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Coding for Level of Difficulty:
Discrete Math Problems of the Week

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:

    Discrete Mathematics, difficulty level 1: Mother's Chores
    A Traveling Salesperson Problem. Tom's mother needs to make several stops around the city and then get home. What is the shortest route without going back to one of the stops?

    In this problem, there is only one concept that needs to be worked on, finding the shortest round trip for the mother to complete her errands. The problem can be solved using prior mathematical 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:

    Discrete Mathematics, difficulty level 2: The Locker Problem
    This classic number theory problem investigates properties of prime numbers, perfect squares, and counting in a problem that involves opening and closing locker doors.

    This problem could be solved by creating a model and simulating the opening and closing of the lockers. The mathematics involved, factors and multiples, is prior knowledge for students in this grade band. The requirement to "discover" a pattern and to determine which lockers are touched exactly twice, however, makes this a level 2 rather than a level 1 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:

    Discrete Mathematics, difficulty level 3: The Fourth of July Parade
    After the parade, the people on the float I was on shook hands with each other. The Mayor came over and shook hands with only the people he knew. How many people did he know if there were 1625 handshakes altogether?

    Although this problem may at first appear to be the traditional handshake problem, it has a slight "twist": the mayor only shakes hands with the people that he knows. Providing an additional challenge, solving the problem requires that students explain mathematically how to account for this "twist."

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:

    Discrete Mathematics, difficulty level 4: First Presidential Veto
    This problem involves the Jefferson, Adams, and Webster apportionment methods.

    This problem is complicated by the fact that students must consider three different apportionment methods. Students could become bogged down in the calculations in any of the methods. An added requirement is that students recognize that the U.S. Constitution guarantees that every state is entitled to at least one representative. Missing this element may lead students astray.

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:

    Discrete Mathematics, difficulty level 5: The Dating Game
    This problem is based on the Stable Marriage Algorithm, which requires students to make the best match possible between a set of girls and a set of boys desiring to date each other.

    This problem requires that students learn about a new concept, the Stable Marriage Algorithm, and understand it well enough to apply it to the problem. The problem also includes many smaller questions that students might overlook.

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