Using Maplets for Teaching Calculus & Precalculus

SCCMT Fall Conference October 25, 2013 Ray Patenaude, South Pointe High School, Rock Hill, SC Doug Meade, University of South Carolina

Session Outline

1. Intro & Examples of Maplets for Calculus 2. Current Uses of Maplets 3. Research using Maplets with Continuity 4. Implications/Advice for using Applets 5. Summary and Q&A

What are they?

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

Example – Reflections of Functions

Example – The Chain Rule

Example – Max/Min Application

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

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

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

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.

Collecting Data

*Image used with permission of student and guardian.

Collecting Data

Questions

1. What understanding did this student demonstrate?

2. What features of this Maplet did the student use to gain this understanding?

Continuity using a Piecewise Function

Second Maplet used in research study.

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



Describe continuity ‘naturally’ using points and endpoints visually or embodied on a graph.

“It’s continuous because from the left, right, and the value of f all meet at this point on the graph” Left/right continuity described by open/closed points on graph “From the left and from the right, the graph goes to two different points.”

blending embodiment & symbolism

Able to describe continuity using limits, graph, and/or both

“There’s no holes or jumps in the graph” “It’s continuous because the values of the left and right limits and the value of f(x) are all the same” “It’s not continuous from the left because the limit does not equal the value of f” “The limit from the left and from the right have the same value”

proceptual-symbolic conceptual-embodied

formal objects based on definitions definitions based on known objects

Primarily discuss continuity in terms of limit definition and function values.

Students’ level of understanding of continuity within David Tall’s Three Worlds model.

Lessons Learned from M4C Research

Ten “take-aways” that can be applied to using applets in teaching mathematics in the classroom.

Lessons Learned from M4C Research

1. 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.

Lessons Learned from M4C Research

2. Maplets kept students engaged - CCSSM practice calling for persistence in problem solving

Lessons Learned from M4C Research

3. ‘Overt’ features students found/reported most helpful: - Check Answer - Hints - Change Answers - Graphs

Lessons Learned from M4C Research

4. ‘Subtle’ features also contributed to understanding: - Layout/organization of problems - Directions - Variety of problems

Lessons Learned from M4C Research

5. Some features were not useful to students:

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

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

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.

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/