Using Novel Tasks in Teaching Mathematics: Three Tensions Affecting the Work of the Teacher (original) (raw)
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Powerful tasks: A contribution to a high level of acting and reflecting in mathematics instruction
Educational Studies in Mathematics, 1993
LEVEL OF ACTING AND REFLECTING IN MATHEMATICS INSTRUCTION ~s'nlhcr. Mathematics instruction contains two conflicting demands: on the one hand, the demand for economical efficiency and for well-developed "motorways" and, on the other hand, the demand that pupils should investigate and discover for themselves and have the freedom to "pave" their own ways. It is argued that tasks with a certain richness and quality offer some steps towards a construefive handling of this dilemma. The author tries to develop some properties of powerful tasks and to sketch the structure and philosophy of one concrete system of powerful tasks for the concept of angle. The main part of this paper presents seven examples of powerful tasks: five from the system of tasks and two with regard to pupils' working with 2-D-graphic systems. The construction of powerful tasks is viewed as a valuable contribution to bringing the theory and practice of mathematics education closer together. BACKGROUND Discussing the importance of tasks in mathematics education has a long tradition. A detailed analysis of the so-called Task Didactics (Aufgabendidaktik), which is one of the marked features of Traditional Mathematics, is given by Lenn6 (1969). Traditional Mathematics was the leading stream of mathematics education in Germany (and in a similar way in Austria) until the middle of this century, and then was progressively displaced by New Mathematics (which in turn is being pushed back more and more). Task Didactics is characterized by a partition of the mathematical subject-matter into specific areas (e.g., fractions, percentages, triangles, quadrilaterals). Each area is determined by a special type of task which is systematically treated, progressing from simple to more complex tasks (combination of simple tasks). Cross-connections (e.g., regarding fundamental ideas or structures) are not worked out in detail. In general, the teacher teaches theories and methods and the pupils have to apply them by solving tasks. How much has this situation changed? Recent empirical research studies, like those of Bromme (1986) or Clark and Yinger (1987), show that even nowadays mathematics teachers plan and organize their instruction on a large scale with the help of tasks. Research with regard to tasks takes different directions. There are many contributions to general considerations about tasks, for example: Wittmann (1984) views teaching units as the integrating core of mathematics education, incorporating mathematical, pedagogical, psychological, and practical aspects in a natural
EXAMINING THE TASKS OF TEACHING WHEN USING STUDENTS' MATHEMATICAL THINKING
Recent research suggests that the examination of students' work may lead to changes in teaching practice that are more effective in terms of students' mathematical learning. However, the link between the examination of students' work and the teachers' actions in the classroom is largely unexamined, particularly at the secondary level. In this paper, I present the results of a study in which teachers had extensive opportunities to examine the development of students' conceptual models of exponential growth in the context of their own classrooms. I describe two related aspects of the practice of one teacher: (a) how she listened to students' alternative solution strategies and (b) how she responded to these strategies in her practice. The results of the analysis suggest that as the teacher listened to her students, she developed a sophisticated schema for understanding the diversity of student thinking. The actions of the teacher supported extensive student engagement with the task and led the students to revise and refine their own mathematical thinking. This latter action reflects a significant shift in classroom practice from the role of the teacher as evaluator of student ideas to the role of students as self-evaluators of their emerging ideas.
This study focuses on the constructions in terms of area and perimeter in equivalent triangles developed by students aged 12 to 15 years-old, using the tools provided by Cabri-Geometry II (Laborde, 1990). Twenty-five students participated in a learning experiment where they were asked to construct: a) pairs of equivalent triangles ‘in as many ways as possible’ and to study their area and their perimeter using any of the tools provided and b) ‘any possible sequence of modifications of an original triangle into other equivalent ones’. As regards the concept of area and in contrast to a paper and pencil environment, Cabri provided students with different and potential opportunities in terms of: a) means of construction, b) control, c) variety of representations and d) linking representations, by exploiting its capability for continuous modifications. By exploiting these opportunities in the context of the given open tasks, students were helped by the tools provided to develop a broader view of the concept of area than the typical view they would construct in a typical paper and pencil environment.
Teacher Actions Bring up Students' Thoughts in Solving Mathematical Problems
Universal Journal of Educational Research, 2019
This article aims to describe the actions taken by the teacher to bring out students' thoughts so that students are able to make strategies in solving mathematical problems. Data obtained from direct observation of the research subject when giving actions to students. The subject of the study was the mathematics teacher who was chosen based on special characteristics that fit the purpose of the study. In this study, there are 2 different characteristics when the teacher gives action to students so that students are able to make strategies in solving problems with the story. First, the teacher gives four actions as follows: (1) the teacher asks students to rewrite the problem using their own words, write down what is known and what is asked in the problem; (2) the teacher asks students to turn problems into symbolic problems; (3) the teacher asks students to create symbolic mathematical problems and use operations and procedures in mathematics appropriately to solve problems; (4) the teacher directs students to check the results of problem solving. Second, the teacher gives the following actions: (1) the teacher asks students to write down what is known and what is asked in the problem; (2) the teacher asks students to turn problems into symbolic problems; (3) the teacher asks students to create symbolic mathematical problems and use operations and procedures in mathematics appropriately to solve problems.
International Group for the Psychology of …, 2004
This research report describes the findings of a study on teachers' ways of interpreting student responses to tasks involving equivalent expressions. The teachers in this study were engaged in model-eliciting activities designed to promote the development of their knowledge and reveal their models (or interpretations) of their students' algebraic thinking about equivalent expressions by creating a library of their students' work. This report focuses on one teacher's model of his algebraic practice. Results showed that this teacher devoted a significant amount of time to the implementation of the algebraic unit. The teacher employed visual strategies for the first time and began to perceive their usefulness in demonstrating the equivalency of two expressions.
Strategies Used by Teachers of Mathematics in the Implementation of Tasks
This article identifies and describes the teaching strategies used by mathematic teachers in implementing tasks. The context of this study was the classroom of three mathematic teachers at elementary and secondary education. This study is framed as a qualitative approach, using observation and interview as the tool for data collection. The results showed that there are varieties of teaching strategies used by the teachers in the implementation of mathematical tasks, classified in the following categories: Pre-instructional, co-instructional and post-instructional proposed by Diaz and Hernandez (2010).
What Mathematic Teachers Say about the Teaching Strategies in the Implementation of Tasks
In this article we will discuss, through the explanations given by teachers who teach Mathematics, the importance of using teaching strategies in the implementation of tasks. Teachers who participated in it belong to the group " Observatory Mathematics Education " (OME-Bahia). This study was framed in a qualitative approach and data were collected through observation and an interview. The interview was conducted taking into account the observation produced through videos where the implementation of mathematical tasks was recorded, serving as support different times where teachers used different teaching strategies in order to take them up again at the time of the interviews. The results showed that in the using of each teaching strategy, there is a particular importance that is assumed by the teacher; that means, their intentions are subject to different variables. 1. Introductión This article was developed in the interest of knowing the importance of using teaching strategies in the implementation of mathematical tasks, through the explanations of teachers. The construction of this proposal was taken from the questions that arose when analyzing the videos referring to the integration of mathematical tasks, where teachers use different teaching strategies; because it is observed that each teacher has a certain intention at the time of using them. The teaching strategies used in the implementation of mathematical tasks, regain importance at the moment the teacher justifies through the different explanations of why, how, when and where to use them, thus recognizing the usefulness of the strategies of teaching through the process of reflection, argumentation, etc. in order to rethink their respective use. In this sense, rethinking the use of teaching strategies is not an easy task, since the pedagogical experience of the teacher necessarily constitutes a pedagogical knowledge. So, " the experience of the teacher's daily work is a pedagogical knowledge, which should be valued, and more than that, used in the service of teaching strategy that provide transformative actions and actions that are not stagnant as often happens " (Stacciarini & Esperidião, 1999). In fact, there are different studies that address the importance of using certain teaching strategies through the explanations expressed by the teacher. For example, in the study, entitled " Alternative approaches for teaching and learning of functions: An analysis of a non-teaching intervention " different teaching strategies are used, such as: the organization of students in small groups, whose explanations given by the teachers show that this strategy seeks to have better contact with students (Geromel & Redling, 2012). And González, Villota and Agredo (2017) state that a teaching strategy is an integral and transversal concept in the field of education. In this way, the aim of this article is to discuss, through the explanations presented by teachers who teach mathematics, the importance of using teaching strategies in the implementation of tasks. Thus, initially we locate the moment where each teacher implements a certain teaching strategy and then presents his/her respective explanation regarding the importance of using it. Some conceptions about the terms that they infer in the understanding of this study are presented.
Journal of Mathematical Modelling and Application, 2014
We describe an experience with Mexican in-service teachers. In this experience the teachers designed didactic proposals to develop students' mathematical knowledge. This proposal was used to teach the same course several times. After each implementation the teachers discussed the experience. We observed the cycles of refinement when the teachers implemented the proposals and discussed the results. In this process the teachers' conceptions about learning and teaching mathematics emerged. The conceptions changed when the teachers shared and discussed the results amongst themselves. We used the Models and Modelling perspective to analyse instructional proposals. The documents produced by the teachers were the base to identify the evolution of ways of thinking about the students' learning process. The teachers described in this paper are studying in a master's program in mathematics education.
Proceedings of the 28th Conference of …, 2006
Research suggests that teachers need to have mathematics content knowledge that allows them to effectively deal with the particular mathematical issues that arise in their everyday practice. This implies the importance of providing teachers with learning opportunities that prepare them to both recognize situations in their practice where these mathematical issues arise and be able to apply their mathematical knowledge to successfully manage these situations. Yet, little research has focused on how such learning opportunities can be effectively promoted in teacher education. In this article we take a step toward addressing this limitation by discussing and exemplifying a special kind of tasks for use in teacher education which we call "teaching-related mathematics tasks." These are mathematics tasks that are connected to teaching and can foster the development of teachers' mathematics content knowledge that is important for teaching.
1995
The persistence of "naive" conceptions relative to many natural phenomena in subjects that have learnt in school a "scientific" interpretation for them, and their difficulty in using school-learnt mathematical models to interpret non-trivial situations raise interesting issues for psychological and educational research. This report analyses some aspects relative to the passage to a geometrical conception of the phenomenon of the Sun's shadows from the "naive" non-geometrical conceptions that most 9/11 year-old students have of this phenomenon. I.Introduction Mathematics plays an important role (in the history of culture and the intellectual maturation of the individuals) for the construction of "scientific" conceptions of phenomena pertaining to a variety of fields (from astronomy to genetics). "Scientific" interpretations based on mathematical models learnt at school, however, appear fragile, and "naive" conceptions not only persist in common culture, but resurface also in cultured people when, in difficult problem situations, mathematical models are for some reason not serviceable. This happens in particular for the case of the geometrical modelisation of the Sun's shadows (Boero, 1985). This is a phenomenon which lends itself particularly well for the study of this subject. Many children manifest, in fact, deeply radicated "naive" (non-geometrical) conceptions of this phenomenon even at a relatively advanced age (9 12 years). At this age the elementary geometrical modelization of the phenomenon is accessible at school as it requires elementary mathematical tools. On the other hand, the phenomenon has been widely used for dozens of years now in different countries in renovating mathematics teaching to introduce different geometrical concepts and motivate geometric activities, and this offers a wide base of experiences for further investigation (