Learning Sciences Research Institute

The purpose of this study is to investigate a pre-service Elementary Mathematics teacher’s one of the TPACK component’s development while she is teaching the concept of Different Views of 3-D figures, namely perspective drawing. Mishra and Koehler's Framework (2006) and the instrument of Niess and her colleagues (2009) allowed the researchers to identify the subcomponents of the teachers’ knowledge, one of which will be explained here; access. Access theme mainly related to student's and teacher’s access to digital technology. A pre-service teacher prepared 4-5 hours lesson plans to teach the concept Different Views of 3-D figures in Geometry. It is aimed to understand the difference, if any, in the TPACK levels of a pre-service mathematics teacher during the implementation of these 4-5 hours lessons and to investigate the development of her TPACK during their practicum. In the light of current literature, the following research question immerged. How does a pre-service elementary mathematics teacher increase students’ access to digital technology while teaching 3-d figures during practicum?

This paper describes a pre-service mathematics teachers’
technological pedagogical content knowledge (TPACK)
during her school experience. This study focuses on how the
participant taught different views of three-dimensional objects
in a private middle school. It is a descriptive case study in
which the data was collected through semi-structured
interviews, observations, lesson plans and corresponding
GeoGebra files. The data analysis showed that there is an
increase in the technological pedagogical content knowledge
of the participant during the practicum. In other words, the
findings of the study reveal that there was an observable
development in the participant’s skills in teaching with
technology during the school experience course which helped
the improvement of her TPACK. It is proposed that the school
experience helped her develop her knowledge of teaching in
dynamic geometry-integrated mathematics classrooms. This
claim and resulting implications for practice are unpacked in
further detail.

Bu çalışma, bir ilköğretim matematik öğretmen adayının, öğretmenlik uygulaması dersi sırasında, geometride üç boyutlu cisimlerin farklı yönlerden görünümleri konusunu anlatırken Teknolojik Pedagojik Alan Bilgisi'ndeki değişimi incelemek için yapılmıştır. Örneklem bir öğretmen adayı olarak Ankara'daki büyük üniversitelerden birinden seçilmiş, katılımcı anlaşmalı özel okulda stajını yaparken gözlemlenmiştir. Çalışmada kullanılan veri toplama araçları görüşmeler, gözlemler, alan notları, GeoGebra dosyaları ve ders planlarıdır. Araştırmanın sonucu düşünüldüğünde, ilköğretim matematik öğretmen adayının Teknolojik Pedagojik Alan Bilgisi'nde (TPAB) öğretmenlik uygulaması dersi boyunca bir gelişim olduğu görülmektedir. Müfredatı teknoloji ile birleştirme ve öğrencilerin teknolojiye erişimini sağlama konusundaki gelişim küçük olsa da, teknoloji aracılığıyla öğretme ve öğrencilerin öğrenmelerini sağlama konusunda öğretmen adayının TPAB seviyesinde önemli bir artış gözlemlenmiştir. Sonuçlar göz önünde bulundurulduğunda, öğretmen eğitimi programlarında öğretmenlik uygulaması dersi sırasında öğretmen adaylarını destekleyici teknoloji derslerinin gerekliliği ortaya çıkmaktadır çünkü matematik sınıflarında etkili bir öğrenme ortamı sağlanabilmesi için öğretmen adaylarının gerekli temel becerilere sahip olması beklenmektedir.

This study attempted to investigate a pre-service mathematics teacher’s Technological Pedagogical Content Knowledge regarding different views of 3-D figures in geometry. Data were collected from a pre-service teacher, enrolled in one of the large public universities in Ankara while she was teaching different views of 3-D figures in seventh-grade level in one of the private schools. Interviews, observations, field notes, GeoGebra files, and lesson plans were the data collection tools used in the study. Data analysis revealed that the participant’s TPACK level increased during school experience course. State differently, there was a development in the pre-service teacher’s TPACK level. While there was just a slight increase in technology integration to the curriculum and access of students to technology levels, there was a significant rise in teaching and learning with technology levels of the participant during school experience course. Findings of the study show that there is a need for complementary technology courses during school experience course in teacher education programs because teacher candidates need necessary competencies in using technology like selection and usage of appropriate technology to provide effective teaching in mathematics classes. Keywords: Pre-service Middle School Mathematics Teacher, Technological Pedagogical Content Knowledge, Different Views of 3-D Figures

This study attempted to investigate four middle school mathematics teachers’ beliefs and goals regarding dynamic mathematics software GeoGebra. Data were collected from four experienced mathematics teachers who work in public schools in four different regions of Turkey (Aegean, Marmara, Mediterranean, and Central Anatolia Regions) and teach to seventh-grade level by integrating GeoGebra into their math classes. Interviews, observation field notes, and surveys were the data collection tools used in the study. Data analysis revealed that a common belief of middle school teachers is that GeoGebra is a useful tool for teaching units which link geometry and algebra. Specifically, the majority of the participants of the study think that the most appropriate units to integrate GeoGebra in the seventh grade are Circle and Circular Regions, Linearity and Analytic Geometry. Furthermore, three fourth of the teachers stated that they incorporate GeoGebra in their classes by mostly using inquiry learning method, which provides a learning environment based more on students’ discovery. In addition to these, particular goals to use GeoGebra mentioned during the interviews were decreasing their workload, facilitating students’ learning, increasing students’ engagement, providing visual representations and saving time even though they listed a number of challenges including classroom management, planning, and technology infrastructure. Findings of the study show that there is a need for in-service training and complementary workshops to handle existing challenges regarding the use of GeoGebra as it could be concluded that mathematics teachers need necessary competencies to use this software effectively into their classes. Since, most of the teachers reported that they would continue integrating GeoGebra even though, for instance, they had difficulty to plan and manage their classroom. In addition to these, all of the teachers also argue that using GeoGebra is time-consuming if technology-related problems, like the power outage and the Internet disconnection, occur. Therefore, it might be suggested to Ministry of National Education to expedite providing necessary technological infrastructure to provide equal opportunities to the students and teachers in different regions. Since, while the teachers from western regions reported that they all have interactive whiteboards and the Internet connection in their middle schools, albeit sometimes problematic, others highlighted the fact that they are still on the waiting list for the Internet connection and using their flash drives to integrate GeoGebra. Keywords: Middle School Mathematics Teachers, Beliefs and Goals, Dynamic Mathematics Software, and GeoGebra

While most of the studies done on GeoGebra has been focused on the dynamic nature of GeoGebra (e.g. Aydin & Monaghan, 2011; Karaarslan et al., 2013) and the effects of GeoGebra on students’ learning (e.g. Bulut & Bulut, 2011; Edwards & Jones, 2006; Icel, 2011), this study explored teachers’ perspectives by describing their goals regarding the mathematics software GeoGebra and their beliefs about the utilisation of this software. Furthermore, instructional strategies and methods while integrating GeoGebra used by those teachers were investigated; and teachers’ favourite topics to integrate GeoGebra are further examined. Thus, the research questions are:

1. What are the middle school mathematics teachers' goals and beliefs about the use of GeoGebra in the seventh-grade middle school mathematics classes in Turkey?
1.1. For what purposes do they use GeoGebra?
1.2. What do they believe about the effectiveness of using GeoGebra?

2. How do middle school mathematics teachers use GeoGebra in the seventh grade middle school mathematics classes in Turkey?
2.1. In which topics do they use GeoGebra?
2.2. Which instructional strategies and methods do they utilise to integrate it?

In the present study, a worksheet having two parts of questions which require spatial reasoning was prepared to assess middle school students’ understanding of two-dimensional representations of three-dimensional shapes. The worksheet had 10 questions, being 5 in each part. In the first part, students were given isometric drawings of five shapes constructed from unit cubes and asked to draw the orthogonal views of those shapes. In the second part, they were provided with the orthogonal views of five shapes constructed from unit cubes and asked to make five isometric drawings. The preliminary results of a study with 199 students show that middle school students performed better in the first part than the second.

How do seventh graders experience two-dimensional representations? Four classes in two schools of the Turkish Republic having 25 to 30 students observed while students were learning two-dimensional representations of shapes constructed from unit cubes. Each class was observed for four lesson hours (40 minutes), and observation notes recorded in five minutes intervals. It is observed that many students had difficulty to understand those representations. Also, many others found it challenging to transfer the shapes in their minds on an activity sheet. Problem-based interviews on students’ answers confirmed the observation results where that visualisation and drawing came up as students’ two main reasons for the difficulty of the concept. After the last teaching session, all students asked to complete a worksheet having orthogonal and isometric drawing questions. They were given two lesson hours to complete the worksheet having 10 questions. It's found that girls and boys internalised some teaching sessions differently. ANOVA results showed that there is no significant effect of gender on students’ performance in orthogonal representations, F(1, 197) = 4.399, p = 0.596, whereas there is a significant effect of gender in isometric drawings, F(1, 197) = 0.282, p = 0.037. Post hoc comparisons using Tukey HSD test indicated that the mean score of girls in isometric drawings (M = 1.51, SD = 1.64) is significantly different than boys (M = 1.09, SD = 1.36), where the girls outperformed the boys.

The present study as a part of design-based research has focussed on the seventh grade students’ understanding of two types of polycubical shape problems. The first type of problems asked students to draw two-dimensional representations (the orthogonal views from the front, left, right and top) of given three-dimensional representations. In the second type, students were asked to construct poly-cubical shapes and then represent them isometrically in a two-dimensional environment where corresponding views from the front, left, right and top were provided. The current study found various types of common errors of students specifically for the first and second type of problems. Redrawing the 3D shape or a part of it, drawing the part only at the very front, swapping the left and right views, drawing the view upside down and drawing squares at the back to another row/diagonally can be listed as the most common errors for the first type; and drawing only one of the views as 3D (mostly the front view), swapping the left and right views, not using the isometric paper properly and drawing a 2D shape or combining the given views as 2D and 3D are found to be the most common errors for the second type. The next iteration of this research will be focussing on designing lessons to overcome the seventh grade students’ errors above-mentioned errors. Keywords: assessment; middle school students’ errors; 2D representations of 3D shapes; 12- to 13-years old Session type: Research paper Duration: 30 minutes

This study is the initial part of a design-based research. The aim was to observe four naturally occurring cases (teachers) to discover the problems related to students’ learning of three-dimensional polycubical shapes more detailly. The first phase of a design-based research can be thought as the analysis of the problem both in the naturally occurring place and in the literature (Bakker & van Eerde, 2014; Herrington, McKenney, Reeves, & Oliver, 2007). In the current case, it is important to pre-determine the possible factors which might affect students’ progression and the process of students’ learning. The next step will be designing geometry lessons according to the determined factors and therefore the needs of the students in order to improve their performance on two-dimensional representations of three-dimensional polycubical shapes. The study was conducted in two public middle schools where two mathematics teachers in each agreed to be a part of the study and invited the researchers to observe their students while they were teaching two-dimensional representations of three-dimensional polycubical shapes during their regular lessons. There were 25 to 30 middle school students, who were 12- to 13-years old, in each of four classrooms. Middle school students were observed for four lesson hours during their regular mathematics lessons when they were learning the topic. Each lesson observation was 40 minutes. 16 lessons of four classes, being 4 lessons in each classroom, were observed as a non-participant observer, sitting at the back side of the classroom and taking field notes in five-minute intervals. A sketch of each classroom was drawn prior to the field notes. The observation protocol had two parts (a descriptive part and a reflective part) and was used to structure the field notes. Copies of the materials (presentations, activity sheets and books) used by the teachers and the students in the class were also collected as additional data. The data were organised according to the emerging themes which were explained in the findings section as three major reasons for students’ poor performance in two-dimensional representations of polycubical shapes.

Four classes in two schools of the Turkish Republic having 25 to 30 students observed while students were learning two-dimensional representations of shapes constructed from unit cubes. Each class was observed for four lesson hours (40 minutes), and observation notes recorded in five minutes intervals. It is observed that many students had difficulty to understand those representations. Also, many others found it challenging to transfer the shapes in their minds on an activity sheet. Problem-based interviews on students’ answers confirmed the observation results where that visualisation and drawing came up as students’ two main reasons for the difficulty of the concept. After the last teaching session, all students asked to complete a worksheet having orthogonal and isometric drawing questions. They were given two lesson hours to complete the worksheet having 10 questions. It's found that girls and boys internalised some teaching sessions differently. ANOVA results showed that there is no significant effect of gender on students’ performance in orthogonal representations, F(1, 197) = 4.399, p = 0.596, whereas there is a significant effect of gender in isometric drawings, F(1, 197) = 0.282, p = 0.037. Post hoc comparisons indicated that the mean score of girls in isometric drawings (M = 1.51, SD = 1.64) is significantly different than boys (M = 1.09, SD = 1.36), where the girls outperformed the boys.

The challenge of learning two-dimensional representations of polycubical shapes has received considerable attention in the last decade (e.g., Fujita et al., 2017; Saralar, Işıksal, & Bostan, 2018; Widder & Gorsky, 2013). For example, in our previous research published in BSRLM Spring 2018 proceedings, we reported average or low performance of middle school students in their use of orthogonal and isometric drawings and exemplified their common mistakes (Saralar, Ainsworth, & Wake, 2018a). Our current study focused on improving these students' learning experiences and understanding of the topic with the help of the RETA (Realistic, Exploratory, Technology-based, and Active) three-dimensional shapes teaching model (For further information about this model see Saralar, Ainsworth, Wake, 2018b). We prepared six lesson plans and an experienced mathematics teacher volunteered to teach these lesson to one of her Grade-7 classes (16 females, 14 males) after making minor changes such as the order and details of the discussion questions. Our findings showed that the planned lessons have the potential to solve the challenge of learning 2D representations. After the intervention, students answered all of the orthogonal drawing questions on the worksheet correctly (Mpre = 14.97, SDpre = 4.88; Mpost = 20, SDpost = .0), and they performed better in the isometric drawing questions (Mpre = 9.07, SDpre = 6.41; Mpost = 17.6, SDpost = 2.63). Keywords: middle school students' learning; two-dimensional representations; three-dimensional shapes, polycubical shapes

It is argued that spatial thinking and geometry are related to each other. This relation can be described as two sets having an intersection which shows issues common to both. Our current work situated in this intersection focuses on improving children’s geometrical drawings. For this purpose, a set of lessons was designed by the researchers and tested with initial samples. The lessons are based on the RETA principles which support realistic, exploratory, technology-enhanced and active learning. This approach was found to be an effective and engaging way of teaching two-dimensional drawings. Consequently, we scaled this approach to include more teachers and students to be able to report how this approach works in mainstream contexts. This study with 205 students in middle schools was the final cycle of our design-based research. The findings confirmed the results of previous cycles and showed that RETA-designed lessons provided more effective instruction than traditional methods. Note: 6-page proceedings was published in July 2019.

Here, we present four different intervention methods that aim to improve numerical and mathematical abilities. These used different specific methods and designs, targeted different sub-domains of mathematics (basic number processing, arithmetic, and geometry), and involved different populations (children and adults, with typical and atypical mathematical performance). Yet crucially, all four intervention programs exploited our understanding of certain aspects of mathematical cognition: one program, Low-Interference Memorization, teaches the multiplication table by capitalizing on the sensitivity of verbal memory to similarity. A second program, the Numeracy Musical Training, stimulates the core numerical cognition systems by exploiting the properties of musical sounds. A third program, the RETA 3D Shapes Teaching Model, teaches orthogonal and isometric drawings through realistic, exploratory, technology-enhanced and active lessons to improve spatial thinking. Last, the Catch Up Numeracy program focuses on assessing primary school children with low mathematical attainment on ten separate components of numeracy, and then giving them half an hour a week of individualized teaching thatfocuses on the components where they show weaknesses.

The problem of teaching 3D shapes has received considerable attention in the last decade and integrating technology into the teaching of 3D shapes has been suggested by many curriculum developers. However, given the plethora of existing tools available for use, teachers can struggle to know and choose them and use these effectively in their classroom. In our previous research, mathematics teachers were observed teaching 3D shapes, interviewed about their teaching choices, and they discussed their challenges in teaching 3D shapes with technology. As a result, a 6-lesson course was designed and then tested with an initial sample of students by the researcher and pre to post scores showed it was successful. The next stage of this design-based research project, which reported in this paper, was to work with a teacher who had not previously been involved as she adapted and adopted the lesson plans for her own classrooms. Interviews, lessons observations and children's test scores were utilised to examine the challenges and the opportunities for the teacher in this approach, how the lessons were enacted by the teacher and how students were helped to learn about 3D shapes using approaches neither they nor the teacher had encountered previously. Extended Summary: Study background The researchers designed, implemented and evaluated a series of lessons based on a new model. This theoretical model is called the RETA Three-dimensional Shapes Teaching Model and it emphasized four design principles for three-dimensional shapes learning in middle school geometry lessons: lessons should be Realistic (refers to the intent to integrate real-life examples and contexts into the lessons), Exploratory (refers to the use of worked examples in lessons that support students in exploring the topic), Technology-enhanced (refers to the strategic use of dynamic geometry tools in teaching) and Active (refers to the learning environments where students themselves have control of the use of tools and manipulatives). The RETA designed lessons were tested with an initial sample of students (Saralar, Ainsworth & Wake, 2018). Researchers used pre (M=5.88, SD=2.85) and post (M=8.75, SD=1.58) scores to detect any improvement due to these lessons. The maximum possible

The problem of learning three-dimensional (3d) geometry received considerable attention in the last decade and using dynamic technology to enhance students’ learning of 3d geometry has been suggested by many curricula including English, American, German and Turkish. However, given the number of studies, many middle school students struggle to learn 3d geometry and mathematics teachers find it challenging to choose these tools wisely so that they can prevent and/or overcome possible errors of students. In our previous research, we observed a number of mathematics teachers when they were teaching 3d geometry in middle school classes and noted the difficulties of both students when learning these with the available tools and methods (Saralar, Ainsworth & Wake, 2018) and teachers when teaching certain objectives (orthogonal and isometric drawings of 3d shapes) from 3d geometry (Saralar, Ainsworth & Wake, 2019). The study reported in this paper focused on classifying students’ errors in orthogonal and isometric drawings of 3d shapes, particularly cube constructions. Consequently, we designed a four-week course to teach orthogonal and isometric drawings with hopes of overcoming observed errors in both orthogonal and isometric drawings by emphasizing four design principles: realistic, exploratory, technology-enhanced and active.

Keywords: middle school students, orthogonal drawings, isometric drawings, 3d shapes, 3d geometry, misconceptions

Günümüzde aktif öğrenme hem akademisyenler tarafından araştırma konusu olarak, hem de uygulamada öğretmenlerimiz tarafından oldukça ilgi görmektedir. Aktif öğrenme, tipik olarak, eğitim araştırmacıları tarafından öğrencilerin materyallerle bilişsel ve anlamlı bir şekilde etkileşim kurmalarını gerektiren öğrenme olarak tanımlanır (Bonwell & Eison, 1991; Chi, 2014). Aktif öğrenmede öğrenciler, konuyu öğretmenlerinden dinlemek yerine, öğretmenin verdiği bir problem üzerine çalışarak keşfederler.

The potential of educational technologies for student understanding and motivation in teaching mathematics is stressed by many researchers (Koehler & Mishra, 2009; Niess, 2008; Polly & Orrill, 2016). Studies argue that maths teachers’ beliefs on this potential have an effect on their teaching methods and thus their beliefs influence whether and how they integrate technologies into their lessons (Ernest, 1989; Saralar, 2016). These beliefs might conceivably be connected to previous experiences and culturally embedded (e.g., Correa et al., 2008; Stipek et al, 2001). For example, Correa et al. (2018) found that Chinese and American teachers have a distinct way of thinking about their students’ learning of mathematical topics. Recently, with the teacher exchange project, England recruited a large number of Chinese mathematics teachers to work in public primary and secondary schools and English-China mathematics teacher exchange extended to for at least two further years, to 2020 (Liang, 2018; National Centre for Excellence in Teaching of Mathematics (NCETM), 2018). Considering the cultural differences, it is important to discuss the beliefs of Chinese and British teachers on technology integration. “There is no one best way to integrate technology into the curriculum. Rather, integration efforts should be creatively designed or structured for particular subject matter ideas in specific classroom contexts” (Koehler & Mishra, 2009, p. 62). Therefore, the aim of this study is to investigate similarities and differences between Chinese and English mathematics teachers’ perceptions of and beliefs regarding technology integration into their lessons. In this paper, we present our literature review findings which created the basis of our study for revealing the differences in the perspectives of British maths teachers and Chinese maths teachers, in terms of technology use.