Preservice Middle School Mathematics 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.

Teacher identities and teacher identity change in pre-service mathematics teachers’ metaphors

ABSTRACT Pre-service middle school mathematics teachers’ descriptions for the metaphors they produced for how they considered themselves as mathematics teachers (actual teacher identities) and how they wanted to be in the future (designated identities) at the end of the third and fourth years of the teacher education programme were analysed via a teacher identity framework focusing on areas of teacher expertise. Participants described their actual teacher identities mostly with the self-referential metaphors in both year levels and expressed the need to improve themselves as mathematics teachers. A hybrid of different expertise areas appeared more in the descriptions of designated teacher identities. There was a decrease in the didactics expert and self-referential metaphors, and an increase in the pedagogical expert and hybrid metaphors from the third year to the fourth year. The increase in the hybrid metaphors revealed that their aim was to become an expert in multiple domains in the future.

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.

MATEMATİK ÖĞRETMEN ADAYLARININ OKUL DIŞI ÖĞRENME ORTAMLARINDA MATEMATİK ÖĞRETİMİNE YÖNELİK BAKIŞ AÇILARI

This study explored pre-service middle school mathematics teachers’ perspectives on teaching mathematics in out-of-school learning environments. The current study designed an online course on out-of-school mathematics education and investigated participants’ perspectives before and after the class. The phenomenology method, one of the qualitative methods, was employed. Participants of the study were 36 2nd and 3rd-year pre-service middle school teachers who enrolled in the course. The data sources of the study were participants’ open-ended responses to questionnaires. Open coding was used to analyze the data. The data analysis was carried out under four dimensions: (1) Identification of out-of-school mathematics education environments, (2) teaching contexts for teaching mathematics in out-of-school learning environments, (3) teaching purposes for teaching mathematics in out-of-school learning environments, and (4) frequency for using out-of-school education and their reasons. The findings of the study pointed to important changes in the participants' perspectives on teaching mathematics in out-of- school learning environments after taking the course. The findings of the study would shed light into studies on teacher education in the emerging field of out-of-school mathematics education.

Examination of Pre-Service Middle School Mathematics Teachers' Noticing Skills Regarding the Argumentation Process in their Lessons

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.

Preservice middle school mathematics teachers’ strategy repertoire in proportional problem solving

ABSTRACT This study examined 91 preservice mathematics teachers’ strategy repertoire that they referred to when solving one direct and one inverse proportion missing-value word problems. When encouraged to provide multiple solutions, the preservice teachers exhibited the ability to solve the two problems using more than one strategy. However, they used a significantly greater number of strategies for solving the direct than for the inverse proportion problem. The most frequently used strategies for the two problems were the cross-multiplication and across-multiplication, respectively, and many of the preservice teachers also used these two strategies as their first strategy. On the other hand, the number of strategies applied by the preservice teachers did not differ significantly according to their first choice of strategies. A key finding of this study was that the preservice teachers possessing less common strategies in their repertoire had a significantly larger strategy repertoire than those who had more common strategies.

Who Is a Mathematics Teacher and What Does a Mathematics Teacher Do?

Preservice middle school mathematics teachers’ (PMTs) beliefs about a mathematics teacher and mathematics teaching were investigated through the metaphors they generated. Third- and fourth-year PMTs (N = 249) at four universities in Türkiye participated in two anonymous metaphor tasks for a mathematics teacher and mathematics teaching. PMTs’ metaphors were first analysed by the framework developed for and used in previous studies exploring teachers’ metaphors (Löfström et al., 2011) with further elaborations. PMTs’ metaphors for a mathematics teacher and mathematics teaching were categorised and compared. Then, deeper analyses of hybrid, self-referential, and contextual metaphors were conducted. Findings revealed that PMTs mostly used didactical expert and self-referential metaphors for the mathematics teacher and hybrid and self-referential metaphors for mathematics teaching. PMTs’ beliefs about mathematics teaching seemed to be more multifaceted than their beliefs about a mathematics teacher. The deeper analyses showed that PMTs associated both the teacher and teaching with positive and/or negative connotations in their self-referential and contextual metaphors. Documenting these beliefs through metaphors allowed us to explore and understand the nature of PMTs’ beliefs and showed that metaphor tasks could be useful for detecting beliefs.

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 Mathematics Teachers’ Ability to Perform the Mathematizing Process: The Cylinder Packing Problem

The present study aims to examine the preservice middle school mathematics teachers’ ability to perform the process of mathematizing and to identify their competencies within this context. For this purpose, the study was conducted with 43 preservice teachers attending a state university. The research method used is descriptive research. As the data collection tool, a real-life problem called "cylinder packing problem" was presented to preservice teachers, with a view to finding the optimal and lowest-cost packing scheme for a hazelnut grower. The solutions presented by the preservice teachers were subjected to content analysis as well as descriptive analysis. The preservice teachers’ competencies were examined to see their level of effectiveness in the use of notations that are indicators of mathematizing, in achieving mathematical results in the process of problem-solving. The research findings indicate that the preservice teachers generally lacked sufficient comprehension of the problem in the real-life contexts and could not perform the requirements of important skills such as using notations, mathematical models and generalization method which are important in making abstract inferences in the mathematizing process. In light of these findings, further studies to contribute to preservice teachers’ mathematical sophistication levels are recommended in order to improve their competencies in mathematizing.

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.

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.

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.

A preservice middle school mathematics teacher s knowledge of student thinking about line graphs

A preservice middle school mathematics teacher s knowledge of student thinking about line graphs

Pre-Service Teachers’ Knowledge Regarding the Area of Triangle

The aim of this study is to investigate the knowledge of pre-service middle school mathematics teachers on possible misconceptions and difficulties held by students regarding the area of triangle. In line with this aim, qualitative case study research design was used to obtain deep information. As the participants of the study, two pre-service middle school teachers were chosen purposively. Data were collected from lesson plans, semi-structured interviews and observations of pre-service teachers’ related lessons. The findings of the study showed that the pre-service teachers specified various misconceptions and difficulties regarding the area of triangle that might be observed among middle school students. When the determined misconceptions and difficulties were reviewed, it attracted attention that most of the misconceptions and difficulties were related to the concept of height. One of the findings of the study differed from other relevant studies in the respect that one of the pre-service teachers stated that students have the perception that a line segment cannot be extended. Difficulties and misconceptions and the reasons of them are discussed accordingly.

Pre-service middle school mathematics teachers’ personal concept definitions of special quadrilaterals

Pre-service middle school mathematics teachers’ personal concept definitions of special quadrilaterals