Understanding and representing the arithmetic averaging algorithm: an analysis and comparison of US and Chinese students' responses

Analysing the responses of 311 sixth-grade Chinese students and 232 sixth-grade US students to two problems involving arithmetic average, this study explored students' understanding and representation of the averaging algorithm from a cross-national perspective. Results of the study show that Chinese students were more successful than US students in obtaining correct numerical answers to each of the problems, but US and Chinese students had similar cognitive difficulties in solving the second task. The difficulties were not due to their lack of procedural knowledge of the averaging algorithm, rather due to their lack of conceptual understanding of the algorithm. There were significant differences between the US and Chinese students in their solution representations of the two average problems. Chinese students were more likely to use algebraic representations than US students; while US students were more likely to use pictorial or verbal representations. US and Chinese students' use of representations are related to their mathematical problem-solving performance. Students who used more advanced representations were better problem solvers. The findings of the study suggest that Chinese students' superior performance on the averaging problems is partly due to their use of advanced representations (e.g. algebraic).     

Instructional Supports for Representational Fluency in Solving Linear Equations with Computer Algebra Systems and Paper‐and‐Pencil

This research addresses the issue of how to support students' representational fluency—the ability to create, move within, translate across, and derive meaning from external representations of mathematical ideas. The context of solving linear equations in a combined computer algebra system (CAS) and paper‐and‐pencil classroom environment is targeted as a rich and pressing context to study this issue. We report results of a collaborative teaching experiment in which we designed for and tested a functions approach to solving equations with ninth‐grade algebra students, and link to results of semi‐structured interviews with students before and after the experiment. Results of analyzing the five‐week experiment include instructional supports for students' representational fluency in solving linear equations: (a) sequencing the use of graphs, tables, and CAS feedback prior to formal symbolic transpositions, (b) connecting solutions to equations across representations, and (c) encouraging understanding of equations as equivalence relations that are sometimes, always, or never true. While some students' change in sophistication of representational fluency helps substantiate the productive nature of these supports, other students' persistent struggles raise questions of how to address the diverse needs of learners in complex learning environments involving multiple tool‐based representations.     

Mathematical Dimensions of Students' Use of Proportional Reasoning In High School Physics

This study examines the mathematical processes used by students when solving physics tasks requiring proportional reasoning. The study investigates students' understanding and explanations of their mathematical processes. A qualitative and interpretive case study was conducted with 6 students from a coeducational urban high school for 5 months. Students were engaged with some high school physics tasks requiring proportional reasoning, during which a hermeneutic dialectic design was used to investigate their processes, understandings, and difficulties. Research techniques such as interviews, dialectical discourses, journal dialogue, and video and audio recordings were employed to generate, analyze, and interpret data. Results of the study indicate that the students employed mathematical proportional reasoning patterns and algorithms which they could not explain. Students also had difficulties translating physics tasks into mathematical statements, symbols, and relations. Students could not perform mathematical operations that were not directly obvious from the physics tasks, and some had difficulty with division. Students did not have adequate understanding of the mathematical processes involved in proportional reasoning.     

Mathematical Thinking Involved in U.S. and Chinese Students' Solving of Process-Constrained and Process-Open Problems

This study examined U.S. and Chinese 6th-grade students' mathematical thinking and reasoning involved in solving 6 process-constrained and 6 process-open problems. The Chinese sample (from Guiyang, Guizhou) had a significantly higher mean score than the U.S. sample (from Milwaukee, Wisconsin) on the process-constrained tasks, but the sample of U.S. students had a significantly higher mean score than the sample of the Chinese students on the process-open tasks. A qualitative analysis of students' responses was conducted to understand the mathematical thinking and reasoning involved in solving these problems. The qualitative results indicate that the Chinese sample preferred to use routine algorithms and symbolic representations, whereas the U.S. sample preferred to use concrete visual representations. Such a qualitative analysis of students' responses provided insights into U.S. and Chinese students' mathematical thinking, thereby facilitating interpretation of the cross-national differences in solving the process-constrained and process-open problems.     

How Students Use Physics to Reason About Calculus Tasks

The present research study investigates how undergraduate students in an integrated calculus and physics class use physics to help them solve calculus problems. Using Zandieh's (2000) framework for analyzing student understanding of derivative as a starting point, this study adds detail to her “paradigmatic physical” context and begins to address the need for a theoretical basis for investigating learning and teaching in integrated mathematics and science classrooms. A case study design was used to investigate the different ways students use physics ideas as they worked through calculus tasks. Data were gathered through four individual interviews with each of 8 ICP students, classroom participant‐observation, and triangulation of the data through student homework and exams. The main result of this study is the Physics Use Classification Scheme, a tool consisting of four categories used to characterize students' uses of physics on tasks involving average rate of change, derivative, and integral concepts. Two of the categories from the Physics Use Classification Scheme are elucidated with contrasting student cases in this paper.     

How can self-regulated learning support the problem solving of third-grade students with mathematics anxiety?

The study compares 140 third-grade Israeli students (lower and higher achievers) who were either exposed to self-regulated learning (SRL) supported by metacognitive questioning (the MS group) or received no direct SRL support (the N_MS group). We investigated: (a) mathematical problem solving performance; (b) metacognitive strategy use in three phases of the problem-solving process; and (c) mathematics anxiety. Findings indicated that the MS students showed greater gains in mathematical problem solving performance than the N_MS students. They reported using metacognitive strategies more often, and showed a greater reduction in anxiety. In particular, the lower MS achievers showed these gains in the basic and complex tasks, in strategy use during the on-action phase of the problem solving process and a decrease in negative thoughts. The higher achievers showed greater improvement in transfer tasks and an increase in positive thoughts towards mathematics. Both the theoretical and practical implications of this study are discussed.     

How Students “Unpack” the Structure of a Word Problem: Graphic Representations and Problem Solving

This research investigated how fourth and fifth grade students spontaneously ‘unpacked’ a word problem when generating a graphic representation to aid in problem solution. Relationships among the type of graphic representation produced, spatial visualization, drawing ability, gender, and problem solving also were examined and described. Instrumentation developed for the study included several math challenge tasks, a spatial visualization task, and a drawing task. For one of the math challenge tasks, students were instructed to draw a picture to assist them with problem solution. These graphic representations generated by students were rated as pictorial or as displaying some level of schematic representation. Schematic representations included germane information from the problem supportive of problem solution. Pictorial representations included expressive and extraneous elements not necessary for problem solution, with no schematic elements. Findings indicated that the majority of students rendered schematic representations, with girls more likely than boys to use schematic representations at a statistically significant level. Students who used schematic visual representations were more successful problem solvers than those pictorially representing problem elements. The more “schematic‐like” the visual representation, the more successful students were at problem solution. Drawing a pictorial representation in the math challenge task also was negatively correlated to drawing skill.     

Exploring college students’ mental representations of inferential statistics

We report a case study that explored how three college students mentally represented the knowledge they held of inferential statistics, how this knowledge was connected, and how it was applied in two problem solving situations. A concept map task and two problem categorization tasks were used along with interviews to gather the data. We found that the students’ representations were based on incomplete statistical understanding. Although they grasped various concepts and inferential tests, the students rarely linked key concepts together or to tests nor did they accurately apply that knowledge to categorize word problems. We suggest that one reason the students had difficulty applying their knowledge is that it was not sufficiently integrated. In addition, we found that varying the instruction for the categorization task elicited different mental representations. One instruction was particularly effective in revealing students’ partial understandings. This finding suggests that modifying the task format as we have done could be a useful diagnostic tool.     

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.     

High‐School Students' Approaches to Solving Algebra Problems that are Posed Symbolically: Results from an Interview Study

A cross‐curricular structured‐probe task‐based clinical interview study with 44 pairs of third year high‐school mathematics students, most of whom were high achieving, was conducted to investigate their approaches to a variety of algebra problems. This paper presents results from three problems that were posed in symbolic form. Two problems are TIMSS items (a linear inequality and an equation involving square roots). The other problem involves square roots. We found that the majority of student pairs used symbol manipulation when solving the problems, and while many students seemed to prefer symbolic over graphical and tabular representations in their first attempt at solving the problems, we found that it was common for student pairs to use more than one strategy throughout the course of their solving. Students' use of graphing calculators to solve the problems is discussed.     

Development of a Geometric Spatial Visualization Tool

This paper documents the development of the Geometric Spatial Assessment. We detail the development of this instrument which was designed to identify middle school students' strategies and advancement in understanding of four geometric concept domains (geometric spatial visualization, spatial projection, cardinal directions, and periodic patterns) after experiencing a carefully designed integrated lunar unit. Previous research with students using this lunar unit showed students making significant gains on lunar‐related concepts (both scientific and mathematical) on a Lunar Phases Concept Inventory (LPCI) ( Lindell & Olsen, 2002 ). Following the administration of single domain assessments, clinical interviews were conducted to ascertain students' problem solving strategies. Results allowed us to select four suitable multiple‐choice items per domain.     

Identifying Fourth Graders' Understanding of Rational Number Representations: A Mixed Methods Approach

This study examined average‐, high‐ and top‐performing US fourth graders' rational number problem solving and their understanding of rational number representations. In phase one, all students completed a written test designed to tap their skills for multiplication, division and rational number word‐problem solving. In phase two, a subset of students sorted cards that showed part‐whole, ratio, quotient, measure, and operator perspectives of rational number representations. Each perspective was shown in numerical notational, word‐problem, and visual formats. The results indicated that top‐performing students scored significantly higher in problem solving and showed more effectively linked rational number representations than the other groups. The results imply that successful rational number problem solving is intertwined with representational knowledge for a wide range of rational numbers and that the bulk of US students do not possess effective skills for working with rational number representations.     

Students' Understanding of the Particulate Nature of Matter

The particulate nature of matter is identified in science education standards as one of the fundamental concepts that students should understand at the middle school level. However, science education research in indicates that secondary school students have difficulties understanding the structure of matter. The purpose of the study is to describe how engaging in an extended project‐based unit developed urban middle school students' understanding of the particulate nature of matter. Multiple sources of data were collected, including pre‐ and posttests, interviews, students' drawings, and video recordings of classroom activities. One teacher and her five classes were chosen for an indepth study. Analyses of data show that after experiencing a series of learning activities the majority of students acquired substantial content knowledge. Additionally, the finding indicates that students' understanding of the particulate nature of matter improved over time and that they retained and even reinforced their understanding after applying the concept. Discussions of the design features of curriculum and the teacher's use of multiple representations might provide insights into the effectiveness of learning activities in the unit.     

Capturing and characterizing students’ strategic algebraic reasoning through cognitively demanding tasks with focus on representations

In mathematics education, it is important to assess valued practices such as problem solving and communication. Yet, often we assess students based on correct solutions over their problem solving strategies—strategies that can uncover important mathematical understanding. In this article, we first present a framework of competencies required for strategic reasoning to solve cognitively demanding algebra tasks and assessment tools to capture evidence of these competencies. Then, we qualitatively describe characteristics of student reasoning for various performance levels (low, medium, and high) of eighth-grade students, focusing on generating and interpreting algebraic representations. We argue this analysis allows a more comprehensive and complex perspective of student understanding. Our findings lay groundwork to investigate the continuum of algebraic understanding, and may help educators identify specific areas of students’ strength and weakness when solving cognitively demanding tasks.     

Exploring Students' Conceptual Understanding of the Averaging Algorithm

Conceptual understanding of arithmetic average includes both an understanding of the computational algorithm and the statistical aspects of the concept. This study focused on the examination of 250 sixth-grade students' understanding of the arithmetic average by assessing their understanding of the computational algorithm. The results of the study showed that the majority of the students knew the “add-them-all-up-and-divide” averaging algorithm, but only about half of the students were able to correctly apply the algorithm to solve a contextualized average problem. Students were able to use various solution strategies and representations to solve the average problem. Those who used algebraic and arithmetic representations were better problem solvers than those who used pictorial and verbal representations. This study not only suggests that the average concept is more complex than the simplicity suggested by the computational algorithm, but also indicates the need for teaching the concept of average, both as a statistical idea for describing and making sense of data sets and as a computational algorithm for solving problems.     

Measuring Sixth‐Grade Students' Problem Solving: Validating an Instrument Addressing the Mathematics Common Core

This article describes the development of a problem‐solving instrument intended for classroom use that addresses the Common Core State Standards for Mathematics. In this study, 137 students completed the assessment, and their responses were analyzed. Evidence for validity was collected and examined using the current standards for educational and psychological testing. Instrument validation findings regarding internal consistency reliability were high, and multiple forms of validity (i.e., content, response processes, internal structure, relationship to other variables, and consequences of testing) were found to be appropriate. The resulting instrument provides teachers and researchers with a sound tool to gather data about sixth‐grade students' problem solving in the Common Core era.     

On high-school students' use of graphic calculators in mathematics

When students are working with hand held technology, such as graphic calculators, we usually only see the outcomes of their activities in the form of a contribution to a written solution of a mathematical problem. It is more difficult to capture their process of thinking or actions as they use the technology to solve the problem. In this paper we report on two case studies that follow the progress of students as they solve mathematical problems. We use software that works in the background of the graphic calculator capturing the students' keystrokes as they use the calculator. The aim of the research studies described in this paper was to provide insights into the working styles of these students. Through a detailed analysis of their graphic calculator keystrokes, interviews and associated written solutions we will discuss the effectiveness of their solution strategies and the efficiency of their use of the technology and identify some barriers to the use of graphic calculators in mathematical problem solving.