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.