Using Maplets for Teaching Calculus & Precalculus SCCMT Fall Conference October 25, 2013

1. Using Maplets for Teaching Calculus & PrecalculusSCCMT Fall ConferenceOctober 25, 2013 Ray Patenaude, South Pointe High School, Rock Hill, SC Doug Meade, University of South Carolina

2. Session Outline • Intro & Examples of Maplets for Calculus • Current Uses of Maplets • Research using Maplets with Continuity • Implications/Advice for using Applets • Summary and Q&A

3. What are they? • Set of over 200 applets designed using Maple software • Provide examples and exercises • Cover topics from Precalculus, Calc. I, II, & III

4. Example – Reflections of Functions

5. Example – The Chain Rule

6. Example – Max/Min Application

7. Current Instructor Uses of M4C • Demonstration • visualize concepts during lecture • Provide Examples for Instruction • Similar to Chain Rule above • Homework Exercises • Assign number of exercises to be completed

8. Current Student Uses of M4C • Lab Exercises • Instructor assign Maplets to be completed prior to, or after lecture • Practice • Many Maplets make student complete one step prior to another • Hints and check feature allow students opportunity to correct

9. Use of Applets for Developing Understanding in Mathematics: A Case Study Using Maplets for Calculus with Continuity Concepts

10. Objectives Determine the features of Maplets for Calculus that promoted understanding of continuity concepts. And, Determine student actions, and strategies developed, while using M4C that promoted understanding of continuity concepts.

11. Collecting Data *Image used with permission of student and guardian.

12. Collecting Data

13. Questions • What understanding did this student demonstrate? • What features of this Maplet did the student use to gain this understanding?

14. Continuity using a Piecewise Function Second Maplet used in research study.

15. axiomatic-formal Basic description of ε-δ to describe discontinuity on graph exercise: “If you draw these epsilon lines here and here, that other point [open] isn’t in there” Picture approximates student drawing  formal objects based on definitions definitions based on known objects “It’s continuous because from the left, right, and the value of f all meet at this point on the graph” “It’s continuous because the values of the left and right limits and the value of f(x) are all the same” blending embodiment & symbolism Left/right continuity described by open/closed points on graph “It’s not continuous from the left because the limit does not equal the value of f” Primarily discuss continuity in terms of limit definition and function values. Describe continuity ‘naturally’ using points and endpoints visually or embodied on a graph. Able to describe continuity using limits, graph, and/or both “From the left and from the right, the graph goes to two different points.” “The limit from the left and from the right have the same value” “There’s no holes or jumps in the graph” proceptual-symbolic conceptual-embodied Students’ level of understanding of continuity within David Tall’s Three Worlds model.

16. Lessons Learned from M4C Research Ten “take-aways” that can be applied to using applets in teaching mathematics in the classroom.

17. Lessons Learned from M4C Research • Teaching/instructing is still necessary. - M4C and most applets are supplemental. - Some students need guidance on use of applets to get the most out of them. - Model the use of applet.

18. Lessons Learned from M4C Research 2. Maplets kept students engaged • CCSSM practice calling for persistence in problem solving

19. Lessons Learned from M4C Research 3. ‘Overt’ features students found/reported most helpful: - Check Answer - Change Answers - Hints - Graphs

20. Lessons Learned from M4C Research 4. ‘Subtle’ features also contributed to understanding: - Layout/organization of problems - Directions - Variety of problems

21. Lessons Learned from M4C Research 5. Some features were not useful to students:

22. Lessons Learned from M4C Research 6. Field test applets/Maplets with students before using with entire class • Students will find trouble spots with applets • Help with your decision to use, modify, or omit from instruction

23. Lessons Learned from M4C Research 7. Students enjoyed working with Maplets for Calculus applets - expressed desire to do these exercises as opposed to textbook exercises

24. Lessons Learned from M4C Research 8. Use ‘Thinking Aloud’ while tutoring students. 9. Especially when starting, allow students to work in pairs. 10. Start small.

25. References Meade, D. B. & Yasskin, P. B. (2008, December). Maplets for calculus: Tutoring without the tutor. Paper presented at the Asian Conference on Technology in Mathematics, Bangkok, Thailand. Retrieved from http://maple.math.sc.edu/maplenet/M4Cfree/pages/publications.html Núñez, R. E., Edwards, L. D., & Matos, J. F. (1999). Embodied cognition as grounding for situatedness and context in mathematics education. Educational Studies in Mathematics, 39, 45-65. Patenaude, R.E. (2013). The use of applets for developing understanding in mathematics: A case study using Maplets for Calculus with continuity concepts. (Doctoral dissertation, University of South Carolina). Tall, D. (2008). The transition to formal thinking in mathematics. Mathematics Education Research Journal, 20(2),5-24. Maplets for Calculus: http://m4c.math.tamu.edu/