Applying Cognition-Based Assessment to Elementary School Students' Development of Understanding of Area and Volume Measurement

As part of a discussion of cognition-based assessment (CBA) for elementary school mathematics, I describe assessment tasks for area and volume measurement and a research-based conceptual framework for interpreting students' reasoning on these tasks. At the core of this conceptual framework is the notion of levels of sophistication. I provide details on an integrated set of levels for area and volume measurement that (a) starts with the informal, preinstructional reasoning typically possessed by students, (b) ends with the formal mathematical concepts targeted by instruction, and (c) indicates cognitive plateaus reached by students in moving from (a) to (b).     

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.     

Simon’s team’s contributions to scientific progress in mathematics education: A commentary on the Learning Through Activity (LTA) research program

I discuss two ways in which the Learning Through Activity (LTA) research program contributes to scientific progress in mathematics education: (a) providing general and content-specific constructs to explain conceptual learning and instructional design that corroborate and/or elaborate on previous work and (b) raising new questions/issues. The general constructs include using instructional design as testable models of learning and using theoretical constructs to guide real-time, instructional adaptations. In this sense, the general constructs promote understanding of linkages between conceptual learning and instruction in mathematics. The concept-specific constructs consist of empirically-grounded, hypothetical learning trajectories (HLTs) for fractional and multiplicative reasoning. Each HLT consists of specific, intended conceptual changes and tasks that can bring them forth. Questions raised for me by the LTA work involve inconsistencies between the stance on learning and reported teaching-learning interactions that effectively led to students’ abstraction of the intended mathematical concepts.     

Exploring the potential role of visual reasoning tasks among inexperienced solvers

The collective case study described herein explores solution approaches to a task requiring visual reasoning by students and teachers unfamiliar with such tasks. The context of this study is the teaching and learning of calculus in the Palestinian educational system. In the Palestinian mathematics curriculum the roles of visual displays rarely go beyond the illustrative and supplementary, while tasks which demand visual reasoning are absent. In the study, ten teachers and twelve secondary and first year university students were presented with a calculus problem, selected in an attempt to explore visual reasoning on the notions of function and its derivative and how it interrelates with conceptual reasoning. A construct named “visual inferential conceptual reasoning” was developed and implemented in order to analyze the responses. In addition, subjects’ reflections on the task, as well as their attitudes about possible uses of visual reasoning tasks in general, were collected and analyzed. Most participants faced initial difficulties of different kinds while solving the problem; however, in their solution processes various approaches were developed. Reflecting on these processes, subjects tended to agree that such tasks can promote and enhance conceptual understanding, and thus their incorporation in the curriculum would be beneficial.     

An Exploration of a Quantitative Reasoning Instructional Approach to Linear Equations in Two Variables With Community College Students

In this exploratory study, we examined the effects of a quantitative reasoning instructional approach to linear equations in two variables on community college students’ conceptual understanding, procedural fluency, and reasoning ability. This was done in comparison to the use of a traditional procedural approach for instruction on the same topic. Data were gathered from a common unit assessment that included procedural and conceptual questions. Results demonstrate that small changes in instruction focused on quantitative reasoning can lead to significant differences in students’ ability to demonstrate conceptual understanding compared to a procedural approach. The results also indicate that a quantitative reasoning approach does not appear to diminish students’ procedural skills, but that additional work is needed to understand how to best support students’ understanding of linear relationships.     

Investigating written expressions of mathematical reasoning for students with learning disabilities

Assessment results from two open-construction response mathematical tasks involving fractions and decimals were used to investigate written expression of mathematical reasoning for students with learning disabilities. The solutions and written responses of 51 students with learning disabilities in fourth and fifth grade were analyzed on four primary dimensions: (a) accuracy, (b) five elements of mathematical reasoning, (c) five elements of mathematical writing, and (d) vocabulary use. Results indicate most students were not accurate in their problem solution and communicated minimal mathematical reasoning in their written expression. In addition, students tended to use general vocabulary rather than academic precise math vocabulary and students who provided a visual representation were more likely to answer accurately. To further clarify the students struggles with mathematical reasoning, error analysis indicated a variety of error patterns existed and tended to vary widely by problem type. Our findings call for more instruction and intervention focused on supporting students mathematical reasoning through written expression. Implications for research and practice are presented.     

Revealing German primary school students’ achievement in measurement

The focus of this study was to investigate primary school students’ achievement in the domain of measurement. We analyzed a large-scale data set (N = 6,638) from German third and fourth graders (8- to 10-year-olds). These data were collected in 2007 within the framework of the ESMaG (Evaluation of the Standards in Mathematics in Primary School) project carried out by the Institute for Educational Quality Improvement (IQB) at Humboldt University, Berlin, Germany. The data were interpreted using a classification scheme based on a conceptual–procedural distinction in measurement competence. The analyses with this classification revealed that grade, gender, and in particular figural reasoning ability are significantly related to overall measurement competence as well as on the sub-competencies of Instrumental knowledge and Measurement sense. The paper concludes with a discussion of the implications of the findings of this study for teaching and assessing measurement.     

Children’s unit concepts in measurement: a teaching experiment spanning grades 2 through 5

We examined ways of improving students’ unit concepts across spatial measurement situations. We report data from our teaching experiment during a six-semester longitudinal study from grade 2 through grade 5. Data include instructional task sequences designed to help children (a) integrate multiple representations of unit, (b) coordinate and group units into higher-order units, and (c) recognize the arbitrary nature of unit in comparison contexts and student’s responses to tasks. Our results suggest reflection on multiplicative relations among quantities prompted a more fully-developed unit concept. This research extends prior work addressing the growth of unit concepts in the contexts of length, area, and volume by demonstrating the viability of level-specific instructional actions as a means for promoting an informal theory of measurement.     

Middle school students’ proportional reasoning in real life contexts in the domain of geometry and measurement

The purpose of this study was to examine middle school students’ proportional reasoning, solution strategies and difficulties in real life contexts in the domain of geometry and measurement. The underlying reasons of the difficulties were investigated as well. Mixed research design was adopted for the aims of the study by collecting data through an achievement test from 935 sixth, seventh and eighth grade students. The achievement test included real life problems that required proportional reasoning, and were related to the measurement of length, perimeter, area and volume concepts. In addition, task-based interviews were conducted on 12 of these students to collect more comprehensive data and to support the findings of the achievement test. Findings revealed that although students were mostly successful in giving correct answers, their reasoning lacked a clear argument of the direct and indirect proportional relationships between the variables and that they approached the problems by superficial characteristics of the problems.     

Ways in which engaging with someone else’s reasoning is productive

Classrooms which involve students in mathematical discourse are becoming ever more prominent for the simple reason that they have been shown to support student learning and affinity for content. While support for outcomes has been shown, less is known about how or why such strategies benefit students. In this paper, we report on one such finding: namely that when students engage with another’s reasoning, as necessitated by interactive conversation, it supports their own conceptual growth and change. This qualitative analysis of 10 university students provides insight into what engaging with another’s reasoning entails and suggests that higher levels of engagement support higher levels of conceptual growth. We conclude with implications for instructional practice and future research.     

Conceptual learning in a principled design problem solving environment

To what extent can instructional design be based on principles for instilling a culture of problem solving and conceptual learning? This is the main focus of the study described in this paper, in which third grade students participated in a one-year course designed to foster problem solving and mathematical reasoning. The design relied on five principles: (a) encouragement to produce multiple solutions; (b) creating collaborative situations; (c) socio-cognitive conflicts; (d) providing tools for checking hypotheses; and (e) inviting students to reflect on solutions. We describe how a problem solving task designed according to the above principles, promoted students' understanding of the area concept. We show that the design afforded the surfacing of multiple solutions and justifications in various modalities (including gestures) and initiated peer argumentation, leading to deep learning of the area concept.     

Integer comparisons across the grades: Students’ justifications and ways of reasoning

This study is an investigation of students’ reasoning about integer comparisons—a topic that is often counterintuitive for students because negative numbers of smaller absolute value are considered greater (e.g., −5 > − 6). We posed integer-comparison tasks to 40 students each in Grades 2, 4, and 7, as well as to 11th graders on a successful mathematics track. We coded for correctness and for students’ justifications, which we categorized in terms of 3 ways of reasoning: magnitude-based, order-based, and developmental/other. The 7th graders used order-based reasoning more often than did the younger students, and it more often led to correct answers; however, the college-track 11th graders, who responded correctly to almost every problem, used a more balanced distribution of order- and magnitude-based reasoning. We present a framework for students’ ways of reasoning about integer comparisons, report performance trends, rank integer-comparison tasks by relative difficulty, and discuss implications for integer instruction.     

Examining preservice teachers' responses to area conservation tasks

The purpose of this work was to explore how elementary preservice teachers responded to area conservation tasks. We administered written pre-assessments, followed by semi-structured interviews with 23 preservice teachers, asking them to respond to and reason with area conservation tasks. Findings highlighted several interesting preservice teachers' struggles when assessing area conservation tasks. In many cases, preservice teachers exhibited struggles similar to students, especially with regards to the justification of their area conservation claims. We provide recommendations to assist preservice teachers in their development of mathematical content knowledge in their teacher education programs, so that in the future they may better plan area lessons that promote procedural fluency from conceptual understanding in area measurement.     

Types of reasoning required in university exams in mathematics

Empirical research shows that students often use reasoning founded on copying algorithms or recalling facts (imitative reasoning) when solving mathematical tasks. Research also indicate that a focus on this type of reasoning might weaken the students’ understanding of the underlying mathematical concepts. It is therefore important to study the types of reasoning students have to perform in order to solve exam tasks and pass exams. The purpose of this study is to examine what types of reasoning students taking introductory calculus courses are required to perform. Tasks from 16 exams produced at four different Swedish universities were analyzed and sorted into task classes. The analysis resulted in several examples of tasks demanding different types of mathematical reasoning. The results also show that about 70% of the tasks were solvable by imitative reasoning and that 15 of the exams could be passed using only imitative reasoning.     

Components of Place Value Understanding: Targeting Mathematical Difficulties When Providing Interventions

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.     

Students’ reasoning about relationships between variables in a real-world problem

This paper reports on part of an investigation of fifteen second-semester calculus students’ understanding of the concept of parametric function. Employing APOS theory as our guiding theoretical perspective, we offer a genetic decomposition for the concept of parametric function, and we explore students’ reasoning about an invariant relationship between two quantities varying simultaneously with respect to a third quantity when described in a real-world problem, as such reasoning is important for the study of parametric functions. In particular, we investigate students’ reasoning about an adaptation of the popular bottle problem in which they were asked to graph relationships between (a) time and volume of the water, (b) time and height of the water, and (c) volume and height of the water. Our results illustrate that several issues make reasoning about relationships between variables a complex task. Furthermore, our findings indicate that conceiving an invariant relationship, as it relates to the concept of parametric function, is nontrivial, and various complimentary ways of reasoning are favorable for developing such a conception. We conclude by making connections between our results and our genetic decomposition.     

Spatial visualization and measurement of area: A case study in spatialized mathematics instruction

The purpose of this study was to explore the influence of spatial visualization skills when students solve area tasks. Spatial visualization is closely related to mathematics achievement, but little is known about how these skills link to task success. We examined middle school students’ representations and solutions to area problems (both non-metric and metric) through qualitative and quantitative task analysis. Task solutions were analyzed as a function of spatial visualization skills and links were made between student solutions on tasks with different goals (i.e., non-metric and metric). Findings suggest that strong spatial visualizers solved the tasks with relative ease, with evidence for conceptual and procedural understanding. By contrast, Low and Average Spatial students more frequently produced errors due to failure to correctly determine linear measurements or apply appropriate formula, despite adequate procedural knowledge. A novel finding was the facilitating role of spatial skills in the link between metric task representation and success in determining a solution. From a teaching and learning perspective, these results highlight the need to connect emergent spatial skills with mathematical content and support students to develop conceptual understanding in parallel with procedural competence.     

Students’ conceptions of mathematics as sensible: Towards the SCOMAS framework

In this article I describe the development of a framework for considering students’ conceptions about the sensible nature of mathematics. I begin by using extant literature on conceptions of mathematics to develop a framework of action-oriented indicators that students’ conceive of mathematics as sensible. I then use classroom data to modify and illustrate the framework. The result is a coding framework, grounded in the literature, which can be used to assess the enacted conceptions of mathematics as sensible of a group of students. This work also provides a conceptual framework, grounded in classroom data, of the dimensions of these conceptions.     

Recursive and explicit rules: How can we build student algebraic understanding?

Differing perspectives have been offered about student use of recursive and explicit rules. These include: (a) promoting the use of explicit rules over the use of recursive rules, and (b) encouraging student use of both recursive and explicit rules. This study sought to explore students’ use of recursive and explicit rules by examining the reasoning of 25 sixth-grade students, including a focus on four target students, as they approached tasks in which they were required to develop generalizations while using computer spreadsheets as an instructional tool. The results demonstrate the difficulty that students had moving from the successful use of recursive rules toward explicit rules. In particular, two students abandoned general reasoning, instead focusing on particular values in an attempt to construct explicit rules. It is recommended that students be encouraged to connect recursive and explicit rules as a potential means for constructing successful generalizations.     

Predetermined accommodations with a standardized testing protocol: Examining two accommodation supports for developing fraction thinking in students with mathematical difficulties

This study examined the effects of two pre-determined accommodations that were provided in a standardized testing. The two accommodations were meant to help students with difficulties in mathematics (SDMs) engage in unit thinking, reasoning, and coordination and consequently improve their ability to process fraction tasks. 23 middle school SDMs took the following tests and were asked to explain their solutions: a baseline fraction test without any accommodation; an annotated test with bolded information and additional simplified explanations; and a warming- up test that involved whole-number multiplicative reasoning tasks followed by the baseline test. Results show that while SDMs were able to construct and coordinate fraction units to solve fraction problems when appropriate accommodations were provided, standardized assessment with a predetermined “one-size-fits -all” accommodation could not meet the specific needs of all students with mathematics learning difficulties.