The Importance of Metaphorical Thinking in the Teaching of Mathematics (original) (raw)
Related papers
5) The centrality of metaphor in the teaching of mathematics
2011
Ingles: One of the challenges of teaching mathematics is that it is not about anything, literally. Mathematical objects (triangles, groups, surds, etc) do not exist in the real world. It is impossible to pick up a circle, although it is fairly easy to pick up a small piece of metal in the shape of a circle. The task of a (constructivist) teacher is to present to the student experiences from which the student may abstract the various aspects of the generalised mathematical CONCEPT and construct them into his own personal concept. In this paper it is argued that metaphoris the principal—perhaps the only—tool at the teacher‘s disposal to achieve this, and that the most important job of the teacher is to select the metaphor for presentation to the student which will most readily help her to construct herown concept. Examples will be presented to show that any given mathematical CONCEPT typically has several metaphors from which the teacher may choose, and also that no single metaphor ca...
Mathematical Thinking and Learning, 2001
The central question addressed in this article concerns the ways in which applied problem situations provide distinctive conditions to support the production of meaning and the understanding of mathematical topics. In theoretical terms, a first approach is rooted in C. S. Peirce's perspective on semiotic mediation, and it represents a standpoint from which the notion of interpretation is taken as essential to learning. A second route explores metaphorical thinking and undertakes the position according to which human understanding is metaphorical in its own nature. The connection between the two perspectives becomes a fundamental issue, and it entails the conception of some hybrid constructs. Finally, the analysis of empirical data suggests that the activity on applied situations, as it fosters metaphorical thinking, offer students' reasoning a double anchoring (or a duplication of references) for mathematical concepts.
Metaphors in mathematical thinking and in research on mathematical thinking: a prop or a trap?
When Rainer Maria Rilke spoke about the “geflügelte Entzücken” (“winged energy of delight”) or the “frühen Abgrund” (“first abysses” of childhood) (Rilke, 1956, 157), he was helping himself with the poets’ favorite device known as metaphor. The use of this special linguistic form, although usually associated with literary context, far exceeds the boundaries of literature and poetry. Scientists are probably as frequent users of metaphors as poets, except that they try to conceal this fact by pushing the tropes into the straitjacket of formal definitions. In this way, they deprive the metaphor of this one feature that poets find most endearing: ambiguity. For instance, geneticists use metaphors when they speak about messenger RNA, as do physicists when they mention chain reaction. In this paper, I reflect on the role of metaphors in two inter-related do-mains: that of creative mathematical thinking and that of research on mathematical thinking. In particular, I aim to show that metaphors shape our thinking, either mathematical or meta-mathematical, and through thinking, they mould our decisions and actions. The focus of these reflections is on one ubiquitous and particularly consequential type of metaphor that can be called metaphor-of-object or MofO, for short. I am guided by the question of MofO's relative strengths and weaknesses. I begin with stressing the all-important constructive role of MofO in the development of mathematics. Later, while reflecting on the way the same metaphor functions in mathematics education research, I give special attention to those of its uses that are potentially harmful. I conclude with an attempt to answer the question of how to utilize metaphors in both mathematics education research and in teaching, so as to make sure they serve as props rather than traps.
Cognitive Metaphors in Teaching Mathematics
Romanian Journal of English Studies, 2011
The paper discusses a number of conceptual metaphors used by teachers as ways of explaining unknown mathematics matters to secondary and high school pupils. The linguistic phenomenon is approached mainly from a cognitive perspective, but brief considerations from an emotional point of view are also made, with the purpose of finding arguments for why they function so efficiently in the teaching-learning process.
International Journal of Secondary Education, 2017
This paper is linked to some work that I have done in my PhD research, which is in progress. It shows metaphors that learners have about mathematics. These metaphors were investigated through a questionnaire with seven open-ended questionnaires. An inductive analysis of the learners' metaphors for mathematics indicated that learners had well developed and complex metaphors about mathematics, mathematics learning, mathematics teaching and the mathematics teacher. Some mental images were formed: Firstly, the notion that mathematics learning is challenging and may involve anxiety; secondly, that perseverance is needed as continuing effort will bring reward; and lastly, that positive or negative mental images of mathematics are related to the perception of and the active versus passive role of the learner. Considering that the metaphors were collected from learners with an extensive engagement with mathematics learning, the endings would imply that these metaphors reflects the students' mental images formed as a result of experiencing mathematics learning through the use of metaphors.
In this paper I present a study investigating ways in which secondary school students (9 th grade) conceive mathematical objects (like functions and variables), in modelling activities. The focus is on how mathematical knowledge is structured in the pupils and what mental dynamics play on activating mathematical thinking. In particular, I analysed students' cognitive processes within the embodied cognition perspective. My aim is to observe how metaphorical thinking and imagery can foster the transition from perception to theory, and the construction and communication of mathematical thinking.
Metaphorical Perceptions of Fourth-Grade Primary Students towards Mathematics Lesson
International Journal of Psychology and Educational Studies, 2020
The aim of this study is to reveal the perceptions of fourth-grade primary students about "mathematics lesson" through metaphors and to investigate whether main categories identified regarding the common characteristics of the metaphors differed by gender. A qualitative research design was employed in the study. The study was carried out in the spring term of the 2018-2019 academic year with the participation of 116 fourth-grade students from three public primary schools in a province which is in the western Black Sea Region. The primary data source of the study is "Mathematics Lesson Metaphors Questionnaire ". The data of the study was obtained by completing the blanks in the sentence 'Mathematics lesson is like ………………. Because ………………………….'. The students were asked to write down a metaphor about the mathematics lesson in the first blank and to explain the reasons why they wrote this metaphor in the second blank. The data were analyzed by content analysis method in this study. As a result of the study it is found that the fourth-grade students developed 64 different metaphors about mathematics. These metaphors are gathered in 13 subcategories under the main categories of mathematical knowledge, principles of mathematics teaching, mathematical skills, affective characteristic towards mathematics. At most metaphors are developed in the category of affective characteristic towards mathematics.
On metaphors in thinking about preparing mathematics for teaching
Educational Studies in Mathematics
This paper explores how different schools of thought in mathematics education think and speak about preparing mathematics for teaching by introducing and proposing certain metaphors. Among the metaphors under consideration here are the unpacking metaphor, which finds its origin in the Anglo-American school of thought of pedagogical reduction of mathematics; the elementarization metaphor, which has its origin in the German school of thought of didactic reconstruction of mathematics; and the recontextualization metaphor, which originates in the French school of thought of didactic transposition. The metaphorical language used in these schools of thought is based on different theoretical positions, orientations, and images of preparing mathematics for teaching. Although these metaphors are powerful and allow for different ways of thinking and speaking about preparing mathematics for teaching, they suggest that preparing mathematics for teaching is largely a one-sided process in the se...
Metaphorical Perceptions of Middle School Students towards Math
International Journal of Instruction, 2018
In every society, the issue of whether an individual can do math has become a central issue. Middle school years, when students' perceptions towards math start to form, have a critical importance for them to develop a positive attitude towards math. The aim of this study is to determine the meaning that the students attributed to the math and to reveal the students' perceptions towards math. Metaphors were also benefited from in order to carry out this aim. The sample of this study was composed of randomly selected 273 students from Yalçın Eskiyapan Middle School located in the district of Keçiören, Ankara in 2015-2016 academic year. When the data obtained from the students were examined, it was seen that the most frequent repeating metaphors were life (38), game (14), book (10), and space (6). The way in which the students explain their metaphors was examined by inductive content analysis. As a result of the analysis, the obtained codes were gathered under four themes as opinions about the content of the math, opinions about the applicability of the math, opinions about the math, and opinions about the teaching of the math. Each theme has two categories, including positive and negative ones.
Using metaphors to unpack student beliefs about mathematics
School Science and …, 2008
This paper reports on the beliefs that ninth and tenth grade students have about mathematics. These beliefs were revealed using contemporary metaphor theory. An analysis of the students' metaphors for mathematics indicated that students had well developed and complex views about mathematics including math as: an Interconnected Structure, a Hierarchical Structure, a Journey of Discovery, an Uncertain Journey, and a Tool. The other common themes that were revealed were that perseverance is needed for success in mathematics and that some students view their role in learning math as active while others view their role as passive.