The Importance of Spatial Visualization and Cognitive Development for Geometry Learning in Preservice Elementary Teachers

Fundamental Statistics in Psychology and Education.

We investigate effects of cognitive style (field dependency) on performance on examination questions that differ in terms of contextual familiarity and degree of structure. We find that field independent students perform significantly better solving unfamiliar questions than do field dependent students, but possess no significant advantage over field dependent students when solving familiar questions, as supported by theory. For unstructured questions, we found no significant performance difference between field dependent and independent students, consistent with the inconclusive nature of current research findings. For structured questions, we found that field independent students performed significantly better than their field dependent counterparts. These results will help educators understand the role of cognitive style on students' ability to function in unfamiliar settings, which is one of the recommendations of the Accounting Education Change Commission. Key words: Cognitive style, field dependence, performance, structured, familiar, examination. Data availability: Data is available from the authors upon request. INTRODUCTION The reasons why some students perform better on different types of examination questions has long intrigued educators. Theories have been put forward to explain why some students will perform better on questions that are "directed" versus "undirected," "structured" versus "unstructured," or "familiar" versus "unfamiliar." In the present study, we are interested in how students perform when faced with accounting examination questions that differ not only in their degree of structure, but also in their context familiarity. By degree of structure, we mean the degree to which the task requirement(s) are spelt out in the question (where an unstructured question would require the student to identify the issues and steps necessary for the solution). By context familiarity, we mean whether the student is likely to recognize the specific accounting situation or not, given that the requisite skills to solve the problem had been taught previously. We investigate whether students' cognitive style affects their performance in examinations that include these question types. Cognitive style, which is examined more completely later, is the way individuals prefer to receive and process information, which in turn affects the way they conceptualize, store, and retrieve information. Early attempts to illuminate these cognitive issues in an accounting setting can be found in the literature of the late 1970s and early 1980s. For example, Shute (1979) investigated student performance on CPA exam questions using a measure of "abstract reasoning" developed by Inhelder and Piaget (1958). He found that students having higher reasoning skills (classified as "formaloperational") performed better than those with lower reasoning skills (classified as "concreteoperational") when faced with both questions that required higher reasoning skills and those requiring more concrete (lower reasoning) skills. The terms formal- and concrete-operational were used by Inhelder and Piaget to describe the complexity of a person's cognitive structures. A concrete-operational person is oriented toward the relatively concrete realities of the world and is generally unable to consider abstract concepts that depart from that reality. Additionally, the ability to perform certain types of logical operations such as hypothesis reasoning, operations of propositional logic, and reasoning about contrary-to-fact situations, is limited in concreteoperational people. Those who have attained the highest level, called formal-operational, are able to understand concepts that depart from concrete reality, and reasoning is no longer limited to extrapolations from sensory experience. They are able to think of many different possibilities and think of what is observed as a special case of the possible (Inhelder and Piaget, 1958). …

Does Cognitive Style Affect Performance on Accounting Examination Questions

An attempt was made to detect an aptitude-treatment interaction between instructional treatment and spatial-visualization ability. Two groups of elementary education majors were given instruction on the fundamental properties of algebraic structures. The treatments given to the groups (verbal and verbal-spatial) were designed to differ as much as possible in the amounts of spatial visualization they required. It was expected that the slope of the regression line for the verbal-spatial group would be greater than that for the verbal group. However, no significant ATI was found. A rationale is provided that explains the present results and predicts that such an ATI will exist for some topics in mathematics but not for others.

The Interaction between Two Instructional Treatments of Algebraic Structures and Spatial-Visualization Ability.

Attribute-Treatment-Interaction Research in Mathematics Education

Aptitude treatment interaction (ATI) studies essentially deal with two related questions: Can we adapt instruction to patterns of individual differences among students? If so, for which students is a particular method of instruction most effective? The general ATI problem as advanced by Cronbach (1957) is well known (e.g., Becker, 1970), and no further discussion of it is presented here. The particular ATI problem investigated in the study reported here dates from 1968. Carry (1968) and Webb (1971) conducted ATI studies to test for an interaction between graphical and analytical instructional treatments in quadratic inequalities and two aptitude variables-spatial visualization and general reasoning. No significant interaction in the hypothesized direction was obtained. Eastman (1972) conducted an ATI study to investigate the problem further and found a significant ATI. The primary purpose of this study was to determine if the ATI found by Eastman (1972), using measures of general reasoning and spatial visualization and the content area of quadratic inequalities, could be shown to exist using the same aptitude measures but the different content area of absolute value equations. The following formal hypotheses were tested using multiple linear regression analysis:

The Interaction of Spatial Visualization and General Reasoning Abilities with Instructional Treatment on Absolute Value Equations.

In recent years the use of manipulative materials for teaching mathematics in the elementary grades has received considerable support from many quarters in the mathematics education community. Modern elementary mathematics methods textbooks have reflected this support, and perhaps hundreds of workshops have been conducted to train in-service teachers in the use of these materials. On the national and state levels, the National Council of Teachers of Mathematics and its affiliated state organizations continue to devote large numbers of sessions at professional meetings to the use of manipulative aids. Despite the growing interest in the use of manipulative materials with elementary school children, the amount of research on the preparation of teachers to use manipulative aids is quite limited. This study submitted to empirical test the assumption that preservice elementary teachers need actual physical experience with manipulative materials to be able to demonstrate numerical and structural properties at the concrete (enactive) and semiconcrete (iconic) levels of abstraction. The results of recent research in this area are equivocal. Although some studies have indicated that the use of manipulative materials with college students can have a positive effect on achievement or attitude toward mathematics (Bluman, 1971; Douthitt, 1972; Hendrickson, 1970; Leitch, 1972; Smith, 1970; Swick, 1959; Unkel, 1971), the results of other studies have shown conflicting conclusions (Boonstra, 1970; Postman, 1971; Carpenter & McLeod, Note 1). Characteristic of the research in this area is the variety of modes of representation used in instruction with manipulative materials. Bruner (1964) has suggested that any domain of knowledge can be represented in three ways:

https://www.jstor.org/stable/pdfplus/748872

The Use of Manipulative Materials and Student Performance in the Enactive and Iconic Modes.

The purpose of this study was to test the assumption that preservice elementary teachers need concrete experience with manipulative aids to demonstrate numerical, structural, and algorithmic properties of arithmetic operations. Thirty‐six students in the experimental group used manipulative aids to work laboratory exercises while thirty‐three students in the control group completed the same exercises without the use of these materials. Dependent measures included achievement scores, interview data, and attitude scores. No significant differences were found on a laboratory test requiring responses in a representational mode, or on a test requiring the manipulation of the aids. No interaction between the mode of instruction and the visualization ability of the subjects was found. Also there were no significant changes in attitude during the experiment. Results suggest that teaching preservice elementary teachers with demonstrations of materials is as effective as the ‘hands‐on’ approach and has the added ec...

The interaction between spatial visualization abilities and the use of manipulative materials in the enactive and iconic modes for preservice elementary teachers

This study investigated the interaction of general reasoning ability and field‐dependence‐independence with mathematical treatments of modulo systems as fields. The treatments differed only in the inclusion or exclusion of three ‘external structure’ variables; feedback of the correct answer to every exercise, explicit objective statements for each section, and an introductory section on the properties of the real numbers. The 104 subjects from four sophomore‐level high school mathematics classes were randomly assigned to one of the two treatments. Students were then tested for achievement and transfer, followed four weeks later with parallel retention tests. General reasoning ability was a significant predictor (p<0.05) of achievement, transfer, and retention for both treatment groups. There were no interactions of field‐dependence—independence with treatments on any of the post‐tests. Further, field‐dependence‐independence was not a significant predictor on any post‐test for either of the treatments. The...

Phillip M. Eastman

Papers

The Interaction of Spatial Visualization and General Reasoning Abilities with Instructional Treatment on Absolute Value Equations.

The Use of Manipulative Materials and Student Performance in the Enactive and Iconic Modes.

The interaction between spatial visualization abilities and the use of manipulative materials in the enactive and iconic modes for preservice elementary teachers

The effects of external structure and cognitive style on learning modulo fields

Brief Reports: Further Study of the Use of Manipulatives with Prospective Teachers