Running to keep up with the lecturer or gradual de-ritualization? Biology students’ engagement with construction and data interpretation graphing routines in mathematical modelling tasks

Through a commognitive lens, we examine twelve first-semester biology students’. engagement with graphing routines as they work in groups, during four sessions of Mathematical Modelling (MM). We trace the students’ meta-level learning, particularly as they fluctuate between deploying graphs for mere illustration of data and as sense-making tools. We account for student activity in relation to precedent events in their experiences of graphing and as fluid, if not always productive, interplay between ritualised and exploratory engagement with graph construction and interpretation routines. The students’ construal of the task situations is marked by efforts to keep up with lecturer expectations which allow for changing degrees of student agency but do not factor in the influence of precedent events. Our analysis has pedagogical implications for the way MM problems are formulated and also foregrounds the capacity of the commognitive framework to trace de-ritualization and meta-level learning in students’ MM activity.     

Facilitated modelling with discrete-event simulation: Reality or myth?

Is it possible for discrete-event simulation to be used in a facilitated workshop environment? Over the last decade there have been various attempts to use simulation in this way, but we argue here that none have been successful in achieving a fully facilitated mode where the model is both developed and used in the workshop. We attempt to use a discrete-event simulation in a facilitated mode as part of a lean improvement workshop in a hospital setting. The model was successfully developed and used within the 3 day period of the workshop. Although the intervention was successful, we still had to build the model in the ‘back-office’, meaning that a fully facilitated mode was not achieved. The paper concludes by discussing how fully facilitated modelling with discrete-event simulation might be made possible; the answer is more about changing mind-sets than about technological challenge.     

Developing the roots of modelling conceptions: ‘mathematical modelling is the life of the world’

A study conducted with 25 Year 6 primary school students investigated the potential for a short classroom intervention to begin the development of a Modelling conception of mathematics on the way to developing a sense of mathematics as a way of thinking about life. The study documents the developmental roots of the cognitive activity, actions and conceptions of both modelling and mathematics that these beginners to modelling displayed. Understanding the conceptions of mathematics that students might hold or be developing and how these can be influenced in early schooling are essential ingredients in any plans for introducing modelling seriously into primary school classrooms. The majority of the students (22/25) were identified as displaying a developing conception of modelling as a way of problem handling. The three other students displayed the developmental roots of a way of understanding the world conception of modelling. These three students also displayed a Modelling conception of mathematics with one showing indications of developing towards a Life conception of mathematics.     

Assessing science students’ attitudes to mathematics: a case study on a modelling project with mathematical software

This article reports the attitudes of students towards mathematics after they had participated in an applied mathematical modelling project that was part of an Applied Mathematics course. The students were majoring in Earth Science at the National Taiwan Normal University. Twenty-six students took part in the project. It was the first time a mathematical modelling project had been incorporated into the Applied Mathematics course for such students at this University. This was also the first time the students experienced applied mathematical modelling and used the mathematical software. The main aim of this modelling project was to assess whether the students’ attitudes toward mathematics changed after participating in the project. We used two questionnaires and interviews to assess the students. The results were encouraging especially the attitude of enjoyment. Hence the approach of the modelling project seems to be an effective method for Earth Science students.     

Predator–prey interaction with harvesting: mathematical study with biological ramifications

In this article, we study a simple predator–prey interaction where predator population is subjected to harvesting. Our qualitative analysis shows different outcomes including switching of stability, oscillations and deterministic extinction. Theoretically observed results have been tested with the parameter values of Paramecium aurelia and its predator Didinium nasutum. Study reveals that harvesting effort and predator’s attack rate may be used as control parameters for the system. Simulation results also indicate that the system may exhibit bistability for some parametric region. Our study also gives the possible answer to the question – why do we frequently observe coexisting predator–prey system in natural system? The study may be helpful to design control strategy for harvested predator–prey system.     

Teachers’ mathematical problem-solving knowledge: In what way is it constructed during teachers’ collaborative work?

This research aims to analyze the type of mathematics problem-solving knowledge for teaching used when working collaboratively in a Lesson Study (LS) process and examine how dialogic interactions contribute to knowledge construction. Five meetings during one LS cycle of a group of eight Swiss primary teachers were video recorded, transcribed and coded with the help of qualitative data analysis software. This analysis is conducted by crossing theoretical frameworks from two different fields in education, namely mathematics education and dialogic analysis. The mixed-method uses quantitative analysis with Markov chains and cross-tables, as well as qualitative analysis at micro-, meso- and macro-levels. This research suggests that participants collectively use their mathematics and their problem-solving content knowledge to focus on pedagogical problem-solving knowledge, that they navigate between different knowledge levels and that the roles of teachers and facilitators are differentiated but are also coequal.     

An in-service primary teacher’s implementation of mathematical tasks: the case of length measurement and perimeter instruction

The purpose of this paper is to examine the cognitive demand levels of tasks used by an in-service primary teacher during length measurement and perimeter instruction and to examine a possible link between these tasks and the teacher’s mathematical knowledge in teaching. For this purpose, a case study approach was used and the data was drawn from classroom observations, semi-structured interviews, and field notes. Specific tasks from length measurement and perimeter instruction were presented and analyzed according to the Mathematical Tasks Framework. Then, how these tasks gave information about the teacher’s mathematical knowledge in teaching in the length measurement and perimeter topics was examined according to the Knowledge Quartet model. According to the findings of the study, the tasks used during length measurement and perimeter instruction were mostly categorized as low-level tasks. In addition, teacher’s mathematical knowledge in teaching affected the implementation of the tasks.     

Semantic contamination and mathematical proof: Can a non-proof prove?

The way words are used in natural language can influence how the same words are understood by students in formal educational contexts. Here we argue that this so-called semantic contamination effect plays a role in determining how students engage with mathematical proof, a fundamental aspect of learning mathematics. Analyses of responses to argument evaluation tasks suggest that students may hold two different and contradictory conceptions of proof: one related to conviction, and one to validity. We demonstrate that these two conceptions can be preferentially elicited by making apparently irrelevant linguistic changes to task instructions. After analyzing the occurrence of “proof” and “prove” in natural language, we report two experiments that suggest that the noun form privileges evaluations related to validity, and that the verb form privileges evaluations related to conviction. In short, we show that (what is judged to be) a non-proof can sometimes (be judged to) prove.     

Is bilevel programming a special case of a mathematical program with complementarity constraints?

Bilevel programming problems are often reformulated using the Karush–Kuhn–Tucker conditions for the lower level problem resulting in a mathematical program with complementarity constraints(MPCC). Clearly, both problems are closely related. But the answer to the question posed is “No” even in the case when the lower level programming problem is a parametric convex optimization problem. This is not obvious and concerns local optimal solutions. We show that global optimal solutions of the MPCC correspond to global optimal solutions of the bilevel problem provided the lower-level problem satisfies the Slater’s constraint qualification. We also show by examples that this correspondence can fail if the Slater’s constraint qualification fails to hold at lower-level. When we consider the local solutions, the relationship between the bilevel problem and its corresponding MPCC is more complicated. We also demonstrate the issues relating to a local minimum through examples.     

Semismooth Newton method for the lifted reformulation of mathematical programs with?complementarity constraints

We consider a reformulation of mathematical programs with complementarity constraints, where by introducing an artificial variable the constraints are converted into equalities which are once but not twice differentiable. We show that the Lagrange optimality system of such a reformulation is semismooth and BD-regular at the solution under reasonable assumptions. Thus, fast local convergence can be obtained by applying the semismooth Newton method. Moreover, it turns out that the squared residual of the Lagrange system is continuously differentiable (even though the system itself is not), which opens the way for a natural globalization of the local algorithm. Preliminary numerical results are also reported.     

Ethnomathematics: Why, and what else?

Ethnomathematics—we all have some notions of what it is, but should it be influencing school mathematics? This paper considers a justification for the influence from a constructivist perspective, and considers the implications of enactivism and other ways of thinking on ethnomathematics. My conclusion is that the influence should be broader—that different cultural groups may have different ways of knowing, that we may be asking the wrong questions, and that we might need to consider ‘ethno-education’     

A ��task-centric approach�� to professional development: enhancing and sustaining mathematics teachers�� ability to implement cognitively challenging mathematical tasks

In this article, we describe a task-centric approach to professional development for mathematics teachers in which teachers?? learning experiences are focused on the selection and implementation of cognitively challenging mathematical tasks. We examined teachers?? selection and implementation of cognitively challenging tasks at three points in time: before and after their participation in the professional development initiative and during a follow-up data collection 2?years later. Data included instructional tasks, samples of student work, and classroom observations, and were compared between the time points to identify changes in teachers?? task selection and implementation and to determine whether these changes were sustained over time. Results indicate that teachers increased and sustained their ability to select high-level instructional tasks and to maintain the level of cognitive demand during instruction. All teachers, however, did not exhibit this pattern. Portraits of teachers who continued to select and enact tasks at a high level are contrasted with those who did not, and factors are identified to account for teachers?? current practices.     

Stochastic programming duality: ∞ multipliers for unbounded constraints with an application to mathematical finance

A new duality theory is developed for a class of stochastic programs in which the probability distribution is not necessarily discrete. This provides a new framework for problems which are not necessarily bounded, are not required to have relatively complete recourse, and do not satisfy the typical Slater condition of strict feasibility. These problems instead satisfy a different constraint qualification called direction-free feasibility to deal with possibly unbounded constraint sets, and calmness of a certain finite-dimensional value function to serve as a weaker condition than strict feasibility to obtain the existence of dual multipliers. In this way, strong duality results are established in which the dual variables are finite-dimensional, despite the possible infinite-dimensional character of the second-stage constraints. From this, infinite-dimensional dual problems are obtained in the space of essentially bounded functions. It is then shown how this framework could be used to obtain duality results in the setting of mathematical finance. Mathematics Subject Classification (2000): 46N10, 49N15, 65K10, 90C15, 90C46Research supported in part by a grant of the National Science Foundation.Received: 9, May 2001     

Proof schemes combined: mapping secondary students’ multi-faceted and evolving first encounters with mathematical proof

In this paper, we propose an enriched and extended application of Harel and Sowder’s proof schemes taxonomy that can be used as a diagnostic tool for characterizing secondary students’ emergent learning of proof and proving. We illustrate this application in the analysis of data collected from 85 Year 9 (age 14–15) secondary students. We capture these students’ first encounters with proof and proving in an educational context (mixed ability, state schools in Greece) where mathematical proof is explicitly present in algebra and geometry lessons and where proving skills are typically expected, and rewarded, in key national examinations. We analyze student written responses to six questions, soon after the students had been introduced to proof and we identify evidence of six of the seven proof schemes proposed by Harel and Sowder as well as a further eight combinations of the six. We observed these combinations often within the response of the same student and to the same item. Here, we illustrate the eight combinations and we claim that a dynamic use of the proof schemes taxonomy that encompasses sole and combined proof schemes is a potent theoretical and pedagogical tool for mapping students’ multi-faceted and evolving competence in, and appreciation for, proof and proving.     

Communicative characteristics of teachers’ mathematical talk with children: from knowledge transfer to knowledge investigation

In-service teachers actively collaborated in a developmental research project. The main aim of the research project was the advancement of one central aspect of teacher professionalism: teachers’ diagnostic competencies. Conditions of understanding and possibilities of enriching teachers’ talk are of special interest because mathematics teaching is particularly affected by speech and communication (Söbbeke and Steinbring in Mathematik für Kinder—Mathematik von Kindern, pp. 26–38, 2004). One research focus was on the support of a productive enhancement of the teachers’ talk with one child. Is the teacher’s talk mainly a kind of knowledge transfer similar to traditional instruction or can it be seen as an investigation of the child’s own views and ideas of elementary mathematical knowledge? These teachers’ talks with one child should offer more reflective communication between teacher and child and result in a changed view of the child’s mathematical understanding. Using an elaborate interpretation based on a theoretical instrument of analysis, called “Forms of teachers’ mathematical Interaction (Formal-In)”, we describe the development from the first diagnostic talk with one child, at the beginning to the last talk at the end of the research project. Using an elaborate analysis of short episodes of teachers’ talk distinguishing the interactive and the epistemological dimensions, we can describe how both dimensions influence each other. The theoretically identified characteristics of teachers’ talk together with compatible video cases can be used in theory-based (in-service) teacher training aimed at enhancing professionalism.