Increasing engineering students' awareness to environment through innovative teaching of mathematical modelling

This article presents the results of two studies on using aninnovative pedagogical strategy in teaching mathematical modellingand applications to engineering students. Both studies are dealingwith introducing non-traditional contexts for engineering studentsin teaching/learning of mathematical modelling and applications:environment and ecology. The aims of using these contexts were:to introduce students to some of the techniques, methodologiesand principles of mathematical modelling for ecological andenvironmental systems; to involve the students in solving real-lifeproblems adjusted to their region emphasizing the aspects ofboth survival (short term) and sustainability (long term); toencourage students to pay attention to environmental issues.On one hand, the contexts are not directly related to engineering.On the other hand, the chances are that many graduates of engineeringwill deal with mathematical modelling of environmental systemsin one way or another in their future work because nearly everyengineering activity has an impact on the environment. The firststudy is a parallel study conducted in New Zealand and Germanysimultaneously with first-year students studying engineeringmathematics. The second study is a case study of the experimentalcourse Mathematical Modelling of Survival and Sustainabilitytaught to a mixture of year 2–5 engineering students inGermany by a visiting lecturer from New Zealand. The modelsused with the students from both studies had several specialfeatures. Analysis of students’ responses to questionnaires,their comments and attitudes towards the innovative approachin teaching are presented in the article.     

From lessons to lectures: NCEA mathematics results and first-year mathematics performance

Given the recent radical overhaul of secondary school qualifications in New Zealand, similar in style to those in the UK, there has been a distinct change in the tertiary entrant profile. In order to gain insight into this new situation that university institutions are faced with, we investigate some of the ways in which these recent changes have impacted upon tertiary level mathematics in New Zealand. To this end, we analyse the relationship between the final secondary school qualifications in Mathematics with calculus of incoming students and their results in the core first-year mathematics papers at Canterbury since 2005, when students entered the University of Canterbury with these new reformed school qualifications for the first time. These findings are used to investigate the suitability of this new qualification as a preparation for tertiary mathematics and to revise and update entrance recommendations for students wishing to succeed in their first-year mathematics study.     

The impact of environmental constraints on short term harvesting: Use of planning tools and mathematical models

Both in New Zealand and Chile there exist important forest industries based mostly on pine plantations. The management of harvesting requires a series of activities such as building access roads, harvesting with skidders on flat terrain and cable logging for steep terrain, transportation from forest origins to destinations, such as a port, pulp plant and sawmills.These harvesting activities imply some potentially harmful environmental impacts such as water sedimentation, erosion and loss of scenic beauty. This has led to the proposal of several mitigating actions, such as creating riparian strips along rivers, avoiding the use of heavy machinery on fragile soils and minimizing road building.The degree of implementation of these mitigating actions differs in Chile and New Zealand. In New Zealand, the 1991 Resource Management Act regulated the implementation of environmental protection measures. In Chile a series of such measures have been implemented, but a set of definite regulations has not been defined yet and there is a need to develop an economic evaluation of the implementation of environmental measures in terms of reduced timber production and high harvesting costs, together with the benefits to the environment. To support decisions at tactical and operational levels, mathematical models have been developed both in Chile and New Zealand and are being used successfully by the timber industries. These models can be modified to consider diverse environmental protection measures. We analyze the implementation of these modifications and how the new models can help evaluate the economic impact of the protection measures. We show some preliminary results.     

The Impact of Teacher Privileging on Learning Differentiation with Technology

This study examines how two teachers taught differentiation using a hand held computer algebra system, which made numerical,graphical and symbolic representations of the derivative readily available. The teachers planned the lessons together but taught their Year 11 classes in very different ways. They had fundamentally different conceptions of mathematics with associated teaching practices,innate ‘privileging’ of representations, and of technology use. This study links these instructional differences to the different differentiation competencies that the classes acquired. Students of the teacher who privileged conceptual understanding and student construction of meaning were more able to interpret derivatives. Students of the teacher who privileged performance of routines made better use of the CAS for solving routine problems. Comparison of the results with an earlier study showed that although each teacher's teaching approach was stable over two years, each used technology differently with further experience of CAS. The teacher who stressed understanding moved away from using CAS, whilst the teacher who stressed rules,adopted it more. The study highlights that within similar overall attainment on student tests, there can be substantial variations of what students know. New technologies provide more approaches to teaching and so greater variations between teaching and the consequent learning may become evident. This revised version was published online in July 2006 with corrections to the Cover Date.     

A Symbolic Dance: The Interplay Between Movement,Notation, and Mathematics on a Journey Toward Solving Equations

This article analyzes the use of the software Grid Algebra with a mixed ability class of 21 nine-to-ten-year-old students who worked with complex formal notation involving all four arithmetic operations. Unlike many other models to support learning, Grid Algebra has formal notation ever present and allows students to “look through” that notation and interpret it either in terms of physical journeys on a grid or in terms of mathematical operations. A dynamic fluidity was found between the formal notation, imagery of movements on a grid, and the process of mathematical operations. This fluidity is interpreted as a “dance” between these three. The significant way in which this dynamic took place reflects the scaffolding and fading offered by the software, which was crucial to the students’ fluency with formal notation well beyond what has been reported from students of that age.     

An investigation of 6th graders’ solutions of Cartesian product problems and representation of these problems using arrays

Two hour-long interviews were conducted with each of 14 sixth-grade students. The purpose of the interviews was to investigate how students solved combinatorics problems, and represented their solutions as arrays. This paper reports on 11 of these students who represented a balanced mix of students operating with two of three multiplicative concepts that have been identified in prior research (Hackenberg, 2007, 2010; Hackenberg & Tillema, 2009). One finding of the study was that students operating with different multiplicative concepts established and structured pairs differently. A second finding is that these different ways of operating had implications for how students produced and used arrays. Overall, the findings contribute to models of students’ reasoning that outline the psychological operations that students use to constitute product of measures problems (Vergnaud, 1983). Product of measures problems are a kind of multiplicative problem that has unique mathematical properties, but researchers have not yet identified specific psychological operations that students use when solving these problems that differ from their solution of other kinds of multiplicative problems (cf. Battista, 2007).     

Using DEA and survival analysis for measuring performance of branches in New Zealand's Accident Compensation Corporation

This paper describes an approach to solving the problem of assessing relative branch performance in the Accident Compensation Corporation (ACC), the New Zealand state-owned, no-fault, personal injury compensation insurance company. The methodology described in this paper is innovative in that it assesses relative performance via a modified data envelopment analysis (DEA) process. Performance is assessed and can be examined in terms that are meaningful to business goals. Performance evaluation has generally aligned well with management preconceptions of performance. DEA results are clustered to find common performance themes that are reviewed for business acceptability and the successful themes are used to measure performance against a Farrell frontier. The methodology is applicable to other situations where there are multiple instances of a unit performing the same or similar functions. Technical analysis as used in this paper is based on survival analysis, DEA, clustering and knowledge of the organisation's business operations and objectives.     

Prior analysis and scheduling of the 2011 Rugby Union ITM Cup in New Zealand

This paper describes work done for the New Zealand Rugby Union (NZRU) in preparation for their most important wholly domestic competition in 2011. This competition had to be played during a shorter timescale than usual because of the 2011 Rugby Union World Cup, and the NZRU were keen to ensure that they could incorporate the format they wanted into this timescale without unfortunate consequences. In addition, they wanted to introduce a novel feature into this tournament. Thus some detailed prior experimental, or ‘what-if’, analysis was necessary. This paper describes this analysis and its results. As a result of this analysis, the NZRU was persuaded to abandon one of its design ideas, but was able to proceed with others, enabling them to announce the detailed format of the competition in the confidence that it would work well in practice. Subsequent scheduling of the competition in this format proved successful, and the resulting schedule is shown together with detailed analysis of its notional costs. The paper demonstrates how important it can be for schedulers to be closely involved in tournament design in advance of the actual scheduling.     

Students’ images of problem contexts when solving applied problems

This article reports findings from an investigation of precalculus students’ approaches to solving novel problems. We characterize the images that students constructed during their solution attempts and describe the degree to which they were successful in imagining how the quantities in a problem's context change together. Our analyses revealed that students who mentally constructed a robust structure of the related quantities were able to produce meaningful and correct solutions. In contrast, students who provided incorrect solutions consistently constructed an image of the problem's context that was misaligned with the intent of the problem. We also observed that students who caught errors in their solutions did so by refining their image of how the quantities in a problem's context are related. These findings suggest that it is critical that students first engage in mental activity to visualize a situation and construct relevant quantitative relationships prior to determining formulas or graphs.     

Multistage stochastic demand-side management for price-making major consumers of electricity in a co-optimized energy and reserve market

In this paper, we take an optimization-driven heuristic approach, motivated by dynamic programming, to solve a class of non-convex multistage stochastic optimization problems. We apply this to the problem of optimizing the timing of energy consumption for a large manufacturer who is a price-making major consumer of electricity. We introduce a mixed-integer program that co-optimizes consumption bids and interruptible load reserve offers, for such a major consumer over a finite time horizon. By utilizing Lagrangian methods, we decompose our model through approximately pricing the constraints that link the stages together. We construct look-up tables in the form of consumption-utility curves, and use these to determine optimal consumption levels. We also present heuristics, in order to tackle the non-convexities within our model, and improve the accuracy of our policies. In the second part of the paper, we present stochastic solution methods for our model in which, we reduce the size of the scenario tree by utilizing a tailor-made scenario clustering method. Furthermore, we report on a case study that implements our models for a major consumer in the (full) New Zealand Electricity Market and present numerical results.     

Using math magic to reinforce algebraic concepts: an exploratory study

An exploratory study was conducted to investigate the use of magic activities in a math course for prospective middle-school math teachers. This research report focuses on a lesson using two versions of math magic: (1) the 5-4-3-2-1-½ Magic involves having students choose a secret number and apply six arithmetic operations in sequence to arrive at a resultant number, and the teacher-magician can spontaneously reveal a student’s secret number from the resultant number; and (2) the Everyone-Got-9 Magic also involves choosing a secret number and applying arithmetic operations in sequence, but everyone will end up with the same resultant number of 9. These magic activities were implemented to reinforce students’ understanding of foundational algebra concepts like variables, expressions, and inverse functions. Analysis of students’ written responses revealed that (1) all students who figured out the trick in the first magic activity did not used algebra, (2) most students could apply what they learned in one trick to a similar trick but not to a different trick, and (3) many students were weak in symbolic representations and manipulations. Responses from a survey and a focus group indicate that students found the magic activities to be fun and intellectually engaging.     

Mathematics learning support and engagement in first year engineering

This paper considers the effects of both free optional mathematics learning support and engagement on the mathematics performance in a foundation mathematics subject of a cohort of engineering students entering university with poor mathematical skills. New engineering students were directed to either a foundation or standard mathematics subject based on the results of a placement test. For students in the foundation subject, it was found that high levels of learning support were associated with greater improvement over the semester. Some form of learning support was used by 57.9% of the students, a reasonably high proportion of the cohort. Some factors for this high level of use of learning support are considered. One possible factor, the placement test, appears to have had a positive effect. Engagement in the subject activities as measured by tutorial attendance and learning management system use was found to have a positive association with final mark. Students who utilized a high level of learning support were more likely to be more engaged with the subject, making it impossible to draw conclusions about improvements being solely due to the use of learning support.     

Towards optimal life cycle management in a road maintenance setting using DEA

This paper focuses on local road aspects of the highway system and aims to assess how efficiently, effectively and economically the 73 Territorial Local Authorities (TLAs) in New Zealand have maintained their respective local road networks from a life cycle perspective. Measures of quality, quantity and cost, together with nondiscretionary measures of environmental factors, are incorporated in Data Envelopment Analysis to evaluate each TLA’s performance in terms of efficiency, effectiveness and economy. TLAs that have high performance on all three measures provide a best practice indication of the optimal maintenance activity mix to undertake. In this study, the best practice mix of expenditure is 59% routine maintenance, 27% resealing and 14% rehabilitation.     

The fractional knowledge and algebraic reasoning of students with the first multiplicative concept

To understand relationships between students’ quantitative reasoning with fractions and their algebraic reasoning, a clinical interview study was conducted with 18 middle and high school students. Six students with each of three different multiplicative concepts participated. This paper reports on the fractional knowledge and algebraic reasoning of six students with the most basic multiplicative concept. The fractional knowledge of these students was found to be consistent with prior research, in that the students had constructed partitioning and iteration operations but not disembedding operations, and that the students conceived of fractions as parts within wholes. The students’ iterating operations facilitated their work on algebra problems, but the lack of disembedding operations was a significant constraint in writing algebraic equations and expressions, as well as in generalizing relationships. Implications for teaching these students are discussed.     

Designing examples to create variation patterns in teaching

This paper uses a post-qualitative philosophical perspective to find new ways of understanding teaching and learning. The paper presents a series of examples that were used in a longitudinal study, with the aim of creating variation patterns that would make it possible for students to discern the use of the four basic arithmetic operations in different situations. The focus of this article is the potential of the examples to systematically create variation patterns that students need to perceive in order to make generalizations. The result demonstrates that well-thought-out examples help identify the correct arithmetic operation in different situations, and provide a basis from which students can discern the connection between text and the use of operation in mathematical example. The result also demonstrates that students develop rhizomatic thinking through the creation of new links between aspects of the object of learning, association and linking of different aspects to each other and the creation of a whole with unique and specific characteristics that cannot be explained by simply adding the characteristics of the individual parts.     

Slope and Line of Best Fit: A Transfer of Knowledge Case Study

This paper brings together research on slope from mathematics education and research on line of best fit from statistics education by considering what knowledge of slope students transfer to a novel task involving determining the placement of an informal line of best fit. This study focuses on two students who transitioned from placing inaccurate to accurate lines of best fit during a task‐based interview. The analysis focuses on describing shifts in slope reasoning that accompanied the change to accurately placed lines, and investigates factors that may have influenced the shift in reasoning. The results have implications for the teaching of both slope and the line of best fit.     

Prospective Elementary Teachers’ Understanding of Order of Operations

This study investigates how well 381 prospective elementary, early childhood, and special education majors solved four arithmetic problems that required using the order of operations. Self‐reported data show these students to be relatively able mathematically and confident in their ability, with no substantial dislike of mathematics. The percentage of answers that were incorrect that is attributable to order of operations ranged from 21.7% to 78.5%. Overall, fewer than half the subjects answered more than two questions correctly. Of those subjects who performed multiplication before addition, which indicates some knowledge of order of operations, 30.9% performed addition before subtraction and 38.0% performed multiplication before division rather than from left to right, which suggests that instead of using the correct order of operations, these students used the common mnemonic PEMDAS or “Please excuse my dear Aunt Sally “literally, performing multiplication before division and performing addition before subtraction, rather than from left‐to‐right. Furthermore, 78.5% of subjects used the incorrect order of operations to compute ?3².     

Mental Computation of Students in a Reform-Based Mathematics Curriculum

Traditional school instruction in mathematics has generally produced students who are poor at mental computation and exhibit a weak sense of number and mathematical operations. In this study, fifth graders who had been in a reform-based mathematics curriculum since kindergarten were given a whole-class test on mental computation problems. Baseline data with students in traditional mathematics curricula were used as a comparison. The students in this reform-based mathematics curriculum performed much higher than the comparison group on all but one problem, and on most problems, this difference was substantial. Additionally, a student preference survey indicated that students in the reform curriculum were more likely to consider the calculator as an option than were the baseline group. They were also more able to recognize problems that did not lend themselves to mental computation. Individual interviews indicated that experiences in the primary grades with “invented” algorithms and discussing alternative solutions led to a better ability to compute mentally and a stronger number sense.     

A decision aid for milk tanker run allocation

This paper reports on a decision aid that recommends solutions to a particular vehicle allocation problem occurring in the New Zealand dairy industry. The decision aid has been developed for use by New Zealand milk tanker schedulers. It is designed to aid them in the allocation of tankers to milk collection routes in order to alleviate pumping bay congestion, which occurs when the tankers return to unload milk at a processing plant. It enables its users to remain in control of the tanker allocation process, while using their own experience and preferences. The paper describes the issues involved and the type of help that the schedulers need. It introduces a typical allocation problem and describes some heuristics for its solution, which are incorporated in the decision aid.     

Large-scale linearly constrained optimization

An algorithm for solving large-scale nonlinear programs with linear constraints is presented. The method combines efficient sparse-matrix techniques as in the revised simplex method with stable quasi-Newton methods for handling the nonlinearities. A general-purpose production code (MINOS) is described, along with computational experience on a wide variety of problems.This research was supported by the U.S. Office of Naval Research (Contract N00014-75-C-0267), the National Science Foundation (Grants MCS71-03341 A04, DCR75-04544), the U.S. Energy Research and Development Administration (Contract E(04-3)-326 PA #18), the Victoria University of Wellington, New Zealand, and the Department of Scientific and Industrial Research Wellington, New Zealand.