# Problem of the Month

**The following information was published by the Noyce Foundation (see copyright below), and explains Problem of the Month. We will usually work on these problems in class, but may send them home for homework from time to time.**

**Why Problem of the Month?**

Problem solving is the cornerstone of doing mathematics. George Polya, a famous mathematician from Stanford, once said, “a problem is not a problem if you can solve it in 24 hours.” His point was that a problem that you can solve in less than a day is usually a problem that is similar to one that you have solved before, or at least is one where you recognize that a certain approach would lead to the solution. But in real life, a problem is a situation that confronts you and you don’t have an idea of where to even start. Mathematics is the toolbox that solves so many problems. Whether it is calculating an estimate measure, modeling a complex situation, determining the probability of a chance event, transforming a graphical image or proving a case using deductive reasoning, mathematics is used. If we want our students to be problem solvers and mathematically powerful, we must model perseverance and challenge students with non-routine problems.

**How should the Problems of the Month be used?**

The Problems of the Month are designed to be used school-wide to promote a problem- solving theme at your school. Each problem is divided into five levels, Level A through Level E, to allow access and scaffolding for the students into different aspects of the problem and to stretch students to go deeper into mathematical complexity. The goal is for all students to have the experience of attacking and solving non-routine problems and developing their mathematical reasoning skills. Although obtaining and justifying solutions to the problems is the objective, the process of learning to problem-solve is even more important. Administrators, teachers, and parents should facilitate and support students in the process of attacking and reasoning about the problems. Students’ self- analysis of how they went about approaching, exploring, and solving the problems is a critical step in the development of becoming a strong problem solver.

The Problem of the Month is structured to provide reasonable tasks for all students in all grades K-12. The Problems of the Month are designed so all learners start at a level of A (either Primary Level A or Level A). Where a learner will be successful and challenged will depend on his or her maturity as a problem solver. It should not be a race to get through the levels; rather a learner should stay at a level and fully investigate that level (or tangents to the problems at that level) before proceeding to the next level. George Polya states, “It is better to solve a problem five ways than to solve five problems.” A learner should be able to fully explain answers and justify solutions before proceeding to the next level. The Problem of the Month’s levels are related around a big idea. Learners work through the levels to deepen their understanding of the big math idea. Often a learner will benefit from additional exploration of a level, using different assumptions or finding related problems to investigate.

Even though all learners should start at level A, there is a general rule of thumb that may apply – although each Problem of the Month is slightly different in access points and levels of complexity, so the rule may not always hold true. With that caveat, a general rule for learners’ success at a given level are: Primary Level A is designed to be accessible to all students and may be challenging for primary students in Kindergarten and 1st grade. Level A may be accessible for some students in 1st grade, and most 2nd and 3rd grade students will be successful working on these tasks. Level B may be the limit of where 3rd and 4th grade students will have success and understanding. Level C may stretch 5th or 6th grade students. Level D may challenge most middle school students, and Level E should be challenging for most high school students. These grade level expectations are just estimates and should not be used as an absolute minimum expectation or maximum limitation for students. Problem solving is a learned skill, and students may need many experiences to develop their reasoning skills, approaches, strategies, and the perseverance to be successful. The Problem of the Month builds on sequential levels of understanding. All students should experience Level A and then move through the tasks in order to go as deeply as they can into the problem. There will be those students who will not have access into even Level A. Educators should feel free to modify the task to allow access at some level.

One caution – the solution is not as important as the process of problem solving. Struggling to get started is a natural part of learning to problem-solve. The educator or parent should not be impatient with the student’s struggle. In fact encouraging and supporting the struggle with some frustration is exactly what the student needs. If a method is shown or told then the problem-solving process ends. Asking good, but not guiding, questions that require students to reflect and focus are most helpful. Having the student carefully read or be read the problem and then having the student restate the problem is often valuable. Having the student talk through the approach or the challenge is also effective in having the student rethink a strategy. Encouragement is the important key. In class, a teacher might have different students share their thinking (approach, not solution) with the class. The teacher should be careful not to assign value to the approaches. If students seem to follow an error in their process, the teacher should pose questions that make the class examine the process to uncover the error. A good problem- solver tries, fails, re-evaluates and tries again.

If some students are able to complete all levels with detailed and accurate solutions and justifications, then the students may be challenged to go deeper into a problem or to extend it into another problem that is more complex. For example, in the Cutting the Cube problem, you may ask the student to formally prove the number of unique hexominoes. You could also extend the problem for them to explore septominoes (7 square figures) and develop number patterns for unique solutions of any nth-omino.

Problem of the Month Guidelines Pages 1 & 2 ©Noyce Foundation 2004.

This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 Unported License (http://creativecommons.org/licenses/by-nc-nd/3.0/deed.en_US).