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.     

Investigating the transformation of a secondary teacher’s knowledge of trigonometric functions

Shulman (1987) defined pedagogical content knowledge as the knowledge required to transform subject-matter knowledge into curricular material and pedagogical representations. This paper presents the results of an exploratory case study that examined a secondary teacher’s knowledge of sine and cosine values in both clinical and professional settings to discern the characteristics of mathematical schemes that facilitate their transformation into learning artifacts and experiences for students. My analysis revealed that the teacher’s knowledge of sine and cosine values consisted of uncoordinated quantitative and arithmetic schemes and that he was cognizant only of the behavioral proficiencies these schemes enable, not the mental actions and conceptual operations they entail. Based on these findings, I hypothesize that the extent to which a teacher is consciously aware of the mental activity that comprises their mathematical conceptions influences their capacity to transform their mathematical knowledge into curricular material and pedagogical representations to effectively support students’ conceptual learning.     

An analytical comparison of students’ reasoning in the context of Inquiry-Oriented Instruction: The case of span and linear independence

In this report we analyze differences in reasoning about span and linear independence by comparing written work of 126 linear algebra students whose instructors received support to implement a particular inquiry-oriented (IO) instructional approach compared to 129 students whose instructors did not receive that support. Our analysis of students’ responses to open-ended questions indicated that IO students’ concept images of span and linear independence were more aligned with the formal concept definition than the concept images of Non-IO students. Additionally, IO students exhibited more coordinated conceptual understandings and used deductive reasoning at higher rates than Non-IO students. We provide illustrative examples of systematic differences in how students from the two groups reasoned about span and linear independence.     

“Rising to the challenge”: using generalization in pattern problems to unearth the algebraic skills of talented pre-algebra students

This study focuses on the generalization methods used by talented pre-algebra students in solving linear and non-linear pattern problems. A qualitative analysis of the solutions of three problems revealed two approaches to generalization: recursive–local and functional–global. The students showed mental flexibility, shifting smoothly between pictorial, verbal and numerical representations and abandoning additive solution approaches in favor of more effective multiplicative strategies. Three forms of reflection aided generalization: reflection on commonalities in the pattern sequence’s structure, reflection on the generalization method, and reflection on the “tentative generalization” through verification of the pattern sequence. The latter indicates an intuitive grasp of the mathematical power of generalization. The students’ generalizations evinced algebraic thinking in the discovery of variables, constants and their mutual relations, and in the communication of these discoveries using invented algebraic notation. This study confirms the centrality of generalizations in mathematics and their potential as gateways to the world of algebra.     

Do students really understand what an ordinary differential equation is?

Differential equations (DEs) are important in mathematics as well as in science and the social sciences. Thus, the study of DEs has been included in various courses in different departments in higher education. The importance of DEs has attracted the attention of many researchers who have generally focussed on the content and instruction of DEs. However, DEs are complex issues that students may find difficulty to understand. The limited research in this literature points to the need for more studies on students’ conceptions, and understanding of DEs and their basic concepts. The objective of this study is to fill this need by revealing the understanding, difficulties and weaknesses of the students who are successful in algebraic solutions, in relation to the concepts of DEs and their solutions. For this purpose, 77 students were asked 13 DE questions (6 of them about algebraic solution, and the rest about interpreting DEs and their solutions). From an analysis of the students’ answers, it was concluded that the students who were quite successful in algebraic solutions, indeed did not fully understand the related concepts, and they had serious difficulties in relation to these concepts.     

Evidence from cognitive neuroscience for the role of graphical and algebraic representations in understanding function

Using traditional educational research methods, it is difficult to assess students’ understanding of mathematical concepts, even though qualitative methods such as task observation and interviews provide some useful information. It has now become possible to use functional magnetic resonance imaging (fMRI) to observe brain activity whilst students think about mathematics, although much of this work has concentrated on number. In this study, we used fMRI to examine brain activity whilst ten university students translated between graphical and algebraic formats of both linear and quadratic mathematical functions. Consistent with previous studies on the representation of number, this task elicited activity in the intra-parietal sulcus, as well as in the inferior frontal gyrus. We also analysed qualitative data on participants’ introspection of strategies employed when reasoning about function. Expert participants focused more on key properties of functions when translating between formats than did novices. Implications for the teaching and learning of functions are discussed, including the relationship of function properties to difficulties in conversion from algebraic to graphical representation systems and vice versa, the desirability of teachers focusing attention on function properties, and the importance of integrating graphical and algebraic function instruction.     

Students’ understanding of algebraic notation: A semiotic systems perspective

The ideas of equivalence and variable are two of the most fundamental concepts in algebra. Most studies of students’ understanding of these concepts have posited a gap between the students’ conceptions and the institutional meanings for the symbols. In contrast, this study develops a theoretical framework for describing the ways undergraduate students use personal meanings for symbols as they appropriate institutional meanings. To do this, we introduce the idea of semiotic systems as a framework for understanding the ways students use collections of signs to engage in mathematical activity and how the students use these signs in meaningful ways. The analysis of students’ work during task-based interviews suggests that this framework allows us to identify the ways in which seemingly idiosyncratic uses of the symbols are evidence of meaning-making and, in many cases, how the symbol use enables the student to engage productively in the mathematical activity.     

Engineering students’ conceptions of the derivative and some implications for their mathematical education

This paper explores Mechanical Engineering students’ conceptions of and preferences for conceptions of the derivative, and their views on mathematics. Data comes from pre-, post- and delayed post-tests, a preference test, interviews with students and an analysis of calculus courses. Data from Mathematics students is used to make comparisons with Mechanical Engineering students. The results show that Mechanical Engineering students’ conceptions of and preferences for the derivative develop in the direction of the rate of change aspects while those of Mathematics students develop in the direction of tangent aspects, and that Mechanical Engineering students view mathematics as a tool and want the application aspects in their course. Students’ developing conceptions, preferences and views with regard to teaching and departmental affiliation are considered and educational implications are suggested for the mathematical education of engineering students.     

Teachers’ conceptions of tangent line

In order to study the conceptions, and their evolutions, of the tangent line to a curve an updating workshop which took place in México was designed for upper secondary school teachers. This workshop was planned using the methodology of cooperative learning, scientific debate and auto reflection (ACODESA) and the conception-knowing-concept model (cK¢) developed mainly by Balacheff. In order to initiate the conceptions reorganization, an initial activity was made in the graphic frame; it seems optimal to start the debates without leading participants to failure. The mathematical core of the workshop was formed by an algebraic method to find tangents to algebraic curves which is close to Descartes’ method. The ACODESA methodology allowed some intense debates mainly concerning the local character of a tangent, generating the teachers’ cognitive unbalance which is a starting point for the refinement or transformation of their conceptions. On the other hand, the cK¢ model allows to understand the conceptions of participants and to analyze the evolution of their knowings.     

Synergies among students’ thinking modes and representation types in linear algebra: employing statistical implicative analysis

In this work, students’ thinking modes and representation types in linear algebra are investigated through statistical implicative analysis techniques. Specifically, our research question considers the implicative relationships between students’ thinking modes and representation types of linear algebra. The participants were 74 undergraduate linear algebra students enrolled in the department of mathematics education of a government university located in western Turkey. The data was collected using six paper-and-pencil tasks, relating to a context of linear equations, matrix algebra, linear combination, span, linear independency–dependency and basis. A document analysis technique was used to analyze the data within a theoretical lens of thinking modes and representation types. To delineate similarity diagrams, hierarchical trees, and implicative models (which will be detailed in the paper), an R version of Cohesion Hierarchical Implicative Classification software was used. According to the results, students’ analytic structural thinking modes on linear combination and span and linear independency significantly imply the use of algebraic and abstract representations. The results also confirm that the notions of linear combination and span and linear dependency/independency are core elements for theoretical thinking and are needed for learning linear algebra.     

What do mathematicians mean by proof? A comparative-judgement study of students’ and mathematicians’ views

We present a study in which mathematicians and undergraduate students were asked to explain in writing what mathematicians mean by proof. The 175 responses were evaluated using comparative judgement: mathematicians compared pairs of responses and their judgements were used to construct a scaled rank order. We provide evidence establishing the reliability, divergent validity and content validity of this approach to investigating individuals’ written conceptions of mathematical proof. In doing so, we compare the quality of student and mathematician responses and identify which features the judges collectively valued. Substantively, our findings reveal that despite the variety of views in the literature, mathematicians broadly agree on what people should say when asked what mathematicians mean by proof. Methodologically, we provide evidence that comparative judgement could have an important role to play in investigating conceptions of mathematical ideas, and conjecture that similar methods could be productive in evaluating individuals’ more general (mathematical) beliefs.     

The role of prediction in the teaching and learning of mathematics

The prevalence of prediction in grade-level expectations in mathematics curriculum standards signifies the importance of the role prediction plays in the teaching and learning of mathematics. In this article, we discuss benefits of using prediction in mathematics classrooms: (1) students’ prediction can reveal their conceptions, (2) prediction plays an important role in reasoning and (3) prediction fosters mathematical learning. To support research on prediction in the context of mathematics education, we present three perspectives on prediction: (1) prediction as a mental act highlights the cognitive aspect and the conceptual basis of one's prediction, (2) prediction as a mathematical activity highlights the spectrum of prediction tasks that are common in mathematics curricula and (3) prediction as a socio-epistemological practice highlights the construction of mathematical knowledge in classrooms. Each perspective supports the claim that prediction when used effectively can foster mathematical learning. Considerations for supporting the use of prediction in mathematics classrooms are offered.     

Exploring Repunits

This article describes a discovery activity which is organized and developed at two different levels of treatment corresponding to the different degree of mathematical maturity of students. Using an 8-digit calculator, students at the first level (7-9th graders) explore patterns which lead them to discover the properties of repunits and their relationships to many mathematical curiosities. At the second level (10-12th graders) students may be interested in further extension exploring why these patterns emerge and offering algebraic justification.     

Conceptions of a sequence formed in secondary schools

The paper is based on research carried out on secondary school students and students commencing their university studies in mathematics. The basic purpose of the research was to investigate the understanding of the concept of a sequence and to determine the sources of the formation of revealed conceptions. To achieve the objectives an expanded set of selected mathematical situations – simple but not quite standard – were investigated and various other research instruments were used. The students’ conceptions were divided into two groups. In the first group a sequence was perceived as a function, in the second it was associated with ordered elements. The diversity of the last set of conceptions was particularly interesting. The students understood the word ‘ordered’ as some kind of relationship between the terms of a sequence, a certain regularity or harmony. In the paper some ways of correcting the conceptions revealed and introducing the concept of a sequence in schools were also discussed.     

Undergraduate students’ initial conceptions of factorials

Counting problems offer rich opportunities for students to engage in mathematical thinking, but they can be difficult for students to solve. In this paper, we present a study that examines student thinking about one concept within counting, factorials, which are a key aspect of many combinatorial ideas. In an effort to better understand students’ conceptions of factorials, we conducted interviews with 20 undergraduate students. We present a key distinction between computational versus combinatorial conceptions, and we explore three aspects of data that shed light on students’ conceptions (their initial characterizations, their definitions of 0!, and their responses to Likert-response questions). We present implications this may have for mathematics educators both within and separate from combinatorics.     

Interactionist perspective on negotiation processes of students’ different understandings during small group work on linear algebra

Student group work represents a central learning setting within mathematics programs at the university level. In this study, a theoretical perspective on collaboration is adopted in which the differences between students’ interpretations of a mathematical concept are seen as an opportunity for individual restructuring processes. This so-called interactionist perspective is applied to student group work on linear algebra. The concepts of linear algebra at the university level are characterized by a versatility of different modes of expression and interpretation. For students of linear algebra, the flexible transitions between the different interpretations of linear algebra concepts usually pose a challenge. This study focuses on how students negotiate their different interpretations during group work on linear algebra and how transitions between interpretations might be stimulated or hindered. Video recordings of eight student groups working on a task that required flexible transition between interpretations of homomorphisms were sampled. The recordings were analyzed from an interactionist perspective, focusing on interaction situations in which the participating students expressed and negotiated different interpretations of homomorphisms. The analyses of students’ interactions highlight a phenomenon whereby differences in students’ interpretations remain implicit in group discussions, which constitutes an obstacle to the negotiation process.     

Changed views on mathematical knowledge in the course of didactical theory development—independent corpus of scientific knowledge or result of social constructions?

The study tries to show one line of how the German didactical tradition has evolved in response to new theoretical ideas and new—empirical—research approaches in mathematics education. First, the classical mathematical didactics, notably ‘stoffdidaktik’ as one (besides other) specific German tradition are described. The critiques raised against ‘stoffdidaktik’ concepts [for example, forms of ‘progressive mathematisation’, ‘actively discovering learning processes’ and ‘guided reinvention’ (cf. Freudenthal, Wittmann)] changed the basic views on the roles that ‘mathematical knowledge’, ‘teacher’ and ‘student’ have to play in teaching–learning processes; this conceptual change was supported by empirical studies on the professional knowledge and activities of mathematics teachers [for example, empirical studies of teacher thinking (cf. Bromme)] and of students’ conceptions and misconceptions (for example, psychological research on students’ mathematical thinking). With the interpretative empirical research on everyday mathematical teaching–learning situations (for example, the work of the research group around Bauersfeld) a new research paradigm for mathematics education was constituted: the cultural system of mathematical interaction (for instance, in the classroom) between teacher and students.     

Prospective teachers’ unified understandings of the structure of identities

United States curriculum standards advise mathematics teachers to teach students to attend to structure and understand how mathematical concepts are related. This requires teachers to have a structural perspective and a coherent, unified understanding of mathematical structures that span curricula. This study explores Prospective Secondary Mathematics Teachers’ (PSMTs) unified understandings of identities and characterizes the structural features of identities that PSMTs attend to. I contribute a theoretical framework of three ways in which PSMTs reason about identities: a do-nothing element, a result of undoing something, and a coordination with inverse, binary operation, and set. I classify the level of coherence of their identity schemas demonstrated as they reasoned about the structural connections among additive, multiplicative, and compositional identities. I illustrate how having unified, coherent understandings of identities can lead PSMTs to reason productively about inverse and identity functions, while having incoherent understandings of identities can lead to inaccurate reasoning about inverse and identity functions. I conclude with teaching implications for fostering PSMTs’ unified understandings of algebraic concepts.