Tuesday, February 6, 2018

MAA IP Guide – Assessment

By Rick Cleary (guest blogger), Babson College

A note from the Editors: This semester Teaching Tidbits will have several posts highlighting the new Instructional Practices Guide (IP Guide) from the Mathematical Association of America (MAA). The MAA has a long tradition of reporting what content should be taught in the mathematics classroom through its Curriculum Guide; now the new IP Guide addresses how things could be taught in the mathematics classroom, how one could to design that experience, and how one could assess that experience. The suggested practices are well grounded in research on student learning. In our first post about the IP Guide, we dive deeper into the Assessment Practices section of the guide. Thanks to Rick Cleary, a lead writer for this section, for providing this post.
The opening statement of the Assessment chapter of the MAA Instructional Practices guide makes the following claim: Effective assessment occurs when we clearly state high-quality goals for student learning, give students frequent informal feedback about their progress toward these goals, and evaluate student growth and proficiency based on these goals. The chapter details some of the ways that effective assessment can be implemented in various types of courses. Many of the same assessment principles apply, whether you are from a big or small school, whether you teach large or small numbers of students, no matter what your lecture/active learning balance, on or off-line, developmental courses through graduate seminars. This portion of the IP Guide is designed to get colleagues thinking and talking about grounding both formative assessments that take place throughout the course and summative assessments at the end of a course in appropriate learning goals.

There is a fine line between assessments that are challenging and assessments that are discouraging. Once students become discouraged, it is hard to get them back on track. For example, traditional lecture-based instruction methods have been associated with traditional summative assessment procedures such as timed exams with questions in very specific formats. Recent research in mathematics education recommends classroom practices that provide ongoing lower stakes assessment to promote student engagement. New technology such as clickers and online polls or quizzes can help faculty provide these types of opportunities. Through vignettes grounded in the experience of the writers, the IP Guide illustrates these developments, providing instructors the tools they need to be creative as they design appropriate and equitable assessments for their courses.

The IP Guide chapter on Assessment provides both a research framework and practical tips needed to implement effective assessments that encourage, rather than discourage, student learning. It considers ways to make assessment consistent with course design and practice to promote effective learning for all students. Rather than seeing assessment as a mandate from an administration or an accrediting agency, the IP guide shows there is great value in creating a positive culture of assessment for students, faculty and departments.

Download a copy of the MAA Instructional Practices Guide today.

Tuesday, January 23, 2018

The One Question Calculus Final

By Lew Ludwig (Editor-in-Chief), Denison University

As the semester begins and we prep for classes, the practice of backward course design is a powerful way to get the most of the learning experience for our students. With this in mind, I thought I would share one of my favorite exam questions for a first semester calculus course, which appears below. I call it “the one question calculus final.” Now of course this is tongue-in-cheek, as the one question has over 15 questions. Nonetheless, this one question tour de force covers the full range of a first semester calc course. To substantiate this bold claim, I found a comprehensive list of typical topics in such a course at Wolfram Mathworld Classroom. The chart below cross-references each alphabetically listed topic with its specific question. While some questions touch on a range of topics, the cross-referencing refers to the primary reference.

Why I like this question:
I really like this question because it requires students to problem-solve, not just memorize a procedure. For example, instead of providing a typical composite function and asking for the derivative, my students must understand that the function in question (f) is a composition, know how to apply the chain rule, then read the graph to fill in the missing values. While I admit that question (m) may be a stretch for a Riemann sum, question (p) helps students realize the a definite integral is just a question about area. I especially like questions such as (j) and (k) that help students intuitively use important results like the intermediate value theorem or the mean value theorem. Finally, I like this question because no piece of technology can do the work for you. I can safely permit graphing calculators during my final without the fear of some CAS (computer algebra system) making short work of my exam.

How I use this question:
Please do not unleash this question on your students without prior exposure! My students have been working with this type of question for the whole semester. The beginning of the semester would focus more on limit questions like (a)-(e). By the second test of the semester, my students work on questions like (f) and (g) to understand the with mechanics of differentiation. By the third test, we get practice with applications of the derivative with questions such as (h)-(l). And by the end of the semester, questions of the type (m)-(p) test students’ understanding of the definite and indefinite integral. To make sure students do not forget prior material, my tests include questions from previous tests. Cognitive psychologists refer to this technique as interleaving.

How I grade this question:
Since this question has so many parts, I only count each sub-question for one point out of a 100-point final exam. Okay, I do ask other questions beside this one! I grade each question as right or wrong, no partial credit. While this may seem harsh, by giving each question a small point value, a student can miss a few of these questions without serious detriment to the overall grade. Moreover, past tests have shown a student’s exam score tracks fairly closely with performance on this question.

How to modify this question:
Of course there is a myriad of ways this question could be modified. For one, change the graph. When I initially developed this question, I would make sketches of the graph by hand. Now the online graphing program Desmos helps me produce graphs that are easy to read and export into LaTex or word processing programs. Students can contribute by creating their own questions for a graph you provide. Or you can turn that on end and have students provide a graph based on questions you provide. However you use it, you will find this focused cumulative approach will help deepen your students understanding of calculus.