Assessment: Students Creating Solution Sets
Written by: Dr. Julian Fleron
The central vision of DAoM courses is for mathematics for liberal arts students to be actively involved in authentic mathematical experiences that
- Are both challenging and intellectually stimulating,
- Provide meaningful cognitive and metacognitive gains, and
- Nurture healthy and informed perceptions of mathematics, mathematical ways of thinking and the ongoing impact of mathematics on STEM fields as well as the liberal arts and humanities.
The majority of class time is spent with students actively engaged in process of mathematical inquiry: exploring, problem solving, (re-) discovering important mathematical ideas, mathematical sense-making, conjecturing, proving, and mathematical communication.
Sharing examples of strong student writing
As these activities are the focus of the class, I want to base my assessment of students' success on exactly this work. This assessment can come through class participation, work on projects, and several other vehicles. Much of it comes through written proofs and solutions that describe/document the students work.
One of two sample student write-ups for Chapter 1 – Fibonacci Numbers from the book Discovering the Art of Mathematics – Number Theory are included here. Selected investigations are graded.
In my course, the students’ written work accounts for 70% of their final course grade.
In this blog I focus on assessment of students work on the detailed prompts that occur throughout the DAoM learning guides.
When using details prompts (as opposed to more open questions or projects), students cover between one half and two dozen investigations in an hour-long class period. A typical chapter – which is generally what I collect to assess – will have 50 investigations answered. It is not feasible to grade every solution for 35 students in each course. Instead, I grade 8 or 10 solutions per assignment. I purposely grade some investigations that are more routine, some that are a bit more challenging, and some that require significant synthesis of earlier work. I point out to students that this sort of selective grading gives a fairly representative sample of their work when I assess work like this every 2 weeks or so throughout the semester.
As described in the handout “Solutions and Proofs – Instruction for Writing and Details of Assessment," there are five grading categories with 5 points available in each category. See the Grading Rubric in the box below for details. Each graded problem receives full credit (indicated by a +), loses one point (indicated by a check), loses two points (indicated by a -) or loses two/three points (indicated by a cross). Points are subtracted from the deficient categories so when the students see their final score they also have an indication of the areas that need attention moving forward. To keep track of which areas points are lost I have a 5 by 6 grid with the scores 5-4-3-2-1-0 across the top and the assessment categories down the side. Five pennies serve as counters and the counters are adjusted as I grade each problem. When all problems are graded the scores easily transferred from this tally sheet to the student’s paper. (This approach was suggested by my former Chair, Catherine Lilly.)
Mathematical Correctness and Completeness Is your solution/proof mathematically correct? Are all of your sentences/claims legitimate? Is your solution/proof complete or does it contain gaps and/or limitations? Depth of Understanding Does your solutions demonstrate an understanding of both the problems at hand and your proposed solution/proof? Are there important aspects of the problem that you have neglected to consider? Justification and Explanation Have you justified your reasoning? Have you clearly explained your thinking? Does your writing compel the reader to believe that you truly understand the problem and have an appropriate solution/proof? Coherence and Clarity Are your solutions coherent and readable? Has your written work clearly expressed the mathematical intent of your solution/proof? Is the identity of objects you refer to clear? Neatness, Organization, Grammar, Spelling, and Effort The presentation and mechanics of your solutions are important. It is also crucial that your solutions indicate that you have expended sufficient effort in solving the problems and presenting the solutions as described here.
Early in the semester it is important to model appropriate proofs/solutions and to have students drafts analyzed and critiqued to illustrate how I will assess the work. See the classroom video on "Sharing examples of strong student writing" at the top of this blog for an example. When doing the actual grading earlier in the semester it is also important to provide more detailed comments for feedback. Grading a chapter write-up (about two weeks of work for a class of 35) takes about 3 hours. Later in the semester students have caught on to what is expected and the grading is significantly easier, taking perhaps 2 hours.
A second sample student write-up for Chapter 1 – Fibonacci Numbers is included here. Selected investigations are graded.
I grade harshly, taking off points for all mistakes. As the semester progresses the quality of the work improves significantly. Realizing that there is a great deal of good work represented by assignments with scores like 17/25 or 18/25, grades are not computed on an absolute scale but on a curve. The assignments given as samples here represent B+/A- work.
It is natural to wonder how much students “share” solutions as this is such a large proportion of their grade. In fact, this is not really an issue. Students come to class (they like it and there is a strict attendance policy), they are engaged in the mathematics, it becomes personal challenge, and they are invested in it. So they take ownership of the entire process – including the written solutions/proofs which they take great care in constructing. Additionally, because explanations are required it is harder for them to read and understand somebody else’s solution than it is to go back to their own notes and craft their own answers.