### CUWS - A Framework for Conceptual Understanding

 Revised Conceptual Understanding Weighting System (CUWS)   Check 1 (Skills): Does this task involve only computational/algebraic skills or memorized facts with no understanding of the concept required? Yes - assign a weight of 0 (Level 0) to the task and skip the remaining checks; No - continue to the next check. Check 2 (Method): Could this task be answered completely using a method that the student or teacher might have developed in prior course work? Yes - assign a weight of 1 (Level 1) to the task and skip the remaining checks; No - continue to the next check. Check 3 (Conceptual Understanding): Does the task involve any of the following? No - assign a weight of 2 (Level 2 - conceptual understanding) to the task; Yes - assign a weight of 3 (Level 3 - reasoning from conceptual understanding) to the task. ·       interpreting the meaning of a mathematical characteristic in a novel setting ·       making connections between different representations (numerical, graphical, symbolic or narrative) of a mathematical characteristic ·       evaluating mathematical statements and providing relevant examples or counterexamples as appropriate·       analyzing sample work on a task and identifying flaws with accompanying reasoning ·       proving or justifying

Background:
While considering how well the exam questions that we had written assessed our students' conceptual understanding, we realized that we were asking ourselves a series of questions about each problem. We decided to try to formalize this questioning process so that we would then have a tool for evaluating the extent to which conceptual understanding is being assessed by a given problem in our course. That tool could then be used to weight the scoring of exam problems, giving us scores for our students (separate from their exam scores) that indicate more specifically their level of conceptual understanding on that exam. In addition, this tool also gives us the ability to track our students' proficiency at specific types of mathematical understanding.

To begin with, we recognized that in every math course there are computational and/or algebraic skills needed for the work in that course. We want our students to become proficient with these skills, so we put some problems on exams that assess these skills. These skills problems, however, do not require any conceptual understanding so our weighting system should assign them a weight of zero. We then began to realize that on some of the problems that we had specifically written to test students' conceptual understanding, students were in fact more likely answering the problem by adapting methods that they had developed previously in class work. Being able to adapt a method in this way demonstrates some conceptual understanding but not as much as when a student must reason directly from the meaning of a concept. Our weighting system should assign these types of problems a weight of one. If no previously developed method can be applied to solve the problem, we assume that students must be reasoning from concepts in order to answer that question. Our weighting system should assign a weight of at least two to these types of problems. If in addition students were required to interpret the meaning of a mathematical characteristic in a novel applied setting, and/or the students were required to make connections between different representations (numerical, graphical, symbolic or narrative) of a mathematical characteristic then students are demonstrating an even deeper understanding of concepts when successfully answering that question. To reflect this, these types of problems should receive a weight of three. This Conceptual Understanding Weighting System (CUWS) weights each problem (weights ranging from 0 to 3) according to the extent to which the problem assesses students' conceptual understanding - depending upon both the characteristics of the problem itself and how a student has previously encountered (in this course) the concept/s being tested by the problem.