## Learning and Mathematics

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### Understanding - Greeno and Riley (1987)

James G. Greeno of Stanford University and Mary S. Riley of San Diego State University examine why younger children seem to lack the ability of older children to solve mathematical word problems. Greeno and Riley distinguish between the ability to do the computation required for problem completion and the ability to identify the question posed by a problem. They dispute the hypothesis that older children's greater facility in solving mathematical word problems results from greater knowledge of possible strategies. Instead, they argue that younger children possess the relevant conceptual knowledge but cannot effectively create a mental representation of the necessary information.

Chapter:

Greeno, J., and Riley, M. (1987). "Processes and Development of Understanding." In R. E. Weinert and R. H. Kluwe (Eds.), Metacognition, Motivation, and Understanding (289-313). Lawrence Erlbaum Associates.

Overview

In order to solve problems successfully, children need to know not only the necessary procedures, but also how to represent the information given in the problem. It is in this representational step, rather than in carrying out the actual calculation, that younger children have difficulty.

For example, suppose a child is given the problem, "Joe is paving his front walk. He needs to place 100 bricks. He has placed 57. How many does he have left to place?" Answering this question takes more than knowing how to subtract 57 from 100; it requires identifying subtraction as the necessary operation, 57 and 100 as the quantities to be subtracted, and 43 as the solution. Younger children frequently do not find the right answer not because they cannot subtract, but because they cannot correctly identify these quantities.

By way of extending Greeno and Riley's ideas, then, one way to aid representation can be through the use of manipulatives and other concrete objects such as blocks or pictures. Individuals who are having difficulty making mental representations of the questions posed by problems can often make connections more easily if, for example, they can make a pile of 100 blocks, remove 57, and count the remainder. With the instructor's help, the connection between the concrete and the mental representations (100 - 57) can be made explicit through questions that require students to reflect on what they do and do not understand about the topic just covered.

Another extension of Greeno and Riley's ideas could involve asking students to write out the steps involved in solving problems, noting where they run into difficulties. This would call their attention to the problems they are experiencing and put them in a position to seek some resolution to their questions.

The steps for such problem solution can be broken down as: 1) identifying what the problem is asking; 2) explaining how one would set the problem up; and 3) finding what needs to be done to calculate the solution. Following completion of the problem, students can be asked to explain their solutions in terms of the question posed. Such a procedure takes apart the process of problem solving and enables the student to see the importance of each of the parts. It can be done first as a paired group exercise, and later as an independent project.

Going through such exercises two or three times should be sufficient to enable students to understand the varied components of problem completion. Following such "intervention," the use of blocks and other concrete representations can gradually be eliminated without sacrificing student understanding.

The way problems are worded can aid or hinder mental representation. For example, suppose a child is given a problem in which there are 7 marbles and 4 children. Asking "How many more marbles than children are there?" poses more representational difficulties for the student than saying, "If each child takes a marble, how many marbles will be left?" The second version incorporates a strategy for solving the problem, matching elements of each set and counting the leftovers; the first version lacks any suggestion of how to solve the problem. While most students probably possess a knowledge of the strategy suggested by the second phrasing, many seem to lack sufficient facility in representation to select it without being given additional linguistic cues; given only the first way of phrasing the question, they cannot identify comparison as the relevant operation and matching as a useful strategy.

Here an application of Greeno and Riley's ideas might include initially presenting each assigned problem in two different formats, and eventually moving on to presentation of problems in one format and asking students to develop their own alternative formats. This sequence of exercises would call students' attention to the question being posed and the relation between its context and its content. It would provide them with a scaffold for comprehending the role of context in problem solution. The requirement that they eventually be able to develop alternative formats for problems would necessitate their developing skills in attending to just what it is that problems are asking.

"On the basis of failures in various tasks, Piaget and his associates concluded that children lack understanding of some very important concepts: conservation of number; class inclusion; seriation; and so on. Evidence for these failures came from performance that is inconsistent with the general concepts. For example, when a child sees two sets with the same number of objects and says one set has more, that performance is inconsistent with the concept of number conservation. On the other side, numerous investigators have shown that children will show performance that is consistent with these principles in other circumstances" (p. 299).

[Thus research efforts undertaken more recently are considered to offer support for Greeno and Riley's hypothesis that the difficulty some individuals have in solving word problems lies in their inability to describe the question being asked, rather than a lack of knowledge of the mathematical concepts necessary to complete the problem.]

"[I]n order to choose an appropriate problem-solving procedure, the child is required to make some kind of translation from the verbal or verbal/pictorial presentation. There is evidence that this translation is the major source of difficulty encountered by young children for these problems, and that the major developmental change that enables older children to perform successfully is acquisition of procedures of understanding that construct representations of problem situations" (p. 311).

[This refers to translating "Joe needs to lay 100 bricks and has laid 57. How many remain to be laid?" into 100 - 57 = 43, the number remaining and the solution to the problem.]

- summarized by Andrea Hall