The idea of interrogating resources and their use in school mathematics practice was mooted some twenty years ago, arguing for an extension of the notion of a resource beyond the physical and material, and examination of their use in practice (Adler, 2000). In this time extensive related work has evolved. In this presentation I will revisit the early work, the trajectory of selected related developments, including relevant work on textbooks, and reflect on its current salience
“Kyozai-Kenkyu” is an essential part of Japanese lesson study and involves task design for classroom lessons. Most of Japanese teachers often refer to textbooks as a main resource in doing “Kyozai-Kenkyu”. However, there are different kinds of use with regard to different approaches to knowledge at stake. In this study, we aim to examine teachers’ paradidactic activities related to “Kyozai-Kenkyu” using mathematics textbooks. For this purpose, some notions of Anthropological Theory of the Didactic (ATD) are adopted for characterizing teachers’ paradidactic praxeologies by analyzing “Kyozai-Kenkyu” about the knowledge of fractional numbers in primary schools. As a result, we identified three types of paradidactic praxeologies (PDP-a, PDP-b, & PDP-c) which are related to different aspects of didactic transposition processes. PDP-a characterizes a straightforward “Kyozai-Kenkyu” which is totally relied on both mathematical and didactic organizations of the textbook. PDP-b characterizes a “Kyozai-Kenkyu” which is partly based on the textbook but some modifications are made on the didactic organization. PDP-c characterizes a “Kyozai-Kenkyu” which slightly refers to the textbook but alternative approaches are adopted on the mathematical and didactic organizations. Accordingly, we discuss teachers’ different attitudes towards mathematics textbooks as a resource.
Textbooks, which still are one of the most used tools in the mathematics classroom worldwide have a substantial impact on the production of students’ subjectivities. In this sense, despite research showing that there is a need in eliminating gender bias in mathematics textbooks, they still reproduce the portrayal of what it means to be a girl/woman or a boy/man in our current society. Therefore, this study provides a comparison between one of the most used 6th-grade mathematics textbooks in Brazil and in the USA in order to unveil the ways gender subjects were presented. We highlighted the similarities and differences in gender inscriptions. Using Discourse Analyses as a framework, our methodological approach consisted of the reading of each textbook and selecting the appeal for gendered characters (professions, occupations, and performed activities). Thus mathematics tasks, exercises/activities as well as images and drawings were the content of analysis. In our interpretation, the systems of images that surrounded representations weaved a discursive logic that exposed the desirable gender practices of girls/women and boys/men. This analysis concluded that textbooks are still major tools to reproduce and create stereotyped gender positions for girls and boys, influencing in the creation of students’ subjectivities in both countries.
In primary schools around the world, mathematics textbooks are the most relevant medium for lesson preparation for the vast majority of teachers. Though the field of textbook research is constantly growing, our knowledge about textbook effects on students learning is still rather constrained. In this contribution, three studies are presented, which address that research gap for the field of primary school arithmetic by using a large-scale longitudinal sample of 1664 students from Grade 1 to 3. The first study shows general effects of the textbooks on the students’ mathematical achievement in arithmetic over the first three years of schooling. Studies 2 and 3 deepen this analysis by focusing on the domains of adaptive strategy choice and the use of arithmetic principles, and explain the effects by the quality of learning opportunities presented in the textbooks.
We utilize four sampling methods (simple random sampling, systematic sampling, stratified random sampling, and cluster sampling) to select pages from ten textbooks, resulting in forty samples of pages. For each sample, we estimate the proportion of pages containing statistics tasks and the number of statistics tasks within each textbook. These estimates are compared with the actual values obtained by examining every page of each textbook. Within these data, stratified sampling produces the least error in the number of statistics tasks, and all confidence intervals for the proportion of pages containing statistics tasks based on stratified sampling contain the actual proportion. We discuss the strengths and limitations of each sampling method, and suggest reasons to use various sampling methods.
Integrating history of mathematics into mathematics textbooks is a commonly accepted idea and practice in modern textbook development in China, and mathematicians are an integral part of mathematical history. So how do Chinese mathematics textbooks fare in this particular aspect? In this study, we investigate how mathematicians are represented in two series of primary and junior secondary school mathematics textbooks currently used in China, one used in Shanghai and the other by People’s Education Press (PEP). We aim to reveal the similarities and differences in the representation of mathematicians in the two series of textbooks, and explore the implications of the findings on mathematics textbook research and development. Our findings show that each series introduced both Chinese and non-Chinese mathematicians, though most of the mathematicians introduced were ancient mathematicians, and there is a higher level of consistency in the distribution of the introductions of mathematicians in both series in terms of mathematics contents and the structures of the chapters. However, the PEP mathematics textbooks introduced considerably more mathematicians than the Shanghai mathematics textbooks, and moreover, the PEP series presented a wider and, in our view, a better distribution of the introductions of mathematicians in terms of the grade levels, the ethnic origins, and the timeline of history.
Teachers are central actors in the interpretation and implementation processes of curricular resources. Holding this central position in the teaching and learning process of mathematics, teacher's interactions with curriculum resources are often studied in different stages and forms of interactions. In this study we focus on means of communication that teachers use to describe the narrative, underlying main characteristics, of a set of curricular resources. This study explores which characteristics do teachers use when they characterize through tagging a set of curricular resources, and how do their description of the curricular resources' narrative make use of these characteristics. The study analyses the tagging and descriptions of 4 individual teaches as part of a professional development program during a MA course. The course focused on analysis of textbooks and curricular materials, and included experimenting in tagging and representing different sets of curricular materials. The tagging included selecting the characteristics to be tagged, and then the use of these characteristics to describe the narrative of the set of curricular resources, composed of 50 different resources (tasks).
In this contribution, we present a praxeological analysis of (introductory) tasks on vectors in German textbooks for secondary schools in the state North Rhine-Westphalia. We motivate our research concern from our interest in transition research from school to university and found a lack of research on vectors in this area. The Anthropological Theory of the Didactic (ATD) has already proven to be suitable for textbook analysis in other studies and was chosen as a framework for our research on vectors in textbooks. With the reconstruction of a praxeological model from the textbooks, an initial analysis on the introduction of vectors with a minimal institutional bias was possible. The reconstructed model builds up from a task-oriented (praxis) perspective by identifying types of tasks at first. Afterwards, a connection is drawn to the knowledge (logos) in form of explanations and justifications the textbooks offer and thereby the reconstructed model is extended and further structured. Applying the model in an analysis of the textbooks allowed us to summarize the textbooks along the model, identify similarities of the textbooks, and systematically describe differences between them. In the end, the results and possible conclusions are discussed.
We present a work in progress of a new method for textbook research based on the Anthropological Theory of Didactics (ATD). Representation of the concept of function is chosen as a particular example to validate how the praxeology and levels of didactic co-determination of ATD can be used for textbook research. Two different Norwegian textbooks published at different times are selected. The analyses of the data via relevant theories contributes to the ongoing discussion and addresses recommendations of recent research focusing on mathematics textbooks.
A challenging task when doing research about the use of mathematics textbooks is the comprehensible description of activity shown by students and their construction of knowledge when using the textbook. As argues Kadunz (2016), the semiotics of Charles S. Peirce seems to be a promising tool for fulfilling this task.
In this paper, we turn our attention to student’s inscriptions (e.g., their drawings, their written signs) when they using a differential and integral calculus textbook to learn the concept of double integral. It will be shown that certain kind of inscriptions are diagrams, in Peirce’s sense, which are valuable means to learn the double integral concept. We analyzed the diagrams of students of an integral and differential calculus lecture of an engineering course in a Brazilian university.
The methodological perspective of the study is the flipped classroom learning environment (Bhagat et al.2016). The students began the study of the double integral concept using the textbook Calculus vol 2 of James Stewart. Findings indicated that the way the textbook introduces the double integral concept favors the students learning process. The invented and transformed diagrams by the students when they used the textbook influenced their learning activity.
In this paper, we aim at identifying inclusive practises in inclusive maths lessons. Based on the textbook “Das Zahlenbuch” and the accompanying artefacts, the lessons should offer occasions for all children to participate in and benefit from the learning situation. In our study, we analyse videographed lessons, considering the ideas of sensitivity of differences, language and materials and connections of content-related and social learning. In this paper, the results of the qualitative analysis of an introductory phase with all children in the field of simple subtraction tasks are discussed.
In Germany, primary school children should learn to not only execute the subtraction algorithm. They should also understand why the algorithm works. But this understanding is hard to achieve because the description of these processes is dominated by many technical terms and phrases like: “Unbundle one ten to ten ones. Reduce the number of tens…” – in this case for the subtraction algorithm with regrouping. Therefore, the German textbook “Das Zahlenbuch” (“The Number Book”, Nührenbörger et al., 2018) has designed language sensitive support material based on a meaning-related vocabulary for understanding the subtraction algorithm. Meaning-related in this context means that we do not focus on fostering technical terms in the first place. Instead we foster phrases and sentences combined with iconic representations for describing how the algorithm works and what “regrouping” means.
Therefore, this paper gives a short insight into the support material and in the individual learning trajectory of the fourth grader Osman who has been fostered with this support material. Osman has already learnt the subtraction algorithm at school. But at the beginning of the support he can hardly explain how the algorithm works. The analyses demonstrate that the language sensitive support seems to be a very promising approach for promoting subtraction with regrouping skills.
Currently, too few students, including those in underrepresented groups, are engaged in making sense of mathematics. Our curriculum and development research goals are to enhance student productive disciplinary engagement and learning through four design principles: problematizing, authority, accountability, and resources (Engle, 2011). Examining student behaviors, participations, and interactions in classroom environments are essential for understanding the extent to which students are engaged in personal thought and the thinking of their peers.
The context for the work is transitioning the problem-based curriculum, Connected Mathematics, to a digital environment. Through design research, we are connecting the (re)design efforts to the learners’ enacted experiences and associated outcomes. These efforts have resulted in a new problem format: (1) the Initial Challenge to problematize the situation, (2) What If...? scenarios to surface the embedded mathematics, and (3) Now What Do You Know? to connect learning to prior and future knowledge.
Initial analysis has focused on small group interactions. We have developed a framework for characterizing problematizing as students work on problems in a highly digital collaborative environment. The findings reveal that while tasks promote a high degree of problematizing in students’ work, a high degree of authority and accountability in students’ small group interactions is associated with mathematically productive problematizing.
Feedback is widely acknowledged as an important influential factor on learning and achievement. The fact that interactive digital learning tools constantly provide feedback to learners’ actions with the contents is indeed one of the most emphasized advantages of learning with digital tools. While there is a growing body of research related to adaptive feedback based on artificial intelligence in education, there still is the need to understand less sophisticated ways of implementing feedback into educational technologies like digital textbooks, which promise to be implementable at a large scale. The study presented in this paper, analyses the effectiveness of a particular combination of different feedback types that are offered in a digital mathematics textbook for the elementary level. The results show a low effectiveness of the different types of feedback. Based on this result, possibilities of developing and evaluating more effective feedback in digital textbooks are outlined.
With the transition from printed to digital textbooks the hope of new forms of textbook concepts is coexistent. On the one hand, this means that due to the digital nature new structural features of textbooks can be realised, but on the other hand, the effects on the learning of mathematics and on the use of textbook elements by the learner is of great research importance. Therefore, this contribution addresses a structural analysis of digital mathematics textbooks as well as an empirical study of student uses of textbook elements with the aim of identifying usages of textbook elements for the learning of mathematics.
Digital textbooks in mathematics lessons offer new ways to assess and analyze students’ engagement in natural school contexts. In this study we ask how learning opportunities are pursued by students when using interactive textbooks on touchscreen devices and whether students can be distinguished based on their interactions with the textbook. Furthermore, it was investigated whether students’ engagement in interactive textbook work is connected to individual characteristics and mathematical achievement.
253 six-graders (110 girls) took part in an intervention study on fractions. Students were taught basic fraction concepts using a recently developed interactive textbook for iPads (eBook). We focus on log data from three writing-to-learn activities in the eBook, i.e., worked on topic; number of words written; time spent on task; wrote correct answers; used mathematical language. Performance was assessed in a paper-based pre- and posttest on fractions.
Cluster analysis based on log data revealed four different groups of students. Analyses showed that achievement in the fraction posttest and gender distribution differed between the four groups.
Our approach offers a new way of assessing students’ engagement in mathematics instruction using interactive textbooks as a measurement device. This offers rich possibilities to measure students’ engagement in natural school contexts, its relation to learning processes, and its role in digitally-supported learning environments.
Cross-cultural studies have inherent challenges as researchers from different cultural backgrounds attempt to make sense of similar-seeming material in unfamiliar contexts and communicate seemingly-obvious aspects of their own culture to outsiders (Clarke, 2013; Osborn, 2004). This contribution explores some of the methodological challenges in a cross-cultural study on teachers’ use of print and digital resources in four regions: Sweden, Finland, the USA, and Flanders (Belgium). All but one of the seven team members are insiders to one of the four contexts and to different extents outsiders to the other contexts. We have surfaced some of the context-specific assumptions about teaching and learning mathematics in our previous studies (e.g., Hemmi & Ryve, 2015; Remillard, Van Steenbrugge, & Bergqvist, 2016) and more continue to emerge.
We focus here on developing team members’ prerequisite understanding, a term used by Andrews (2007) to relate to the alignment of insider and outsider lenses to facilitate a cross-cultural team’s growing intersubjectivity. Our expanding process of developing prerequisite understanding required us to step back and develop processes and instruments including addressing language issues, writing context descriptions including descriptions of curriculum programs in context, creating case descriptions of individual participants as an early introduction to teachers within contexts, and developing common understandings of teacher interviews through lengthy conversations between insider-outsider pairs. Through these preliminary steps, we hope to build a foundation for analyzing resource use with the perspective of insiders taking on outsider views and vice versa.
Note
This study is funded by the Swedish Research Council (2016-04616).
References
Andrews, P. (2007). Negotiating meaning in cross‐national studies of mathematics teaching: Kissing frogs to find princes. Comparative Education, 43(4), 489-509.
Clarke, D. (2013). The validity-comparability compromise in crosscultural studies in mathematics education. Paper presented at the Proceedings of the Eighth Congress of the European Society for Research in Mathematics Education.
Hemmi, K., & Ryve, A. (2015). Effective mathematics teaching in Finnish and Swedish teacher education discourses. Journal of Mathematics Teacher Education, 18(6), 501-521.
Osborn, M. (2004). New methodologies for comparative research? Establishing ‘constants’ and ‘contexts’ in educational experience. Oxford Review of Education, 30(2), 265-285.
Remillard, J. T., Van Steenbrugge, H., & Bergqvist, T. (2016). A cross-cultural analysis of the voice of six teacher's guides from three cultural contexts. Paper presented at the AERA annual meeting, Washington, DC.
In our previous studies, we have examined the content of Finnish teaching materials (Hemmi, Krzywacki, & Koljonen, 2017) and what potential mathematics classroom they construe (Koljonen, Ryve & Hemmi, 2018). However, when examining how one Swedish teacher interact with one Finnish teaching material while planning and implementing teaching did not show that kind of mathematics classroom (Koljonen, 2017). So, in order to deepen our understanding of the cultural practices in two neighboring countries, I intend to investigate, how mathematics lessons at elementary school level in Finland and Sweden are structured and what kind of teaching and questioning strategies, the teachers use in order to achieve their educational goals.
The empirical data available for this study comes from 24 video-recorded mathematics lessons and 1 complementary audio-recorded semi structured interview with each of the 8 strategically selected “locally competent” mathematics teachers from 4 classes in Sweden, and 4 in Finland. I analyzed the teaching material in use before digging into the video and interview data.
The data analysis revealed that the Swedish lessons display versions of the individualized learning pedagogy where students predominately work at their own pace during 10 of the 12 video-recorded Swedish mathematics lessons and, where the participation in the whole class relate to low inference interaction and with minimal variations. This is in contrast with the Finnish lessons, which are displaying a form of differentiated teaching pedagogy where teacher lead activities, which often involve student concurrent classroom participation during 11 of the 12 Finnish video-recorded lessons. These Finnish classrooms also contain a substantial variation of various activities as well as question types the teachers were using in their classrooms.
The results add to knowledge about classroom practice at elementary-school level, in both the Swedish and the Finnish cultural-educational context, where teachers are using an originally Finnish teacher guide.
References
Hemmi, K., Krzywacki, H., & Koljonen, T. (2017). Investigating Finnish teacher guides as a resource for mathematics teaching. Scandinavian Journal of Educational Research.
Koljonen (2017). Finnish teaching materials in the hands of a Swedish teacher: The telling case of Cecilia. In T. Dooley & G. Gueudet (Eds,). Proceedings of the Tenth Congress of the European Society for Research in Mathematics Education (pp. 1626-1633). Dublin, Ireland: DCU Institute of Education & ERME.
Koljonen, T., Ryve, A., & Hemmi, K. (2018). Analysing the nature of potentially constructed mathematics classrooms through teacher guides – the case of Finland. Research in Mathematics Education, 20(3), 295-311.
I will reflect on design research work with instructional resource on Fractions as Measures across small-scale studies conducted with teachers and teacher-researchers in Mexico, Australia, and South Africa. The resource, designed to support teacher learning and classroom practice, has been initially developed and trialled in multiple classroom design experiments in Mexico. By making adaptations to the resource in different cultural contexts, the collaborating researchers aim to make visible and understand functions served by resource design features. I will also discuss how institutional contexts in which participating teachers work profoundly shape the nature of research studies in these contexts.
We recently carried out a project-based students and teacher exchange programme between Switzerland and Japan (called PEERS project). A group of student-teachers designed a Grade 4 mathematics lesson together through online and face-to-face communication, and implemented separately in each country. We found the explicit differences, albeit the collaborative design, in the implemented lessons, in particular in the structure of lesson and teacher’s way of validating pupil’s answers.
In this paper, we investigate the cultural factors that yield the differences in teaching practices through the analysis of the resources in these two countries. An analysis is conducted on the resources—including mathematics textbooks, lesson plans, and national or state curricula—student-teachers used and developed during the collaborative design and implementation of a mathematics lesson in the context of exchange programme. The results of analysis show the tight relationship between the teacher’s resources and the teaching practices in the classroom, in particular the fact that the idea of problem solving shared through resources differs each other to a large extent, and affects the lesson organization in each country.
For example, the Swiss mathematics textbook (Vaud State) consists of the assemblage of different problem-situations without explicit mathematical knowledge for pupils to learn; the objectives of lesson identified in the lesson plan of the Swiss group focus on the process of problem solving given in the state curriculum. The problem solving for Swiss group is an idea of focusing on the process of resolution, not so much on the acquisition of a specific mathematical knowledge. This is probably one of the reasons why the lesson implemented by the Swiss student-teacher did not allocate time for the institutionalization.
In contrast, a chapter of the Japanese textbook consists of the amalgam of different elements such as problem-situations, summaries of specific mathematical knowledge to learn, and exercises. The Japanese student-teachers tried to make explicit in the lesson plan a specific mathematical knowledge as an objective, although they adopted, as a problem-situation of the designed lesson, the one from the Swiss textbook, wherein the mathematical knowledge was not clear enough. This is due to the structure of Japanese problem solving lesson (Stigler & Hiebert, 1999), including summary of mathematical content, which Japanese group tried to follow in their lesson. The idea of problem solving for Japanese group remains in the structure of lesson and in teaching and learning practices instead of the objective of mathematics teaching.
References
Stigler, J. W., & Hiebert, J. (1999). The teaching gap. Best ideas from the worlds teachers for improving education in the classroom. New York: The Free Press.
Reading mathematics is difficult for many reasons such as dense text explaining abstract concepts. Many undergraduate mathematicians still struggle to read mathematics proficiently suggesting that not enough is being done at school or college to support learning this important skill. Although research in this area is increasing, research into the most widely used reading material in the mathematics classroom – the textbook – is relatively scarce. This project aimed to explore whether mathematics textbooks are fulfilling their potential by conducting two investigations into the design and use of a specific college-level textbook. Firstly, we interviewed authors of the textbook to investigate intentions behind design decisions and then we recorded students’ eye-movements as they read the textbook. We thematically analysed the interviews and the resultant themes helped to generate specific research questions for the eye-tracking investigation. Consequently, we found that authors tended to correctly predict how the textbook would be read even when the reading strategies were not representative of best practice. This poster recognises how better communication between educators, researchers, authors and publishers could improve a textbook’s potential as a facilitator of mathematical reading. Additionally, we explore differences between how students read and processed specific elements of the textbook’s design and discuss how better design might inspire more efficient reading.
In Japan (maybe, also around other countries), one has pointed out that elementary students’ ability and skill for explanation in mathematics lacked. For this reason, we should analyze our own mathematics textbooks because it works as curriculum. (Notes: In Japan, all textbooks are authorized by the Ministry of Education).
Our theoretical framework is framework for mathematical explanation in elementary school that is proposed by Miyazaki (1995). The purpose of Miyazaki(1995) is to improve teaching and learning proof and proving in junior high school students; for this purpose, he also construct the framework because explanation in elementary school will connect with proof. Miyazaki(1995) identify and distinguish 4 levels of explanation in elementary school mathematics; level 1 is lowest, and 4 is highest. Our methodology is qualitative analysis by the framework. Our object is “formula for the area of a parallelogram” in the fifth grade (Shimizu, Funakoshi, Negami, Teragaito et al., 2015, pp.124-125), because there are 4 explanations for one problem; since the number of explanations for one problem is most, it is expected that the intention of the textbook appears.
As a result, we can pointed out there are explanation of level 2, 3, and 4 are written in it (there are two level 2); however, its order is not from 2 to 4, but from 4 to 2. Thus, we can say the textbook has not paid attention to level (or quality) of explanation.
On the other hand, actual students may explain as it, level 2 or 3. Miyazaki’s (1995) framework does not refer how to improve it or reveal conception of students who explain as level 1, 2 or 3. Thus, we found necessity of another types of theoretical frameworks for it.
The purpose of this research is to clarify the measures being used in Japanese elementary and junior high school’s textbooks to foster children’s understanding and usage of mathematical representations.
First, a discussion were conducted on the kinds of representations required as well as the process of its development based on the national curriculum and achievement test. Both of them possess an explicit process of the continuity between elementary school and junior high school. As grade level increases, subject matter changes from arithmetic to mathematics as representations become more formal and mathematical, as well as more logical and systematic.
Next, measures used by the textbooks were considered. As students progress through grade levels, their understanding and usage of representations, in the same manner as their textbooks, becoming more formal, for instance, from Manipulative Representation and Illustrative Representation to more Linguistic Representation, as well as the usage of more Symbolic Representation. Additionally, measures of representations are conducted not only through a textbook’s speech bubbles or comments, but also by preparation and usage of mathematical notes and reports that guide students in the development of their understanding and usage of representations. Certain textbook companies develop students’ representations by including content focused on knowledge and usage of vocabulary necessary for students to represent themselves logically.
Purpose of this study is to describe an ideology of era of modernization in mathematics education by our theoretical framework. According to Chevallard (2015), our current ideology is ‘paradigm of visiting monuments’, on the other hand ‘paradigm of questioning the world’ should replace it. Here, we focus on the term ‘paradigm’ which is proposed by Kuhn (1970) because it was critiqued by Lakatos’s (1978) notion of ‘research program’. Base on it, we constructed our hypothetical framework which consists unseparatable three levels of ‘hard core’ and ‘protective belt’: ‘rational level’, ‘belief level’, and ‘society level’; ‘rational level’ is identical with Lakatos’s (1978) research program.
Our methodology is philosophical consideration for mathematics textbook because it reflects social and ideological conditions and constraints in era of modernization. The reason for paying attention to modernization is that, it is one of most important educational movements and, in Japan, the curriculum clearly stated that the purpose is to foster students’ attitude for first time.
As a result of consideration, we can describe that modernization attacked only protective belt in ‘belief level’ and ‘society level’; for example, it changed the contents and arrangement of mathematical knowledge in curriculum. Thus, modernization can not change hard core so it becomes new protective belt for monumentalism.
In this study, we analyse Japanese digital mathematics textbook in senior high school with focusing on students’ algebraic skills or notions. For learning algebra, generalization plays essential role; especially, Dörfler (1991) pointed out that to use algebraic letters relates with students’ activities. For instance, according to Fujii (2000), when students can not consider about relationships between mathematical objects, they can not use algebraic letters well; Dörfler (1991) said students abstract these relationships through their own activities.Thus, our theoretical framework is Dörfler (1991), and methodology is qualitative analyses of the digital textbook; we focus on unit of quadratic function because it is compulsory and core unit of learning algebra in senior high school.
As a result, we can point out the layout of digital textbook is same as paper textbooks; however, each problem has an icon to access the digital tools. From our theoretical view, almost of all these tools are for validating (and/or justifying) explanation by sentences. In other words, we found that these tools do not allow students’ activities (in sense of Dörfler (1991)). Based on this result, we discuss nature of digital tools and propose concrete tools and theoretical perspective for digital tool which realize Dörfler's (1991) activities.
The purpose of this paper is to analyze mathematics textbooks used in elementary school math classes in Japan. In Japan, textbooks are prepared based on the Course of Study prepared by the Ministry of Education (including guidelines). This study indents to analyze explanations of how to multiply decimals in the fifth grade of elementary school. In the Japanese calculation guidance, two explanations are recommended: one for writing expressions, and the other for calculating them. For these two kinds of explanations, two different publishers were compared and analyzed. The analysis was conducted using a modified and reinforced version of the TIMSS framework (Valverde et al. 2002). The results revealed that, while the subject being explained remained unchanged, there were differences in the following areas. First, in terms of the explanation of the grounds on which the formula was set up, Publisher T had an explicit activity explaining the process using a proportional relationship, while Publisher K did not use the proportional relationship. In addition, in the explanation regarding how to calculate, Publisher T used a number line, along with a verbal explanation of how to perform the calculation. Meanwhile, Publisher K used only a verbal explanation. Based on the above, it became clear that in the teaching of how expressions are written and calculated, the type of formula, and the content of the explanation differ, depending on the publisher.
A considerable amount of time has elapsed since ICT was introduced into the class in Japan and it is recognized that it is comprehensively effective, but it is in progress in mathematics department. In this article, research trends related to mathematics education and technology are described as follows: (1) timing and scene of utilization in the class, (2) what kind of ability of the child to improve, (3) how to use commercial software, self-made software, (4) class design, The four points were organized. In classes, methods of using PCs in accordance with conventional learning processes are being studied, but research on the evaluation method is in short supply. It is not enough to clarify what kind of ability a child is able to develop, and it is necessary to combine the functions and properties of PC with self-made software after clarifying the aim and purpose of the lesson. As a future topic, I showed that desktop PC study from arithmetic education is desirable for PC to contribute to mathematics education.
This paper aims to create an analytical framework for empirical studies seeking to understand “proof by coincidence,” a type of indirect proof. Although little is known about teaching and learning proof by coincidence, all junior high school mathematics textbooks in Japan include this method of proof. To hypothesize the potential problems of teaching proof by coincidence, we investigated the variety of ways in which Japanese junior high school textbooks present the Converse of the Pythagorean Theorem (CPT), which is usually proven by coincidence. We found that the figures and emphatic points used to present the Pythagorean Theorem (PT) and its converse are often identical. It is noteworthy that we are not sure whether the differences in the ways textbooks present the PT and the CPT necessarily contributes to the resolution of students’ confusion, while we can conduct empirical research on students’ process of understanding of proof by coincidence from this point of view. Further research is therefore required to determine what influence this editing policy may have on students learning the proof by coincidence method.
Given the international evidence that students have trouble in understanding proof in mathematics (e.g. Hanna & de Villiers, 2012), the appropriateness of the content of textbooks is one crucial factor (Fan et al., 2018). Here, we focus on the introduction of proof in Japanese mathematics textbooks at the junior high school level. In one of these textbooks, authorized by the Ministry of Education in Japan, a proof is defined as ‘an explanation of a statement based on properties that we already know to be true’. If these ‘already known to be true statements’ are treated ambiguously or unclearly in the textbooks, this might underlie some of the difficulties in the teaching and learning of proofs. Hence our research question is: in Japanese junior high school mathematics textbooks, what is the relationship between the set of geometrical properties as assumptions in the definition of a proof, and the geometrical properties that appear in the proofs? For our analytic framework, we define ‘correspondence’ as the consistency between facts for students and geometrical properties, and ‘coherence’ as the deductive consistency between geometrical properties. We find that there is a ‘gap’ of adopting ‘correspondence’ and the ‘coherence’. One way to overcome this gap is to design textbooks for geometry that provide the sections for the proof structure, ‘justification’ and ‘systematization’ as the functions of proofs.
In this paper, we consider the solution based on the universal mathematical idea in Japanese mathematics textbooks. We consider the conceptual understanding in the multiplication of the fraction. Teachers teach it in the fifth graders of elementary school. The conclusion is applying to numerators and denominators each other. It is the mere procedural understanding. In order to make this conceptual understanding, the area of the square is shown in the textbook. For example, the right figure illustrates 1/2×3/4. The Japanese fourth grader in elementary schools learn decimal × decimal. The teachers urge conceptual understanding based on the number line. However, that teachers do not use that number line in this scene. Because it is not used in textbooks. It can be expected that conceptual understanding is urged to examining the area of the square on the other hand. The relation of such Textbook and Student, Teacher, and Mathematical knowledge is as being shown in Rezat's model. So, in the multiplication of the decimal and fraction, it is concluded that the method of urging conceptual understanding is not related.
It is useful and meaningful to used diagrams for making sense of calculation and solving word problems. But it is difficult for pupils to interpret and use the diagrams (NIER, 2014). In response to this situation in Japan, we need to reconsider using diagrams in teaching/leaning of calculations from the cross-sectional perspectives for the development of number concepts and meaning of calculation. So, the purpose of this study is to examine and consider the features and classification of diagrams in textbooks used in teaching/leaning of calculations in elementary school. As a method on this study, based on the theory of mathematical expressions by Nakahara (1995) and a process of solving word problems by Mayer (1992) as a theoretical framework, we are analysed and considered the diagrams used in textbooks. In this paper, we will cover teaching/leaning of addition and subtraction of integers of Japanese textbooks used from grade 1 to grade 3. As conclusions of this study, we have two points as below: (1) to suppose the new classification of the diagrams and (2) the difference of the quality of diagrams according to pupils’ development.
This work presents results of a research, in development, which aims to investigate how Geometry is present in the other branches of Mathematics in some textbooks. We make a cut here, focusing on fraction tasks in textbooks destined to the Final Years of Elementary Education. We are looking at two books in the sixth year (because they contain most of the fraction content) of two different collections. First, we selected the chapters dealing with the fractions notions, and then separated the tasks in which Geometry appears. We analysed the tasks based on Arcavi (2003). The results show that Geometry appears in tasks that work the correspondence between fractional numbers and points of the numerical line and in the fraction representation through the decomposition of figures or geometric solids. Thus, the visual representations of Geometry can aid in the process of teaching and learning by making visible purely algebraic properties.
The use of mathematics textbooks makes an impact on teaching and learning in the classroom. Because of the important role of textbooks, it can be presumed that digital mathematics textbook also have an effect on didactic aspects of the organisation of lessons and teaching. Since 2019, the project KomNetMath surveys the influence of a digital mathematic textbook regularly used by teachers and students in grade 10 and 11 at German schools. The goal is to examine the actual use of the digital textbook and the impact on teachers’ and students‘ attitude towards the use of digital textbooks as well as its influence on the progress of mathematical competencies. In this contribution, we present the concept of the digital textbook, the theoretical background, the research questions and the research design.
Textbooks may help education innovation as they can efficiently steer and support many teachers to enact renewed curriculum intentions in classroom processes. At the same time textbooks may also hinder real innovation as they reduce the opportunities of teachers to (re)design the curriculum, or to develop curriculum design capacity, by an overdose of detailed scripts that reduce teachers to technical slaves. E-textbooks are heralded to be interactive and to support teachers in their everyday classroom practices, as well as in their curriculum (re-)design, and innovative and collaborative work with colleagues. However, access to useful subject-didactical resources does not always lead to curriculum renewal and innovative practices.
In this presentation I suggest a promising middle road: educative materials, focusing on a limited number of essential characteristics for curriculum renewal. Such materials may:
It is argued that such educative materials are best designed and piloted by small teams of mixed composition: teachers, teacher educators, curriculum designers, and researchers. The development process is said to be iterative, with gradually shifting emphasis in quality criteria: from relevance, to consistency, practicality, and effectiveness.
The aim of this research is to design age-appropriate curriculum to nurture children’s mental operations ability and to clarify these effects through concrete classroom-based lessons.
I designed two lesson plans for grade4 students that nurtured children’s mental operations, and taught these in two classes. Two of the lesson plans were as follows.
1. The group 1 lesson was the same order in the textbook. The first lesson was looking for two sides of the frat rectangular pattern, which would overlap with each other when assembled. The next lesson was to look for the missing one face of position that flat cube pattern could be assembled correctly.
2. The group 2 lesson was the reverse order of group 1.
I observed what type of mental strategies were used by children in these two courses of lessons.
Results
Group 1
The first lesson, the children searched for overlapping edges to join sides of the same length. The next lesson, I observed children who used a mental operation strategy. Finally, children found the four locations of the one remaining face. However, they did not apply the strategy of joining sides that they used for rectangles.
Group 2
The first lesson, I observed that children used mental strategy of mind operation and manipulation to do this. The strategy children used was to establish one bottom face to start and then to manipulate the remaining four sides to find the upper face. In the next lesson, children applied a strategy that involved setting out a starting bottom face and manipulating the remaining four sides to ensure that the sides would overlap with each other when it was assembled as a rectangle.
Conclusion
The results of the study found that the strategy of mental operation, which was used to assemble cubes could be applied to imaginarily construct the more difficult rectangular flat pattern. In addition, the children who had the experience of using this mental strategy for other problems and recognizing its usefulness, began to aggressively use it for other similar problems and showed that they had acquired the skill of spatial ability.
Textbooks are different in many ways. Not only the mathematical content differs even more the presentation and handling of mathematics show great varieties.
To explore the field and to identify the differences with statistical methods, an instrument for analysing textbooks was developed. This instrument focuses on the three main structural elements of textbooks: tasks, pictures and explanations. Typical aspects for analysing textbooks, such as number and type of tasks, are supplemented by results from research about the three structural elements. These are aspects like mathematical modeling or open-ended tasks. Overall, the instrument can give information about 140 characteristics of these three structural elements. It can be used for frequency analysis, such as the number or amount of tasks that require mathematical argumentation. Additionally the correlation of two characteristics can be shown, for example the relationship between the type of explanations and the mathematical content. The instrument makes it possible to analyse and compare textbooks from different countries, decades, publishers, school types and so on. In addition, the effect of pedagogical, didactical or mathematical influences on textbooks, like the New Math in the 1960th, can be analysed.
First assumptions (based on 30.000 records) can be made as a result of the analysis of 14 different 7th grade textbooks from Germany published between 1940 and 2011.
The concept of vector is a central part of mathematics and physics at school. Three approaches to the concept of vector can be distinguished: arrow classes, n-tuples and vector space axioms. In order to develop adequate conceptions of vectors, different facets of the concept should be presented to the students and the representation in the subjects mathematics and physics should be coordinated. The method of textbook analysis was chosen to investigate this relationship with the help of a deductively developed system of categories. These categories are based on the theoretical framework using qualitative content analysis according to Mayring (2000). The following question was focused: How are vectors represented in German mathematics and physics textbooks?
To be used in Mexican public schools, all mathematics textbooks have to pass an allegedly rigorous review process by Ministry of Public Education (Secretaría de Educación Pública). Nevertheless, various revisions of approved mathematics textbooks reveal many deficient features, going from orthographical, mathematical and contextual errors to inadequate learning sequences. The aim of this initial exploration study was to find out what can secondary-school students (N = 155) take from an inadequate learning sequence and an erroneous fact-like statement in the mathematics textbook they use. The inadequate learning sequence was related to positive and negative numbers. Its inadequacy comes from (a) artificial context; (b) unclear questions regarding concepts and calculations; and (c) unnecessary complicated drawing tasks. Erroneous fact-like statement, related to the age of young persons should have in order to work legally, contains the affirmation that “four of ten” is 37 %. The results show that students’ performances are influenced negatively by unclear questions and unnecessary complication of the drawings tasks. Although many students were able to detect the erroneous percentage value in the statement, they differ greatly in their argumentative skills. The students with poor skills only say “something is wrong with the percentage”, while those with good skills affirm “the percentage is erroneous because “four in ten” is not 37 % but 40 %”.
Ratio, rate and proportional relationships are arguably the most important topics in middle grades mathematics curriculum before algebra. However, many teachers find these topics challenging to teach and students find them difficult to learn.
We examined the Japanese national course of study published by the Ministry of Education and one of the most widely used elementary and lower secondary school mathematics textbook series. Both vertical and horizontal analyses (Charalambous et al, 2010) were conducted, examining when and what specific topics are discussed and how they are treated in the Japanese curriculum materials.
Highlights of the findings are listed below.
• Japanese curriculum discusses comparisons of 2 quantities in Grade 5, both 2 quantities with the same measurement unit and 2 quantities with different measurement units.
• Ratio is discussed in Grade 6 in the Japanese curriculum.
• A proportional relationship is initially defined as the relationship of 2 co-varying quantities such that when one quantity becomes 2, 3, 4, … times as much, the other also becomes 2, 3, 4, … times as much. In Grade 6, the definition is extended: 2 quantities are in a proportional relationship when one quantity becomes m times as much, the other also becomes m times as much, where m is a positive rational number. Finally, in Grade 7, m is extended to the entire rational numbers.
This paper summarized domestic main models of degree of difficulty of curriculum and textbook from two aspects：the quantification of the component factors and the proportion of the component factors. Based on this, we propose five points the future research on models of degree of difficulty of curriculum and textbook needs to focus on: the concept definition in the study of models of degree of difficulty of curriculum and textbook, clarifying influence factors of the degree of difficulty of curriculum and textbook, the scientificity of quantification of the component factors and the proportion of the component factors, the reliability and validity of the research on degree of difficulty of curriculum and textbook, constructing more scientific and complex model of degree of difficulty of curriculum and textbook.
The capacity needed for using existing resources was explored and its potential components were identified previously in the context of elementary teachers using various curriculum programs in the United States. This study further explored the capacity needed for productive resource use in a mathematics methods course, focusing on preservice teachers’ learning to use existing resources productively.
Data were gathered from 15 preservice teachers in the course, including videotapes of their teaching, photos of student solutions, surveys with open-ended questions, and reflection papers on the teaching experience. The data were examined using Sleep’s (2009) conceptualization of teachers’ work of steering instruction toward the mathematical point.
Teaching toward the mathematical points of the lesson was a very challenging task for the preservice teachers. They attended to and managed multiple purposes, spent instructional time on the intended mathematics, and opened up and emphasized key mathematical ideas, as in Sleep’s framework. Careful preparations for the lesson did not result in productive teaching; many struggled to probe and develop students’ thinking to the desired level. Overall, the result leaves the course instructor in particular and mathematics educators in general a challenging task of articulating the complexity of preparing preservice teachers to use existing resources productively.
Semiotics raises an increasing interest in Mathematical
Education since Semiotics provides an array of methods and concepts to understand
the nature of mathematical discourse. Signs are carriers of cultural conventions and
Semiotics helps to understand the way in which individuals think and communicate
with signs in a cultural context. Duval's theory of semiotic representations
characterizes Mathematical learning in terms of the ability of students to handle
several representation registers of mathematical concepts. Using as an example a
concept of higher mathematics, in this work we propose as criteria for the analysis of
textbooks the existence of activities that aim at students making conversions between
representations of the concept among the diverse representation registers.
The utilization of textbooks by mathematics teachers has been the subject of many recent studies; students’ opinions, however, have not received such attention. The study presented in this paper aims to investigate both students’ and their teacher’s ways of and reasons for using the textbook, with an emphasis on the vertices of the Socio-Didactical Tetrahedron. The findings indicate that the beliefs about being a teacher and about being a student strongly influence textbook utilization. Also, the students’ use of the textbook is influenced by the teacher’s intentions. Here the extension of the didactical tetrahedron to a socio-didactical tetrahedron proved to be very valuable due to the social factors involved in textbook use.
As already known well, for learning geometry, students start from their own experience and concrete shapes; on the other hand, students have to leave from them for developing their geometric thinking (cf. van Hiele, 1958).
When we seen textbooks as (implemented) curriculum, textbook must help students’ this process.
For this background, our research question is: Are Japanese elementary school textbooks appropriate for developing students’ geometric thinking from 1st to 6th grade, especially leaving from their own experiences and shapes?
Our theoretical framework is “praxeology (Chevallard, 2015; Bosch & Gascón, 2014)” for revealing the character of knowledge in textbook.
Here, we focus on theory[Θ] in praxeology ; because, Θ is fundamental principle for justifying praxis- (tasktype and technique).
Our methodology is quantitative analysis of geometrical knowledge in Japanese elementary textbook (1st - 6th grade); ratio of Θ as following 5 categories.
Θ of geometric knowledge in the textbook is classified into four categories: i) empirical (shapes and/or visual), ii)measurement, iii) definition, property, and/or theorem, iv) mathematical operation. In addition, some knowledge has no “theory” because students’ activities themselves are purpose (e.g. making animals by using boxes); we identified them as X) which is fifth category of knowledge.
As a result, the developing process from 1st grade to 5th grade is appropriate; because ratio of i), ii), and X) gradually decrease, and iii) and iv) gradually increase. However, 6th grade has specific character; i) has obviously increased than the 5th grade because they start to learn properties of solid figures.
In the Undergraduate Teaching and Learning in Mathematics with Open Software and Textbooks (UTMOST, Beezer et al., 2018) project we study the use of open source computational resources in the teaching and learning of mathematics at undergraduate level. The project gathers (a) real-time, individualized viewing data from three dynamic undergraduate textbooks authored in textbooks authored in PreTeXt (Calculus, Linear Algebra, and Abstract Algebra); (b) ongoing surveys of users’ descriptions of the textbook use; (c) users’ questionnaires (beliefs and attitudes towards mathematics, technology, teaching, and learning); and student performance (tests of knowledge and grades). The textbooks have been enhanced with additional features such as WeBWorK, Javascript, Geogebra, and include interactive sections in which students can submit answers to Reading Questions, and instructors can see the responses in real time.
Viewing data indicate which sections are being accessed by users, but are insufficient for knowing what users are actually doing when they view the textbook. For this reason, every two weeks we ask users for descriptions of what they do during their own viewing of the textbook via a survey that is sent to 6 to 9 courses every semester (each with an average of 20 students). Each survey generates between 120 and 300 responses that are synchronized with the viewing data and analysed for patterns. We have turned into natural language processing programs that facilitate the identification of patterns in the responses. In the presentation we will address issues of identification, classification, and integration, that we have faced and how we have managed them in the current project.
Proportional reasoning is an important skill that requires a long process of development and is a cornerstone at the middle school level. One of the reasons why students cannot demonstrate this skill at the desired level is the learning opportunities provided by the textbooks. The aim of this study examines the extent to which selected Turkish, Singaporean and Canadian middle school mathematics textbooks provide students opportunities to experience proportional reasoning. In the study, 4 developmental shifts were used. These shifts processes have been developed in the form of adept at forming ratios, reasoning with proportions, and understanding rates. The ratio and rate units of the textbooks were evaluated as 4 main content structures as narratives, tasks, examples, and representations. Shifts were graded as “none-weak-medium-strong “according to the frequency of taking part in the content structures. As a result of the study, it was determined that the selected Turkish and Canadian textbooks presented one shift strongly, and Singaporean textbooks presented two shifts strongly. The results show that the selected textbooks in Singapore provide the strongest opportunity for proportional reasoning.
In mathematics teaching and learning, it is important for students to learn not only mathematical contents but also roles and functions of the contents. Functions of a hierarchical classifications are particularly related to mathematical definition and defining. The purpose of this paper is to clarify functions of a hierarchical classification of quadrilaterals in Japanese textbook. From the perspective of the five functions of a hierarchical classification of quadrilaterals (de Villiers, 1994), geometry in the latest Japanese elementary and middle school textbooks were analysed. Three functions: simplification of deductive structure, a useful conceptual schema during problem solving, a useful global perspective, were identified related to contexts of problem solving. On the other hand, the rest two functions: economical definitions and formulations of theorems, alternative definitions and new propositions, were not identified in textbooks, because Japanese textbooks don't explicitly deal with exclusive definitions or equivalent definitions of quadrilaterals. The results of analysis imply there is a limitation that contradictory definitions must not be presented in textbooks, and that textbooks can present only a part of defining processes. This paper suggests teachers should pay attention to the limitation of textbooks and try to teach authentic defining activity beyond textbooks.
The integer numbers structure and the idea of equivalence are elementary in the mathematical construction of the ordered field of the rational numbers. Hence, the concept of equivalence should not be absent in the Elementary School´s classrooms and textbooks, and it should be well constructed with the students in the sense that they are able to answer the question ‘How can one decide if two fractions a/b and c/d are equivalent?’ and explain their answer. However, this discussion does not appear in many 6th grade Brazilian textbooks. This paper is motivated by this fact and is based on the belief that proofs should be present in the classrooms, which was the focus of the Topic Study Group 18 entitled Reasoning and Proof in Mathematics Education, in the International Congress on Mathematical Education (ICME 13), in 2016.
We present a ‘proof that explains’ (Hanna, 1990) for the characterization for two given fractions to be equivalent which we consider is adequate for 6th grade students. Then we compare textbooks up to the 6th grade from six other countries, focusing mainly on the questions: Is a complete characterization for equivalent fractions clearly presented to the students? Is equivalence used in the comparison, addition and subtraction of fractions?
Only partial characterizations were found, and in most of the analyzed books a complete characterization is not even suggested. We conclude that an important opportunity of developing students’ mathematical thinking is lost.
Our work is part of a thesis that deals with university teachers’ practices, particularly the relation between research and teaching practices of university professors. We study this issue from the lens of interactions with resources (including textbooks). We choose graph theory as an object of study. Graph theory is a contemporary mathematical field, taught in several academic paths (computer sciences, mathematics, engineering education, etc.) and no consensus has been made on the choice of concepts and notions to be included in a university course (West, 2001). We seek to characterize the impact of the epistemological specificities of graph theory along with the institutional context on the resources designed in the tertiary level.
We considered seven graph theory textbooks, in particular contents related to eulerian and hamiltonian paths. We identified that the teaching of graph theory allowed different kinds of connections: connecting different concepts, different registers of representation (Duval 2006), and different topic areas. Based on these results, we use here the concept of connectivity (Gueudet, Pepin, Restrepo, Sabra, & Touche, 2018) to analyze a chapter of a graph theory course for future engineers in France. Finally, we present a discussion of findings and their implications.
Duval, R. (2006). A cognitive analysis of problems of comprehension in a learning of mathematics. Educational Studies in Mathematics, 61(1 - 2), 103-131.
Gueudet, G., Pepin, B., Sabra, H., Restrepo, A., & Trouche, L. (2018). E-textbooks and connectivity: proposing an analytical framework. International Journal for Science and Mathematics Education, 16(3), 539-558.
West, D. B. (2001). Introduction to graph theory ( second edition ed.). Pearson education.
In the arena of mathematics teaching and learning at undergraduate level, we strive to assess and evaluate the use of dynamics textbooks authored in PreTeXt. We seek to investigate how mathematical content in a textbook is taken up by instructors as they plan and teach their lessons, and by students who participate in those lessons. We analyze textbook raw content to identify competencies afforded by it, and textbook usage reported by the users via periodic surveys and automatic real-time textbook viewing data (heat maps).
To analyze the textbook raw content, we apply manual and automatic coding, using 25% of the entire textbook as training data. This is followed by automatic coding using natural language processing techniques that will derive a set of competencies for the rest of the textbook.
Underlying this investigation are two frameworks: the didactical tetrahedron (Rezat & Strasser, 2012) that models the mediating role of textbooks in instruction (Cohen, Raudenbush, & Ball, 2003) and competencies (Niss, 2011) which refers to a set of mathematical competencies that are afforded by the textbook, instructors’ actions and resources, and activated by the students.
In the United States, teaching mathematics for social justice has the potential to make studying advanced mathematics more accessible to students coming from low income and historically underrepresented ethnic groups. Gutstein’s (2005) work on teaching students to “read” and “write” the world with mathematics laid the foundation for the framework that will be used in this middle school textbook analysis. The purpose of this textbook analysis was to determine the nature of “mathematical power” (Gutstein 2005) within the sections focused on ratios and rates in the “Mathematics in Context” (Holt, Rinehart & Winston, 2006) curriculum. This was done by searching for evidence of the development of “mathematical power” (as defined by Gutstein) in conjunction with alignment to the National Council of Teachers of Mathematics (NCTM) publication Essential Understandings of Ratios, Rates, and Proportions for Middle School Teachers (Lobato, et. al 2010). The analysis of sampled sections focused on evidence of this author’s operationalization of the five Essential Understandings (EUs) of ratios and rates along with an operationalization of Gutstein’s working definition of mathematical power. Results suggest that Gutstein’s characterization of the mathematical power in Mathematics In Context (MiC) is accurate and evidence of EUs suggest that the written curriculum allows for the potential to develop an essential understanding of ratios and rates.
To understand current state of the research on the representation of textbooks, 170 articles on mathematical representation are randomly selected to explore the research objects in recent decades, with the result of only 10.6% papers focusing on mathematics textbooks. Then, those papers are analyzed in terms of research questions, theoretical frameworks and research methods. Results show that 1）there are four main research problems, and most of the studies presented or compared the characterization of the mathematics textbooks. 2）There are four main theoretical frameworks, and most of the studies focus on different types of representation and different types of problems. 3）59% articles use the document analysis method and 29% articles use comparative research method. The trend of the representation about textbooks are studied at last.
The Dutch reform movement towards Realistic Mathematics Education (RME) started at the end of the 1960s. As an alternative for the then prevailing mechanistic approach, tasks, lessons and longitudinal teaching sequences were developed together with suggestions for helpful contexts and didactical models. Over the years, RME ideas were laid down in many studies, covering both general ideas on the teaching of mathematics, and elaborations in specific subdomains. Simultaneously, these ideas were implemented in succeeding generations of mathematic textbooks. A systematic analysis of RME documents and textbooks with the focus on addition and subtraction revealed that Dutch textbooks to a certain degree have evolved along the lines of RME, but also deviated from RME ideas.
In this paper, the contents of Korean elementary school math textbooks before and after the revision were analyzed in order to present the trend of teaching method of ‘division of fractions’ in South Korea. Fraction division can be easily solved by using 'invert and multiplying' algorithm, It is easy for students to perform calculation mechanically without understanding. But in fact, it is very difficult for students to understand why they should use ‘the invert’ in fraction division. The division of fractions should understand the calculation process using complex fraction concepts. Therefore, the understanding of students will vary depending on the order in which fraction concepts are presented and what math materials are used.
South Korea is one of the East Asian countries that maintains the highest ranking in international assessments such as PISA and TIMSS, using a single state-authored textbook that applies to the national curriculum. Also, Korea recently revised its curriculum in 2015. In this study, I compared two Korean elementary school math textbooks that were applied with the revised curriculum in 2009 and 2015. To provide framework of the teaching method of fraction division, this study focus on the concept(meaning) of fractions and the visual models used in textbooks that were analyzed with the changes in Korea elementary school math textbooks.
PreTeXt [1] is an authoring and publishing system, used primarily (but not exclusively) to create and distribute undergraduate mathematics textbooks. There are presently about fifty textbooks authored with PreTeXt, mostly published with open licenses. A key part of the design is that authors create source material in a very structured form. This allows us to replicate that structure within the electronic versions produced--in ways invisible to the reader, but such that it is possible to very accurately observe how a reader interacts with their book. See [2] for more details.
This will be a very hands-on workshop. Participants will need to bring their Internet-connected laptop equipped with a recent Chrome or Firefox web browser. Then, each participant will author a small textbook, and by the end of the workshop simultaneously produce an online version and a print version of their book.
Even if you do not have plans to write a textbook, or do not even know anybody else who does, this workshop will give you a concrete introduction to the value derived by authoring scholary documents in a structured way, and the implications for collecting data about textbook use.
[1] pretextbook.org
[2] K.L. O’Halloran, R.A. Beezer, D.W. Farmer, A New Generation of Mathematics Textbook Research and Development. ZDM Mathematics Education, Special Issue: Recent Advances in Mathematics Textbook Research and Development. Gert Schubring & Lianghuo Fan (eds.), 50(2), June 2018.
In the last decade，project-based learning (PBL) has been increasingly applied in discipline education; However, there is really few PBL application research has been applied in mathematics education. Therefore, our team have continuously conducted theoretical and practical researches on PBL application and developed PBL textbook during the past five years.
Our theoretical research mainly explored how to apply PBL philosophy in mathematics textbook and its possible effect. Firstly, we clarified the standards for implementation and evaluation of PBL textbook; Secondly, we concluded the learning model of PBL textbook; Thirdly, we clarified the application path, influence effect and challenges of PBL textbook; Fourthly we standardized the general path of project-based textbook design.
Practically, we mainly explored the textbook development and instruction of PBL. First of all, we clarified the functions and roles, general layout, as well as design principles of project-based textbook in mathematics, which provided a unified standard for the follow-up systematic design. Secondly, based on the current curriculum standards and textbooks, we systematically developed project-based mathematics textbook in middle school，which has the following features: 1) design project theme based on key concepts; 2) Take the task implementation as the overt plot while the knowledge development as the covert plot; 3) Pay attention to the authenticity of situation that allowed students to immersed in; 4) Focus on the mathematical learning process and gain more experience during mathematics activity. Thirdly, we prompted a series of research to apply mathematics PBL textbook at the experiment schools. And we concluded that the PBL textbook was a very good fit for middle school students’ cognitive style and could meet their learning requirements. Moreover, it had significant positive impact not only on the development of students' mathematical abilities in problem solving, exploring and innovation, but also on some non-intellectual perspectives, such as their learning attitudes and interests in mathematics.
As a result, we propose the following suggestions for the functions and implementation of PBL textbook in mathematics:
On one hand, PBL textbook could be used as an auxiliary form of learning material. It could be used as a mean to intervene mathematical learning process of gifted students or low learners.
On the other hand, in PBL textbook development, the relationship between knowledge learning and project activities, as well as the contextualization and mathematicization of learning content should be considered. Meanwhile, it is also necessary to deal with the contradiction between PBL and standardized test, and the relationship between PBL and traditional instruction in mathematics; moreover, teacher training such, as knowledge accumulation, belief, and etc. should be strengthen.
When textbook research is conducted as design research, two aims are systematically combined: (a) research-based design of teaching-learning arrangements and (b) design-based research for a deeper understanding of the initiated learning processes. In the Kosima-project, 14 years were spent on enhancing and understanding students’ learning processes, with a specific focus on mathematization and active knowledge organization. Initiating rich processes of mathematization is a central aim for all mathematics education. However, many obstacles appear for these processes to generate solid and sustainable mathematical knowledge. By Didactical Design Research, obstacles in students’ learning pathways can be systematically identified in the Kosima-project and then overcome.
Our study in France concerns two very different digital platforms and their use by secondary school mathematics teachers. The “Digital Educational Resources Bank” (DERB) is an official platform that has been financed by the ministry of education, to support a curriculum reform starting in September 2016. The DERB contains complete mathematical contents for grade 6 to 9. It offers some elementary “bricks”, which can be static texts, slides, interactive exercises, mental maps, videos etc. It also provides tools to the teacher to build learning trajectories for his/her students (called “modules”), which can be different for different students (and this coincides with central institutional recommendations about differentiation). It is hence likely to influence the way teachers associate different resources, an aspect of documentation work (Gueudet & Trouche 2009) which was previously under the responsibility of the teacher him/herself. Nevertheless, using DERB is not compulsory, and not supported by the schools; as a consequence its use seems still quite limited, and restricted to the downloading of some isolated “bricks”. At the same time, selected secondary schools are engaged in an experiment concerning the use of “Pearltrees education”, a platform allowing to build structured collections of resources and to share them with students and colleagues. Through the platform the teachers can have access to many digital resources available on the Internet, or in collections opened by colleagues (individual resources are called “Pearls”). The teachers can build their own collections - that we interpret as resource systems (Trouche, Gueudet & Pepin 2018) - on the platform, they are supported by the school’s administration. Hence the platform is likely to foster the development of collective resource systems for the teachers engaged in the experiment. In this communication we analyze these platforms in terms of connectivity (Gueudet, Pepin, Sabra, Restrepo & Trouche 2018). Moreover these two contrasting cases lead us to discuss the consequences, in terms of teachers’ resource systems, of the affordances and constraints of a digital platform, and the importance of the institutional environment.
References
Gueudet, G., Pepin, B., Sabra, H., Restrepo, A. & Trouche, L. (2018). E-textbooks and connectivity: proposing an analytical framework. International Journal for Science and Mathematics Education, 16(3), 539-558.
Gueudet, G., & Trouche, L. (2009). Towards new documentation systems for teachers? Educational Studies in Mathematics, 71(3), 199-218.
Trouche, L., Gueudet, G., & Pepin, B. (2018). Documentational Approach to Didactics. In S. Lerman (Éd.), Encyclopedia of Mathematics Education (p. 1-11). Cham: Springer International Publishing.
Professional learning and sharing opportunities that are meaningful, focused on mathematics teaching and learning, flexible, and low-cost (or free) are difficult to find in the USA. For mathematics teachers working in small towns or rural school districts, such collaborative professional learning opportunities are particularly limited. The Ohio Mathematics Teacher Hubs (OMTH) Project provides a digital space for math teachers and mathematics intervention specialists to discuss issues and share ideas with colleagues from other schools and districts and university mathematics education faculty. The project takes as foundational the assertion that teachers learn when choosing, transforming, discussing, implementing, and revising resources (Gueudet, Pepin, & Trouche, 2012). The project utilizes the documentational approach of didactics (Gueudet, Pepin, & Trouche, 2012) to study the consequences, actual and potential, of the use of digital platforms on teachers’ work. Central to the documentational approach is documentational genesis, with its dialectical processes involving both a teacher’s shaping of the resource and her teaching practice being shaped by the resource (Gueudet, Pepin, & Trouche, 2012). Documentational genesis plays a significant role due to the project’s focus on teachers’ work on and with resources, both individually and collectively. The project functions as a network, allowing participants to share experiences and expertise, promoting collaboration as ‘hubs’ or collectives work toward the common goal of improving mathematics education. The project integrates three platforms to connect teachers and connect with teachers: a) a free web hosting service that makes the project website accessible (https://mathteacherhubs.weebly.com/); b) a file storage and sharing service (i.e. Google Drive); and c) a communication platform (i.e. Google Hangouts). The project was initiated in October 2018 and several existing international projects serve as models (e.g. Maths Hubs; Math MOOC UniTO). As the project develops, activities are allowed to arise naturally through interactions with individual and groups of teachers (i.e. collectives/hubs)—as needs and interests are identified and clustered. Project data is generated through a variety of activities, including: synchronous and asynchronous online discussions, teacher educator-embedded professional learning experiences, and instruction and assessment planning sessions. Project findings are subsequently dispersed throughout network. The presentation will describe the project design, discuss preliminary results from the project’s first eight months, and reflect on modifications that have occurred as the project’s structure continues to develop. Gueudet, G., Pepin, B., & Trouche, L. (Eds.) (2012). From textbooks to ‘lived’ resources: Mathematics curriculum materials and teacher documentation. New York: Springer.
In the Netherlands several digital platforms are offered to mathematics teachers; some commercially by textbook publishers, others initiated by the Ministry, and again others by groups of teachers. Moreover, they all offer different kinds of resources for teachers, while some provide opportunities of working together at a distance, and possibilities for sharing resources. An exploratory survey study (Kock & Pepin, 2019) has suggested that at present the use of digital platforms by mathematics teachers is limited, in particular of platforms that allow teachers to construct and share their own resources, individually or collectively.
In this study, using the theoretical frame of connectivity (Gueudet, Pepin, Restrepo, Sabra, & Trouche, 2018), we analyse the affordances and constraints of the most commonly used not-for-profit platforms for mathematics teachers, at primary and secondary level. First, we analyse the platforms in terms of resource offer and use for teaching. Second, using the notion of ‘teacher design’ (Brown, 2009), we analyse the potential of those platforms for teacher design work, individually or collectively.
Two of the platforms we will analyze are Math4all (www.math4all.nl) and Wikiwijs (www.wikiwijs.nl). Math4all offers a selection of mathematics classroom materials and resources. Teachers can select, arrange and share resources offered on the platform, but cannot create or import resources themselves. Wikiwijs has been available since 2009, and was started on the initiative of the minister of Education, to stimulate the development and use of open educational resources. On the platform teachers can find resources, and develop, save and share their own resources. According to its website, Wikiwijs aims to support schools and teachers to assemble an optimal mix of educational materials.
The platforms provide different opportunities to teachers for connecting and extending their resources systems. The analysis will highlight these differences, which will be discussed in the context of secondary education in the Netherlands.
References
Brown, M. W. (2009). The teacher-tool relationship: Theorizing the design and use of curriculum materials. In J. T. Remillard, B. A. Herbel-Eisenmann, & G. M. Lloyd (Eds.), Mathematics teachers at work: Connecting curriculum materials and classroom instruction (pp. 17-36). New York: Routledge.
Gueudet, G., Pepin, B., Restrepo, A., Sabra, H., & Trouche, L. (2018). E-textbooks and connectivity: Proposing an analytical framework. International Journal of Science and Mathematics Education, 16(3), 539-558.
Kock, Z.-J., & Pepin, B. (2019). Secondary school mathematics teachers’ selection and use of resources. Poster presented at the Eleventh Congress of the European Society for Research in Mathematics Education (CERME11), Utrecht, the Netherlands.
Currently, a new type of digital platforms are being implemented in the Danish compulsory schools. Among other things, the platforms provide teachers a template in which they can design lessons and distribute them to their students. A distinct characteristic of these platforms is that they are mandatory for teachers to use, and that the interfaces in which teachers design lessons is based on learning objectives from a legislative curriculum, which also is mandatory for teachers to use. This curriculum have been discussed heavily, and there are great variation among teachers in to what extent they subscribe to the form and content of it. By integrating this curriculum and providing interfaces for teachers to use it, the platforms contribute in what Remillard and Heck (2014) refer to as curriculum translations. Curriculum translation involve re-representations and/or reformulations of the curriculum, which may be of significance for teachers’ perceptions of how to use the curriculum.
Building on interviews and observations of mathematics teachers’ usage of these platforms from Danish compulsory schools, this paper seeks to investigate the relation between mathematics teachers’ documentational work and the digital platforms, and the role that the platforms’ translations of the curriculum standards play in this process. We will study this matter by drawing on a combination of the documentational genesis (Gueudet & Trouche, 2009) and Remillard’s (2005) concept of curriculum voice. The documentational approach (Gueudet & Trouche, 2009) is designed to study teachers’ selection, design and appropriation of resources. It draws on inspiration from the instrumental approach (Guin, Ruthven & Trouche, 2005) and Adler’s (2000) concept of resources (Gueudet & Trouche, 2009). It considers teachers’ work with resources to be a dialectic process, where teachers’ usages (and knowledge) and the resources used mutually affect each other (Gueudet & Trouche, 2009). Based on Adler (2000), resources are defined as “a range of other human and material resources, as well as mathematical, cultural, and social resources” (Adler, 2000, 210). A document is thus considered as the product of combined resources, usages and knowledge (Gueudet, Soury-Lavergne & Trouche, 2012). We combine this approach with Remillard’s (2005) concept of curriculum voice, which involve how authors of the curriculum communicate to the teacher in for example grammatical structures that enforce authority. In this case, we consider the curriculum voice as emerging from and relationship between the curriculum and the platforms’ translation of it. The aim of this paper is thus to provide a better insight in how platforms may mediate curricula (and other resources) and the implications of this for teachers’ documentation work.
Recent years have seen an increased effort to incorporate problem posing into school mathematics at different educational levels around the world. Curriculum has historically been viewed as a powerful agent for instructional change. Given the potential positive impact of including problem-posing activities in mathematics classrooms, it is useful to consider how curriculum might support such activities. In this plenary, I will first present a historical analysis of mathematical problem-posing activities in both China and the United States over the past several decades. Unfortunately, this analysis shows that only a very small proportion of the mathematical activities included in Chinese and U.S. curricula were problem-posing activities. Thus, the call to integrate problem-posing activities in mathematics classrooms is not well aligned with existing curricular resources and there is a need to help teachers develop problem-posing resources and implement problem posing in mathematics classrooms. I will then present findings from a longitudinal study investigating the impact of teachers’ learning to teach mathematics using problem posing on students’ thinking and affect. I will end by discussing methodological issues related to historical analyses of curricula and a longitudinal study of teachers’ professional learning.
In this article, we examine and compare how the procedures for solving one-variable linear equations and inequalities are presented in the narrative of Brazilian, Portuguese and Spanish textbooks. We analyze the main resources used, the relationships between the procedures of solving equations and inequalities, and the presence of reasoning-and-proving in the exposition of content. Results include the fact that all authors use the (two-plate) scales as a resource to illustrate some property in order to solve a specific equation, however, this feature is not used as a reasoning-and-proving opportunity, merely as an empirical example. Only one author uses scales to solve inequalities. We also verified that the relationship between solving procedures for equations and for inequalities is little explored in these textbooks. Moreover, we present suggestions for the use of (two-plate) scales with two different goals: firstly, as a tool to illustrate properties of both equations and inequalities, thus as an instrument to compare their solving procedures, and, secondly, as a generalization tool, providing, as proposed by Stylianides (2009), a reasoning-and-proving activity.
It is widely recognized that textbooks play an important role in teaching and learning of mathematics. Therefore, the way in which textbooks address gender equity is expected to have a significant influence on learning outcomes. However, when gender stereotyping is often investigated by textbook researchers, gender equity is rarely a concern when selecting and organizing teaching contents during textbook design. The aim of this study is to identify whether male and female middle school students have different needs in their actual mathematics learning. It is believed that the findings of this study would provide insightful information for the curriculum development and textbook design.
In the study, a total of 1000 male and 1000 female middle school students were randomly selected from a mathematics online-question-answering system. Based on the mathematics topics covered, the questions posed by the 2000 students were first coded aligning to the PEP mathematics textbook chapters and sections. Using a multiple regression model, the differences on learning needs between male and female students were then quantified. It is found that female students have a larger demand for consultation in terms of number of chapters as well as sections than their male peers. Female students tend to need more help on the topics related to function and plane geometry, while male students need more help on statistics. Moreover, the average difficulty level of the topics on which female students have significantly larger demands is much higher than that on which male students have significantly larger demands (8.13 vs. 7.30). This work closes with discussions about the potential factors causing these gender differences and educational implications for teaching and learning of mathematics, particularly for textbook and learning resource developers.
Interactions between teachers and textbooks are an active process and have an impact on mathematical lessons. International studies show that historical snippets are one way to including the history of mathematics in textbooks. Teachers must handle these tasks, but in which way they use these historical snippets? The aim of this explorative survey is to identify teachers’ language about tasks concerning the history of mathematics. Which connotations do teachers have about these textbook tasks? Do teachers identify the same benefits of the history of mathematics in education as researchers do? This paper proposal will give an insightful view of a pilot study on prospective teachers’ language about historical snippets. The survey uses an online instrument, presents teachers three tasks concerning history of mathematics, and records teachers’ connotations about these tasks. The results indicate different views on historical snippets and, on more than one occasion, that the benefits of history of mathematics are missing. Therefore, teachers need an introduction to handle these tasks occurring in every-day mathematical textbooks.
Mathematical and linguistic features of word problems have been investigated with respect to their potential difficulties in various studies. In the current study, the transition of grade 4 (primary school) to grade 5 (secondary school) is studied for identifying changing demands in these features. For this, the study focuses word problems using the basic rules of arithmetic in different German grade 4 and grade 5 textbooks. By a corpus linguistic approach, similarities among the features can be revealed as well as differences depending on the grade and the type of rule of arithmetic. The results of the study concerning these features and changing demands are relevant in order to design word problems for teaching-learning arrangements, which prepare students to cope with the word problems in German grade 5 textbooks.
Report on the main ideas, the design, the practical and theoretical components and the realisations of the workshop ‘Writing Maths Textbooks’ which the author developed on behalf of the GIZ (Deutsche Gesellschaft für Internationale Zusammenarbeit) for the Republic of Yemen to teach future textbook writers how to write Maths Textbooks. Because of acts of war the workshop was delivered in 2016 per skype. A second face to face realisation of the workshop took place 2018 at the Hanoi National University of Education (HNUE) in Vietnam.
Results from earlier research found that textbook use by teachers of mathematics was largely tacit and not deliberate, making for un-intimate teacher-textbook relationships, and therefore low levels of teacher capacity for pedagogic design. The change from textbooks as primary curricular resources for teaching to daily scripted lesson plans begs a question about the kinds of relationships teachers forge with the scripted lesson plans. The present article explores the affordances and constraints of the scripted lesson plan, and how one teacher mobilises the scripted lesson plan in her lesson. The analysis, from a socio-cultural perspective, illuminates the teacher’s PDC-in-action, and sheds light on how context helps us to understand and reflect on the notion of PDC.
Present setcentric and pre-setcentric math are challenged by
post-setcentric math seeing math, not as a goal, but as a means to develop
the mastery of Many children bring to school.
Asked “How old next time?”, a 3year old says “Four” showing 4 fingers; but
protests if held together 2 by 2: “That is not four, that is two 2s”, thus
describing what exists, and with units: the total is bundles of 2s, and 2
of them. Children thus develop both word- and number-sentences with a
subject, a verb and a predicate. The outside total exists as a natural
fact, but the inside predication it chosen and can be changed: T = 4 1s = 2
2s = 1Bundle 1 3s = 0Bundle less 1 5s, etc.
Post-setcentric textbooks allow children to develop further their mastery
of Many by counting and recounting totals before adding them; and to number
instead of being taught about numbers. A textbook thus can be based upon
the following ‘research’ questions.
• The digit 5 is an icon with five sticks. Does this apply to all digits?
• How to count fingers in different sequences and bundles?
• How can a calculator predict a recounting result?
• What to do with the unbundled singles?
• How to recount in the same or in another unit?
• How to recount between tens and icons?
• How to recount the sides in a block halved by its diagonal?
• How to perform and reverse next-to and on-top addition?
This paper seeks to explore the value given to technology in the Chilean school mathematics textbooks. In recent years, digital media has been incorporated as an important part of the school mathematics textbook. This phenomenon has unfolded forms of producing a particular reality, as well as, conducting—and sometimes delineating and shaping—students’ and teachers’ ways of thinking and acting. That is, the use of technology is affecting how students and teachers experience and perceive school mathematics. Within the materials provided by the Chilean Ministry of Education, it is possible to identify a gap between the use and representation of technology—its value—and the technology—actual technological devices—needed for the proper implementation of the activities suggested in these materials. The identified gap increases in schools with a high level of vulnerability. We contend the need of problematizing both the use of technology as part of the content of school mathematics tasks (i.e. as context for and activity) and the use of digital materials as a way of improving students achievement given that, more often than expected, digital school mathematics textbooks do not help—and even obstruct—teacher´s practices.
This paper presents an examination about mathematical textbooks for high school students from China, Japan and the United States at both macro and micro levels, paritcularly with respect to general features, the trigonometric content included, the sequencing of trigonometric content, the structure of trigonometric content, presentation features and the requirement of problems. We have found the following results: The American textbook is visually attractive and many pictures in full colour. In contrast, the other two textbooks are plain. Meanwhile, both Chinese and American textbooks are convenient for students to read. These three countries value the trigonometry highly, but the Chinese textbook includes less trigonometric content than the other two countries. American and Chinese textbooks present trigonometric application after trigonometric function, while Japanese textbook presents these topics much earlier. The Chinese and Japanese textbooks have more emphasis on content instruction and less emphasis on problems when compared to the American textbook. Almost of the textbooks emphasize the development of students’ procedural skills. Several suggestions are proposed at the end.
Curricula around the world make more and more use of goals trying to capture different
kind of processes for the students to master. For mathematics education in Denmark,
these ambitions have been described in terms of a set of mathematical competencies.
However, bringing such competencies into the actual teaching practices has proved
challenging. Matematrix is a Danish mathematics textbook system for grades k-9
designed to support the mathematics teachers in facing this challenge. In this paper,
I – as one of the designers and authors of the textbooks – present one of the key
elements in this endeavour: A three-dimensional content and objectives model combining
mathematical competencies, mathematical core concepts and grade level. Following that,
I exemplify the use of the model at three different levels of textbook design: The
structuring of the content for the books in general, the focal points for each chapter
in the various books and the development of tasks for a specific chapter.
This paper focuses on troubling the distribution and market strategies of school mathematics textbooks in a context embedded in neoliberal policies and assumptions about education. And so, it raises a set of questions about the dynamic of the production, distribution, and selection of the official mathematics textbooks for Chilean schools and the market that unfolds simultaneously of non-official mathematics textbooks. Some of these non-official textbooks—meaning these are not distributed by the Chilean Ministry of Education—are been sold at highest rates in bookstores and, because of it, are considered to be of better quality. The paper grasps how circulating narratives of ‘if you pay more, you will have a better product—a better textbook—and thus better opportunities in life’ within neoliberal-based market strategies govern the selecting of some textbooks in schools. This has led to some schools deciding not to use the official mathematics textbooks that are distributed for free by the Chilean Ministry of Education, and rather asking parents to buy the other “better” and expensive textbooks for their children’s education.
In this workshop, Professor Jinfa Cai (Editor-in-Chief for JRME) will first describe the overall process of the review process at the JRME Editorial Office, and then he will discuss the common mistakes to avoid when submitting a manuscript for publication. He is open to individual consultation and discussion of possible studies participants are preparing for publication.