Our understanding of the importance of processes and practices in student achievement has grown dramatically in recent years, both in mathematics education and education more broadly. As a result, at the K-12 level explicit practice standards are given in the Common Core Mathematics Standards  and the Next Generation Science Standards  alongside content standards. At the postsecondary level, studies regarding student learning and achievement have revealed the importance of many key practices, and accessible sources exist on this topic [3, 4, 5]. Further, we understand now that not all advanced postsecondary mathematics students are well-served by the same curriculum; for example, pre-service high school mathematics teachers need to develop unique ways of practicing mathematics compared to math majors with other emphases [6, 7]. As discussed by Elise Lockwood and Eric Weber in the previous post on this blog, mathematicians generally appreciate these issues; for readers unfamiliar with mathematical practice standards, their article is a nice introduction to this topic.
All of this leads us to the following question:
Given the breadth of both content and practices required for students to deeply learn and understand mathematics, what are effective techniques we can use at the postsecondary level to gauge student learning?
This is an important question for us to reflect on, and one that will never be completely resolved. The stereotypical assessment structure in math courses, especially at the service level, are homework problems (often collected and graded using multiple-choice online homework systems), midterm exams, and a final exam. While these can be useful components of an overall assessment structure for a course, these assessments alone often do not serve students and instructors as well as they could. The main reason for this is that these methods typically assess only procedural mastery and “basic” conceptual understanding, without assessing any aspects of the mathematical processes and practices of students.
I’d like to share some methods of assessment that have been either useful in my own classes or thought-provoking and inspiring. The common theme of these methods is that they are informed by both content and practices, even when they are primarily focused on assessing content. In my personal reflections on the question above, I’ve found the MAA Notes volumes regarding assessment practices in undergraduate mathematics [8, 9] to be helpful sources of ideas and inspiration; readers familiar with those documents will likely notice their influence in the list below (PDF versions of both of these are available for free at the MAA website). I also found reading Alan Tucker’s recent article in the American Mathematical Monthly  regarding the history of undergraduate programs in the United States to be thought-provoking in this context, as it provides a sense of how our current curriculum, which is closely related to our assessment methods, developed.
Allow submission of revised work, grading both the mathematics and the depth of the revision. I like to reward “honest, productive failure” on the part of my students. While we often assign students exercises on their homework, that is, problems that should be reasonably straightforward given a basic understanding of the course content, it is also good to give students hard problems that they might not succeed with at first. When I give students problems such as this, problems that I don’t expect any of them to solve, I often allow students to revise and resubmit their work after an initial round of grading. Then, when I re-grade their work, I can give credit to students for both their mathematical content and for the depth of their revision, for the degree to which they tried out new ideas and sought to determine the reasons for their initial failure. This allows me one way to reward persistence and self-monitoring, which are important practices. It is a good idea to not accept “first” submissions after the initial due date, so that students take their initial work seriously in order to have their revised work re-graded.
Assign frequent quizzes rather than infrequent long exams. I’ve recently been happily experimenting with giving 10-15 minute quizzes on a weekly basis, rather than 50-minute exams every 4-6 weeks. The class time spent with students taking tests ends up being roughly equivalent in either situation. My experience has been that by assessing student content knowledge more frequently, students feel that they are receiving better feedback regarding their progress and I am able to more quickly identify and respond to student misunderstandings.
Provide brief peer discussion time a few minutes into quizzes and exams. One of my regular complaints about quizzes and exams is that students might simply miss an obvious idea that would unlock the door allowing them to succeed. Outside of a classroom setting, people are rarely given a task and then required to work completely in isolation; typically it is important to work as part of a team, and to be able to make individual contributions effectively in this context. Allowing students to think about a quiz or exam problem for 3-4 minutes, then giving students a minute or two for discussion with their peers, better mirrors the reality of the mathematical world while still requiring individual students to do the bulk of the work on a problem.
Assign short essays. Essays are one of the best tools that mathematics faculty have to motivate students to reflect on their own processes and practices. Whether requiring students to write personal reflections about their performance in a class, having students critically analyze passages in their textbook, or having students compare and contrast different problems on homework assignments, writing forces students to step back and think about what they are doing in ways that they might not otherwise recognize that they should. The most important practical aspect of using essays in math classes is to identify and clearly communicate a grading rubric to students well before any essays are submitted for grading. Many grading rubrics for general student writing exist (university writing centers are good local resources in this regard), while others have been developed specifically for mathematical writing at various levels [11, 12].
Assign long-term projects, both expository and open-ended. Long term projects, such as writing an extensive (10+ page) paper about a major theorem, collaboratively writing a textbook wiki-style with classmates, creating a series of instructional videos, and creating course portfolios, can be incredibly empowering for students. Much like essay writing, long-term projects force students to think at a meta-cognitive level about their own work in a larger context than only considering one homework problem at a time. I’ve assigned 15-page papers regularly in my history of mathematics course, and collected course portfolios in both linear algebra and problem solving for teachers courses. I feel that these activities have had very positive effects on my students.
Use in-class student presentations of proofs and examples, problem solutions, etc. While I haven’t used these methods in my classes (yet), I have a positive memory regarding a graded presentation I gave in my college geometry class as an undergraduate. These activities are common in classes taught via Inquiry-Based Learning (IBL) methods , and various workshops exist to train faculty in IBL techniques ; I’ve heard uniformly positive reports from students regarding their experiences in IBL-style courses. My feeling is that one of the best aspects of using student presentations in class is that it makes errors and mistakes public, which often leads to the counterintuitive result that students feel more comfortable with their mathematical abilities (since they realize that they aren’t alone in needing to persist through mistakes and error-correction during mathematical work).
These are only a few ideas regarding assessment techniques. As the references below illustrate, a wealth of interesting and inventive methods exist for assessment, inspired by both student content knowledge and student processes and practices.
 Bain, Ken. What the Best College Students Do. Belknap Press, 2012.
 Ambrose, Susan A., Bridges, Michael W., DiPietro, Michele, Lovett, Marsha C., and Norman, Marie K. How Learning Works: Seven Research-Based Principles for Smart Learning. Jossey-Bass, 2010.
 Dweck, Carol S. Mindset: The New Psychology of Success. Ballantine Books, 2007.
 Conference Board of the Mathematical Sciences (2012). The Mathematical Education of Teachers II. Providence RI and Washington DC: American Mathematical Society and Mathematical Association of America. http://cbmsweb.org/MET2/
 Shulman, Lee S. “Those Who Understand: Knowledge Growth in Teaching.” Educational Researcher, Vol. 15, No. 2, (Feb. 1986), pp 4-14
 Assessment Practices in Undergraduate Mathematics, Bonnie Gold et al., editors. Mathematical Association of America, 1999. Available in PDF at: http://www.maa.org/publications/ebooks/assessment-practices-in-undergraduate-mathematics
 Supporting Assessment in Undergraduate Mathematics, Mathematical Association of America, 2006. Available in PDF at: http://www.maa.org/publications/ebooks/assessment-practices-in-undergraduate-mathematics
 Tucker, Alan. “The History of the Undergraduate Program in Mathematics in the United States.” The American Mathematical Monthly, Vol. 120, No. 8 (October), pp. 689-705.
 Grading Rubric, MA 310, University of Kentucky, Spring 2014. http://ms.uky.edu/~braun/LinkedMaterial_AMSBlog/Sample_Rubric.pdf
 Crannell, Annalisa. “Writing in Mathematics.” https://edisk.fandm.edu/annalisa.crannell/writing_in_math/