The Mathematical Nature of Reasoning-and-Proving Opportunities in Geometry Textbooks

International calls have been made for reasoning-and-proving to permeate school mathematics. It is important that efforts to heed this call are grounded in an understanding of the opportunities to reason-and-prove that already exist, especially in secondary-level geometry where reasoning-and-proving opportunities are prevalent but not thoroughly studied. This analysis of six secondary-level geometry textbooks, like studies of other textbooks, characterizes the justifications given in the exposition and the reasoning-and-proving activities expected of students in the exercises. Furthermore, this study considers whether the mathematical statements included in the reasoning-and-proving opportunities are general or particular in nature. Findings include the fact that the majority of expository mathematical statements were general, whereas reasoning-and-proving exercises tended to involve particular mathematical statements. Although reasoning-and-proving opportunities were relatively numerous, it remained rare for the reasoning-and-proving process itself to be an explicit object of reflection. Relationships between these findings and the necessity principle of pedagogy are discussed.     

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.     

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.     

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.     

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.     

The quality of non-textual elements in mathematics textbooks: an exploratory comparison between South Korea and the United States

As an exploratory investigation, this study aims to analyze non-textual elements in some Korean and US mathematics textbooks using a conceptual framework whose components include accuracy, connectivity, contextuality, and conciseness. By analyzing three US textbooks and three Korean ones, the study not only shows patterns in the use of non-textual elements in mathematics textbooks in different contexts but also provides insights into how to assess the quality of non-textual elements in mathematics textbooks, which I hope will contribute to the provision of more meaningful and productive learning opportunities to school children. Overall, the results from this study show that there is significant difference across topics and textbooks, which implies different opportunities to learn through non-textual elements. This study makes a unique contribution to the conceptualization of non-textual elements in mathematics education and has implications for textbook analysis and curriculum development.     

Teachers' ways of listening and responding to students' emerging mathematical models

In this paper, I present the results of a case study of the practices of four experienced secondary teachers as they engaged their students in the initial development of mathematical models for exponential growth. The study focuses on two related aspects of their practices: (a) when, how and to what extent they saw and interpreted students' ways of thinking about exponential functions and (b) how they responded to the students' thinking in their classroom practice. Through an analysis of the teachers' actions in the classroom, I describe the teachers' developing knowledge when using modeling tasks with secondary students. The analysis suggests that there is considerable variation in the approaches that teachers take in listening to and responding to students' emerging mathematical models. Having a well-developed schema for how students might approach the task enabled one teacher to press students to express, evaluate, and revise their emerging models of exponential growth. Implications for the knowledge needed to teach mathematics through modeling are discussed.     

Mathematics standards and curricula in the Netherlands

This paper addresses the question of what mathematics Dutch students should learn according to the standards as established by the Dutch Ministry of Education. The focus is on primary school and the foundation phase of secondary school. This means that the paper covers the range from kindergarten to grade 8 (4~14 years olds). Apart from giving an overview of the standards, we also discuss the standards' nature and history Furthermore, we look at textbooks and examination programs that in the Netherlands both have a key role in determining the intended mathematics curriculum. In addition to addressing the mathematical content, we also pay attention to the way mathematics is taught. The domain-specific education theory that forms the basis for the Dutch approach to teaching mathematics is called “Realistic Mathematics Education” Achievement scores of Dutch students from national and international tests complete this paper. These scores reveal what the standards bring us in terms of students' mathematical understanding. In addition to informing an international audience about the Dutch standards and curricula, we include some critical reflections on them.     

The Impact of Professional Noticing on Teachers' Adaptations of Challenging Tasks

This study investigates how teacher attention to student thinking informs adaptations of challenging tasks. Five teachers who had implemented challenging mathematics curriculum materials for three or more years were videotaped enacting instructional sequences and were subsequently interviewed about those enactments. The results indicate that the two teachers who attended closely to student thinking developed conjectures about how that thinking developed across instructional sequences and used those conjectures to inform their adaptations. These teachers connected their conjectures to the details of student strategies, leading to adaptations that enhanced task complexity and students' opportunity to engage with mathematical concepts. By contrast, the three teachers who evaluated students' thinking primarily as right or wrong regularly adapted tasks in ways that were poorly informed by their observations and that reduced the complexity of the tasks. The results suggest that forming communities of inquiry around the use of challenging curriculum materials is important for providing opportunities for students to learn with understanding.     

A comparative analysis of linear functions in Korean and American standards-based secondary textbooks

This paper compares sections on functions and linear functions from two Korean textbooks and an American standards-based textbook (University of Chicago School Mathematics Project [UCSMP] Algebra) to understand differences and similarities among these textbooks through horizontal and vertical analyses. We found that these textbooks provide different opportunities to learn (OTL). UCSMP Algebra places strong emphasis on real-life applications of linear functions rather than on pure mathematics and mathematical algorithms. Also, compared to Korean textbooks, UCSMP Algebra offers more OTL for students to solve, explain, and reason about higher level cognitively demanding mathematics problems than Korean secondary textbooks. Contradictory results, compared to previous studies about East Asian mathematics textbooks indicate the need for further study to compare secondary textbooks from more East Asian countries. Also, with the results of this study, we need to understand the results of international assessments more carefully.     

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.     

From the Written to the Enacted Curricula: The Intermediary Role of Middle School Mathematics Teachers in Shaping Students' Opportunity to Learn

In this paper is reported the extent of textbook use by 39 middle school mathematics teachers in six states, 17 utilizing a textbook series developed with funding from the National Science Foundation (NSF‐funded) and 22 using textbooks developed by commercial publishers (publisher‐generated). Results indicate that both sets of teachers placed significantly higher emphasis on Number and Operation, often at the expense of other content strands. Location of topics within a textbook represented an oversimplified explanation of what mathematics gets taught or omitted. Most teachers using an NSF‐funded curriculum taught content intended for students in a different (lower) grade, and both sets of teachers supplemented with skill‐building and “practice” worksheets. Implications for documenting teachers' “fidelity of implementation” ( National Research Council, 2004 ) are offered.     

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.     

Investigating Elementary Mathematics Curricula: Focus on Students With Learning Disabilities

The purpose of this study was to investigate three elementary mathematics curricula to examine the accessibility for students with learning disabilities (LD) with regards to challenges associated with working memory. We chose to focus on students' experiences when finding the area of composite shapes due to the multiple steps involved for solving these problems and the potential for these problems to tax working memory. We conducted a qualitative analysis of how each curriculum provided opportunities for students with LD to engage with these problems. During our analysis, we focused on instruction that emphasized visual representations (e.g., manipulatives, drawings, and diagrams), facilitated mathematical conversations, and developed cognitive and metacognitive skills. Our findings indicated a need for practitioners to consider how each curriculum provides instruction for storage and organization of information as well as how each curriculum develops students' thinking processes and conceptual understanding of mathematics. We concluded that all three curricula provide potentially effective strategies for teaching students with LD to solve multi‐step problems, such as area of composite shapes problems, but teachers using any of these curricula will likely need to supplement the curriculum to meet the needs of students with LD.     

Exploring Mathematics in Imaginative Places: Rethinking What Counts as Meaningful Contexts for Learning Mathematics

This paper explores what happens when students engage with mathematical tasks that make no attempt to be connected with students' everyday life experiences. The investigation draws on the work of educators who call for a broader view of what might count as real and relevant contexts for studying mathematics. It investigates students' experiences with two imaginative tasks and reports on the students' intellectual and emotional engagement. This engagement is examined and described in terms of the character and quality of the class and group discussions generated. Findings suggest that students can indeed engage productively with mathematics when it is explored in imaginative settings and that such contexts can help students support and sustain their engagement with the mathematics in the task.     

Elements of Design‐Based Science Activities That Affect Students' Motivation

The primary purpose of this study was to examine the ways in which a 12‐week afterschool science and engineering program affected middle school students' motivation to engage in science and engineering activities. We used current motivation research and theory as a conceptual framework to assess 14 students' motivation through questionnaires, structured interviews, and observations. Students reported that during the activities they perceived that they were empowered to make choices in how to complete things, the activities were useful to them, they could succeed in the activities, they enjoyed and were interested in the hands‐on activities and some presentations, they felt cared for by the facilitators and received help when they were stuck or confused, and they put forth effort. Based on our examination of data across our three data sources, we identified motivating opportunities that were provided to students during the activities. These motivating opportunities can serve as examples to help both formal and informal science educators better connect motivation theory to practice so that they can create motivating opportunities for students. Furthermore, this study provides a methodological example of how students' motivation can be examined during the context of authentic science and engineering instruction.     

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.     

Components of Place Value Understanding: Targeting Mathematical Difficulties When Providing Interventions

Place value understanding requires the same activity that students use when developing fractional and algebraic reasoning, making this understanding foundational to mathematics learning. However, many students engage successfully in mathematics classrooms without having a conceptual understanding of place value, preventing them from accessing mathematics that is more sophisticated later. The purpose of this exploratory study is to investigate how upper elementary students' unit coordination related to difficulties they experience when engaging in place value tasks. Understanding place value requires that students coordinate units recursively to construct multi‐digit numbers from their single‐digit number understandings through forms of unit development and strategic counting. Findings suggest that students identified as low‐achieving were capable of only one or two levels of unit coordination. Furthermore, these students relied on inaccurate procedures to solve problems with millennial numbers. These findings indicate that upper elementary students identified as low‐achieving are not to yet able to (de)compose numbers effectively, regroup tens and hundreds when operating on numbers, and transition between millennial numbers. Implications of this study suggest that curricula designers and statewide standards should adopt nuances in unit coordination when developing tasks that promote or assess students' place value understanding.