Teachers’ conceptions of mathematical fluency

Fluency in mathematics is defined in various forms, such as computational fluency, procedural fluency, and mathematical fluency (Keiser, 2012; Kilpatrick, Swafford, & Findell, 2001; Sullivan, 2011) ‘procedural’ often leave teachers interpreting fluency as simply being able to follow a set formula or to quickly compute mathematics. This narrow belief may affect the way teachers teach mathematics and what they expect their students to be able to do to be fluent. This study explored primary teachers’ conceptions of mathematical fluency; including how they define mathematical fluency and what features they associate with the term. Exploration of teacher conceptions can provide insight into what teachers think and how this may affect their teaching. A qualitative approach was taken and data were collected via an on-line questionnaire (n = 42) and semi-structured interviews (n= 17). Thematic analysis of the questionnaire and interview data focused on the words teachers used to define mathematical fluency and how they described fluency in their students. A theoretical framework of teacher conceptions was applied, highlighting internal and external factors that influence teachers’ conceptions. Rich descriptions of fluency shared by teachers was captured through the analysis process. Findings revealed that teachers held mainly contemporary views of mathematics and of how students learn mathematics. Teachers believed that students did not truly have mathematical fluency if they could not also apply, and demonstrate or communicate their understanding of concepts. Teachers spoke of students having ‘fluidity’ and ‘flexibility’ in their ways of thinking, that students were able to move beyond errors that caused others to be ‘stuck’. A case is made for the reframing of mathematical fluency based on these findings. Adopting a view of fluency as an amalgamation of conceptual understanding and strategic competence, making it synonymous with mathematical proficiency

Elementary school teachers\u27 beliefs about teaching and learning mathematics : selected case studies in Taiwan.

Mathematics curriculum innovation has been launched in Taiwan recently in order to reflect the changing needs of the 21st century. The underlying assumptions of reform are: a learner-centered approach, emphasis on confluent education, and a problem-solving & reasoning approach. Research has revealed that teachers\u27 beliefs can negatively interact with curriculum reform. On the other hand, some studies document that beliefs have little effect on instructional behavior. Therefore, this study attempts to investigate three questions: (1) what are the teachers\u27 beliefs about the teaching and learning of mathematics in Taiwanese elementary schools and in what ways are teachers\u27 beliefs congruent with the ongoing trend of reform; (2) what is the general picture of teachers\u27 mathematical instructional practices in Taiwanese elementary schools and in what ways are these instructional practices congruent with the ongoing trend of reform; and (3) what is the relationship between teachers\u27 beliefs and their instructional practices? Basically, this study combines qualitative and quantitative methods in collecting and analyzing data. That is, teacher interviews and questionnaires were administered in order to understand teachers\u27 beliefs about teaching and learning mathematics while observational checklists and naturalistic field observations were used to portray instructional behavior. The major findings of this study are: (1) Elementary school teachers\u27 beliefs tend to hold with the traditional absorption learning theory and seem incongruent with the undergoing curriculum reform. (2) The instructional practices tend to reflect a traditional teacher-centered classroom and also seem incongruent with the launched reform. (3) Teachers\u27 beliefs about teaching and learning play a vital role in shaping their instructional behavior; the situational constraints merely play a minor role. In light of the above findings, some implications such as teacher education were drawn to broaden teachers\u27 beliefs

Justifications-on-demand as a device to promote shifts of attention associated with relational thinking in elementary arithmetic

Student responses to arithmetical questions that can be solved by using arithmetical structure can serve to reveal the extent and nature of relational, as opposed to computational thinking. Here, student responses to probes which require them to justify-on-demand are analysed using a conceptual framework which highlights distinctions between different forms of attention. We analyse a number of actions observed in students in terms of forms of attention and shifts between them: in the short-term (in the moment), medium-term (over several tasks), and long-term (over a year). The main factors conditioning students´ attention and its movement are identified and some didactical consequences are proposed

Problem Solving, Beliefs About Mathematics, And The Long Arm Of Examinations.

A ZJER journal article.The recent, almost global, shift in emphasis from computation towards problem solving skills in mathematics education curricula has opened up fresh areas of research. As one of the factors now widely acknowledged as having a tremendous influence on the course and quality of the problem solving process, beliefs about mathematics have been the subject of a number of studies including the present one. The main objective of this study was to explore and uncover the kinds of beliefs Zimbabwean secondary school students hold concerning the nature of mathematics, the leaning of mathematics, and the doing of mathematics. The study focused on Form 4 students (11th graders, typically 16 years old) and used in-depth individual interviews of 10 students (4 of them males) to gather data. A preliminary survey was used to structure the interviews, and video-taped observations of classroom sessions were done to explore the relationship between the beliefs and the context in which most of the mathematics is learned. Analysis uncovered 46 beliefs. The nature of the beliefs suggests that the students simultaneously and mostly subconsciously hold two distinct views of mathematics. The views, which can be characterized as "discipline" mathematics and "examination” mathematics, overlap to varying degrees in different individuals and have conflicting characteristics in some aspects. Furthermore, the views appear to be strongly influenced and dominated largely by an evaluation effect originating from the practice and culture of summative national examinations and, to some extent, by the nature of the mathematics curriculum and a lack of exposure to genuine problem solving activities in the students’learning experiences

UNDERGRADUATE STUDENTS’ PROOF CONSTRUCTION ABILITY IN ABSTRACT ALGEBRA

The opinion of mathematics education expert toward the necessity of introducing mathematical proof to be thought at all levels was increased. Number of mathematics teacher in America conducted intensive discussion about whether mathematics proof should be included or excluded in mathematics curriculum. Teachers agree on the importance of proof and on the necessity for students to develop the skills needed to construct proofs. However many students of all levels of education face serious difficulties with constructing mathematical proof. Whereas, the limitedness on proving ability would influence on learning other advanced mathematics such as real analysis, abstract algebra, and others. That condition would hamper the development of students’ reasoning and others mathematical thinking abilities. The objective of developing proof methodology was to improve students’ ability on understanding mathematical proof, and proof constructing of mathematical statements. Some approaches had been developed, among them was concept of generic proof. Generic proof method of example level was explained of a concepts in general based on a specific example or case. The purpose of this paper is to categorizing and describing the different types of processes that undergraduate students use to construct proofs. This study involved 87 undergraduate students and two kinds instruments those proof reading test and a proof construction test. Keywords: mathematical proof, geometr

Computer Aided Phenomenography: The Role of Leximancer Computer Software in Phenomenographic Investigation

The qualitative research methodology of phenomenography has traditionally required a manual sorting and analysis of interview data. In this paper I explore a potential means of streamlining this procedure by considering a computer aided process not previously reported upon. Two methods of lexicological analysis, manual and automatic, were examined from a phenomenographical perspective and compared. It was found that the computer aided process - Leximancer - was a valid investigative tool for use in phenomenography. Using Leximancer was more efficacious than manual operation; the researcher was able to deal with large amounts of data without bias, identify a broader span of syntactic properties, increase reliability, and facilitate reproducibility. The introduction of a computer aided methodology might also encourage other qualitative researchers to engage with phenomenography

Developmental Understanding of the Equals Sign and Its Effects on Success in Algebra

For some students, the equals symbol is viewed as a directive to carry out a procedure, instead of a symbol expressing mathematical equivalence. The purpose of this study was to develop and to pilot a questionnaire to measure students’ understandings of relational equivalence as implied by their interpretations and use of the equals symbol. The results of this questionnaire were compared with student testing data with the goal of determining a correlation between understanding of symbolic equivalence and success in a typical algebra course. It was found that students who demonstrated an ability to define and articulate an appropriate meaning for the equals symbol scored significantly higher on an end-of-course test and on a state achievement test. However, this study also found that students who can define or articulate an appropriate meaning for the equals symbol may not necessarily be able to demonstrate a working knowledge or understanding of the symbol’s appropriate uses. Another significant conclusion found was that quality instructional practices contribute to students performing at a seemingly higher level, with respect to symbolic relational equivalence, than those shown in previous studies