Problem-solving ability is an important aspect of intelligence and a common skill that is evaluated in research on how children think. Most of this existing research focuses on accuracy of problem-solving, but the proposed research focuses on how children solve problems. Strategies children use to solve simple addition problems will be examined to determine the effect of the presence or absence of manipulatives. Children will be randomly assigned to two conditions: manipulatives and no-manipulatives. Strategy discovery, generalization, and evolution will be examined as children in each condition solve 288 addition problems over a twelve-week period. Problems will be presented on a computer monitor in the format of, "How much is m + n?". Children will be allowed as much time as necessary to solve the three types of problems given: problems with small addends, problems with large addends, problems with one small and one large addend. The proposed research will examine strategy discovery, generalization, and evolution in a microgenetic (dense sampling over an extended time period) framework allowing investigation of the effects of manipulatives and problem difficulty on strategies used to solve simple addition problems.

Introduction

Problem-solving ability is an important aspect of many definitions of intelligence (e.g., Detterman, 1987, Sternberg, 1990) and a common skill that is evaluated in research on how children think (Siegler, 1991). Problem-solving strategies have been investigated in order to determine how children reach problem solutions and to understand the conditions that facilitate effective problem solving. Bjorklund (1989) defines a strategy as an "effortful mental operation aimed at facilitating the accomplishment of a goal" (p. 207). Moreover, he notes that differences in strategy use may explain developmental as well as individual differences. The types of strategies used and when they are used has been the focus of much recent research on how thinking develops (Siegler, 1991).

The investigation of strategy development during childhood has focused on covert and overt strategies. The study of covert strategies has been influenced by theory and research on verbal processes in adults and has concentrated mainly on the use of rehearsal, free recall, and imagery strategies in laboratory tasks. The study of overt strategy use has examined verbal processes (e.g., Ornstein & Naus, 1985) as well but has also included strategies involving the external representation of the environment. Very young children use external orientation (touching, pointing, etc.) in tasks requiring memory for the location of objects in the environment. For instance, during a hide-and-seek game, children 18 to 24 months of age will look at the hiding place, point to it, hover near it, and even peek at it during a delay interval (DeLoache, Cassidy, & Brown, 1985). Three year old children will look at a target, point to it, or touch it as a means of remembering the location of a hidden object (Wellman, Ritter, & Flavell, 1975). Preschool children will also manipulate to-be-remembered objects, and this manipulation may increase recall accuracy (Baker-Ward, Ornstein, & Holden, 1984; Fletcher & Bray, submitted).

Older children and adults also use external memory strategies. For example, many adults strategically place their briefcases and other items in a special place (next to the door) in order to remember to take them to work (Intons-Peterson & Fournier, 1986). In aging adults, the use of external aids for remembering has been shown to be widespread, and there are many commercially produced items to assist older adults with memory difficulties (e.g., pill organizers and timers; Petro, Herrmann, Burrows, & Moore, 1991). These examples illustrate that across the life span, people use external strategies.

The study of external representation in the area of math has centered on the examination of finger-counting strategies in children still mastering basic math facts (Baroody, 1992; Bray, Huffman, Ward & Hawk, 1994; Siegler & Jenkins, 1989). When asked to solve simple addition problems, children as young as kindergarten-age externally represented addends on their hands by counting them out on their fingers or by using manipulatives (colored chips, blocks, etc.) to aid in the counting process (Baroody, 1992; Bray et al., 1994, Carpenter & Moser, 1982; Siegler & Jenkins, 1989). Counting with manipulatives has been studied less than finger counting, and very little is known about the relations among the use of finger counting and the use of manipulatives.

Manipulatives are objects designed to appeal to a child's senses: they can be touched, handled and moved (Yeatts, 1991). Manipulatives have been widely used in early elementary education because they are thought to provide a means of external representation which involves the child in learning, thereby improving performance. They concretely represent otherwise abstract numbers, and this external representation is thought to aid children in counting tasks.

The use of manipulatives has been investigated in school-based studies (Bolduc, 1970; LeBlanc, 1968; Steffe & Johnson, 1970). Steffe and Johnson ((1970) found that first grade children who had manipulatives available during a problem-solving task scored significantly higher than those who did not have manipulatives available. Bolduc (1970) showed that first grade children presented with problems accompanied by a visual aid found them significantly easier to solve than those presented without a visual aid. LeBlanc (1968) found that subtraction problems presented with physical or pictorial aids were significantly easier than those presented without an aid. Steffe, Thompson, and Richards (1982) found that first grade children needed tangible objects to count correctly.

Results of other school-based studies demonstrate conflicting findings about the use of manipulatives. Fennema (1972b) found that second grade children learning a repeated addition approach to multiplication (symbolic) performed better than those who learned a multiplication strategy based on manipulating Cuisenaire rods, which are prefabricated, colored, wooden bars manufactured by a manipulatives company. Moody, Abell, and Bausell (1971) found no significant differences on tests of learning, transfer and retention between third grade children learning multiplication via a symbolic approach versus those who used a manipulative approach.

In a literature review of 15 studies on math instruction at an elementary school level, Fennema (1972a) found that four studies, three of which were conducted with first-graders, reported significant differences favoring the use of manipulatives. Seven reported no significant differences between groups given a manipulative treatment and groups given no-manipulative treatment. Three reported mixed results, and one reported a significant difference in favor of the no manipulative treatment. On the basis of the results cited above, proponents of the no manipulative approach agree, though, that manipulatives appear to have some effectiveness in children in the first grade (Friedman, 1978).

Because the focus of nearly all the studies described above has been on accuracy of performance, they offer little guidance in understanding the effects of manipulatives on how children attempt to solve problems. They are largely observational in nature and are mostly conducted by teachers, education students, and even manufacturers of manipulative materials in environments that are not well-controlled. These studies provide no clear picture of how the presence of manipulatives influences strategy use while children are mastering basic math facts.

Several controlled studies have been conducted by researchers but these have also focused on the relationship between manipulatives and accuracy with little or no attention to the effects of manipulatives on strategy use. Carpenter and Moser (1982) found that first grade children who had manipulatives available while solving large-number addition problems achieved a higher percentage of accuracy than those who did not have manipulatives available. Carpenter and Moser (1982) examined the pattern of strategy use during three test sessions over the first year of school and found a general progression from less sophisticated to more sophisticated strategy use, but the contribution of manipulatives to changes in the use of strategies across the year were not investigated. Starkey and Gelman (1982) found that preschool children were still able to solve simple math problems in which objects were added to or taken away from arrays that were screened from the child's view. Children used both covert and overt strategies, including using fingers to represent the screened array and counting out loud, but the focus of this study, like the other, was on accuracy of problem solution and not on how manipulatives might influence strategy use.

Although not focusing on accuracy, Fuson (1982) conducted an in-depth structural analysis of the counting-on solution procedure for solving addition problems. "Counting-on" is a strategy in which counting begins at the word for the first addend and continues the number of counts represented by the second addend; for example, on the problem "5 + 3", verbalization would be "five, six, seven, eight". Counting-on is a more efficient strategy than the "shortcut sum" strategy (also called "counting all") described in Table 1 (page 33) where the sum is determined by counting the total number of entities comprised by the two addends; for "5 + 3", the verbalization would be "one, two, three, four, five, six, seven, eight".

Fuson's analysis focused on data collected by Secada and Fuson (1979) which investigated how 6-8 year-old children devise the counting-on strategy. Addends of eight large-addend addition problems were presented to the children in two ways: by a numeral written on one card and by a row of squares drawn on another card. Two conditions of card placement were used, and children were given both conditions in a counterbalanced order. In Condition A, both cards containing the row of squares were displayed. In Condition B, the card that contained the row of squares for the second addend only was visible, while the card for the first addend was face down. The numeral card for each addend was visible at all times. Five of 28 children used both the counting-all and the counting-on strategies. They used counting-on in Condition B, where all squares were not visible. They used counting-all in Condition A, where all squares were visible. Thus, the complete pictorial representation of the addends appeared to cause the children to use a less efficient strategy than when a partial pictorial representation of only one addend was available. They concluded that despite the cumbersome nature of the counting-all strategy, children who knew more efficient strategies (e.g., counting-on) would still elect to use it under certain circumstances.

The pictorial representation of the addends can be viewed as external support for the task of addition. The findings of Secada and Fuson (1979), then, are intriguing because they seem to show that external support, while aiding accuracy, can somehow impede the general progression from selecting less sophisticated to selecting more sophisticated strategies. Even so, this study did not examine the effects of manipulatives on the "counting-on" strategy or any other strategy.

So, while strategy use has been examined in studies of addition strategies, the level of analysis has been general and the focus has largely been on accuracy of performance. Little is known about strategy use at a more refined level of change. For example, math facts are usually acquired during a relatively brief period of weeks or months, but we know nothing concerning the effects of manipulatives on strategy use during this period. However, developmentalists interested in problem-solving such as Siegler (Siegler & Richards, 1982; Siegler & Jenkins, 1989) and Bray (Bray et al., 1994) have more precisely focused on the interface between the problem solver and the task environment during periods of rapid change in strategy use and acquisition of math facts or other types of strategies involving external representation. They have examined the task demands in a variety of domains to investigate strategies children construct to cope with task-demands (discovery), the way children generalize strategy use from one situation to another (generalization), and the changes in the way children select among different strategies over time (evolution).

A strategy is said to be discovered when it is used for the first time (Siegler & Jenkins, 1989). Both strategy discovery and generalization can involve sudden, seemingly discontinuous changes that may lead to dramatic insights or may occur through a series of smaller realizations that gradually culminate in a breakthrough (Siegler & Jenkins, 1989). Via intensive data collection over an extended period of time during which preschool children were expected to discover addition strategies, Siegler and Jenkins (1989) observed that discovery of a new strategy is often accompanied by long pauses and slow counting, culminating in a longer solution time overall than on other problems. Though discovery of a new strategy is often thought to be due to an inability to correctly solve a problem using an existing strategy (e.g., Gholson, Levine & Phillips, 1972), Siegler and Jenkins (1989) did not find this to be true: new strategies were discovered when children perceived a need to be more efficient in their problem solving, not when the children needed to be more accurate.

Once a child had discovered a strategy, s/he rarely used it in the period immediately following the discovery. Siegler and Jenkins (1989) postulate that this is because discovery may only be the first step in understanding the strategy and generalizing its use to other problems. Particularly difficult addition problems resulted in generalization of strategies that had previously been discovered on easier problems. It seemed that when the need for efficiency arose and discovery of a more efficient strategy had previously occurred, generalization of that strategy to the difficult problem took place (Siegler & Jenkins, 1989).

In a similar study, Bray et al. (1994) examined the number of different strategies discovered over a 12-week period. Children's discoveries appeared to evolve according to a hypothesized continuum (see Table 1, page 33) where less sophisticated strategies were apparently discovered early in the study and more sophisticated strategies were apparently discovered later.

The work of Siegler and Jenkins (1989) and Bray et al. (1994) represents a shift in focus from exploring accuracy in problem-solving to exploring strategy discovery, generalization, and evolution in problem-solving at a detailed level and at a time of rapid change in strategy use. Both studies examined children's strategy use on addition problems because the study of addition is one domain where diverse strategies are often observed and reported and because addition strategies change rapidly over a relatively short period of time (Fuson, 1982; Siegler, 1987). The availability of multiple strategies is crucial to the study of change in strategy selection over time: when there is only one way to solve a problem (as in many laboratory-based studies of strategy development), there is no strategy evolution to study; when there are multiple ways to solve a problem, any developmental increases in the sophistication and usefulness of strategies can be delineated.

Several variables could be expected to influence strategy discovery, generalization, and evolution in the solving of simple addition problems, one of which is the presence of manipulatives. Though manipulatives have been studied in relation to accuracy, the effect of manipulatives on strategy change in the solving of addition problems has yet to be systematically examined. Providing continuous viewing of the addition problem until it is solved may also be an important situational variable influencing strategy discovery, generalization and evolution. Children in Bray et al. (1994) often lost track of the problem (briefly displayed on a computer monitor) during the counting process inquiring, "What was the problem again?".

The level of difficulty presented by the problem may also affect strategy discovery, generalization, and evolution. Addition problems comprised of large addends, for example, may provoke discovery of a more efficient strategy or generalization of an efficient strategy previously used on a problem with small addends (e.g., Siegler & Jenkins, 1989). Problems in which both addends are less than five (small addend), one addend is less than five and the other is between 6 and 9 (large addend), and one addend is between 1 and 9 and the other is between 12 and 29 (challenge) will be used so that differing levels of problem complexity can be examined.

The study of strategy discovery, generalization and evolution requires a unique experimental design called a microgenetic approach. The methodology of the proposed research relies on Siegler and Jenkins' (1989) definition of microgenetic: dense sampling over an extended period of time coupled with an intensive trial-by-trial analysis. The proposed research will examine strategy discovery, generalization, and evolution in a microgenetic framework allowing investigation of questions about the effects of different levels of situational support and problem difficulty on strategies used to solve simple addition problems such as: Does the presence of manipulatives aid discovery? Does displaying the addition problem continuously offer enough external support so that strategies are discovered or evolve sooner than would be expected according to prior research? (Although the exposure variable will be held constant in the proposed study, the results will be compared to those of Bray et al. (1994) who used an exposure time of 2.5 seconds per problem.) Are strategies discovered more often on small-addend problems? Are they generalized more often on more difficult large-addend and/or challenge problems?

Subjects. The subjects will be 24 4 ½ - to 6-year old children attending a structured kindergarten program in the Birmingham area.

Materials. As in the study by Bray et al. (1994), pretest and experimental sessions will involve number-fact ("How much is 5 + 3?") problems using small addends. In the experimental session, large addend and "challenge" problems will also be included. The small addend problems will consist of all possible pairs of digits from 1 to 5 (excluding ties such as 2 + 2); the large addend problems will consist of all pairs with the digits 1 to 5 for one addend and the digits 6 to 9 as the other. The challenge problems will consist of pairs with digits 1 to 9 as one addend and the digits 12 to 29 as the other addend. Many researchers (Groen & Parkman, 1972; Levine, Jordan & Huttenlocher, 1992) use sums less than nine, but sums greater than nine are important to encourage strategy evolution (Baroody, 1992; Bray et al., 1994; Siegler & Jenkins, 1989). In all cases, the first addend will be the larger number on half of the problems presented during each session. For the small and large addend problems, each pair will be used approximately equally often across sessions. No particular pair will be repeated within a session. The manipulatives will be forty plastic bears of the same color.

Subjects and the experimenter will be seated on a comfortable, bright red "math mat" on the floor of a room provided by the school. The math mat is designed to appeal to kindergarteners by invoking a "fun" atmosphere which engages them in the task and fosters a positive attitude toward solving the math problems. The mat also offers an unobstructed view of the children's counting behaviors as there is no table or chair under which bears or fingers can be hidden. A laptop computer will generate problems which the children will view on a video monitor. A Panasonic AG-180 video camera will be used to videotape each session. The camera will be placed 75 cm to the right of the child, providing a view of the child, the math mat, and the computer screen.

Subject Consent. Consent forms will be sent home with children to be signed by parents/guardians. Those who return consent forms will be eligible for participation in the pretest phase of the proposed research. This research has been approved by the Institutional Review Board of the University of Alabama at Birmingham.

Procedure. Each child will be tested individually in a room provided by their school. The child and the experimenter will be seated on the "math mat". All sessions will last approximately 10 to 15 minutes each.

Pretest Sessions. All eligible participants will be screened during two pretest sessions, lasting approximately 15 minutes each. Manipulatives will not be available in the pretest sessions. Measures for the pretest sessions include highest count (counting task in which the child is asked to count to 50) because it reflects an advanced conception of the number sequence, magnitude estimation (a task in which the child identifies the larger digit of two numbers displayed on a computer monitor; e.g., "Which is larger, 3 or 6?") because it includes a component of numeral recognition as well as a conceptual understanding of the number line, and 24 small addend addition problems. Bray et al. (1994) found that "highest count" and "magnitude estimation" tasks were good predictors of addition accuracy (r = .54, r=.62, respectively, p < .01).

The first pretest session will consist of the highest count task followed by twelve small addend problems simultaneously generated by the computer and read aloud by the experimenter, e.g., "3 + 5". The problem will be continuously displayed on the monitor until the child offers an answer. The amount of time it takes to answer each problem (latency) and the responses the children give will be recorded by the experimenter. Participants will be told that "You can do anything you want to get the right answer. You can just say the right answer if you know it, or you can use your fingers, or do whatever you want to do." Participants will also be told that they do not have to answer the same way every time as long as they try the best they can to get the right answer. The child will receive a colorful sticker after each correct answer.

After the child responds with an answer to the given problem, the experimenter will ask the child how s/he figured out the answer. (Siegler & Jenkins, 1989). If the child's response is ambiguous, the experimenter will probe to obtain an unambiguous response. For example, if the experimenter observes the child counting on his/her fingers, but the child does not report counting, the experimenter will ask, "What were you doing with your fingers". If the child reports counting, s/he will be asked how they counted. Prior research has not provided a clear distinction between retrieval and guessing. Children in Bray et al. (1994) often said they "knew it" when it seemed obvious that they were guessing. The proposed research includes a probe to be used on all retrieved (covert) answers. For example, when the child says they figured out the problem because they "already knew it", the experimenter will ask, "Did you know it or did you guess?". Once the child's answer is unambiguous, the experimenter will offer accuracy feedback, e.g., "right" or "wrong", and give the child a sticker if the answer was right or offer words of encouragement if the answer was wrong. The experimenter will then proceed with the next problem. This trial-by-trial query provides accurate measures of unobserved counting strategies and retrieval (Bray et al., 1994; Siegler & Jenkins, 1989).

The second pretest session will consist of 36 consecutive magnitude estimation trials, each problem displayed continuously on the monitor until the child offers an answer, followed by 12 more small addend problems.

The exclusion criteria will be based on performance in the pretest sessions. Because prior research has shown that children who show strategy evolution are relatively accurate problem solvers on small addend problems given in the pretest (Bray et al., 1994), the following exclusion criteria have been developed: children who demonstrate a high count of less than 20 or who miss more than 50% of the magnitude estimation problems or whose accuracy falls below 68% or above 88% on the pretest sessions will not participate in the experimental sessions. Children who demonstrate the use of the "min" strategy during either pretest session will also be excluded in order to keep the proposed research comparable to the research of Siegler and Jenkins (1989) and Bray et. al (1994).

Experimental Sessions. The experimental sessions will be conducted in a manner similar to the small addend portion of the pretest sessions except that children will be randomly assigned to one of two conditions. In the manipulatives condition, the children will have manipulatives available for all experimental sessions; children in the no-manipulatives condition will not have manipulatives available in any of the sessions.

All children will receive a mix of 6 small addend, 3 large addend, and 3 challenge problems during each experimental session. Subjects will be given no strategy instruction. There will be two sessions per week for 12 weeks. There will be 12 problems in each session for a total of 288 problems per child. Each session will begin with a review of the instructions, which will be the same as in the pretest session (see page 15) except that children in the manipulatives condition will be told, "You can do anything you want to get the right answer. You can just say the right answer if you know it, you can use your fingers, you can use these chips, or do whatever you want to do." Responses will be scored as they were in the pretest.

In both conditions, problems will be displayed until the child offers an answer in an effort to uniformly provide an environment in which strategy evolution can occur without impediment.

Follow-Up. After experimental sessions are completed and preliminary data analysis has been conducted, the experimenter will meet with the teacher to obtain teacher observations and anecdotal information about children who participated in the study. Teachers will be provided with informational packets about the experimental findings.

Design. A mixed factorial design of manipulatives (present vs. absence), and problem type (small addend, large addend, challenge) will be used with the last factor varying within-subjects. Subjects will be randomly assigned to one of the two manipulative conditions after the pretest sessions are conducted.

Data Reduction. A microgenetic analysis, focusing on detailed, trial by trial analyses of the children's videotapes will be conducted. Levels of addition strategy used will be identified and categorized in a manner similar to that shown in Table 1 (page 33). Videotapes of children will be viewed and scored by trained raters with a minimum interrater reliability of .90. problem-solving accuracy will be computed as a simple percentage of correct answer/288 problems. Discovery will be defined as the first time a strategy is used. Generalization will be computed as the proportion of trials on which a strategy was used after the first time it was discovered. Strategy evolution will be analyzed by computing the number of different strategies discovered over time. For example, if a child used "sum" on 29 problems (10%), "shortcut sum" on 57 (20%), min on 86 (30%), and retrieval on the remaining 116 problems (40%), s/he would have shown an evolution in strategy use by using a range of 4 of the 6 strategies delineated in Table 1 (page 33).

Measures for finger-counting strategies will include the proportion of trials on which children counted on their fingers to arrive at a sum (e.g., sum, short-cut sum, finger recognition). In the manipulative condition, the same "manipulative-counting" strategies are possible (with the exception of "finger recognition") and will be included in the analyses comparing the manipulative conditions. Combinations of finger-counting and manipulative-counting strategies will be similarly categorized and scored.

Where children's strategies can be observed via the videotape, strategies will be scored based on the observed strategy, even if the child reports conflicting strategy use. When strategies are unobservable, strategies will be scored based on the child's report of strategy use. This scoring protocol is similar to the procedures used by Bray et al. (1994) and Siegler and Jenkins (1989).

. Microgenetic methodology uses non-traditional analyses and small sample sizes (with large numbers of observations) focusing on patterns and consistencies across subjects. It incorporates traditional statistical methodology wherever possible but is not driven by traditional statistical analysis. The goal of the data analysis in the proposed microgenetic study is to examine the course of strategy discovery, generalization, and evolution in solving addition problems where manipulatives are present or absent.

Whether the discovery trial of a particular strategy occurs sooner in the manipulatives condition than in the no-manipulatives condition will be analyzed. The trial number (1-288) on which a child discovered a particular strategy will be analyzed with a t-test for independent groups comparing the means for the two manipulative conditions. Whether solution times are longer on discovery trials, as they were in Siegler and Jenkins' (1989) study, will be analyzed with a t-test comparing mean solution times on the discovery trials of the manipulatives group with mean solution times on the discovery trials of the no-manipulatives group. Regression analyses similar to those conducted by Siegler and Jenkins (1989) will be performed to determine if any aspects of the pretest performance were related to the number of sessions it took to discover a particular strategy, such as min. The analysis will examine predictors of the session of the first use of a particular strategy. The predictors will be: number of problems correct on the magnitude estimation task, number of problems correct on the highest count task, percent correct on the combined first and second pretest small-addend problems, and the percentage of correct trials where the retrieval strategy was used on the small-addend problems of the pretest sessions. Age, sex, and total number of problems will also be included as predictor variables. A Chi-square analysis will be conducted where the number of children who discover each strategy in the manipulatives condition will be compared to the number who discover the same strategy in the no-manipulatives condition.

Similarly, generalizations of each particular strategy will be analyzed. For each strategy, the proportion of trials on which it is used after discovery will be analyzed for the manipulatives vs. the no-manipulatives condition with a t-test for independent groups. The other analyses will parallel those described for strategy discovery.

Strategy evolution will be measured in analyses of the number of different strategies used at least once over the course of the 24 sessions. Similar to Siegler and Jenkins (1989) and Bray et al. (1994), two sets of correlations will be computed to determine if the progression of strategy evolution moves from use of less sophisticated strategies toward the use of strategies that are more advanced as is hypothesized by the continuum outlined in Table 1 (page 33).

One set of correlations will be between percent of a particular strategy used on small addend problems in the first 12 sessions and percent of the same strategy used on small addend problems in the last 12 sessions; these correlations will help to establish a range of strategy use for each child. Across sessions, the least advanced strategy tends to decrease while a concomitant increase in the most advanced strategy is observed (Bray et al., 1994; Siegler & Jenkins, 1989). A 25% decrease in the least advanced strategy use with a concomitant increase of 25% in the most advanced strategy will be the basis for a Chi-square analysis between children in the manipulatives and the no-manipulatives condition; the number of children who fit this predicted pattern will be compared to determine the effect of condition.

To ascertain whether the data confirm or disconfirm the hypothesized order of the continuum shown in Table 1 (page 33), correlations between pairs of different strategies used in the first and second half of the study will be computed. Siegler and Jenkins (1989) found higher correlations between adjacent strategies listed in Table 1 than between those which were farther apart, thus supporting the hypothesized ordering of strategies.

Another set of correlations will be between early strategy use on small addend problems and later strategy use on large addend and challenge problems; these correlations will allow examination of strategy evolution from easier problems to harder problems. A further interest is in whether there is an increase in strategy evolution due to condition. A t-test will analyze the effect of manipulatives on strategy evolution. The other analyses will parallel those described for strategy discovery.

To examine how well the conditions (manipulatives, no-manipulatives) and the strategies within each condition (see Table 1, page 33) influence accuracy of recall, the relationship between strategy use and accuracy will be analyzed in a hierarchical regression analyses with accuracy as the criterion variable. Separate analyses will be conducted for finger-counting strategies only, manipulative-counting strategies only, and both strategies. The strongest relationships are hypothesized to exist in variables listed early in the order, so the following order of entry for the predictor variables will be used in each of the three analyses: manipulative condition, mean proportion of trials with sum, short-cut sum, count-from-first addend, min, finger recognition, and retrieval strategies. Similar sets of regression analyses will be conducted for each manipulative condition.

As a further examination of the effect manipulatives have on problem-solving, overall accuracy will be analyzed with a t-test for independent groups comparing the means for the two manipulative conditions.

The study conducted by Bray et al. (1994) has a very similar trial structure that can be compared to the proposed study. Bray et al. (1994) did not include the manipulatives variable and utilized a 2.5 second exposure time for the display of each problem. Analysis of situational support provided by the continuous problem display in the proposed study will involve contrasts between measures of strategy discovery, generalization, and evolution as described herein for the manipulatives and no-manipulatives conditions and data for comparable children in Bray et al. (1994). Situational support will be analyzed in a one-way ANOVA with separate analyses conducted for the measures of strategy discovery, generalization, and evolution.

Expectations. It is expected that both the nature and frequency of strategy use and evolution will be affected by the presence of manipulatives. More specifically, accuracy and strategy evolution of young children is expected to be aided by the situational support (external representation) provided by the presence of manipulatives, such that children in the manipulatives condition will be more accurate and demonstrate more strategy evolution than children in the no-manipulatives condition. This would support previous research which demonstrated that the presence of manipulatives led to a higher frequency of counting strategies (Carpenter & Moser, 1982). Whether more counting strategies will be used on a particular problem type (small or large addend or challenge problems) is not yet known. Previous research has shown that frequency of counting strategy increases with problem difficulty (Carpenter & Moser, 1982). Whether Secada and Fuson's (1979) finding regarding less demonstration of strategy progression due to the presence of external support will be borne out with the tangible manipulatives remains to be seen.

The additional situational support provided to all subjects by continuous display of the problem should aid their performance by increasing their accuracy. The continuous display is expected to result in a reduced working memory load, thereby offering more of an opportunity for children to show strategic behavior(s) (Brainerd, 1983).

Siegler and Crowley (1994) and Bray et al. (1994) have shown that problem difficulty can encourage strategy generalization and evolution. It appears that children are content to use current strategies until there is a need to devise more efficient ones. In the proposed research, it is expected that the more difficult large addend and challenge problems will encourage children to develop more efficient strategies than the ones they use on the small addend problems and/or to generalize previously discovered strategies.

The focus of this research is not only on whether solutions to addition problems are correct but on the underlying basis of how strategies are discovered and generalized in the process of problem-solving. Neither is the focus exclusively on the "min" strategy as it has been in the recent work of Siegler and Jenkins (1989). The min strategy, as well as all other finger counting and manipulative counting strategies, will be investigated.

The framework of strategy change in the proposed research, including concepts of discovery, generalization, and evolution, is more refined than in any other aspect of strategy change examined to date, whether overt or covert and should, therefore, make a meaningful contribution to the strategy development literature as well as the literature on math education.

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Strategy	Typical Use of Strategy to Solve "5+3"
Sum	Put up 3 fingers, count "1,2,3". Put up 5 fingers, count "1,2,3,4,5". Begin counting again at 1, "1,2,3,4,5,6,7,8".
Shortcut Sum	Count "1,2,3,4,5,6,7,8" perhaps simultaneously as putting up one finger for each count.
Count from first addend	Say "3,4,5,6,7,8" or "4,5,6,7,8" perhaps simultaneously putting up one finger for each count.
Min	Count from larger addend by saying, "5,6,7,8" or "6,7,8" perhaps simultaneously putting up one finger for each count.
Finger Recognition	Put up 3 fingers, put up 5 fingers, say "8" without counting.
Retrieval	Say an answer and explain it by saying, "I just knew it".