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Reflections on Mathematics and Democracy

Matthew Peeples  

As those familiar with Boy and Girl Scout merit badges know, the idea is a very simple one. Scouts advance in rank by earning badges; they earn badges on their own by demonstrating skills in an area (such as knowledge of farm equipment, fishing, archery).

Earning merit badges motivates Scouts because when they achieve a goal, they have shown that they have learned valuable skills or knowledge through their own efforts.

I created the Calculus Merit Badge program to motivate my calculus students at the Naval Academy Preparatory School (NAPS) in Newport, Rhode Island, to learn key concepts in preparation for the U.S. Naval Academy calculus I validation exam. In the test, the students must demonstrate a rigorous understanding of calculus concepts typically encountered in a freshman-level course.

The exam measures student aptitude in such concepts as limits, the definition of the derivative, differentiation rules for various functions, implicit differentiation, graphing, tangent lines and approximations, related rates, critical points, maximums and minimums, antiderivatives and the definite integral, the fundamental theorem of calculus, and areas under curves.

To be successful, students need to actively learn the concepts over time. The traditional methods of lecture and note taking, combined with homework assignments, followed by quizzes or tests, were not working.

A typical classroom of advanced-level students consists of twenty to twenty-five students. In this setting, the learning is passive. Note-taking usually involves copying definitions and examples, and perhaps a proof or two. Most homework consists of problem sets from the back of the textbook section or from an Internet-based system such as Webassign. Quizzes and tests follow up on homework, and students are often conditioned by these methods to simply remember the examples from the textbook.

Another problem with these methods is that the goal of giving student feedback is often undermined by the instructor’s need to grade large amounts of homework in addition to creating quizzes, tests, and lectures. At NAPS, instructors do not have teaching assistants to handle such tasks or to provide extra hours helping students. As a result, the instructor often cannot encourage deep learning because there is only limited time available in the classroom to address homework questions before moving on to the next set of topics.

NAPS maintains a period for extra instruction at the end of each academic day and employs supplemental instructors and tutors to provide help during evening study hours. But students often underuse the office hours, extra instruction periods, and supplemental instruction we have available to them. They must also attend office visits in other academic disciplines as well as fulfill military and athletic commitments.

Five Badges to Earn

To encourage students to engage with the material, I set up the Calculus Merit Badge program. In addition to the normal work on the syllabus (Webassign homework, quizzes, and tests), the project required students to earn badges corresponding to the different areas on the validation exam. Cloth badges similar to the scouting badges are difficult to obtain and cannot be applied to naval uniforms. Instead, I opted to use stickers, each with a different mathematician’s picture. I printed off sheets of them using mailing labels, which turned out to be a good choice. Below are the pictures I used as badges.

The badges corresponded to items in the calculus curriculum:

- Differentiability and Limits Merit Badge (Cauchy Badge)
- Basic Derivative Rules (Fermat Badge)
- Implicit Derivatives and Inverse Functions Merit Badge (Leibniz Badge)
- Applications of Derivatives Merit Badge (Newton Badge)
- Antiderivatives and the Definite Integral Merit Badge (Riemann Badge)

Each badge had three levels of accomplishment. Students completed a level by demonstrating certain activities associated with each area (as shown below) to mathematics faculty or tutors. The mathematician signed the form certifying that the student had become (A) Knowledgeable, (B) Proficient, and (C) Mastery at each level. (An example of these signature sheets is below.)

  1. At the Knowledgeable level, students had to show they could remember the basic rules associated with the area of concern to the satisfaction of the mathematician.
  2. At the Proficiency level, students needed to demonstrate they could solve basic problems associated with the area of concern, chosen by the mathematician.
  3. At the Mastery level, students had to independently demonstrate an example or explanation.

To earn a grade of 75 percent for the project, students had to complete all the tasks from the lists given at the Knowledgeable and Proficiency levels. Higher grades could be earned for completing Mastery-level problems. The project grade was included as 10 percent of each student’s overall grade for the course.
The deadline for obtaining all signatures was the last class day before the validation exam. Since there were so many problems offered for each badge, the task of completing all the assigned work was very difficult to obtain at the last minute. Successful students earned badges during the term rather than attempting to complete them all during the last week of classes.

In practice, the program was found to be helpful. Of the fifty students assigned to the advanced math program, half of them earned a score that would be considered as validation for the calculus I course offered. Motivated students managed to complete all five badges. Twelve out of the fifty students obtained all signatures early; thirty-four students earned all signatures by the deadline day.

Explicit Expectations

As we went through this inaugural program, I learned that I needed to be explicit with fellow instructors as well as with students about how badge requirements must be met.

For example, for students to earn a Knowledgeable rank, they had to literally recite rules or definitions from memory; it was not enough for students to complete a problem from the set to demonstrate this requirement. I also had to make clear that for Mastery, students were expected to choose an example independently and demonstrate the concept to the satisfaction of the instructor.

Students were not being properly challenged in some cases because I had not made my expectations plain to my colleagues. To overcome this problem in my next attempt, I intend to spend extra time with the tutor explaining my expectations and to put them in writing. Having these expectations on paper is also a good strategy for ensuring that students who tend to procrastinate will not forget the project instructions over the term.

Overall, the results were mostly positive. Since the students were required to demonstrate an understanding of the concepts, a large percentage of them sought extra instruction during my office hours. Students who might not otherwise go to supplemental instructors or tutors sought out their help both to understand the concepts better and to obtain signatures for completed work.

In class, students were recognized for each achievement, and a small ceremony was conducted each week in which badges were awarded. Student proudly displayed badges on notebooks and were overheard bragging about them in hallway conversations with classmates.

This project can be a way to motivate students to complete work throughout the term that they might otherwise find tedious.

Matthew Peeples is a member of the Mathematics Department at the Naval Academy Preparatory School, Newport, Rhode Island;


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