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Mathematical Notation, Culture, Computation, and Future Directions

Mathematical Notation, Culture, Computation, and Future Directions. Some notational systems:. Some things are easier to use than others. XIV + VIII ??. 14 + 8 22. The utility of symbols.

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Mathematical Notation, Culture, Computation, and Future Directions

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  1. Mathematical Notation, Culture, Computation, and Future Directions

  2. Some notational systems:

  3. Some things are easier to use than others XIV + VIII ?? 14 + 8 22

  4. The utility of symbols “... a sixfold number increased by twelve, which is divided by the difference by which the square of the number exceeds three.” (From http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Diophantus.html)

  5. The utility of symbols “... a sixfold number increased by twelve, which is divided by the difference by which the square of the number exceeds three.” or (12 + 6n)/(n2-3) (From http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Diophantus.html)

  6. Algebra Culture matters: "Clearly, al-Khwarizmi conceived algebra as a way of solving pressing practical problems of the Islamic Empire" 4x+7=24 Notation matters: “Computational skill became encoded in syntactically defined rules on a symbol system” (Kaput, Noss, and Hoyles 2001)

  7. Calculus Newton Leibniz Notation was very important: one notational system won out over the other because it was easier to use and to teach. Calculus made new types of work possible.

  8. A small example ’ P N (w)w Q D A A Q N ’ ’ ’ w N ’ P o

  9. A small example • Task: • Please translate the following using the rules shown: wNP’ NpAD (Not a real problem)

  10. A small example N’ PP NP (N) N Q Det AP AP Q NP N’ N’ N’ N N N’ PP Pro

  11. *Kaplan, J. P. (1994). English Grammar: Principles and Facts. New Jersey: Prentice-Hall. A small example* N’ PP NP (N) N Q Det AP AP Q NP N’ N’ N’ N N N’ PP Pro NP=noun phrase, N=noun, Pro=pronoun, PP=prepositional phrase, AP=adjective phrase, etc.

  12. A small example From http://www.ifi.unizh.ch/groups/CL/volk/SyntaxVorl/Chomsky.html

  13. Computation • Do computers make mathematics knowledge unnecessary? • Do they obscure access/understanding of mathematics? • Do they shelter students from computational drudgery and allow access to higher level mathematics?

  14. Hope for the future… “New representational infrastructures become possible that enable the reconstitution of previously constructed knowledge through, for example, the new types of visually editable graphs and immediate connections between functions and simulations and/or physical data of the type developed and studied in the SimCalc Project.” (Kaput & Schorr, in press)

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