Coding for Level of Difficulty:
For a full explanation, see A Rubric for Coding Problem Difficulty, from: Renninger, K. A. & Feldman-Riordan, C. (in preparation). "Technology as a tool for developing students' mathematical thinking." (The help of Crystal Akers and Alice Henriques in clarifying this coding scheme is gratefully acknowledged.)
Coding of problem difficulty focuses on the mathematical challenges represented by the problem, the difficulty of the mathematical concept, and the difficulty of mathematical calculations for students at a given level of problem solving. The rating scale consists of 5 levels of difficulty, wherein a Level 5 problem is a very difficult problem for students in a given grade band.
Level 1. Only one concept needs to be worked on; the mathematics is rudimentary and represents prior knowledge rather than something new. Example:
A rectangle is divided into four rectangles with areas 45, 25, 15, and x. Find x.
This problem asks the student to find the factors of three values and to identify common factors. For geometry students, finding common factors should be prior knowledge.
Level 2. Either a) the concept is clearly stated within the problem, and the mathematics is challenging for students at the given level of the PoW, or b) the concept requires some "stretching" for students at this level, and the mathematics is based on prior knowledge. (Note: Problems that require attention to explanation are likely to be found at Level 3, rather than Level 2, because of the difficulty involved in explaining mathematical understanding.) Example:
Given the surface area of three sides, find the dimensions and volume of a rectangular box.
The math concepts in this problem should be easy, since they involve finding common factors from the surface area, and then using these to find the volume of a box. The problem is difficult, however, because the students are asked to identify and explain their solution path.
Level 3. The problem (a) contains a "twist" or additional problem requirement that students in this grade band may overlook even though they can complete the problem accurately, and (b) requires discourse knowledge of mathematical concepts and basic mathematical ability appropriate to students at this level. (Note: At the elementary level, multiple parts within a problem make what may initially appear to be a Level 3 problem into a Level 4 problem.) Example:
Two congruent circles are drawn, and four congruent chords are drawn, two in each circle, all perpendicular to the diameter through both circles. The distance between the two furthest chords is 20, and the distance between two chords of the same circle is 8. What's the area of one of the circles?
In this problem, students may overlook the need to explain why the congruent chords are equidistant from the centers of their respective circles. In addition, the student needs to figure out how to get the information needed to provide the area of a circle, since this information is not provided.
Level 4. The problem includes the elements listed in difficulty Level 3 and contains an algorithm new to students in this grade band; students may miss the problem by getting bogged down in the math but not by missing the concept; students may not finish the problem or may not attempt all parts of the problem. Example:
Take any triangle ABC. Construct D and E as the midpoints of BC and AB, respectively. Now construct DF and EF, where F is any point on AC. How are the areas of the triangles related to the area of the quadrilateral?
It is unlikely that students in this grade band will expect that the two areas may be the same. They also need to provide a proof and a clear explanation, adding to the difficulty of the problem. Here students could get bogged down in the math, or not know how to reach the correct conclusion.
Level 5. The problem includes the elements listed in difficulty Level 3 and requires discourse knowledge of mathematical concepts and mathematical ability above the level of the students in this grade band; or it contains a concept, theorem, or algorithm that a rater familiar with this mathematics topic does not recognize. Example:
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