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:
What dimensions for a given page will use the least amount of paper?
Students in this grade band should have the mathematics ability to solve this problem. The concept is clearly stated within the problem and no twist is necessary to solve it.
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:
Find the maximum volume of a rectangular prism constructed from an 8.5 in. by 11 in. piece of paper.
In this problem, the concept of finding the maximum volume of the box is clearly stated. The mathematics in the problem is challenging for students, however, because they need to create a function for the volume of a three-dimensional object that is being constructed from a two-dimensional object.
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:
Prove that the area of a parabola is 2/3 the product of its width and height.
Students may be challenged by this problem. Although it does not contain a true "twist," it does have an additional problem requirement that students are likely to overlook: that the formula describes the area of any given parabola.
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:
Find the measurement of a spike's angle, given the star's perimeter.
In this problem, the student must be careful to include the three points rather than just the perimeter of a disk when calculating the perimeter of the shuriken. It is a problem in which the student may get bogged down with the math, or not know what to do with all the information given (i.e., the density of the titanium).
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:
Use trigonometry and calculus to find the probability of a needle touching a line.
In this problem, the student is challenged to use prior knowledge of probability, trigonometry, and calculus together in new ways, pushing the mathematical ability required for the problem above the level of students in this grade band. As a result, students may have difficulty figuring out how to set up the problem.
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