Common Core State Standards in the Middle Grades: What's New in the Geometry Domain and How Can Teachers Support Student Learning?

The Common Core State Standards for Mathematics (CCSSM) is a primary focus of attention for many stakeholders' (e.g., teachers, district mathematics leaders, and curriculum developers) intent on improving mathematics education. This article reports on specific content shifts related to the geometry domain in the middle grades (6–8) mathematics curriculum. The methodology employed allows for comparisons of content across multiple standards documents. We report on some dramatic changes with regards to the geometry content taught in the middle grades. We found 52% of the middle grades geometry CCSSM learning expectations will be new to the respective grade level at which they are taught in at least six of the eight states analyzed in this study (57% in grade 6, 50% in grade 7, and 50% in grade 8). We also highlight three areas that represent “new” geometry content at the middle grades based on our analysis of CCSSM and pre‐CCSSM state standards.     

How Common Is the Common Core? A Global Analysis of Math Teaching and Learning

The U.S. educational system is undergoing rapid and substantial changes with many states grappling with the adoption of the Common Core State Standards in Mathematics (CCSSM). Important research questions remain unanswered regarding the potential efficacy of the CCSSM to improve student math performance compared with students around the globe. This article utilized TIMSS 2007 8th‐grade math assessment results and curricular frameworks to (1) measure the degree of overlap between the CCSSM and TIMSS standards, and (2) use this finding to create a predictive model to determine the potential efficacy of the CCSSM in improving the U.S. 8th‐grade student math performance compared with six culturally matched, TIMSS‐assessed countries, provinces, and states. Comparisons of CCSSM and TIMSS‐assessed jurisdictions show that the CCSSM holds many items in common with TIMSS‐assessed jurisdictions, but lacks rigor in some key areas. The CCSSM deficiencies include algebraic knowledge and problem solving at the 8th‐grade level, and are a significant detractor from the CCSSM when compared with TIMSS.     

Middle School Mathematics Teachers’ Perceptions of the Common Core State Standards for Mathematics and Its Impact on the Instructional Environment

This study examines Middle School Mathematics Teachers’ (MSMTs') (N = 1,241) perceptions of the Common Core State Standards for Mathematics (CCSSM) and its impact on the instructional environment. A total of eight factors appear in the data. These factors include professional support, teachers’ use of district‐adopted and non‐district‐adopted curricular resources, influence of CCSSM assessment and teacher evaluation on the instructional environment, influence of the CCSSM on classroom instruction, teachers’ planning practices, and perceived rigor of the CCSSM. The data suggest that teachers’ use of digital resources located online was disconnected from their district‐adopted resources suggesting a lack of curricular coherence. MSMTs note that the CCSSM had caused them to incorporate more reform‐oriented practices such as exploration and productive struggle into their daily instruction. MSMTs also perceive that the CCSSM includes new content that is more rigorous than previous state standards. Sampled MSMTs state that while state assessments will measure CCSSM content, they are less likely to include more complex problems or the standards for mathematical practice. Teachers are more likely to read teacher resources than student textbook activities online and to use digital resources for remediation instead of inquiry activities. Over one‐third of MSMTs wanted more CCSSM professional development.     

Using Productive Disposition to Differentiate Between Students' Level of Precision When Critiquing a Peer's Work

This study examined the productive disposition of pre‐algebra students who demonstrated similar knowledge of the focal content but varied in other academic behaviors expected in the Common Core State Standards for Mathematics (CCSSM). Specifically, the study considered students' attention to precision when critiquing a peer's work. The comprehensive definition of productive disposition used included task values (interest, utility), an ability belief (efficacy), three personal achievement goals, and negative emotions. As hypothesized, the 61 students who provided a more precise critique reported higher productive disposition (in particular, significantly higher mastery‐approach personal achievement goals and less frequent negative emotions) than the 79 students who provided a basic critique. These findings illustrate how productive disposition can inform assessments of mathematical competence within the CCSSM recently implemented across the United States.     

Representations of Nature of Science in Selected Histories of Science in the Integrated Science Textbooks in China

This study aimed to examine the representations of nature of science (NOS) in the eight histories of science selected from three series of integrated science textbooks used in junior high school in China. Ten aspects of NOS were adopted in the analytical framework. It was found that NOS had not been well treated in the selected histories of science in the three series of science textbooks. Specifically, it was found that the empirical and inferential aspects were treated better than other aspects, and discrepancies existed among the three series of science textbooks and among the eight histories of science in terms of dealing with the target NOS aspects. Implications for addressing NOS in historical materials in science textbooks were discussed in the final part of this paper.     

How is cultural influence manifested in the formation of mathematics textbooks? A comparative case study of resource book series between Shanghai and England

Focusing on issues about the development of mathematics textbooks from a cultural perspective, this study examined a widely-used curriculum resource series, One Lesson One Exercise, published in China, and its adapted English series, published in the UK, to explore how cultural influence is manifested in the two series of resource books. For the study we established a conceptual framework classifying culture into six types in relation to people’s beliefs, values and ways of interacting about them, for data collection and analysis. The results indicate that there exist considerable differences between the Chinese and the English series that are related to cultural factors. It appears that, to a large extent, culture plays an essential role in the development of mathematics textbooks. Concerning the different types of culture, the results show that most adaptations between the Chinese series and the English series are related to ‘ways of behaving and customs’ and ‘artifacts, flora and fauna’, followed by ‘identities’ and ‘geography’, and the least are related to ‘organisations’ and ‘history’. Based on the study, we argue that the relevance and importance of culture to the development of mathematics textbooks must not be underestimated, and more research in this direction is needed.     

Field-based hypotheses on advancing standards for mathematical practice

The Common Core State Standards in Mathematics (CCSSM, 2010) are organized around two types of standards: the standards for mathematical content and standards for mathematical practice. The central goal of this paper is to present cognitive and instructional analyses of standards for mathematical practice through a discussion of field-based activities with inservice secondary mathematics teachers and students. A potential value of the study is that it provides researchers with specific field-based hypotheses on advancing standards for mathematical practice.     

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.     

An inexact parallel splitting augmented Lagrangian method for large system of linear equations

Parallel iterative methods are powerful in solving large systems of linear equations (LEs). The existing parallel computing research results focus mainly on sparse systems or others with particular structure. Most are based on parallel implementation of the classical relaxation methods such as Gauss-Seidel, SOR, and AOR methods which can be efficiently carried out on multiprocessor system. In this paper, we propose a novel parallel splitting operator method in which we divide the coefficient matrix into two or three parts. Then we convert the original problem (LEs) into a monotone (linear) variational inequality problem (VI) with separable structure. Finally, an inexact parallel splitting augmented Lagrangian method is proposed to solve the variational inequality problem (VI). To avoid dealing with the matrix inverse operator, we introduce proper inexact terms in subproblems such that the complexity of each iteration of the proposed method is O(n²). In addition, the proposed method does not require any special structure of system of LEs under consideration. Convergence of the proposed methods in dealing with two and three separable operators respectively, is proved. Numerical computations are provided to show the applicability and robustness of the proposed methods.     

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.     

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.     

Computational uncertainty and the application of a high-performance multiple precision scheme to obtaining the correct reference solution of Lorenz equations

The computational uncertainty principle (CUP) is applied to explain the experimental formulae of the critical time of decoupling for Lorenz equations (LEs). We apply the multiple precision (MP) library in obtaining the long-time solution of LEs, and based on the classic Taylor scheme, we developed a high-performance parallel Taylor solver to do the computation. The new solver is several hundreds times faster than the reported solvers developed in MATHEMATICA software, and it has the ability to yield longer solutions of LEs, up to t ∼ 10⁴ LTU (Lorenz time unit). Further, we notice that the two computation processes with different precisions or orders will produce the reliable correct reference solutions before they have a significant difference. According to this property we propose an approach for maintaining the correct numerical solution. The new solver and the solution validation approach are used to identify and correct an erroneous solution reported in a previous study.     

Opportunities to Learn: Inverse Relations in U.S. and Chinese Textbooks

This study, focusing on inverse relations, examines how representative U.S. and Chinese elementary textbooks may provide opportunities to learn fundamental mathematical ideas. Findings from this study indicate that both of the U.S. textbook series (grades K-6) in comparison to the Chinese textbook samples (grades 1–6), presented more instances of inverse relations, while also containing more unique types of problems; yet, the Chinese textbooks provided more opportunities supporting meaningful and explicit learning. In particular, before presenting corresponding practice problems, Chinese textbooks contextualized worked examples of inverse relations in real-world situations to aid in sense making of computational or checking procedures. The Chinese worked examples also differed in representation uses especially through concreteness fading. Finally, the Chinese textbooks spaced learning over time, systematically stressing structural relations including the inverse quantities relationships. These findings shed light on ways to support students’ meaningful and explicit learning of fundamental mathematical ideas in elementary school.     

Preservice mathematics teachers' conceptions and enactments of modeling standards

Mathematical modeling has been highlighted recently as Common Core State Standards for Mathematics (CCSSM) included Model with Mathematics as one of the Standards for Mathematical Practices (SMP) and a modeling strand in the high school standards. This common aspect of standards across most states in the United States intended by CCSSM authors and policy makers seems to mitigate the diverse notions of mathematical modeling. When we observed secondary mathematics preservice teachers (M‐PSTs) who learned about the SMP and used CCSSM modeling standards to plan and enact lessons, however, we noted differences in their interpretations and enactments of the standards, despite their attendance in the same course sections during a teacher preparation program. This result led us to investigate the ways the M‐PSTs understood modeling standards, which could provide insights into better preparing teachers to teach mathematical modeling. We present the contrasting ways in which M‐PSTs presented modeling related to their conceptions of mathematical modeling, choices of problems, and enactments over an academic year, connecting their practices to extant research. We consider this teaching and research experience as an opportunity to make significant changes in our instruction that may result in our students enhanced implementation of mathematical modeling.     

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.     

A Longitudinal Study of Middle and Secondary Level Science Textbook Vocabulary Loads

Middle and secondary science textbooks have long been a primary support for instruction, often leading to heavy emphasis on domain‐specific vocabulary. A longitudinal study was conducted to compare vocabulary loads of science textbooks published since 2010 to results of two previous studies going back to 1983. In each study, textbooks chosen represent convenience samples of commonly used middle and secondary science textbooks. Vocabulary emphasis in science textbooks remains high, and may present a problem for struggling readers and English language learners (ELL). Some vocabulary loads were found to approach or even exceed recommended limits for vocabulary instruction in modern foreign language courses. Also, science vocabulary can present an additional challenge for native English speakers and ELLs since many terms have no common colloquial English or native language equivalents. Recommendations for how to promote effective science vocabulary development are provided. Overall, over three decades there has been little change in heavy emphasis on vocabulary in middle and secondary science textbooks.     

Comparing the development of the multiplication of fractions in Turkish and American textbooks

This study analyzed the methods used to teach the multiplication of fractions in Turkish and American textbooks. Two Turkish textbooks and two American textbooks, Everyday Mathematics (EM) and Connected Mathematics 3 (CM), were analyzed. The analyses focused on the content and the nature of the mathematical problems presented in the textbooks. The findings of the study showed that the American textbooks aimed at developing conceptual understanding first and then procedural fluency, whereas the Turkish textbooks aimed at developing both concurrently. The American textbooks provided more opportunities for different computational strategies. The solutions to most problems in all textbooks required a single computational step, a numerical answer, and procedural knowledge. Furthermore, compared with the Turkish textbooks, the American textbooks contained a greater number of problems that required high-level cognitive skills such as mathematical reasoning.     

Approximate and efficient calculation of dominant Lyapunov exponents of high-dimensional nonlinear dynamic systems

This short communication presents an efficient method for calculating dominant Lyapunov exponents (LEs) of high-dimensional nonlinear dynamic systems based on their reduced-order models obtained from the linear model reduction theory. Mathematical derivation shows that the LEs of the reduced-order models correspond to the dominant LEs of the original systems. Two numerical examples are provided to demonstrate the effectiveness of the method.     

On the incomplete block designs derivable from the association schemes

Blackwelder (1969) has given two methods of constructing balanced incomplete block (BIB) designs from the association matrices of association schemes with two and three associate classes. In this note these two methods are incorporated in a general method, and the existence of a series of BIB designs is shown by the generalized method. In addition, a remark about partially balanced incomplete block (PBIB) designs with respect to the method is made.