high quality in his/her teaching. This ability is also the one that I need to develop for my teaching.

1. Reflective thinking to Halliday's article

Halliday’s paper: Sociolinguistic aspects of mathematical education
As a person who speaks English as a second language, I really understand the importance of a language, especially when you are far away from your home country, using a second language to study, work, communicate with local people in a foreign country. If you do not have enough proficiency in English, for example, in Canada, you will feel to be isolated by the local society. Language is not only a tool that we can draw on to express our ideas explicitly or interpret others’ opinions but also is an art, for example, how to use the exact words to convey your meaning precisely, how to use the language to show our respect, politeness without breaking local people’s customs when we talk to them. These are my personal experiences in the use of English in Canada. Although I had been learning English for more than 10 years when I was still in Shanghai, sometimes I found the English I learnt in China is different from the English I’m using now. So, in my view, to learn a second language needs a good language context, the same as the one in which we learn Chinese.

In this article, Halliday conferred the old word “register” with new meaning in a language context. He defined: “A register is a set of meanings that is appropriate to a particular function of language, together with the words and structures which express these meanings.” To be honest, this is the first time that I know the meaning of register in a language context. I think I need more readings to help me understand the meaning of “register” further.

Halliday mentioned that the development of a new register of mathematics will involve the introduction of new “thing-names” through reinterpreting existing words, creating new words, borrowing words from other languages, calquing, etc. This let me think how we should use the language in our math teaching. In my previous teaching, I sometimes employed new or reinterpreted words which are in accordance with my students’ daily language, the prevalence of the society, etc, as my classroom verbal interaction with my students.

I’m interested in knowing that Chinese favors calquing as meaning to creating new words in imitation of another language. I suddenly realized that it’s exactly true. In Chinese, we have a lot of exotic words imported from western countries. As for these exotic words, we use Chinese words which sound like the same pronunciation of English, such as cement (shui men ting), chocolate (qiao ke li), email (yi mei er), etc.

What I’m impressed with reading is the part of talking about the uniqueness of the mother tongue. When I speak mandarin or Shanghai dialect, I can express the same thing in different ways. That’s exactly true. When my students had difficulty in understanding one way of interpreting math concepts, I can immediately use several other alternatives with the same meaning to explain them. However, when I use English to teach math, I have to admit that I really have encountered the dilemma the author described in his paper. Halliday said: “the teacher who is forced to teach in a language other than his mother tongue has at his command only one way of saying something.” As a teacher, I know that language is very vital in our teaching. If I can not speak a second language freely, it’s hard for me to develop the ability to say the same thing in different ways, to predict what the other person is going to say or add new verbal skills.

Finally, I agree to the author’s opinion: the more informal talk goes on between teacher and learner around the concept, relating to it obliquely through all the modes of learning that are available in the context, the more help the learner is getting in mastering it. Mathematics teaching definitely involves lots of language conveying the understanding of concepts and problems. That means language is a key factor leading to students’ efficient final understanding of math knowledge. Therefore, it’s necessary for us to study the relationship between language and math education. The benefit from the study is how we can use language to facilitate and enhance students’ math learning.

Self-introduction

Hi, everyone,
I'm Julia Lan Dai. I come from Shanghai, China. I had been teaching mathematics at a senior high school in Shanghai for consecutive ten years before I immigrated to Canada. I landed in Vancouver in August of 2006, I have been living here for a year and a half. I am a student from CUST department with mathematics education in major. This is the last course for my master's degree. Since English is a second language to me, this course involving studying the language between language (English) and math education will be challenging to me. Hope everybody can succeed in our pursuit.
Anyway, nice to meet you all.
Cheers,
Julia