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Tommy Ratliff, Wheaton College, Norton MA
Reading, writing, and discussing mathematics are three activities I stress in my calculus classes. Of these, structuring my courses so reading is meaningful has been the most difficult: in particular, getting students to read the text prior to each class.
My expectations are fairly modest: I want students to be familiar with the terminology and have a rough idea of the basic concepts of each section. Since very few of my students have experience reading a mathematics text, I give them well-defined goals for each reading assignment and reward their efforts. In addition, my time spent grading needs to be manageable.
This past year, I posted reading assignments for Calculus I on my home page. Two sample assignments from Volume 1 of Ostebee/Zorn were:
Section 2.2 Estimating Derivatives: A Closer Look
Section 4.4 Newton's Method: Finding Roots
Last fall, I gave short two-minute (or so I hoped) quizes at the beginning of class, asking one of the reading questions. As the semester went on, I became increasingly dissatisfied with the results. Quiz time stretched to five or ten minutes, and I saw very little difference in the students' preparation. My hope had been that the quizzes would provide motivation for completing the reading, but instead the quizzes themselves became the focus.
This spring I had students email me their answers to reading questions prior to each class. I was able to quickly review their answers and gain some idea of areas that needed extra attention. To keep my in-box manageable, students gave their messages a specific subject line which I used to filter their assignments into a separate mail box.
I was very pleased with the results. The class average was 23 out of 28 assignments, which means that 19 out of 23 students in the class answered questions for each assignment. As a result, the class as a whole was better prepared for each class meeting than any other calculus class I have taught.
One of the most surprising results was that the students seemed more willing to write complete sentences and express complete thoughts when using email than with handwritten work. The lack of symbols on the keyboard was a decided advantage. For example, in a handwritten assignment, many students would explain Newton's Method by giving the formula x_{n+1} = x_{n} - f(x_{n})/f'(x_{n}) and hope that would be enough. However, the difficulty of entering mathematical notation via the keyboard, combined with the ease of editing their answers, made most students prefer using complete sentences to writing a few cryptic comments. In this case, technology, in particular, the limitations of technology, encourages the students' mathematical development.
For example, one student's response to the questions from February 17 was:
1. The term "locally linear" means that at a certain point in a function, if you zoom in enough, the function will appear to be a straight line, even if the function is a curve. This allows us to estimate the slope of a curve to give us the derivative.
2. The derivative of f(x)=|x| does not exist at 0 because that equation is not locally linear. No matter how much you zoom in on the graph, the kink will always be at the origin. You can't find the slope at this point, therefore there is no derivative of f(0) for this function.
While the last sentence is not technically correct, it is clear this student had put in much thought.
From April 7, one student responded with:
1. The purpose of Newton's Method is to simply and efficiently find a root of some function. It is also used because oftentimes it is impossible to find roots algebraically.
2. The idea behind Newton's Method is that you start with 'one approximation to a root' and then find another better approximation. In Newton's Method you find successive approximations to eventually find the root, r.
Another responded with:
1.The purpose of Newton's Method is to approximate the root of a function that can not be easily solved algibraically (sic).
2. I really don't understand exactly how Newton's method works or the idea behind it.
The last response, and others like it, gave me worthwhile information on what to expect in class. I spent less time than in previous semesters defining basic terms and discussing basic examples. In their course evaluations, students indicated that these assignments were a lot of work, and even tedious at times, but most felt the readings were very useful for understanding the course material.
To keep the evaluation manageable, I graded the assignments on a binary scale (1 or 0). This took me about 15 minutes per assignment. (All three responses above scored a 1.) These assignments counted 5% of their final grade, which was just enough so most students did them, whle not being overly worried about the grading.
The downside for the instructor is theare startup costs in creating assignments. One needs to plan the semester on a day-by-day basis, if one is going to write the assignments before the semester begins (which I strongly suggest). However, the assignments are fairly low maintenance once they are written, and email submission is the best method I have found for getting the students to read the text.
I plan on using email submission of reading assignments this coming year in both Calculus I and II, and I hope to gather some hard data on the effectiveness of the assignments. If you have any questions, please feel free to email me at tratliff@wheatonma.edu.
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