Analyzing student understanding in linear algebra through mathematical activity

In this paper we characterize students’ conceptions of span and linear (in)dependence and their mathematical activity to provide insight into their understanding. The data under consideration are portions of individual interviews with linear algebra students. Grounded analysis revealed a wide range of student conceptions of span and linear (in)dependence. The authors organized these conceptions into four categories: travel, geometric, vector algebraic, and matrix algebraic. To further illuminate participants’ conceptions of span and linear (in)dependence, the authors developed a categorization to classify the participants’ engagement into five types of mathematical activity: defining, proving, relating, example generating, and problem solving. Coordination of these two categorizations provides a framework that proves useful in providing finer-grained analyses of students’ conceptions and the potential value and/or limitations of such conceptions in certain contexts.     

Construction of a geometrical concept within a dialectical learning environment

The goals of the study were to design and investigate a teaching-learning environment that encourage freedom and autonomy of pre-service teachers in constructing their own new geometrical concept, and to analyze the dialectic process of the concept construction in the designed environment. A dialectic process of the participants’ defining activity emerged from the necessity to resolve the tensions between hypothesizing the concept’s examples and the appropriate critical attributes. Such a process created a vivid learning trajectory, in which learners examined logically the ways in which the examples, the critical attributes and the definition match. In this way, the three elements of the mathematical concept cannot play a passive role, or be neglected, and it appeared that prototypical examples were not created, so that no example is more dominant than others. The groups constructed different concepts, but with full harmony between its definition, example space, and concept-critical attributes.     

Developing and refining a framework for mathematical and linguistic complexity in tasks related to rates of change

We present and develop a preliminary framework for describing the relationship between the mathematical and linguistic complexity of instructional tasks used in secondary mathematics. The initial framework was developed through a review of relevant literature. It was refined by examining how 4 ninth grade mathematics teachers of linguistically diverse groups of students described the linguistic and mathematical complexity of a set of tasks from their curriculum unit on linear functions. We close by presenting our refined framework for describing the interaction of linguistic complexity and mathematical complexity in curriculum materials, and discuss potential uses of this framework in the design of more accessible classroom learning environments for linguistically diverse students.     

How Emergent Models May Foster the Constitution of Formal Mathematics

This article deals with the role that so-called emergent models can play in the process of constituting formal mathematics. The underlying philosophy is that formal mathematics is something that is, or should be, constituted by the students themselves. In the instructional design theory for realistic mathematics education, models always have been employed to foster a process in which formal mathematics is reinvented by the students themselves. This article describes how the use of models became more and more explicated over time and developed into the notion of emergent models. The design of an instructional sequence, which deals with flexible mental computation strategies for addition and subtraction up to 100, is taken as an instance for elaborating what is meant by emergent models and what role they play in fostering the constitution of formal mathematics. The analysis shows that there are 3 interrelated processes. First. at a more holistic level, there is a global transition in which “the model” initially emerges as a model of informal mathematical activity and then gradually develops into a model for more formal mathematical reasoning. Second, the transition from “model of” to “model for” involves the constitution of anew mathematical reality that can be denoted formal in relation to the original starting points of the students. Third, in the series of instructional activities, there is not 1 model, but the model actually is shaped as a series of signs, in which each new sign comes to signify activity with a previous sign in a chain of signification.     

Building on mathematical events in the classroom

Mathematical events from classrooms were used as stimuli to encourage mathematical discussion in two groups of mathematics teachers at the secondary level. Each event was accompanied by an analysis of mathematics that would be useful to the teacher in such a situation. The Situations, mathematical events and analyses, were used originally to create a framework describing the Mathematical Proficiency for Teaching at the Secondary Level, and then they were used with both Prospective and Practicing teachers to validate the framework. Teachers involved in the validation research claimed that the process was instructional. The process is explained, and teachers’ quotes provide evidence that the experience provoked changes in teachers’ understanding of mathematics. This process, which builds on mathematical events from the classroom, holds potential as a professional development experience that helps teachers expand their expertise in teaching mathematics.     

New directions in differential equations: A framework for interpreting students' understandings and difficulties

The purpose of this paper is to offer a framework for interpreting students' understandings of and difficulties with mathematical ideas central to new directions in differential equations. These new directions seek to guide students into a more interpretive mode of thinking and to enhance their ability to graphically and numerically analyze differential equations. The framework reported here is the result of investigating in depth six students' understandings through a series of task-based individual interviews and classroom observations. The two major themes of the framework, the function-as-solution dilemma theme and students' intuitions and images theme, extend previous research on student cognition at the secondary and collegiate level to the domain of differential equations and reflect the increased recognition of situating analyses of student learning within students' learning environment. For new areas of interest such as differential equations, mapping out students' understandings of important mathematical ideas can be an important part of curricular and instructional design that seeks to refine and build on students' ways of thinking.     

Leveraging interdiscursivity to support elementary students in bridging the empirical-deductive gap: the case of parity

Research has recognized deductive reasoning as challenging but not impossible for young mathematics learners. In this paper, we present a learning environment developed to assist elementary-school students to bridge the empirical-deductive gap in the context of parity of numbers. Using the commognitive framework, we construe the empirical-deductive gap as part of a broader divide between two discourses that abide by different rules of a “mathematical game”: a discourse on specific numbers and a discourse on numeric patterns. Interdiscursivity is leveraged as a mechanism for instructional design, where students’ familiar routines with specific numbers are teased out and advanced to make sense in the new discourse. We mobilize this mechanism to create opportunities for students to play an active role in recognizing issues with empirical reasoning and generating deductive arguments to establish the validity of universal statements. The environment is illustrated with a small group of 8-year-olds who learned to justify deductively that “odd + odd = even”.     

Towards an integrated theory of mathematics conceptual learning and instructional design: The Learning Through Activity theoretical framework

We discuss the theoretical framework of the Learning Through Activity research program. The framework includes an elaboration of the construct of mathematical concept, an elaboration of Piaget’s reflective abstraction for the purpose of mathematics pedagogy, further development of a distinction between two stages of conceptual learning, and a typology of different reverse concepts. The framework also involves instructional design principles built on those constructs, including steps for the design of task sequences, development of guided reinvention, and ways of promoting reversibility of concepts. This article represents both a synthesis of prior work and additions to it.     

How Emergent Models May Foster the Constitution of Formal Mathematics

This article deals with the role that so-called emergent models can play in the process of constituting formal mathematics. The underlying philosophy is that formal mathematics is something that is, or should be, constituted by the students themselves. In the instructional design theory for realistic mathematics education, models always have been employed to foster a process in which formal mathematics is reinvented by the students themselves. This article describes how the use of models became more and more explicated over time and developed into the notion of emergent models. The design of an instructional sequence, which deals with flexible mental computation strategies for addition and subtraction up to 100, is taken as an instance for elaborating what is meant by emergent models and what role they play in fostering the constitution of formal mathematics. The analysis shows that there are 3 interrelated processes. First. at a more holistic level, there is a global transition in which "the model" initially emerges as a model of informal mathematical activity and then gradually develops into a model for more formal mathematical reasoning. Second, the transition from "model of" to "model for" involves the constitution of anew mathematical reality that can be denoted formal in relation to the original starting points of the students. Third, in the series of instructional activities, there is not 1 model, but the model actually is shaped as a series of signs, in which each new sign comes to signify activity with a previous sign in a chain of signification.     

The teacher’s role in guiding children’s mathematical ideas toward meeting lesson objectives

This paper investigates the classroom interactive pattern, in which the teacher aims to introduce new mathematical content to children by focusing on their mathematical thinking. First, by drawing on the results of studies on the features of social interaction patterns in mathematics classrooms, we develop a framework that we call a “guided focusing pattern,” composed of four phases. Next, we use this framework and Sfard’s (J Res Math Educ 31(3):296–327, 2000) theory of focal analysis to examine the social interaction occurring in a series of mathematics lessons conducted by an experienced teacher. In the ten consecutive lessons that we analyzed, the guided focusing pattern was salient; the teacher introduced key mathematical content to children while offering support and guidance in a variety of forms within each phase and when transitioning to the next phase. On the basis of the results, we highlight the teacher’s key instructional actions that facilitate the pattern of progressing through the mathematical content as closely linked to and guided by her lesson objectives.     

Local Instruction Theories as Means of Support for Teachers in Reform Mathematics Education

This article focuses on a form of instructional design that is deemed fitting for reform mathematics education. Reform mathematics education requires instruction that helps students in developing their current ways of reasoning into more sophisticated ways of mathematical reasoning. This implies that there has to be ample room for teachers to adjust their instruction to the students' thinking. But, the point of departure is that if justice is to be done to the input of the students and their ideas built on, a well-founded plan is needed. Design research on an instructional sequence on addition and subtraction up to 100 is taken as an instance to elucidate how the theory for realistic mathematics education (RME) can be used to develop a local instruction theory that can function as such a plan. Instead of offering an instructional sequence that “works,” the objective of design research is to offer teachers an empirically grounded theory on how a certain set of instructional activities can work. The example of addition and subtraction up to 100 is used to clarify how a local instruction theory informs teachers about learning goals, instructional activities, student thinking and learning, and the role of tools and imagery.     

How students interpret and enact inquiry-oriented defining practices in undergraduate real analysis

This study investigates the influence of inquiry-oriented real analysis instruction on students’ conceptions of the situation of mathematical defining. I assess the claim that inquiry-oriented instruction helps acculturate students into advanced mathematical practice. The instruction observed was “inquiry-oriented” in the sense that they treated definitions as under construction. The professor invited students to create and assess mathematical definitions and students sometimes articulated key mathematical content before the instructor. I characterize students’ conceptions of the defining situation as their (1) frames for the classroom activity, (2) perceived role in that activity, and (3) values for classroom defining. I identify four archetypal categories of students’ conceptions. All participants in the study valued classroom defining because it helped them understand and recall definitions. However, students in only two categories showed strong acculturation to mathematical practice, which I measure by the students’ expression of meta-mathematical values for defining or by their bearing mathematical authority.     

Developing shift problems to foster geometrical proof and understanding

Meaningful learning of formal mathematics in regular classrooms remains a problem in mathematics education. Research shows that instructional approaches in which students work collaboratively on tasks that are tailored to problem solving and reflection can improve students’ learning in experimental classrooms. However, these sequences involve often carefully constructed reinvention route, which do not fit the needs of teachers and students working from conventional curriculum materials. To help to narrow this gap, we developed an intervention—‘shift problem lessons’. The aim of this article is to discuss the design of shift problems and to analyze learning processes occurring when students are working on the tasks. Specifically, we discuss three paradigmatic episodes based on data from a teaching experiment in geometrical proof. The episodes show that is possible to create a micro-learning ecology where regular students are seriously involved in mathematical discussions, ground their mathematical understanding and strengthen their relational framework.     

Local Instruction Theories as Means of Support for Teachers in Reform Mathematics Education

This article focuses on a form of instructional design that is deemed fitting for reform mathematics education. Reform mathematics education requires instruction that helps students in developing their current ways of reasoning into more sophisticated ways of mathematical reasoning. This implies that there has to be ample room for teachers to adjust their instruction to the students' thinking. But, the point of departure is that if justice is to be done to the input of the students and their ideas built on, a well-founded plan is needed. Design research on an instructional sequence on addition and subtraction up to 100 is taken as an instance to elucidate how the theory for realistic mathematics education (RME) can be used to develop a local instruction theory that can function as such a plan. Instead of offering an instructional sequence that "works," the objective of design research is to offer teachers an empirically grounded theory on how a certain set of instructional activities can work. The example of addition and subtraction up to 100 is used to clarify how a local instruction theory informs teachers about learning goals, instructional activities, student thinking and learning, and the role of tools and imagery.     

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.     

Learning Trajectories,Lesson Planning,Affordances, and Constraints in the Design and Enactment of Mathematics Teaching

Recent reform efforts in mathematics education have stimulated a focus on learning trajectories. At the same time, a global increase in high-stakes testing has influenced instructional practices. This study investigated how four fourth grade teachers within a school planned and enacted lessons to understand what mediated their planning and teaching decisions. Findings reveal that three of these teachers, who were veteran teachers, used a testing trajectory approach with decisions mediated by preparing students for high-stakes tests. The fourth teacher, a novice, attempted to use a learning trajectory approach to support student understanding. Results reveal that high-stakes testing played a crucial role in teachers' instructional decisions. Based on the findings, we provide a framework for a testing trajectory approach that the veteran teachers used to make instructional decisions. Further research is needed to understand how to support teachers to prepare students for testing using effective teaching practices.     

The onto-semiotic approach to research in mathematics education

In this paper we synthesize the theoretical model about mathematical cognition and instruction that we have been developing in the past years, which provides conceptual and methodological tools to pose and deal with research problems in mathematics education. Following Steiner’s Theory of Mathematics Education Programme, this theoretical framework is based on elements taken from diverse disciplines such as anthropology, semiotics and ecology. We also assume complementary elements from different theoretical models used in mathematics education to develop a unified approach to didactic phenomena that takes into account their epistemological, cognitive, socio cultural and instructional dimensions.     

Defining belief functions using mathematical morphology – Application to image fusion under imprecision

We address in this paper the problem of defining belief functions, typically for multi-source classification applications in image processing. We propose to use mathematical morphology for introducing imprecision in the mass and belief functions while estimating disjunctions of hypotheses. The basic idea relies on the similarity between some properties of morphological operators and properties of belief functions. The framework of mathematical morphology guarantees that the derived functions have all required properties. We illustrate the proposed approach on synthetic and real images.     

Response to Mahmood and Mahmood (2015)

In the article published in the International Journal of Mathematics Education in Science and Technology in 2015, Mahmood and Mahmood suggested an explanation for defining 0! as 1. In this response, I argue that their reasoning is flawed.     

Pre-service mathematics teachers become full participants in inquiry investigations

Research is described concerning the effectiveness of inquiry-based laboratory environments created in US mathematics/science education programme courses. Laboratory projects were conducted using a framework that allowed pre-service teachers to explore, analyse, and communicate ‘investigable’ realms of physical phenomena. Goals were for pre-service teachers to experience the value of learning in an inquiry-enhanced environment and to engage in contextualized mathematics so they would utilize this instruction in their future classrooms. It is proposed that inquiry-based laboratories are needed within the mathematics classroom in order to allow students the opportunity to contextualize, to connect to other disciplines, and to experience mathematical concepts. Pre-service teachers were expected to pursue conjectures, collect data, think critically, and communicate findings. This qualitative research shows how the use of inquiry can complement the learning of mathematical content and educational strategies for pre-service teachers. Results provide detailed information for teacher educators regarding instructional design of contextualized mathematical inquiry.