Instructional decisions about the nature and extent of teacher input in mathematics classrooms typically lie on a continuum somewhere between teachers directly telling students how to carry out mathematical procedures and free exploration situations where teachers provide little or no guiding input. Making decisions about instructional input is an area of practical concern for many teachers. Some worry that by providing too much information they may deprive their students of opportunities to reason and build knowledge of mathematical relationships. Others struggle with concerns that their students who freely explore mathematical situations with little direction may fail to reach established curriculum goals in a timely manner.
The challenge of instructional decision-making has been the subject of much research and scholarly consideration (Cobb, Yackel, & Wood, 1991; Lampert, 1985). Several researchers have identified characteristics for appropriate teacher intervention in mathematics classrooms. Chazan and Ball (1995) pointed out that it is appropriate for a teacher to contribute to and shape a classroom discussion by inserting substantive mathematical comments aimed at moving students away from entrenched disagreement or to provoke useful disagreement. Rittenhouse (1998) suggested that teachers might explicitly teach vocabulary, rules and conversational norms associated with a developing mathematical discourse, and conventional notation for a distinction that students already have made. Brousseau (1997) proposed a learning theory based upon strategic problem situations selected by a teacher to engage learners and facilitate their development of culture-specific mathematics knowledge and in which learners reason without teacher inp ut to produce knowledge intended to be useful beyond the immediate situation. But many teachers remain uncertain about judging the appropriateness of their input in the development of students' mathematics knowledge.
How can teachers insert substantive mathematical comments and teach notational conventions without impairing opportunities for learners to develop meaning from their instructional experiences? How can they insure the continuation of learning when students appear unable to progress? To address these concerns, one must consider the questions of what understanding in mathematics means and how teachers' inputs influence students' understandings. According to Lesh (1979), the ability to translate from one mode of representation to another is an important way by which students make mathematics meaningful. He identified five nondistinct representational modes: real-world situations, manipulative models, pictures, spoken symbols, and written symbols. Linking mathematics knowledge constructed in one representational format to another is valued as an indication of students' understanding of mathematics (Lesh, Landau, & Hamilton, 1983). Kaput (1989) described representational systems in which learners use their knowledg e of mathematics developed in one representational format to give meaning and to extend their knowledge of mathematics in a different representational format. When teachers assist learners to develop links between representations, they help their students to learn mathematics. By insuring that those links connect a "known" representational format to a less familiar format, teachers maximize the opportunities for reasoning and building knowledge of mathematics relationships.
Tracing its roots to Vygotsky's theory of intellectual development (1962, 1978, 1983), the notion of scaffolding may potentially provide an instructional mechanism through which students might be enabled to create links between representations. Wood, Bruner, and Ross (cited in Wood, 1989) describe "scaffolding functions" through which a more knowledgeable individual helps novices extend their competence beyond levels of their individual capability. Scaffolding is a term used by many to refer to the language-based guidance provided to a novice learner by a more knowledgeable individual (e. …