Crossroads in Mathematics

Appendix:

Problem Solving

Most visitors to the program thought that the heart of our project was group learning. . . . But the real core was the problem sets which drove the group interaction. One of the greatest challenges that we faced and still face today was figuring out suitable mathematical tasks for students that not only would help them to crystallize their emerging understanding . . . , but that also would show them the beauty of the subject. (Treisman, 1992, p. 368)

Problems provide opportunities for students to learn and do mathematics. The mathematical tasks faculty present to students shape what and how they learn and are critical in assessment of their learning. Within a traditional instructional sequence, problems are typically given to students after they have learned some piece of mathematics as a way to apply what they have learned. However, problems can also be used as a catalyst for learning. They can be the driving force leading to student discovery and invention. In a course driven by problems, "mathematics is what you have left over after you have invented ways to solve a problem and reflected on those inventions" (A. Selden and J. Selden, 1994, p. 5, paraphrasing Robert B. Davis).

This appendix presents a variety of examples of problems aimed at capturing the spirit and vision of the standards in this document. The problems are arranged in order of increasing difficulty and complexity. Most offer story lines. The level of realism varies, with some problems using genuine data.

A variety of content strands are represented, with an emphasis on material that traditionally has not been highlighted in introductory college mathematics. Brief remarks address the nature and role of each problem. The solutions given or suggested often span several levels of mathematical sophistication. Hence, many of the problems could be used in a variety of courses or within a single class in which students' backgrounds vary significantly. Such problems should help the reader see how the same standards can and should apply to all mathematics taught in college below the level of calculus.

The problems assume students have access to modern technology. Calculator and computer technology is becoming increasingly ubiquitous, portable, affordable, and user-friendly. This technology gives students access to certain mathematical ideas and problems at an earlier stage in the students' development than is possible without technology. Technology opens the door to exploration and experimentation because many examples can be investigated in a brief span of time. Learning mathematics empirically by discovery, with carefully sequenced tasks and the guidance of knowledgeable faculty, is a much more viable instructional option with technology than without.

The problems can accommodate a variety of teaching and learning styles. However, it is generally assumed that the mathematics classroom is a place where students are "engaged in collaborative, mathematical practice--sometimes working with others in overt ways, and always working collaboratively with peers and with the teacher in a sense of shared community and shared norms for the practice of mathematical thinking and reasoning" (Silver, 1994, p. 316).

Many of the problems can best be solved by students working in small groups. Some may be used as classroom or homework exercises. The more extensive problems may be used as laboratory projects. To get students to organize their thoughts and express them in writing, individual or group reports can be required. Students will generally write better reports if they are given detailed directions. Here is a sample set of directions for a group laboratory report:

All of the work on this lab should be done collaboratively. You will be asked to grade your own effort as well as the efforts of the other members of your group. Within the lab report clearly indicate the primary author for each section. Authorship should be shared fairly among the group members. Each lab report should have three parts: (a) Begin with a paragraph of introduction giving an overview of the nature and purposes of the lab in your own words. (b) The introduction should be followed by a write-up of each activity--what you did, how you did it, and what conclusions you reached. Include tables and graphs as appropriate. Show your work and indicate your thought processes in an organized fashion. © Each report should end with a paragraph of summary, conclusions, and reflections.

Producing, Gathering, and Using Genuine Data

Problem solving can be made more meaningful to students if they are given opportunities to produce, gather, and use data. Gathering and producing data and all of the related issues form an important area of statistics. One need not go into all of these issues in detail, however, to use such data throughout introductory college mathematics. It is best to use genuine data, that is, real data from real sources. These can be gathered by students through computer and calculator gathering devices, library research, and surveys.

Modern technology makes obtaining real-time data much easier. This can be done by linking probes to computers or graphing calculators via special data collection devices that are readily available. Probes, or sensors, can measure temperature, light intensity, voltage, motion, sound, acidity, and other variables of scientific interest. These data can be instantly transferred into the memory of a computer or graphing calculator and used in data analysis and modeling.

Library research and surveys are also excellent ways to obtain real data. Newspapers, magazines, and simple 15- to 20-item class surveys that include such variables as height, shoe size, and gender offer a rich source of authentic data for statistical analysis. Surveys can build a sense of class unity and identity. They also offer faculty the opportunity to introduce students to the ideas of random samples and bias in data.