Investigating of Teachers’ Mathematical Conceptions and
Pedagogical Content Knowledge in Mathematics
Hsin- Mei E. Huang
Department of Elementary Education,
Abstract
This study examined relationships within primary school teachers’ knowledge of school mathematics, cognition about children’s learning, and knowledge of instructional practice among fifth and sixth grades teachers. Teachers (N=201) completed structured questionnaires which evaluated their cognition about children’s learning difficulties, knowledge of instructional practice, and mathematical concepts in the fifth and sixth grades mathematics curriculum. Results indicated that the prominent children’s learning difficulty was in understanding abstract mathematical concepts. The primary knowledge of instructional practice that was suggested from teachers was to engage in problem solving in cooperative small groups. However, the teacher’s mathematical knowledge did not significantly affect their cognition of children’s learning difficulties and knowledge of instructional practice. Providing more inservice education for teachers to develop more understanding of mathematical knowledge and epistemology is needed for further research.
Theoretical Framework and Objective of the Research
Teachers are a link in the chain of influence from reform to teaching and learning events. Furthermore, how the mathematics reform is implemented can be influenced by teachers’ pedagogical content knowledge (Knapp, 1997), for instance: subject matter knowledge, knowledge about students’ learning, as well as knowledge about mathematical instruction (Even & Tirosh, 1995; Shulman, 1986). The knowledge of what makes the learning of specific topics easy or difficult, and the method of teaching for understanding are vital aspects of teachers’ cognition that relates to teachers’ beliefs about pedagogical practice (Fennema, Franke, Levi, Jacobs & Empson, 1996; Swafford, Jones & Thornton, 1997) in the classroom.
Conceptions of mathematical knowledge is a critical aspect of teachers’ knowledge before they are able to help students learn it (Swafford, et al., 1997). As, fractions, averages, decimals, number lines, volume, area and geometry are important topics in the mathematics curriculum in the higher grades of elementary school. What are teachers’ conceptions of mathematical knowledge on the aforementioned topics? Several researchers have examined teachers’ beliefs about the mathematics curriculum and instruction in general (e.g.: Collier, 1972; Skemp, 1978), as well as analyzed teachers’ mathematical knowledge within a specific topic area (e.g.: Lampert, 1986; Swafford, et al., 1997). However, previous studies have not adequately taken into account mathematical problems which arose in daily mathematical learning situations when analyzing teachers’ pedagogical content knowledge. Moreover, some studies found teachers who acquired more mathematical knowledge facilitated their students’ learning and thereby improved problem solving performance (e.g., Carpenter, Fennema & Franke,1996; Swafford, et al., 1997). Few studies reported how teachers’ mathematical knowledge influenced their cognition of children learning difficulties and knowledge of instructional practice. The empirical data regarding children’s learning difficulties, as identified from teachers’ cognition, is less compelling. Identifying and examining teachers’ pedagogical content knowledge and the relationship within the knowledge that teachers have is critical to improve the implementation of reform.
The aims of this study were to investigate the fifth and sixth grades teachers’ mathematical conceptions and cognition about children’s learning difficulties, as well as knowledge of instructional practices in daily mathematical classes. Furthermore, the study also explored the possibility of a relationship among teachers’ mathematical conceptions, cognition about children’ learning difficulties and knowledge of instructional practice. The research questions included: (a) What are the main components of teachers’ cognition about children’s learning difficulties as well as knowledge of instructional practice? (b) Is there a relationship between teachers’ mathematical knowledge and their cognition of children’s learning difficulties ? ( c) How do teachers’ mathematical knowledge affect their knowledge of mathematics instructional practice ?
Method
Subjects
The subjects in
the study involved 201 fifth-and-sixth grade elementary school teachers from 39
public schools located in the east, south , north and central areas of
Instrument
In a prior study, eight open-ended questions were designed to investigate three aspects of knowledge: basic mathematical conceptions, children’s learning difficulties, and knowledge of instructional practice. The mathematical problems, eight in total, included: fractions, averages, decimals, number lines, volume, area, parallelism and, figures and shapes problems. Opinions written in response to the open-ended questions were collected and categorized from 27 teachers who taught the higher grade in public primary schools. In addition, statements from many of the teachers through informal interviews were obtained. Both teachers’ cognition of children’s learning difficulties and knowledge of instructional practice were categorized by two instructors after examining all the descriptions on the answering sheets. The categories of teachers’ cognition of children’s learning difficulties included: (a) Difficulty in Understanding mathematical concepts: mathematical concepts are abstract and difficult for children to understand; (b) Difficulty in Operating objects: much mathematical knowledge is difficult to understand by operating objects or materials; ( c) Imitating the Solution without understanding: children were able to imitate solutions or algorithms from teachers, without understanding the meaning. The categories of teachers’ knowledge of instructional practice included: (a) instructing directly by Oral Explanation and demonstration; (b) enabling children to communicate and engage in problem solving in Cooperative Small Groups; (c) demonstrating the solution steps in a procedure, then having their students to Repeatedly Practice the steps.
Next, we developed a questionnaire that consisted of eight different mathematical problems as used in the prior study, and each problem had three subscales: (a) Scale of basic mathematical conceptions of specific topic as in the prior study. The subscale consisted of 21 items to which the teachers responded with a yes/no choice and one number line problem. A high score on this subscale indicated that the teacher had high mathematical knowledge performance, whereas a low score on the subscale reflected that the teacher had low mathematical knowledge performance. (b) Scale of children’s learning difficulties, which consisted of three items described in the categories above. ( c) Scale of instructional practice knowledge, which consisted of the three items described in the categories above. The second and the third subscale consisted of 24 and 27 items, respectively. Both scales were designed using a 4-point Likert scale. The validity of both scales was .78 to .98, respectively, and the reliability of the retest was .43 to .87, respectively.
Results and Discussion
Teachers’ Mathematical Knowledge, Cognition of Children’s Learning Difficulties, and Knowledge of Instructional Practice. The full score of the teachers’ mathematical knowledge subscale was 27. The mean and standard deviation of scores on the subscale was 17.8 and 3.71, respectively. If a teachers’ score was higher than or equal to 18, s/he would be defined in the high mathematical knowledge group. If a teachers’ score was lower than 18, s/he would be defined in the lower mathematical knowledge group. About 47.3%(n=95) of the teachers had mathematical knowledge higher than the average score (mean=21.22, SD=2.00), and 52.7%(n=106) of the teachers had mathematical knowledge lower than the average score (mean=14.82, SD=1.76).
Children’s learning difficulties identified from teachers’ cognition included three main categories: (a) Difficulty in Understanding abstract mathematical concepts; (b) Difficulty in Operating concrete objects; (c) Imitating Solution without understanding. Knowledge of instructional practice suggested from teachers included three main categories: (a) Oral Explanation and demonstration; (b) Cooperative problem solving in Small Groups. ( c) Repeatedly Practicing solution steps.
The Relationships Between Teachers’ Mathematical Conceptions and Cognition of Children’s Learning Difficulties. The data was analyzed with a two (high versus low mathematical knowledge groups) by three (DU, DO, and IS, the three categories of children’s learning difficulties) two-way mixed-design ANOVA. The results indicated no significant interaction effect between teachers’ mathematical knowledge and cognition of children’s learning difficulties. Different level of teachers’ mathematical conceptions did not significantly affect cognition of children’s learning difficulties, F(2, 398)=1.17. Similarly, simple effects of teachers’ mathematical knowledge did not reach statistical significance, F(1, 199)=.97. However, there was significant simple effects of teachers’ cognition about children’s learning difficulties, F(2, 398)=72.58, p<.001. Teachers had higher cognition of Difficulty in Understanding abstract mathematical concepts and Imitation Solution without understanding rather than Difficulty in Operating objects in children’s mathematical learning.
The Relationships Between Teachers’ Mathematical Knowledge and Knowledge of Instructional Practice. The data was analyzed with a two (high versus low mathematical knowledge groups) by three (OE, CSG, and RP, the three categories of instructional practice knowledge) two-way mixed-design ANOVA. The interaction effects between teachers’ mathematical knowledge and knowledge of instructional practice did not reach statistical significance, F(2, 398)=.61; as did simple effects of teachers’ mathematical knowledge, F(1,199)=.24. However, there was significant simple effects of instructional practice knowledge, F(2,398)=8.67, p<.001. The results illustrated that teachers had higher agreement with Cooperative Small Groups instruction and Repeatedly Practice solution steps rather than with Oral Explanation and demonstrative instruction.
The findings illustrated that more than half of the teachers’ conceptions of mathematical knowledge were lower than the average score. It seems that a few of the fifth and sixth grades teachers may not have sufficient subject matter knowledge that related to practical learning situations. The results also found that teachers with different levels of mathematical knowledge may not directly affect their cognition of children’s learning difficulties and knowledge of instructional practice. Teachers who had a limited view of how mathematical knowledge is acquired could alienate themselves from the reasonability of the subject matter structure in their teaching (Lampert, 1986). Such a phenomenon might partially explain the results described above that teachers’ mathematical knowledge did not affect their cognition of children’s learning difficulties and knowledge of instructional practice . In addition, the information from teachers’ interviews indicated a tendency to heavily depend on mathematics instructional handbooks in classroom teaching. When teaching, teachers are more likely to incorporate mathematical ideas made on the basis of guiding booklets and subject matter knowledge, and perhaps disregard children’s learning difficulties.
Moreover, teachers had higher agreement in Cooperative Small Groups work rather than in Oral Explanation. Perhaps the reform recommendations of the New Curriculum and constructivism approach had influenced parts of teachers’ beliefs. On the other hand, the results also found teachers who preferred to have their students Repeatedly Practice rather than work in Cooperatively Small Groups. The reasons, as suggested from the information obtained during teachers’ interviews are twofold. Firstly, mathematical concepts are known as a more formal abstract system than other subjects. It’s a challenge for children to connect mathematical symbols and operations in problem solving (Lampert, 1986). Some teachers have their students engage in drills and repetitive practice as an alternative method for mathematical instruction, especially for students learning abstract concepts. Secondly, teachers believe that students must master some computation skills and fear that many problems are beyond the discussion abilities of their students (Carpenter, Fennema, Peterson, Chiang, & Loef, 1989).
Conclusion
Teachers’ efforts to put the ideas and recommendations of mathematics reform into practice have been effected by the teachers’ knowledge about mathematics and pedagogical content knowledge (Knapp, 1997). The results of this study also reflect such views. For the purpose of enhancing teachers effectively apply what they learned about children in their classroom teaching, more research-based knowledge on mathematics learning and inservice education program for teachers are needed to enrich teachers’ knowledge about school mathematics and challenge teachers’ beliefs of drill-and-practice.
Acknowledgement
This research was supported by a grant from the
national Science Council under grant No. 85-2513 S 081B001. The opinions do not
reflect the views of the foundation. We would also like to express our thanks
to Dr. Tien-Chen.
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