5 ms·

In my experience, everything is easy if you DON'T try to learn it using analogies and metaphors.

cool_shit 10y ago
Terrible advice! I tell all of my students to find as many analogies as they can. You can never lose by increasing the number of ways in which you understand something. Analogies, some may argue, are at the heart of mathematics. > Mathematics is the art of giving the same name to different things Poincare. > The art of doing mathematics consists in finding that special case which contains all the germs of generality Hilbert. > The vast majority of us imagine ourselves as like literature people or math people. But the truth is that the massive processor known as the human brain is neither a literature organ or a math organ. It is both and more. John Green. > Sometimes I think that creativity is a matter of seeing, or stumbling over, unobvious similarities between things - like composing a fresh metaphor, but on a more complex scale. David Mitchell. There is an ounce of truth in what you say -- metaphors can be abused to draw false conclusions. This does not mean one should cower from using them.
- ankurdhama 10y ago
  Everyone has their own philosophy of mathematics. To me, the core idea is the idea of representations, such that those representations allow you to extract patterns (aka similarity or commonality) from various cases.
  - nicklaf 10y ago
    I can't help but indulge a bit and now ask whether or not we can't extract a common pattern from the two representations of mathematical cognition offered by yourself and GP. :-)
    - ankurdhama 10y ago
      Not here but may be on "meta hacker news".
  - fmoralesc 10y ago
    Aren't metaphors and analogies representations that allow us to extract patterns?
    - ankurdhama 10y ago
      To me, metaphors and analogies are about trying to learn some new concept(s) in terms of already known and familiar concepts.
      - nicklaf 10y ago
        As an opposing view, I offer you the following analogy, by mathematician Charles C. Pugh, which he draws between conceptions of metaphor as understood in natural versus mathematical language: From "Real Mathematical Analysis" 1st edition, p. 9: Metaphor and Analogy In high school English, you are taught that a metaphor is a figure of speech in which one idea or word is substituted for another to suggest a likeness or similarity. This can occur very simply as in "The ship plows the sea." Or it can be less direct, as in "his lawyers dropped the ball." What gives a metaphor its power and pleasure are the secondary suggestions of similarity. Not only did the lawyers make a mistake, but it was their own fault, and, like an athlete who has dropped a ball, they could not follow through with their next legal action. A secondary implication is that their enterprise was just a game. Often a metaphor associates something abstract to something concrete, as "Life is a journey." The preservation of inference from the concrete to the abstract in this metaphor suggests that like a journey, life has a beginning and an end, it progresses in one direction, it may have stops and detours, ups and downs, etc. The beauty of a metaphor is that hidden in a simple sentence like "Life is a journey" lurk a great many parallels, waiting to be uncovered by the thoughtful mind. Metaphorical thinking pervades mathematics to a remarkable degree. It is often reflected in the language mathematics choose to define new concepts. In his construction of the system of real numbers, Dedekind could have referred to A|B as a "type-two, order preserving equivalence class", or worse, whereas "cut" is the right metaphor. It corresponds closely to one's physical intuition about the real line. See Figure 3. In his book, Where Mathematics Comes From, George Lakoff gives a comprehensive view of metaphor in mathematics. An analogy is a shallow form of metaphor. It just asserts that two things are similar. Although simple, analogies can be a great help in accepting abstract concepts. When you travel from home to school, at first you are closer to home, and then you are closer to school. Somewhere there is a halfway stage in your journey. You know this, long before you study mathematics. So when a curve connects two points in a metric space (Chapter 2), you should expect that as a point "travels along the curve," somewhere it will be equidistant between the curve's endpoints. Reasoning by analogy is also referred to as "intuitive reasoning." Moral: Try to translate what you know of the real world to guess what is true in mathematics.
tchaffee 10y ago
The latest research on how we learn and how the brain works places great importance on analogy as a learning tool.
- ankurdhama 10y ago
  Any references to such research?
  - tchaffee 10y ago
    - Baddeley, Alan, Michael W. Eysenck, and Michael C. Anderson. Memory. NY: Psychology Press, 2009. - Cat, Jordi. "On Understanding: Maxwell on the Methods of Illustration and Scientiﬁc Metaphor" Studies In History and Philosophy of Science Part B32, no. 3 (2001): 395-441 - Derman, Emanuel. Models of Behaving Badly. New York, NY: Free Press, 2011 - Foer, J. Moonwalking with Einstein. NY: Penguin, 2011 - Lutzen, Jesper. Mechanistic Images in Geometric Form. NY: Oxford University Press, 2005 - Maguire, E.A., D.G. Gadian, LS. Johnsrude, C.D. Good, J. Ashburner, R.SJ. Frackowiak, and CD. Frith. “Navigatioanelated Structural Change in the Hippocampi of Taxi Drivers.“ Proceedings of the National Academy of Sciences 97, no. 8 (2000): 4398-403 - Maguire, E.A., ER. Valentine, J.M. Wilding, and N. Kapur. “Routes to Remembering: The Brains Behind Superior Memory." Nature Neurosciencee, no. 1 (2003): 90-95 - Rocke, AJ. Image and Reality Chicago, IL: University of Chicago Press, 2010 - Solomon, Ines. “Analogical Transfer and 'Functional Fixedness' in the Science Classroom." Journal of Educational Researd 87, no. 6 (1994): 371-77
taejo 10y ago
I think that is a very "mathematician" thing to say; my own experience of doing mathematics is largely in accord. A little experience of teaching mathematics shows that this doesn't work for most people.
- ankurdhama 10y ago
  IMHO the best way to teach any concept is to start with a problem or question that can eventually be solved using the concept. Start with the question in general terms but don't provide the answer, let people think about it, make it engaging conversation and slowly and steadily build the concept that leads to the solution.

Terrible advice! I tell all of my students to find as many analogies as they can. You can never lose by increasing the number of ways in which you understand something. Analogies, some may argue, are at the heart of mathematics. > Mathematics is the art of giving the same name to different things Poincare. > The art of doing mathematics consists in finding that special case which contains all the germs of generality Hilbert. > The vast majority of us imagine ourselves as like literature people or math people. But the truth is that the massive processor known as the human brain is neither a literature organ or a math organ. It is both and more. John Green. > Sometimes I think that creativity is a matter of seeing, or stumbling over, unobvious similarities between things - like composing a fresh metaphor, but on a more complex scale. David Mitchell. There is an ounce of truth in what you say -- metaphors can be abused to draw false conclusions. This does not mean one should cower from using them.

ankurdhama 10y ago
Everyone has their own philosophy of mathematics. To me, the core idea is the idea of representations, such that those representations allow you to extract patterns (aka similarity or commonality) from various cases.
- nicklaf 10y ago
  I can't help but indulge a bit and now ask whether or not we can't extract a common pattern from the two representations of mathematical cognition offered by yourself and GP. :-)
  - ankurdhama 10y ago
    Not here but may be on "meta hacker news".
- fmoralesc 10y ago
  Aren't metaphors and analogies representations that allow us to extract patterns?
  - ankurdhama 10y ago
    To me, metaphors and analogies are about trying to learn some new concept(s) in terms of already known and familiar concepts.
    - nicklaf 10y ago
      As an opposing view, I offer you the following analogy, by mathematician Charles C. Pugh, which he draws between conceptions of metaphor as understood in natural versus mathematical language: From "Real Mathematical Analysis" 1st edition, p. 9: Metaphor and Analogy In high school English, you are taught that a metaphor is a figure of speech in which one idea or word is substituted for another to suggest a likeness or similarity. This can occur very simply as in "The ship plows the sea." Or it can be less direct, as in "his lawyers dropped the ball." What gives a metaphor its power and pleasure are the secondary suggestions of similarity. Not only did the lawyers make a mistake, but it was their own fault, and, like an athlete who has dropped a ball, they could not follow through with their next legal action. A secondary implication is that their enterprise was just a game. Often a metaphor associates something abstract to something concrete, as "Life is a journey." The preservation of inference from the concrete to the abstract in this metaphor suggests that like a journey, life has a beginning and an end, it progresses in one direction, it may have stops and detours, ups and downs, etc. The beauty of a metaphor is that hidden in a simple sentence like "Life is a journey" lurk a great many parallels, waiting to be uncovered by the thoughtful mind. Metaphorical thinking pervades mathematics to a remarkable degree. It is often reflected in the language mathematics choose to define new concepts. In his construction of the system of real numbers, Dedekind could have referred to A|B as a "type-two, order preserving equivalence class", or worse, whereas "cut" is the right metaphor. It corresponds closely to one's physical intuition about the real line. See Figure 3. In his book, Where Mathematics Comes From, George Lakoff gives a comprehensive view of metaphor in mathematics. An analogy is a shallow form of metaphor. It just asserts that two things are similar. Although simple, analogies can be a great help in accepting abstract concepts. When you travel from home to school, at first you are closer to home, and then you are closer to school. Somewhere there is a halfway stage in your journey. You know this, long before you study mathematics. So when a curve connects two points in a metric space (Chapter 2), you should expect that as a point "travels along the curve," somewhere it will be equidistant between the curve's endpoints. Reasoning by analogy is also referred to as "intuitive reasoning." Moral: Try to translate what you know of the real world to guess what is true in mathematics.

tchaffee 10y ago

The latest research on how we learn and how the brain works places great importance on analogy as a learning tool.

ankurdhama 10y ago
Any references to such research?
- tchaffee 10y ago
  - Baddeley, Alan, Michael W. Eysenck, and Michael C. Anderson. Memory. NY: Psychology Press, 2009. - Cat, Jordi. "On Understanding: Maxwell on the Methods of Illustration and Scientiﬁc Metaphor" Studies In History and Philosophy of Science Part B32, no. 3 (2001): 395-441 - Derman, Emanuel. Models of Behaving Badly. New York, NY: Free Press, 2011 - Foer, J. Moonwalking with Einstein. NY: Penguin, 2011 - Lutzen, Jesper. Mechanistic Images in Geometric Form. NY: Oxford University Press, 2005 - Maguire, E.A., D.G. Gadian, LS. Johnsrude, C.D. Good, J. Ashburner, R.SJ. Frackowiak, and CD. Frith. “Navigatioanelated Structural Change in the Hippocampi of Taxi Drivers.“ Proceedings of the National Academy of Sciences 97, no. 8 (2000): 4398-403 - Maguire, E.A., ER. Valentine, J.M. Wilding, and N. Kapur. “Routes to Remembering: The Brains Behind Superior Memory." Nature Neurosciencee, no. 1 (2003): 90-95 - Rocke, AJ. Image and Reality Chicago, IL: University of Chicago Press, 2010 - Solomon, Ines. “Analogical Transfer and 'Functional Fixedness' in the Science Classroom." Journal of Educational Researd 87, no. 6 (1994): 371-77

taejo 10y ago

I think that is a very "mathematician" thing to say; my own experience of doing mathematics is largely in accord. A little experience of teaching mathematics shows that this doesn't work for most people.

ankurdhama 10y ago
IMHO the best way to teach any concept is to start with a problem or question that can eventually be solved using the concept. Start with the question in general terms but don't provide the answer, let people think about it, make it engaging conversation and slowly and steadily build the concept that leads to the solution.