Predictions were determined prior to analysis and were based on the literature on arithmetic development. The sample of selections used for prediction analysis was compared with a sample collected under specific conditions.
CHAPTER ONE
STATEMENT OF THE RESEARCH PROBLEM
Therefore, it is equally important to study familiar domains, such as basic arithmetic, an important aspect of the "other half of cognitive development" (Siegler, 1987, p. 731). As a caution, it is noted that Sherman (1999) warns that cognitive theorists tend to jump to the educational implications of their research without empirical testing under realistic classroom conditions.) The debate between these two developmental models is the central theme of this thesis.
THE STAIRCASE METAPHOR OF COGNITIVE DEVELOPMENT - JEAN PIAGET AND THE NEO-
PIAGETIANS
- Piaget's General Theory
- The Child's Conception of Number
- Criticisms of Piaget's Theory of Numerical Development
- The neo-Piagetians
An integral aspect in the development of the concept of number is the concept of conservation (Piaget. Thus, language plays an important role in the development of the concept of number in a number of different ways.
CHAPTER THREE
THE OVERLAPPING WAVES METAPHOR OF COGNITIVE DEVELOPMENT - ROBERT SIEGLER
AND OTHERS
The Strategy
However, according to Siegler and Jenkins (1989), children's verbal reports of how they solved the problem were often inconsistent with their use of the min strategy. As a result, Siegler and Jenkins concluded that verbal reports are accurate descriptions of the method used to solve the problem.
Variation and Selection
However, explaining how a child chooses between alternatives has proven a difficult task for theorists. The child should spontaneously create new and different ways to solve problems. It is insufficient to merely record this wide range of solutions to the problem.
The Overlapping Waves Metaphor of Cognitive Development
New strategies are created that add to the child's repertoire and become new waves in the model. Siegler's (1996) metaphor of overlapping waves emphasizes the variability of strategy use, which is in direct contrast to Halford's (1987) neo-Piagetian model (see Chapter Two), which suggests that there are structural constraints that determine the strategies a child uses. able to use.
The Strategy of Retrieval
Anderson (1974, as cited in Anderson & Reder, 1999) argues that the organization of human memory is associative. Being able to solve problems using a backup strategy accurately means that the child is able to develop maximal associative distribution.
Choosing between the Existing Backup Strategies
Siegler and Shipley (1995) developed the Adaptive Strategy Choice Model (ASCM) as a modification of the earlier associative models. Feature data refers to the efficiency of specific classes of problems, such as simple addition problems with a large difference between the sizes of the additions.
The Discovery of New Strategies
The metacognitive system consists of an attention spotlight, strategy change heuristics, and goal sketch filters. Siegler and Jenkins (198Y) first introduced the goal sketch which, they argue, governs the construction of new strategies. This goal outline specifies the objectives that a functioning strategy must meet before it is included in the repertoire of existing strategies.
Siegler and Jenkins originally argued that the target sketch drives the discovery process, but it appears to be little more than a standard by which new possibilities are judged. The outline of the goal is similar to Chomsky's, that children are not trying to think about certain wrong grammatical ideas.
Other Variables that may be Associated with Strategy Discovery
However, in Siegler and Jenkins' (1989) study, most children discovered the short-sum strategy before discovering the minus strategy. The difference between the two positions is the point at which the child begins to reverse the order of the two additions. Fewer resources are required to use the strategy and more attention is paid to coding the problem.
Perhaps Siegler would now state that inarticulate behavior is the result of the competing interaction between associative and metacognitive systems. Accounting for how the child benefits from the change is only one aspect of the puzzle.
THE RELATIONSHIP BETWEEN CONCEPTUAL UNDERSTANDING AND PROCEDURAL
KNOWLEDGE IN CHILDREN'S ADDITION
The Relationship between Conceptual and Procedural Knowledge
Both views recognize that young children have an advanced conceptual understanding of early developing numerical domains, whether this understanding precedes or follows procedural competence ( Rittle-Johnson & Siegler, 1998 ). It is possible that this declining conceptual understanding is related to a decline in the child's enthusiasm for mathematics.). However, the relationship between children's conceptual understanding and their procedural knowledge has been less extensively studied.
Children's conceptual understanding was assessed by their ability to use conceptual clues from the previous problem. If we think of addition as a type of rapid counting, then early conceptual understanding of addition may be a consequence of the ability to count.
The Staircase versus the Overlapping Waves
The monolithic nature of these depictions seems unlikely to have arisen from any deep conviction that all children develop through the same path of change. Siegler (1995) also found that learning appears to be enhanced by having children explain the reasoning of others. Computers, on the other hand, are capable of performing a procedure in exactly the same way over and over again.
According to Siegler (1994), cognitive variability has several aspects: it occurs between children of different ages; between different children of the same age; within the same child when faced with similar problems; within the same child when the same problem is presented to more than one child. He points out that variability in simple addition is especially pronounced during the trial immediately before the discovery of a new strategy and during the trial when this new strategy is first attempted (Siegler, 1994; Siegler & Jenkins, 1989). This last point should be obvious, as using a new strategy should increase variability. Thus, there is a proximal link between variability and strategy discovery.
Summary
CHAPTER FIVE
AIMS AND METHOD
- The Aims of the Study
- The Research Method
- Statistical Analysis
- Observer Agreement Reliability
Answer any questions they may have that may include repeating the instructions.] The first amount is. Another requirement was that participants not use a retrieval strategy for more than half of the problems presented during the pretest trials. The model used for the pilot study was used for the first phase of the main study.
The first phase of the study involved exposing the children to a variety of single-digit addition problems, which varied in complexity. Each of the strategies was also coded based on whether it was performed in a covert (coded a) or overt (coded b) manner.
CHAPTER SIX
RESULTS
The Descriptive Statistics
Although using decomposition (strategy 9) always resulted in the correct answer, it was only used 4 times. Therefore, according to Table 6-C, the shortcut sum strategy is assigned a value of 1, the count from first strategy a value of 2, the min strategy a value of 3, and the decomposition strategies. Decomposition is assigned a value of 5 because two additional conceptual advances separate this strategy from the min strategy.
This complexity index can now be used as one of the scales in a scatterplot plotting the complexity of the strategy versus the average time for each of the strategies. This diagram, Figure 6-A, provides an indication of the benefits associated with discovering the more advanced strategies, and thus reveals something about how the principle of least effort drives strategy acquisition.
Conceptual Complexity vs Average Time
The Inferential Statistics
Since the V value is an indication of how well the table fits the mode, it can therefore be concluded that the least effort hypothesis results in a much better fit of the collected data than the min model. The prediction indicating that children will find answers to problems with a problem size less than or equal to seven is the least effective of the predictions in reducing the error rate. Finally, it is important to consider an alternative hypothesis, one that states that the previously used strategy is the best predictor of subsequent strategy choice.
This correlation is 0.37, meaning that prior strategy choice accounts for 13.9% of the variance. Therefore, since R2 and V can be compared, it is reasonable to conclude that the least effort model appears to fit the data collected better than the idea that children's choices are governed by their most recently used strategy.
CHAPTER SEVEN
DISCUSSION
Review of the Descriptive Results
In other words, the child can master only some of the possible strategies that are immediately accessible. The most primitive of the strategies used (with the exception of the guessing strategy) was the 'shortcut sum' strategy, which resulted in the slowest and least accurate recording of all the arithmetic strategies used. For example, the shorter sum strategy involves counting from one to the total of the two additions.
Therefore, the time required to solve a problem should best be related to the number of 'counts' indicated by the size of the sum. However, an interesting departure from this pattern is that sum-squared and lag resulted in a slightly higher correlation score than the sum-lag correlation for strategies in which problem size would be expected to be the best predictor of solution time.
Review of the Inferential Results
For example, if children are not sufficiently proficient with the best backup strategy for the problem presented, they are likely to turn to the next best one. Recovery may be related to the amount of exposure the child with additional problems has and therefore reflects individual differences. Three children reported getting the answer to the problem 10+3, which was not predicted.
Perhaps these problems are similar to equalization problems, which are easy to solve and therefore more likely to be rebuilt. Effect of problem size. refers to the observation that latency increases with problem size.).
General Discussion
According to Siegler, at this point the child is able to reverse the order of the strategies and uses a combination between the shortcut sum and the min strategies. Resnick (1977) suggests this possibility. (Unfortunately, their methodology precluded the possibility of solving the problem.) The data collected for this thesis suggest that both of these positions may be correct. The child does not need to count out the part of the total represented by the first addition (step 3), but rather can start at the number indicated by the first addition (x) and count with the rest of the total that is indicated by the second addition (Step 4).
Reversing the order of the additions before applying the count-to-cardinal principle will certainly reduce the processing when performing the shortcut sum strategy. Siegler suggests that the shortcut sum strategy serves as a transition strategy instead of the count-from-first strategy.
REPRISE
Given that the evidence supports the idea of structural constraints, it seems likely that these structural constraints dictate the sequence of strategic discoveries. He further argues that backup strategies are discovered in a sequence that corresponds to increasing information processing demands and decreasing completion times. Perhaps the reason the debate between the two traditions continues is that both views are partially correct.
It is possible that the variability emphasized by Siegler occurs around an orderly underlying structural progression compatible with neo-Piagetian theories. Finally, if we accept that children's conceptual levels determine the strategies they discover, then the extent of the strategy arsenal reflects the depth of their number concept.
Thelanguage ojmathematics: Making the invisible visible.New York: W.H. The Math Genius: Why Everyone Has It, But Most People Don't Use It. Preschool origins of international differences in mathematical competence: The role of number naming systems.
