...educational experiences different from those the United States, could also potentially benefit from CGI. Thirty-five grade 2 Zimbabwean students' mathematics problem solving attempts were assessed using the 14 CGI problem types. Their solution strategies were consistent with findings in previous research. Most of the children were at the direct modeling stage in their development and they had difficulty solving the more complex problems where modeling is not as effective. Cognitively Guided Instruction appears to offer considerable benefits for elementary school children in Zimbabwe.

Keywords: mathematics problem solving, Cognitively Guided Instruction, Zimbabwe

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The National Council of Teachers of Mathematics (2000) in their visionary document on the curriculum and instruction of mathematics, Principles and Standards for School Mathematics, has made a strong case for the necessity of teachers helping students develop a deep understanding of mathematics. Contributions to this deep understanding come in part from developing good number sense and problem solving ability. The importance of problem solving is emphasized by its prominence as one of the ten standards at all levels from PK--12 in the NCTM document.

Mathematics performance by students in the United States, as indicated by recent national and international mathematics assessment data (NAEP, 1992; TIMSS, 1996), has not always been to the entire satisfaction of American educators and has supported the need for change in mathematics curriculum and pedagogy. Although scores by American fourth graders have been satisfactory compared to other countries participating in the TIMSS study, results of seventh and eighth graders leave something to be desired. One should however not conclude that the middle school must bear the total burden of responsibility. In order for middle school students to have the number sense and problem solving proficiency required at this level, a solid foundation must be laid in the elementary grades.

Considerable research in the area of elementary school children developing a deep understanding of the mathematics they are learning has resulted in a number of promising findings. Cognitively Guided Instruction (CGI), developed by Carpenter, Fennema, and others (1999) at the University of Wisconsin, Madison, is a well-proven, successful approach based on such findings. Children in CGI classrooms have shown remarkable development in mathematical understanding particularly regarding number, operations, and authentic problem solving.

To further investigate the appropriateness of CGI, particularly in a setting considerably different from previous investigations, the author focused on a mixed-ability grade 2 classroom in Zimbabwe. Furthermore, the author's extensive experience with education in Zimbabwe (Fast, 2000) suggested a need for a different approach to teaching mathematics in the elementary school.

The author's observation in Zimbabwean classrooms over an eight-year period indicated that direct instruction was the preferred approach at all levels. Students' procedural knowledge in mathematics was admirable but their conceptual understanding was often limited. Problem solving on O Level and A Level mathematics exams was therefore rather challenging for many students. Developing a better conceptual understanding of the mathematics, beginning in the (Fast, 2000) elementary school, is as much a necessity in Zimbabwe as it is in America.

Theoretical Background

Cognitively Guided Instruction (CGI) developed from an extensive mathematics research project at the University of Wisconsin Madison. This learning approach has been highly successful for developing solving ability and number sense with mainstream as well as minority elementary school children in the United States (Carpenter, et al., 1999; Ghaleb, 1992;

Hankes, 1998; Hankes & Fast, 2002; Villasenor, 1991). It is also recognized as an approach that complies with national mathematics reform standards (NCTM, 2000).

A major attribute of CGI is its focus on helping teachers learn about the relation between the structure of elementary level mathematics and children's thinking of that mathematics. The goal of this approach is that teachers will be able to understand how their students learn mathematics concepts and that this knowledge will inform instruction (Carpenter, 1985; Carpenter & Fennema, 1992; Fuson, 1992).

CGI research is based on a detailed analysis of content domains. Basic addition, subtraction, multiplication, and division problem situations are separated into several classes which are distinguished by different mathematical relationships. This scheme provides a framework for systematically generating a complete taxonomy of mathematical word problems that distinguishes between problems in terms of difficulty (Carpenter, et al., 1999).

In the past we have typically only given children mathematics problems involving the four operations at the lowest development levels. These problems were ones where the children were required to find the result of adding, subtracting, multiplying or dividing two quantities with the answer (unknown quantity) following the equal sign. An example might be: "Tom had 8 apples. Mary gave him 6 more apples. How many apples does Tom have now?" The number sentence or equation for this is simply 8 + 6 = ___. In CGI language, this type of problem is referred to as a Join: Result Unknown (JRU) Problem.

The CGI approach to teaching mathematics does not stop here but provides children with more challenging problems as they are able to do them. The JRU problem becomes cognitively more difficult if it is changed to: 'Tom had 8 apples. Mary gave him some more apples and now he has 14 apples. How many apples did Mary give him?' The number sentence now becomes 8 + ___ = 14. Instead of simply adding the two given numbers, even though the action in the situation is one of joining or adding, the student must carefully analyze the situation to determine what is known and what is to be found. One way of solving this would be to undo the joining/addition operation described in the given situation. That is, the child could perform the inverse operation for addition, which of course is subtraction. Traditionally these kinds of questions were reserved for algebra classes in secondary education and were written with a variable as, in this case, 8 + n = 14. In CGI language, this type of problem is known as a Join: Change Unknown (JCU) Problem.

The most difficult of these "joining" problem types is where the first number in the number sentence is unknown and this is referred to as a Join: Start Unknown (JSU) Problem. Using the previous context, this would be: "Tom has some apples and Mary gave him 6 more. Now Tom has 14 apples. How many apples did Tom have before Mary gave him any?" The number sentence for this would be ___ + 6 = 14. The varying difficulty levels of these different problem types allow teachers to challenge all students at the appropriate developmental levels. Giving JCU and JSU problems provides children with genuine problem solving situations where they must think carefully about the situation. That is, the problem is genuine in that it is not immediately obvious what one should do to solve it. It also helps prepare children for the higher levels of mathematical thinking necessary for the mathematics they will be learning in the future. Furthermore, this material fits well with the NCTM standards which require algebraic thinking at the elementary levels in mathematics (NCTM, 2000).

The CGI taxonomy of problem types is also valuable because it provides a framework to identify the developmental cognitive processes that children use to solve problems. When children begin to solve problems, they concretely represent the relationships in the problem. Over time, a process of abstraction occurs so that the concrete strategies are transformed into counting strategies and finally the use of facts or derived facts.

Referring to the examples above, most children in grade one can do a Join: Result Unknown Problem by directly modeling the situation with concrete objects. That is, the child would count out 8 apples (or some representative manipulative of apples such as unifix cubes), then count out another 6 apples, push them together, and count the total number.

In the Join: Change Unknown Problem, direct modeling can still be utilized but some direct modelers may experience difficulties. The problem situation cannot be modeled directly in the order in which it is described. That is, since the "change" is unknown, the child does not immediately know what to join to the initial quantity but may use a trial and error approach to obtain the required result which is given.

When children develop into the next strategy level, they typically use a counting strategy where the child says "8", then counts up to 14 putting up one finger for each number as the child counts 9, 10, 11, 12, 13, 14. The child will then have 6 fingers showing which is the number of apples he was given.

The Join: Start Unknown Problem is the most difficult of the joining problems because the child is not given how many objects to lay out or how many fingers to count since the number of items in the beginning of the situation is unknown. Again, reversal ability would facilitate the solving...

NOTE: All illustrations and photos have been removed from this article.



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