Exploring mathematical connections of pre-university students through tasks involving rates of change

This paper reports the results of a research exploring the mathematical connections of pre-university students while they solving tasks which involving rates of change. We assume mathematical connections as a cognitive process through which a person finds real relationships between two or more ideas, concepts, definitions, theorems, procedures, representations or meanings or with other disciplines or the real-world. Four tasks were proposed to the 33 pre-university students that participated in this research; the central concept of the first task is the slope, the last three tasks contain concepts like velocity, speed and acceleration. Task-based interviews were conducted to collect data and later analysed with thematic analysis. Results showed most of the students made mathematical connections of the procedural type, the mathematical connections of the common features type are made in smaller quantities and the mathematical connection of the generalization type is scarcely made. Furthermore, students considered slope as a concept disconnected from velocity, speed and acceleration.     

Young students’ subjective interpretations of mathematical diagrams: elements of the theoretical construct “frame-based interpreting competence”

During the last few decades several studies have showed that mathematical visual aids are not at all self-explanatory. Nevertheless, students do make sense of those representations spontaneously and—as a matter of course—cannot avoid their own sense-making. Further, the function of visual aids as “re-presentation” of a given structure is complemented through an epistemological function to explore mathematical structures and generate new meaning. But in which way do socially learned interpreting schemes (frames) influence children’s subjective interpretations of mathematical diagrams? The CORA project investigates which frames can be reconstructed in young pupils’ interpretations of visual diagrams. This paper presents central ideas, theoretical background and—by means of short sequences from pre- and post-interviews—first aspects of “frame-based interpreting competence”. We describe children’s subjective frames in a range between “object-oriented” (focus on the diagram’s visible elements) and “system-oriented” (focus on relation between those elements).     

Children's approaches to area measurement through different contexts

This study focuses on 12 years old children's approaches to area measurement in a project environment. These approaches are not explored through a specific set of mathematical tasks. The tasks, here, are defined through researchers' and children's interactions in a classroom. The children by working in small groups are asked to make a proposal about the location and the form of an area which would be given to them for their leisure activities. This environment defines different contexts where the children act and consider different aspects of the area measurement. These aspects are identified and compared among the three groups of children. The study has shown that the concept of area measurement carries different cultural dimensions for the children. Moreover, the children use those elements of the concept which fit in with their personal experience and the tasks they have to face.     

Mathematical modeling and computer simulation of a robotic rat pup

In this paper, we describe the mathematics and computer implementation of a robotic rat pup simulation. Our goal is to understand neurobehavioral principles in a mammalian model organism—the Norway rat pup (Rattus norvegicus). Our approach is unique in that animal, simulation, and robot studies occur in parallel and inform each other. Behavior is dependent on the nervous system, body morphology, physiology, environment, and the interactions among these elements. Autonomous robotics hardware models and their associated simulations allow the possibility of systematically manipulating variables in each of these elements in ways that would be impossible using live animals. Specifically, we describe the development and validation of a Newtonian-dynamics-based simulation of a robotic rat pup, including mathematical formulation and computer implementation. The computer simulation consists of three distinct components that interact to simulate robotic behavior: (1) dynamics of the robotic rat pup itself, including sensors and actuators, (2) environmental coupling dynamics of the robot arena with the robotic rat pup, and (3) the robot control algorithms as implemented on the physical robot. The mathematical formulation, software implementation, model identification, model validation, and an application example are all described.     

Symmetry and bifurcation in vestibular system

The vestibular system in almost all vertebrates, humans included, controls balance by employing a set of six semicircular canals, three in each inner ear, to detect angular accelerations of the head. Signals from the canals are transmitted to neck motoneurons and activate eight corresponding muscle groups. These signals may be either excitatory or inhibitory, depending on the direction of acceleration. McCollum and Boyle have observed that in the cat the network of neurons concerned possesses octahedral symmetry, a structure deduced from the known innervation patterns (connections) from canals to muscles. We re-derive the octahedral symmetry from mathematical features of the probable network architecture, and model the movement of the head in response to the activation patterns of the muscles concerned. We assume that connections among neck muscles can be modeled by a ‘coupled cell network’, a system of coupled ODEs whose variables correspond to the eight muscles, and that network also has octahedral symmetry. The network and its symmetries imply that these ODEs must be equivariant under a suitable action of the octahedral group. Using results of Ashwin and Podvigina, we show that with the appropriate group actions, there are six possible spatiotemporal patterns of time-periodic states that can arise by Hopf bifurcation from an equilibrium corresponding to natural head motions. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Paradox,programming, and learning probability: A case study in a connected mathematics framework

Formal methods abound in the teaching of probability and statistics. In the Connected Probability project, we explore ways for learners to develop their intuitive conceptions of core probabilistic concepts. This article presents a case study of a learner engaged with a probability paradox. Through engaging with this paradoxical problem, she develops stronger intuitions about notions of randomness and distribution and the connections between them. The case illustrates a Connected Mathematics approach: that primary obstacles to learning probability are conceptual and epistemological; that engagement with paradox can be a powerful means of motivating learners to overcome these obstacles; that overcoming these obstacles involves learners making mathematics—not learning a “received” mathematics and that, through programming computational models, learners can more powerfully express and refine their mathematical understandings.     

The influence of semi-rigid connections on the shakedown of elasto-plastic frames

The main structural elements of steel framed multi-storey structures are the columns, the beams and their connections. The assumption has been widely applied in the past that the connections are either rigid or pinned. The actual behaviour of the connections is however somewhere between these limits, they are semi-rigid. The aim of this paper is to analyze the effect of the semi-rigid connections on the shakedown of steel structures under multi-parameter static loading. In the analysis, to control the plastic behavior of the structure, bound on the complementary strain energy of the residual forces is applied. The formulation of the problem yields to nonlinear mathematical programming. The results of the numerical calculation show that the semi-rigid connections can influence significantly the magnitude of the shakedown parameter. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Representations in applying functions

As part of a large research project—Heuristic Education of Mathematics: developing and investigating strategies to teach applied mathematical problem solving—inquiries were made into the question of the transfer of knowledge and skills from the area of functions to real-world problems. In particular, studies were made of the translation processes from one representation of a problem into another representation. Surprisingly, students often used informal methods not taught in their lessons. After a full year of teaching mathematics, including a lot of applied problem solving, a shift from informal methods to the analytical (expert) solution method was identified. There were also significant differences among the learning results of three instructional design conditions. This research was extended to consider implications of the use of the graphic calculator. Casual use of the graphic calculator diminished the application of analytical methods, but integrated use brought about an enrichment of solution methods.     

Hyperbolastic modeling of wound healing

A new mathematical model for wound healing is introduced and applied to three sets of experimental data. The model is easy to implement but can accommodate a wide range of factors affecting the wound healing process. The data sets represent the areas of trace elements, diabetic wounds, growth factors, and nutrition within the field of wound healing. The model produces an explicit function accurately representing the time course of healing wounds from a given data set. Such a function is used to study variations in the healing velocity among different types of wounds and at different stages in the healing process. A new multivariable model of wound healing capable of analyzing the effects of several variables on accelerating the wound healing process is also introduced. Such a model can help to formulate appropriate strategies to treat wounds. It also would enable us to evaluate the efficacy of different treatment modalities during the inflammatory, proliferative, and tissue remodeling phases.     

What part of the concept of acceleration is difficult to understand: the mathematics, the physics, or both?

In this paper, details of student difficulties in understanding the concept of acceleration and the mathematical and physical/intuitive sources of these are delineated by utilizing the teaching experiment methodology. As a result of the study, two anchoring analogies are proposed that can be used as a diagnostic tool for students’ alternative conceptions. These can be used in teaching to highlight the peculiarity of acceleration concept. This study portrays how seeing acceleration as ‘rate of change’ of a quantity (velocity) and recognizing the consequences of such a definition are hindered in certain ways which in turn negatively affect learning the concept of force. This is also an example that illustrates that a rather “simple” mathematical concept (i.e., rate of change) for the expert can become a complex phenomenon when embedded in a physical concept (i.e., acceleration) which is consistently found to be as a misconception among learners at various levels that is widely occurring and very resistant to change.     

Coupled modified KdV equations,skew orthogonal polynomials,convergence acceleration algorithms and Laurent property

In this paper, we show that the coupled modified KdV equations possess rich mathematical structures and some remarkable properties. The connections between the system and skew orthogonal polynomials, convergence acceleration algorithms and Laurent property are discussed in detail.     

Mathematics and molecules: Exploring connections via programming

This article examines the self-directed activity of two students who learned about molecular structure by writing computer programs. The students wrote programs to display the solution of a mathematics problem and then extended their programs to represent several classes of organic molecules. In the course of this activity, the students learned the standard system for naming organic molecules while maintaining a sense of ownership of their project. We discuss ways to enhance the mathematical connections to chemistry education.     

Word problems in mathematics education: a survey

Word problems are among the most difficult kinds of problems that mathematics learners encounter. Perhaps as a result, they have been the object of a tremendous amount research over the past 50 years. This opening article gives an overview of the research literature on word problem solving, by pointing to a number of major topics, questions, and debates that have dominated the field. After a short introduction, we begin with research that has conceived word problems primarily as problems of comprehension, and we describe the various ways in which this complex comprehension process has been conceived theoretically as well as the empirical evidence supporting different theoretical models. Next we review research that has focused on strategies for actually solving the word problem. Strengths and weaknesses of informal and formal solution strategies—at various levels of learners’ mathematical development (i.e., arithmetic, algebra)—are discussed. Fourth, we address research that thinks of word problems as exercises in complex problem solving, requiring the use of cognitive strategies (heuristics) as well as metacognitive (or self-regulatory) strategies. The fifth section concerns the role of graphical representations in word problem solving. The complex and sometimes surprising results of research on representations—both self-made and externally provided ones—are summarized and discussed. As in many other domains of mathematics learning, word problem solving performance has been shown to be significantly associated with a number of general cognitive resources such as working memory capacity and inhibitory skills. Research focusing on the role of these general cognitive resources is reviewed afterwards. The seventh section discusses research that analyzes the complex relationship between (traditional) word problems and (genuine) mathematical modeling tasks. Generally, this research points to the gap between the artificial word problems learners encounter in their mathematics lessons, on the one hand, and the authentic mathematical modeling situations with which they are confronted in real life, on the other hand. Finally, we review research on the impact of three important elements of the teaching/learning environment on the development of learners’ word problem solving competence: textbooks, software, and teachers. It is shown how each of these three environmental elements may support or hinder the development of learners’ word problem solving competence. With this general overview of international research on the various perspectives on this complex and fascinating kind of mathematical problem, we set the scene for the empirical contributions on word problems that appear in this special issue.

     

The Lappo-Danilevskii method and trivial intersections of radicals in lower central series terms for certain fundamental groups

In this paper, it is proved that the intersection of the radicals of nilpotent residues for the generalized pure braid group corresponding to an irreducible finite Coxeter group or an irreducible imprimitive finite complex reflection group is always trivial. The proof uses the solvability of the Riemann—Hilbert problem for analytic families of faithful linear representations by the Lappo-Danilevskii method. Generalized Burau representations are defined for the generalized braid groups corresponding to finite complex reflection groups whose Dynkin—Cohen graphs are trees. The Fuchsian connections for which the monodromy representations are equivalent to the restrictions of generalized Burau representations to pure braid groups are described. The question about the faithfulness of generalized Burau representations and their restrictions to pure braid groups is posed.     

The L-Functions and Modular Forms Database Project

The Langlands Programme, formulated by Robert Langlands in the 1960s and since much developed and refined, is a web of interrelated theory and conjectures concerning many objects in number theory, their interconnections, and connections to other fields. At the heart of the Langlands Programme is the concept of an L-function. The most famous L-function is the Riemann zeta function, and as well as being ubiquitous in number theory itself, L-functions have applications in mathematical physics and cryptography. Two of the seven Clay Mathematics Million Dollar Millennium Problems, the Riemann Hypothesis and the Birch and Swinnerton-Dyer Conjecture, deal with their properties. Many different mathematical objects are connected in various ways to L-functions, but the study of those objects is highly specialized, and most mathematicians have only a vague idea of the objects outside their specialty and how everything is related. Helping mathematicians to understand these connections was the motivation for the L-functions and Modular Forms Database (LMFDB) project. Its mission is to chart the landscape of L-functions and modular forms in a systematic, comprehensive, and concrete fashion. This involves developing their theory, creating and improving algorithms for computing and classifying them, and hence discovering new properties of these functions, and testing fundamental conjectures. In the lecture I gave a very brief introduction to L-functions for non-experts and explained and demonstrated how the large collection of data in the LMFDB is organized and displayed, showing the interrelations between linked objects, through our website www.lmfdb.org. I also showed how this has been created by a worldwide open-source collaboration, which we hope may become a model for others.     

Making Connections: Elementary Teachers' Construction of Division Word Problems and Representations

If teachers make few connections among multiple representations of division, supporting students in using representations to develop operation sense demanded by national standards will not occur. Studies have investigated how prospective and practicing teachers use representations to develop knowledge of fraction division. However, few studies examined primary (K‐3) teachers' learning of contextual division problems, making connections among representations of division, and resolving the ambiguity of representing quotients with remainders. A written post‐course assessment provided evidence that most teachers created partitive division word problems, used a set model without splitting the remainder, and wrote equations with limited success. Post‐course written reflections demonstrated that many teachers developed pedagogical knowledge for helping students make connections among multiple representations, and mathematical knowledge of unit fractions. These findings suggest two areas that have implications for mathematics teacher educators who design professional development courses to facilitate teachers' learning of mathematical content and pedagogical knowledge of division and fraction relationships.     

The role of metacognitive monitoring in explaining differences in mathematics achievement

The relationship between practised monitoring activities and performance, especially in mathematics was examined within three nested studies. The first study deals with problems of faulty term rewritings submitted to three groups of subjects—10th to 13th graders, differing in their mathematical performance—whose task was to find the mistakes. Moreover, a questionnaire on the practice and appreciation of monitoring activities was developed. The third study, first, repeats the first study with a similar population and secondly adds interviews with some of the subjects while solving additional items concerning faulty term rewritings. Studies 1 and 3 show similar success in finding mistakes and in the replies to the questionnaire within the various groups. Furthermore, the third study points up that the subject’s answers do neither predict the practised monitoring nor the success in the test. However, the success correlates significantly with the practised monitoring. For a deeper understanding concerning the role of metacognition in explaining performance, the second study examined two of the groups who had already been involved in the first study. These were assigned some problems of a matrices test as used in cognitive psychology. While trying to solve the problem, their eye movements were recorded by means of an eye-tracker. Afterwards they had to justify their solutions in an interview. The eye movements were analysed, the verbal comments classified. Again, the groups differ in their problem solving success, dependant on the quality of the monitoring practised. Altogether, the results of the three studies elucidate the importance of practised metacognitive monitoring activities not only for success in school algebra, but furthermore the ability and the willingness to do it is deeper anchored in a person than just a trained behaviour for school algebra.     

The location of public facilities intermediate to the journey to work

Location—allocation models typically locate facilities with respect to points to be served, for example to the homes of potential patrons. Certain types of facility, however, are employed by persons who travel to the facility from their homes and continue their journey to another location. Child care facilities are an example of this pattern of patronage, with parents dropping children off at a centre en route to work. The paper presents a discrete-space location—allocation model minimizing the diversion of patrons' journeys to work. The problem reduces to the structure and combinatorial dimensions of the simple P-median problem. The model is applied to the transit worktrip patterns of single parents in Edmonton, Canada. The facilities generated by the model tend to central locations in the city where workplaces are concentrated and transit connections are efficient. The model provides a compromise between ones minimizing home-facility travel times and facility-workplace travel times.     

Kinematic and dynamic analysis of the Gantry-Tau,a 3-DoF translational parallel manipulator

This paper deals with the mathematical modeling of kinematics and dynamics of the 3-degrees-of-freedom Gantry-Tau manipulator. Compared to many other parallel robots, Gantry-Tau offers a large accessible workspace and high stiffness. The kinematics of Gantry-Tau is presented which includes inverse kinematics formulation for the position, velocity and acceleration of the mechanism. Also, based on the obtained Jacobin matrices, singular configurations of the robot are studied. Afterwards, the equations of the inverse dynamic model of the Gantry-Tau are obtained through two different methods, i.e., virtual work and Newton–Euler. Finally, a case study is performed to verify the correctness of the derived models and investigate their computational efficiency.     

Developing Student Understanding: Contextualizing Calculus Concepts

This study looked at the practice of one high school teacher who provided students with concrete examples from their physics class to give them a contextually rich environment in which to explore the abstractions of calculus. Students discovered connections between the physics concepts of position, velocity, and acceleration and the calculus concepts of function, derivative, and antiderivative. The qualitative study sought to describe several critical aspects of understanding: students' ability to explain concepts and procedures, to apply concepts in a physics context, and to explore their own learning. It included 32 seniors at a large, urban, comprehensive, religious school in a midwestern stale. Samples of student work and reflections were collected by the teacher, as well as by students in individual portfolios. The teacher kept a reflective journal. This study suggests that making connections between calculus and physics can yield deep understandings of semantic as well as procedural knowledge.