PSYCHOLOGY STUDENTS’ CONCEPTIONS ON THE TEACHING AND LEARNING OF ARITHMETIC: AN ANALYSIS OF REPORTS ON WORK SESSIONS WITH CHILDREN

Alvaro Buenrostro FES Zaragoza, UNAM, México alvaroba@servidor.unam.mx

Olimpia Figueras Cinvestav, IPN, México dfiguera@mailer.main.conacyt.mx

Abstract: In the study we are now reporting, research was performed on the conceptions that psychology students uphold concerning the changes in arithmetical knowledge of first and second grade-school pupils, as well as on the role that such psychology students assign to the didactic activities they carry out with the children. Through a qualitative analysis of reports from the work sessions, that were written by the students, three major tendencies were identified in the conceptions on change regarding arithmetical knowledge. One of them focused on the strategies, a second one on the assessment of answers, and the last one on the explanation of change. In the matter of the role assigned to didactic activities, it was found that, from the students’ perspectives, such activities may serve for a consolidation of the acquired knowledge, for the promotion of new knowledge, or for the evaluation of such knowledge.

Buenrostro and Figueras (1999) have suggested the need that educational psychologists be provided with both the theoretical and the practical tools to face the problem of a low school performance in mathematics, and particularly in arithmetic. Obtaining information about the conceptions upheld by psychology students about the various aspects related to the teaching and learning of arithmetic, is an important step towards a proper orientation of the teaching that these students receive. One way to approach an understanding of these conceptions is to examine the writings that the students produce as part of their training in the intervention process that they carry out with children in the early grade-school years.

The purpose of the present research was to inquire into the conceptions of a group of psychology students, with respect to:

· the way they conceived changes in arithmetical knowledge of first and second grade pupils which were considered low-performance children, and

· the role they assigned to the didactic activities used to promote such changes.

Theoretical Perspectives

Thompson’s (1992) position is adopted here, concerning the notion of conception as “a more general mental structure, encompassing beliefs, meanings, concepts, propositions, rules, mental images, preferences, and the like” (p. 130); and also, his statement relating to the aspects that are included in a teacher’s conception concerning the teaching of mathematics as “desirable goals of the mathematics program, his or her own role in teaching, the students’ role, appropriate classroom activities, desirable instructional approaches and emphases, legitimate mathematical procedures, and acceptable outcomes of instruction...” (p. 135). The research being reported in this article must be placed along with the studies analyzing the written reports of teachers, as a means to understand their beliefs (Gellert, 1999; Kaplan, Rosenfeld & Appelbaum, 1999).

On the other hand, a concern is shared with various researchers about whether teachers, and, in our case, educational psychologists, are becoming aware of the various thought processes, and of the actions and justifications that children carry out when facing situations of a mathematical nature. Our starting point is that the decisions made both by teachers and psychologists in their respective realms will be better based if these persons have acquired a greater knowledge of these issues. Through the application of Cognitively Guided Instruction, Carpenter, Fennema, Franke, Levi, and Empson (1999, p. 105) have found that “learning to understand the development of children’s mathematical thinking could lead to fundamental changes in teachers’ beliefs and practices, and that these changes were reflected in students’ learning”. Other researchers have highlighted the need for teachers use of the flexible interview (Ginsburg, Jacobs, & López, 1998) or the clinical interview methods (Hunting, 1997) as tool that permits uncovering children’s thinking about mathematics. Doig and Hunting (1995), and Buenrostro and Figueras (1999) emphasize the importance that in programs focused on the understanding of the processes of the children’s mathematical thinking, participants thereof carry out a practical and direct work with children.

In order to analyze the students’ conceptions, we will lean on several concepts from authors like Watzlawick (1989) and Keeney (1987). For the latter, epistemology on a sociocultural level, “is tantamount to the study of the way in which persons or person systems get to know things, and of the manner in which they think they know things” (Keeney, 1987, p. 27). The punctuations they establish, (i.e., the issues which are relevant to them), “create various realities, in the strict sense of the word” (Watzlawick, 1976, p. 75). Thus, beliefs are placed within a frame with a specific sense. To reframe these beliefs implies a change in their framework in order to give them a different sense.

The Context of the Research

This research is performed within the framework of the application of a teaching model (see Buenrostro & Figueras, 1999) wherein the psychology students from the Facultad de Estudios Superiores Zaragoza of the Universidad Nacional Autónoma de México, are carrying out work sessions, twice a week, with children from the first and second primary grades that have been reported by their teachers as pupils with a low school performance. In these sessions, a first assessment is made of the pupils' arithmetical knowledge and, based on this, various didactic activities are carried out with the purpose of enriching their knowledge and promoting an improvement of the children’s school performance. An average of ten students participate, with ten children and a professor in educational psychology who provides advise to the students concerning the instruction that they provide the children.

Modes of Inquiry and Data Collection

The research has a qualitative and reconstructive nature. Attention is focused on the comprehension of the processes, conceptions, and actions by the participants. Therefore, we concur with Gellert (1999) when he asserts, in reference to the purposes of an investigation on the conceptions of mathematics teachers: “The aim is to reconstruct the intentions and strategies of the actions of the people under study. The focus is on understanding and not on predicting” (p. 28). What we try to do, then, is to understand the psychology students’ conceptions through a reconstruction exercise which permits to arrive not to a precise and representative image of such conceptions, but to the identification of certain cognitive and action tendencies. In order to attain this, an analysis was performed of the contents of the reports that the students prepare for each of the work sessions that they hold with the children. In the reports, the purposes to be attained in the session are specified, a description is drawn, and an assessment is made both of the didactic activities and the strategies used by the children; also, suggestions are made for the following session. The analysis is based on the procedure employed by Gellert (1999) to get to know the convictions of future teachers about various aspects of mathematics teaching through the analysis of such teachers’ diaries. The analysis of the reports was done in three clearly designed stages: 1) the performance of successive revisions of the reports, with an aim to derive specific categories; 2) a new revision, now guided by the categories which were identified in the former stage; and 3) the specification of tendencies in the students’ conceptions.

Results and Discussion

Regarding the way in which the students conceive the changes in the children’s arithmetical knowledge, three major tendencies are seen which can be characterized as follows:

Focus on the Strategies

A change is considered to have been promoted when the child makes use of a strategy that differs from the one used before to solve the situation posed. The following statements are representative of this tendency: “a change occurred in the type of strategy, for instead of counting everything, he started with the greatest addend”; “as opposed to former sessions where he counted the cubes in a bar one by one, he now counted each bar while saying ten, twenty, thirty, etc.” It is important to note that the assertions corresponding to this tendency contain specific descriptions of the actions performed by the children, and that belong in a student’s conceptual framework allowing him/her to identify such actions as significant and indicative of a change in the children’s knowledge levels.

Focus on the Evaluation of the Answers

In this tendency, more than describing change, judgement is passed about it: “an improvement was seen in the solution of problems...”; “the child’s performance was good”. Or else, the change is qualified as right or wrong: “she was asked to solve the last problem, which was correctly solved”; “the child answered the activities correctly”. One additional element touches on the help provided by the psychology student: “he solved most of the problems without any help from the instructors”. As it can be seen, it is difficult within this tendency to realize in what way the change took place, for what seems to matter is the result in the posed situations, or the final answer.

Focus on the Explanation of the Change

Here, the child’s action, which is not described, serves to infer an internal cognitive process that explains the action: “the child understood how the numbers should be read and written”; “she realized that it was easier to count by twos.”

These tendencies reflect three different ways of punctuating the children’s behavior that would seem to stem from different observational perspectives. In the first of these, there is an intent to describe the child’s actions and to frame them within a context of meaningfulness that could allow the student to interpret them as part of the construction processes of the children’s arithmetical knowledge, and which, in this case, are indicative of some progress in the children’s arithmetical thinking. In the other two tendencies, the children’s actions are taken as reference points, so that, starting from such actions, an evaluation can be made of them, or a number of inferences can be drawn with respect to internal processes that account for such actions. In both cases, the description of the actions is left aside, and the evaluation is done in terms of how “correct” the answers are, or on what the improvement has been concerning the actions. In the case of the inferences, it is the internal status that is responsible for the children’s actions.

With respect to the way in which the students conceive the didactic activities that are carried out to propitiate a change in the children’s arithmetical knowledge, our findings are that these are closely related to the purpose of the activity. Thus, the activities can serve to

· Give continuity to teaching: “the same mechanics for activities will be followed”.

· Consolidate the acquired knowledge: “problems on... will be presented in order to reinforce the child’s knowledge”.

· Increase the level of difficulty: “problems with a higher degree of complexity will be posed to her”.

· Facilitate learning: “simple multiplication problems will be posed to him”.

· Evaluate the acquired knowledge: “we will try to analyze whether the child is already able to solve any kind of problem”.

It is interesting to observe how the purpose of the activities is conceived in different ways, not necessarily excluding each other, and how these different approaches reflect three conceptions as to the purposes of teaching: the consolidation of knowledge acquired; the promotion of new knowledge; and the evaluation of such knowledge. Apparently, in the first two cases above, it is considered that the children’s behaviors are appropriate, and that it is therefore necessary to continue to apply a certain set of activities, with an aim to consolidate the knowledge that such activities had promoted. In the third and fourth cases, the intention is that the child progresses, either through proposing situations which, in some way, lead him/her to more complex resolution forms, or through a graduation of activities going from the simple to the complex. And finally, activities are conceived as an instrument to evaluate the children’s knowledge.

Conclusions and Implications

The psychology students’ conceptions concerning the changes in the children’s arithmetical knowledge pertain to different punctuation processes; this leads to conceive the change from various standpoints. If one of the purposes of the programs of the initial formation of psychologists and teachers consists in linking such programs with the children’s mathematical thought processes, then it is important that the faculty in charge of these programs are clear about the students’ conceptions so that, if the case arises, they can help them understand the children’s behavior from different perspectives than those they have used before, and thus open new ways for them to conceive such a behavior. In other words, a reframing of the situation ought to be promoted –i.e., a change in the manner of understanding an assertion or a behavior, to attribute a different sense to these-- which could allow the students to become aware of those aspects in the children’s behavior that are relevant in the process of structuring arithmetical knowledge.

References

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