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.
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.
1 comment:
Julia --
It looks like this article really "spoke to you" and addressed many of your own interests and observations!
I agree with you (and David Tall) that mathematics teachers need to become very familiar with student difficulties related to inappropriate meanings carried by met-befores. Addressing what features can be carried forward into learning about a new topic and what must be left behind would be very helpful for students who struggle with the metaphors we set up in mathematics classes.
I agree, too, that making connections among different areas of mathematics is key.
It's very interesting to see the similarities between the British and the Chinese exam systems, and the similar problems they may cause. (By the way, did you know that Britain adopted its national exam system in imitation of the civil service exams in China?)
I'm not sure that I agree with the idea that there are genetic set-befores that make some people good at math and others not. I think this can work as a circular, self-fulfilling prophecy -- the teacher feels a student is not doing well and tells the student "it's all in your genes" (an assumption that is completely without scientific evidence, since the teacher is not a geneticist, and there has been no "math gene" identified as far as I know!) Then the student feels even more discouraged, since there is no way that one can influence one's genetic heritage. The student gives up, the teacher gives up on the student, and mathematics is treated as a field only available to a small, elite "priesthood". That is a profoundly undemocratic way of thinking about math, and it also wastes the time, talents and potential of a huge number of people who might enjoy and contribute to mathematics. I think it's exactly the wrong way to go!
Thanks Julia!
Susan
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