Algebra & Math of Change and Variation
From Kaput Center Wiki
Participants: Teresa, John, Corey, Walter, SH, Sara & Luis
Q: opportunities
Q: obligations
Q: cautions
[edit] Discussion that occurred in the group
SH: where is algebra and MCV?
John: What is math.? (in the upper corner of the drawing) Two very distinct pieces:
- first 2 years: serves social sciences or science and for your "in" into math; explain how things are done, how you solve a problem; application and algorithm driven
- second two years: proof and reasoning
SH: axiomatic proof is irrelevant for a lot of math majors at university
John: look at what mathematicians do, we get a different picture from what is taught
"Math is integrated" and coherent
Walter: the Bob Moses argument; this claim of college readiness; focus on doing "stuff" earlier Math of Change is thought of as an end of high school course - we want to push it earlier, i.e. early algebra
NCTM: strands
- first part of strand: shouldn't think of the courses as a layer cake - should be more like strands of learning; they emerge during the course of your education
- second part of strands: insist kids view mathematics as reasonable; students should feel ownership; can make sense of it and justify
Both things would scaffold. See the mathematics as deeply integrated
SH: what has happened: Q: ask elementary school teacher, what is proof? A: reasoning
Mathematicians argument: paper, pencil, written rigorous proof
SH and Luis: what does a proof look like in a new medium? In math dept: Computational proof different from proof in topology
Walter: ask students to participate in proof a lot earlier
John: mistake to suppose that a student in first year at university will be ahead Walter: you are saying university is behind John: yes
Walter: In TX reform - move the first two years of university math earlier; pushing the layer cake back
Theresa: use of technology to promote algebraic reasoning: ask two students if two mathematical statements are equal; can try many numbers in spreadsheet, looks like it works - is that acceptable as an argument for proof? At university level, what is acceptable?
John: in first level: question would not be asked, in second level: would be unacceptable
Walter: there is a way of making a continuity In Theresa's example: NCTM standards trying to argue that this will prompt the question of: is this now always true? For an example: Judah's example of isosceles triangles; invites meta-cognition that mathematicians should accept
John: these are not the same as deductive reasoning
SH: mathematicians can have a conversation with mathematics; co-action idea: principle that is conceptually embodied in mathematicians mind; that is why they are a mathematician Role of technology - isn't to make country more productive, what are the qualities of mathematics that make us fall in love with mathematics, reasoning, inviting, generating structure; not a prosthetic device; way to develop proceptual thinking to categorize the world around them
SH: What do we want our children to be able to do? Not what do we want them to learn. Mathematicians: conjecture needs to be formalized; formalisims not a part of the world for a 5-year old
Walter: Peirce's semiotics - a mathematician works dynamically with symbols; mistake to think that static symbols work in mathematicians mind statically. All thought is in signs (Peirce) Mistake: infer the role of a static inscription with sign and symbol
SH: T. Deacon (work with animals) dynamic occurs when we see the flexibility between icon, index, symbol
SH: Euler wrote a large amount of work after he went blind. He dictated his papers. Isn't about static symbols on a piece of paper. Mathematics as a social practice rather than only a notational practice
Luis: method and norm induction and deduction
dynamic environment - border between methodology and deductivity is blurred; you have new ways of exploring; new world of plasticity that is inaccessible when working in a static medium; new way of exploring leads to new way of producing knowledge; co-action mentioned again.
Walter questions if Luis is disagreeing with what he said.
Luis: no. This co-action is producing this new environment with new mathematical phenomena emerging that had been hidden in a static medium; experiencing mathematics in a different way
Walter: instead of saying "new," your relationship with signs and symbols _________
Walter: kids feel kinship with mathematicians - and this is important SH: what about kinship with mathematics educator?
Walter: Peirce - abduction between deduction and induction
SH: Newton's Principia was dynamic; Focus on understanding, learning will follow
Walter: we can show we know something by recapitulating it (Vygotsky)
- We can't push powerful (loved) mathematical topics, modes, ideas to later int he curriculum (No Gates)
- Includes the piecing together styles of exploration via cases, technology mediated investigation, "abductive" insight, etc.
Walter
- There are natural affinities between mathematicians and reform educators that SHOULD BE THE BASIS FOR A RESPONSE TO COLLEGE READINESS.
