Preservice Middle School Mathematics Teachers Research Articles

Quadrilaterals hierarchical classification and properties of the diagonals: A study with pre-service mathematics teachers

The aim of this research is to investigate the main characteristics of the processes of constructing a hierarchical classification of quadrilaterals and identifying properties of their diagonals by pre-service middle and secondary school mathematics teachers (PMTs), when involved in whole-class discussions with the teacher educators, after having solved tasks focused on those topics. Data collection comprised video recordings of the sessions and PMTs’ written work, analyzed qualitatively. Findings indicate that constructing the hierarchical classification of quadrilaterals involved the PMTs in prototypical judgments, and dichotomous comparisons. In comparison the identification of the diagonals’ properties was influenced more by definitions and logical relationships, reflected in judgments and comparisons. It was concluded that participation in these processes, with whole-class discussions and development of schemes to illustrate inclusion relationships, may assist PMTs with prototypical phenomenon and dichotomous comparisons, benefiting their future teaching practice.

Investigation of Pre-service Middle School Mathematics Teachers Habits of Mind in Pattern Generalization Problems

In this study, it was aimed to determine pre-service middle school mathematics teachers’ habits of mind. The method of the study was determined as a case study. Participants are 23 junior pre-service middle school mathematics teachers. Data were collected with four open-ended linear and quadratic pattern problems. The clinical interviews were conducted with six of the pre-service teachers. In the analysis of the data, content analysis was carried out using habits of mind framework for pattern generalization. The results of the study show that similar habits of mind revealed by pre-service teachers in problem situations. The pre-service teachers had more difficulty in balancing exploration and reflection compared to the other. It was found that the pattern types of the problems affected the habits of mind. It is suggested that more studies need to be done to reveal and foster the quality of algebra and algebraic thinking.

Ortaokul Matematik Öğretmeni Adaylarının Derslerindeki Argümantasyon Sürecine İlişkin Fark Etme Becerilerinin İncelenmesi

Bu çalışmanın temel amacı ortaokul matematik öğretmeni adaylarının matematik derslerinde argümantasyon sürecini destekleyecek soru ve eylemleri nasıl fark ettiklerini incelemektir. Durum çalışması ile desenlenen bu araştırmanın katılımcıları, araştırmaya gönüllü olarak katılan dört ortaokul matematik öğretmeni adayıdır. Çalışmanın verilerini her bir öğretmen adayının gerçek sınıf ortamında gerçekleştirdiği derslerin video kayıtları ve öğretmen adayları ile derslerine ilişkin yapılan bireysel görüşmelerin video kayıtları oluşturmaktadır. Öğretmen adaylarının ders anlatımları argümantasyon sürecini destekleyici soru ve eylemler açısından, görüşmeler ise ders anlatımlarında argümantasyon sürecine ilişkin fark ettikleri durumlar açısından analiz edilmiştir. Analiz sonuçları öğretmen adaylarının derslerinde oluşturabildikleri argümantasyon süreçleri kadar oluşturma olasılıkları olan argümantasyon süreçlerinin de farkında olduğunu göstermiştir. Fark etme bileşenleri bağlamında öğretmen adayları derslerinde en fazla dikkat etme bileşenine ilişkin görüş belirtmişlerdir. Ulaşılan sonuçlar ayrıca öğretmen adaylarının dikkat ettikleri durumlar kapsamlı olmasına rağmen bu durumları yeterince yorumlayamadıklarını ve karşılık veremediklerini göstermektedir.

Supporting teacher identity development of preservice middle school mathematics teachers in informal learning environments

ABSTRACT This study focused on the reflections of preservice middle school mathematics teachers who participated in informal learning environments (science and mathematics camps) as assistant teachers. It was aimed to explore whether participation in these informal settings contributed to their teacher identity development. The findings revealed that preservice teachers developed a better understanding of learner-centered teaching and learner characteristics and reflect better on their existing teacher identities based on their experiences in the science and mathematics camps. Furthermore, their reflections revealed that they became more confident about their teaching capabilities and motivated to begin teaching. Therefore, it is possible to claim that the experiences in the science and mathematics camps cognitively, affectively, and emotionally contributed to the participant preservice teachers’ teacher identity development.

Components of collective argumentation in geometric construction tasks

This study aims to examine the components of collective argumentation of pre-service middle school mathematics teachers during geometric construction activities. To scrutinize this issue, case study research was utilized. The participants were 14 pre-service middle school mathematics teachers who worked collectively by forming four groups. During the data collection process, the groups worked on four geometric construction tasks by using compass-straightedge or GeoGebra. The findings presented that the collective argumentation processes were depicted by means of eleven components. In more detail, the six components of Toulmin’s argument model which are data, warrant, claim, backing, rebuttal, and qualifier were insufficient to represent collective argumentation. Instead of claim, the term conclusion was used in this study since the associated data and warrant were provided in the argumentation. The collective argumentation processes of the groups involved not only the mentioned six components but also the five additional components, which were named conclusion/data, target conclusion, guidance, challenger, and objection. The new components might be used while investigating the argumentation process in other disciplines.

Preservice Middle School Mathematics Teachers’ Definitions of Algebraic Expression and Equation

Using correct definitions of the mathematical concepts is crucial for learning and teaching of any mathematical content. Being able to make mathematically correct definition of the concepts is an indicator of teachers’ content knowledge. The purpose of this study is to determine how preservice middle school mathematics teachers define the concept of algebraic expression and equation. The participants of this case study were 35 preservice middle school mathematics teachers. The data were collected through written exam and semi-structured interviews. Written exam includes two questions asking preservice teachers to define equation and algebraic expression and write an example of each. Only 9 of participants correctly defined algebraic expression. Preservice teachers’ definitions of algebraic expression were classified under three themes which are expressions containing unknown, expressions containing equality, and mathematical expressions. Two themes arose from preservice teachers’ definition of equation: expressions with unknown, and expressions with equality.

Representation of Algebraic Reasoning in Sets through Argumentation

The purpose of the current study is to examine the ways in which preservice middle school mathematics teachers (PMSMT) apply and represent algebraic reasoning in their solution processes for the problems in the concept of sets. This model provides detailed information about the reasoning made through the process of the solution of set problems. The study group of this case study was composed of 20 preservice mathematics teachers. The data were collected through written documents and whole class discussions. Based on the findings of the study, three ways to represent algebraic reasoning in sets emerged; context-based representation of algebraic reasoning, generalization-based representation of algebraic reasoning and formulization-based representation of algebraic reasoning. These ways determined based on the argumentations that they formed. They produced different warrants since they reasoned differently.

What Is Mathematical Understanding and How Is It Achieved? Preservice Middle School Mathematics Teachers’ Views and Reflections in Their Teaching Practices

This research was designed to determine the views of preservice middle school mathematics teachers on what mathematical understanding is and how to achieve it, as well as the reflections of these views in their teaching practices. The participants included six preservice teachers studying in the final year of the Middle School Mathematics Teaching Program at a state university in Turkey. The research was carried out during the Teaching Practice course. The data consisted of the lesson plans prepared by the preservice teachers, video recordings of the lessons in which they put those plans into practice, field notes taken during lesson observations, and transcripts of one-on-one interviews with the preservice teachers after the lessons. Participants related mathematical understanding to interpreting situations or problems encountered in daily life mathematically, representing mathematical ideas, answering every question related to the subject, and developing a positive attitude toward mathematics. They began their lessons with daily life examples and then reminded the students of their prior knowledge, presented definitions of concepts, and worked with students on sample exercises using different representations during the lessons. These views and practices are discussed in the context of preservice teachers’ designs of learning environments that promote mathematical understanding.

Emerging Proof Productions of Freshmen in Euclidean Geometry Proof Tasks between Conjecturing and Proving

ABSTRACT This study was conducted to examine the effectiveness of proof tasks in the transition from conjecture to proof in Euclidean geometry on freshmen’s proof schemes. In line with this aim, the proof schemes of the freshmen who performed conjecture-proof and theorem-proof tasks were compared. The freshmen were composed of 109 pre-service middle school mathematics teachers who are enrolled in their first year of undergraduate education. The study was modeled as a multiple-case study. Fifty-three freshmen performed conjecture-proof tasks in Case-1, and fifty-six freshmen performed theorem-proof tasks in Case-2. The video recordings, including the written proof reports, reflection papers, and proof explanations of the freshmen, were used as data collection tools in the tasks. The proof schemes were used as construct maps to evaluate the proofs of freshmen and analyzed using Winsteps Rasch software. The proof schemes in freshmen’s proof tasks were evaluated by the Wright Map and supported with direct quotations from the proofs. In the study, it was observed that the proof schemes of freshmen who made proofs based on their own conjectures were mostly empirical and analytical proof schemes, while the proof schemes of freshmen who made proof of the presented theorem were generally external and empirical proof schemes.

Preservice Middle School Mathematics Teachers’ Development of Flexibility and Strategy Use by Geometric Thinking in Dynamic Geometry Environments

The current study, aimed to examine the development of flexibility and strategy use through instructional sequence encouraging the preservice middle school mathematics teachers’ geometric thinking and problem-solving performance on dynamic geometry environment. Mixed-method research design was used in the study. The study was conducted with 46 preservice middle school mathematics teachers selected by criterion sampling strategy. The data were collected through the tests including open-ended and multiple-choice questions. In the quantitative part of the study, it was attempted to identify whether the tasks affect problem solving skills and geometric thinking. The said individuals participated in six-week instructional sequence designed by the tasks of geometric constructions performed by GeoGebra. For the qualitative part, document analysis technique was used in order to illustrate the ways of this effect in detail and illustrate the development of flexibility and strategy use. It was observed that tasks by DGE improved the preservice middle school mathematics teachers’ scores of problem-solving and geometric thinking. Also, they solved the problems by representing the properties of higher thinking levels than the levels they had before participating in the instructional sequence enacted by GeoGebra. It is believed that this could encourage the improvement of their flexibility and strategy use.

Teknolojik Pedagojik Alan Bilgisinin Gelişimi: Dijital Öykülemenin Etkisi

It may be important to provide preservice mathematics teachers with opportunities to acquire necessary knowledge and skills about digital stories and use them effectively in their classrooms by enhancing their professional development. Hence, the study focused on how professional development workshops about digital storytelling impacted 60 preservice middle school mathematics teachers’ (PMSMTs) technological pedagogical content knowledge. The present study was conducted in the context of four professional development workshop series focusing on digital storytelling based on four main objectives. In the study designed based on multiple case study, the data were collected via videotapes of the workshops, interviews and artefacts produced by the preservice middle school mathematics teachers. The data were analysed using the framework formed through the study. The findings of the study showed that these workshops enhanced the development of the preservice middle school mathematics teachers on technological pedagogical content knowledge in relation to digital storytelling. They represented the content knowledge, technological content knowledge, and technological pedagogical content knowledge in their lesson plans designed by using their digital stories. Moreover, specialized technological pedagogical content knowledge for digital storytelling could be formed to enhance their preparation of the lessons using these stories.

Examining pre-service mathematics teachers’ argumentation-supported lesson plans and their noticing during planning

This research examined how pre-service mathematics teachers (PMTs) plan to support argumentation in their lessons about the hierarchical relationships among the quadrilaterals and what they notice during planning. The participants of this case study were eight pre-service middle-school mathematics teachers. Data consisted of the PMTs’ argumentation-supported lesson plans and video recording of the focus group meeting conducted to discuss their plans. The results revealed that the PMTs preferred to use various questions and actions that support the argumentation process and these generated different argumentation sections that mostly included data, claim and warrant. The PMTs’ statements about argumentation indicated in the focus group meeting were related to attending, responding and interpreting components of noticing, respectively. The PMTs generally provided evidence for their attention mostly to student and mathematical content while planning argumentation-supported lessons. The study presents a novel view of illustrating the PMTs’ argumentation-supported lesson plans and their noticing about it.

An Investigation of Pre-service Middle School Mathematics Teachers’ Discussion Skills in the Context of Lesson Study

The purpose of the current study is to investigate how pre-service teachers have used five practices for conducting productive mathematical discussions (anticipating, monitoring, selecting, sequencing, and connecting) in teaching statistics throughout their participation in lesson study. Researchers employed a qualitative research method. The participants of the study were three pre-service teachers as they carried out three lesson studies. In the data analysis, the framework developed by Stein et al. (2008) was used. The results revealed that the pre-service teachers did not include all practices in their lesson plans and implementations at the beginning. As the lesson study progressed, the pre-service teachers attempted to include anticipating, monitoring, selecting, and connecting practices. However, the difficulties in terms of the sequencing practice were seen to be continuing. In light of the obtained findings, it was suggested that lesson study should be integrated into teacher training programs in Turkey.

Pre-service Middle School Mathematics Teachers’ Descriptions of Definite Integral and Indefinite Integral

The importance of calculus in mathematics, mathematics education, and other disciplines and the necessity of developing students’ conceptual understanding regarding integral, which is one of the major concepts in calculus course, are among the issues emphasized by researchers. Thus, the purposes of this study were to examine how pre-service middle school mathematics teachers describe definite integral and indefinite integral and also to what extent they can see the relation between definite integral and indefinite integral. For these purposes, 173 pre-service middle school mathematics teachers were asked to answer three questions. According to the findings, the concepts pre-service middle school mathematics teachers mentioned while describing both definite and indefinite integral are similar which are bound, notation, mathematical formula, example, area, volume, antiderivative, calculation process, the form of result (number, function, algebraic expression or unknown), and the constant c. It was also seen that mostly mentioned concept is bound for each type of integral. However, the minority of them presented evidence regarding the relation between definite integral and indefinite integral in their responses.

Pre-service Middle School Mathematics Teachers’ (Mis)conceptions of Definitions, Classifications, and Inclusion Relations of Quadrilaterals

The purpose of this study is to examine pre-service middle school mathematics teachers’ (mis)conceptions related to definitions, classifications, and inclusion relations of convex quadrilaterals through a case study research design. The participants of the study were 20 pre-service middle school mathematics teachers who attended a must course, “The Methods of Teaching Mathematics” in a public university in Ankara, Turkey. Purposive sampling strategy was used to select the participants. The data were collected through an achievement test before and after taking the methods of teaching mathematics course. The findings of the study revealed that pre-service teachers had misconceptions in hierarchical classification of quadrilaterals, dual inclusion relations (e.g., rhombus-trapezoid), and difficulty in formulating minimal definitions of quadrilaterals before taking the course. However, the findings also indicated that the course had a positive impact for eliminating those misconceptions and supporting their conceptions. Several suggestions were made based on these findings.

Investigating Preservice Middle School Mathematics Teachers’ Competencies in Statistics and Probability in Terms of Various Variables

In this study, probabilities of preservice middle school mathematics teachers’ possession of four fundamental cognitive skills required for learning and teaching statistics and probability topics were examined by using the log-linear cognitive diagnostic model, which is one of the cognitive diagnostic models. Moreover, the probabilities of preservice teachers’ possession of these skills were investigated according to gender, university ranking, and grade level variables. Hence, it was examined whether there was a significant relationship between the probabilities of having each skill and these variables. A Statistical Reasoning Test, which was developed by Arican and Kuzu in 2019, measured preservice teachers’ possession of four critical skills was used in collecting the data. These four skills included representing and interpreting data, drawing inferences about populations based on samples, selecting and using appropriate statistical methods to analyze data, and understanding and applying basic concepts of probability. In the 2016-2017 academic year, the test was applied to 456 preservice teachers selected from four different universities in Turkey, and probabilities of their possession of each attribute were calculated. Later, the relationship between the preservice teachers’ test scores and gender was examined by using the Mann-Whitney U test, and the relationship between their test scores and ranking of the attended university and grade level were examined using the Kruskal Wallis-H test. Although probabilities of the preservice teachers’ possession of these four skills did not significantly differ according to gender, some significant differences were detected for university ranking and grade level variables.

An inquiry-based design research for teaching geometric transformations by developing mathematical practices in dynamic geometry environment

This study reports the classroom mathematical practices proposed in an instructional sequence grounding on the principles of inquiry-based learning (IBL) through the use of the dynamic geometry software (DGS) in learning to teach geometric transformations. A 5-week instructional sequence consisting of IBL activities that were guided by the hypothetical learning trajectory (HLT) was designed. The participant 23 preservice middle-school mathematics teachers (PMSMTs) used the geometer’s sketchpad in the sequence. The qualitative data obtained from whole-class discussions, classroom artifacts, and teacher candidates’ products were analyzed by the three-phase methodology of Rasmussen and Stephan based on the Toulmin argumentation model. The findings revealed that three mathematical practices emerged at the end of the 5-week instructional sequence: (1) an in-depth understanding of geometric transformations, (2) operational definitions of geometric transformations and the identification of the relationship among them, and (3) the properties and attributes of geometric transformations. The development of the PMSMTs’ understanding of geometric transformations in the instructional sequence was explained by the systematic nature of classroom mathematical practices. Hence, it was concluded that the PMSMTs’ knowledge and understanding of transformation geometry were enhanced through the instructional sequence based on the specific HLT.

Exploring Self-Efficacy Beliefs Within the Context of Teaching Mathematics with Concrete Models

The purpose of this study was to investigate pre-service mathematics teachers’ self-efficacy beliefs about using concrete models in teaching mathematics. Participants were 31 junior pre-service middle school mathematics teachers. Six instructional sessions based on using concrete models were carried out during a three week period. Data were collected by a survey on pre-service teachers’ efficacy beliefs about using concrete models and semi-structured interviews. Data analysis demonstrated that the instruction had positive contributions on the pre-service teachers’ self-efficacy beliefs. In addition, results revealed that pre-service teachers had confidence in themselves about using concrete models both as learners and as teachers. Moreover, they believed that using concrete models would have positive consequences in teaching process and students’ learning. However, pre-service teachers had relatively low efficacies about classroom management.

Mathematical Practices in a Social Learning Environment Guided by the Hypothetical Learning Trajectory for Quadrilaterals

The purpose of the current article is to test and revise the hypothetical learning trajectory designed for teaching quadrilaterals by reporting the classroom mathematical practices emerged in a social learning environment developing preservice middle school mathematics teachers' understanding of quadrilaterals. Ten preservice teachers participated in five-week instructional sequence guided by the hypothetical learning trajectory designed by problem-based learning strategy about quadrilaterals. This teaching experiment study by a case study focused on their learning and understanding of quadrilaterals through argumentations. Hence, the qualitative data collected through whole class and peer group discussions, field notes, written worksheets, group meetings, and interviews were analysed by three-phase methodology of Rasmussen and Stephan (2008) based on Toulmin's model of argumentation (1969). Three mathematical practices emerged through the five-week instructional sequence: reasoning on definitions of quadrilaterals, reasoning on relationship between quadrilaterals and reasoning on properties of quadrilaterals. Moreover, it was observed that preservice teachers developed their learning and understanding of quadrilaterals by the hypothetical learning trajectory.

Preservice Middle School Mathematics Teachers’ Understanding of Triangle Inequality through Collective Argumentation

The purpose of the present study was to examine how preservice middle school mathematics teachers develop the understanding and reasoning of triangle inequality through collective argumentation. Data collection process was based on whole class and peer group discussions and written documents. The data including the transcriptions of the discussion processes with the written documents were analyzed by using Toulmin’s model of argumentation. Through this collective argumentation process, they attained the knowledge and understanding of triangle inequality by suggesting and challenging their geometrical ideas about the concept and they developed and constructed their knowledge and understanding of this concept. It was found that the participants improved their knowledge and understanding of triangle inequality by argumentation through criticizing their mathematical ideas.