Do Small Groups Enhanced by Technology Affect Learning in Calculus?

Joy Becker

Math

Project Summary

I investigated the question ≥What is the effect of small groups and a technology-enhanced curriculum on students learning in a calculus course?≤ The project centered on how groups and technology affect studentsπ understanding of concepts and problem-solving skills. I conducted this project during the Fall 2003 semester in my Calculus & Analytic Geometry II course and repeated the study during the Spring 2004 semester in my Calculus & Analytic Geometry I course. The students were assigned to homework groups, which met about once a week during class. They also used graphing calculators and completed computer labs using the program Maple. Project details are explained below in the ≥Key Learning Activities≤ section.

Course Context

Students in math classes often have lots of homework questions and are seldom confident in their answers unless they have access to the solutions. I have found that this is true even with math majors in a calculus course. While I am happy to answer homework questions, if I devote too much time to answering homework questions, too much new material gets left out. The purpose of the homework groups is to set aside some class time for students to discuss homework problems. This is something I strongly encourage the students to do on their own, outside of class, but by having the groups meet in class, I was able to observe their interactions and some of the effects these groups had on the studentsπ learning. Because both of these calculus courses meet five days a week, the groups met during class on Wednesday to break up the week.

Another area where students struggle in the math discipline is in communicating mathematics. Besides getting help with homework questions, my hope was that by participating regularly in groups, students would increase their communication skills. Like any other discipline, there is a lot of discipline-specific terminology in calculus. One of my goals for the homework groups was to increase the studentsπ comfort level in communicating mathematics. Being able to communicate and explain ideas clearly often helps students achieve a deeper understanding of the material. By participating in these groups, students were also exposed to the problem-solving techniques of others, which might be different from their own. I believe that this also increases their level of understanding of concepts.

We use graphing calculators and the program Maple regularly in our calculus sequence. As instructors we feel these technological tools are helpful for students, particularly in visualizing concepts. With this project, I also wanted to study the impact of technology on student learning. For instance, do students feel technology helps them understand concepts, or does it get in the way of learning?

Key Learning Activities

Students were placed into homework groups at the beginning of the semester. These group assignments were based on studentsπ proficiency with the material in their previous math course and their comfort level with technology. (Both were self-rated.) I assigned students to groups so that each group included a range of skill levels in both areas. I changed the groups once during the semester and assigned the new groups based on my observations. The homework groups met almost every Wednesday throughout the semester. Students were able to ask me questions, too, but I strongly encouraged them to consult with their group members before asking me for help. The students evaluated themselves and their group members, and I gave each student a group participation score based on the following rubric.

Homework Group Rubric

5	Regular attendance Participate‹ask questions and try to answer other group membersπ questions Prepared‹have most of the assignment done, questions ready to ask
4	Fairly regular attendance Ask some questions, attempt to answer some questions Somewhat prepared‹part of the assignment is done, a few questions ready
3	Periodic attendance Ask few questions, no attempt to answer questions Minimal preparation‹a few problems done, not ready to ask questions
2	Periodic attendance Donπt ask or answer questions Barely prepared‹hardly any problems done, no questions ready
1	Poor attendance Donπt ask or answer questions Not prepared‹havenπt looked at the assignment
0	Never attend and/or not at all prepared

In terms of technology, students used their graphing calculators almost daily throughout the semester. Four to six Maple labs were assigned, and these were completed individually or in pairs. The Maple labs were mainly extensions of concepts covered in class, although some students also used Maple to check homework problems.

Students completed two surveys, one at midterm and one at the end of the semester. They were asked to rate how the homework groups and technology affected both their understanding of the material and their problem-solving skills. Students were also asked to provide comments and to describe their general reactions to both the groups and technology.

Key Findings

Research question: What is the effect of small groups and a technology-enhanced curriculum on students learning in a calculus course? The following findings are based on my own observations and reflections, as well as student ratings and comments from surveys.

• Interaction and communication among students increased. Students seemed more comfortable sharing ideas, not only in the small group environment, but also with the entire class.
• Students gained confidence in their math skills and problem-solving abilities. I observed that many students in these two classes were more likely than my previous students to try a new approach to a problem, rather than immediately ask for help when getting stuck.
• Graphing calculators are valuable for visualizing concepts. Students are very comfortable with graphing calculators and find them extremely helpful in understanding ideas through visualization.
• There is a gap between the perceived benefits of Maple and those of graphing calculators. The comfort level of students using Maple is significantly lower than studentsπ comfort level with graphing calculators. Therefore, many students donπt think Maple is as beneficial as the graphing calculators.

Evidence of Student Learning

Homework groups:

• More than 85% of students reported an increase in understanding of the course material due to participation in the homework groups.
• More than 50% of students reported improved problem-solving skills from participating in the homework groups.

Technology:

• More than 65% of students reported an increase in understanding of the course material due to the use of technology.
• More than 60% of students reported improved problem-solving skills due to the use of technology.

Select student comments:

• ≥Not only did I have to figure it out for myself, I also had to explain it to other people, which made me know it better.≤
• ≥It helps to have things explained in many ways and to explain it to others.≤
• ≥If the group is stuck, youπre able to generate more ways of computing it (than by yourself).≤
• ≥I think the ability to conduct a small forum of mathematical discussion was great.≤
• ≥Graphing functions, sequences, etc. provided a much-needed visual to illustrate the ideas.≤
• ≥We are headed towards a future full of technology and computers, and learning some applications of the technology is important.≤
• ≥Visual images are nice.≤

My Observations:

Students in these two courses seemed more interactive with each other, both in small groups and with the entire class than in some of the other calculus courses I have taught. This was particularly true in the fall section of Calculus II, and it might be due more to the types of students I had rather than the group work. However, I do feel that participating regularly in the small groups helped the students become more comfortable in communicating mathematics, and as a result, students were more active in their learning.

When asked what would make the homework groups more useful, the overwhelming response was ≥if everyone looked at the homework problems before coming to class.≤ While I stressed this throughout the semester and included a preparation component on the grading rubric, that wasnπt enough motivation for every student. I plan to consider a better way to motivate students to come to class prepared. I had thought about collecting the homework the same day the groups met, but I wanted to give students time to write up their solutions neatly after discussing problems with their group. Perhaps more grade weight needs to be given to preparation, in terms of points. However, I dislike the idea of always using grades to motivate learning. At some point, I hope the students will take more responsibility for their learning, including coming to class prepared. Motivation is definitely something I need to spend more time reflecting on, and I will probably make some changes the next time I use these groups.

Another fairly common comment was that students would like more time in these groups. While I donπt feel that I can use more class time for the groups, I did encourage students to meet with each other outside of class. Some students did this regularly and found it quite helpful. They generally didnπt meet in the assigned groups, but I think in many ways, self-selected groups work better for meetings outside of class.

In terms of technology, there were several computer issues that affected studentsπ attitudes toward Maple. This was the first semester that the program was accessed through a key server, which means that only a certain number of licenses are available at any given time. While the license number restrictions were not a problem, the access was, particularly in the fall. The computers in the lab were extremely slow, which was very frustrating to the students and me. This spring, almost all of my students had laptops, and there were not nearly as many technology problems. Even so, students tend to view Maple as a ≥black box≤ which spits out the answer, if you can type the command correctly. I try to limit the syntax problems by providing examples, but many students still struggle with the details. I think that their comfort level with Maple will increase as they use it more in subsequent courses. I still believe Maple is a valuable tool, but maybe as instructors we need to spend more time introducing it and explaining how it works. Students are used to graphing calculators, but Maple tends to be a mystery.

Concluding Remarks

This has been an extremely valuable project. By participating, I have become much more aware of how my students learn. Although there is still a gap between what I think I am teaching and what they are actually learning, I hope that this gap is growing smaller due to my renewed focus on learning. I plan to investigate similar questions in the future and generate new ideas as I branch into online learning.