*By Jerry Dwyer, professor in the Department of Mathematics & Statistics and Director of the Science, Technology, Engineering and Mathematics Center for Outreach, Research & Education (STEM-CORE) at Texas Tech University.*

This essay describes the changes that have taken place in my teaching philosophy and practice over the past 30 years or so. I have always loved teaching and the satisfaction of explaining concepts to others. However, my understanding and love of the profession has greatly increased over the years, with some pivotal moments emphasizing when those changes occurred. I present these reflections as a possible encouragement to others who may wonder about their teaching and how to make it more satisfying and enjoyable.

**Lecturing**

I had a standard undergraduate and graduate education in the mathematical sciences. This was in Europe but was probably similar to that of many in the United States. All classroom instruction was in the form of lectures with little or no feedback or student interaction. All learning was tested in a final end-of-year exam worth 100% of the grade. This exam was as much a test of memorization as of any deep understanding.

I obtained reasonably good grades through this system but I was never confident that I understood enough. I didn’t really think this was a good system and I didn’t think I was learning very well. I didn’t really enjoy the process and it was more of an endurance test than a learning experience. Deep down I loved mathematics but I wished there was a way that I could understand more and maintain that love rather than simply survive an endurance test.

My first teaching assignment as a graduate instructor was a course in numerical methods to a class of about 100 engineering students. I taught this class in exactly the same manner as I had been taught. I gave a final exam that required memorization. I graded it and gave the best grades to those who could reproduce more of their transcribed notes. I didn’t think I was doing a bad job. I continued this pattern of traditional lecturing for several years. Most courses were applied or engineering based. I didn’t reflect much on what I was doing. Students got the grades they expected and I got the teaching evaluations I expected.

I moved to the United States where continuous assessment and mid-term exams had reduced the impact of the all consuming final exam. I began to see myself in the role of a coach and facilitator of learning. But my overall approach and lecturing style hadn’t changed. However, somewhere in the back of my mind there was a nagging suspicion that there could be a better way and this was occasionally confirmed by encounters with previous students who frankly admitted they hadn’t understood any of the material I presented.

**Beyond Lecturing**

For about 15 years my academic focus was applied mathematics and computing with collaboration with engineers, geologists, and the automobile industry. While this was rewarding interdisciplinary work part of me missed doing pure mathematics. I also noticed more articles about the declining state of math education in the US. So I made a decision to focus more on educational issues. This coincided with some evolving career opportunities and the development of the role of an outreach mathematician (Conway, 2001; Dwyer 2001).

I began to more seriously reflect on teaching and began to read about teaching. A pivotal moment for me was an article describing the processes of learning math in an elementary school classroom (Ball, 1993). This deep analysis of how students processed information and communicated their understanding impressed me greatly. Suddenly teaching was no longer about the transmission of information. It was about the far more exciting venture of interacting with students as they grasped to understand concepts. It became an intellectual challenge to find multiple ways of developing that student understanding.

The first change I implemented was to introduce some group work into the classroom. The advantages and disadvantages of group work are well documented (eg. Morris and Hayes, 1997). This was well received by students. They enjoyed working with one another and learning from one another. They had more time to assess a problem and grapple with it than in a traditional lecture setting. I also found it much more enjoyable to walk around and discuss problems with students rather than just write on the board. This leads to an important change in class preparation. The emphasis is no longer on learning the material and preparing an impeccable set of notes to deliver. The emphasis now is on developing diverse ways of delivering and presenting material and motivating students to work with the material. It was my hope that the classroom become an interactive learning laboratory and not just a location for transcription.

Around that time I became aware of service learning, whereby students perform some service in the community which complements and enhances their classroom learning. In my pre-service teachers’ classes this was best implemented by having students work as tutors in local K-12 schools where they gained increased understanding of elementary school mathematics through explaining the concepts to younger students. The college students liked this option and all of the participants reported increased learning as a result of the service learning activities. This also showed to me that student learning can take place in venues beyond the traditional classroom setting (Dwyer, 2005).

I have firmly believed that quality of learning is far more important than quantity. That means there should be no rush to “cover” all of the material. It is better to fully understand two chapters than to have read (but not understood) five chapters. It seems to me that very few students understand as much as we think they do. From that realization I decided that I would make an effort to keep the syllabus as short as possible and only proceed to the next topic when I feel that all students have some grasp of the material. I think that students enjoy going slowly and really enjoy the fact that they understand as they go along. The amount of material covered is less but the students should leave the classroom with a greater sense of satisfaction than if they had covered several topics with limited understanding. A related point is the observation that a student question is not time consuming; rather it is an opportunity for increased learning.

My teaching still included a combination of lecture and group work. But another change had taken place in my approach to lecturing. Rather than preparing detailed notes I often leave a problem untouched until I am in the classroom. Then I work through the problem with student assistance where possible. This may be risky if I find I am unable to complete the problem in time. But the big advantage lies in the fact that students see me “doing” math on the board. They don’t see an artificial canned presentation but an actual mathematician doing math. Surely this is a nice opportunity to display our real life work and passion, and too many instructors miss this opportunity.

**Assessment of Learning**

I had changed some of the instruction methodology but I still retained traditional exams. My thinking on this issue was also evolving and some reading led to a critical question: how do we know that a student’s writing indicates anything about that student’s understanding of the concept? The area of assessment is very broad but for now my only reaction was a clear understanding that student memorization and reproduction in an exam didn’t tell me anything except that the student had memorized well. So I decided to allow students to bring their books and notes to an exam. Of course, this was welcomed by the students, but they did realize that they could not take advantage of the book if they hadn’t prepared their understanding ahead of time. There are arguments for and against open book exams and when each is appropriate. In my case I could only see the increased emphasis on understanding that resulted from this change.

I was still unsure whether these exams reflected an accurate assessment of learning. I also noticed that students were anxious about exams and despite the relaxed classroom atmosphere they were still hampered by test anxiety. Over a number of years I sought to reduce that anxiety by decreasing the weighting of final exams and increasing numerous alternative methods of assessment. There have been instances where I have given no exams. That increases the challenge to find alternatives. Sometimes I have included the students in the development of their own exams. This has again been well received by students who can relax more and not feel exam pressure. But it doesn’t work for all students as some lack any motivation apart from exams. Ironically this is most prevalent among pre-service teaching students (Ha, 2006) and education majors. As a result my current assessments are a mix of homework, student projects, student essays, in-class work, take home exams, and some judicious open book in-class quizzes.

**Conclusion**

In summary, I believe any instructor can make their teaching more rewarding and productive by considering the following: (a) think deeply about what you want your students to learn; (b) incorporate multiple instruction/facilitation strategies in the classroom and beyond; (c) focus on quality rather than quantity; (d) *Do* mathematics in the classroom rather than *present *mathematics; (e) think deeply about how to assess student learning. * *

My goal is to have my classroom be an open laboratory of learning. I have learned through service learning, group work, and alternative assessments that learning can be achieved in multiple ways. I believe I have made progress in that direction through the strategies described above. I am now more definite about the role of coach and motivator. But I don’t think we should ever stop reflecting on our teaching and improvements can always be made. I do know that I enjoy my teaching much more now than ever before. I approach each class session with a sense of excitement and a desire to facilitate student learning and to pass on my own enthusiasm for mathematics. It is my hope that this essay encourages the reader to reflect on their own teaching and consider ways in which it can be more effective and enjoyable.

**References**

- Ball, D. L. (1993). With an eye on the mathematical horizon: Dilemmas of teaching elementary school mathematics.
*Elementary School Journal, 93*(4), 373-397. - Conway, J. B. (2001). Reflections of a Department Head on Outreach Mathematics.
*Notices of the AMS*48(10), 1169-1172. - Dwyer, J. F. (2001). Reflections of an Outreach Mathematician,
*Notices of the AMS*, 48 (10), 1173-1175. - Dwyer, J. F. (2005). K-12 Math Tutoring as a Service-Learning Experience for Elementary Education Students, in
*Mathematics in Service to the Community*, MAA Notes. - Ha, A. (2006). Alternative assessment in pre-service teachers’ geometry course.
*MA Report*, Texas Tech University. - Morris, R. and Hayes, C. (1997). Small Group Work: Are group assignments a legitimate form of assessment? In Pospisil, R. and Willcoxson, L. (Eds),
*Learning Through Teaching*, 229-233. Proceedings of the 6th Annual Teaching Learning Forum, Murdoch University, February 1997. Perth: Murdoch University. http://lsn.curtin.edu.au/tlf/tlf1997/morris.html

I especially liked these two sentences…”I have firmly believed that quality of learning is far more important than quantity. That means there should be no rush to “cover” all of the material. It is better to fully understand two chapters than to have read (but not understood) five chapters”. I see this as the biggest problem in elementary and middle school math…this rush, rush, rush. How much better for everyone involved if teachers were allowed to go at a much slower pace, with young students really understanding what they are doing, building a really strong conceptual foundation upon which to build. I would posit that with the slower pace early on, we would probably end up at the same place in high school, with students who are much better prepared for the higher math classes. I would love to know if there is any research out there on that topic.

I was wondering if you teach mostly upper level students. As someone who teaches university freshman I am very hesitant to include any kind of group work. Since the university draws from many school systems I have a huge array of different backgrounds per student. So, having people work together can create the same feeling as in high school group work where you have one person that knows how to do everything and then the student who would find it challenging, but now feels like they’re dumb because they don’t know how to do the problem like their peer. This is, honestly, how I felt when I did group work in High School. I can certainly see implement group work in higher levels, as serious students interested in the discipline should only make it that far, but I feel like adding it into Freshman/Sophomore level might not work. Just thought I would add something I’ve thought about, because generally my students are relieved to hear there will be no math group work like they experienced in High School.

I don’t know what courses the author of the article has used group work in, but I’ve used some group work techniques successfully in lower-division courses. I’ve found that the key is to use group work in a more directed fashion in this context. I’ll be posting an article on this blog about some of these techniques in early July. In the meanwhile, you might find the article “Does Active Learning W0rk?” by Prince interesting: http://onlinelibrary.wiley.com/doi/10.1002/j.2168-9830.2004.tb00809.x/abstract

Thanks for the comments. Most of my classes have been in two categories. Some are the proof type courses such as Advanced Calculus and its prerequisite Introduction to Proof. These are typically populated by math majors from sophomore to senior level. The other category includes specialized courses for pre-service middle school teachers. So it is reasonable to suggest that these groups may be a little more open to group work. However I have used group work in some lower division courses also and it has been well received.

I think it may be related to a broader question about the atmosphere in a classroom. Over on the MAA blog MathEd Matters there is a posting about making mistakes and providing students the freedom to make mistakes. If we can create such an atmosphere in a classroom where there is genuine exploration and freedom to make mistakes them we may minimize the fear of anyone feeling dumb. But we also need to be intentional in how we form the groups and how different personalities interact. Finally, we do need to note that everyone doesn’t learn in the same way and that there is a minority (?) for whom the group work is not optimal for their learning.

This is a great post and brings up many important issues. Relevant to the discussion is the recent work of Sandra Laursen et al. (University of Colorado) about implementing inquiry-based learning (IBL) in the mathematics classroom:

http://www.colorado.edu/eer/research/steminquiry.html

The measures used were broad, not content-specific, to accommodate the variety of courses and sites. Data includes 300 hours of classroom observation, 1100 surveys, 110 interviews, 220 tests, and 3200 academic transcripts, gathered from over 100 course sections at 4 campuses over 2 years. In “Assessing long-term effects of inquiry-based learning: A case study from college mathematics” (2013), the authors state the following:

“Our study indicates that the benefits of active learning experiences may be lasting and significant for some student groups, with no harm done to others. Importantly, covering less material in inquiry-based sections had no negative effect on students’ later performance in the major.”

In my experience implementing IBL, I’ve been able to cover all of the required topics, but it’s nice to know there is some research/data to fall back on if the content pressure becomes too severe.