A model for a widespread implementation of inquiry-based learning

Innovative teaching practices such as inquiry-based learning (IBL) have long been topics of discussion amongst mathematics and science educators. However, it is not easy to change day-to-day teaching on a large scale. The relevant question of how to promote a widespread uptake of IBL in day-to-day teaching therefore needs more consideration. In order to ensure such uptake of IBL in a variety of different contexts, a model including dissemination and implementation strategies needs to be designed. In this paper, we present the design of a focused and flexible model for dissemination and implementation as developed within the international project PRIMAS, funded by the EU under Framework 7. The design of this model is rooted in design research. We will outline and explain the complexity of the model, including its theoretical basis, its iterative approach for evaluation and refinement, and its intended contributions to research.     

Conceptualizing inquiry-based education in mathematics

The terms inquiry-based learning and inquiry-based education have appeared with increasing frequency in educational policy and curriculum documents related to mathematics and science education over the past decade, indicating a major educational trend. We go back to the origin of inquiry as a pedagogical concept in the work of Dewey (e.g. 1916, 1938) to analyse and discuss its migration to science and mathematics education. For conceptualizing inquiry-based mathematics education (IBME) it is important to analyse how this concept resonates with already well-established theoretical frameworks in mathematics education. Six such frameworks are analysed from the perspective of inquiry: the problem-solving tradition, the theory of didactical situations, the realistic mathematics education programme, the mathematical modelling perspective, the anthropological theory of didactics, and the dialogical and critical approach to mathematics education. In an appendix these frameworks are illustrated with paradigmatic examples of teaching activities with inquiry elements. The paper is rounded off with a list of ten concerns for the development and implementation of IBME.     

Challenges and opportunities for the implementation of inquiry-based learning in day-to-day teaching

In this paper, we analyse the conditions and constraints which might favour, or on the contrary hinder, a large-scale implementation of inquiry-based mathematics and science education, on the basis of our work within the PRIMAS project in 12 European countries. As a complement to the approach through the analysis of teachers’ beliefs and practices (see Engeln et al. in ZDM Int J Math Educ 45(6), this issue, 2013), we tackle this issue from a systemic institutional perspective. Indeed, in our approach, we consider teachers as actors of institutions, representing some disciplines, embedded in a school system, sharing some common pedagogical issues, in relation to society. Our sources of information are easily accessible public documents. With a theoretical background from Chevallard’s anthropological theory of didactics, we organized our analysis according to four levels of institutional organization that co-determine both content and didactical aspects in the teaching of mathematics and sciences: society, school, pedagogy and disciplinary. Our approach is systemic in the sense that we do not focus on teachers as individuals, nor on the curricula, the organization of teachers’ training or the textbooks themselves. Rather, we trace the way the conditions and constraints are operative, provide the main results of our analysis and draw out a few perspectives according to our four levels of didactical determination. Finally, in the conclusion, we reflect on the limits and potential of our analysis.     

Implementation and outcomes of inquiry-based learning in mathematics content courses for pre-service teachers

This mixed-methods study describes classroom characteristics and student outcomes from university mathematics courses that are based in mathematics departments, targeted to future pre-tertiary teachers, and taught with inquiry-based learning (IBL) approaches. The study focused on three two-term sequences taught at two research universities, separately targeting elementary and secondary pre-service teachers. Classroom observation established that the courses were taught with student-centred methods that were comparable to those used in IBL courses for students in mathematics-intensive fields at the same institutions. To measure pre-service teachers' gains in mathematical knowledge for teaching, we administered the Learning Mathematics for Teaching (LMT) instrument developed by Hill, Ball and Schilling for in-service teacher professional development. Results from the LMT show that pre-service teachers made significant score gains from beginning to end of their course, while data from interviews and from surveys of learning gains show that pre-service teachers viewed their gains as relevant to their future teaching work. Measured changes on pre-/post-surveys of attitudes and beliefs were generally supportive of learning mathematics but modest in magnitude. The study is distinctive in applying the LMT to document pre-service teachers' growth in mathematical knowledge for teaching. The study also suggests IBL is an approach well suited to mathematics departments seeking to strengthen their pre-service teacher preparation offerings in ways consistent with research-based recommendations.     

Implementation of inquiry-based learning in day-to-day teaching: a synthesis

This synthesis is designed to provide insight into the most important issues involved in a large-scale implementation of inquiry-based learning (IBL). We will first turn to IBL itself by reflecting on (1) the definition of IBL and (2) examining the current state of the art of its implementation. Afterwards, we will move on to the implementation of IBL and look at its dissemination through resources, professional development, and the involvement of the context. Based on these theoretical reflections, we will develop a conceptual framework for the analysis of dissemination activities before briefly analyzing four exemplary projects. The aim of our analysis is to reflect on the various implementation strategies and raise awareness of the different ways of using and combining them. This synthesis will end with considerations about the framework and conclusions regarding needed future actions.     

Teacher's reflections on experimenting with technology-enriched inquiry-based mathematics teaching with a preplanned teaching unit

According to previous studies, inquiry-based mathematics teaching enhances learning. However, teachers need support in implementing this type of teaching. In this study, a high school teacher was given a short preplanned inquiry-based mathematics teaching unit that included activities with GeoGebra. The teacher was interviewed after every lesson to explore her reflections after teaching. I analyzed how the teacher described the differences between her regular teaching style and the teaching unit and the pros and cons of the teaching unit. The teacher reflected on the roles of the teacher and students, depth of students’ knowledge, her stance toward the teaching unit, constraints for using this type of teaching approach, and challenges in guiding the students. The results give insights to what kind of reflections on technology-enriched inquiry-based mathematics teaching it is possible to initiate with short preplanned teaching units.     

A three-year perspective on conceptions of and orientations to learning mathematics of prospective teachers and first year university students

The paper reports a compilation of results from three studies conducted over three years to determine students' conceptions of mathematics, and orientations they follow in learning the subject. Respondents were 459 first year mathematics students from four universities and one teacher college. Results indicated that more than half the sample reported mathematics to be a subject made of numbers and formulae that could be memorized. This suggests a shallow emphasis when learning the subject, with no intention to understand. However, most students passed their examinations. It was concluded that there was no statistically significant relationship between examinations results and students' learning orientations. It is recommended that lecturers should foster students' meta-learning capabilities and an awareness of different learning strategies.     

Teaching and learning mathematics in the collective

In this paper we analyse and explore teaching and learning in the context of a high school mathematics classroom that was deliberately structured as highly interactive and inquiry-oriented. We frame our discussion within enactivism—a theory of cognition that has helped us to understand classroom processes, particularly at the level of the group. We attempt to show how this classroom of mathematics learners operated as a collective and focus in particular on the role of the teacher in establishing, sustaining, and becoming part of such a collective. Our analysis reveals teaching practices that value, capitalize upon, and promote group cognition, our discussion positions such work as teaching a way of being with mathematics, and we close by offering implications for teaching, educational policy, and further research.     

A new perspective on implementation by voting trees

Voting trees describe an iterative procedure for selecting a single vertex from a tournament. They provide a very general abstract model of decision‐making among a group of individuals, and it has therefore been studied which voting rules have a tree that implements them, i.e., chooses according to the rule for every tournament. While partial results concerning implementable rules and necessary conditions for implementability have been obtained over the past 40 years, a complete characterization of voting rules implementable by trees has proven surprisingly hard to find. A prominent rule that cannot be implemented by trees is the Copeland rule, which singles out vertices with maximum degree. In this paper, we suggest a new angle of attack and re‐examine the implementability of the Copeland solution using paradigms and techniques that are at the core of theoretical computer science. We study the extent to which voting trees can approximate the maximum degree in a tournament, and give upper and lower bounds on the worst‐case ratio between the degree of the vertex chosen by a tree and the maximum degree, both for the deterministic model concerned with a single fixed tree, and for randomizations over arbitrary sets of trees. Our main positive result is a randomization over surjective trees of polynomial size that provides an approximation ratio of at least 1/2. The proof is based on a connection between a randomization over caterpillar trees and a rapidly mixing Markov chain. © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 39, 59–82, 2011     

Assessment of learning in university mathematics

The important issues in assessment practices for university undergraduates are identified. The way in which assessment can be used to enhance student learning, the impact of external factors on assessment methods and the barriers that inhibit change are discussed. The paper also discusses the various ways in which changes in assessment practices have have been implemented and studies that have been carried out to gauge the effect of different methods of assessment.     

A support learning programme for first-year mathematics*

ABSTRACT

This article is a follow-up to an earlier paper on the mathematics support learning tutorial programme (SLT), an intervention programme at The University of Queensland that targets students considered to be at risk of failing Calculus and Linear Algebra I, the first tertiary level mathematics subject at The University of Queensland. The first paper (Hillock, P., Jennings, M., Roberts, A., & Scharaschkin, V. (2013). Amathematics support programme for first-year engineering students. International Journal of Mathematical Education in Science and Technology, 44(7), 1030–1044) reported on the inaugural programme implemented in 2012. This article provides an update of the progress of the SLT since 2012. We provide statistics for the subsequent 12 semesters to Semester 2, 2018 and describe the evolution of the SLT since its implementation. Statistical analysis of the additional data and student feedback indicate that the SLT continues to have a positive impact on student learning, with weak students making significant gains from attending the programme.     

Isetl: A programming language for learning mathematics

This paper gives a brief history of the development of an approach to help students learn mathematical concepts at the post-secondary level. The method uses ISETL, a programming language derived from SETL, to implement instruction whose design is based on an emerging theory of learning. Examples are given of uses of this pedagogical strategy in abstract algebra, calculus, and mathematical induction. © 1996 John Wiley & Sons, Inc.     

Understanding the complexities of student motivations in mathematics learning

Student motivation has long been a concern of mathematics educators. However, commonly held distinctions between intrinsic and extrinsic motivations may be insufficient to inform our understandings of student motivations in learning mathematics or to appropriately shape pedagogical decisions. Here, motivation is defined, in general, as an individual's desire, power, and tendency to act in particular ways. We characterize details of motivation in mathematical learning through qualitative analysis of honors calculus students’ extended, collaborative problem solving efforts within a longitudinal research project in learning and teaching. Contextual Motivation Theory emerges as an interpretive means for understanding the complexities of student motivations. Students chose to act upon intellectual-mathematical motivations and social-personal motivations that manifested simultaneously. Students exhibited intellectual passion in persisting beyond obtaining correct answers to build understandings of mathematical ideas. Conceptually driven conditions that encourage mathematical necessity are shown to support the growth of intellectual passion in mathematics learning.     

The use of concrete learning objects taken from the history of mathematics in mathematics education

This study aimed to reveal the effects of teaching with concrete learning objects taken from the history of mathematics on student achievement. Being a quasi-experimental study, it was conducted with two grade 8 classes in a secondary school located in Trabzon. The experimental group consisted of 27 students and the control group consisted of 25. Data were collected by using worksheets, an achievement exam and written opinion forms. The data from the achievement exam were analysed by using the Mann-Whitney U-test while the data from written opinion forms were analysed through content analysis. The Mann–Whitney U-test results showed a significant difference between the mean ranks of the experimental and control groups in favour of the former. Findings from the written opinion forms suggested that the students found the activities to be instructive and fun, enjoyed using concrete models in their classes, and learned from discovering the rules. It was also found that students had previously not engaged in similar activities and had only experienced the history of mathematics through the life stories and works of mathematicians and the representation of ancient numbers at the beginning of each unit.     

An analysis of history of mathematics research literature in Turkey: the mathematics education perspective1

Research in history of mathematics gained momentum in the past two decades in Turkey. The present paper aims to describe the patterns in the history of mathematics research in Turkey and to analyse the research in Turkey using a mathematics education framework. The qualitative paradigm and a case study design are used in the study. The obtained data were analysed by using the document analysis technique with the help of a content analysis. The study group which is comprised of twenty-two postgraduate theses at master's or doctoral level were purposefully selected from the higher education council postgraduate theses database. Findings indicate a dearth of research in the area and that most of the theses are done in the area of mathematics education. Moreover, the focus, in general, was on attitudinal variables, and cognitive aspects seemed to be ignored.     

Problem solving in the mathematics classroom: the German perspective

In Germany, problem solving has important roots that date back at least to the beginning of the twentieth century. However, problem solving was not primarily an aspect of mathematics education but was particularly influenced by cognitive psychologists. Above all, the Gestalt psychology developed by researchers such as Köhler (Intelligenzprüfungen an Anthropoiden. Verlag der Königlichen Akademie des Wissens, Berlin, 1917; English translation: The mentality of apes. Harcourt, Brace, New York, 1925), Duncker (Zur Psychologie des produktiven Denkens. Springer, Berlin, 1935), Wertheimer (Productive thinking. Harper, New York, 1945), and Metzger (Schöpferische Freiheit. Waldemar Kramer, Frankfurt, 1962) made extensive use of mathematical problems in order to describe their specific problem-solving theories. However, this research had hardly any influence on mathematics education—neither as a scientific discipline nor as a foundation for mathematics instruction. In the German mathematics classroom, problem solving, which is according to Halmos (in Am Math Mon 87:519–524, 1980) the “heart of mathematics,” did not attract the interest it deserved as a genuine mathematical topic. There is some evidence that this situation may change. In the past few years, nationwide standards for school mathematics have been introduced in Germany. In these standards, problem solving is specifically addressed as a process-oriented standard that should be part of the mathematics classroom through all grades. This article provides an overview on problem solving in Germany with reference to psychology, mathematics, and mathematics education. It starts with a presentation of the historical roots but gives also insights into contemporary developments and the classroom practice.     

A guide to the establishment of a successful mathematics learning support centre

In October 1996 the Mathematics Learning Support Centre at Loughborough University was established as part of the Department of Mathematical Sciences. The purpose of the Centre was to provide a range of resources and services for students, over and above those normally available. Almost immediately it proved to be an asset to Loughborough and it is now supported by all faculties and caters for students throughout the university. This article describes the philosophy behind its development and details a wide range of practical issues. As such, it is a case study of one particular support centre. The author hopes that the information contained herein will be of interest and help to those considering developing supplementary ways of supporting the learning of mathematics in their own institutions.