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.     

Solution representations of percentage change problems: the pre-service primary teachers’ mathematical thinking and reasoning

Solution representations can reveal how problem solvers communicate mathematical thinking and reasoning in problem-solving process. The present study examined the solution representations used by 20 pre-service teachers for the percentage change problems. The pre-service teachers were invited to solve a combination of simple and complex percentage change problems. The score for the majority of simple problems was 75% or above, but the score for the complex problems was below 75%. The highest percentage error occurred when the pre-service teachers encountered a percentage greater than 100% in the percentage change problems. Irrespective of their level of mathematics qualifications, the equation approach demonstrating two-step problem-solving process was the predominant strategy adopted by the pre-service teachers. The equation approach imposes low cognitive load and, therefore, is more accessible and efficient than the unitary approach. A few pre-service teachers used the unitary approach. The findings indicate that the pre-service teachers possessed relevant mathematical knowledge for percentage change problems. Furthermore, the inclusion of the equation approach in mathematics textbooks would provide an alternative perspective regarding the teaching and learning of percentage change problems.     

Pre-service Teachers’ Developing Conceptions about the Nature and Pedagogy of Mathematical Modeling in the Context of a Mathematical Modeling Course

Adopting a multitiered design-based research perspective, this study examines pre-service secondary mathematics teachers’ developing conceptions about (a) the nature of mathematical modeling in simulations of “real life” problem solving, and (b) pedagogical principles and strategies needed to teach mathematics through modeling. Unlike other studies that have focused on single-topic and lesson-sized research sites, a course-sized research site was used in this study. Having been through several iterations over three teaching semesters, the 15-week long course was implemented with 25 pre-service secondary mathematics teachers. Findings revealed that pre-service teachers developed ideas about the nature of mathematical modeling involving what mathematical modeling is, the relationship between mathematical modeling and meaningful understanding, and the nature of mathematical modeling tasks. They also realized the changing roles of teachers during modeling implementations and diversity in students’ ways of thinking. The researchers’ conceptual development, on the other hand, involved realizing the critical aspect of the “teacher role” played by the instructor during modeling implementations, and the need for more experience of modeling implementations for pre-service teachers.     

Elementary school teachers’ growth in inquiry-based teaching of mathematics

This article reports on a self-directed, school-based, practice-based professional development (PD) experience aimed at helping elementary school teachers to develop knowledge and expertise in inquiry-based teaching of mathematics. It discusses the characteristics of the self-directed orientation of this PD that supported the teachers’ learning, the nature of the inquiry-based knowledge they constructed, and the impact on their teaching. It highlights the centrality of agency, practical knowledge, and situated learning in this PD approach. The findings suggest that this approach can help mathematics teachers who want to be the architect of their own learning to transform their classrooms in meaningful and desirable ways.     

Using narrative inquiry for investigating the becoming of a mathematics teacher

This article presents narrative inquiry as a method for research in mathematics education, in particular the study of how pre-service teachers’ views of mathematics develop during elementary teacher education. I describe two different, complementary approaches to applying narrative analysis, one focusing on the content of a narrative, the other focusing on the form. The examples discussed are taken from interviews with and teaching portfolios compiled by four pre-service teachers. In analysing the content of the students’ narratives, I use emplotment to construct a retrospective explanation of how one pre-service teacher’s own experiences at school were reflected in the development of her mathematical identity. In analysing the form of the narratives, I also look at how the students told their stories, using linguistic features, for example, to identify core events in the accounts. This particular focus seems to be promising in locating turning points in the trainees’ views of mathematics.     

Teachers attending to student reasoning: Do beliefs matter?

We present the results of a quasi-experimental study of pre-service elementary teachers' learning to recognize students' mathematical reasoning from classroom videos. Researchers examined the nature of participants’ beliefs regarding mathematics education. We found that pre-service elementary teachers whose beliefs were consistent with NCTM Process Standards (NCTM, 2000), or that transitioned in the direction of consistency with the Standards, regarding the teaching and learning of mathematics, were more successful in recognizing students' reasoning than those whose beliefs were generally inconsistent. Predictive Analytics and Generalized Linear Regression modeling were used to quantify the magnitude of experimental pre-service teachers’ reasoning growth and combined pre/post study assessment reasoning success in contrast to that of the comparison groups. The resulting model explained nearly 90% of the variability in success on the reasoning assessment, showing that beliefs do indeed matter for recognition of reasoning.     

A coloured window on pre-service teachers’ conceptions of rational numbers

In undergraduate mathematics courses, pre-service elementary school teachers are often faced with the task of re-learning some of the concepts they themselves struggled with in their own schooling. This often involves different cognitive processes and psychological issues than initial learning: pre-service teachers have had many more opportunities to construct understandings and representations than initial learners, some of which may be more complex and engrained; pre-service teachers are likely to have created deeply-held–and often negative–beliefs and attitudes toward certain mathematical ideas and processes. In our recent research, we found that pre-service teachers who used a particular computer-based microworld, one emphasising visual representations of and experimental interactions with elementary number theory concepts, overcame many cognitive and psychological difficulties reported in the literature. In this study, we investigate the possibilities of using a similarly-designed microworld that involves a set of rational number concepts. We describe the affordances of this microworld, both in terms of pre-service teacher learning and research on pre-service teacher learning, namely, the helpful “window” it gave us on the mathematical meaning-making of pre-service teachers. We also show how their interactions with this microworld provided many with a new and aesthetically-rich set of visualisations and experiences.     

Facilitating parental engagement in school mathematics and science through inquiry-based learning: an examination of teachers’ and parents’ beliefs

This study examined teachers’ and parents’ beliefs on the implementation of inquiry-based modeling activities as a means to facilitate parental engagement in school mathematics and science. The study had three objectives: (a) to describe teachers’ beliefs about inquiry-based mathematics and science and parental engagement; (b) to describe parents’ beliefs about inquiry-based mathematics and science and their engagement in inquiry-based problem solving; and (c) to explore the impact of an inquiry-based learning environment comprising a model-eliciting activity and Twitter. The research involved three sixth-grade teachers and 32 parents from one elementary school. Teachers and parents participated in workshops, followed by the implementation of a model-eliciting activity in two classrooms. Three teachers and six parents participated in semi-structured interviews. Teachers reported positive beliefs on parental engagement in the mathematics and science classrooms and the potential positive role of parents in implementing innovative problem-solving activities. Parents expressed strong beliefs on their engagement and welcomed the inquiry-based modeling approach. Based on the results of this aspect of a four-year longitudinal design, implications for parental engagement in inquiry-based mathematics and science teaching and learning and further research are discussed.     

Constructs of engagement emerging in an ethnomathematically-based teacher education course

This paper presents a case study, in which we apply and develop theoretical constructs to analyze motivating desires observed in an unconventional, culturally contextualized teacher education course. Participants, Israeli students from several different cultures and backgrounds (pre-service and in-service teachers, Arabs and Jews, religious and secular) together studied geometry through inquiry into geometric ornaments drawn from diverse cultures, and acquired knowledge and skills in multicultural education. To analyze affective behaviors in the course we applied the methodology of engagement structures recently proposed by Goldin and his colleagues. Our study showed that engagement structures were a powerful tool for examining motivating desires of the students. We found that the constructivist ethnomathematical approach highlighted the structures that matched our instructional goals and diminished those related to students’ feelings of dissatisfaction and inequity. We propose a new engagement structure “Acknowledge my culture” to embody motivating desires, arising from multicultural interactions, that foster mathematical learning.     

Game Design as an Interactive Learning Environment for Fostering Students'' and Teachers'' Mathematical Inquiry

Many learning environments, computer-based or not, have been developed for either students or teachers alone to engage them in mathematical inquiry. While some headway has been made in both directions, few efforts have concentrated on creating learning environments that bring both teachers and students together in their teaching and learning. In the following paper, we propose game design as such a learning environment for students and teachers to build on and challenge their existing understandings of mathematics, engage in relevant and meaningful learning contexts, and develop connections among their mathematical ideas and their real world contexts. To examine the potential of this approach, we conducted and analyzed two studies: Study I focused on a team of four elementary school students designing games to teach fractions to younger students, Study II focused on teams of pre-service teachers engaged in the same task. We analyzed the various games designed by the different teams to understand how teachers and students conceptualize the task of creating virtual game learning environment for others, in which ways they integrate their understanding of fractions and develop notions about students' thinking in fractions, and how conceptual design tools can provide a common platform to develop meaningful fraction contexts. In our analysis, we found that most teachers and students, when left to their own devices, create instructional games to teach fractions that incorporate little of their knowledge. We found that when we provided teachers and students with conceptual design tools such as game screens and design directives that facilitated an integration of content and game context, the games as well as teachers' and students' thinking increased in their sophistication. In the discussion, we elaborate on how the design activities helped to integrate rarely used informal knowledge of students and teachers, how the conceptual design tools improved the instructional design process, and how students and teachers benefit in their mathematical inquiry from each others' perspectives. In the outlook, we discuss features for computational design learning environments. This revised version was published online in August 2006 with corrections to the Cover Date.     

Experienced engineers becoming mathematics teachers: preliminary perceptions of mathematics teaching

Engineers who choose to change careers and become mathematics teachers are a specific group as far as their mathematics learning in the context of engineering and their previous work experience are concerned. Regarding mathematics, they mainly engaged in applied mathematics associated with engineering, which is a highly practical field. This research explores experienced engineers’ perceptions of mathematics teaching-related topics, before starting their studies in a pre-service mathematics teacher preparation programme. This research explores their perceptions of mathematics as a discipline, mathematics teaching and mathematical understanding. The qualitative research involves three mechanical engineers, two industrial management engineers, and an electrical engineer. Semi-structured interviews were conducted before the beginning of the programme, and analysed qualitatively. The participants view engineering as an applied and changing discipline while perceiving mathematics as closed, rigorous, accurate, systematic, theoretical and as a tool for engineering. They mostly address general features of mathematics teaching while expressing a more multifaceted view of mathematical understanding. Due to the specific characteristics of the participants, this study may contribute to planning mathematics teacher preparation programmes for engineers.     

Challenging Preservice Teachers' Mathematical Understanding: The Case of Division by Zero

Preservice elementary school teachers' fragmented understanding of mathematics is widely documented in the research literature. Their understanding of division by 0 is no exception. This article reports on two teacher education tasks and experiences designed to challenge and extend preservice teachers' understanding of division by 0. These tasks asked preservice teachers to investigate division by 0 in the context of responding to students' erroneous mathematical ideas and were respectively structured so that the question was investigated through discussion with peers and through independent investigation. Results revealed that preservice teachers gained new mathematical (what the answer is and why it is so) and pedagogical (how they might explain it to students) insights through both experiences. However, the quality of these insights were related to the participants' disposition to justify their thinking and (or) to investigate mathematics they did not understand. The study's results highlight the value of using teacher learning tasks that situate mathematical inquiry in teaching practice but also highlight the challenge for teacher educators to design experiences that help preservice teachers see the importance of, and develop the tools and inclination for, mathematical inquiry that is needed for teaching mathematics with understanding.     

Incorporating the dynamic mathematics software GeoGebra into a history of mathematics course

The purpose of this study was to investigate pre-service teachers’ views about the history of mathematics course in which GeoGebra was used. The qualitative research design was used in this study. The participants of the study consisted of 23 pre-service mathematics teachers studying at a state university in Turkey. An open-ended questionnaire was used as a data collection tool. Qualitative data obtained from the pre-service teachers were analyzed by means of content analysis. As a result, it was determined that GeoGebra software was an effective tool in the learning and teaching of the history of mathematics.     

Using dynamic software in mathematics: the case of reflection symmetry

This study was carried out to examine the effects of computer-assisted instruction (CAI) using dynamic software on the achievement of students in mathematics in the topic of reflection symmetry. The study also aimed to ascertain the pre-service mathematics teachers’ opinions on the use of CAI in mathematics lessons. In the study, a mixed research method was used. The study group of this research consists of 30 pre-service mathematics teachers. The data collection tools used include a reflection knowledge test, a survey and observations. Based on the analysis of the data obtained from the study, the use of CAI had a positive effect on achievement in the topic of reflection symmetry of the pre-service mathematics teachers. The pre-service mathematics teachers were found to largely consider that a mathematics education which is carried out utilizing CAI will be more beneficial in terms of ‘visualization’, ‘saving of time’ and ‘increasing interest/attention in the lesson’. In addition, it was found that the vast majority of them considered using computers in their teaching on the condition that the learning environment in which they would be operating has the appropriate technological equipment.     

A changing trajectory of proof learning in the geometry inquiry classroom

The purpose of this study is to explore how students changes through learning to construct mathematical proofs in an inquiry-based middle school geometry class in Korea. Although proof has long been considered as one of the most important aspects of mathematics education, it is well-known that it is one of the most difficult areas of school mathematics for students. The geometry inquiry classroom (GIC) is an experimental class designed to teach geometry, especially focusing on teaching proof, based on students’ own inquiry. Based on a 2-year participant observation in the GIC, this study was conducted to address the following research question: how has students’ practice of mathematical proof been changed through their participation in the GIC? The in-depth analysis of the classroom discourse identified three stages through which the students’ practice of mathematical proof was transformed in the GIC: ‘emergent understanding of proof’, ‘proof learning as a goal-oriented activity’, ‘experiencing proof as the practice of mathematics’. The study found that as learning evolved through these stages, so the mathematics teacher’s role shifted from being an instructor to a mediator of communication. Most importantly, this research showed that the GIC has created a learning environment where students develop their competence in constructing meaningful mathematical proof and grow to be ‘a human who proves’, ultimately ‘a person who playfully engages with mathematics’.     

Mathematics and science teacher training program improves inquiry practices for teachers

With the call for increasing the level of inquiry-based instruction in mathematics and science classrooms, it is imperative for teacher education programs to prepare teachers with these skills. This study draws from classroom observation data to determine the influence a unique teacher certification program has on key aspects of teacher practice using a series of Kruskal-Wallis and Wilcoxon Rank Sum tests. Findings indicate that teachers participating in the program have a statistically significant level of growth in their use of inquiry practices related to Classroom Culture. In addition, by the fourth semester of their involvement in the program, participating teachers employ higher levels of inquiry instruction than a comparison group of teachers for almost every factor assessed.     

Pre-service teachers' point of views about learning history of mathematics: a case study in Turkey

This study examines pre-service teachers’ points of view about learning history of mathematics during their undergraduate education. An open-ended questionnaire was administered to one hundred and twenty pre-service teachers, during the fall semester of the 2013–14 academic year. The participants indicated that learning history of mathematics could increase their content knowledge as they understand how formulas, theories and relations were developed over time. In addition, it could develop them intellectually as they learn life stories of mathematicians. Also, it could help them to hold the attention of students, and answer some of the why questions. Particularly, they reported using history of mathematics knowledge while teaching Geometry and Numbers.     

The Kissing Triangles: The Aesthetics of Mathematical Discovery

Papert's (1978) appeal to reconsider the power and possibilities of the aesthetic in mathematics learning is often ignored in mathematics education research. This paper begins with the premise, put forth by Dewey (1934), that the aesthetic structures many dimensions of inquiry and experience. In the same way that using particular paintings, musical compositions, or even everyday experiences has been instrumental to attempts by philosophers to understand the aesthetic dimensions of meaning and experience in artistic domains, I propose that analysing a particular encounter with mathematics may help reveal the nature and role of the often nebulous responses of elegance, beauty, and `fit' to which mathematicians lay claim in their mathematical activity. To achieve this, I draw on and adapt the defining features of the aesthetic character of experience set forth by the aesthetician Beardsley (1982). This, in turn, sheds light on the role thataesthetics can play in mathematical inquiry and experience, and provides initial categories and conjectures that can be used to investigate the potential roles of aesthetics in mathematics learning contexts.This revised version was published online in September 2005 with corrections to the Cover Date.     

Why it is important for in-service elementary mathematics teachers to understand the equality .999… = 1

Researchers conducted semi-structured interviews with in-service fifth grade teachers. The purpose of these interviews was to examine teachers’ reactions to arguments that .999… = 1. Previously reported results indicate that some pre-service elementary school teachers possess misunderstandings about mathematical issues related to decimals with single repeating digits. This research investigates whether some in-service teachers possess misunderstandings about mathematical issues related to .999…. This paper reports on one instance of a teacher whose responses indicate that the teacher's sense of number and sense of measurement are intertwined, resulting in fragile understanding of repeating decimals. These data present evidence that teachers continue to develop repeated decimal understandings and misunderstandings throughout their careers, and that the curriculum, everyday experience, and perceptions of student learning combine to form or reinforce these understandings. Because decimals with a single repeating digit (e.g. .333… and .666…) are an integral part of the elementary mathematics curriculum, we argue that it is important that in-service elementary mathematics teachers have a clear understanding of concepts related to the concept of infinity as they emerge through the study of the equality .999… = 1.     

Knowledge and confidence of pre-service mathematics teachers: the case of fraction division

To make teacher preparation and professional development effective, it is important to find out possible deficiencies in teachers’ knowledge as well as teachers’ own perceptions about their needs. By focusing on pre-service teachers’ knowledge of fraction division in this article, we conceptualize the notion of pre-service teachers’ knowledge in mathematics and pedagogy for teaching as containing both teachers’ perceptions of their preparation and their mathematics knowledge needed for teaching. With specific assessment instruments developed for pre-service middle school teachers, we focus on both pre-service teachers’ own perceptions about their knowledge preparation and the extent of their mathematics knowledge on the topic of fraction division. The results reveal a wide gap between sampled pre-service middle school teachers’ general perceptions/confidence and their limited mathematics knowledge needed for teaching fraction division conceptually. The results suggest that these pre-service teachers need to develop a sound and deep understanding of mathematics knowledge for teaching in order to build their confidence for classroom instruction. The study’s findings indicate the feasibility and importance of conceptualizing the notion of teachers’ knowledge in mathematics and pedagogy for teaching to include teachers’ perceptions. The applicability and implications of this expanded notion of teachers’ knowledge is then discussed.