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.