A Learning Progression for Elementary Students’ Functional Thinking

In this article we advance characterizations of and supports for elementary students’ progress in generalizing and representing functional relationships as part of a comprehensive approach to early algebra. Our learning progressions approach to early algebra research involves the coordination of a curricular framework and progression, an instructional sequence, written assessments, and levels of sophistication describing students’ algebraic thinking. After detailing this approach, we focus on what we have learned about the development of students’ abilities to generalize and represent functional relationships in a grades 3–5 early algebra intervention by sharing the levels of responses we observed in students’ written work over time. We found that the sophistication of students’ responses increased over the course of the intervention from recursive patterning to correspondence and in some cases covariation relationships between variables. Students’ responses at times differed by the particular tasks that were posed. We discuss implications for research and practice.     

Planning District‐Wide Professional Development: Insights Gained From Teachers and Students Regarding Mathematics Teaching in a Large Urban District

This paper describes the mechanism used to gain insights into the state of the art of mathematics instruction in a large urban district in order to design meaningful professional development for the teachers in the district. Surveys of close to 2,000 elementary, middle school, and high school students were collected in order to assess the instructional practices used in mathematics classes across the district. Students were questioned about the frequency of use of various instructional practices that support the meaningful learning of mathematics. These included practices such as problem solving, use of calculators and computers, group work, homework, discussions, and projects, among others. Responses were analyzed and comparisons were drawn between elementary and middle school students' responses and between middle school and high school responses. Finally, fifth‐grade student responses were compared to those of their teachers. Student responses indicated that they had fewer inquiry‐based experiences, fewer student‐to‐student interactions, and fewer opportunities to defend their answers and justify their thinking as they moved from elementary to middle school to high school. In the elementary grades students reported an overemphasis on the use of memorization of facts and procedures and sparse use of calculators. Results were interpreted and specific directions for professional development, as reported in this paper, were drawn from these data. The paper illustrates how student surveys can inform the design of professional development experiences for the teachers in a district.     

A third grader's way of thinking about linear function tables

This paper is inscribed within the research effort to produce evidence regarding primary school students’ learning of algebra. Given the results obtained so far in the research community, we are convinced that young elementary school students can successfully learn algebra. Moreover, children this young can make use of different representational systems, including function tables, algebraic notation, and graphs in the Cartesian coordinate grid. In our research, we introduce algebra from a functional perspective. A functional perspective moves away from the mere symbolic manipulation of equations and focuses on relationships between variables. In investigating the processes of teaching and learning algebra at this age, we are interested in identifying meaningful teaching situations. Within each type of teaching situation, we focus on what kind of knowledge students produce, what are the main obstacles they find in their learning, as well as the intermediate states of knowledge between what they know and the target knowledge for the teaching situation. In this paper, we present a case study focusing on the approach adopted by a third grade student, Marisa, when she was producing the formula for a linear function while she was working with the information of a problem displayed in a function table containing pairs of inputs-outputs. We will frame the analysis and discussion on Marisa's approach in terms of the concept of theorem-in action (Vergnaud, 1982) and we will contrast it with the scalar and functional approaches introduced by Vergnaud (1988) in his Theory of Multiplicative Fields. The approach adopted by Marisa turns out to have both scalar and functional aspects to it, providing us with new ways of thinking of children's potential responses to functions.     

Investigating functional thinking in the elementary classroom: Foundations of early algebraic reasoning

This paper examines the development of student functional thinking during a teaching experiment that was conducted in two classrooms with a total of 45 children whose average age was nine years and six months. The teaching comprised four lessons taught by a researcher, with a second researcher and classroom teacher acting as participant observers. These lessons were designed to enable students to build mental representations in order to explore the use of function tables by focusing on the relationship between input and output numbers with the intention of extracting the algebraic nature of the arithmetic involved. All lessons were videotaped. The results indicate that elementary students are not only capable of developing functional thinking but also of communicating their thinking both verbally and symbolically.     

Second graders articulating ideas about linear functional relationships

In this paper, we explore the ideas that second grade students articulate about functional relationships. We adopt a function-based approach to introduce elementary school children to algebraic content. We present results from a design-based research study carried out with 21 second-grade students (approximately 7 years of age). We focus on a lesson from our classroom teaching experiment in which the students were working on a problem that involved a linear functional relationship (y = 2x). From the analysis of students’ written work and classroom video, we illustrate two different approaches that students adopt to express the relationship between two quantities. Students show fluency recontextualizing the problem posed, moving between extra-mathematical and intra-mathematical contexts.     

Singaporean students' mathematical thinking in problem solving and problem posing: an exploratory study

This study explored Singaporean fourth, fifth, and sixth grade students' mathematical thinking in problem solving and problem posing. The results of this study showed that the majority of Singaporean fourth, fifth, and sixth graders are able to select appropriate solution strategies to solve these problems, and choose appropriate solution representations to clearly communicate their solution processes. Most Singaporean students are able to pose problems beyond the initial figures in the pattern. The results of this study also showed that across the four tasks, as the grade level advances, a higher percentage of students in that grade level show evidence of having correct answers. Surprisingly, the overall statistically significant differences across the three grade levels are mainly due to statistically significant differences between fourth and fifth grade students. Between fifth and sixth grade students, there are no statistically significant differences in most of the analyses. Compared to the findings concerning US and Chinese students' mathematical thinking, Singaporean students seem to be much more similar to Chinese students than to US students.     

A Multi‐Method Investigation of Mathematics Motivation for Elementary Age Students

This paper presents the results of a multi‐method study examining elementary students with high self‐reported levels of mathematics motivation. Second‐ through fifth‐grade students at a Title One school in the southeastern United States completed the Elementary Mathematics Motivation Instrument (EMMI), which examines levels of mathematics motivation across three subscales: (a) Math Anxiety, (b) Self‐Efficacy, and (c) Value of Math. Results from this quantitative phase were used to identify a sample for a qualitative phase examining how students who report high levels of motivation perceive mathematics. The resulting qualitative phase utilized a phenomenological design to explore mathematics motivation for a particular set of students in a fifth‐grade setting. Findings indicate that elementary students with high mathematics motivation value mathematics as a present and future oriented discipline and value teachers that deemphasize testing as a measure of success.     

Student Initiatives in Urban Elementary Science Classrooms

Student initiatives play an important role in inquiry‐based science with all students, including English language learning (ELL) students. This study examined initiatives that elementary students made as they participated in an intervention to promote science learning and English language development over a three‐year period. In addition, the study examined whether student initiatives were related to other domains of classroom practices. The study involved 70 third‐, fourth‐, and fifth‐grade classrooms with ELL students in six urban elementary schools. Results indicated that students generally made few, low‐quality initiatives. Student initiatives were generally not related to the other domains of classroom practices for grades 3 and 4, whereas initiatives were significantly related to almost all the other domains for grade 5. These results contribute to the knowledge base for fostering ELL students' initiatives in science classrooms.     

Preservice Teacher Beliefs About Proofs

This article describes a study of backgrounds, beliefs, and attitudes of teachers about proofs. Thirty preservice elementary teachers enrolled in a mathematics content course and 21 secondary mathematics teachers in an abstract algebra course were surveyed. The study explored four issues: preservice teachers' experiences/exposure to proof, their beliefs about what constitutes a proof and the role of proof in mathematics, and their beliefs about when proof should be introduced in grades K-12. Results of the survey are described as a means for discussing the backgrounds and beliefs future teachers hold with regard to teaching proofs in their own classrooms. Finally, a short collection of sample explorations and questions, which could be used to encourage the thinking and writing of proofs in grades K-12, is provided. One of these questions was posed to 215 secondary students; examples of their reasoning and a discussion of the various techniques employed by the students are included.     

School mathematics and creativity at the elementary and middle-grade levels: how are they related?

In this study, we evaluated students’ creativity, as expressed in the solution methods of three problems for groups of students in different grades. Posing the same problems to students of similar (advanced) mathematical abilities in different grades allowed us to look for possible connections between creativity and mathematical knowledge. The findings indicate that at the elementary school level, the number of solution methods and creativity scores increased with age. The collective methods space of the eighth graders seemed to narrow almost exclusively to algebraic methods, but the increase in the number of solutions was renewed in the ninth grade.     

Fractions as Subtraction: An Activity‐Oriented Perspective from Elementary Children

A sample of third‐, fourth‐, and fifth‐grade student responses to the question “What is a fraction?” were examined to gain an understanding of how children in upper elementary grades make sense of fractions. Rather than measure children's understanding of fractions relative to mathematically conventional part–whole constructions of fractions, we attempted to understand children's actions and processes. A small but nontrivial group of children used subtraction (takeaway and removal) as a framework for understanding how fractions were created and written. An analysis of the content of their responses as well as a comparison of the performance of these children with that of children who used other ways of describing fractions suggests that the use of subtraction may be a reasonable (or at least not harmful) way for children to begin to access concepts related to fractions. Also, this study suggests that attention to children's understanding through the lens of children's activity might reveal ways of thinking and insights that are masked when we compare children's thinking in more structured research settings.     

Additive relations and function tables

We present work with a second grade classroom where we carried out a teaching experiment that attempted to bring out the algebraic character of arithmetic. In this paper, we specifically illustrate our work with the second graders on additive relations, through the children’s work with function tables. We explore the different ways in which the children represented the information of a problem in the form of a self-designed function table. We argue that the choices children make about the kind of information to represent or not, as well as the way in which they constructed their tables, highlight some of the issues that children may find relevant in their construction of function tables. This open-ended format pointed to how they were understanding and appropriating tables into their thinking about additive relations.     

Teaching Algebraic Expressions to Young Students: The Three‐day Journey of “a+ 2”

This classroom scholarship report is based on the teaching experience using Davydov's mathematics curriculum, which was developed in the former Soviet Union. While “from arithmetic to algebra” is the normally accepted instructional sequence in school mathematics, Davydov's curriculum is laid out “from algebra to arithmetic,” focusing on algebraic thinking from the very beginning of the elementary grades. The purpose of this report is not to provide a definitive conclusion about which curriculum or sequence is better nor to address which instructional strategy is right in all circumstances. Rather, it is to explore how primary grade students develop their own conceptual understanding while confronting difficulties met within a specific context. This report provides actual classroom episodes from working with a group of first graders and describes dynamic interactions between the teacher and children while they discuss the use of algebraic expressions and understand the meaning behind them.     

Assessments accompanying published textbooks: the extent to which mathematical processes are evident

Assessments accompanying published textbooks are often used by teachers in the USA as a primary means to evaluate students’ mathematical knowledge. In addition to assessing content knowledge, assessments should provide insight into students’ ability to engage with mathematical processes such as reasoning, communication, connections, and representations. We report here an analysis of the extent to which the assessments accompanying published textbooks in the USA at the elementary, middle grades, and high school levels provide opportunities for students to engage with these mathematical processes. Results indicate that in elementary grades, communication, connections, and graphics are not consistently emphasized across grade levels and publishers. In middle grades, students are rarely asked to record their reasoning or translate among representational forms of a concept. In high school geometry, students are given many opportunities to interpret and create graphics, but the same is not true for algebra. With the exception of connections, the results suggest that inconsistent emphasis is placed on the mathematical processes within assessments accompanying commercial textbooks in the USA.     

Student Competence in Understanding the Matter Concept and Its Implications for Science Curriculum Standards

Using the US national sample from the 1995 Third International Mathematics and Science Study (TIMSS), this study examined students' competence levels in understanding the matter concept at grades 3, 4, 7, 8 and high school graduation, and compared them to the expectations in the US national science education standards. It was found that third‐grade students were developing understanding on mixtures, and fourth‐grade students were developing understanding on separating mixtures; seventh‐ and eighth‐grade students were only at the beginning level of differentiating chemical properties from physical properties; they were not ready for the particulate model of chemical change. High school physical science specialization students were still at the developing level of understanding kinetic and atomic models of chemical and physical changes; they may not be able to master those theories. The findings suggest that the Benchmarks for Science Literacy and Atlas for Science Literacy may have overestimated the competences of elementary, middle school, and high school students.     

Exploring the Use of New Representations as a Resource for Teacher Learning

An important goal of mathematics education reform is to support teacher learning. Toward this end, researchers and teacher educators have investigated ways in which teachers learn about mathematical content, pedagogical strategies, and student thinking as they implement reform. This study extends such work by examining how one elementary school and one high school teacher learned from students' interpretations of new conceptually based representations contained in instructional materials aligned with the Principles and Standards for School Mathematics ( National Council of Teachers of Mathematics, 2000 ). Results indicated that teaching with new representations provided a rich context for teacher learning at both the elementary and high school level, and three dimensions were identified along which such learning occurred. The results suggest that pedagogical content knowledge with respect to representations is an important facet of teacher cognition that should be studied in greater depth.     

The use of interactive visualizations to foster the understanding of concepts of calculus: design principles and empirical results

Calculus and functional thinking are closely related. Functional thinking includes thinking in variations and functional dependencies with a strong emphasis on the aspect of change. Calculus is a climax within school mathematics and the education to functional thinking can be seen as propaedeutics to it. Many authors describe that functions at school are mainly treated in a static way, by regarding them as pointwise relations. This often leads to the underrepresentation of the aspect of change at school. Moreover, calculus at school is mainly procedure-oriented and structural understanding is lacking. In this work, two specially designed interactive activities for the teaching and learning of concepts of calculus based on dynamic geometry software are presented. They accentuate the aspect of change and the object aspect of functions using a double stage visualization. Moreover, the activities allow the discovery and exploration of some concepts of calculus in a qualitative-structural way without knowing or using curve-sketching routines. The activities were used in a qualitative study with 10th grade students of age 15–16 in secondary schools in Berlin, Germany. Some pairs of students were videotaped while working with the activities. After transcribing, the interactions of the students were interpreted and analyzed focusing to the use of the computer. The results show how the students mobilize their knowledge about functions working on the given tasks, and using the activities to formulate important concepts of calculus in a qualitative way. Also, some important epistemological obstacles can be detected.     

Understanding STEM‐focused elementary schools: Case study of Walter Bracken STEAM Academy

Increasingly, STEM focused high schools are used prepare students for college STEM majors and launch them into STEM careers. Yet a new focus on STEM education at the elementary levels suggests that the importance of STEM education is much broader than a preparation for workforce needs in high school or college. This paper describes a case study designed to articulate the mission and design of an effective and nationally recognized STEM‐focused elementary school. As described through the six most impactful components of STEM‐focused elementary school design at Walter Bracken STEAM Academy, the case study emphasizes the school's strong and inclusive school leadership, with staff organized into grade level groups empowered to innovate and honing their teaching practices. External partnerships are leveraged to broaden student learning opportunities. Students at Bracken engage in active learning opportunities and multidisciplinary lessons where STEM is used as a way of thinking and as a way to coherently combine content into active learning opportunities that are engaging for learners. By organizing the structural components of an exemplary STEM‐focused elementary school, we hope to deliver actionable reforms for elementary schools wanting to increase their STEM‐focused offerings.     

Concept Development of Decimals in Chinese Elementary Students: A Conceptual Change Approach

The aim of this study was to examine the concept development of decimal numbers in 244 Chinese elementary students in grades 4–6. Three grades of students differed in their intuitive sense of decimals and conceptual understanding of decimals, with more strategic approaches used by older students. Misconceptions regarding the density nature of decimals indicated the progress in an ascending spiral trend (i.e., fourth graders performed the worst; fifth graders performed the best; and sixth graders regressed slightly), not in a linear trend. Misconceptions regarding decimal computation (i.e., multiplication makes bigger) generally decreased across grades. However, children's misconceptions regarding the density and infinity features of decimals appeared to be more persistent than misconceptions regarding decimal computation. Some students in higher grades continued to use the discreteness feature of whole numbers to explain the distance between two decimal numbers, indicating an intermediate level of understanding decimals. The findings revealed the effect of symbolic representation of interval end points and students' responses were contingent on the actual representations of interval end points. Students in all three grades demonstrated narrowed application of decimal values (e.g., merchandise), and their application of decimals was largely limited by their learning experiences.     

The Cumulative and Residual Impact of a Systemic Reform Program on Teacher Change and Student Learning of Science

This longitudinal, five‐year study of teachers and students who had participated in a systemic reform program in science explored if (1) teacher change in practice realized during a three‐year program is sustained one, two, and three years following the program, (2) student performance on state science assessments two years following studying with teachers at this school still demonstrated significant differences from students who attended the control school, and (3) student performance continued to be enhanced for both White and Minority students. Student achievement was assessed using the Discovery Inquiry Test in Science during sixth through eighth grades and the Ohio Graduation Test was used in 10th grade. The same students completed the test in grades 6–8 and 10th grade. Students from the Program school significantly outperformed students who attended the control school on the 10th grade state assessment in science. Findings in this study revealed the ability for sustained, whole‐school, professional development programs to have a cumulative and residual impact on teacher change and student learning of science.