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.     

The parametric nature of two students’ covariational reasoning

Researchers have argued that covariational reasoning is foundational for learning a variety of mathematics topics. We extend prior research by examining two students’ covariational reasoning with attention to the extent they became consciously aware of the parametric nature of their reasoning. We first describe our theoretical background including different conceptions of covariation researchers have found useful when characterizing student reasoning. We then present two students’ activities during a teaching experiment in which they constructed and reasoned about covarying quantities. We highlight aspects of the students’ reasoning that we conjectured created an intellectual need that resulted in their constructing a parameter quantity or attribute, a need we explored in closing teaching episodes. We discuss implications of these results for perspectives on covariational reasoning, students’ understandings of graphs and parametric functions, and areas of future research.     

On students’ conceptions of arithmetic average: the case of inference from a fixed total

There is more to understanding the concept of mean than simply knowing and applying the add-them-up and divide algorithm. In the following, we discuss a component of understanding the mean – inference from a fixed total – that has been largely ignored by researchers studying students understanding of mean. We add this component to the list of types of reasoning needed to understand mean and discuss student responses to tasks designed to elicit this component of reasoning. These responses reveal that inference from a fixed total reasoning is rare even in advance high school students.     

The effect of quantitative reasoning on prospective mathematics teachers’ proof comprehension: The case of real numbers

We report a mixed-methods research study investigating the effect of quantitative reasoning on prospective mathematics teachers’ comprehension of a proof on real numbers. Nineteen prospective mathematics teachers engaged in quantitative reasoning while developing real numbers as rational number sequences in a series of instructional activities. All participants completed a proof comprehension assessment prior to and upon completion of the instruction. Six of the prospective mathematics teachers also participated in semi-structured interviews after the post-test. Results showed a significant difference in proof comprehension performance between the pre- and post-tests. Moreover, results from the interviews showed that prospective teachers reasoned quantitatively on the proof comprehension dimensions. Results suggest that engaging in quantitative reasoning during instruction may help to develop proof comprehension, particularly in situations involving the analysis of proofs entailing properties of the real number system. We recommend embedding quantitative reasoning in teacher education and professional development programs to facilitate mathematics teachers’ proof comprehension and proving activities.     

Algorithmic contexts and learning potentiality: a case study of students’ understanding of calculus

The study explores the nature of students’ conceptual understanding of calculus. Twenty students of engineering were asked to reflect in writing on the meaning of the concepts of limit and integral. A sub-sample of four students was selected for subsequent interviews, which explored in detail the students’ understandings of the two concepts. Intentional analysis of the students’ written and oral accounts revealed that the students were expressing their understanding of limit and integral within an algorithmic context, in which the very ‘operations’ of these concepts were seen as crucial. The students also displayed great confidence in their ability to deal with these concepts. Implications for the development of a conceptual understanding of calculus are discussed, and it is argued that developing understanding within an algorithmic context can be seen as a stepping stone towards a more complete conceptual understanding of calculus.     

How social interactions within a class depend on the teacher’s assessment of the students’ various mathematical capabilities

In this contribution, we will address from aclinical point of view the issue of the interrelations between the knowledge acquiring processes and the social interactions within a class of mathematics: a) how can the knowledge that is to be acquired determine the kind of social relationship established during a didactic interaction, and b) reciprocally, how can the social relationship already established within the class influence each and every student’s acquisition of knowledge?     

Calculus students’ understanding of the vertex of the quadratic function in relation to the concept of derivative

The purpose of this study was to gain insight into 30, first year calculus students’ understanding of the relationship between the concept of vertex of a quadratic function and the concept of the derivative. APOS (action-process-object-schema) theory was applied as a guiding framework of analysis on student written work, think-aloud and follow up group interviews. Students’ personal meanings of the vertex, including misconceptions, were explored, along with students’ understanding to solve problems pertaining to the derivative of a quadratic function. Results give evidence of students’ weak schema of the vertex, lack of connection between different problem types and the importance of linguistics in relation to levels of APOS theory. A preliminary genetic decomposition was developed based on the results. Future research is suggested as a continuation to improve student understanding of the relationship between the vertex of quadratic functions and the derivative.     

The value of combining forecasts in inventory management – a case study in banking

Managing inventories in the face of uncertain stochastic demand requires an investment in safety stocks. These are related to the accuracy in forecasting future demands and the noise in the demand generation process. Reducing the demand forecasting error can free up capital and space, and reduce the operating costs of managing the inventories. A leading bank in Hong Kong consumes more than three hundred kinds of printed forms for its daily operations. A major problem of its inventory control system for the forms management is to forecast the monthly demand of these forms. In this study the idea of combining forecasts is introduced and its practical application is addressed. The individual forecasts come from well established time series models and the weights for combination are estimated with Quadratic Programming. The combined forecast is found to perform better than any of the individual forecasts.     

Poverty,inequality and mathematics performance: the case of South Africa’s post-apartheid context

South Africa’s recent history of apartheid, its resultant high levels of poverty and extreme social and economic distance between rich and poor continue to play-out in education in complex ways. The country provides a somewhat different context for exploring the relationship between SES and education than other countries. The apartheid era only ended in 1994, after which education became the vehicle for transforming society and a political rhetoric of equity and quality education for all was prioritized. Thus education focused on redressing inequalities; and major curriculum change, with on-going revisions, was attempted. In this sense engagement with SES and education became foregrounded in policy, political discourse and research literature. Yet for all the political will and rhetoric little has been achieved and indicators are that inequality has worsened in mathematics education, where it is particularly pronounced. This paper proposes that continued research confirming poverty–underachievement links, which suggest an inevitability of positive correlations, is unhelpful. Instead we should explore issues of disempowerment and agency, constraints and possibilities, and the complex interplay of factors that create these widely established national statistics while simultaneously defying them in particular local contexts. Such research could shift the focus from a discourse of deficit and helplessness towards a discourse of possibilities in the struggle for equity and quality education for all.     

The Turán number for the edge blow-up of trees: The missing case

The edge blow-up of a graph is the graph obtained from replacing each edge of it by a clique of the same size where the new vertices of the cliques are all different. Wang, Hou, Liu and Ma determined the Turán number of the edge blow-up of trees except one particular case. Answering a problem posed by them, we determined the Turán number of this particular case.     

A model for developing students’ example space: the key role of the teacher

Our research addresses the role of examples to foster the students’ development of the mathematical concepts, and of their mathematical ways of thinking. We consider the notion of example space introduced by Watson and Mason (Mathematics as a constructive activity: learners generating examples, 2005), particularly when it is not formed by a simple juxtaposition of examples, rather it is endowed by a certain structure. Such a structure is provided by the semiotic actions and by the theoretic and logical dimensions of the mathematical activities. However, the formation of structured example spaces is far from being an automatic process. In this paper, we focus on the genesis of examples and on the role of the teacher in helping the students to structure their examples spaces through the so-called cognitive apprenticeship method. We point out that the genesis of examples is often accomplished within a complex cyclic dynamics, the “cycle of examples production and modification”. We illustrate it by means of two emblematic episodes from a classroom discussion. We show that the teacher’s intervention can be crucial in helping the students to modify a wrong example, to generate the right one for the task and to start the long-term process of building up the structure of their own space of examples.     

On the content-dependence of prospective teachers’ knowledge: a case of exemplifying definitions

In this article, we demonstrate that prospective teachers’ content knowledge related to defining mathematical concepts is dependent on content area. We use the example of generation (a research tool we developed in a previous study) to investigate prospective teachers’ knowledge. We asked prospective secondary mathematics teachers to provide multiple examples of definitions of concepts from different areas of mathematics. We examined teacher-generated examples of concept definition and analysed individual and collective example spaces, focusing on their correctness and richness. We demonstrate differences in prospective teachers’ knowledge associated with defining mathematical concepts in geometry, algebra and calculus.     

‘Negative of my money,positive of her money’: secondary students’ ways of relating equations to a debt context

We interviewed 40 students each in grades 7 and 11 to investigate their integer-related reasoning. In one task, the students were asked to write and interpret equations related to a story problem about borrowing money from a friend. All the students solved the story problem correctly. However, they reasoned about the problem in different ways. Many students represented the situation numerically without invoking negative numbers, whereas others wrote equations involving negative numbers. When asked to interpret equations involving negative numbers in relation to the story, students did so in two ways. Their responses reflect distinct perspectives concerning the relationship between arithmetic equations and borrowing/owing. We discuss these findings and their implications regarding the role of contexts in integer instruction.     

The role of definition in students’ understanding with particular reference to the concavity of a function

In this study, the effect of different definitions of concavity on students’ understanding was investigated. Students enrolled in the calculus-II course in the science and engineering faculties in Anadolu University, Turkey were divided into two groups and each group was given a different definition of concavity. At the end of the study, both groups’ understanding was observed to depend on the definition. Another observation is that both groups can apply the second derivative test as well as each other.     

Identifying authority structures in mathematics classroom discourse: a case of a teacher’s early experience in a new context

We explore a conceptual frame for analyzing mathematics classroom discourse to understand the way authority is at work. This case study of a teacher moving from a school where he is known to a new setting offers us the opportunity to explore the use of the conceptual frame as a tool for understanding how language practice and authority relate in a mathematics classroom. This case study illuminates the challenges of establishing disciplinary authority in a new context while also developing the students’ sense of authority within the discipline. To analyze the communication in the teacher’s grade 12 class in the first school and grade 9 class early in the year at the new school, we use the four categories of positioning drawn from our earlier analysis of pervasive language patterns in mathematics classrooms—personal authority, discourse as authority, discursive inevitability, and personal latitude.     

Broadening the sense of ‘dynamic’: a microworld to support students’ mathematical generalisation

In this paper, we seek to broaden the sense in which the word ‘dynamic’ is applied to computational media. Focussing exclusively on the problem of design, the paper describes work in progress, which aims to build a computational system that supports students’ engagement with mathematical generalisation in a collaborative classroom environment by helping them to begin to see its power and to express it for themselves and for others. We present students’ strengths and challenges in appreciating structure and expressing generalities that inform our overall system design. We then describe the main features of the microworld that lies at the core of our system. In conclusion, we point to further steps in the design process to develop a system that is more adaptive to students’ and teachers’ actions and needs.     

The probabilistic nature of McShane’s identity: planar tree coding of simple loops

We present here a probabilistic interpretation of McShane’s identities. These identities actually describe a probability on the space of simple loops.     

The opaque nature of generic examples: The structure of student teachers’ arguments in multiplicative reasoning

The study aims to explore the structural aspects of generic examples, to get better insight into what makes them potentially opaque for learners. We have analyzed 27 written arguments, for which student teachers (grades 1–10) were asked to use a generic example to prove a given statement in multiplication. Using Toulmin’s framework, we developed five categories of arguments based on their structure: examples, empirical arguments, leap arguments, embedded arguments, and other arguments. Also, we conclude that none of the student teachers provided arguments that we recognize as complete generic examples. The results bring us to a discussion about features of generic examples making them difficult to come to grips with, having implications for how teacher educators can support student teachers’ learning to prove. From this, we propose a definition of generic examples that attends to the criteria suggested in previous research, yet, emphasizing their structural nature.