Student understanding of area and volume in Selected calculus concepts

1. Student understanding of area and volume in Selected calculus concepts Thesis research Fall 2009 to present

2. Student understanding of volume of revolution Fall 2009

3. Fall 2009 Research • Research questions: • How do students think about a volume of revolution problem as they are working on it? • How does the thinking of students who understand volume of revolution differ from the thinking of students who don’t? • Participants: four fall 2009 MAT 127 students • Methodology: audio-recorded task-based interview

4. To refresh your memory:

5. Emergent Categories • Correct answer; visualized a slice but not final solid • Correct answer; visualized a slice and final solid • Correct answer; no visualization • Incorrect answer; visualized a slice but not final solid • Incorrect answer; visualized a slice and final solid • Due to the small number of subjects, the researcher thinks it possible that a study with a greater sample size might yield a few more categories, those categories being various permutations of the above.

6. Summary of Coded Data

7. Findings • Students visualize a slice and/or the final solid while working on a volume of revolution problem • All students visualized a slice • Over a total of eight tasks, students visualized the solid in four cases • Cognitive variability (Siegler, 2003): using multiple thinking strategies when solving problems of the same type (p. 293) • Some students visualized the solid in one problem but not the other

8. Continued • Students generally were able to express volume as a sum of an infinite number of infinitessimally thin slices, but struggled with finding the correct radii to use in integration. • The apparent difference between students who answered the tasks correctly and students who did not was the visualization of a slice, used to find the radii, but not visualizing the final solid.

9. My findings made me wonder • We know that students have difficulties reading information from graphical representations; ie, what we [mathematicians] see in a graph is different from what they [students] see. Does this account for the trouble in finding the radius? • Apart from the issue of what to integrate and why, how do calculus students understand area and volume? Does this affect their understanding of calculus concepts using area and volume, such as Riemann sums and volume of revolution?

10. Calculus Students’ understanding of area and volume Current research (Fall 2010)

11. Discussion of Reading • I had two questions while reading this paper.One: Does it state somewhere in this paper what the definition of array is? I realize that this is out of a book of some sort so maybe the definition is somewhere else within the book? ARRAY – an order or an arrangement – like a Rubik’s cube • Two: One of the categories they observed during their research that students feel into was named Category D. Where "Students explicitly use the formula LxWxH with no indication that they understand it in terms of layers." Can one make the assumption that students don't understand them in terms of layers? My guess is that the researchers probed the students, asking questions to reveal if they saw L, W, H as layers or if they were just multiplying. Battista, M. and Clements, D. (1998). Students' understanding of three-dimensional cube arrays: findings from a research and curriculum development project.

12. Discussion of Reading • I think that introducing 3-d cubes in an early age is very important because it is a big part of the way of thinking... • I agree. I really liked the article and I also think it is important to introduce 3-D to younger kids. I think there are a lot of things that could be introduced in lower grade levels. • Also, I can't help but think that spatial visualization issues that never get addressed in lower levels must come back to haunt students later on in organic chemistry where it's all about considering 2-D structures on a page and making them 3-D mentally and manipulating them. . . Battista, M. and Clements, D. (1998). Students' understanding of three-dimensional cube arrays: findings from a research and curriculum development project.

13. Discussion of Reading • I enjoyed reading this article and I am looking forward to the class discussion to learn more about what pieces of this article you took to bridge the gap from its focus on elementary content to calculus content. What are the requirements to be able to generalize claims from differing levels of content? My reply: There is not much research (any?) about CALC student understanding of area and volume! What a perfect idea for a study! Battista, M. and Clements, D. (1998). Students' understanding of three-dimensional cube arrays: findings from a research and curriculum development project.

14. Rationale • Anecdotal evidence from University of Maine calculus professors suggests that calculus students do not understand area and volume. • Example: the volume of a classroom • Perhaps if students do not understand the concept of area and the concept of volume, they do not understand the applications of integration that involve area and volume

15. Mini Lit Review • In order to understand area and volume, students must understand and be able to represent the structure of space and the tools to measure it in some way (Example from my reading) • Students often use length measures for measurement of other spatial content (area, volume, angle) • Students do not tend to understand the units associated with various spatial measures • Students need to understand making subdivisions; that those subdivisions are equal units; and if they are, a count represents the measure Lehrer, R. (2003). Developing understanding of measurement. In J. Kilpatrick, W. G. Martin, & D. E. Schifter (Eds.), A Research Companion to Principles and Standards for School Mathematics. Reston, VA: National Council of Teachers of Mathematics.

16. Mini Lit Review: Area Measure • Students treat length as a space-filling attribute • Students will mix units freely (ie: give them squares, triangles, circles and ask them to find the area of a large rectangle – they will use all 3) • Students will fill a shape’s area with the units that most resemble the shape. (Not shown: grid) Lehrer, R. (2003). Developing understanding of measurement. In J. Kilpatrick, W. G. Martin, & D. E. Schifter (Eds.), A Research Companion to Principles and Standards for School Mathematics. Reston, VA: National Council of Teachers of Mathematics.

17. Mini Lit Review: Area • Resemblance: students want the unit to look like/be a similar shape to the form • To understand area, students must be able to restructure the plane • Students have trouble seeing the area of a form to be a composition of other forms • Students see square units as things to be counted, not as subdivisions of a plane Lehrer, R. (2003). Developing understanding of measurement. In J. Kilpatrick, W. G. Martin, & D. E. Schifter (Eds.), A Research Companion to Principles and Standards for School Mathematics. Reston, VA: National Council of Teachers of Mathematics.

18. Mini Lit Review: Volume • Students now have to be able to coordinate units of measure in three dimensions • Students try to: count faces or subdivisions of faces; count subdivision of layers and multiply by layers; multiply area*height or area*area Lehrer, R. (2003). Developing understanding of measurement. In J. Kilpatrick, W. G. Martin, & D. E. Schifter (Eds.), A Research Companion to Principles and Standards for School Mathematics. Reston, VA: National Council of Teachers of Mathematics.

19. Researchable questions • What do calculus students understand about area? • What do calculus students understand about volume? • How do calculus students explain the idea of a Riemann sum? • How do calculus students explain the idea of finding the volume of a solid by using a Volume of Revolution?

20. Methodology • Give written tasks to a couple (2-4) calc classes • Students can give permission to be contacted for interviews • Analyze written tasks; make categories • Contact 1-2 students from each category for interviews • Interviews will be over the written task + a couple other problems (array of cubes; Riemann sum; VoR)

21. Try my tasks!

22. Find the area of the rectangle. 12 cm Find the area of the circle. 4 cm 5 in

23. What does area mean? What type of things can we use area to measure? What unit is area expressed in?

24. Find the volume of the box. Find the volume of the cylinder. 8 in 3 in

25. What does volume mean? What type of things can we use volume to measure? What unit is volume expressed in?

26. What does this computation tell you?

27. I would like to use my time to talk about my tasks and my researchable questions. • What do calculus students understand about area? • What do calculus students understand about volume? • How do calculus students explain the idea of a Riemann sum? • How do calculus students explain the idea of finding the volume of a solid by using a Volume of Revolution?