The mathematics textbook at tertiary level as curriculum material – exploring the teacher's decision-making process

This paper reports on a study about how the mathematics textbook was perceived and used by the teacher in the context of a calculus part of a basic mathematics course for first-year engineering students. The focus was on the teacher's choices and the use of definitions, examples and exercises in a sequence of lectures introducing the derivative concept. Data were collected during observations of lectures and an interview, and informal talks with the teacher. The introduction and the treatment of the derivative as proposed by the teacher during the lectures were analysed in relation to the results of the content text analysis of the textbook. The teacher's decisions were explored through the lens of intended learning goals for engineering students taking the mathematics course. The results showed that the sequence of concepts and the formal introduction of the derivative as proposed by the textbook were closely followed during the lectures. The examples and tasks offered to the students focused strongly on procedural knowledge. Although the textbook proposes both examples and exercises that promote conceptual knowledge, these opportunities were not fully utilized during the observed lectures. Possible reasons for the teacher's choices and decisions are discussed.     

The State of Proof in Finnish and Swedish Mathematics Textbooks—Capturing Differences in Approaches to Upper-Secondary Integral Calculus

Students’ difficulties with proof, scholars’ calls for proof to be a consistent part of K-12 mathematics, and the extensive use of textbooks in mathematics classrooms motivate investigations on how proof-related items are addressed in mathematics textbooks. We contribute to textbook research by focusing on opportunities to learn proof-related reasoning in integral calculus, a key subject in transitioning from secondary to tertiary education. We analyze expository sections and nearly 2000 students’ exercises in the four most frequently used Finnish and Swedish textbook series. Results indicate that Finnish textbooks offer more opportunities for learning proof than do Swedish textbooks. Proofs are also more visible in Finnish textbooks than in Swedish materials, but the tasks in the latter reflect a higher variation in nature of proof-related reasoning. Our results are compared with methodologically similar U.S. studies. Consequences for learning and transition to university mathematics, as well as directions for future research, are discussed.     

Requirements in mathematics textbooks: a five-dimensional analysis of textbook exercises and examples

Mathematics textbooks play a very important role in mathematics education and textbook tasks are used by students for practice to a large extent. Since the nature of the tasks may influence the way students think it is important that the textbooks provide a balance of a variety of tasks. The analyses of the requirements in textbook tasks contain the usual dimensions of content, cognitive demands, question type and contextual features. The aim of this study is to embed a new fifth dimension into the framework: mathematical activities. This addresses the question of what a student should do in a particular textbook task: to represent, to compute, to interpret or to use argumentation. The analysis encompassed more than 22,000 tasks from the most commonly used Croatian mathematics textbooks in the 6th, 7th and 8th grade. The results show that the textbooks do not provide a full range of task types. There is an emphasis on computation, while argumentation and interpretation activities, reflective thinking and open answer tasks are underrepresented. The study revealed that incorporating mathematical activities into the multidimensional framework of textbook tasks may help to better understand the opportunities to learn which are afforded students by using mathematics textbooks.     

A Comparative Analysis of the Addition and Subtraction of Fractions in Textbooks from Three Countries

In this paper, we report on a comparison of the treatment of addition and subtraction of fractions in primary mathematics textbooks used in Cyprus, Ireland, and Taiwan. To this end, we use a framework specifically developed to investigate the learning opportunities afforded by the textbooks, particularly with respect to the presentation of the content and the textbook expectations as manifested in the associated tasks. We found several similarities and differences among the textbooks regarding the topics included and their sequencing, the constructs of fractions, the worked examples, the cognitive demands of the tasks, and the types of responses required of students. The findings emphasized the need to examine textbooks in order to understand differences in instruction and achievement across countries. Indeed, we postulate that within a given country there may exist a recognizable “textbook signature.” We also draw on the results and the challenges inherent in our analysis to provide suggestions and directions for future textbook analysis studies.     

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

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

Electronic vs. paper textbook presentations of the various aspects of mathematics

Based in part on our work in adapting existing paper textbooks for secondary schools for a digital format, this paper discusses paper form and the various electronic platforms with regard to the presentation of five aspects of mathematics that have roles in mathematics learning in all the grades kindergarten-12: symbolization, deduction, modeling, algorithms, and representations. In moving to digital platforms, each of these aspects of mathematics presents its own challenges and opportunities for both curriculum and instruction, that is, for the content goals and how they connect with students for learning. A combination of paper and electronic presentations may be an optimal solution but some difficulties with such a complex solution are presented.     

Interpretations of National Curricula: the case of geometry in textbooks from England and Japan

This paper reports on how the geometry component of the National Curricula for mathematics in Japan and in one selected country of the UK, specifically England, is interpreted in school mathematics textbooks from major publishers sampled from each country. The findings we report identify features of geometry, and approaches to geometry teaching and learning, that are found in a sample of textbooks aimed at students in Grade 8 (aged 13–14). Our analysis raises two issues which are widely recognised as very important in mathematics education: the teaching of mathematical reasoning and proof, and the teaching of problem-solving. In terms of the teaching of mathematical reasoning and proof, our evidence indicates that this is dispersed in the textbook in England while it is concentrated in geometry in the textbook in Japan. In terms of the teaching of mathematical problem-solving and modeling, our analysis shows that it is more concentrated in the textbook from England, and rather more dispersed in the textbook from Japan. These findings indicate how important it is to consider ways in which these issues can be carefully designed in the geometry sections of future textbooks.     

Approach to mathematics in textbooks at tertiary level – exploring authors’ views about their texts

The aim of this article is to present and discuss some results from an inquiry into mathematics textbooks authors’ visions about their texts and approaches they choose when new concepts are introduced. Authors’ responses are discussed in relation to results about students’ difficulties with approaching calculus reported by previous research. A questionnaire has been designed and sent to seven authors of the most used calculus textbooks in Norway and four authors have responded. The responses show that the authors mainly view teaching in terms of transmission so they focus mainly on getting the mathematical content correct and ‘clear’. The dominant view is that the textbook is intended to help the students to learn by explaining and clarifying. The authors prefer the approach to introduce new concepts based on the traditional way of perceiving mathematics as a system of definitions, examples and exercises. The results of this study may enhance our understanding of the role of the textbook at tertiary level. They may also form a foundation for further research.     

A new generation of mathematics textbook research and development

This paper adopts a multimodal approach to the latest generation of digital mathematics textbooks (print and online) to investigate how the design, content, and features facilitate the construction of mathematical knowledge for teaching and learning purposes. The sequential organization of the print version is compared to the interactive format of the online version which foregrounds explanations and important mathematical content while simultaneously ensuring a high level of connectivity and coherence across hierarchical layers of mathematical knowledge. For example, mathematical content in the online version is linked to definitions, theorems, examples and exercises that can be viewed in the original context in which the material was presented, and the content can also be linked to mathematics software. Significantly, the development process for the new generation of mathematics textbooks involves using a ‘design neutral’ markup language so that the books are simultaneously published as both print books and online books. In this development process, the structure of the chapters, sections, and subsections with their various elements are explicitly marked-up in the master document and preserved in the output format, giving rise to new methodologies for large-scale analysis of mathematics textbooks and student use of these books. For example, tracking methodologies and interactive visualizations of student viewings of online mathematical textbooks are identified as new research directions for investigating how students engage with mathematics textbooks within and across different educational contexts.     

Cognitive opportunities in textbooks: the cases of grade four and eight textbooks in Israel

The cognitive domain in mathematics, defined as thinking and understanding in the process of learning mathematics, is a main focus of curricula in many countries. This study explores breadth and depth of understanding as addressed in mathematics textbooks certified as aligned to Israeli national mathematics curricula. We compare opportunities for students to engage with mathematics requiring different types and levels of understanding provided by the tasks in mathematics textbooks. Comparison of two fourth grade and two eighth grade mathematics textbooks showed significant differences in the opportunities to learn in the cognitive domain that each provides. These differences can be quantified; the quantification defines the cognitive demand of the textbook. The cognitive demand of the four textbooks varies. This reveals a potential source of inequity in students’ opportunities to learn mathematics. Results should prompt discussion around standardization and alignment of textbooks to the cognitive goals of the curriculum.     

Standing on each other’s shoulders: A case of coalescence between geometric discourses in peer interaction

In this paper we draw on the commognitive theory to examine novice students’ transition from familiar mathematics meta-rules to less familiar ones during peer interaction. To pursue this goal, we focused on a relatively symmetric interaction between two middle-school students given a geometric task. During their dyadic problem-solving, the students transitioned from configural procedures to deductive ones. We found that this transition included an interactive coalescence pattern in which one student “borrowed” her partner’s configural sub-procedures and built on them to develop a new deductive procedure. Furthermore, we found that during their peer interaction, the students oscillated between configural, coalesced and deductive procedures. Several patterns in the students’ interpretation of the task-situation contributed to these oscillations. We discuss the contribution of our findings to commognitive research, to geometry learning research and to peer learning research.     

Reasoning-and-Proving in School Mathematics Textbooks

Despite widespread agreement that the activity of reasoning-and-proving should be central to all students' mathematical experiences, many students face serious difficulties with this activity. Mathematics textbooks can play an important role in students' opportunities to engage in reasoning-and-proving: research suggests that many decisions that teachers make about what tasks to implement in their classrooms and when and how to implement them are mediated by the textbooks they use. Yet, little is known about how reasoning-and-proving is promoted in school mathematics textbooks. In this article, I present an analytic/methodological approach for the examination of the opportunities designed in mathematics textbooks for students to engage in reasoning-and-proving. In addition, I exemplify the utility of the approach in an examination of a strategically selected American mathematics textbook series. I use the findings from this examination as a context to discuss issues of textbook design in the domain of reasoning-and-proving that pertain to any textbook series.     

The textbook-in-use: students’ utilization schemes of mathematics textbooks related to self-regulated practicing

This paper presents a qualitative study on how students make use of their mathematics textbooks for practicing. The study was carried out in two German secondary schools with 74 students (44 in 6th and 30 in 12th grade). Students’ utilization of textbooks for practicing is analyzed using the theoretical framework of instrumental genesis. The results indicate that students’ choices of contents from the book for practicing can be categorized into three utilization schemes: position-dependent practicing, block-dependent practicing, and salience-dependent practicing. In terms of position-dependent practicing the relative position of the textbook’s contents to teacher-mediated sections guides the students’ choice. Block-dependent practicing relates to the use of contents from the book that belong to particular blocks. Finally, salience-dependent practicing is a utilization scheme of the book where students’ choice is guided by perceptual salience of the book contents. These findings both show how textbook users are influenced by the way mathematics is presented in textbooks and provide insights into students’ conceptions of practicing mathematics.     

The finite-to-finite strand of a learning progression for the concept of function: A research synthesis and cognitive analysis

In this paper we report validation efforts around the finite-to-finite strand of a provisional learning progression (LP) for the concept of function. We regard an LP as an empirically-verified account of how student understandings form over time and in response to instruction. The finite-to-finite strand of the LP was informed by literature on students’ thinking and learning related to functions as well as the Algebra Project’s curricular approach, which is designed for students who are traditionally underserved by mathematics education. Developing and validating an LP is a multi-step, cyclic process. Here we report on one step in this process, an item and response analysis. Data sources include 680 students’ responses to 13 multipart computer-delivered tasks. Results suggest that revisions to the items, associated scoring rubrics, and in some instances the LP are warranted. We illustrate this task, rubric, and LP revision process through an item analysis for a selected task.     

Promoting a concept of fraction-as-measure: A study of the Learning Through Activity research program

Promoting deep understanding of fraction concepts continues to be a challenge for mathematics education. Research has demonstrated that students whose concept of fractions is limited to part-whole have difficulty with advanced fraction concepts. We conducted teaching experiments to study how students can develop a measurement concept of fractions and how task sequences can be developed to promote the necessary abstractions. Building particularly on the work of Steffe and colleagues and aspects of the Elkonin-Davydov curriculum, we focused on fostering student reinvention of a measurement concept of fractions. As a study of the Learning Through Activity research program, our goal was to promote particular activity on the part of the students through which they could abstract the necessary concepts.     

Using Student Reasoning to Inform the Development of Conceptual Learning Goals: The Case of Quadratic Functions

Despite the proliferation of mathematics standards internationally and despite general agreement on the importance of teaching for conceptual understanding, conceptual learning goals for many K-12 mathematics topics have not been well-articulated. This article presents a coherent set of five conceptual learning goals for a complex mathematical domain, generated via a method of systematic empirical analysis of students' reasoning. Specifically, we compared the reasoning of pairs of students who performed differentially on the same task and inferred the pivotal intermediate conceptions that afforded one student deeper engagement with the task than another student. In turn, each pivotal intermediate conception framed an associated conceptual learning goal. While the empirical analysis of student reasoning is typically used to understand how students learn, we argue that such analysis should also play an important role in determining what concepts students should learn.     

Problems Identifying Independent and Dependent Variables

This paper discusses one step from the scientific method—that of identifying independent and dependent variables—from both scientific and mathematical perspectives. It begins by analyzing an episode from a middle school mathematics classroom that illustrates the need for students and teachers alike to develop a robust understanding of independent and dependent variables. It then outlines four rationales (two from science and two from mathematics) for identifying independent and dependent variables. Finally, it reports the results of a textbook analysis that used these rationales to examine the extent to which typical mathematics textbook problems support or supplant a sensible view of independent and dependent variables. The findings indicate that often, mathematics textbook problems misrepresent the sense‐making aspect of identifying independent and dependent variables, possibly setting students up to develop misconceptions about this step from the scientific method.     

Reducing the graphical user interface of a dynamic geometry system

Graphical user interfaces (GUIs) of dynamic or interactive geometry software (DGS) allow users to interact with the DGS by using a computer mouse. Clicking on a GUI icon performs an action like choosing a construction tool or manipulating an object. For novices, it may be difficult to recognize and recall the icons needed for a task. Learning mathematics and learning the use of a dynamic geometry system at the same time could lead to cognitive overload. Several DGS systems try to solve this problem by offering different GUIs: expert users can choose between a wide range of icons, while for novice users only the most basic icons are presented. By preselecting a specific set of icons, a teacher can adapt a DGS to create a tool, which meets specific pedagogical demands. Two experiments were conducted to investigate the effects of reducing GUIs of a DGS. In experiment 1, which was carried out with full and reduced interfaces of the DGS Cinderella, the eye movements and gaze points of the users were recorded by an eye tracker. The time taken by users to find given icons in different types of interfaces was measured. In experiment 2, students measured the angle sums of polygons using the DGS Cinderella with a full or a reduced interface. No significant effects of GUI reduction were found in both experiments. The results are discussed and ideas for future research are presented.     

Creative and algorithmic mathematical reasoning: effects of transfer-appropriate processing and effortful struggle

Two separate studies, Jonsson et al. (J. Math Behav. 2014;36: 20–32) and Karlsson Wirebring et al. (Trends Neurosci Educ. 2015;4(1–2):6–14), showed that learning mathematics using creative mathematical reasoning and constructing their own solution methods can be more efficient than if students use algorithmic reasoning and are given the solution procedures. It was argued that effortful struggle was the key that explained this difference. It was also argued that the results could not be explained by the effects of transfer-appropriate processing, although this was not empirically investigated. This study evaluated the hypotheses of transfer-appropriate processing and effortful struggle in relation to the specific characteristics associated with algorithmic reasoning task and creative mathematical reasoning task. In a between-subjects design, upper-secondary students were matched according to their working memory capacity.

The main finding was that the superior performance associated with practicing creative mathematical reasoning was mainly supported by effortful struggle, however, there was also an effect of transfer-appropriate processing. It is argued that students need to struggle with important mathematics that in turn facilitates the construction of knowledge. It is further argued that the way we construct mathematical tasks have consequences for how much effort students allocate to their task-solving attempt.