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.     

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.     

Opportunity to learn problem solving in Dutch primary school mathematics textbooks

In the Netherlands, mathematics textbooks are a decisive influence on the enacted curriculum. About a decade ago, Dutch primary school mathematics textbooks provided hardly any opportunities to learn problem solving. In this study we investigated whether this provision has changed. In order to do so, we carried out a textbook analysis in which we established to what degree current textbooks provide non-routine problem-solving tasks for which students do not immediately have a particular solution strategy at their disposal. We also analyzed to what degree textbooks provide ‘gray-area’ tasks, which are not really non-routine problems, but are also not straightforwardly solvable. In addition, we inventoried other ways in which present textbooks facilitate the opportunity to learn problem solving. Finally, we researched how inclusive these textbooks are with respect to offering opportunities to learn problem solving for students with varying mathematical abilities. The results of our study show that the opportunities that the currently most widely used Dutch textbooks offer to learn problem solving are very limited, and these opportunities are mainly offered in materials meant for more able students. In this regard, Dutch mainstream textbooks have not changed compared to the situation a decade ago. A textbook that is the Dutch edition of a Singapore mathematics textbook stands out in offering the highest number of problem-solving tasks, and in offering these in the materials meant for all students. However, in the ways this textbook facilitates the opportunity to learn problem solving, sometimes a tension occurs concerning the creative character of genuine problem solving.     

Solution representations of percentage change problems: the pre-service primary teachers’ mathematical thinking and reasoning

Solution representations can reveal how problem solvers communicate mathematical thinking and reasoning in problem-solving process. The present study examined the solution representations used by 20 pre-service teachers for the percentage change problems. The pre-service teachers were invited to solve a combination of simple and complex percentage change problems. The score for the majority of simple problems was 75% or above, but the score for the complex problems was below 75%. The highest percentage error occurred when the pre-service teachers encountered a percentage greater than 100% in the percentage change problems. Irrespective of their level of mathematics qualifications, the equation approach demonstrating two-step problem-solving process was the predominant strategy adopted by the pre-service teachers. The equation approach imposes low cognitive load and, therefore, is more accessible and efficient than the unitary approach. A few pre-service teachers used the unitary approach. The findings indicate that the pre-service teachers possessed relevant mathematical knowledge for percentage change problems. Furthermore, the inclusion of the equation approach in mathematics textbooks would provide an alternative perspective regarding the teaching and learning of percentage change problems.     

Selection of content in high school mathematics textbooks: an international comparison

As a component of the ongoing development of the mathematics curriculum in China, we compare the country’s high school mathematics textbooks with those of several other countries. We base our analysis on the assumption that textbooks, as primary printed teaching resources, are key tools for interpreting educational policy. In this paper, we compare what content is selected and how that content is presented in series of high school textbooks from China, France, Germany, Japan, Russia, the United Kingdom, and the United States according to four core domains: algebra, geometry, statistics and probability, and calculus. We then discuss the implications of the analysis for the reform of the high school mathematics curriculum in China, particularly as it applies to the development of textbooks within that process of reform. The comparative results provide us with the opportunity to recognise the distinguishing features of the content and presentation of the Chinese mathematics curriculum and lead to some suggestions for future curriculum development.     

Conceptualizing and organizing content for teaching and learning in selected Chinese, Japanese and US mathematics textbooks: the case of fraction division

In this study, selected Chinese, Japanese and US mathematics textbooks were examined in terms of their ways of conceptualizing and organizing content for the teaching and learning of fraction division. Three Chinese mathematics textbook series, three Japanese textbook series, and four US textbook series were selected and examined to locate the content instruction of fraction division. Textbook organization of fraction division and other content topics were described. Further analyses were then conducted to specify how the content topic of fraction division was conceptualized and introduced. Specific attention was also given to the textbooks’ uses of content constructs including examples, representations, and exercise problems in order to show their approaches for the teaching and learning of fraction division. The results provide a glimpse of the metaphors of mathematics teaching and learning that have been employed in Chinese, Japanese, and US textbooks. In particular, the results from the textbook analyses demonstrate how conceptual underpinnings were developed while targeting procedures and operations. Implications of the study are then discussed.     

Mathematics textbooks and their use in English, French and German classrooms:

After a through review of the relevant literature in terms of textbook analysis and mathematics teachers' user of textbooks in school contexts, this paper reports on selected and early findings from a study of mathematics textbooks and their use in English, French and German mathematics classrooms at lower secondary level. The research reviewed in the literature section raises important questions about textbooks as representations of the curriculum and about their role as a link between curriculum and pedagogy. Teachers, in tunr, appear to exercise control over the curriculum as it is enacted by using texts in the service of their own perceptions of teaching and learning. The second and main part of the paper analyses the ways in which textbooks vary and are used by teachers in classroom contexts and how this influences the culture of the mathematics classroom. The findings of the research demonstrate that classroom cultures are shaped by at least two factors: teachers' pedagogic principles in their immediate school and classroom context; and a system's educational and cultural traditions as they develop over time. It is argued that mathematics classroom cultures need to be understood in terms of a wider cultural and systemic context, in order for shared understandings, principles and meanings to be established, whether for promotion of classroom reform or simply for developing a better understanding of this vital component of the mathematics education process.     

Computerized proof techniques for undergraduates

The use of computer algebra systems such as Maple and Mathematica is becoming increasingly important and widespread in mathematics learning, teaching and research. In this article, we present computerized proof techniques of Gosper, Wilf–Zeilberger and Zeilberger that can be used for enhancing the teaching and learning of topics in discrete mathematics. We demonstrate by examples how one can use these computerized proof techniques to raise students' interests in the discovery and proof of mathematical identities and enhance their problem-solving skills.     

What does it mean to critique? An exploration of the types of critiques promoted in seventh-grade mathematics curriculum and assessment materials

While research in mathematics education has shown that mathematics assessments are highly consequential, traditional assessments often lag behind advancements in instructional methods. One such advancement is the promotion of mathematical habits of mind such as students' abilities to critique others' reasoning. This study explored the use of student work embedded in seventh-grade curriculum-based mathematics assessment tasks as a mechanism for critiquing others' thinking. The researcher investigated the prevalence and nature of student work in assessment tasks as compared to textbook tasks from five seventh-grade, Common Core State Standards for Mathematics-aligned curriculum series. The text analyses findings revealed that while there were multiple critique types in student work across both assessment and textbook tasks, there were substantial differences between students' opportunities to make sense of someone else's mathematical thinking in curriculum-based assessments as compared to the student textbooks. These findings reinforced prior curriculum and assessment research that found assessment often lags behind instructional methods.     

Integrating algebra and proof in high school mathematics: An exploratory study

Frequently, in the US students’ work with proofs is largely concentrated to the domain of high school geometry, thus providing students with a distorted image of what proof entails, which is at odds with the central role that proof plays in mathematics. Despite the centrality of proof in mathematics, there is a lack of studies addressing how to integrate proof into other mathematical domains. In this paper, we discuss a teaching experiment designed to integrate algebra and proof in the high school curriculum. Algebraic proof was envisioned as the vehicle that would provide high school students the opportunity to learn not only about proof in a context other than geometry, but also about aspects of algebra. Results from the experiment indicate that students meaningfully learned about aspects of both algebra and proof in that they produced algebraic proofs involving multiple variables, based on conjectures they themselves generated.     

Integrating Algebra and Proof in High School: Students' Work with Multiple Variables and a Single Parameter in a Proof Context

In the United States, researchers argue that proof is largely concentrated in the domain of high school geometry, thus providing students a distorted image of what proof entails, which is at odds with the central role that proof plays in mathematics. Despite the centrality of proof, there is a lack of studies addressing how to integrate proof into other mathematical domains. In this article, we discuss a teaching experiment designed to integrate algebra and proof in the high school curriculum. Algebraic proof was envisioned as the vehicle that would provide high school students the opportunity to learn not only about proof in a context other than geometry but also about aspects of algebra. Results from the experiment indicate that students meaningfully learned about aspects of both algebra and proof in that they produced algebraic proofs involving multiple variables and a single parameter, based on conjectures they themselves generated.     

Preservice teachers learning to teach proof through classroom implementation: Successes and challenges

Proof and reasoning are central to learning mathematics with understanding. Yet proof is seen as challenging to teach and to learn. In a capstone course for preservice teachers, we developed instructional modules that guided prospective secondary mathematics teachers (PSTs) through a cycle of learning about the logical aspects of proof, then planning and implementing lessons in secondary classrooms that integrate these aspects with traditional mathematics curriculum in the United States. In this paper we highlight our framework on mathematical knowledge for teaching proof and focus on some of the logical aspects of proof that are seen as particularly challenging (four proof themes). We analyze 60 lesson plans, video recordings of a subset of 13 enacted lessons, and the PSTs’ self- reported data to shed light on how the PSTs planned and enacted lessons that integrate these proof themes. The results provide insights into successes and challenges the PSTs encountered in this process and illustrate potential pathways for preparing PSTs to enact reasoning and proof in secondary classrooms. We also highlight the design principles for supporting the development of PSTs’ mathematical knowledge for teaching proof.     

A changing trajectory of proof learning in the geometry inquiry classroom

The purpose of this study is to explore how students changes through learning to construct mathematical proofs in an inquiry-based middle school geometry class in Korea. Although proof has long been considered as one of the most important aspects of mathematics education, it is well-known that it is one of the most difficult areas of school mathematics for students. The geometry inquiry classroom (GIC) is an experimental class designed to teach geometry, especially focusing on teaching proof, based on students’ own inquiry. Based on a 2-year participant observation in the GIC, this study was conducted to address the following research question: how has students’ practice of mathematical proof been changed through their participation in the GIC? The in-depth analysis of the classroom discourse identified three stages through which the students’ practice of mathematical proof was transformed in the GIC: ‘emergent understanding of proof’, ‘proof learning as a goal-oriented activity’, ‘experiencing proof as the practice of mathematics’. The study found that as learning evolved through these stages, so the mathematics teacher’s role shifted from being an instructor to a mediator of communication. Most importantly, this research showed that the GIC has created a learning environment where students develop their competence in constructing meaningful mathematical proof and grow to be ‘a human who proves’, ultimately ‘a person who playfully engages with mathematics’.     

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.     

How well aligned are common core textbooks to students’ development in area measurement?

Among dozens of factors that influence mathematics teaching in the elementary classroom, textbooks endure as a significant contributor to the conversation. While teachers have many considerations while lesson planning, the textbook often forms an important launch point in determining what to include in lessons and how to do so. It follows that discrepancies between textbooks and research‐recommended pathways for learning may lead to concerns or issues with pacing in the classroom. To explore this idea further, this study examined the alignment between three popular Common Core–aligned textbooks series and learning trajectories with respect to the topic of area measurement. Our findings indicated key differences in the ways textbooks presented area lessons and research‐recommended ways of learning area topics, including a lack of appropriate area topic coverage in early grades and a mismatch of timing of concepts in later grades. The results indicated that the standards‐based textbooks examined may lack attention to important topics in the pacing of area instruction, and suggest the need to inform both preservice and inservice teachers about the gap between textbook lessons and area learning trajectories so that development steps in area learning trajectory can be included in lesson plans.     

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.     

Validating proofs and counterexamples across content domains: Practices of importance for mathematics majors

Validating proofs and counterexamples across content domains is considered vital practices for undergraduate students to advance their mathematical reasoning and knowledge. To date, not enough is known about the ways mathematics majors determine the validity of arguments in the domains of algebra, analysis, geometry, and number theory—the domains that are central to many mathematics courses. This study reported how 16 mathematics majors, including eight specializing in secondary mathematics education, who had completed more proof-based courses than transition-to-proof classes evaluated various arguments. The results suggest that the students use one of the following strategies in proof and counterexample validation: (1) examination of the argument's structure and (2) line-by-line checking with informal deductive reasoning, example-based reasoning, experience-based reasoning, and informal deductive and example-based reasoning. Most students tended to examine all steps of the argument with informal deductive reasoning across various tasks, suggesting that this approach might be problem dependent. Even though all participating students had taken more proof-related mathematics courses, it is surprising that many of them did not recognize global-structure or line-by-line content-based flaws presented in the argument.     

Investigating textbooks as crucial interfaces between culture, policy and teacher curricular practice: two contrasted case studies in France and Norway

This paper reports on an investigation of mathematics curriculum documents, commonly used textbooks and teacher ‘curricular practice’ with respect to educational traditions in France and Norway. The study has helped to develop a deeper understanding of (1) educational traditions in France and Norway; (2) the ways the educational traditions permeate the system, from policy documents through textbooks into the classroom; and (3) the connections between the worlds of policy, textbooks and teacher curricular practice in mathematics. Results show that French and Norwegian curricular documents and practices were influenced by egalitarian values, albeit differently interpreted and ‘lived’ in each country. In terms of mathematics, using a focus on grade 6 geometric transformations has shown that French textbooks and teacher curricular practices emphasized theoretical properties and mathematical reasoning, whereas the Norwegian counterparts stressed practical and inquiry-based activities. It is argued that mathematics textbooks can be regarded as a crucial interface between culture, policy and curricular practice, and as a pivotal resource in teachers’ resource system for curricular practice, even in times of digitization.     

Content and Organization of Teacher's Manuals: An Analysis of Japanese Elementary Mathematics Teacher's Manuals

In this paper, results from a study that analyzed the content and organization of teacher's manuals for elementary school mathematics from Japan and the United States are presented. Studies have shown that the nature of mathematics instruction in Japan is different from instruction commonly observed in the United States. Moreover, other scholars have noted that elementary school teachers, both in the United States and Japan, rely heavily on textbooks to teach mathematics. Thus, teacher's manuals accompanying textbook series may be a contributing factor to this difference. The results of the analysis showed that there are some significant differences in the way Japanese teachers' manuals are prepared from those of the US series. The findings suggest that curriculum developers should critically reflect on how to prepare teacher's manuals so that they become useful resources for teachers.