Concepts of Calculus for Middle Level Teachers
Helpful Hints from Past Instructors
What follows is a set of questions and answers written by an instructor of this course upon its completion. The questions were designed to provide advice and assistance to future instructors.
- Describe the course content as actually accomplished in the course you taught. Note: This may differ in some ways from the curriculum materials given to the participants.
This calculus course is an exploration or discovery-type course built around 18 problems sets which lead the participants through exercises conveying the basic concepts of calculus. A course outline is listed in the syllabus and copies of the worksheets are included in the notebook. After each problem set was completed, it was discussed as a class -- with participants sharing solutions to the problems. The only addition to the course (that is not specifically identified in the notebook) was the inclusion of periodic reviews in which the instructors summarized the main ideas underlying the content of the worksheets completed to that point in time. According to the teacher participants, these reviews were extremely important in helping them make sense of the concepts they had learned.
The schedule (as it played out) is as follows:
Day 1:PS 0 –4
Day 2: recap of 0 – 4, PS 5– 8
Day 3:recap of 5 – 8, PS 9–12, (video was shown this day)
Day 4: finish 12, recap of 9 – 12, PS 13–16
Day 5: recap of 13 – 16, PS 17–19
If a group of participants completed a problem set early, they were encouraged to begin looking at the homework problems related to that section rather than moving on to later problem sets. This way everyone worked through the material at about the same pace – making it easier for facilitators to manage the class, and at the same timeminimizing idle time for the “fast” groups.
- Describe your approach to homework assignments and any reflections or recommendations for change that you believe will be helpful to future instructors. (E.g., Which problems were assigned? Were problems submitted by groups or individuals? Which problems were presented to the class? Etc.).
Each evening problems from the corresponding section/s on the homework list, along with a reading assignment (when appropriate) were assigned for homeworkas indicated on the daily schedule (see Daily Assignments in the notebook/handouts folder). Of these problems, participants were required to submit a written solution to some of the problems each day. On some occasions, the participants were to consider one additional problem in order to be prepared to discuss it in small groups. The remaining problems that were assigned served as optional presentation problems, solutions to which participants had the opportunity to present to the class the next morning. Note that the solutions to many of the problems in the text book can be found in the back of the book. The problems which were assigned and discussed are indicated on the Daily Assignments handout. Due to circumstances surrounding the first group of teachers (in cohort 1) the amount of homework assigned is minimal. We believe additional homework problems could be required in subsequent course offerings.
- Identify the prerequisite skills/knowledge needed for participants to succeed in this course. (In particular, identify any topics covered in preceding Math in the Middle courses which were foundational to concepts covered in this course.) We welcome your reflections on the preparation of the participants you taught, their work ethic and the success that they had in your course.
Students were very successful in this course! The student work completed in Math 802T (Functions, Algebra and Geometry) was KEY to participant success in this course.
The section on exponential functions and the related homework handout (entitled Derivatives and Differences) built upon experiences in Math 807T, where participants were introduced to the basics of exponential functions and their uses.
A homework assignment on regression lines (see the handout) demonstrates how the regressions lines are determined using optimization methods of calculus, thus building on their experience in Statistics 892.
- What insights can you offer future instructors of this course regarding the content and clientele? For example, which topics were difficult for the participants? Which were more readily grasped? Were there methods of presenting topics that worked particularly well for middle school teachers? Are there cautions you want to pass along to future instructors?
When the nature of the course was first introduced (specifically that the worksheets were designed for them to explore/discover the principle concepts of calculus on their own) several of the participants expressed concern – that they would not learn the material as thoroughly as if they were in a lecture-based course (they wanted lecture notes!!!!). While our original intent was to draw out the main ideas of calculus from the participants strictly via class discussion as each problem set was completed, we instead decided to add to these discussions lecture-style reviews after the completion of several problem sets. (These took place approximately once each day, usually first thing in the morning.) While this adjustment eased the concern of those who wanted a lecture-based course (and were cited as extremely valuable in the evaluations), it also allowed the course to continue as was intended --- and by the end of the course many were praising its exploratory nature and expressing gratitude that they had been “forced” to work through the materials on their own.
Familiarity with technology is extremely beneficial for instructors of this course – particularly graphing calculators. While many of the graphics are already included in the worksheets to minimize the necessity of graphing calculators, the sections on integration are greatly enhanced by their use. Programs which display rectangles used in Riemann sums were utilized and greatly valued by instructors and participants alike.
Problem Set 7 (on exponential functions) and Problem Set 12 (area and summation notation) were the only problem sets which werecompleted by the class with the aid of an instructor. Instead of assigned as an exploration, the instructor led the participants through the problem set --- using the written materials as lecture notes.
A word of caution: This course does not provide sufficient background to prepare a middle level teacher to teach calculus. Much of the material considered standard in a first semester calculus course is omitted in order to focus on basic concepts.
- Indicate the duties of the assistant instructors (master teachers, graduate students, etc.) for the course you taught. Do you have any recommendations for changes in how they are used?
Assistant instructors were asked to select one or two of the problem sets for which they would lead the discussion following their completion during the course.
Additional duties for assistant instructors:
-lead small group problem sessions
-grade daily homework (and provide feedback)
-keep record of participants who give presentations (both in small groups and before the entire class)
-provided assistance on homework problems in the evenings (at the hotel where participants were staying)
Assistant instructors should be provided with a copy of the course notebook prior to the beginning of the course.
- Identify the supplies needed for the course (scissors, VCR, poster paper, calculators, etc.).
VCR/DVD player for the movie, “Newton’s Dark Secrets”, by NOVA
Overhead projector or ELMO
Daily homework folders for participants with score sheet
Name tags and/or tents
Evaluation forms
Several white/chalk board markers and erasers
Rulers
Scissors
Poster paper (for presentations)
Markers
Graphing calculators
10. We welcome any additional comments, suggestions or other reflections you might have
regarding your experience teaching a Math in the Middle course.
At the end of the course we played Calculus Jeopardy as a final, fun activity. We used the categories Integration, Derivatives, Limits and Miscellaneousand asked questions which served to review concepts throughout the course. Caution: It is easy to let this activity run too long – be sure to limit the amount of time, the number of questions or the number of turns each team may take to complete the game in a reasonable amount of time.