In this section, I consider planning - both in the contexts of planning individual lessons and the course as a whole.
Course Design
Non-content learning goals that I particularly value for introductory level courses, and which I have incorporated into my teaching this semester are:
At the end of the course, the student will be able to:
- • Read level-appropriate mathematics, construct questions, and apply reading strategies to learn from reading.
- • Work with peers, asking questions and offering help as needed.
- • Make sense of problems and persevere in solving them.
- • Look for and express regularity in repeated reasoning.
I value the first goal because I want students to be independent learners and able to work with each other, and for this they need to have skills and practice in mathematical communication. The second goal I value because students consolidate and digest information better by talking it over with peers. By making communication a goal of the class, I am reminded to regularly set aside class time for it. The last two goals are what I see as the purpose of english majors taking calculus. These courses, in the way that they are taught help students to build pattern-recognition. Although professors usually see pattern-matching as a secondary skill, it is extremely useful in lots of different contexts and professions. Similarly, persevereness is an important skill in many contexts, and even the best students need persevereness to be successful in calculus.
When deciding on course policies, I try to think about what will be useful for students. For example, because students were not checking their homework against homework solutions, this semester I delayed homework quizzes to be after solutions were released. I explicitly told students that they should be studying by checking their homework against the solutions. However, students were not very responsive to this, and eventually I abandoned these homework quizzes half-way into the semester, because they were not fulfilling their purpose and were taking up too much class-time.
Until this semester, I did not think much about how much work and complication students were able to handle. In this semester's iteration of calculus, students were overloaded with assignments, and had trouble keeping tracking of all of the work. Moreover, there was enough work that we didn't have time to discuss it all in class. As a result, lots of little assignments that individually seemed useful were considered busywork by the students.
I need to think further about how much I can ask the students before they get overwhelmed.
Example: Calc 1 Syllabus
This syllabus was created for a differential calculus class that has class MWF, recitation T and lab H. The non-content standards come in large part from the the National Council of Teachers of Mathematics process standards for problem solving, reasoning, proof, communication, and connections. I have added to this objectives for group work and reading mathematics, as I think these are also important skills to develop in a math course.
Class Design
To prepare for a class, I usually review the material in the textbook, and consider what students will be expected to demonstrate in the end, what they will find challenging, and what I care about them taking away. I look at reading quiz questions (that they are supposed to solve twice a week) and at Good Questions (Just in Time Teaching for calculus) to find conceptual weak spots that students probably need more work on. This usually helps give me an idea of the material that I want to cover and the objectives.
I also review old notes and reflections I wrote about them, to remember if the activities I used in the past seemed to have been helpful or not. As I write my lecture plan, I build activities/pauses/questions around the places that students find confusing.
Building in a few activities, pauses, and questions means that I often have to eject material, and abandon the 10th example so that students have enough time for discussion and peer-teaching. One thing that I struggle with is building lectures so that there is `enough' content. Students sometimes are unhappy that I haven't shown them all of the techniques that they are required to be able to do on the exam - but often we spend that time reinforcing a concept. Overall, I think that my balance has been better since I started thinking about having just two or three objectives per 50 minute class, and trimming to make sure that both my lectures and my activities refinforced these objectives. Planning is much faster once I know what my objectives are, since I can build the class around them.
In the future, I think my notes will include stated daily objectives. This will both help my prep go faster when a course is repeated, and will help me organize the day into a more cohesive whole. As I tend to lecture slowly, for a while I included lots of time-guidelines, which in class I ignored, and I was always hurrying to make it through the material. Now, I prefer to include just a few guides along the way - beginning, 15 minutes for concept A, 20 minutes for skill B, ending. This gives me permission to skip activities or material as necessary, and focuses my attention on hitting the main goals of the class.
More thought: Could you suggest videos to students for the more mechanical computational parts so that they could hear someone explain the technique?
A Teaching Plan
The following is a typed up version of a plan I have used in my Calc class. This class is supposed to wrap up a discussion of `continuity' definitions and introduce the Intermediate Value Theorem. Although this is a revised version of a plan that I had used previously, further revisions would be necessary if I used this plan in the future.
This demonstrates (1) setting learning objectives (2) an attempt to choose activities that will accomplish theses learning objectives (3) time budgetting (4) choosing questions.
This also demonstrates the importance of considering what students will be confused about. I had set the first activity to point out the differences in the definitions, but even though the form suggested that definitions were subtle, and the function 1/x was highlighted for special consideration, students charged through with their high-school definitions. This activity might have been MUCH more successful if I had given them a list of true statements and asked them to justify why they were true.
