The National Academies have just released a report that should be of interest to readers of this column: Reaching Students: What research says about effective instruction in undergraduate science and engineering.  It is based on their earlier report, Discipline-Based Education Research: Understanding and Improving Learning in Undergraduate Science and Engineering (the DBER Report), which was the subject of my Launchings column in December 2012, Mathematics and the NRC Discipline-Based Education Research Report. The new report illustrates the insights and recommendations from DBER with current examples and presents practical suggestions for improving classroom instruction.
Before I get into the many things I like about this report, I will start with its one glaring fault: It completely ignores undergraduate mathematics education. Like the DBER Report itself, it reads as if mathematicians have never thought about effective classroom practice. Based as it is on the DBER Report, this is perhaps not surprising. It is still disappointing.
Nevertheless, there is a lot that mathematicians can learn from this report. The many examples that describe actual classroom practice include:
- Facilitation of reflective learning (p. 6)
- Use of peer-led team learning (p. 18)
- Effective use of clickers in large classes (p. 22)
- Effective use of learning goals (p. 37)
- Methods for identifying the ideas that are most misunderstood by or confusing to students (p. 67)
- Assessment in active learning classes (p. 124)
- Effective faculty professional development (p. 196)
- The Association of American Universities efforts to improve undergraduate STEM education (p. 203)
This report discusses the role of lecturing, both its strengths and its weaknesses. More importantly, it talks about strategies for making lectures more interactive. It looks at assessment as more than measuring what questions students can answer, describing how to use it—especially student writing—to understand student reasoning, misconceptions, and misunderstandings.
It also deals with the challenges of changing one’s pedagogy and the obstacles that we all face, recognizing the difficulty in finding the time and energy required to adapt one’s approach to teaching. The advice includes: start with whatever is comfortable for you, use proven materials that others have developed, take advantage of the support that is available (there are many small grants specifically designed to ease the adoption of such practices ), and share the effort with interested colleagues.
The report also tackles the issue of coverage, one of the most frequently cited reasons for sticking with lectures. As the report accurately states, “What really matters is how much content students actually learn, not how much content an instructor presents in a lecture.” (p. 160) Moreover, as I have found in my classes, helping students learn how to think about mathematics, how to read it, how to wrestle with it, how to tackle unfamiliar and challenging problems, means helping them learn how to learn it on their own. As we succeed in these goals, there will much content that can be assigned to them to learn through reading or online resources rather than by taking up precious contact time.
Noah Finkelstein of CU-Boulder makes exactly this point, “You must be willing to move away from the idea that teaching is the transmission of information and learning is the acquisition of information, to the notion that teaching and learning are about enculturating people to think, to talk, to act, to do, to participate in certain ways.” (p. 31)
This enculturation enables students to use what they have learned in our classes. As the report states in the chapter on Using Insights from Research on Learning to Inform Teaching, “expertise consists of more than just knowing an impressive array of facts. What truly distinguishes experts from novices is experts’ deep understanding of the concepts, principles, and procedures of inquiry in their field, and the framework for organizing this knowledge.” (their italics, p. 58)
Helping students develop this kind of expertise is difficult, but we know that active learning approaches are much more effective than simply watching an expert produce the solution in a flawless flow.
The report ends with a summary of lessons (pp. 212–213), from which I have chosen and paraphrased four:
- Begin by understanding how students learn. 
- Start small with the changes that make the most sense and are easily implemented.
- Establish challenging goals for what students will learn and use them to guide both your instructional strategies and your assessments.
- Draw on the research, materials, and support structures that are already available.
I hope that this report will sit in the reading room of every math department and at hand for every mathematician who cares about teaching.
References: Kober, N. (2015). Reaching Students: What research says about effective instruction in undergraduate science and engineering. Washington, DC: The National Academies Press. www.nap.edu/catalog/18687/reaching-students-what-research-says-about-effective-instruction-in-undergraduate
 One example of a source of small grants for the teaching of undergraduate mathematics is the Academy of Inquiry Based Learning, www.inquirybasedlearning.org.
 Two of the best resources for this are:
Ambrose, S.A., Bridges, M.W., DiPietro, M., Lovett, M.C., and Norman, M.K. (2010). How Learning Works: Seven research-based principles for smart teaching. San Francisco, CA: Jossey-Bass.
National Research Council. (2005). How Students Learn: Mathematics in the Classroom. M.S. Donovan and J.D. Bradford, Editors. Washington, DC: The National Academies Press.