Words and their meanings may present the greatest obstacle to mathematical problem solving for students. Reading for understanding and writing for explanation are at the foundation of successful learning in all disciplines. It is quite clear that as a high school teacher, I depend greatly on my colleagues teaching in younger grades to help students learn to read and write. Every year I rediscover how much I, too, need to coach and encourage basic communication skills in my students. If any data drives this point dramatically, it is the recent “Reading at Risk” survey which found that “fewer than half of all Americans over 18 read novels, short stories, plays or poetry . . . and the pace at which the nation is losing readers, especially young readers, is quickening.”1
In other words, I cannot just “teach math” with the assumption that all can speak, read and write with ease any more than I can assume that all my matriculating ninth graders can solve problems using fractions, decimals, percents, measurement and graphing. Our students vary greatly in all aspects of their development and understanding. Figuring out where each individual stands along numerous continua of conceptual and expressive development ought to be a shared task of teachers of all grades working together to develop and repeatedly teach useful “tools” for analyzing, thinking about and solving problems (and not only mathematical ones).
Word problems represent the “real world” of money, measurement, information, consequences and arguments. At all levels, mathematics curricula should include and embrace word problem solving. Such exercises not only review and reinforce computational skills, but also they challenge students to shift from verbal situations to patterns, tables, graphs, equations and back. Building problem sets, which explore key arithmetic concepts and relate symbols to the concrete realities of students’ lives, can moderate the abstract and remote nature of “algebra” for many students.
In this unit, I look at a few types of algebra problems and discuss ways of solving them that might work on a range of grade levels. First, I briefly discuss the Vertical Team2 concept and its value for mathematical problem solving. Second, I introduce the idea of “problem space”, a phrase frequently used by our seminar leader, Professor Howe. Third, I propose that along with our curricular sequencing we look for and practice deliberate techniques that work each year for progressively more complex and intricate problems. Students thus will be building their own “toolkits” for algebraic problem solving skills, which they can expect to master and apply as they advance through the grades. Finally, I present and discuss several model problem sets designed for classroom activities in developing algebraic problem solving.