Students’ understanding of the structure of deductive proof

While proof is central to mathematics, difficulties in the teaching and learning of proof are well-recognised internationally. Within the research literature, a number of theoretical frameworks relating to the teaching of different aspects of proof and proving are evident. In our work, we are focusing on secondary school students learning the structure of deductive proofs and, in this paper, we propose a theoretical framework based on this aspect of proof education. In our framework, we capture students’ understanding of the structure of deductive proofs in terms of three levels of increasing sophistication: Pre-structural, Partial-structural, and Holistic-structural, with the Partial-structural level further divided into two sub-levels: Elemental and Relational. In this paper, we apply the framework to data from our classroom research in which secondary school students (aged 14) tackled a series of lessons that provided an introduction to proof problems involving congruent triangles. Using data from the transcribed lessons, we focus in particular on students who displayed the tendency to accept a proof that contained logical circularity. From the perspective of our framework, we illustrate what we argue are two independent aspects of Relational understanding of the Partial-structural level, those of universal instantiation and hypothetical syllogism, and contend that accepting logical circularity can be an indicator of lack of understanding of syllogism. These findings can inform how teaching approaches might be improved so that students develop a more secure understanding of deductive proofs and proving in geometry.     

The relationship between biology teachers’ understanding of the nature of science and the understanding and acceptance of the theory of evolution

ABSTRACT

Despite the importance of the theory of evolution (TE) to scientific knowledge, a number of misconceptions continue to be found among biology teachers. In this context, the first objective of this study was to identify the impact of professional development programme (PDP) on teachers’ understanding of nature of science (NOS) and evolution and on the acceptance of this theory. Its second objective was to study the relationship among these variables. Three instruments were used to quantify these variables: the Views of the Nature of Science Version D (VNOS D+), the Assessing Contextual Reasoning about Natural Selection (ACORN), and the Measure of Acceptance of Theory of Evolution (MATE). The results indicate that the PDP had a positive impact on teachers, significantly improving their understanding of the NOS and natural selection, as well as their acceptance of the TE. Furthermore, a positive correlation between the understanding of the NOS obtained by teachers in the first part of the PDP and the understanding and acceptance of evolution that these teachers showed at the end of the programme was determined. However, no relationship between an understanding of the NOS and gains in the understanding and acceptance of evolution was found.     

Exploring pre-service secondary teachers’ understanding of care

Care in teaching has been widely investigated; however, little research has sought secondary pre-service teachers’ understandings of caring and their potential responsibility to care for students. Accordingly, semi-structured interviews were employed with four focus groups, involving 12 (2 male and 10 female) participants. Data were analysed within a qualitative paradigm using Interpretative Phenomenological Analysis (IPA) and inter-rater reliability. Five overlapping themes were identified, and although many tensions around the ambiguities of establishing appropriate boundaries to care were shown, caring was deemed central by the participants to being effective teachers. Overall, pre-service secondary teachers agreed that neither pedagogy nor discipline strategy would be effective without care. Training implications and suggestions for future research conclude this paper.     

Effects of interactive teacher-generated drawings on students’ understanding of plate tectonics

Instructional Science - We investigate an interactive teacher-generated drawing strategy in which the teacher constructs a drawing with the help of the students. The students contribute their ideas...     

The effects of explanation-driven inquiry on students’ conceptual understanding of redox

ABSTRACT

The study that is the subject of this paper tested the effects of EDI (explanation-driven inquiry) on students’ redox conceptual understanding and their misconceptions. Two classes of 119 10th grade high school students were involved. Two groups of students in the same school were chosen. One class that was taught by EDI teaching was set as the treatment group and another class that received conventional teaching was set as the control group. Students’ conceptual understanding was measured by a standardised instrument, and the students’ misconceptions were compared. The analysis of covariance showed that EDI teaching had significant effects on students’ redox conceptual understanding, and the results of the chi-square test demonstrated that students’ conceptual understanding level was improved by using EDI teaching. Also, after comparing the two groups of students’ misconceptions of redox, the results indicated that the students’ misconceptions changed toward deeper conceptual understanding in the treatment group that used EDI. Finally, some suggestions were made on how to implement EDI in the classroom.     

Developing understanding of pupil feedback using Habermas’ notion of communicative action

The focus of this article is to explore the notion of pupil feedback and the possible ways in which it can be understood and developed using Jürgen Habermas’ theory of Communicative Action. The theoretical position adopted is framed within the concept of assessment for learning, and is particularly related to the notion of assessment as learning within AfL. Furthermore, the paper is located within a social constructivist perspective. Jürgen Habermas’ theory of Communicative Action enables us to recognise that feedback, and more importantly the interpretation of feedback, cannot be a one-way process. Without recognition of pupil interpretation, its very purpose (to alter the learning gap) is compromised. This paper offers new ways of exploring feedback, which recognise complexity and the importance of interpretation and relationships in shared negotiated communicative contexts. It further contributes to the ways in which assessment and learning are understood and intersect.     

Scaffolding young children’s understanding of symbolic objects

Symbolic objects are cognitive tools children must learn to master very early in life. A crucial factor in cognitive development is the instruction or informational support children receive in social contexts. While numerous studies have investigated the relation between instruction and development in various domains, like language or symbolic play, still little is known concerning the role of this factor in the comprehension of symbolic objects. The purpose of this article is to illustrate with an overview of the results of the authors’ research how instruction scaffolds the comprehension and use of symbolic objects. In this framework instruction is conceived as a process that mediates the skills that children develop and meanings they build. Some educational implications are discussed.     

The effect of distributed practice on students’ conceptual understanding of statistics

In this study the effect of the reduced distribution of study activities on students’ conceptual understanding of statistics is investigated in a quasi-experiment. Conceptual understanding depends on coherent and error free knowledge structures. Students need time to construct such knowledge structures. A curriculum reform at our university resulted in statistics courses which were considerably shortened in time, thereby limiting students’ possibility to distribute study activities. Independent samples of students from before and after the reform were compared. To gauge conceptual understanding of statistics, students answered open ended questions in which they were asked to explain and relate important statistical concepts. It was shown that the reduction of distributed practice had a negative effect on students’ understanding. The finding that condensed courses make it more difficult for students to reach proper understanding of the subject matter is of interest for anyone who is engaged in reforming curricula or designing courses.     

Students’ images and their understanding of definitions of the limit of a sequence

There are many studies on the role of images in understanding the concept of limit. However, relatively few studies have been conducted on how students’ understanding of the rigorous definition of limit is influenced by the images of limit that the students have constructed through their previous learning. This study explored how calculus students’ images of the limit of a sequence influence their understanding of definitions of the limit of a sequence. In a series of task-based interviews, students evaluated the propriety of statements describing the convergence of sequences through a specially designed hands-on activity, called the ɛ–strip activity. This paper illustrates how these students’ understanding of definitions of the limit of a sequence was influenced by their images of limits as asymptotes, cluster points, or true limit points. The implications of this study for teaching and learning the concept of limit, as well as on research in mathematics education, are also discussed.     

Educators’ understanding of the mediation role in the classroom

Abstract

This study investigated educators’ understanding of the mediation role in the classroom. A quantitative research approach was used in a survey of a sample of 261 participants. To this end, a questionnaire was used for collecting data. The Chi-square test was used to analyse data and to test the hypotheses of the study. The findings indicated that educators differed significantly in the extent to which they understand the concept mediation. The findings also indicated that gender and teaching phase had a significant influence on educators’ understanding of the concept mediation. The findings further indicated that educators differed significantly in their understanding of the processes and competences involved in mediation. The findings also revealed that teaching phase had a significant influence on educators’ understanding of the processes and competences involved in mediation.     

Student teachers’ understanding of multicultural education in public schools

Abstract

As schools in South Africa become increasingly multicultural, pressure is being put on them to meet the needs of all learners. The South African Schools Act, No 84 of 1996, affirms the recognition of all forms of diversity in public schools. It seems clear from media reports, however, that diversity in schools is not readily accepted and that teachers are largely held accountable for racial prejudice in schools. This raises the question whether teachers fully understand multicultural education as a means to adequately manage diversity in schools. Against this background, a project was undertaken to establish what student teachers’ understanding of multicultural education is, and what they regard as the most important aims of multicultural education. Data were obtained by the use of semi-structured questions serving as a self-report instrument. The data collected were consolidated and categorised into themes. It became clear from the data that most respondents showed a seriously deficient conception of the nature, aims and material manifestations of multicultural education. It is therefore recommended that teacher education programmes should be scrutinised to ascertain whether they are providing the right kind of knowledge content to prepare student teachers to function competently in multicultural schools.     

An analysis of Grade 11 learners’ levels of understanding of functions in terms of APOS theory

This article reports on a study of six Grade 11 learners’ levels of understanding of concepts related to the function definition and representation. Task-based clinical interviews were used to elicit the learners’ interpretations and reasoning when working with these function-related concepts. Indicators for Action–Process–Object–Schema (APOS) theory conception levels were formulated and used to categorise learners’ written and interview responses into conception levels of understanding of the function concept. According to the South African Curriculum Assessment Policy Statement (CAPS) curriculum, Grade 11 learners are expected to be operating at the object and schema levels after instruction. however, the results indicated that learners were operating at the action and process levels as their understanding was characterised by vague definitions of function-related concepts and memorisation of procedures for translating between symbolic and graphical representations.     

Assessing students’ understanding of models of biological processes: a revised framework

Models are very important tools when learning and communicating about science. Models used in secondary school biology education range from concrete scale models, such as a model of a skeleton, to abstract concept-process models, such as a visualisation of meiosis. Understanding these concept-process models requires a profound understanding of the concept of models and how they are used in biology. This study evaluates an existing framework for its use in assessing students’ understanding of biological concept-process models. Four additions were required to extend the applicability of the framework to concept-process models. We were also able to give an indication of students’ current level of understanding of these models, showing room for improvement in all aspects of understanding. Since concept-process models have a central place in many scientific disciplines, it is important that students have a deep understanding of the nature, application and limitations of these models. The current study contributes to assessing the way students reason with concept-process models. Knowing how to improve students’ view on the use of concept-process models in biology may lead to higher scientific literacy.     

What predicts adult readers’ understanding of STEM texts?

The current study examined the relations among key variables that underlie reading comprehension of expository science texts in a diverse population of adult native English readers. Using Mechanical Turk to sample a range of adult readers, the study also examined the effect of text presentation on readers’ comprehension and knowledge structure established after reading. In Study 1, ratings of situational interest, select reading background variables, and select measures of readers’ knowledge structure accounted for significant variance in comprehension. In Study 2, the knowledge structure metrics of primacy, recency, and node degree as well as several text ratings were found to be comparable across text presentation formats. Participants who read the text sentence-by-sentence obtained higher scores on measures of comprehension and provided higher ratings of situational interest than those who received the whole paragraph text at once. Knowledge structure measures for the sentence-by-sentence and paragraph formats were similar (68% overlap). We discuss implications for future research examining factors that underlie the successful comprehension of science texts.     

Investigating the effectiveness of a POE-based teaching activity on students’ understanding of condensation

This article reports on the development of a Predict–Observe–Explain, POE-based teaching strategy to facilitate conceptual change and its effectiveness on student understanding of condensation. The sample consisted of 52 first-year students in primary science education department. Students’ ideas were elicited using a test consisting of five probe questions and semi-structured interviews. A teaching activity composed of three Predict–Discuss–Explain–Observe–Discuss–Explain (PDEODE) tasks was employed, based on students’ preconceptions identified with the test. Conceptual change in students’ understanding of condensation was evaluated via a pre-, post-, and delayed post-test approach and students’ interviews. Test scores were analyzed using both qualitative and quantitative methods. The findings suggested that the strategy helps students to achieve better conceptual understanding for the concept of condensation and enables students to retain these new conceptions in their long-term memory.     

Profound understanding of emergent mathematics: broadening the construct of teachers’ disciplinary knowledge

We discuss the teachers’ disciplinary knowledge of mathematics in this article, arguing two main points as we report on a 2-year study involving 22 practicing teachers. First we argue that teachers’ knowledge of mathematics might be productively construed as a complex evolving form, a significant dimension of which is tacit knowledge. Second, based on our first point, we suggest that this knowledge is better understood as a learnable disposition than a domain to be mastered.