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.
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.
1 comment:
Julia --
Thanks for a very thoughtful blog! I like your example of the visual sign that helps younger kids see that the sums of consecutive odd numbers are always squares. These were the kind of visual, iconic, geometric proofs the Greeks used in ancient times -- something we've lost to some extent with the symbol systems of algebra.
Well done!
Susan
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