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For children with persistent mathematics difficulties, research and practice espouses that an altered kind of mathematics instruction is necessary due to sustained performance differences. Yet, a critical issue in mathematics education rests in the question of why research locates the problem within these children. In this paper, we challenge a longstanding assumption about the type of mathematics children with low achievement in mathematics "need" along with how these children are positioned in terms of mathematical thinking and reasoning. Our aim in this work is to identify ways of reasoning evident in the partitioning activity of 43 fifth-grade children as they solved equal sharing situations independent of instruction over ten sessions. Results reveal three themes of reasoning that show a resemblance between these children's reasoning and existing frameworks of reasoning in equal sharing problems found in prior research among children who did not show low achievement in mathematics. We discuss the results in terms of the problem of a continued conceptualization of low achieving students' need for specific kinds of teaching and learning experiences and/or detached instructional experiences in school. We advocate for an increase in research that examines how teachers can support participation of these children in mathematics classrooms such that children might develop powerful mathematics conceptions.

The purpose of this study was to explore the relationship between instructional modality used for teaching fractions and third- and fourth-grade students' responses and strategies to open-response fraction items. The participants were 155 third-grade and 200 fourth-grade students from 17 public school classrooms. Students within each class were randomly assigned to two instructional treatment groups: a virtual manipulatives representations (VMR) instruction group and a physical manipulatives and textbook representations (PMTR) instruction group. A conversion mixed methods analysis was used to examine quantitative and qualitative data. The quantitative analysis showed achievement outcomes were the same for both groups. The qualitative analysis revealed shifts in learning that were otherwise hidden with solely quantitative achievement results. Specifically, the results indicated VMR group success in understanding fractions as relationships and PMTR group success in maintaining conceptualization of the whole. Overall, the results of this study corroborate previous research indicating the importance of both types of instructional modalities, showing that virtual manipulatives and physical manipulatives are effective instructional tools with positive effects on student learning. The study expands existing research by offering an opportunity to explore the nuances of students' fractions understanding and provide a window into students' shifts in fraction learning.

The purpose of this study was to determine variables that predict performance when virtual manipulatives are used for mathematics instruction. This study used a quasi-experimental design. This design was used to determine variables that predict student performance on tests of fraction knowledge for third- and fourth-grade students in two treatment groups: classroom instruction using texts and physical manipulatives (CI), and computer lab instruction using virtual fraction applets (VM). The Pre-test, Post-test 1, and Post-test 2 measured learning and retention of fraction concepts. Observation ethograms documented representation use. The results revealed that fewer demographic predictors of student performance (e.g., socio-economic status, English language learner status, and gender) exist during fraction instruction when virtual manipulatives were used. When instructors used virtual manipulatives, there was an equalizing effect on achievement in third and fourth grade classrooms, in that fewer demographic factors were influential for VM groups compared to CI groups.

Elementary pre-service teachers report high levels of mathematics anxiety (MA), but the construct less widely addressed is their mathematics teaching anxiety (MTA). This study investigated the frequency with which MA stemming from prior experiences leads to MTA. Fifty-three elementary pre-service teachers' written reflections were analyzed, using a Framework for MA and MTA. The results showed that one-third of the pre-service teachers reported high prior mathematics anxiety but did not experience MTA, suggesting the relationship between MA and MTA is not always linked. Teacher educators must address this relationship to support successful mathematics teaching experiences for pre-service teachers. (Contains 1 table and 1 figure.)

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.

The purpose of this study was to explore relationships between app affordances and user abilities in second graders' interactions with mathematics virtual manipulative touchscreen tablet apps. The research questions focused on varying manifestations of affordance-ability relationships during children's interactions with mathematics virtual manipulative touchscreen tablet apps. Researchers qualitatively analyzed video recordings and ethograms from clinical interviews of 33 second-grade children. Each 45-min clinical interview involved one child interacting with two sequences of mathematics virtual manipulative touchscreen tablet apps: one sequence focusing on place value concepts and the other sequence focusing on skip counting concepts. Results provided evidence of Moyer-Packenham and Westenskow's ("Int J Virtual Pers Learn Environ" 4(3):35-50, 2013) five affordance categories of virtual manipulatives. Approach to and degree of affordance access varied depending on a child's corresponding ability, and some children modified their affordance access as their ability changed. Results also indicated that outcomes of accessing an affordance also related to a child's ability. Context also influenced affordance access. These results imply that affordance-ability relationships are multifaceted. Overall, these results imply that is important to consider affordance-ability relationships in relation to mathematics education technology.

This paper focuses on understanding the role that affordances played in children's learning performance and efficiency during clinical interviews of their interactions with mathematics apps on touch-screen devices. One hundred children, ages 3 to 8, each used six different virtual manipulative mathematics apps during 30-40-min interviews. The study used a convergent mixed methods design, in which quantitative and qualitative data were collected concurrently to answer the research questions (Creswell and Plano Clark 2011). Videos were used to capture each child's interactions with the virtual manipulative mathematics apps, document learning performance and efficiency, and record children's interactions with the affordances within the apps. Quantitized video data answered the research question on differences in children's learning performance and efficiency between pre- and post-assessments. A Wilcoxon matched pairs signed-rank test was used to explore these data. Qualitative video data was used to identify affordance access by children when using each app, identifying 95 potential helping and hindering affordances among the 18 apps. The results showed that there were changes in children's learning performance and efficiency when children accessed a helping or a hindering affordance. Helping affordances were more likely to be accessed by children who progressed between the pre- and post-assessments, and the same affordances had helping and hindering effects for different children. These results have important implications for the design of virtual manipulative mathematics learning apps.

The purpose of this study was to identify and describe students' use of number sense as they solved story problem tasks. Three 8- and 9-year-old students participated in clinical interviews. Through a process of holistic and qualitative coding, researchers used the number sense view as a theoretical framework for exploring how students' number sense was reflected in their numerical reasoning and verbal explanations of strategies for solving story problems. The results suggest that students' coordination of number sense constructs affords flexibility in problem-solving and deeper understanding of place value. Implications for classroom practice and mathematics education research are discussed.

The literature seems limited in what is known about conceptual processes that underlie evolution of students with learning disabilities (SLD) conceptions of fractions. This exploratory study examines how a foundational scheme of unit fractions (1/n) may evolve through the mathematical activity of two fifth grade girls. We analyze data segments from episodes conducted during a teaching experiment grounded in the activity of iterating estimates of one person's equal share. Our findings include four distinct conceptual stages: (1) No Conception of the Nature of Adjustment to the Magnitude of a Unit Fraction, (2) Evolving Anticipation of the Nature of Adjustment but not of its Relative Amount, (3) Anticipation of the Nature of Adjustment with an Evolving Partial Amount, and (4) a Dual Anticipation of the Nature and Amount of Adjustment. Findings demonstrate each girl was able to use her constructed scheme to successfully solve and reason about novel problems. We discuss the need for more research to confirm the findings from this study, while offering a conjecture of the possibilities for more SLDs to advance their conceptions of fractions in future interventions.

This research study examined the perceptions of 24 parents of rising 5th-grade students with mathematics learning difficulties as part of a 10-week summer mathematics tutoring experience. During the summer tutoring program, parents observed their children participating in mathematics learning experiences during one-to-one tutoring sessions. At the conclusion of the summer tutoring services, parents reflected on observations of their own children in surveys and interviews. Results indicated that parents' observations of the tutoring impacted their perceptions and changed the support they provided to their children during and after the tutoring program. Parents reported changes in attitudes and beliefs about mathematics, gaining insights into their child's mathematical understanding, and changes in their methods and practices as mathematics supporters outside of the tutoring sessions.

The purpose of this study was to deconstruct the relationship between visual static models and students' written solutions to fraction problems using a large sample of students' solutions. Participants in the study included 162 third-grade and 209 fourth-grade students from 17 different classrooms. Students' written responses to open-ended tasks were examined to determine common solutions and errors when using visual static models. The results indicate that (a) common students errors relate to how students interpret the given model or their own model of the situation, and (b) students' flexibility with visual static models is related to successful written solutions. Students with errors generally demonstrated a lack of flexibility in interpreting their own and the given visual static models. Researchers hypothesize that students' exposure to varied mathematical representations influences their ability to flexibly use static visual representations. They recommend that students have a solid understanding of real-world mathematics situations in order to successfully create and interpret visual static models of mathematics.

The study reported here examined virtual manipulatives as an instructional treatment in 17 third- and fourth-grade classrooms. Students were randomly assigned to two treatment groups: texts and physical manipulatives, and virtual manipulatives. Results revealed no significant differences in achievement between the treatments. Additional results showed that objective ability predicted fraction achievement; virtual manipulative use can be modulated by test question type (e.g., symbolic vs. pictorial); percentage of class time using representations differed between VM and PM classrooms; and percentage of class time spent using representation types differed, potentially providing differential opportunities to learn.

This paper is a meta-analysis that synthesizes the findings from 66 research reports examining the effects of virtual manipulatives on student achievement. Of the 66 reports, 32 contained data yielding 82 effect size scores with effects of virtual manipulatives on student achievement. The 66 reports also contributed to a conceptual analysis of affordancesthat promote mathematical learning. The results of the averaged effect size scores yielded a moderate effect for virtual manipulatives comparedwith other instructional treatments. There were additional large, moderate, and small effects when virtual manipulatives were compared with physical manipulatives and textbook instruction, and when the effects were examined by mathematical domains, grade levels, and study duration. The results of the conceptual analysis revealed empirical evidence that five specific interrelated affordances of virtual manipulatives (VMs) promoted mathematical learning. These five specific researcher-reported affordances included focused constraint (i.e., VMs focus and constrain student attention on mathematical objects and processes), creative variation (i.e., VMs encourage creativity and increase the variety of students'solutions), simultaneous linking (i.e., VMs simultaneously link representations with each other and with students' actions), efficient precision (i.e., VMs contain precise representations allowing accurate and efficient use), and motivation (i.e., VMs motivate students to persist at mathematical tasks)

Intense focus on student achievement results in mathematics and science has brought about claims that K-12 teachers should be better prepared to teach basic concepts in these disciplines. The focus on teachers' mathematics and science content knowledge has been met by efforts to increase teacher knowledge through funded national initiatives focusing on mathematics and science. The purpose of the present study was to look across projects in the National Science Foundation's Math and Science Partnership Program to determine how partnerships developed processes for measuring growth in teacher content knowledge. Pre- and post-testing was the most common process for measuring growth in content knowledge, with 63% of the mathematics and 78% of the science teachers showing significant gains in content knowledge. A notable difference was found between teacher outcomes when the Learning Mathematics for Teaching instrument was used in comparison with the use of other instruments measuring teacher content knowledge growth. Results revealed two pathways for promoting teacher content knowledge growth: content explicit, where the goal of growth in teacher content knowledge was explicit in the activity, and content embedded, where the goal of growth in teacher content knowledge was embedded in the activity. As a result of the analysis, a framework demonstrating the interrelationships among processes and pathways was developed.

Over the last 50 years, there has been ample research dedicated to mathematics anxiety in contexts of teaching and learning. However, there has been less attention focused on pre-service teachers' anxieties about teaching mathematics in classroom settings. This study analyzed pre-service teachers' reflections at the conclusion of an elementary mathematics field experience in order to determine common themes surrounding anxiety-related events based on mathematics practice-teaching experiences. Through qualitative analysis of pre-service teachers' reflections using open and axial coding, three categories and ten themes surrounding elementary pre-service teachers' anxiety-related events based on mathematics practice-teaching experiences emerged. Some of the themes presented were ones that increased pre-service teachers' anxieties for teaching mathematics, whereas other themes reflected aspects of the field experience that decreased anxiety for teaching mathematics. Based on the themes revealed in this study, suggestions for strategies and materials that could be developed for university mathematics methods courses are discussed. Novel perspectives for considering mathematics teaching anxiety, such as locus of control and future time perspective, are shared. (Contains 1 table and 1 figure.)

No person, place, or thing can capture the attention of a class of sixth graders like "man's best friend." To prompt students' interest in a series of lessons on proportional relationships, the authors brought in a unique teaching aid--a dog. A family dog was used to supply the measurements for scatter plots and variables so that students could explore relationships among data sets--not to mention paws and tails. Students used dog data and scatter plot applets to explore various relationships in a lesson called Canine Conjectures. A visit by a live dog provided the motivation, while electronic technologies allowed data to be efficiently processed, freeing students to focus on reasoning, observing patterns, decision making, and problem solving. (Contains 5 figures.)

Elementary pre-service teachers report high levels of mathematics anxiety (MA), but the construct less widely addressed is their mathematics teaching anxiety (MTA). This study investigated the frequency with which MA stemming from prior experiences leads to MTA. Fiftythree elementary pre-service teachers' written reflections were analyzed, using a Framework for MA and MTA. The results showed that one-third of the pre-service teachers reported high prior mathematics anxiety but did not experience MTA, suggesting the relationship between MA and MTA is not always linked. Teacher educators must address this relationship to support successful mathematics teaching experiences for pre-service teachers. [ABSTRACT FROM AUTHOR]