A r c h i v e d  I n f o r m a t i o n

Toward a New Science of Instruction: Programmatic Investigations in Cognitive Science and Education--August 1993

What Do Children Know About Numbers? What Do They Need To Be Taught?

Research on mathematical intuitions--that is, a grasp of basic numerical properties and relationships that children develop without formal instruction--indicates that such intuitive knowledge has an authority all its own. For example, preschool children know the difference between larger and smaller objects or amounts. They know that drinking milk from a glass reduces the amount inside the glass and that pouring milk into the glass increases its contents. They know that a quantity remains constant if nothing is added or subtracted. They know that if an object is cut into pieces, the pieces taken together contain the same amount of material as the uncut object. And most of them know both how to count and how to use counting to quantify sets of objects. Children regard such knowledge as self-evident. This is an important characteristic of intuitive knowledge: It requires no analysis or justification.

What preschool children do not know, and what formal instruction has traditionally been assigned to teach them, are the rules, symbols, and language of mathematics and how to use them in calculations. And it is in the course of formal schooling that children too often begin to lose their sense of authority and confidence about mathematical thinking--though they may not realize that is what they engaged in before entering the classroom.

NRCSL work on the intuitive knowledge of young children, led by Lauren Resnick, has thus investigated the gap between conceptual and procedural understanding from a different perspective than Ohlsson's. In this case, the gap exists between mathematical intuitions that develop easily and almost universally and the difficulties that arise in the learning of formal mathematical notations and procedures in school. Like Ohlsson's work, Resnick's research asks why arithmetic is so hard; but it also asks why, if children enter school with strong and accurate intuitions about basic mathematical relationships, those intuitions do not help them to learn school math.

Resnick hypothesizes that the classroom emphasis on manipulating formal symbols discourages children from applying--or trusting--their informally developed understanding. She further suggests that the reason for the discontinuity is that children's intuitions arise directly from their real-world experiences, whereas formal instruction in math requires them to reason about abstractions such as numbers and operators that they cannot experience directly. Support for these hypotheses is provided by some of the "buggy algorithms" that young learners use in attempts to solve arithmetic problems--systematic routines that look right in terms of procedural rules but that fail to connect the manipulation of symbols with what the symbols represent.

A common "bug" analyzed by Resnick is called "borrow-from-zero." When a student tries to solve a non-canonical subtraction problem using the regrouping method, 

        9                   6 Ø 2      -4 3 7       -----        2 6 5
the bug appears in the middle column. The student writes 9 correctly but fails to continue borrowing from the next column to the left. This procedure obeys rules of calculation (e.g., in borrowing one must cross out and rewrite the numeral to the left of the column that is incremented), but it fails to conserve the total quantity in the minuend. Thus, the mechanical operations appear to break down at the point where their connection to the meaning of the problem is lost an error that reflects the gap between intuitive and formal processes of learning.

In addressing the question of how to close the gap and permit intuitive knowledge to support formal learning, Resnick identifies four kinds of mathematical reasoning that emerge in developmental sequence, and she proposes, in this and other lines of work, some principles that can help teachers build upon and sustain their students intuitions during formal instruction.

The four kinds of mathematical reasoning that Resnick discusses are the basis for a theory of "layers" of mathematical knowledge. This theory describes children's progress from intuitive understanding rooted in their knowledge of the physical world to an ability to reason about abstract entities in formal instruction. The most elementary layer of understanding is the "mathematics of protoquantities," in which children understand and can predict the effects of changes in amounts of material but do not engage in counting or quantifying. They may talk about a big house or many cookies or more juice, but it is not until they begin to develop understanding at the second layer, the "mathematics of quantities," that they talk about 3 houses or 8 ounces of juice. Here, they ascribe meaning to numbers and measurements in the context of dealing with actual objects and materials. In the "mathematics of numbers," numbers now make sense not only as adjectives but also as separate, abstract entities that can be manipulated and acted upon. Children can add 3 and 5, for example, without having to understand them in terms of, say, 3 cars and 5 buses; and they realize that 3 is less than 5 without having to compare sets of objects. This ability to conceptualize abstractions develops further in the fourth and final layer, the "mathematics of operators," in which not only numbers but operations and relations can be reasoned about. By this stage of understanding, children know that operations themselves can be manipulated; that, for example, if one adds 4 to a quantity and then subtracts 1 from the answer, the result is the same as if one had simply added 3.

These four layers, or types of mathematical thinking, do develop sequentially, but Resnick cautions that the late ones do not supersede those that go before. To the contrary, a student might engage in all four types of thinking while solving a single problem in arithmetic. What is most important about the sequential character of the four levels of reasoning is that each needs to be fully grasped in order to support and interact with the next. If classroom instruction can identify the levels at which students are able to perform and can guide them through all aspects of all four types of reasoning in mathematics, then, presumably, the troublesome discontinuity between intuitions and symbolic manipulations can close.

Other lines of work by Resnick suggest that children might proceed more easily along this cognitive continuum if instruction can provide a link between the familiar forms of learning through which their intuitive knowledge of numbers and amounts develops. Children acquire their intuitions by observing and interacting with the world around them. Their explorations are not structured or bound by rules but instead are spontaneous and inventive. In research on math as an ill-structured rather than a highly organized discipline, Resnick has proposed that arithmetic classrooms permit much the same kind of experimentation and discovery that go on before a child enters school. She recommends that students be encouraged to talk about math, work together to solve problems, use finger counting and any other physical aids they can devise, and compare multiple methods of solving the same problem.

Resnick's work suggests that students will discover and internalize mathematical principles for themselves if they have opportunities to handle and experiment with quantities and sets of objects, invent procedures for solving problems, talk about their own and other children's solutions, observe their classmates methods of investigating numbers, and explore real-life situations involving arithmetic. Such activities help students to expand their knowledge and become more aware of their processes of learning until, with guidance and demonstrations from their teachers--but little direct instruction--they begin to link their experiences and their intuitive methods with the formal mathematical symbols and notations that can express numerical insights more precisely.

According to Resnick, such instructional innovations reduce the discouraging sense of abstractness that too abrupt a transition from intuitions to formality can generate. By continuing children's exploration of the physical world and gradually helping children to express their findings in mathematical language, the referents remain visible. In fact, says Resnick, "an explosion of interpretations" becomes possible when mathematical statements are seen as referring not just to numbers and operations but to all the actual things in the world that those abstract entities might represent. Easing the transition to formal learning in this way might guide children with less difficulty through all four layers of mathematical reasoning that Resnick describes as essential to a complete understanding of school math.
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