Assessing Zimbabwean children's mathematics problem solving for Cognitively Guided Instruction.

Abstract

Cognitively Guided Instruction (CGI) has been highly effective in helping elementary school children in America develop number sense and mathematics problem solving ability. This study attempted to determine if children in Zimbabwe, a developing country with cultures and educational experiences very different from those in 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 of the JSU Problem or the child may again use a trial and error approach. If the child knows some addition facts and is able to derive other facts from these, then the child might say, "I know that 4 + 6 = 10, but I need 14 altogether so I need to start with 4 more and 4 + 4 is 8 so Tom had 8 apples before Mary gave him any." If the child knows the fact, 8 + 6, directly, then the child can use this to immediately solve the problem if it is understood.

By carefully observing the solution strategies the child is using, the teacher can pose developmentally appropriate problems which do not frustrate the child while at the same time assisting in the cognitive development by providing challenging problems that lead the child into the next developmental stage. The choice of number size in these problems also assists in making the transition from one stage to another. For example, if the numbers in the previous JRU Problem are changed to 37 and 2 and the child is a direct modeler, the child may be encouraged to count on 2 from 37 rather than model by counting out 37 cubes, 2 cubes, putting them together, and counting all of them again.

The problem types and strategies mentioned above will be discussed in further detail in the following sections of this paper. Altogether, 14 different problem types derived from previous research by Carpenter, et al. (1999) are described in the Method section. These problem types formed the core of the interview tasks given the children. The three main strategy types used by children, and derived from work by Carpenter, et al. (1999), are direct modeling, counting, and facts/derived facts. Direct modeling is indicative of the earliest level of cognitive development in solving mathematics problems. This is followed by counting strategies at the next level and then facts/derived facts at the most advanced level. These will also be discussed in greater detail in succeeding sections of the paper.

Summarily, research-based CGI provides the teacher with a coherent analysis of the structure of the mathematics as well as the developmental strategies that children use when acquiring the ability to solve mathematical problems. This allows teachers to make informed instructional decisions based on each child's developmental level as indicated by the problem types the child is able to solve. Children are thus able to acquire a deep understanding of the mathematics because they are proceeding at their own pace and doing what makes sense to them. The teacher's role is one of a skilled facilitator of children's learning by posing appropriate problems and asking questions which provide scaffolding for the child in building mathematics knowledge.

The Research Problem

In the United States, some studies with diverse cultural groups such as Native American (Hankes, 1998; Hankes & Fast, 2002), African American (Carey, et al., 1994), Hispanic (Villasenor, 1991), and Lebanese (Ghaleb, 1992) have also shown positive results with CGI in elementary mathematics. It cannot however be inferred that children outside the United States in developing countries, would necessarily exhibit the same developmental patterns in mathematical problem solving as their American counterparts who belong to diverse cultural groups.

This researcher, therefore, questioned the generalizability of such findings to children from other countries where the cultures are distinctly different from those in the United States. In particular, the mathematical developmental stages and problem solving strategies (described by CGI research) of children from developing countries is largely unknown. The researcher's previous experience with mathematics education in Zimbabwe (Fast, 2000) led him to suspect that there could be significant differences between previous research findings in the United States and what might be uncovered in a developing country such as Zimbabwe. The factors which lead to such differences are numerous and varied, and they were not the focus of this research, but in order to consider adoption of the CGI approach to teaching mathematics in Zimbabwe and other developing countries, it is necessary to first establish the basic tenets on which this approach is grounded, in reference to the children in the developing country in question.

Research Questions

Could a problem solving approach be utilized at the grade two level in Zimbabwe to have children construct a deeper understanding of mathematics? How do grade two children in Zimbabwe attempt to solve the various CGI problem types when encouraged to use any method they want? Do they use strategies similar to grade two children in previous studies? Are they able to solve the problems? What is the potential effectiveness of the CGI approach to teaching elementary mathematics in Zimbabwe?

Specifically do grade 2 children in Zimbabwe:

1) Use a direct modeling strategy for solving problems where this strategy is effective?

2) Use a counting strategy for solving problems where this strategy is effective, indicating that the child is at a more advanced cognitive level?

3) Use facts or derived facts as a strategy for solving problems where this strategy is effective, indicating that the child is at an advanced cognitive level?

4) Use any other problem solving strategies which are substantially different from those documented in previous research on CGI?

5) Consistently utilize one problem solving strategy indicative of a specific developmental level or do they revert to an earlier developmental strategy such as direct modeling in some problem situations when they have exhibited proficiency in a more advanced strategy such as counting?

6) Exhibit approximately the same distribution of direct modelers, counters, users of facts and derived facts, indicated by their problem solving strategies, as their counterparts at the same grade-age level in the United States?

7) Experience difficulties with the same problems types as grade 2 children in previous studies?

Method

The researcher interviewed a class of thirty-five grade 2 children of varying abilities in an urban Zimbabwe school to assess their thinking and problem solving strategies as they attempted to solve various mathematics problems, thereby attempting to determine the potential effectiveness of the CGI approach to teaching mathematics in Zimbabwe. All interviews were videotaped and analyzed to determine each participant's developmental performance based on CGI guidelines.

Specifically, the researcher interviewed each student individually in a room apart from the classroom. Each student was asked to solve problems according to the protocol (See Appendix A). The students were asked some preliminary questions such as their names and ages. The researcher then briefly explained the purpose of the interview. The researcher placed a set of 40 cubes and a paper and pencil in front of the student and explained that these could be used at any time to solve the problems. The student was then asked to perform a few counting tasks, (Appendix A, Part A) to help determine the size of the numbers to be used in the interview. These straightforward tasks also helped in putting the child at ease in preparation for the more demanding task of problem solving.

The interview then proceeded to Stage I of the problem solving (Appendix A, Part B). This involved asking the student three problems which could be solved by direct modeling and are thus indicative of the first developmental level as defined in CGI research. These three problems were given to all the participating students. If the student solved all these problems, the interview proceeded to Stage II that consisted of another five problems. These problems are at a more advanced developmental level. If the child could not solve Problem 8, which is indicative of a pivotal point in the child's development, the interview was usually terminated. Otherwise, the interview proceeded to Stage III where the most advanced thinking is required. Some variations occurred in this protocol based on the researcher's interpretation of the responses of the individual participant. The 14 problems are given below. The numbers given in the problems were utilized with most of the participants although the researcher also had a set of smaller and a set of larger numbers available to be given in each problem depending on the ability of the child (see Appendix A). The problem types are indicated with an explanation of the classifications including the difficulty levels of the problem and how the problems might be solved.

The problems were devised to portray situations that would readily be understood by grade two children in Zimbabwe--the possession and sharing of sweets, better known as candies in America. Also, the names of the two children involved in the story problems, Chipo and Tendai, are common Zimbabwean names.

The Problems

Problem 1 Join: Result Unknown (JRU).

Chipo had 7 sweets. Tendai gave her 8 more. How many sweets does Chipo have now? Previous research has shown that this is one of the easiest problems because children can directly model the action in the problem by making two sets of objects and joining them. It is like the typical addition problem traditionally given in classrooms.

Problem 2 Separate: Result Unknown (SRU).

Tendai had 12 sweets. He gave 7 sweets to Chipo. How many sweets does Tendai have left? This is also an easy problem because children can directly model the action in the problem by making a set of objects and separating a given number of objects from that set. It is like the typical subtraction problem traditionally given in classrooms.

Problem 3 Part-Part-Whole: Whole Unknown (PPW: WU).

Chipo has 4 red sweets and 5 yellow sweets. How many sweets does she have altogether? This problem is very similar to the Join: Result Unknown Problem 1 because again, the two given sets of objects can be directly modeled and it is like the typical addition problem given in traditional classrooms. This problem, however, is different from Problem 1 in that the joining action is not explicitly stated which can make it more difficult for younger children. Children may simply respond by saying, "Chipo has 4 red sweets and 5 yellow sweets," because the action to be modeled is not specified.

Problem 4 Multiplication (M).

Chipo has 3 bags of sweets. There are 4 sweets in each bag. How many sweets does she have?

This is a problem involving the operation of multiplication and even though children have not formally studied this concept in the classroom, they are often able to directly model the problem situation, that is, making a given number of sets of objects with the same number of objects in each set. This is not unlike the typical addition (JRU) problem. The only difference is that the child must usually make more than two sets of objects and the sets must contain the same number of objects.

Problem 5 Division: Measurement (D:M).

Tendai wants to put 12 sweets in bags with 3 sweets in each bag. How many bags will he need?

This is a division problem in which the size of the equal sets that the given set is distributed into are known, but the number of such sets formed is not known. Again, children should be able to directly model this situation with counters but if they have never had this experience before, they may find it difficult to do so. They must take out of the given set of objects equally-sized sets of objects until all the objects are distributed, and then count how many sets have been formed.

Problem 6 Division: Partitive (D:P).

Chipo has 15 sweets to give to 5 friends. How many sweets will each friend get?

This is another division problem but it is different from Problem 5 in that the number of sets into which the given set is to be distributed is known but the number of objects in each set is not known. Again, this problem can be solved by direct modeling with counters but the modeling action is very different from that in Problem 5. Here, the objects are placed one at a time in each of the given number of sets until all the objects have been distributed. The child must then count the number of objects in each set. The action is similar to dealing cards in a typical card game such as poker or bridge.

Problem 7 Compare: Difference Unknown (CDU).

Tendai has 8 sweets. Chipo has 11 sweets. How many more sweets does Chipo have than Tendai?

This is often a difficult problem for young children and although an adult may see this problem as simple subtraction, the child does not recognize it as such since no action is specified and the young child therefore typically does not know what to do with the numbers. There may also be a language difficulty here involving the terminology "more than". Typically, this problem is solved by making the two sets of objects and then matching them up in a one-to-one correspondence and seeing how many are left over in the greater set.

Problem 8 Join: Change Unknown (JCU).

Tendai has 8 sweets. How many more sweets does he need to have 14 sweets altogether?

The problem can be solved by direct modeling but the solution can also involve the notion of inverse operation which is a fairly advanced concept for grade two children. Also, the action described in the problem is that of joining and consequently children who do not listen carefully to the problem situation may simply add the two given numbers.

Problem 9 Separate: Change Unknown (SCU).

Chipo had 12 sweets. She gave some to Tendai. Now she has 7 sweets left. How many sweets did Chipo give to Tendai?

This problem is similar to Problem 8 in that the change is unknown, but dissimilar in that the given action is separating instead of joining.

Problem 10 Part-Part-Whole: Part Unknown (PPW:PU).

Tendai has 13 sweets. 6 are red and the rest are yellow. How many yellow sweets does Tendai have?

This is a Part-Part Whole Problem like Problem 3 but because a part is unknown rather than the whole, children have more difficulty with this problem. Direct modeling can be used to solve the problem.

Problem 11 Join: Start Unknown (JSU).

Chipo has some sweets. Tendai gave her 7 more sweets. Now she has 16 sweets. How many sweets did Chipo have to start with?

The Start Unknown problems are usually very difficult for children who are at the direct modeling stage because they are generally unable to model the situation since the number of objects in the starting set is unknown. The inverse operational thinking which could be useful in solving this problem is often not well developed at the grade 2 level.

Problem 12 Separate: Start Unknown (SSU).

Tendai has some sweets. He gave 9 to Chipo. Now he has 7 sweets. How many sweets did Tendai have to start with?

This problem is also a Start Unknown problem like Problem 11 and the same comments apply.

Problem 13 Compare: Quantity Unknown (CQU).

Chipo had 4 sweets. Tendai has 9 more sweets than Chipo. How many sweets does Tendai have?

This is another compare problem like Problem 7 which children find difficult. The comparative kind of thinking necessary to solve these problems is often not well developed in children at this level.

Problem 14 Compare: Referent Unknown (CRU).

Tendai has 9 sweets. He has 6 more than Chipo. How many sweets does Chipo have?

This type of compare problem is considered to be among the most difficult in this set of problem types.

Each problem was read aloud to the child and reread as many times as the child requested. If the child asked for specific information given in the problem, the interviewer reread the entire sentence containing the information. If the child's solution was not readily understood, the interviewer asked the child to explain what he or she had done.

The researcher coded each child's responses by reviewing his interview notes and carefully analyzing the videotapes. For each problem type, there are a small number of easily-identified solution strategies that children tend to use which provided the primary categories for coding. If a strategy did not conform to one of the typical characterizations, detailed notes were made about what the child did. Responses were coded both in terms of the strategy used and whether the solution attempt was successful. For a child's solution attempt to be coded as successful, the child had to use a strategy that resulted in a correct answer or would have resulted in a correct answer if there was no counting error. Furthermore, the child had to complete the solution to the problem and report an answer that was off by no more than one or two.

The videotaped interviews of each child were observed two times by the researcher to validate the accuracy of the initial coding in the interview. Coding results were also checked for reliability by randomly selecting three videotaped interviews, having the interview responses coded by an independent evaluator familiar with CGI, and comparing the coded evaluations of the researcher and the independent evaluator.

Findings

Strategies Used

In all the problem types attempted by a significant number of children, direct modeling was the dominant strategy (see Table 1). The only exception was Problem 14 where a dominant strategy could not be determined because this problem was attempted by only four children. Two of these children used facts and two used modeling. Considering individual participants, 26 out of the 35 children utilized modeling with cubes as their dominant strategy (see Appendix B Table 3).

Direct modeling can be done in various ways using counters other than cubes. Fingers and tally marks were also used by a few children. There were three participants out of the 35 who used direct modeling with fingers as their dominant strategy. There were also three children who seemed to prefer using paper and pencil to model the problem situations by drawing circles or tally marks to represent the sweets (see Appendix B Table 3).

The second main strategy type is counting which has been well documented as being used by children as they advance beyond the direct modeling stage (Carpenter, et al., 1999). This strategy was hardly evident at all in the Zimbabwean children's problem solving attempts (see Table 1). Only Participant 1 used a counting strategy for Problem 1, and Participant 15 used it for Problem 1 and Problem 7 (see Appendix B Table 3).

The third main strategy type is using facts or derived facts. This strategy was used by quite a few students for certain problems, especially when the problems contained smaller numbers (see Table 1). Only one student however, Participant 13, used this as a dominant strategy (see Appendix B Table 3).

Inappropriate Operations

Sixteen children at times just added the two numbers given to them in the problem. They apparently did not understand or listen to the actual problem. After these children were reminded to listen carefully to the problem situation, a few of them started to correctly model the situation with cubes or fingers and obtain the correct result. For example, Participant 24, in the first part of the interview, tended to always add the numbers given in the problem. Later however, the child thought about the given situations more carefully and was able to solve both Change Unknown problems which required separating/subtraction. The child even managed to solve the difficult Compare: Quantity Unknown problem. Participant 16, after just adding the two numbers in each of the first six problems, which resulted in only two correct answers (Problems 1 and 3), surprisingly changed strategy and correctly solved the Join: Change Unknown Problem 8.

An unusual case was Participant 7. For the three problems given, the child simply put the two numbers in the problem together to create a new number with the first number as the ten's digit and the second as the one's digit. For example, in the first problem with given numbers 7 and 8, the child said the answer was 78.

Miscounting

As with many children previously observed using a direct modeling strategy, there is a tendency to miscount the number of cubes primarily because the children do not set them out in a line but have them bunched up. Consequently some cubes are counted twice or not at all. For example, Participant 11 used the correct modeling strategy to solve problems 1 and 3 but a miscount occurred in both cases.

Ability to Solve Only One Type of Division Problem

Partitive Division problems are generally considered to be more difficult for children to model than Measurement Division problems because in Partitive Division the child must model or imagine the number of groups first and then start distributing the counters to them one at a time. Sometimes the children attempt a trial and error method to obtain the same number of objects in each group. In the Measurement Division Situation, the child can just count out the objects to form each group until all the objects are utilized. In this study, however, considering participants who were given both problems, participants 6, 28, and 32 were able to solve the Measurement Division problem but not the Partitive Division problem, while conversely, Participants 5, 12, 15, 22, 34, and 35 were able to solve the Partitive Division problem but not the Measurement Division problem.

Ability to Solve Multiplication but not Division Problems

There were a number of students who were able to solve the multiplication problem but experienced difficulties with one or both of the division problems. There was only one participant who could not solve the multiplication problem but managed to solve both division problems.

Language Difficulties

Since most of the participants' first language was not English and yet the problems were presented orally in this language, there were some instances where children were clearly disadvantaged by language. For example, Participant 8 misunderstood the number 13 as 30. In these cases, numbers that could more clearly be understood had to be substituted. This would lead one to believe that there may also have been numerous other terms and descriptions of situations that the students did not comprehend well due to language difficulties.

Exceptional Abilities

Some children performed exceptionally well in their problem solving attempts. One case in point is Participant 5, who was able to successfully solve the first 10 problems except the Measurement Division Problem, and this might be attributed to the use of the larger numbers 24 and 6. One could suppose this to be the case because the child was able to solve the Partitive Division Problem correctly by first making 5 groups of 2 then adding 1 more to each group to make 5 groups of 3.

Participant 12 also performed exceptionally well on these problems. Unlike most other children in this sample, the child predominantly used paper and pencil to directly model the number of objects by drawing them. This was the only child to whom all fourteen problems were given since the child was still making reasonable progress on even the more difficult problems. Ten of the problems were solved.

Unlike most other children in the study, Participant 13 responded to the majority of the problems by stating the answer as a fact. Problems 1 to 9 were solved successfully with the exception of the Multiplication Problem.

Participant 15 solved the first nine problems except for the Measurement Division Problem. This child also made use of facts and derived facts as in Problem 3 which was answered as a fact (4 + 5 = 9) while Problem 4 was answered as a derived fact by saying 4 + 4 = 8 and 8 + 4 = 12. The latter indicates an advanced level of thinking.

Participant 22 also performed exceptionally well and obtained ten successes out of the thirteen problems attempted. This was the greatest number of successes obtained by an individual in the sample and was only matched by one other child, Participant 12.

Other participants who did well were Participant 8 who was able to solve eight out of eleven problems given; Participant 32, who solved eight of the thirteen problems given; and Participant 35, who needed some time to adjust to the situation, incorrectly responded to Problem 1, but ultimately scored nine correct out of thirteen.

Successes by Problem Type

The greatest success occurred in the first three problems, given to all the participants, and Problem 9, given to only fourteen participants. It is not surprising that the children performed well on the first three problems since previous research has shown that children find these to be the easiest of the fourteen problems. Problems 1 and 3 can be done by simple addition (joining), and since children in this research had the most practice in this area through their prior classroom work, they performed well. Problem 3, which does not explicitly state the action to be modeled, is considered to be a little more challenging than Problem 1 where the action to be modeled is explicit. Somewhat surprisingly, Problem 3 resulted in an even slightly higher success rate than Problem 1.

Problem 2, which involves separating and is traditionally referred to as a subtraction problem, was not solved quite as well as the two addition problems, but nevertheless 71% of the children solved it.

It was quite surprising that the children who were given the Separate Change Unknown problem 9 were able to solve it in 13 out of 14 cases. Since this was considered to be one of the more difficult problems, it was not given to all the children, especially when they had already experienced difficulty with preceding problems in the order they were normally given from easiest to most difficult.

One-half of the students who were given the Multiplication Problem were able to solve it, mostly by modeling with cubes, even though they had not had any formal instruction in multiplication. The division problems proved to be more difficult for the Zimbabwean children than the Multiplication Problem, and of the two division problems, the Measurement Division was more difficult than the Partitive Division problem. In the Partitive Division Problem, children would sometimes start with two objects in each set and then distribute the remaining objects, or they would use trial and error to make the necessary adjustments when the sets were of unequal size after distribution.

The children found both Change Unknown problems 8 and 9 to be easier than expected since these are considered to be somewhat more difficult problems than the ones preceding them. Although these problems can be solved using the notion of inverse operation, which is a fairly advanced concept for grade 2 children, the Zimbabwean children typically used direct modeling to solve them.

The Start Unknown problems 11 and 12 are difficult, as previously indicated. Consequently, these problems were not given to most children and only three of the fifteen attempts were successful.

The comparative kind of thinking necessary to solve the Compare problems 7, 13 and 14 is often not well developed in children at the grade two level. Consequently these problems, especially the latter two, have been found in prior research to be of the most difficult problems in this set of problem types. Only six of thirty children could solve Problem 7. Problems 13 and 14 were only given to the most capable children and only five of thirteen attempts were successful.

In total, 177 problems were solved of the 300 that were attempted for an overall success rate of .59. Problem success data for each participant is given in Table 4 (see Appendix B).

Summary, Discussion, and Conclusions

In almost all cases, the grade two Zimbabwean children participating in this study attempted to solve the problems with direct modeling strategies using the counters which were provided for them. Some children also used their fingers for directly modeling the objects in the problem situations and a few represented the objects with tally marks or circles on the paper provided. However, by far, the majority of children used the unifix cubes for direct modeling.

Very few children, if any, used what is referred to in CGI as a counting strategy. In the cases where this might have occurred, the children concealed their fingers to the extent that it was not clear to the researcher whether the children were using their fingers as a direct modeling strategy or a counting strategy. When asked to show how they were finding the answers with their fingers, the children seemed reluctant to do so.

In a few cases, some children immediately responded to problems by stating the answers as facts but this occurred mostly with the easier problem types such as the Join Result Unknown or the Part--Part Whole: Whole Unknown, particularly when the given numbers were small. In very few cases, children utilized derived facts. One example occurred in the Measurement Division Problem which required dividing 12 into groups of 4. One child responded with the correct answer of 3 and explained the thinking as "4 + 4 = 8 and 8 + 4 = 12 so the answer is 3." Facts or derived facts were not used by any children for the more difficult problem types.

Although most of the children in this study would be categorized as direct modelers since this was their dominant problem solving strategy, some could do a few of the more challenging problems such as the JSU problem that can usually be solved only by children who are at a more advanced cognitive developmental level. Perhaps these children were at the more advanced level but they were not using the higher level strategies of counting and facts/derived facts usually observed in advanced level children in previous studies (Carpenter, et al., 1999).

The Compare Difference Unknown Problem was difficult for the Zimbabwean children. The few children who solved it did so with direct modeling, but not in the one-to-one correspondence matching method usually observed in previous studies (Carpenter, et al., 1999). Instead, those who solved this problem directly modeled the larger given set, separated the smaller given set, and then counted the cubes remaining. In other words, they solved this as a separating problem even though the problem is stated as a separating or "take away" situation is. Direct Modelers often are not able to solve a compare problem requiring them to find the difference. Mathematically advanced students might interpret both compare and separate situations as simply subtraction problems.

The majority of children were consistent in their strategy usage. In most cases, this was direct modeling. Some of the children utilized facts for easier problems with small numbers but then reverted to direct modeling for more difficult problems, or if the numbers were larger.

The children in this research were nearing the middle of their second grade but they were still mostly at the direct modeling stage. In the United States, teachers who use Cognitively Guided Instruction have found that children at this grade level adopt the more cognitively advanced strategies of counting and using facts or derived facts (Carpenter, et al., 1998). This facilitates the children's ability to solve the problems that involve more complex thinking and problems with larger numbers. A CGI approach would likely also benefit Zimbabwean children in developing greater problem solving ability.

In the United States, grade 2 children in CGI classrooms are better able to solve the problem types utilized in this study than children in non-CGI classrooms. When Cognitively Guided Instruction is not utilized, children in the United States also experience considerable difficulty with the nontraditional kinds of problems such as the Change Unknown, Start Unknown, or Compare problems (Carpenter, et al., 1998). Traditionally, problems given to children at this grade level were typically addition or subtraction problems following the Result Unknown format. These are the easiest of the problems given in this study and most children could solve them.

In conclusion, the majority of the 35 grade 2 Zimbabwean children in this study approached the solutions of most of the problems given to them by direct modeling with counters. Like children at this level in previous research in the United States (Carpenter, et al., 1999), the easier traditional Result Unknown problems were solved by a high percentage of the children whereas the more complex problems were often not solved, or not given by the researchers because of children's failure to solve problems of less difficulty. Experience with the more challenging non-traditional problems appropriate for the child's developmental level, and the more advanced problem solving strategies of counting and derived facts, would likely be of benefit to the Zimbabwean children in developing their mathematical ability in general and their problem solving ability in particular. Consequently, this research suggests that Cognitively Guided Instruction could be highly beneficial for elementary school children in Zimbabwe.

References

Carey, D.A., Fennema, E., Carpenter, T. P., & Franke, M.L. (1994). Cognitively Guided Instruction: Towards equitable classrooms. In W. Secada, E. Fennema, & L. Byrd (Eds.), New directions in equity for mathematics education (pp. 93-125). New York: Teachers College Press.

Carpenter, T.P. (1985). Learning to add and subtract: An exercise in problem solving. In E.A. Silver (Ed.), Teaching and learning mathematical problem solving; Multiple research perspectives (pp. 17-40). Hillsdale, NJ: Erlbaum.

Carpenter, T.P., & Fennema, E. (1992). Cognitively Guided Instruction: Building on the knowledge of students and teachers. In W. Secada (Ed.), Curriculum reform: The case of mathematics in the United States. Special issue of the International Journal of Educational Research (pp. 457-470). Elmswood, NY: Pergamon Press.

Carpenter, T.P., Fennema, E., Franke, M.L., Jacobs, V., & Fennema, E. (1998). A longitudinal study of intervention and understanding in children's multidigit addition and subtraction. Journal for Research in Mathematics Education 29 (1), 3-20.

Carpenter, T.P., Fennema, E., Franke, M.L., Levi, L., & Empson, S.B. (1999). Children's mathematics: Cognitively Guided Instruction. Portsmouth, NH: Heinemann.

Fast, G. R. (2000). "Africa My Teacher!": An expatriate's perspectives on teaching mathematics in Zimbabwe. Anthropology and Educations Quarterly 31 (1).

Fuson, K.C. (1992). Research on whole number addition and subtraction. In D. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 243-275). New York: McMillan.

Ghaleb, M.S. (1992). Performance and solution strategies of Arabic-speaking second graders in simple addition and subtraction word problems and relation of their performance to their degree of bilingualism. Unpublished dissertation. University of Wisconsin Madison.

Hankes, J.E. (1998). Native American pedagogy and cognitive based mathematics instruction. New York: Garland Publishing, Inc.

Hankes, J.E. & Fast, G.R. (2002). Investigating the correspondence between Native American pedagogy and constructivist-based instruction. In J.K. Hankes & G.R. Fast (Eds.), Changing the faces of mathematics: perspectives on indigenous people of North America. Reston, VA: National Council of Teachers of Mathematics.

National Assessment of Educational Progress (NAEP). Washington D.C.: National Center for Education Statistics. (1992). Author.

National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author.

The Third International Mathematics and Science Study (TIMSS). Washington D.C.: National Center for Education Statistics (1996). Author.

Villasenor, A. (1991). Teaching the first grade mathematics curriculum from a problem-solving perspective. Unpublished doctoral dissertation. University of Wisconsin-Milwaukee.

Appendix A

Interview Protocol for Zimbabwean Children

Materials

50 cubes; paper and pencil

Interview setting

Conduct interviews in an area where there will be no distraction. Sit across from the child at a low table that is comfortable for the child.

Introduction

Put the child at ease by asking his/her name, age, grade, etc. Continue by saying: 'I am trying to understand how children think about counting and mathematics. I will be asking you some math questions. Is this okay? Will you answer some questions?'

If the child is willing, continue to Part A.

Part A Counting Tasks

These warm-up activities prepare the child for problem solving. The child's success or lack of success with these tasks will help the interviewer determine the appropriate numbers sizes for each problem asked in Part B.

1. Put out a pile of 23 cubes. Say: 'How many cubes are in this pile?' If a child is unable to count accurately, put out a pile of 12 and repeat.

2. Say: 'Count from 19 to 25.' If a child is unable to do this, ask them to count from 8 to 12. If a child is unable to do this, ask to count from 3 to 5.

3. Say: 'Count Backwards from 15.' If a child is unable to do this, ask to count down from 8.

Part B Problem Solving Tasks

Things to remember:

* Whenever possible use the child's name in the word problem.

* After a child has solved a problem using the cubes, direct the child to push the cubes to the side and carefully listen to the next problem. Do not allow children to play with the cubes.

* Three sets of varying number sizes are available for you to use. Base your decision regarding which set of numbers to use on what you observed about the child's counting ability in Part A. If you are uncertain of the child's ability, begin with the middle set of numbers. If the child has difficulty, switch to lower numbers. If the child solves as a fact, switch to higher numbers.

* The following 14 word problems have been separated into three stages. Finish each stage that you begin with a child--just because a child has failed one problem within a stage doesn't mean that he/she will fail to solve all the problems within that stage. However, if a child fails all problems within a stage, do not continue to the next stage.

Begin the interview by saying,

"You may solve the problem by making drawings or you may use these blocks (indicate the cubes) or you may solve the problem in your head."

If a child solves a problem quickly without the use of paper/pencil or cubes, say: 'You did that very quickly. Can you tell me what you were thinking when you solved it? How did you do it?'

If the child responds that he/she just knew it', accept this to be a fact response and continue on. If the child responds frequently with accurate fact recall, increase the size of number that you are using.

Read each problem as many times as the child wishes.

Never show a child how to solve a problem. This is not a teaching experience. We want to find out what the child is able to do without instruction. However, you may prompt the child by saying, 'Perhaps you could use these cubes to help' or 'Maybe you could make a drawing to help you solve it.'

Always end an interview positively with a problem that the child can solve.

Stage I

Read each question utilizing one set of numbers given below the problem. Choose the set according to the child's ability. The first number in the set goes in the first blank. The second number in the set goes in the second blank.

1. Chipo had ___ sweets. Tendai gave her ___ more. How many sweets does Chipo have now? (JRU) (3,2) (7,8) (12,14)

If a child does not appear to understand question #1, prompt by repeating the problem and saying: 'Can you show me Chipo's 3 sweets?' 'Can you show me the 2 sweets Tendai gave her?'

2. Tendai had ___ sweets. He gave ___ sweets to Chipo. How many sweets does Tendai have left? (SRU) (8,5) (12,7) (24,13)

3. Chipo has ____ red sweets and ____ yellow sweets. How many sweets does she have altogether? (PPW-WU) (4,5) (7,6) (22,14)

Even if the child solves only one of these problems, continue to Stage #2. If the child is not able to solve all three problems in Stage #1, discontinue the interview.

End the session by asking the child to do something that he/she is able to do.

Stage II.

4. Chipo has ___ bags of sweets. There are ___ sweets in each bag. How many sweets does she have? (M) (2,3) (3,4) (5,5)

If the child has difficulty modeling this with cubes, suggest to the child that the paper and pencil can be used to make a drawing to solve the problem.

5. Tendai wants to put ___ sweets in bags with ___ sweets in each bag. How many bags will he need? (D-M) (4,2) (12,3) (24,6)

6. Chipo has ___ sweets to give to ___ friends. How many sweets will each friend get? (D-P) (6,3) (15,5) (24,4)

7. Tendai has ___ sweets. Chipo has ___ sweets. How many more sweets does Chipo have than Tendai? (CDU) (4,6) (8,11) (12,24)

8. Tendai has ___ sweets. How many more sweets does he need to have ___ sweets altogether? (JCU) (3,7) (8,14,) (18,26)

If the child is unable to solve problem #8, repeat the problem using smaller numbers. If the child is still unable to solve after the second attempt, discontinue the interview. End the session by asking the child a question that he/she will be able to solve.

Stage III.

9. Chipo had ___ sweets. She gave some Tendai. Now she has ___ sweets left. How many sweets did Chipo give to Tendai? (SCU) (12,7) (24,18) (31,22)

10. Tendai has ___ sweets. ___ are red and the rest are yellow. How many yellow sweets does Tendai have? (PPW-PU) (13,6) (26,12) (35,26)

11. Chipo had some sweets. Tendai gave her ___ more sweets. Now she has ___ sweets. How many sweets did Chipo have to start with? (JSU) (7,16) (14,23) (28,36)

12. Tendai has some sweets, He gave ___ to Chipo. Now he has ___ sweets. How many sweets did Tendai have to start with? (SSU) (9,7) (17,9) (26,17)

13. Chipo had ___ sweets. Tendai has ___ more sweets than Chipo. How many sweets does Tendai have? (CQU) (4,9) (16,23) (33,18)

14. Tendai has ___ sweets. He has ___ more than Chipo. How many sweets does Chipo have? (CRU) (9,6) (15,8) (27,19)

Always end the interview with a problem that the child is able to solve successfully. Thank the student for helping you and give a treat.

Appendix B

Table 3 Strategies by Student and by Problem Type

                            Problem # and Type
                    3                               10
          1    2    PPW  4   5   6   7    8    9    PPW  11   12   13
Student   JRU  SRU  -WU  M   DM  DP  CDU  JCU  SCU  -PU  JSU  SSU  CQU

 1        Cf   Mc   Mc   F   Mc  Mc  Mc   Mc   Mc   Mc   NR
 2        Mc   NC   Mc   NC
 3        Mf   Mc   Mc   Mc  Mc      NC   Mc
 4        Mc   Mc   Mc   F           Mc
 5        Mc   Mc   F    Mc  NC  Mc  Mf   Mf   Mc   Mc   NC
 6        Mc   Mc   Mc   F   Mc  Mc  NC   Mc
 7        O    O    O
 8        Mc   Mc   Mc   Mc  Nc  Mc  Mc   Mc   Mc   Mc   Mc
 9        Mc   Mc   Mc   Mc      Mc  Mc   Mc
10        Mf   Mf   Mf   Mf          Mf   Mf
11        Mc   Mc   Mc   Mc
12        Mp   Mp   Mp   Mp  Mp  Mp  Mp   Mc   Mc   Mp   Mc   Mp   Mp
13        Mc   F    F    F   Mc  F   F    F    F    F    F
14        Mc   Mc   Mc   Mp  Mp      Mp   Mp
15        Cc   Mc   F    Fd  NC  Mc  Cf   Mc   Mc   NC   NC        NC
16        Mc   Mc   Mf   Mc  Mc      Mc   Mc
17        Mf   Mf   Mf   Mf  Mc  Mf  NR   Mf
18        Mc   Mc   Mc   Mc  Mc  Mc  Mc   Mc   Mc   Mc   Mc        NC
19        Mc   Mc   Mc   Mc  Mc  Mc  Mc   Mc
20        Mc   Mc   Mc   Mc          NC   F
21        Mc   Mc   Mc   Mc
22        Mc   Mc   Mc   Mc  Mc  Mc  Mc   Mc   Mc   Mc   Mc        Mc
23        Mc   Mc   Mc   F           Mc   F
24        Mc   Mc   Mc   F   F   Mc  Mc   Mc   Mc        Mc        Mc
25        Mf   Mf   Mf   Mf  NC  Mf  Mf   Mf   Mc   Mc   Mc        Mc
26        Mc   Mc   Mc   Mc  Mc  Mc  NC   NC
27        Mc   Mc   Mc   Mc  Mc  Mc  Mc   Mc
28        Mc   Mc   F    F   F   NC  Mf   Mf
29        Mc   Mc   Mc   Mc  Mc      Mc   Mc
30        Mc   Mp   Mp   Mp  F   NC  Mc   NC
31        Mf   Mf   Mc   Mc          Mc   Mc   Mc   Mc
32        Mc   Mc   Mc   Mc  Mc  Mc  Mc   Mc   Mc   Mc   Mc   Mc   Mc
33        Mp   Mp   Mc   Mc  Mc           Mc
34        Mc   Mc   Mc   Mc  Mc  Mc  Mp   Mp   Mc   Mc             Mp
35        Mp   Mp   Mc   Mc  Mc  Mc  Mc   Mc   Mc   Mc   Mc        Mc
Dominant  Mc   Mc   Mc   Mc  Mc  Mc  Mc   Mc   Mc   Mc   Mc   --   Mc
Strategy

          14   Dominant
Student   CRU  Strategy

 1             Mc
 2             Mc
 3             Mc
 4             Mc
 5             Mc
 6             Mc
 7             O
 8             Mc
 9             Mc
10             Mf
11             Mc
12        Mp   Mp
13             F
14             Mp
15             Mc
16             Mc
17             Mf
18             Mc
19             Mc
20             Mc
21             Mc
22        F    Mc
23             Mc
24        F    Mc
25             Mf
26             Mc
27             Mc
28             --
29             Mc
30             Mp
31             Mc
32             Mc
33             Mc
34             Mc
35        Mc   Mc
Dominant  --   Mc
Strategy

Note: See Table 1 coding key.

Table 4 Successes by Student by Problem

                              Problem                   Total
Student  1  2  3  4  5  6  7  8  9  10  11  12  13  14  Successes

 1       1  1  1  1  0  0  1  1  1  0   0                7
 2       0  1  1  0                                      2
 3       1  1  1  1  0     0  1                          5
 4       1  0  1  0        0                             2
 5       1  1  1  1  0  1  1  1  1  1   0                9
 6       1  1  1  1  1  0  0  1                          6
 7       0  0  0                                         0
 8       1  1  1  1  0  0  1  1  1  0   1                8
 9       1  1  1  1     0  0  1                          5
10       1  1  1  0        0  1                          4
11       1  0  1  0                                      2
12       1  1  1  1  0  1  0  1  1  1   1   0   0   1   10
13       1  1  1  0  1  1  1  1  1  0   0                8
14       1  1  1  1  1     0  1                          6
15       1  1  1  1  0  1  1  1  1  0   0       1        9
16       1  0  1  0  0     0  1                          3
17       1  1  1  0  0  0  0  0                          3
18       1  1  1  1  0  0  0  1  1  1   0       0        7
19       1  1  1  0  0  0  0  0                          3
20       1  1  1  0        0  1                          4
21       1  0  1  0                                      2
22       1  1  1  1  0  1  0  1  1  1   1       1   0   10
23       0  1  1  0        0  0                          2
24       1  0  1  0  0  0  0  1  1      0       1   0    5
25       1  1  1  1  0  0  0  1  1  1   0       0        7
26       1  1  1  0  0  0  0  0                          3
27       1  1  1  1  1  1  0  0                          6
28       1  1  1  1  1  0  0  0                          5
29       1  0  1  0  0     0  0                          2
30       1  0  1  0  0  0  0  0                          2
31       1  1  1  0        0  1  1  1   0                6
32       1  0  1  1  1  0  1  1  1  1   0   0   0        8
33       1  0  0  0  0        0                          1
34       1  1  1  1  0  1  0  1  0  0           0        6
35       1  1  1  1  0  1  0  1  1  1   0       0   1    9

         Total  Success
Student  Given  Rate

 1       11      .64
 2        4      .50
 3        7      .71
 4        5      .40
 5       11      .82
 6        8      .75
 7        3     0
 8       11      .73
 9        7      .71
10        6      .67
11        4      .50
12       14      .71
13       11      .73
14        7      .86
15       12      .75
16        7      .43
17        8      .38
18       12      .58
19        8      .38
20        6      .67
21        4      .50
22       13      .77
23        6      .33
24       12      .42
25       12      .58
26        8      .38
27        8      .75
28        8      .63
29        7      .29
30        8      .25
31        9      .67
32       13      .62
33        6      .17
34       11      .55
35       13      .69

Note: 1=Success 0=Unable to Solve

Gerald R. Fast

University of Wisconsin Oshkosh 

Table 1 Frequency of Strategy Use By Problem Type

                  Problem # and Type
                    3                               10
          1    2    PPW  4   5   6   7    8    9    PPW  11   12   13
Strategy  JRU  SRU  -WU  M   DM  DP  CDU  JCU  SCU  -PU  JSU  SSU  CQU

Mc        24   24   24   19  16  15  16   18   13   10    9   1    5
Mf         5    4    4    3   0   2   4    5    0    0    0   0    0
Mp         3    4    2    3   2   1   3    2    0    1    0   1    2
Cc         1    0    0    0   0   0   0    0    0    0    0   0    0
Cf         1    0    0    0   0   0   1    0    0    0    0   0    0
Cp         0    0    0    0   0   0   0    0    0    0    0   0    0
F          0    1    4    8   3   1   1    3    1    1    1   0    0
O          1    1    1    0   0   0   0    0    0    0    0   0    0
NC         0    1    0    1   4   2   5    2    0    1    3   0    2
Total     35   35   35   34  25  21  30   30   14   13   13   2    9

          Problem # and Type
          14
Strategy  CRU  Total

Mc        1    195
Mf        0     27
Mp        1     25
Cc        0      1
Cf        0      2
Cp        0      0
F         2     26
O         0      3
NC        0     21
Total     4    300

Note. Mc -- Direct Modeling using cubes
Mf -- Direct Modeling using fingers
Mp -- Direct Modeling using paper
F -- Facts or derived facts
O -- Other strategy
NC -- Not Clear
Cc -- Counting using cubes
Cf -- Counting using fingers
Cp -- Counting using paper

Table 2 Successes by Problem Type

                     Problem # and Type
                            3
             1       2      PPW:   4      5      6      7      8
             JRU     SRU    WU     M      DM     DP     CDU    JCU

Number of     32     25     33     17      6      8      6      21
  Successes
Number of     35     35     35     34     25     21     30      30
  Attempts
Success         .91    .71    .94    .50    .24    .38    .20     .70
  Rate

                   Problem # and Type
                    10
             9      PPW:   11     12   13    14
             SCU    PU     JSU    SSU  CQU   CRU  Total

Number of    13      8      3     0    3     2    177
  Successes
Number of    14     13     13     2    9     4    300
  Attempts
Success        .93    .62    .23  0     .33   .5     .59
  Rate