Mathematics Curriculum Materials Research Articles

ОСОБЛИВОСТІ ФОРМУВАННЯ ІНФОРМАЦІЙНО-КОМУНІКАЦІЙНОЇКОМПЕТЕНТНОСТІ УЧНІВ НА УРОКАХ МАТЕМАТИКИ

To delineate and substantiate the nuances of developing information and communication competence in students during mathematics lessons, we conducted an extensive analysis of state educational regulations, including the Law of Ukraine "On Education," the New Ukrainian School Concept, the Concept of Digital Competence Development, the State Standard of Basic Secondary Education, and mathematics curriculum materials. We also scrutinized sources available on the Internet related to international educational monitoring studies and synthesized both international and domestic pedagogical experience. The study asserts that the formation of information and communication competence in students during mathematics education arises from the demands of the school mathematics curriculum. Mandatory learning outcomes in the mathematical education sector encompass requirements characterizing students' information and communication competence. The process of learning mathematics should facilitate not only the overall development of students' personalities but also the acquisition of a system of mathematical knowledge, improvement of problem-solving skills, and an understanding of the importance of mathematics as the language of science, technology, and innovation. Additionally, students should recognize the significance of information and communication technologies in effectively addressing mathematical challenges. The integrated use of information technologies in the educational process, especially in the context of distance and blended learning, contributes to the development of students' information and communication competence. This is contingent upon the methodically sound selection of educational content, didactic materials, and teaching strategies by educators. It is elucidated that the primary means of cultivating students' information and communication competence lies in the specialized system of educational tasks and examples present in school mathematics textbooks. These materials are designed to foster students' personal qualities, creativity, critical thinking, and their ability to generate and articulate their thoughts.

Relating Students’ Early Algebraic Thinking to Mathematics Curriculum Materials

This study investigated the trends in algebraic thinking of Korean elementary school students through the students’ correct answer rates for the early algebra-related items in the 4th-grade assessment of TIMSS 2011, TIMSS 2015, and TIMSS 2019, and explored these trends in relation to the changes in early algebra-related content in the Korean mathematics curriculum and textbooks. As a result of the study, with regards to finding unknowns or unknown operations and representing problem situations as expressions, the correct answer rates of the students participating in TIMSS 2019 decreased compared to those participating in TIMSS 2015. The change in the correct answer rates was related to the decrease of relevant standards in the mathematics curriculum and tasks in mathematics textbooks.
 Meanwhile, for tasks about identifying a recursive pattern and representing a covariational relationship, the relevance of the mathematics curriculum materials to the changes in students’ algebraic thinking were not revealed due to the limitation of the small number of related TIMSS items. Nevertheless, it is worth noting that the correct answer rates of students participating in TIMSS 2019 for recognizing recursive patterns and finding appropriate values decreased slightly. Finally, this study makes suggestions for the learning and teaching of early algebra based on the observed results.

Assessment for Learning: Factors Influencing Utilization of Learning Targets and Criteria for Success in Teaching and Learning Mathematics

Assessment for learning is described as a process by which key information is communicated to learners through sharing learning targets and criteria for success in the teaching and learning process. Effective utilization of learning targets and criteria for success serves as the foundation for all other formative assessment strategies. This study assessed factors that influenced secondary school mathematics teachers' utilization of learning targets in line with formative assessment in the teaching and learning process. The study employed a mixed-methods research approach and consisted of 59 respondents drawn from twelve private and public secondary schools in Arusha City and Kinondoni Municipal in Tanzania. The data were collected through interviews, classroom observations, and questionnaires. Qualitative data from interviews were analyzed using thematic analysis. Quantitative data from classroom observation were analyzed statistically using nonparametric techniques such as Mann -Whitney U test and Kruskal-Wallis H test. The key instrument was the classroom observation schedule adapted from Oswalt (2013), which consisted of five Likert scales. Results show that learning targets were the most utilized aspect of formative assessment during teaching and learning mathematics. Analysis indicated that the demographic characteristic that influenced teachers' utilization of learning targets was teachers' gender. While teaching experience, training status, school types, qualifications, number of periods, and classroom size did not statistically influence teachers’ utilization of learning targets. The study recommends teachers' training on the utilization of learning targets in line with formative assessment. In addition, the Ministry of Education, Science, and Technology (MoEST) and Tanzania Institute of Education (TIE), in particular, should develop textual mathematics curriculum materials to support teachers with knowledge and skills regarding the utilization of learning targets and criteria for success in line with formative assessment.

Teachers as redesigners of curriculum to teach mathematics through problem posing: conceptualization and initial findings of a problem‑posing project

Problem posing, the process of formulating problems based on a given situation, is an essential practice in mathematics and other disciplines. Although this is acknowledged in policy documents, problem posing is neither substantively nor consistently included in the school mathematics curriculum. In this paper, we first comment on the state of problem posing in school mathematics and discuss three recommendations to improve its integration into curriculum materials and classroom practice. These recommendations present a low barrier to entry for teachers and students and require only minor changes to common mathematics classroom activities and curriculum materials. Based on the three recommendations, as well as the features of effective teacher professional development, a program was created to investigate longitudinally the impact of problem-posing professional development on teachers’ conceptions of problem posing and their design of lessons to teach mathematics using problem posing, as well as on students’ learning. Initial findings are presented, including the significant changes in teachers’ conceptions of problem posing and teachers’ design of lessons to teach mathematics using problem posing, as well as the impact on students’ learning.

Profiles of Elementary Teachers’ Use of Mathematics Curriculum Materials and the Influence of Teacher Expertise

The National Council of Teachers of Mathematics has consistently emphasized the importance of curricular coherence in mathematics education. However, the predominance of the Internet has led to a lack of consistency in the use and quality of curricular materials. We drew on teachers’ self-report of their use of curriculum materials and conducted a Latent Class Analysis to examine patterns in 56 elementary teachers’ selection, use, and perceptions of materials for teaching mathematics, including the role that teacher expertise may play in these patterns. Findings indicated that the most salient difference between classes was at least one digital resource reported—the Blended class—and no digital resources reported—the Non-Digital class. Teacher expertise was not a significant covariate. Future research should examine the role of policies on teachers’ curricular decisions. Implications include how specialist programs may support positive patterns of curriculum selection and use.

Like the layers of an onion: curricular noticing as a lens to understand the epistemological features of the Philippine K to 12 secondary mathematics curriculum materials

As major sources of information, curriculum materials (CMs) have generally served teachers and students by providing instructional and learning support. The impact of CMs on teachers’ pedagogical practice and learners’ engagement and success had been fairly documented. However, just as the use of CMs is influenced by teachers’ orientations and learners’ identity with the subject matter, the same can be said for textbook writers who attend, interpret and respond to curriculum content and standards. Guided by Dietiker, Males, Amador and Earnest (2018)’s curriculum noticing construct, this study examined the epistemological features of Philippine secondary school mathematics textbooks of the Department of Education. Four textbooks, one from each grade level, were analyzed via a directed qualitative content analysis. Following the inductive approach, significant statements from the texts were culled and organized through coding, categorizing and abstraction using a repertory grid. Member checking was done to validate the findings of the study. Interestingly, the study afforded the development of the Onion Model of Curricular Noticing Dynamics in Mathematics Textbooks. The model reveals how CM writers frame the mathematics curriculum in the design and development of textbooks. Specifically, the encountering frame lays emphasis on their interaction with the mandated curriculum through an examination of curricular elements. In the sense-making frame, writers reflect on four curricular questions leading to an understanding of the curriculum and the context in which it operates. Finally, the operationalizing frame captures the unique moves observed by textbook writers in developing the materials. The emerged model can serve as a valuable means in achieving heightened consciousness among textbook writers as they develop materials that promote teacher learning and respond to learners’ needs, interests and orientations.

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

AbstractWhile 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.

Analysis of Pre-Service Teachers' Interpretation and Utilization of Non-Textual Elements in Mathematics Curriculum Materials

This study explores how pre-service teachers (PSTs) view, interpret, and utilize non-textual elements (NTEs) in mathematics curriculum. Fifty-two PSTs, who enrolled in a mathematics methods course at a Midwestern university in the U.S., engaged in a three-part task that consisted of evaluations and modifications of NTEs in the sample mathematics curriculum materials. We ascertain what PSTs consider to be the strengths and weaknesses of NTEs, how they define the primary goals of NTEs, and how they would work to modify or adapt existing NTEs with effective teaching in mind. By using the Curricular Noticing Framework, we can better understand how PSTs recognize opportunities within mathematics curriculum and gain a deeper understanding regarding how PSTs’ prior experiences may affect their curricular-attending habits, which has consequences for their future teaching. Findings indicate that PSTs understand NTEs to be simply a support for traditional mathematics curriculum, rather than tools on their own. Also, they tend to prefer NTEs that are familiar to them. From our findings, we draw implications for teacher educators who support PSTs’ interpretation and utilization of NTEs.

Creative thinking in mathematics curriculum: An analytic framework

Current approaches in mathematics education promote the teaching of creative thinking (CT) to develop a deep conceptual understanding of mathematics, and many nations are including explicit CT learning goals in their curriculum. The goals alone are not enough; appropriate curriculum materials should be used in the classroom if students are to acquire CT skills. This paper describes the development of a framework to analyze CT in primary school mathematics curriculum materials. It then applies the framework to primary school math textbooks in the context of the Israeli curriculum. The framework includes nine categories under three overarching themes: lateral thinking, divergent thinking and convergent-integrative thinking. Although the CT themes are covered, the textbooks provide much more opportunity to engage with CT in mathematics in grade one than in any other grade, but the curriculum sets the same CT objectives for all students. Further, the textbooks have emphases that do not necessarily correspond with the emphasis in the official curriculum. The proposed framework permits the assessment of curriculum materials, an important mediating factor between curriculum and instruction. It can show which activities and materials promote the development of students’ CT and which do not.

What is the point? Examining how curriculum materials articulate mathematical goals and how teachers steer instruction

This study examined how authors of four different elementary mathematics curriculum materials articulate the mathematical goals of the lesson and orient the activities toward the goals, as well as the extent to which teachers steered instruction toward the goals during enactment. Through document analysis, we identified four ways in which authors support teachers in identifying and orienting the lesson toward the goals. Analysis of the enacted lessons of 8 teachers suggests a correlation between the depth to which the goals were communicated across the four categories and the extent to which teachers steered the instruction towards the goals. The authors conclude with recommendations to curriculum developers for communicating mathematical goals in ergonomic ways.

Curriculum spaces for connecting to children’s multiple mathematical knowledge bases

Elementary mathematics curriculum materials can serve as a lever for instructional change. In this paper, we promote a particular kind of instructional change: supporting teachers in learning to integrate children’s multiple mathematical knowledge bases (MMKB), including children’s mathematical thinking and children’s home and community-based mathematical funds of knowledge, in instruction. A powerful means of supporting pre-service teachers in integrating children’s MMKB in instruction may be to scaffold teachers’ noticing of potential spaces in elementary mathematics curriculum materials for connecting to children’s MMKB and then developing practices for leveraging these spaces during instruction. We focus on existing and potential spaces in written curriculum materials, or curriculum spaces, so as to better support teachers in enacting curriculum that opens spaces for connecting to children’s MMKB.

Middle school mathematics teachers’ orientations and noticing of features of mathematics curriculum materials

We report findings on teachers’ noticing of features in the teacher resources of mathematics curriculum programs. Based on prior analysis, we selected teachers using one of two curriculum types: delivery mechanism or thinking device. The participating teachers and the curriculum programs aimed to align with the Common Core Standards for Mathematics, and thus, they ostensibly held a common aim for instruction. We analyzed 147 lesson planning interviews with 20 middle school mathematics teachers. We found that teachers attended to similar features of teacher resources; however, patterns for interpreting and planning decisions varied based on teachers’ orientations and curriculum type.

Promoting teacher learning: a framework for evaluating the educative features of mathematics curriculum materials

Educative curricula, curriculum materials that intentionally foster teacher professional development, can serve as a site for teacher learning through their use in daily instructional practices. The present article introduces a framework, Teacher Learning Opportunities in Mathematics Curriculum Materials (TLO-Math), for designing and evaluating mathematics curriculum materials’ educative features according to seven theoretically based variables: (1) mathematics content knowledge for teaching, (2) teacher knowledge of student thinking in mathematics, (3) teacher knowledge of disciplinary discourse in mathematics, (4) teacher knowledge of assessment in mathematics, (5) teacher knowledge of differentiated instruction in mathematics, (6) teacher knowledge of technology use in mathematics, and (7) teacher knowledge of mathematical community. Each variable is illustrated with a definition, guiding questions, discipline-specific literature, and examples from two sets of elementary mathematics curriculum materials. The development of the TLO-Math framework is a critical first step for further study of the use of mathematics curriculum materials as sites for teacher learning.

Knowledge of curriculum embedded mathematics: exploring a critical domain of teaching

This paper proposes a framework for identifying the mathematical knowledge teachers activate when using curriculum resources. We use the term knowledge of curriculum embedded mathematics (KCEM) to refer to the mathematics knowledge activated by teachers when reading and interpreting mathematical tasks, instructional designs, and representations in mathematics curriculum materials. The KCEM framework is situated within existing research on content-specific teacher knowledge. Through analysis of elementary mathematics teacher’s guides from the USA, we identified elements of curriculum resources teachers interact with when using them to plan instruction. These findings were complemented by interviews of teachers using curriculum guides to plan lessons, uncovering how they interacted with different elements of their guides. From this analysis, we propose four overlapping dimensions of KCEM: foundational mathematical ideas, representations and connections among these ideas, problem complexity, and mathematical learning pathways. Using an excerpt from one elementary curriculum guide and one of the teacher interviews on using this guide to plan a lesson, we illustrate how these dimensions might be activated as teachers read and use their guides.

Fidelity of implementation: bringing written curriculum materials into the equation

Fidelity of curriculum implementation (FOI) is an important area of research because of the critical role it plays in understanding how and why curriculum materials work and how they can be improved. This analysis explores written features within the Math Trailblazers curriculum that may influence the ways teachers implement mathematics curriculum materials. In particular, we examine FOI data from prior research in order to identify features within the materials themselves that may influence why teachers adhere to the intended curriculum to varying degrees. This paper reports on our analysis of examining five whole number lessons previously analysed for level of FOI alongside an analysis scale of written curriculum features. Our purpose in this analysis is to examine features of the written curriculum that potentially mediate teachers’ implementation of the materials in line with the intended curriculum. In doing so, our analysis further emphasises the critical role teachers play in curriculum implementation, and thus may provide some insight for curriculum developers as they consider ways in which to design the written curriculum that increase the likelihood that teachers will adhere to the intended curriculum.

Developing Addition Strategies: Preservice Teachers' Learning From Standards-Based Curriculum Materials

Despite the prevalence of mathematics curriculum materials in elementary classrooms, most current mathematics methods texts provide little or no support for preservice teachers (PSTs) learning to use curriculum materials. To meet this need, we have designed and studied several modules intended to provide PSTs with opportunities to learn about and from the use of curriculum materials. This article describes our research related to 1 of these modules–Addition Starter Sentences. Our results examine the nature of PSTs' developing content knowledge and pedagogical content knowledge, evidenced through their interactions with and reflections onStandards-based curriculum materials. We conclude with implications for mathematics teacher education research and practice.

From the guest editors

Carlos Castilo-Chavez is a Regents Professor, a Joaquin Bustoz Jr. Professor of Mathematical Biology, and a Distinguished Sustainability Scientist at Arizona State University. His research program is at the interface of the mathematical and natural and social sciences with emphasis on (i) the role of dynamic social landscapes on disease dispersal; (ii) the role of environmental and social structures on the dynamics of addiction and disease evolution, and (iii) Dynamics of complex systems at the interphase of ecology, epidemiology and the social sciences. Castillo-Chavez has co-authored over two hundred publications (see goggle scholar citations) that include journal articles and edited research volumes. Specifically, he co-authored a textbook in Mathematical Biology in 2001 (second edition in 2012); a volume (with Harvey Thomas Banks) on the use of mathematical models in homeland security published in SIAM's Frontiers in Applied Mathematics Series (2003); and co-edited volumes in the Series Contemporary Mathematics entitled '' Mathematical Studies on Human Disease Dynamics: Emerging Paradigms and Challenges'' (American Mathematical Society, 2006) and Mathematical and Statistical Estimation Approaches in Epidemiology (Springer-Verlag, 2009) highlighting his interests in the applications of mathematics in emerging and re-emerging diseases. Castillo-Chavez is a member of the Santa Fe Institute's external faculty, adjunct professor at Cornell University, and contributor, as a member of the Steering Committee of the '' Committee for the Review of the Evaluation Data on the Effectiveness of NSF-Supported and Commercially Generated Mathematics Curriculum Materials,'' to a 2004 NRC report. The CBMS workshop '' Mathematical Epidemiology with Applications'' lectures delivered by C. Castillo-Chavez and F. Brauer in 2011 have been published by SIAM in 2013.

Examining calculator use among students with and without disabilities educated with different mathematical curricula

This study assessed if students with and without disabilities used calculators (fourfunction, scientific, or graphing) to solve mathematics assessment problems and whether using calculators improved their performance. Participants were sixth and seventh-grade students educated with either National Science Foundation (NSF)-funded or traditional mathematics curriculum materials. Students solved multiple choice and open-ended problems based on items from the State’s released previous assessments. A linear mixed model was conducted for each grade to analyze the factors impacting students’ self-reported calculator use. Chi Square tests were also performed on both grade’s data to determine the relationship between using a calculator and correctly solving problems. Results suggested only time as a main factor impacting calculator use and students who self-reported using a calculator were more likely to answer questions correctly. The results have implications for practice given the controversy over calculator use by students both with and without disabilities.

Exploring complex issues: Insights from classrooms, courses and conversations

As researchers we are continually developing our knowledge and understanding ofmathematics education and, indeed, of the subject itself. In this issue we explore some ofthe current concerns we face as a community of mathematics educators. The challenges areundoubtedly complex. Here I would like to share some insights from classrooms, coursesand conversations.Over the years mathematics curriculum materials have drawn considerable attention andraised some interesting issues particularly in the way teachers have adapted them forclassroom use. Unlike many studies Jeffrey Choppin’s work benefits from being longitu-dinal. In Learned adaptations: Teachers’ understanding and use of curriculum resources,he discusses three teachers’ (grades 6–8) evolving implementation of curriculum materialshaving observed their use on at least three occasions. While there were some similarities intheir practice he explores marked differences in how they revised their tasks and how theseadaptations related to student sense making. Significantly he provides an example how oneof the teachers adopted curriculum resources in ways that built on and developed thedesigners’ original intentions.Managing whole group discussions is notoriously difficult and little understood. In theirarticle—Extending students’ mathematical thinking during whole group discussions—Nesrin Cengiz, Kate Kline and Theresa Grant closely examine eight episodes taken fromsix elementary classrooms. They explain that the teachers, having recognised opportunitiesto build new connections or address misconceptions, engaged in ‘a skilful dance’ in orderto extend student thinking. While the authors acknowledge the need for further work on thetopic they succeed in providing considerable insight into, ‘…much needed detail on whatextending students’ mathematical thinking looks like and which instructional actions areinstrumental in achieving the potential for extending episodes.’Teachers’ resources—particularly in terms of time and energy—are often limited, andtherefore, it is important that the professional development they are offered is of thehighest quality. In their article Kelli Nipper and her colleagues explore the relationshipbetween teachers’ expectations and actual experiences of a 1-week professional

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.