Lampert, M. (1986). Knowing, doing, and teaching multiplication. Cognition and Instruction, 3(4), 305-342.
Lampert describes experimental lessons and explains her view of the teacher's role in the classroom: to help students make explicit their ideas about analyzing and solving problems, to act as referee in arguments about the reasonableness of competing ideas, and "to sanction students' intuitive use of mathematical principles as legitimate." She requires that students decide in their own way whether something is mathematically reasonable, and sees instruction in mathematics content as inseparable from "building a culture of sense-making in the classroom, wherein teacher and students have a view of themselves as responsible for ascertaining the legitimacy of procedures by reference to known mathematical principles."
Lampert points out that because the social organization of the traditional elementary school classroom consists of large groups under one teacher, a sequential, standard curriculum within grade levels, and an assessment system that clearly distinguishes between right and wrong answers, it's easiest to teach mathematics primarily or even exclusively as computation, which then comes to be considered 'basic', the only kind of knowledge appropriate to teach directly, and the necessary foundation for principled knowledge.
She continues that it's often assumed--incorrectly--that if principled knowledge is to be acquired at all, "the brighter students will ascertain the principles that underlie the procedures without those principles ever having been the focus of instruction" (p. 312).
Lampert says that to counter this view, she almost always follows up an answer that has been offered without an explanation with a question intended to reveal "how the student figured it out." She does this for two reasons: 1) the way the student responds will show how the answer was arrived at and whether it's warranted, and 2) this approach will "develop a habit of discourse in the classroom in which work in mathematics is referred back to the knower to answer questions of reasonability. This habit needs to be developed because, in the traditional culture of classroom interaction, students have learned to rely on the authority of a book or a teacher to 'know' if their answers are right or wrong rather than asking themselves whether either the answer or the procedure they use to arrive at it makes sense" (p. 317).
Lampert questions whether and how the formal language of mathematical principles should be used in describing what students know and how they use what they know, and argues that "it is important not to create a separation between symbols and what they represent too early in the student's academic career, before the idea that mathematics is a system of principles that make sense has had time to be considered seriously" (p. 338).
[In discussions about double-digit multiplication], "students used principled knowledge that was tied to the language of groups to explain what they were seeing. They were able to talk meaningfully about place value and order of operations to give legitimacy to procedures and to reason about their outcomes, even though they did not use technical terms to do so. I took their experimentations and arguments as evidence that they had come to see mathematics as more than a set of procedures for finding answers. Such discussions would not have been possible without their having learned a symbolic language for representing the steps of the procedure in ways that were meaningfully tied to operations on quantities" (p. 337).
"The experimental lessons described in this article stand as evidence that fourth graders can do mathematics and think mathematically. These students have the capacity to gather information, to organize it strategically, to generate and test hypotheses, and to produce and evaluate solutions. They can talk about what they are thinking, they can listen to and appreciate another student's procedures or way of understanding something, and they can invent problem-solving procedures that are both useful and sensible...
"What sort of help do children need from adults in order to do these things and to be confident of their ability to do them? I would suggest that they need to be asked questions whose answers can be 'figured out' not by relying on memorized rules for moving numbers around but by thinking about what numbers and symbols mean. They need to be treated like sense-makers rather than rememberers and forgetters. They need to see connections between what they are supposed to be learning in school and things they care about understanding outside of school, and these connections need to be related to the substance of what they are supposed to be learning. They need to learn to do computation competently and efficiently without losing sight of the meaning of what they are doing and its relation to solving real problems" (p. 340).
"One [lesson] is on coin problems such as: 'Using only two kinds of coins, make $1.00 using 19 coins.' Constructing solutions to these kinds of problems requires students to use knowledge about mathematical principles that derives its legitimacy from the realm of trading money; coins are so familiar to children that even when these kinds of problems are worked verbally or using paper and pencil, they have a concrete quality about them" (p. 316).
"Finding the monetary value of combinations of coins was an important exercise in working in a context where the order of operations matters; because of the familiarity with how money works, everyone knows that you multiply first and then add to get the value of the coins. This familiarity gave students the opportunity to do mathematics confidently in an area where they would later be introduced to more abstract forms. Because of their intuitive knowledge about how coins worked, they would be unlikely to make the error of adding first and then multiplying, whereas this is a common error when students are using procedural knowledge alone to multiply" (p. 318).
"The second set of lessons uses stories and illustrations to link familiar computational procedures using numbers and arithmetical symbols with legitimate procedures for separating large numbers of groups into parts, counting the quantities in each part, and recomposing to find the total..." (p. 316).
"In my class, the most obvious result from our work on the coin problems was the initiative that many students displayed on other occasions for searching out different decomposition and recomposition strategies when faced with the task of counting groups of groups. This might be thought of as evidence that they had learned that there is a variety of legitimate ways to decompose numbers, operate on them, and recompose to count a total, which might be taken as an indication of principled knowledge" (p. 321).
"After three or four lessons... in which I used students' stories to do drawing and numerical symbolization representing the decomposition process on the board, I constructed assignments in which the children would do their own stories, numerical representations, and drawings on paper with decreasing amounts of teacher direction. In some of those assignments, students were directed to 'find the total' according to whatever decomposition and recomposition method they chose and then to find it again using a different method...
"Some of the children became quite interested in showing many different ways in which they could decompose one of the factors to find the partial products. They were using the language and drawings we had practiced to make a bridge between their intuitive knowledge about how concrete objects can be grouped for counting and the meaning of procedures using arithmetic symbols. By rewarding them for inventing reasonable procedures rather than for simply finding the correct answer, I was able to communicate a broader view of what it means to know mathematics and to learn something from what they were doing about how they would use mathematical principles in a concrete context. This 'game' of inventing multiple decompositions was an approach to doing mathematics that the class had derived from the lessons on coin problems" (p. 330).
-- summarized by Maria Ong Wenbourne