Carl Lutzer: Teaching
  • Have you ever seen a YouTube video that was so delightful, you felt an overwhelming urge to share it with your friends? That joy of sharing is the feeling I get from teaching.

  • At the university level, teaching is more than the delivery of facts and development of a skills. It's about nurturing new ways of thinking and habits of mind, facilitating the internalization of ideas and relationships, and helping people to blend skills and concepts in creative problem solving. And at the pinnacle, with the practical achieved and the creative expressed, teaching can be about the discovery of beauty.

  • I've recently adopted a phrase, adapted from the musical Hamilton, that I use as a basic principle to guide my development of lesson plans: Talk less. Teach more.

    Fundamentally, I believe that we construct knowledge from experience. So at the undergraduate level, especially in lower division classes, my lessons should begin with an activity in which students examine a simple situation that isolates and exhibits an important kernel of truth. The more this introductory activity can include or reference a common real-world experience, the better. These preliminary experiences provide a framework for further discussion, and help students to develop an intuitive understanding of mathematical ideas.

  • We often speak of teaching and learning separately, but without learning there has been no teaching. I don't mean to play a semantic game, but to express succinctly what I believe is a central idea: the instructor and the student are a team, engaged together in an enterprise of creative growth, conceptual enrichment, and skill development.

    As part of this team, my task is to design and decide on activities that I believe will help students to achieve learning objectives, much in the same way that a soccer coach designs small-sided games by which players can internalize fundamental ideas and skills.

  • A colleague of mine named Michael Starbird once asked, "Are we teaching a way of doing, or a way of thinking?" Mike put these goals in opposition as a rhetorical device because he was trying to make a point about the larger goals of education, but they are not mutually exclusive, and ideally we should strive for both. I work hard to make sure that when students leave my courses, they are equipped with new concepts, new ways of solving problems, and the tools to do it.

  • I recognize that learning mathematical methods is an important part of STEM education, but I firmly believe that the most important thing I can do for my students is to help them develop their own understanding of concepts and the relationships among them.

    My emphasis on clear thinking and understanding concepts frustrates the #@$!! out of some students, especially those who have been taught that Process is the centerpiece of mathematics. Students with that kind of preparation often focus on the methods of mathematics, and are looking for training. While I provide relevant demonstration of skills, and exercises so that students can develop their own ability, the skill set is neither the starting nor the stopping point in my classes. I tend to begin each new topic with a discussion of "why we care," often in the context of practical motivation, and I conclude by helping students to interpret what we've found, and how it's related to other things we know.

  • I work diligently to provide a positive educational environment in which students can succeed in meeting high standards. This means a lot of little things. I answer my students' questions thoroughly, and take each question seriously. I make myself as available as possible, and make my office spacious and welcoming. I greet students with a smile, and I often provide home-baked cookies! (As my grandmother used to say, "If itta moves, feed it. If it donna move, put tomato sauce on it.") I am prolific with email, so that my students are always informed about what's coming next, and I maintain an on-line archive where they can find full solutions to quizzes, weekly summaries of the class, etc.

  • My efforts to help students succeed are balanced by my expectation that students do their part, too (we're a team, remember). That means arriving to class on time, reading ahead for each class (as directed), actively engaging in class by asking and answering questions, and coming to office hours (or making appointments for help). In short, I expect students to participate fully in their education. I expect them to move away from high-school notions of learning and responsibility, toward healthy, productive professionalism.

  • I'm known as a demanding teacher. Here are a few of the reasons:

    • Questions: If students have questions, I expect them to ask so that I can help. Questions are not an interruption but, to the contrary, are part of the learning process. If students aren't asking questions, something is wrong.

    • Nightly work: I expect that students either have questions about the previous day's discussion, or that they are comfortable with the ideas and capable with the basic techniques. This means they must work every night to be prepared for the next day. I don't spend multiple days belaboring a point.

    • Homework: I expect students to do it, and do it well. There's no point in practicing shoddy work or making half-baked arguments. I tend to assign a mix of skill exercises and concept exercises; some of them are plainly difficult, but I don't leave students out in the cold. I expect them to work in groups (because learning happens better that way) and I am happy to help out if they get stuck.

    • Quizzes: I usually give a quiz each week. This helps students keep up with the course and gives them some periodic feedback from me. The quizzes are usually announced ahead of time in class, and I often send out email reminders. The questions are relatively straight-forward. They ask students to apply the ideas and skills we've covered in class and in homework (or workshop).

    • Tests: Former students often say that my tests are extensive and difficult, but fair. They cover all of the concepts and techniques, and occasionally I will ask students to combine ideas instead of relying on just one.

      There is usually a set of questions that form the body of the test, and I expect average students to complete these successfully. Then there are a few questions that are a little more difficult and, lastly, one or two that students should be able to answer if they've really mastered the course content. Difficult questions are not weighted heavily, since they are intended only to separate "A" from "B" papers, and I award partial credit throughout a test.

      Tests in upper-division courses are slightly different, depending on the level of the course. These usually ask students to demonstrate both the ability to calculate and facility with proofs.

    • Grades: My job description changes at the end of the term. Instead of being a mathematics "coach," concerned with the development of my "players" (students), I am an evaluator and my job is to recommend students for different levels of accreditation. That accreditation is a measure of the degree to which a student has mastered the course content, as indicated by the assessment tools used throughout the term (the tests and quizzes, etc.).

      I spend a lot of time determining final grades. I use a student's overall percentage as a guide, and a lower bound. For example, a student who earns an overall percentage of 78% will receive at least a letter grade of "C" and may receive a letter grade of "C+" or even "B-." What makes the difference? A lot of things. Here are two:

      • Do her grades ascend or descend? There's a significant difference in the level of mastery between a student who earns scores of 85, 80, 77, 74 (on the final) and one who earns scores of 74, 77, 80, 85 (on the final).

      • Did the student ever, on his own, earn a "B" on any test, or were the test scores consistently high grades of "C"?

      • Is her score on the final exam consistent with the grade? (For example, an overall grade of 82 is probably a B- if the student scored 75 on the final, but the same overall score of 82 is probably indicating a grade of B if the student earned a grade of 85 on the final.)

      • Did he care about his own grade? (Did he attend class, actively participate, and hand in homeworks on time?)

      Many questions have to be answered, and the decision between a lower and higher letter grade is often a difficult one to make. I don't make it lightly.

  • I care about my students and I want them to do more than just graduate from RIT. I want them to finish their education with the knowledge and skills they will need to be successful, with confidence in their futures, and pride in what they've accomplished.

Last update: January, 2018