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:
Aidan is how many days younger than T.J.?
The concept is clearly presented in this problem. Solving the problem only involves prior knowledge about the number of days in a year.
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:
Charlene needs to add some items to her winter collection. Help her determine where she should shop.
The concept in this problem is easily presented, but the math is challenging because the students have to calculate the percentage reduction before shipping and handling from one of the store orders. There is no "twist" to the problem.
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:
How many half gallon cartons of ice cream can fit in the freezer?
This problem is appropriate for students in this grade band but it contains a twist. Because you have to fit whole boxes into the freezer, students cannot just divide the volume of the individual boxes into the volume of the freezer and answer with a decimal.
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:
Help the poor farmer predict his revenue (if he can sell all his fruit).
The expected discourse knowledge of mathematical concepts and the basic mathematics ability required to solve this problem is appropriate to students in this grade band, but they may get bogged down in the many calculations it requires. The twist in this problem is that students have to deal with many different fractions of a row.
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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