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HCC class lecture 2 items

HCC class lecture 2 items. John Canny 1/24/05. Administrivia. The class swiki is up. Office hours this week: 3-4 pm Tuesday 4-5 pm Wednesday. Kinds of knowledge. Things you might overhear: That was a beautiful proof… It’s a very elegant result… Mathematics is an art…

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HCC class lecture 2 items

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  1. HCC classlecture 2 items John Canny1/24/05

  2. Administrivia The class swiki is up. Office hours this week: 3-4 pm Tuesday 4-5 pm Wednesday

  3. Kinds of knowledge Things you might overhear: • That was a beautiful proof… • It’s a very elegant result… • Mathematics is an art… But these adjectives are out of place. Surely the only sensible ones are e.g. correct, complete, short, long,… Why do we use the first kind of adjective?

  4. Kinds of knowledge The readings contrast various kinds of knowledge – knowledge of facts and figures vs. “knowledge with understanding” Knowledge with understanding included: Conditionalization (contextualization) Relating to existing knowledge Applying in multiple contexts Thinking ahead (meta-cognition) about how this concept might be useful in future.

  5. Kinds of knowledge The readings also underscore that learning doesn’t stop once we “have” the concept. What happens afterwards? We gradually “automate” the concept – make it pre-attentive so we can “fluidly” use it. Structures fuse into “chunks” that are immediately perceived. We prioritize the new concept relative to others. Once we know how and when to use it, we place it in our bag of “big ideas” with a priority.

  6. Discussion point 1 Contrast knowledge of mathematics with the skill to do mathematics. Mathematics (predicate logic) allows meta-mathematical knowledge to be represented. i.e. Allowable steps in a proof can be represented mathematically. Is math a good language for describing math skill?

  7. Microgenesis The readings only briefly mentioned “microgenesis” – the evolution of an individual concept during learning. But much of the readings cover various “steps” that happen when we acquire a concept with understanding.

  8. Discussion point 2 Pick a “concept” that many of you share (e.g. proof of the halting theorem, or understanding of compilation etc.) List some important steps that occur during micro-genesis. Give the time frame for those steps.

  9. Aside Pretty much everything you read is “abstract knowledge”. Your ability to understand it depends on your ability to connect it to existing knowledge, preferably first-hand. So don’t forget to use examples from personal experience in the discussion.

  10. Discussion groups An ideal group size is 4-5 people. As the group grows, its more likely to have margins. Remember you have to summarize your group’s discussion at the end. Elect a note-taker.

  11. Discussion points • Contrast knowledge of mathematics with the skill to do mathematics. Is math a good language for describing math skill? • List some important steps that occur during micro-genesis. Give the time frame for those steps.

  12. Reading for next time Several chapters from “The Psychology of the Child” by Piaget and Inhelder. Volunteer? Piaget’s classic description of “ontogenesis” in young children. Contrast this with the map of “micro-genesis” that you just built.

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