1 / 10

Teaching Mathematical Reasoning Across the Curriculum Discrete Math

Teaching Mathematical Reasoning Across the Curriculum Discrete Math. Joseph E. Hollingsworth Indiana University Southeast Computer Science Department jholly@ius.edu This research is funded in part by NSF Grant DUE-1022191 . Reasoning Across the Curriculum. Question

ophira
Télécharger la présentation

Teaching Mathematical Reasoning Across the Curriculum Discrete Math

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Teaching Mathematical Reasoning Across the CurriculumDiscrete Math Joseph E. Hollingsworth Indiana University Southeast Computer Science Department jholly@ius.edu This research is funded in part by NSF Grant DUE-1022191

  2. Reasoning Across the Curriculum Question • Can we explicitly connect discrete math topics with reasoning about software correctness?

  3. Examining CS2013 Strawman Draft • KA:Discrete Structures (DS/Proof Techniques) • Core-Tier1 Topic:Induction over natural numbers • Learning Outcomes: • Outline the basic structure of each proof technique described in this unit. [Application] • Apply each of the proof techniques correctly in the construction of a sound argument. [Application] • Level of Mastery:Knowledge, Application, Mastery

  4. Unsatisfying for CS Undergrad Core-Tier1 Topic: Induction over natural numbers Prove using induction:

  5. 1st Some Comments • I am not the first to say what appears next. • What appears next can be done. • It’s not impossible. • Is meant to be a CS2013 “Knowledge” topic in a discrete math class.

  6. Goal: Connect Induction with Reasoning About Software Correctness • Example: Compute the sum of two integers (not the summation from above) intsum(int j, int k) // requires j >= 0 // ensures result = j + k { if (j == 0) { return k; } else { j--; int r = sum(j, k); return r + 1; } }

  7. Base Case intsum(int j, int k) // requires j >= 0 // ensures result = j + k { if (j == 0) { return k; // Assume: (j = 0) ^ (result = k) // Confirm ensures: result = 0 + k } else { ... } }

  8. Inductive Assumption int sum(int j, int k) // requires j >= 0 // ensures result = j + k { if (j == 0) { ... } else { j--; int r = sum(j, k); return r + 1; // Assume: r = (j – 1) + k } }

  9. Induction Step int sum(int j, int k) // requires j >= 0 // ensures result = j + k { if (j == 0) { ... } else { j--; int r = sum(j, k); return r + 1; // Assume: (r = (j – 1) + k) ^ (result = r + 1) // Confirm ensures: result = j + k } }

  10. Final Comments Question: • What about connecting induction to reasoning about correctness of software at the CC2013 “Application” level? Answer: • The example above was at the “Knowledge” level, i.e., where we show students the connection. • In later courses, e.g., data structures or analysis of algorithms, move to the “Application” level for induction on software. Reference: • Long, T.J., Weide, B.W., Bucci, P., and Sitaraman, M.,"Client View First: An Exodus From Implementation-Biased Teaching," Proceedings 30th SIGCSE Technical Symposium on Computer Science Education, ACM, March 1999, 136-140.

More Related