# «PROCEEDINGS OF THE 15TH ANNUAL CONFERENCE ON RESEARCH IN UNDERGRADUATE MATHEMATICS EDUCATION EDITORS STACY BROWN SEAN LARSEN KAREN MARRONGELLE ...»

2. What do instructors think of using ALEKS as the sole provider of course topics?

3. What are the benefits and downsides to online homework systems and traditional homework formats?

4. How could ALEKS be used in an activity-based, collaborative course where the focus is on conceptual understanding over procedural mastery?

**References:**

Falmagne, J-C., Cosyn, E., Doignon, J-P., and Thiery, N. (2000). The Assessment of Knowledge in Theory and in Practice. Retrieved from http://www.aleks.com/about_aleks/Science_Behind_ALEKS.pdf Hagerty, G. & Smith, S. (2005). Using the Web-Based Interactive Software ALEKS to Enhance College Algebra. Mathematics and Computer Education, 39(3), 183-194.

Hauk, S. & Segalla, A. (2005). Student perceptions of the web-based homework program WeBWorK in moderate enrollment college algebra courses. Journal of Computers in Mathematics and Science Teaching, 24(3), 229-253.

Hirsch, L. & Weibel, C. (2003). Statistical evidence that web-based homework helps. FOCUS, 23(2), 14.

15TH Annual Conference on Research in Undergraduate Mathematics Education 587 Oshima, C. (2010). Integrating of Learning Software into Introductory Level College Mathematics Courses. Retrieved from http://archives.math.utk.edu/ICTCM/VOL22/C008/paper.pdf Stillson, H. & Alsup, J. (2003). Smart ALEKS…or not? Teaching Basic Algebra Using an Online Interactive Learning System. Mathematics and Computer Education, 37(3), 329-336.

Taylor, J. (2008). The effects of a computerized-algebra program on mathematics achievement of college and university freshmen enrolled in a developmental mathematics course. Journal of College Reading and Learning, 39(1), 35–53.

Zerr, R. (2007). A quantitative and qualitative analysis of the effectiveness of online homework in first-semester calculus. Journal of Computers in Mathematics and Science Teaching, 26(1), 55-73.

This presentation will share initial observations and data collected from a pilot classroom study in which groups of multivariable calculus students made physical measurements and drew actual curves on real, tangible surfaces to construct geometric mathematical objects fundamental to the course and discover their properties. Students completed short group activities focusing on relationships between functions and level curves, properties of gradient vectors and directional derivatives, and solutions to optimization problems before other symbolic representations or procedures were discussed in lecture. Initial self-reported data suggests working with the surfaces helped students visualize functions. It also appears the activities helped students develop strong connections between the geometric, symbolic, and verbal representations of multivariable calculus concepts. Collected data suggests the surfaces helped uncover students’ single variable misconceptions which hindered their new understandings. The goal of this presentation is to receive feedback for the design of a rigorous phase two study of this project.

Key words: Multivariable Calculus Visualization, Classroom Research, Geometric Representation The basic ideas of multivariable calculus, those of level curves, gradient vectors, directional derivatives, and optimization problems with constraints, can be seen as direct generalizations of fundamental single variable calculus ideas through the use of geometry. The author is aware of various studies addressing the effective use of visualization technology to help students understand multivariable calculus concepts, but the implementation of technology in this setting produces two problems: Due to the necessity of projecting a three-dimensional shape onto a two-dimensional screen, students are unable to interact with the real object in a way that mimics the way they worked with graphs of one-dimensional functions. Secondly, although visualization technology is wonderful, it can restrict the ability of students to conduct selfdirected explorations of multivariable calculus ideas without additional technical help. The former issue is subtle, while the latter can prevent students from being allowed to explore and find answers to fundamental questions important for the full understanding of multivariable calculus concepts until such concepts like coordinate systems and multivariable functions have been defined.

In Class Activities In Fall 2009, the author of this paper used real, tangible surfaces and short mini activities to allow groups of students in a multivariable calculus course the opportunity to discover the important features and concepts related to multivariable functions. Pictures and descriptions of the surfaces are included in Figure 1. The first activity, occurring 10 minutes into the very first class, required students to identify the relationship between their surface and level curves. The second activity focused on understanding the dot product as a project. In the third activity, 15TH Annual Conference on Research in Undergraduate Mathematics Education 589 students constructed the gradient vector and discovered its geometric properties by measuring slope on the surface in two perpendicular directions. Students also discovered the relationship between directional derivatives and gradient vectors in the fourth activity, and they discovered the geometric relationship between a constraint and the gradient of a function for problems typically solved by the method of Lagrange multipliers.

Collected Data The author of this paper has collected data from the in-class activities and student exams, as well as self-reported data from students collected at the conclusion of the pilot study. For each activity, students self-reported the main point of the activity. In addition, they described (a) something that was still not clear and (b) something which they better understood as a result of doing the activity. Despite using the activities to introduce ideas before formal lecture or discussion in class, very few students reported having trouble understanding the point of the activities. Furthermore, most students were able to describe the geometric properties of gradient, directional derivatives, and level curves at the start of the lecture intended to introduce those properties. As these activities were designed to help students discover these concepts, the author of this paper is very interested in designing assessment activities which can investigate the level of understanding of these students on later exams or in later activities.

The author has collected anecdotal evidence suggesting students uncover misconceptions about single variable calculus ideas, like derivatives and functions, using the surfaces. The reliance upon the dy/dx notation for derivative is troublesome in a setting where y and x are now independent variables. In order to understand directional derivatives, one group physically changed the x and y coordinate system so that, instead of lying flat beneath the surface, the x direction lay tangent to the surface and the y direction was oriented perpendicular to the surface.

(See Figure 2.) This group held firm to the notion that a derivative was dy/dx, instead of a more general notion dg/dx for partial derivatives of the function g. Additional troubles occur when trying to generalize the notion of negative slope. Students are reluctant to recognize that a negative directional derivative indicates the surface function is decreasing in that direction. On a more positive note, most students are able to discover and explain the geometric relationship characterizing the solutions to optimization problems subject to a constraint, typically solved by the method of Lagrange Multipliers, after the 20 minute lab activity.

Student Self-Reported Feedback The surface activities and minilabs were used during the first five weeks of the course, after which students (n = 36) were asked to self-report on how the activities and surfaces influenced their learning. Students were asked questions about how working with the surfaces helped them visualize (10 questions) and understand (8 questions) various multivariable calculus concepts.

Students were allowed to indicate that the surfaces (A1) provided no help, (A2) provided a bit of help, (A3) provided some help, or (A4) provided a lot of help. Of the 36 respondents, 25 indicated that working with the surfaces really helped them visualize gradient vectors while only 2 said the surfaces helped a bit. No students said the surfaces provided no help. In regards to visualizing solutions to Lagrange multiplier problems, 35 of the 36 students indicated that the surfaces helped some (16) or a lot (19). The surfaces were least helpful for helping students visualize second order and mixed partial derivatives, with only 24 of the 36 students indicating the surfaces helped some or a lot. Additional results are listed in Table 3.

In term of understanding concepts, students also indicated that working with the surfaces helped them understand the relationship between gradient vectors and level curves. Overall, 590 15TH Annual Conference on Research in Undergraduate Mathematics Education 77% of students indicated that working with the surfaces and level curve boards helped them understand how to match level curves with surface features. 91% of the students said that working with the surfaces helped them understand slope in different directions on a surface, and 80% of students indicated that working with the surfaces provided some or a lot of help as they connected ideas of directional derivatives and slopes in various directions.

In general, students appreciated the design of the mini-labs and being able to explore the concepts using the functions. As one student said, working in groups with the mini-labs “was a good chance to bounce ideas off of each other and [get] us more involved in what was going on.

You weren’t just being told what to do. We had to figure it out on our own.” When asked “How did working with the surfaces help your ability to visualize and work with

**multivariable functions?", one student replied:**

“Being able to see all three dimensions at once rather than interpret the height from a 2D curve was really helpful. It allowed me to spend less time thinking about the vertical components of the graph and more time on solving and learning from the problem.” Many students indicated the best features of the surfaces were that they liked having a visual representation of what the graph looks like is very helpful. As one student said, “working with the surfaces helped with the learning of ideas in the course because it contained problems that are key ideas, and by doing the lab I better understand the ideas.” Students repeatedly commented on the value of being able to see how vectors compared to the level curves for a surface, and how being able to draw on the surfaces helped them.

Audience Questions

**One student summed up the effect of using the surfaces by saying:**

“I did not understand gradient until working with the surfaces.” The author of this paper would like to know how working with the surfaces actually changed the student’s ability to expand upon calculus ideas into multivariable calculus concepts.

The author of this paper is not a trained mathematics education researchers, and is unfamiliar with resources and studies focused upon the student understanding of multivariable calculus

**concepts. The author is looking for feedback specifically on:**

How to make sense of the data collected during this pilot study in regards to the connection between a student’s calculus and multivariable calculus understandings.

How to design assessment activities which investigate the actual level of student comprehension of these concepts as a result of the in-class activities.

How to design and implement a rigorous study as phase two of the project.

15TH Annual Conference on Research in Undergraduate Mathematics Education 591 Figure 1: Each surface is constructed of wood, with a dry-erase finish. Each of the six models have a base of 10”x10”, and stands roughly 4”-6” tall. Two dry-erase whiteboards, one engraved with a rectangular coordinate system and the other engraved with level curves, are associated with each surface.

Figure 2: Confusion about a derivative for the one-dimensional case (dy/dx) perhaps blocked the ability of one group from connecting the partial derivative (dg/dx) with the correct quantities dg, dy, and dx when working with the surface. Instead of measuring vertical rise, dg, this group used dy to measure the rise perpendicular to the surface and dx to measure the run parallel to the surface.

Evan Fuller1, Keith Weber2, Juan Pablo Mejia-Ramos2, Kristen Lew2, Philip Benjamin2 1- Montclair State University 2- Rutgers University In undergraduate mathematics courses, proofs are regularly employed to convey mathematics to students. However, research has shown that students find proofs to be difficult to comprehend.

Some mathematicians and mathematics educators attribute this confusion to the formal and linear style in which proofs are generally written. To address this difficulty, some researchers have suggested that students be exposed to generic proofs. We report preliminary results of a study that employs a recent model of proof comprehension to assess the extent to which reading a generic proof improves student understanding over reading a traditional proof.

Key words: students understanding of proof, generic proof, proof at the undergraduate level.

1. Introduction In advanced mathematics courses, proofs are a primary way that teachers and textbooks convey mathematics to students (e.g., Weber, 2004). However, researchers note that students find proofs to be confusing or pointless (e.g., Harel, 1998; Porteous, 1986; Rowland, 2001) and undergraduates cannot distinguish a valid proof from an invalid argument (Selden & Selden, 2003; Weber, 2010). Some mathematicians and mathematics educators attribute students’ difficulties in understanding proofs to the formal and linear style in which proofs are written (e.g., Thurston, 1994; Rowland, 2001).