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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.