An historical example of mathematical modelling: the trajectory of a cannonball

Classroom considerations of the concept and processes of mathematical modelling can do much to strengthen students’ problem solving skills. A systematic exposure to the techniques of mathematical modelling helps students formulate problems, re‐think those problems in mathematical terms, appreciate possible solution constraints and seek solutions that are realistic within the scope and conditions of the problem. While many mathematical modelling situations can be found in today's world, there are special pedagogical values in examining existing mathematical models that have an historical basis. Such an examination should reveal the mechanics of a modelling situation and how a model evolves or is refined to meet ever increasing human demands for accuracy or practicality. The trajectory of a cannonball provides such a modelling example. This topic captures the imagination of students and supplies the basis for a variety of classroom discussions and problem solving encounters.     

Contextual approach in teaching mathematics: an example using the sum of series of positive integers

In this paper, a contextual approach to teaching Mathematics at the pre-university level is recommended, and an example is illustrated. A context in the form of a real common mathematical problem is presented to the students. Different approaches to tackle the problem (from topics within and outside the syllabus) can be elicited from students. The insight obtained from the various methods of solving the problem can be used to deepen students’ learnt concepts and to enhance concepts to be learnt later in the curriculum.     

How students use the ‘see similar example’ feature in online mathematics homework

Homework is one of students’ opportunities to learn mathematics, but we know little about what students learn from homework. This study employs the instructional triangle and didactic contract to explore how students used the ‘see similar example’ feature in an online homework platform and how that use reflected their learning goals. Findings indicate students used similar examples to troubleshoot, to check if they were on the right track, and to see the form of the answer. Students also sought to unpack the reasoning in solution steps, used solutions as templates for solving their own problems, and sometimes copied answers. One student did a ‘see similar example’ problem for more practice. Students’ goals included completing the homework, maximizing their score, and understanding the content. This research lays groundwork for future work characterizing what students learn from homework and how features that provide students with similar examples help or hinder their learning.     

Multiple-solution problems in a statistics classroom: an example

The mathematics education literature shows that encouraging students to develop multiple solutions for given problems has a positive effect on students’ understanding and creativity. In this paper, we present an example of multiple-solution problems in statistics involving a set of non-traditional dice. In particular, we consider the exact probability mass distribution for the sum of face values. Four different ways of solving the problem are discussed. The solutions span various basic concepts in different mathematical disciplines (sample space in probability theory, the probability generating function in statistics, integer partition in basic combinatorics and individual risk model in actuarial science) and thus promotes upper undergraduate students’ awareness of knowledge connections between their courses. All solutions of the example are implemented using the R statistical software package.     

There exist 6n/13 ordinary points

In 1958 L. M. Kelly and W. O. J. Moser showed that apart from a pencil, any configuration ofn lines in the real projective plane has at least 3n/7 ordinary or simple points of intersection, with equality in the Kelly-Moser example (a complete quadrilateral with its three diagonal lines). In 1981 S. Hansen claimed to have improved this ton/2 (apart from pencils, the Kelly-Moser example and the McKee example). In this paper we show that one of the main theorems used by Hansen is false, thus leavingn/2 open, and we improve the 3n/7 estimate to 6n/13 (apart from pencils and the Kelly-Moser example), with equality in the McKee example. Our result applies also to arrangements of pseudolines.The research of J. Csima was supported in part by NSERC Grant A4078. E. T. Sawyer's research was supported in part by NSERC Grant A5149.     

Richardson数对后台阶流动熵产的影响

后台阶流动是研究伴随有传热现象的分离流动的常用模型．虽然Richardson数的改变会明显影响分离流动的流动和传热特性,但是迄今为止关于Richardson数对后台阶流动熵产影响的研究依然很少．基于求解熵产方程,第一次系统研究Richardson数对后台阶流动熵产的影响．对于求解熵产方程所需的速度和温度等变量,通过格子Boltzmann方法来得到．通过上述工作可以发现,后台阶流动中熵产和Bejan数的分布随Richardson数变化显著．总熵产数是Richardson数的单调减函数而平均Bejan数是Richardson数的单调增函数．     

Developing Interdisciplinary Units: A Strategy Based on Problem Solving

While the benefits of the interdisciplinary unit are well documented, it presents a complex challenge to teachers in the natural and social sciences, mathematics, and humanities. Teachers must become active curriculum designers who shape and edit the curriculum according to students' needs. This paper describes knowledge for teachers as curriculum designers and a framework for interdisciplinary unit development. The framework addresses a metacurricular process (problem solving) that will be the unit centerpiece, the development of this central process related to the learner, and the tasks that teach explicit learning and thinking skills attached to the central process. An example of the framework in action is also described. As the faculty and curriculum coordinators for an innovative summer academy for minority students in northern Arizona have used this framework, they have evolved from a group that created a good idea to interest students with parallel subject development in separate classrooms to humanities/mathematics/science teams united in one team/classroom, in which content is integrated through the actions of the problem solving process.     

Learning to be an opportunistic word problem solver: going beyond informal solving strategies

Informal strategies reflecting the representation of a situation described in an arithmetic word problem mediate students’ solving processes. When the informal strategies are inefficient, teaching students to make way for more efficient ways to find the solution is an important educational issue in mathematics. The current paper presents a pedagogical design for arithmetic word problem solving, which is part of a first-grade arithmetic intervention (ACE). The curriculum was designed to promote adaptive expertise among students through semantic analysis and recoding, which would lead students to favor the more adequate solving strategy when several options are available. We describe the ways in which students were taught to engage in a semantic analysis of the problem, and the representational tools used to favor this conceptual change. Within the word problem solving curriculum, informal and formal solving strategies corresponding to the different formats of the same arithmetic operation, were comparatively studied. The performance and strategies used by students were assessed, revealing a greater use of formal arithmetic strategies among ACE classes. Our findings illustrate a promising path for going past informal strategies on arithmetic word problem solving.

     

A case study of dynamic visualization and problem solving

This paper reports an example of a situation in which university students had to solve geometrical problems presented to them dynamically using the interactive computerized environment of the ‘MicroWorlds Project Builder’. In the process of the problem solving, the students used ten different solution strategies. The unsuccessful strategies were then classified into three main categories: distracting, reducing and confusing. One student group had to solve the same problem in its non-dynamic version. The results received from both groups were compared and analysed. Analysis of the solution strategies and the process of the categorization revealed that the percentage of success in both groups was similar and in the case of the given problem, the dynamic visual mode of the problem distracted the students’ attention away from proper handling of the solution of the problem.     

High school students' use of representations in physics problem solving

Findings from physics education research strongly point to the critical need for teachers’ use of multiple representations in their instructional practices such as pictures, diagrams, written explanations, and mathematical expressions to enhance students' problem‐solving ability. In this study, we explored use of problem‐solving tasks for generating multiple representations as a scaffolding strategy in a high school modeling physics class. Through problem‐solving cognitive interviews with students, we investigated how a group of students responded to the tasks and how their use of such strategies affected their problem‐solving performance and use of representations as compared to students who did not receive explicit, scaffolded guidance to generate representations in solving similar problems. Aggregated data on students' problem‐solving performance and use of representations were collected from a set of 14 mechanics problems and triangulated with cognitive interviews. A higher percentage of students from the scaffolding group constructed visual representations in their problem‐solving solutions, while their use of other representations and problem‐solving performance did not differ with that of the comparison group. In addition, interviews revealed that students did not think that writing down physics concepts was necessary despite being encouraged to do so as a support strategy.     

The impact of a conceptual model-based mathematics computer tutor on multiplicative reasoning and problem-solving of students with learning disabilities

The study explored the impact of Please Go Bring Me-COnceptual Model-based Problem Solving (PGBM-COMPS) computer tutoring system on multiplicative reasoning and problem solving of students with learning disabilities. The PGBM-COMPS program focused on enhancing the multiplicative reasoning and problem solving through nurturing fundamental mathematical ideas and moving students above and beyond the concrete level of operation. This is achieved by taking advantages of the constructivist approach from mathematics education and explicit conceptual model-based problem solving approach from special education. Participants were three elementary students with learning disabilities (LD). A mixed method design was employed to investigate the effect of the PGBM-COMPS program on enhancing students’ multiplicative reasoning and problem solving. It was found that the PGBM-COMPS program significantly improved participating students’ problem solving performance not only on researcher developed criterion tests but also on a norm-referenced standardized test. Qualitative and quantities data from this study indicate that, in addition to nurturing fundamental concept of composite units, it is necessary to help students to understand underlying problem structures and move toward mathematical model-based problem representation and solving for generalized problem solving skills.     

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.     

Students’ and Teachers’ Conceptual Metaphors for Mathematical Problem Solving

Metaphors are regularly used by mathematics teachers to relate difficult or complex concepts in classrooms. A complex topic of concern in mathematics education, and most STEM‐based education classes, is problem solving. This study identified how students and teachers contextualize mathematical problem solving through their choice of metaphors. Twenty‐two high‐school student and six teacher interviews demonstrated a rich foundation for these shared experiences by identifying the conceptual metaphors. This mixed‐methods approach qualitatively identified conceptual metaphors via interpretive phenomenology and then quantitatively analyzed the frequency and popularity of the metaphors to explore whether a coherent metaphorical system exists with teachers and students. This study identified the existence of a set of metaphors that describe how multiple classrooms of geometry students and teachers make sense of mathematical problem solving. Moreover, this study determined that the most popular metaphors for problem solving were shared by both students and teachers. The existence of a coherent set of metaphors for problem solving creates a discursive space for teachers to converse with students about problem solving concretely. Moreover, the methodology provides a means to address other complex concepts in STEM education fields that revolve around experiential understanding.     

The Road To Digitopolis: Perils of Problem Solving

How students solve problems is a topic of central concern both to educational researchers and to math/science teachers: What is the nature of good and poor problem solving? How can students improve their problem-solving capacities? Teachers are in a unique position to witness problem solving in action, and to draw connections between the classroom experiences of their students and the findings of research. This article presents an instance of problem solving (drawn from a popular children's book) annotated with references to current research in cognition and education. The annotations explore issues such as the effect of performance anxiety on problem solving, how problem solvers handle the experience of confusion, and the role of self-monitoring and metacognition in problem solving.     

Minimalism as a Guiding Principle: Linking Mathematical Learning to Everyday Knowledge

Studies report that students often fail to consider familiar aspects of reality in solving mathematical word problems. This study explored how different features of mathematical problems influence the way that undergraduate students employ realistic considerations in mathematical problem solving. Incorporating familiar contents in the word problems was found to have only a limited impact. Instead, removing contextual constraints from the problem goal was found to motivate students to validate their problem solving in terms of their everyday experiences. Based on these findings, what determines the authenticity and relevance of a mathematical problem seems to be whether the problem allows students to freely reconstruct the problem situation by making use of their imagination and everyday experiences. In short, the basic principle seems to be “less is more”; that is, fewer constraints in problem goals could function to help students personally associate problem solving with their everyday experiences.     

Minimalism as a Guiding Principle: Linking Mathematical Learning to Everyday Knowledge

Studies report that students often fail to consider familiar aspects of reality in solving mathematical word problems. This study explored how different features of mathematical problems influence the way that undergraduate students employ realistic considerations in mathematical problem solving. Incorporating familiar contents in the word problems was found to have only a limited impact. Instead, removing contextual constraints from the problem goal was found to motivate students to validate their problem solving in terms of their everyday experiences. Based on these findings, what determines the authenticity and relevance of a mathematical problem seems to be whether the problem allows students to freely reconstruct the problem situation by making use of their imagination and everyday experiences. In short, the basic principle seems to be “less is more”; that is, fewer constraints in problem goals could function to help students personally associate problem solving with their everyday experiences.     

Network flows with age dependent decay rates

This paper considers the problem of maximizing the output flow in a multicommodity network in which flow entering an arc experiences a decay rate which is a function of three factors: the arc, the commodity, and the age of the commodity as it enters the arc. An arc-chain linear programming formulation of the problem is given. The algorithm for solving the problem involves a novel column generation scheme for basis entry embedded in the revised simplex algorithm. An efficient algorithm for generating, at each iteration, such a column is provided and illustrated with a numerical example.     

Comparing German and Taiwanese secondary school students’ knowledge in solving mathematical modelling tasks requiring their assumptions

Mathematization is critical in providing students with challenges for solving modelling tasks. Inadequate assumptions in a modelling task lead to an inadequate situational model, and to an inadequate mathematical model for the problem situation. However, the role of assumptions in solving modelling problems has been investigated only rarely. In this study, we intentionally designed two types of assumptions in two modelling tasks, namely, one task that requires non-numerical assumptions only and another that requires both non-numerical and numerical assumptions. Moreover, conceptual knowledge and procedural knowledge are also two factors influencing students’ modelling performance. However, current studies comparing modelling performance between Western and non-Western students do not consider the differences in students’ knowledge. This gap in research intrigued us and prompted us to investigate whether Taiwanese students can still perform better than German students if students’ mathematical knowledge in solving modelling tasks is differentiated. The results of our study showed that the Taiwanese students had significantly higher mathematical knowledge than did the German students with regard to either conceptual knowledge or procedural knowledge. However, if students of both countries were on the same level of mathematical knowledge, the German students were found to have higher modelling performance compared to the Taiwanese students in solving the same modelling tasks, whether such tasks required non-numerical assumptions only, or both non-numerical and numerical assumptions. This study provides evidence that making assumptions is a strength of German students compared to Taiwanese students. Our findings imply that Western mathematics education may be more effective in improving students’ ability to solve holistic modelling problems.     

Identifying Fourth Graders' Understanding of Rational Number Representations: A Mixed Methods Approach

This study examined average‐, high‐ and top‐performing US fourth graders' rational number problem solving and their understanding of rational number representations. In phase one, all students completed a written test designed to tap their skills for multiplication, division and rational number word‐problem solving. In phase two, a subset of students sorted cards that showed part‐whole, ratio, quotient, measure, and operator perspectives of rational number representations. Each perspective was shown in numerical notational, word‐problem, and visual formats. The results indicated that top‐performing students scored significantly higher in problem solving and showed more effectively linked rational number representations than the other groups. The results imply that successful rational number problem solving is intertwined with representational knowledge for a wide range of rational numbers and that the bulk of US students do not possess effective skills for working with rational number representations.     

Note on “Some models for deriving the priority weights from interval fuzzy preference relations”

Deriving accurate interval weights from interval fuzzy preference relations is key to successfully solving decision making problems. Xu and Chen (2008) proposed a number of linear programming models to derive interval weights, but the definitions for the additive consistent interval fuzzy preference relation and the linear programming model still need to be improved. In this paper, a numerical example is given to show how these definitions and models can be improved to increase accuracy. A new additive consistency definition for interval fuzzy preference relations is proposed and novel linear programming models are established to demonstrate the generation of interval weights from an interval fuzzy preference relation.