A Beginner's Guide to Teaching Mathematics in the Undergraduate Classroom

Suzanne Kelton

• Publisher: Routledge
• Year: 2020
• ISBN:
  9781000282924
• Edition: 1
• Topic: Pedagogía
• Subtopic: Enseñanza de matemáticas

This practical, engaging book explores the fundamentals of pedagogy and the unique challenges of teaching undergraduate mathematics not commonly addressed in most education literature.

Professor and mathematician, Suzanne Kelton offers a straightforward framework for new faculty and graduate students to establish their individual preferences for course policy and content exposition, while alerting them to potential pitfalls. The book discusses the running of day-to-day class meetings and offers specific strategies to improve learning and retention, as well as concrete examples and effective tools for class discussion that draw from a variety of commonly taught undergraduate mathematics courses. Kelton also offers readers a structured approach to evaluating and honing their own teaching skills, as well as utilizing peer and student evaluations.

Offering an engaging and clearly written approach designed specifically for mathematicians, A Beginner's Guide to Teaching Mathematics in the Undergraduate Classroom offers an artful introduction to teaching undergraduate mathematics in universities and community colleges. This text will be useful for new instructors, faculty, and graduate teaching assistants alike.

5

Challenges and Opportunities within Commonly Taught Courses

Each course you teach will come with its set of challenges and opportunities. In this chapter, I will discuss a selection of standard courses. For the entry-level material, I will address some common issues that might arise, through the use of specific examples. While your course may not cover these exact skills, the examples should illustrate a general teaching concern which extends to the material (or typical audience) for the course. There are also a variety of handouts provided for in-class activities. For the upper-level content, it is assumed that you have gained some experience prior to teaching the related course, so the focus shifts in those sections to general techniques you might employ.

It should be noted that this is not intended to be a comprehensive list and, depending on your institution, your audience may have a differing skill set than is presumed. Due to the variety in learning objectives and desired learning outcomes that might be employed across institutions, I have not suggested what specific content you should deem essential in each course. For guidance on this, you should turn to your department chair and colleagues, as well as the general discussion offered in Chapter 1.

Discussion Sections

If you are a graduate teaching assistant, you may have been assigned to teach a discussion section (or “fourth hour”) of a course in which you generally answer homework questions rather than teach new course content. This is a nice way to ease into teaching if you have not taught before, since you will practice presenting math problems and gain insight as to where students get confused, but you may not be responsible for constructing any lessons. These sections present their own set of challenges.

In graduate school, I was once assigned to conduct a discussion section for a differential equations course with an emphasis on physics, which contained material which I had never seen before. The students in the course were quite sharp and the combination of their ability and my lack of exposure was more than a little daunting. For me to succeed in that setting, it was essential that I prepare well. I read each section in the text that the students were covering and completed all of the assigned homework. While this was sufficient for my purposes, if you find you are unsure of the material you are to cover, contact the professor teaching the course to address any questions you may have. You may find it helpful to attend the lecture as well.

In another discussion section I conducted, students were unhappy with the instruction from the professor for the course. They felt quite lost in the class at times and would occasionally look to me to illuminate a week's worth of material. I learned quickly to come to class with a list of relevant formulas, definitions, and other such very brief notes on the major concepts involved in the assignment. Having a high degree of comfort with the material covered in the assignment is extremely important in any discussion section, but especially so if you find the class often requires a significant amount of help on the basic concepts. Be prepared in any case to address common areas of confusion in the material.

Even if the class adores the professor for the course and has learned the material well, you may still face some difficulties in reviewing the assignment. Any variation on your part from the notation or method used by the professor may be met with dismay. Students in the first year or two of undergraduate study often fail to recognize when methods are equivalent and minor differences may confuse them. Check the text to see if the students' book uses the notation and method you would naturally use. If it differs, you may want to check with the professor teaching the course to see what was presented in class. Consider asking students as you start a problem if this notation agrees with what they have seen in class. If students do question your notation or methodology, attempt to switch to what they have seen in class. Your goal is to be cohesive with the professor and to help students learn the material as originally presented whenever possible.

You can learn a lot from teaching discussion sections. Perhaps the most important lesson is to be prepared for a wide variety of questions. Regardless of the environment in the class, the students' skill level, or your comfort with the material, do not underestimate the value of your readiness to handle the specific homework problems, as well as the material in general.

Algebra – Is It Too Late?!

By “algebra,” I mean the skills hopefully learned in high school algebra courses. Unfortunately, the skills possessed in this area by some high school graduates can be appalling to those teaching mathematics at the undergraduate level. The question, which begs to be asked, is “Is it too late to learn algebra skills in college?”

It is the rare undergraduate student who is actually incapable of learning this material with the right amount of dedication, time, and materials, paired with the right teacher for that student's learning style. However, students possessing extremely sub-standard algebra skills upon reaching college may feel a future in math is unlikely. Therefore, a responsible prioritizing of their workloads may require that they limit the amount of time spent attempting to master skills they struggled with at length in the past. This is especially true when the material in question is not material in the current course but presumed background knowledge.

The reality is that it takes much more work to deprogram incorrect notions about algebra than to learn it correctly in the first place. It can require extensive drill work to accomplish this deprogramming – something you generally do not have time for in the semester. So, we reach the question: “What do I do as an instructor faced with teaching students who lack basic algebra skills?”

It depends on the focus of the course at hand. If you are actually teaching a remedial math course, which is intended to patch up these holes in students' mathematical backgrounds, then you must go back to basics and go slowly. Remember, it takes time to uproot all of the weeds of misconception and replant the knowledge correctly.

… it is important for instructors to recognize that a single correction or refutation is unlikely to be enough to help students revise deeply held misconceptions. Instead, guiding students through a process of conceptual change is likely to take time, patience, and creativity.

(Ambrose et al., 2010: 27)

You may need to take a very discussion-intensive approach with in-class worksheets and group work, students working at the board and plenty of repetition of the processes at hand. This type of course can be just as rewarding as those you might find more mathematically interesting; the challenge is pedagogical, not mathematical. The teacher in you will be working overtime, using a few notes the mathematician left behind. While this may not be your dream assignment,...