Solutions to the Equation [x]x = n

I came upon the following question (Posamentier and Salkind 1996) while looking for nonroutine problems to give to my first-year algebra students:

Bugs, Planes, and Ferris Wheels: A Problem-Centered Curriculum

Mathematical power includes the ability to explore, conjecture, and reason logically; to solve nonroutine problems; to communicate about and through mathematics; and to connect ideas within mathematics and between mathematics and other intellectual activity.

News from the Net: Math Fundamentals

The Math Fundamentals Problem of the Week, located at mathforum.org/funpow, features nonroutine problems to challenge upper-elementary-grade students to think and communicate mathematically. Posted every two weeks during the school year, the problems focus on numbers, operations, and measurement but also may include introductory geometry, probability, and data.

The Thinking of Students: The 100th Letter

Solving nonroutine problems with multiple approaches is one of the goals promoted by the NCTM's Curriculum and Evaluation Standards for School Mathematics (1989). I present two problems that seem to fit the bill, since my seventh-grade students worked on them in various ways.

The Changing Role of the Mathematics Teacher

This case-study research investigated changing teacher roles associated with two teachers' use of innovative mathematics materials at Grade 6. Using daily participant observation and regular interviews with the teachers and the project staff member responsible for providing in-school support, a picture emerged of changing teacher roles and of those factors influencing the process of change. One teacher demonstrated little change in either espoused beliefs or observed practice over the course of the study. The second teacher demonstrated increasing comfort with posing nonroutine problems to students and allowing them to struggle together toward a solution, without suggesting procedures by which the problems could be solved. He also increasingly provided structured opportunities for students' reflection on activities and learning. Major influences on this teacher's professional growth appeared to be the provision of the innovative materials and the daily opportunity to reflect on classroom events in conversations and interviews with the researcher.

Exploring Patterns in Nonroutine Problems

For the past several years, we have been examining ways to introduce notions of algebra to our heterogeneously grouped middle school students. Through our experiences of designing instructional activities that integrate nonroutine, nontraditional problems. we found a meaningful way to involve students in exploring and formalizing patterns, conjecturing about the patterns they identify, verbalizing relationships between and among elements in patterns, and eventually generalizing and symbolizing the relationships—all essential components of algebraic thinking (Silver, in press).

Connecting Research to Teaching: Affective Responses to Problem Solving

The vision of the mathematics classroom that IS presented 1n the Natwnal Council of Teachers of Mathematics's Curriculum and Eualuation Standards for School Mathematics (1989) has inspired many of us to want to change the way in which we teach. We want to pose challenging problems, to see our students work cooperatively, and to have productive discussions with students about significant mathematical ideas. But as Ball and Schroeder have pointed out, that vision is “much more difficult to realize than to endorse” (1992, 69). We will encounter many difficulties as we move toward that ideal classroom of the future; getting students to respond positively to nonroutine problems or other tasks that require higher-orderthinking skills is one difficulty that teachers often face. Research suggests that students' affective reactions to nonroutine problems can be a source of both difficulty and support as we work to reform mathematics classrooms.

Ideas

The “IDEAS” section for this month focuses on combinations, an important part of discrete mathematics probability. from the point of view of the NCTM's Curriculum and Evaluation Standards (1989). As we continually strive to make mathematics an area for problem solving. we see the need to investigate questions from problem situations and to connect mathematics to the outside world. Students are encouraged to discover all the combinations for a given problem; they should experience estimating, eliminating, collecting data in an organized manner, and drawing conclusions. Active involvement with a set of crayons allows all students to attack nonroutine problems. Forming generalizations — or mathematical statements to explain the phenomena — can extend the upper-grade activities. The use of such higher-level-thinking skills as synthesis, analysis, and evaluation replaces working on tedious worksheets and memorizing rules and algorithms. Students explore relationships, discover ways to accomplish tasks, and make predictions about outcomes without being presented with prescribed formulas.