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Old May 6th, 2010, 05:23 AM   #1
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Twinkies & Chips—Sound Bites & Bumper Stickers

Twinkies & Chips—Sound Bites & Bumper Stickers



A steady diet of Twinkies and chips will give us a fat gut while a steady diet of sound bites and bumper stickers will give us a fat head.



Political parties and television commercials often depend upon well-crafted sound bits and bumper stickers to motivate a naive population while the opposition tries to use rational argumentation; one need not be a rocket scientist to recognize which side will generally win this contest.



Early in our institutional education system we learn arithmetic. We learn to add, subtract, multiply, and divide. We learn to calculate without understanding.



This mode of education follows us throughout our formal education system. We learn to develop answers devoid of understanding. We do this because, in a society focused upon maximizing production and consumption, most citizens need only sufficient education to perform mechanical type operations; that is perhaps why our electronic gadgets fit so well within our culture.



If we think about this situation we might well say that this form of education best serves our needs. It is efficient and quick. However, beyond the process of maximizing production and consumption we are ill prepared to deal with many of life’s problems because we have learned only how to develop answers that are “algorithmically friendly”.



In grade school we are taught to manipulate numerals (symbols) not numbers (concepts). We are taught in grade school not ideas about numbers but automatic algorithmic processes that give consistent and stable results when dealing with symbols. With such capability we do not learn meaningful content about the nature of numbers but we do get results useful for a culture of production and consumption.



We have a common metaphor Numbers are Things in the World, which has deep consequences.
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Old May 6th, 2010, 07:09 PM   #2
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I saw this on tv a couple days ago and thought it was a pretty good concept for math Maya Mathematical System - Maya World Studies Center
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Old May 7th, 2010, 01:06 AM   #3
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Quote:
Originally Posted by coberst
Early in our institutional education system we learn arithmetic. We learn to add, subtract, multiply, and divide. We learn to calculate without understanding


Not true. We don't learn to calculate until we first learn to understand. Perhaps it's been too long for you to remember your primary education.



We don't really learn our math facts (in our head addition and subtraction) until we first learn viscerally that when you have 7 apples and take away 3, you're left with 4, and that if you have 4 apples and someone gives you 3, you now have 7.



We don't really know how to multiply or divide until we understand completely that 3 groups of 7 is equal to 21, and that from 21 apples we can make 3 groups of 7 or 7 groups of 3.



Subtraction is just a variation on addition, and multiplication and division are just variations on addition and subtraction. All other advanced math is just more complicated versions of.... addition and subtraction, which are two sides of the same coin.



If you missed that key concept in your education, you missed out on how nearly everything... physics, philosophy, language, music and art... are based on mathematics... which is based on the difference between 0 and 1....
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Old May 7th, 2010, 03:58 AM   #4
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Quote:
Originally Posted by Zack
I saw this on tv a couple days ago and thought it was a pretty good concept for math Maya Mathematical System - Maya World Studies Center


You might find this to be interesting:



Arithmetic is Object Collection



It is a hypothesis of SGCS (Second Generation Cognitive Science) that the sensorimotor activity of collecting objects by a child constitute a conceptual metaphor at the neural level leading to a primary metaphor that ‘arithmetic is object collection’. The arithmetic teacher attempting to teach the child at a later time depends upon this already accumulated knowledge. Of course, all of this is known to the child without the symbolization or the conscious awareness of the child.



The pile of objects became ‘bigger’ when the child added more objects and became ‘smaller’ when objects were removed. The child easily recognizes while being taught arithmetic that 5 is bigger than 3 and 3 is littler than 7. The child knows many entailments, many ‘truths’, resulting from playing with objects. The teacher has little difficulty convincing the child that two collections A and B are increased when another collection C is added, or that if A is bigger than B then A+C is bigger than B+C.



At birth an infant has a minimal innate arithmetic ability. This ability to add and subtract small numbers is called subitizing. (I am speaking of a cardinal number—a number that specifies how many objects there are in a collection, don’t confuse this with numeral—a symbol). Many animals display this subitizing ability.



In addition to subitizing the child, while playing with objects, develops other cognitive capacities such as grouping, ordering, pairing, memory, exhaustion-detection, cardinal-number assignment, and independent order.




Subitizing ability is limited to quantities 1 to 4. As a child grows s/he learns to count beyond 4 objects. This capacity is dependent upon 1) Combinatorial-grouping—a cognitive mechanism that allows you to put together perceived or imagined groups to form larger groups. 2) Symbolizing capacity—capacity to associate physical symbols or words with numbers (quantities).



“Metaphorizing capacity: You need to be able to conceptualize cardinal numbers and arithmetic operations in terms of your experience of various kinds—experiences with groups of objects, with the part-whole structure of objects, with distances, with movement and location, and so on.”



“Conceptual-blending capacity. You need to be able to form correspondences across conceptual domains (e.g., combining subitizing with counting) and put together different conceptual metaphors to form complex metaphors.”



Primary metaphors function somewhat like atoms that can be joined into molecules and these into a compound neural network. On the back cover of “Where Mathematics Comes From” is written “In this acclaimed study of cognitive science of mathematical ideas, renowned linguist George Lakoff pairs with psychologist Rafael Nunez to offer a new understanding of how we conceive and understand mathematical concepts.”



“Abstract ideas, for the most part, arise via conceptual metaphor—a cognitive mechanism that derives abstract thinking from the way we function in the everyday physical world. Conceptual metaphor plays a central and defining role in the formation of mathematical ideas within the cognitive unconscious—from arithmetic and algebra to sets and logic to infinity in all of its forms. The brains mathematics is mathematics, the only mathematics we know or can know.”



We are acculturated to recognize that a useful life is a life with purpose. The complex metaphor ‘A Purposeful Life Is a Journey’ is constructed from primary metaphors: ‘purpose is destination’ and ‘action is motion’; and a cultural belief that ‘people should have a purpose’.



A Purposeful Life Is A Journey Metaphor

A purposeful life is a journey.

A person living a life is a traveler.

Life goals are destinations

A life plan is an itinerary.



This metaphor has strong influence on how we conduct our lives. This influence arises from the complex metaphor’s entailments: A journey, with its accompanying complications, requires planning, and the necessary means.



Primary metaphors ‘ground’ concepts to sensorimotor experience. Is this grounding lost in a complex metaphor? ‘Not by the hair of your chiney-chin-chin’. Complex metaphors are composed of primary metaphors and the whole is grounded by its parts. “The grounding of A Purposeful Life Is A Journey is given by individual groundings of each component primary metaphor.”




The ideas for this post come from Philosophy in the Flesh. The quotes are from Where Mathematics Comes From by Lakoff and Nunez
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