First‐year secondary school mathematics students’ conceptions of mathematical proofs and proving

The aim of this study is to investigate students’ conceptions about proof in mathematics and mathematics teaching. A five‐point Likert‐type questionnaire was administered in order to gather data. The sample of the study included 33 first‐year secondary school mathematics students (at the same time student teachers). The data collected were analysed and interpreted using the methods of qualitative and quantitative analysis. The results have revealed that the students think that mathematical proof has an important place in mathematics and mathematics education. The students’ studying methods for exams based on imitative reasoning which can be described as a type of reasoning built on copying proof, for example, by looking at a textbook or course notes proof or through remembering a proof algorithm. Moreover, they addressed to the differences between mathematics taught in high school and university as the main cause of their difficulties in proof and proving.     

Teaching students with Down syndrome in regular classrooms in Ghana: views of secondary school mathematics teachers

ABSTRACT

The recent development of making secondary school education free in Ghana has raised concerns about the level of preparedness of teachers to teach students with diverse needs in one classroom. Significantly, mathematics is one of the core areas that the Ghanaian government has prioritised, and it has institutionalised mechanisms to encourage participation by many students. Accordingly, this qualitative study aimed to document the level of preparedness of mathematics teachers to support the teaching of students with Down syndrome in secondary school classrooms. Twenty-seven mathematics teachers from 14 schools, made up of 18 males and nine females, took part in the study. We found that participants were in favour of implementation of inclusive education. However, regarding the prospect of teaching students with Down syndrome, most of the participants thought that the regular secondary school classroom is not a suitable environment for these students to access education, especially due to a number of challenges. The need for the government to support schools with appropriate teaching materials and facilities is discussed extensively.     

Institutional and personal meanings of mathematical proof

Although studies on students’ difficulties in producing mathematical proofs have been carried out in different countries, few research workers have focussed their attention on the identification of mathematical proof schemes in university students. This information is potentially useful for secondary school teachers and university lecturers. In this article, we study mathematical proof schemes of students starting their studies at the University of Córdoba (Spain) and we relate these schemes to the meanings of mathematical proof in different institutional contexts: daily life, experimental sciences, professional mathematics, logic and foundations of mathematics. The main conclusion of our research is the difficulty of the deductive mathematical proof for these students. Moreover, we suggest that the different institutional meanings of proof might help to explain this difficulty. This revised version was published online in July 2006 with corrections to the Cover Date.     

Investigating peer-assessment strategies for mathematics pre-service teacher learning on formative assessment

Formative assessment practices for secondary mathematics have been advocated as valuable for students, but difficult for teachers to learn. There have been calls in the literature to increase the emphasis on formative assessment in mathematics teacher preparation courses. This study explored the use of peer-assessment strategies for helping pre-service secondary mathematics teachers (PSTs) cultivate formative assessment principles and practices for assessing school students. Twenty-seven PSTs participated in a peer-assessment cycle comprised of: sourcing a rich mathematics task; constructing an assessment rubric for it; and collecting and analysing a selection of secondary student responses to the task. Each PST then provided written and verbal feedback to a peer on his/her rubric and student solution assessments. We draw on theoretical conceptions of Teacher Assessment Literacy in Practice to characterize the PSTs’ perceptions of their experience of formative assessment processes for learning to assess school students, in terms of cognitive and affective dimensions of their conceptions of assessment. The cohort evidenced a wide range of levels of confidence with the various aspects of formative assessment practices but on average less confidence in assessing school student task responses themselves than in assessing peer work. In addition to highlighting specific changes to different types of assessment knowledge, the PSTs also evidenced an awareness of shifts in their attitudes, in coming to view student task responses with more appreciation and humility.

     

Connecting Science and Mathematics: The Nature of Proof and Disproof in Science and Mathematics

Disagreements exist among textbook authors, curriculum developers, and even among science and mathematics educators/researchers regarding the meanings and roles of several key nature-of-science (NOS) and nature-of-mathematics (NOM) terms such as proof, disproof, hypotheses, predictions, theories, laws, conjectures, axioms, theorems, and postulates. To assess the extent to which these disagreements may exist among high school science and mathematics teachers, a 14-item survey of the meanings and roles of the above terms was constructed and administered to a sample of science and mathematics teachers. As expected, the science teachers performed better than the mathematics teachers on the NOS items (44.1 versus 24.7%, respectively) and the mathematics teachers performed better than the science teachers on the NOM items (59.0 versus 26.1%, respectively). Nevertheless, responses indicated considerable disagreement and/or lack of understanding among both groups of teachers concerning the meanings/roles of proof and disproof and several other key terms. Therefore it appears that these teachers are poorly equipped to help students gain understanding of these key terms. Classroom use of the If/and/then/Therefore pattern of argumentation, which is employed in this paper to explicate the hypothesis/conjecture testing process, might be a first step toward rectifying this situation.     

Proofs produced by secondary school students learning geometry in a dynamic computer environment

As a key objective, secondary school mathematics teachers seek to improve the proof skills of students. In this paper we present an analytic framework to describe and analyze students' answers to proof problems. We employ this framework to investigate ways in which dynamic geometry software can be used to improve students' understanding of the nature of mathematical proof and to improve their proof skills. We present the results of two case studies where secondary school students worked with Cabri-Géeomèetre to solve geometry problems structured in a teaching unit. The teaching unit had theaims of: i) Teaching geometric concepts and properties, and ii) helping students to improve their conception of the nature of mathematical proof and to improve their proof skills. By applying the framework defined here, we analyze students' answers to proof problems, observe the types of justifications produced, and verify the usefulness of learning in dynamicgeometry computer environments to improve students' proof skills.     

Fostering empirical examination after proof construction in secondary school geometry

In contrast to existing research that has typically addressed the process from example generation to proof construction, this study aims at enhancing empirical examination after proof construction leading to revision of statements and proofs in secondary school geometry. The term “empirical examination” refers to the use of examples or diagrams to investigate whether a statement is true or a proof is valid. Although empirical examination after proof construction is significant in school mathematics in terms of cultivating students’ critical thinking and achieving authentic mathematical practice, how this activity can be fostered remains unclear. This paper shows the strength of a particular kind of mathematical task, proof problems with diagrams, and teachers’ roles in implementing the tasks, by analysing two classroom-based interventions with students in the eighth and ninth grades. In the interventions, the tasks and the teachers’ actions successfully prompted the students to discover a case rejecting a proof and a case refuting a statement, modify the proof, properly restrict the domain of the statement by disclosing its hidden condition, and invent a more general statement that was true even for the refutation of the original statement.     

Research mathematicians as teacher educators: focusing on mathematics for secondary mathematics teachers

We believe that professional mathematicians who teach undergraduate mathematics courses to prospective teachers play an important role in the education of secondary school mathematics teachers. Thus, we explored the views of research mathematicians on the mathematics that should be taught to prospective mathematics teachers, on how the courses they teach can serve teachers in their work with school students, and on the changes they would implement if their courses were designed specifically for prospective teachers. We constructed profiles of the four mathematicians based on their responses to a clinical interview. We employed the construct of mathematics teacher-educators’ triad in the reflective analysis of our findings and extended the construct based on the results of this study. In conclusion, we commented on potential ways to draw stronger connections between university mathematics and the mathematics taught in schools.     

Mathematics‐teachers’ Knowledge Bases: implications for teacher education

Understanding the knowledge bases of mathematics teachers is an important task in working towards the construction of adequate models for: (i) teacher education and development, and (ii) teacher operations in the classroom. To date, little systematic attention has been focused on this task. The primary aim of this study is to obtain a view from the field of mathematics teacher knowledge with respect to content knowledge in mathematics, content‐specific pedagogical knowledge in mathematics and curriculum knowledge relevant to teaching tasks. This study has used data obtained from a survey of primary teachers and secondary mathematics teachers. Analysis of the results has indicated that less than half of the teachers in the study believed that they were sufficiently prepared in mathematics content, and that almost two‐thirds of the teachers in the sample believed that their level of knowledge in contemporary teaching methodologies in mathematics is not sufficient for their role as school teachers. Key differences emerge between the primary and secondary sectors and also within the secondary sector. Implications for preservice and in‐service mathematics teacher education are drawn.     

The Notion of Proof in the Context of Elementary School Mathematics

Despite increased appreciation of the role of proof in students’ mathematical experiences across all grades, little research has focused on the issue of understanding and characterizing the notion of proof at the elementary school level. This paper takes a step toward addressing this limitation, by examining the characteristics of four major features of any given argument – foundation, formulation, representation, and social dimension – so that the argument could count as proof at the elementary school level. My examination is situated in an episode from a third-grade class, which presents a student’s argument that could potentially count as proof. In order to examine the extent to which this argument could count as proof (given its four major elements), I develop and use a theoretical framework that is comprised of two principles for conceptualizing the notion of proof in school mathematics: (1) The intellectual-honesty principle, which states that the notion of proof in school mathematics should be conceptualized so that it is, at once, honest to mathematics as a discipline and honoring of students as mathematical learners; and (2) The continuum principle, which states that there should be continuity in how the notion of proof is conceptualized in different grade levels so that students’ experiences with proof in school have coherence. The two principles offer the basis for certain judgments about whether the particular argument in the episode could count as proof. Also, they support more broadly ideas for a possible conceptualization of the notion of proof in the elementary grades.     

Proofs produced by secondary school students learning geometry in a dynamic computer environment

As a key objective, secondary school mathematics teachers seek to improve the proof skills of students. In this paper we present an analytic framework to describe and analyze students' answers to proof problems. We employ this framework to investigate ways in which dynamic geometry software can be used to improve students' understanding of the nature of mathematical proof and to improve their proof skills. We present the results of two case studies where secondary school students worked with Cabri-Géeomèetre to solve geometry problems structured in a teaching unit. The teaching unit had theaims of: i) Teaching geometric concepts and properties, and ii) helping students to improve their conception of the nature of mathematical proof and to improve their proof skills. By applying the framework defined here, we analyze students' answers to proof problems, observe the types of justifications produced, and verify the usefulness of learning in dynamicgeometry computer environments to improve students' proof skills.This revised version was published online in September 2005 with corrections to the Cover Date.     

Effects of response options on the mathematics performance of secondary students with emotional or behavioral disorders

ABSTRACT

This study examined the effects of three response options (traditional responding, response cards, and response systems) on the mathematics performance, participation, and time on-task of secondary students with emotional or behavioral disorders (EBD). A three-way crossover design was utilized to evaluate the efficacy of response options in secondary mathematics classrooms. Thirty-three students with EBD attending an urban high school and their teachers served as participants. Results indicated that the use of response cards (white boards) or response systems (ActivResponders) significantly increased students’ mathematics performance, participation, and time on-task when compared to traditional responding. Further, the use of response cards resulted in significant increases in performance and response accuracy over the increases found when using response systems. Social validity data indicated that students and teachers felt they benefited from the use of response cards and systems. Limitations, discussions, and implications for practice and future research are presented.     

Holistic school pedagogy and values: Finnish teachers’ and students’ perspectives

The purpose of the study was to identify the components of holistic school pedagogy as identified by a sample of Finnish secondary school teachers and students from two schools. Holistic pedagogy concerns the development of the whole student and acknowledges the cognitive, social, moral, emotional and spiritual dimensions of education. The data were gathered in spring 2008 and include 19 interviews of Finnish secondary school teachers of different subjects and 37 interviews of their students. The teachers and students came from two schools that both emphasize mathematics in their curricula. Analysis of the teachers’ and students’ interviews revealed three different components in holistic school pedagogy. These components were values and worldviews, field-invariant pedagogical components (which are important regardless of the content or the subject) and field-dependent pedagogical components (subject- or context-specific issues, such as pedagogical content knowledge of mathematics or the type of school). Holistic school pedagogy also emphasized the importance of the whole school community for the best pedagogical practices in schools.     

Relationship between sociometric types and academic achievement in a sample of compulsory secondary education students / Relación entre tipos sociométricos y rendimiento académico en una muestra de estudiantes de Educación Secundaria Obligatoria

Abstract

The aim of this study was to analyse the relationship between sociometric types, behavioural categories and academic achievement in a sample of 1,349 compulsory secondary education students (51.7% boys), ranging in age from 12 to 16 years. The students’ sociometric identification was performed by using the Programa Socio and academic performance was measured by school marks provided by teachers in the subjects of Spanish language, mathematics and average academic performance. The results show that sociometric types were significant predictors of academic achievement, as students who were rated positively by their peers (popular, leaders, collaborators and good students) were more likely to have high academic achievement (in mathematics, Spanish language and average academic achievement) than students rated negatively by peers (rejected-aggressive, rejected-shy, neglected and bullies).     

Providing written feedback on students’ mathematical arguments: proof validations of prospective secondary mathematics teachers

Mathematics teachers play a unique role as experts who provide opportunities for students to engage in the practices of the mathematics community. Proof is a tool essential to the practice of mathematics, and therefore, if teachers are to provide adequate opportunities for students to engage with this tool, they must be able to validate student arguments and provide feedback to students based on those validations. Prior research has demonstrated several weaknesses teachers have with respect to proof validation, but little research has investigated instructional sequences aimed to improve this skill. In this article, we present the results from the implementation of such an instructional sequence. A sample of 34 prospective secondary mathematics teachers (PSMTs) validated twelve mathematical arguments written by high school students. They provided a numeric score as well as a short paragraph of written feedback, indicating the strengths and weaknesses of each argument. The results provide insight into the errors to which PSMTs attend when validating mathematical arguments. In particular, PSMTs’ written feedback indicated that they were aware of the limitations of inductive argumentation. However, PSMTs had a superficial understanding of the “proof by contradiction” mode of argumentation, and their attendance to particular errors seemed to be mediated by the mode of argument representation (e.g., symbolic, verbal). We discuss implications of these findings for mathematics teacher education.     

Preservice teachers’ knowledge of proof by mathematical induction

There is a growing effort to make proof central to all students’ mathematical experiences across all grades. Success in this goal depends highly on teachers’ knowledge of proof, but limited research has examined this knowledge. This paper contributes to this domain of research by investigating preservice elementary and secondary school mathematics teachers’ knowledge of proof by mathematical induction. This research can inform the knowledge about preservice teachers that mathematics teacher educators need in order to effectively teach proof to preservice teachers. Our analysis is based on written responses of 95 participants to specially developed tasks and on semi-structured interviews with 11 of them. The findings show that preservice teachers from both groups have difficulties that center around: (1) the essence of the base step of the induction method; (2) the meaning associated with the inductive step in proving the implication P(k) ⇒ P(k + 1) for an arbitrary k in the domain of discourse of P(n); and (3) the possibility of the truth set of a sentence in a statement proved by mathematical induction to include values outside its domain of discourse. The difficulties about the base and inductive steps are more salient among preservice elementary than secondary school teachers, but the difficulties about whether proofs by induction should be as encompassing as they could be are equally important for both groups. Implications for mathematics teacher education and future research are discussed in light of these findings.

THE PRACTICALITY OF IMPLEMENTING CONNECTED CLASSROOM TECHNOLOGY IN SECONDARY MATHEMATICS AND SCIENCE CLASSROOMS

Connected classroom technology (CCT) is a member of a broad class of interactive assessment devices that facilitate communication between students and teachers and allow for the rapid aggregation and display of student learning data. Technology innovations such as CCT have been demonstrated to positively impact student achievement when integrated into a variety of classroom contexts. However, teachers are unlikely to implement a new instructional practice unless they perceive the practical value of the reform. Practicality consists of three constructs: congruence with teacher’s values and practice; instrumentality—compatibility with the existing school structures; and cost/benefits—whether the reward is worth the effort. This study uses practicality as a framework for understanding CCT implementation in secondary classrooms. The experiences of three science teachers in their first year implementing CCT are compared with matched-pair mathematics teachers. Findings suggest that despite some differences in specific uses and purposes for CCT, the integration of CCT into regular classroom practice is quite similar in mathematics and science classrooms. These findings highlight important considerations for the implementation of educational technology.     

PREDICTING MATHEMATICS ACHIEVEMENT: THE INFLUENCE OF PRIOR ACHIEVEMENT AND ATTITUDES

Achievement in mathematics is inextricably linked to future career opportunities, and therefore, understanding those factors that influence achievement is important. This study sought to examine the relationships among attitude towards mathematics, ability and mathematical achievement. This examination was also supported by a focus on gender effects. By drawing on a sample of Australian secondary school students, it was demonstrated through the results of a multivariate analysis of variance that females were more likely to hold positive attitudes towards mathematics. In addition, the predictive capacity of prior achievement and attitudes towards mathematics on a nationally recognised secondary school mathematics examination was shown to be large (R ²  =  0.692). However, when these predictors were controlled, the influence of gender was non-significant. Moreover, a structural equation model was developed from the same measures and subsequent testing indicated that the model offered a reasonable fit of the data. The positing and testing of this model signifies growth in the Australian research literature by showing the contribution that ability (as measured by standardised test results in numeracy and literacy) and attitude towards mathematics play in explaining mathematical achievement in secondary school. The implications of these results for teachers, parents and other researchers are then considered.     

Supporting the transition to secondary school: The voices of lower secondary leaders and teachers

ABSTRACT

Background: The primary-secondary transition is recognised as a challenging time for students, and poor transition processes can negatively affect the students’ development. School professionals play an important role in enhancing the students’ transition experience, but international literature calls for more research concerning their perspective on this transition.

Purpose: The aim of this study was to investigate what lower secondary school leaders and teachers in Norway emphasise when supporting the primary to lower secondary school transition.

Methods: A qualitative single case study approach was used. The participants were ten form teachers, their team leader and the principal (n = 12) within one lower secondary school. These were the individuals overseeing the transition process on behalf of a cohort of students who transferred to their school in August 2017. Data were collected through observations and focus group interviews. The data were transcribed and analysed qualitatively, inspired by the constant comparative method of analysis.

Findings and conclusion: Framed by their own experiences, the leaders and teachers emphasised ensuring predictability, establishing a safe psychosocial learning environment, giving the students time to learn to be lower secondary school students, and collaboration at the school level and with the families. These efforts are largely in line with what the research recommends. The findings indicate, however, that the teachers need more support during this process. The article concludes that a closer dialogical interaction with colleagues at the primary and secondary levels, parents and students could support the leaders and teachers to promote an even better transition.     

The Status of Proving Among US Secondary Mathematics Teachers

This report examines teachers’ self-espoused attitudes and beliefs on proving in the secondary mathematics classroom. Conclusions were based on a questionnaire of 78 US mathematics teachers who had completed at least 2 years of teaching mathematics at the secondary level. While these teachers placed importance on proving as a general mathematical skill, when they discuss their own classrooms, procedural skill consistently surpasses proof-related activities in importance for a majority of high school teachers. Furthermore, teachers labeling their own past experiences in proving as causing anxiety are predictably more likely to put less value on proving. Interestingly, the quantity of past college mathematics courses is a reverse predictor indicating that further study should consider how students perceptions of proving change as they pass through a mathematics major.