Lesh, R. (1985). Processes, skills, and abilities needed to use mathematics in everyday situations. Education and Urban Society, 17(4), 439-446.
Lesh criticizes traditional textbooks and teaching methods, saying that their one-step problems rarely exercise students' skills, and that they do not reflect real-life mathematical situations.
Lesh believes that there should be as much--or more--emphasis in the mathematics classroom on understanding mathematical concepts and possible mathematical relationships as on accurate computation. Students should be taught to recognize situations in which their mathematical skills can be utilized.
Mathematizable situations in the classroom, such as wallpapering a wall or balancing a class budget, cause students to engage in multiple mathematics processes and to learn how mathematical concepts are related to one another in a useful and meaningful way. Such experiences also require students to talk and think about mathematics with one another and with the teacher.
"A goal of the project was to identify important processes, skills, and understandings that are needed by students to use mathematical ideas in everyday situations. A substantial part of the project consisted of students working together in small groups on problems that might reasonably occur in the normal lives of the students and their families: balancing a checkbook, planning a vacation within a budget, wallpapering a room, estimating distances using a map, and so on" (p. 439).
"Getting a collection of isolated concepts in a youngster's head (e.g., measurement, addition, multiplication, decimals, proportional reasoning, fractions, negative numbers) does not guarantee that these ideas will be organized and related to one another in some useful way; it does not guarantee that situations will be recognized in which the ideas are useful or that they will be retrievable when they are needed" (pp. 439-440).
"The 'back to basics' movement that is currently influencing many of the nation's schools is often uninformed and misdirected. Results of National Assessment Tests, for example, show that 'Johnny can add; computation with whole numbers is far from a lost art' (Carpenter et al., 1975: 457). In fact, although there are plenty of students who do poorly on computation problems, results from most large-scale testing programs show that there are fewer of them today than at any time in the past. Today's youngsters run into difficulty in making inferences, solving problems, evaluating the reasonableness of results, using references to 'look up' what they need to know, and so on. It is the complex skills, not the basic skills that are deteriorating. What we need is to get back to complexity, where thinking is required in addition to simply knowing some isolated fact or procedure. In realistic situations in which mathematics is used, question asking, information gathering, and trial-answer evaluating are often more important than simple answer giving. Real problems usually require more than simple one-step solutions" (p. 441).
[On a problem about individuals' overall performances in sports events, in
which multiple pieces of information were given, including scores on
running and jumping and qualitative comments from the coaches, the
students] "seemed driven to do some messy calculation with the numbers,
sensible or not. In fact, most of the students were quite skillful at the
arithmetic of *numbers*, but many real problems involve the arithmetic of
*quantities* -- a skill that mathematics textbook authors tend to assume is
covered in science, and science textbook authors tend to assume is covered
in mathematics" (p. 444).
"In problems that require multi-step solution procedures, it is important
to *plan* what you are going to do before doing it; *monitor* what you are
doing *while* you are doing it; and *evaluate* the sensibility of your
results. These skills are seldom practiced in simple one-step textbook
problems... The students who participated in our project did improve in
their ability to deal with problems [like the one described above], which
we believe are similar to the kinds they will meet in everyday situations,
job situations, or later mathematics courses. To some extent, our students
improved because we worked with them individually on some specific skills:
graphing, measuring, estimating, and so forth. They probably also got
better on problems that require planning, organizing, and recording simply
because our problems required them to practice and use these skills.
Further, they were forced to reorganize their mathematical ideas because
our problems usually involved more than a single concept. The more
organized character of their knowledge should benefit them greatly in
future mathematics courses" (p. 445).
Carpenter, T. P., T. G. Coburn, R. E. Kays, and J. W. Wilson (1975).
Results and implications of the NAEP mathematics assessment: secondary
school. Mathematics Teacher, 68, 453-470.
Summary by Maria Ong