Resnick, L. B. (1988). Treating mathematics as an ill-structured discipline. In R. I. Charles & E. A. Silver (Eds.), The teaching and assessing of mathematical problem solving (pp. 32-60). Hillsdale, NJ: Lawrence Erlbaum Associates.
Resnick states that mathematics is traditionally taught as a "well-structured discipline"; i.e., students learn that for certain types of problems there are particular rules, such as formulas, to be discovered, and that following these rules will allow them to arrive at (a single) appropriate answer. Resnick argues, however, that students with this mindset miss opportunities to conceptualize mathematics and find meaning in their learning. She says that in order to foster the development of more meaningful, flexible, and inventive problem solving, mathematics should be taught as an "ill-structured discipline": a domain that invites more than one rigidly defined interpretation of a task.
One way of presenting an ill-structured discipline in the classroom is to encourage students to talk about mathematics--to engage them in dialogue with other students, journals, you as the teacher, etc. Teachers and students alike are to express their mathematical questions and discoveries in "plain English" as much as possible, in order to close the gap between classroom mathematics and the mathematics in students' everyday lives, fostering better conceptual and meaningful understanding, and deterring students from simply manipulating symbolic rules for formal mathematical expression.
Direct Quotes (and some comments):
"[W]hen we teach problem solving, we often present stereotyped problems and look for rules that students can use to decide what the right interpretation of the problem is--so that they can find the single appropriate answer... One result of this common way of teaching mathematics is that many children come to think of mathematics as a collection of symbol manipulation rules, plus some tricks for solving rather stereotyped story problems [such as adding when they read the word "altogether" or multiplying when they read the word "each"]. They do not adequately link symbolic rules to mathematical concepts--often informally acquired--that give symbols meaning, constrain permissible manipulations, and link mathematical formalisms to real-world situations. Widespread indications of this problem include children's... general inability to use mathematical knowledge for problem solving" (p. 32).
"In its purest form, the formalist position would deny any necessary relationship between mathematics and physical reality. But viewed from another perspective, if mathematical statements really had no meaning beyond their relationship to other statements in the same formal system, mathematics could not be used to describe patterns or relationships in the world or to draw inferences of new, not yet observed patterns and relations. Mathematics would be merely an intricate game, entrancing to those who loved it, but of no general value to society. Like music, it would be valued for emotion and aesthetic qualities, without reference and 'utility.' But mathematics is useful. It helps us describe and manipulate real objects and real events in the real world. Mathematical expressions, therefore, must have some reference" (pp. 33-34).
"We encounter an explosion of interpretations... when we include as potential referents for mathematical statements the actual things in the world to which abstract mathematical entities can be reliably mapped--what we might term 'situations that we can mathematize.' If we are willing to treat mathematizable situations as in some sense the potential referents of mathematical statements... we must recognize that there is no single meaning for a mathematical expression and no single reason the relationships it expresses are true" (p. 34).
"We usually regard mathematics problem solving, or at least the part that treats real world problems, as a process of building a mathematical interpretation of a situation and then using a formal, fully determined system to manipulate relationships that have been 'mathematized' by this interpretation...[T]reating mathematizable situations as [having] potential referential meaning of mathematical expressions... opens mathematics to interpretation and construction of a kind that is an inherent part of all language use" (p. 34).
"... many students have relevant, informal knowledge that they do not normally draw upon in thinking about formal expressions...[I]nstruction focused on the task of interpreting mathematics expressions as mathematizations of possible real-world situations seems essential to their development as mathematics problem solvers" (p. 38).
"At the heart of the suggestion that we teach mathematics as an ill-structured discipline lies the proposal that talk about mathematical ideas should become a much more central part of students' mathematical experience than it now is. This will inevitably entail greater use of ordinary language, rather than the specialized language and notation of mathematics, in mathematics classrooms" (p. 53). [In other words, teachers should encourage dialogue revolving around mathematical ideas, including debates, multiple interpretations, and small group work. As dialogue becomes a more important part of learning mathematics and considering mathematical concepts, the language in the mathematical classroom should include more everyday language; "under current teaching conditions students have little opportunity to develop a vocabulary that expresses their implicit knowledge of mathematical concepts" (p. 53).]
"When students themselves generate linguistic expressions of mathematical arguments, we move closer to natural language discussion of mathematics. A recent book by Eleanor Wilson Orr (1987), a teacher who has for many years required students studying algebra and geometry to develop informal, natural language justifications for problems they work, describes some of the difficulties that may be encountered in such a program. Orr's book documents the ways in which some students' language may fail to encode precisely key mathematical relationships. These relationships include distinctions between distances and locations, between directions of movement, and between quantities and differences among quantities. Orr is concerned that some students may be particularly poor at expressing these mathematical relationships linguistically... The success and difficulties of a mathematics teaching program that has been grounded in natural language expression will suggest many new questions for systematic investigation" (pp. 53-54).
"Considerable evidence now exists... that story problems do not
effectively stimulate out-of-school contexts in which mathematics
is used...[T]he language of story problems is highly specialized and
functions as almost quasi-formalism, requiring special linguistic
knowledge and distinct effort on the part of the student to build a
representation of the situation described. Furthermore, this
representation, once built, is a stripped down and highly schematic
one that does not share the material and contextual cues of a real
situation" (p. 56). [In other words, story problems are not as "real
world" context-oriented as we like to believe. In fact, they require
of the student special formulaic, mathematical knowledge, such as
interpretation of formal structure of the story problem and
identifying key words for particular operations. Story problems
become, in and of themselves, part of the students' collection of
symbol manipulation rules.]
"If we are to engage students in contextualized mathematics
problem solving, we must find ways to create in the classroom
situations of sufficient complexity and engagement that they
become mathematically engaging contexts in their own
right...Some of Lesh's (1985) extended and not fully defined
problems, for example, can be thought of not as stories containing
mathematics problems, but as settings in which planning a project
(e.g., wallpapering a room) engages a substantial amount of
mathematical knowledge and strategy... Computerized simulation
environments can also provide settings for highly contextualized mathematical activity" (pp. 56-7). [Wallpapering a room and computerized simulation environments are examples of efforts to develop more contextualized problems for classroom use. Such contextualized problems help to integrate conceptual and problem->solving activities.]
Lesh, R. (1985). Processes, skills, and abilities needed to use
mathematics in everyday situations. Education and Urban Society,
Orr, E. W. (1987). Twice as less. New York: Norton.
-- Summarized by Maria Ong Wenbourne, Swarthmore '94