Reasoning-and-Proving in School Mathematics Textbooks

Despite widespread agreement that the activity of reasoning-and-proving should be central to all students' mathematical experiences, many students face serious difficulties with this activity. Mathematics textbooks can play an important role in students' opportunities to engage in reasoning-and-proving: research suggests that many decisions that teachers make about what tasks to implement in their classrooms and when and how to implement them are mediated by the textbooks they use. Yet, little is known about how reasoning-and-proving is promoted in school mathematics textbooks. In this article, I present an analytic/methodological approach for the examination of the opportunities designed in mathematics textbooks for students to engage in reasoning-and-proving. In addition, I exemplify the utility of the approach in an examination of a strategically selected American mathematics textbook series. I use the findings from this examination as a context to discuss issues of textbook design in the domain of reasoning-and-proving that pertain to any textbook series.     

The Impossibility of Inverted Reasoners

An ‘inverted’ reasoner is someone who finds the inferences we find easy, inversely difficult, and those that we find difficult, inversely easy. The notion was initially introduced by Christopher Cherniak in his book, Minimal Rationality, and appealed to by Stephen Stich in The Fragmentation of Reason. While a number of difficulties have been noted about what reasoning would amount to for such a reasoner, what has not been brought out in the literature is that such a reasoner is in fact logically impossible. This is what I hope to demonstrate in this paper.     

Resource Review: Supporting the New Students in Higher Education Today

Abstract

It would be nice if there were a demonstrable relationship between the price that a college or university charges and the product that it delivers. Institutions with superior market positions most often make that claim, whether true or not, when defending their higher prices—stating that what the student pays for is both a better and a different educational experience. Those vaunted differences ought to include what faculty do before a course begins, how they prepare for the daily challenge of the classroom, how they administer their courses and assess their students' progress, and how frequently they interact with students outside of the classroom.

The National Center for Postsecondary Improvement (NCPI) is supported under the Educational Research and Development Center program, agreement number R309A60001, CFDA 84.309A, as administered by the Office of Educational Research and Improvement (OERI), U.S. Department of Education. The findings and opinions expressed by NCPI do not necessarily reflect the position or policies of OERI or the U.S. Department of Education. The Institute for Research on Higher Education retains the copyright for this column.     

The MIME project at Loughborough — a first report†

This paper describes the experiences of the first months of the MIME Project (Micros In Mathematics Education) at Loughborough. The early difficulties and setbacks are mentioned as well as the initial successes.

The first part of the paper examines the educational philosophy behind the establishment of the project and details its aims, objectives and scope. Consideration is given to how these aims were put into practice and what revisions to them were necessitated by experience.

The second part traces the story of the production of a unit from the author's early ideas, through the program development, to the finished product. Emphasis is given to the way the educational objectives are achieved by careful design of the program. The testing and evaluation of a unit is outlined.

The third part reviews the practical programming difficulties that have been encountered; the strengths and weaknesses of the BBC micro are discussed and problems of the speed of display and of storage limitation are highlighted. The need for assembly language is shown and the case is put for hardware/software independence.

The fourth part looks at the different modes of use of the units. Demonstrations are given of use as a teacher aid in the classroom and as self‐paced instruction.

Finally, conclusions are drawn from the early experiences of the MIME team and the lessons they have learned, and future developments are outlined.     

Affine and eclipsing multifunctions

In this paper, we want to compare two classes of multifunctions which can be used as approximating multifunctions in differentiability theory: affine and eclipsing multifunctions. We show how the notion of eclipsing multifunctions is an extension of affine multifunctions, and what kinds of difficulties arise in this extension.     

University Students' Reading of Their First-Year Mathematics Textbooks

This article reports the observed behaviors and difficulties that 11 precalculus and calculus students exhibited in reading new passages from their mathematics textbooks. To gauge the effectiveness of these students' reading, we asked them to attempt straightforward mathematical tasks, based directly on what they had just read. The students had high ACT mathematics and high ACT reading comprehension test scores and exhibited much of the Constructively Responsive Reading of good readers as described in the reading comprehension research literature. However, they were not effective readers of their mathematics textbooks. We discuss some reasons for this that might not be easily discernible among the variety of task-working difficulties of less able students. Finally, we pose questions for future research and suggest some implications for teaching.     

Some experiments with interval methods for two-point boundary-value problems in ordinary differential equations

Methods of interval mathematics are used to find upper and lower bounds for the solution of two-point boundary-value problems at discrete mesh points. They include interval versions of shooting and of finite-difference techniques for linear and non-linear differential equations of second order, and of finite-difference methods for Sturm-Liouville eigenvalue problems.Good results are obtained whenever the difficulties of dependency-width can be avoided, and particularly for the finite-difference method when the associated matrix is anM matrix.     

On Restarting the Arnoldi Method for Large Nonsymmetric Eigenvalue Problems

The Arnoldi method computes eigenvalues of large nonsymmetric matrices. Restarting is generally needed to reduce storage requirements and orthogonalization costs. However, restarting slows down the convergence and makes the choice of the new starting vector difficult if several eigenvalues are desired. We analyze several approaches to restarting and show why Sorensen's implicit QR approach is generally far superior to the others. Ritz vectors are combined in precisely the right way for an effective new starting vector. Also, a new method for restarting Arnoldi is presented. It is mathematically equivalent to the Sorensen approach but has additional uses.

     

Understanding kindergarten teachers’ perspectives of teaching basic geometric shapes: a phenomenographic research

Geometry is one of the disciplines children involve within early years of their lives. However, there is not much information about geometry education in Turkish kindergarten classes. The current study aims to examine teachers’ perspectives on teaching geometry in kindergarten classes. The researchers inquired about teachers’ in-class experiences in geometry and asked a series of questions such as “what are the benchmarks in your kindergarten class?”; “what kind of tools and materials you use to teach geometry in your class?”; “what shape do you teach first in your kindergarten class?”; “what do you expect to hear when you asked your students ‘what is square’?”; “how do you teach rectangular?”. The study utilized one of the qualitative research methods, namely phenomenography, to collect the data and analyze the data. The study involved with eight kindergarten teachers who work in different schools in central Kutahya, Turkey. The researchers collected data by conducting face-to-face half-structured interviews. The findings of this phenomenographic research showed that kindergarten teachers have some difficulties in teaching geometry and have lack of knowledge and skills in teaching geometry in kindergarten classes.     

An analysis of the amount and characteristics of writing prompts in Grade 3 mathematics student books

Curriculum guidelines and professional organizations’ recommendations lack details about how often and how much students should write in mathematics and what characteristics should define their writing. This study presents an analytic framework that addresses how often students are prompted in student mathematics books to write, how much they may be encouraged to write, and the characteristics of the writing prompts. Consequently, 2,095 writing prompts in student books across 10 comprehensive Grade 3 resources were analyzed. Findings indicate a marked variation in how often and how much students are positioned to write. Most prompts have students explain what they did to solve a problem and why about number concepts, with most pressing for procedures. The greatest percentage of prompts had students write about their own solutions and do not urge them to include specific writing features.     

Multiscale Analysis and Data Networks

The empirical finding of self-similarity in data network traffic over many time scales motivates the need for analysis tools that are particularly well adapted for identifying structures in network traffic. These structures span a range of time scales or are scale-dependent. Wavelet-based scaling analysis methods are especially successful, both collecting summary statistics from scale to scale and probing the local structure of packet traces. They include both spectral density estimation to identify large time-scale features and multifractal estimation for small time-scale bursts. While these methods are primarily statistical in nature, we may also adapt them to visualize the “burstiness” or the instantaneous scaling features of network traffic. This expository paper discusses the theoretical and implementation issues of wavelet-based scaling analysis for network traffic. Because data network traffic research does not consist solely of analysis, we show how these wavelet-based methods may be used to monitor and infer network properties (in conjunction with on-line algorithms and careful network experimentation). More importantly, we address what types of networking questions we can and cannot investigate with such tools.     

Peeking Inside the Black Box: Visualizing Statistical Learning With Plots of Individual Conditional Expectation

This article presents individual conditional expectation (ICE) plots, a tool for visualizing the model estimated by any supervised learning algorithm. Classical partial dependence plots (PDPs) help visualize the average partial relationship between the predicted response and one or more features. In the presence of substantial interaction effects, the partial response relationship can be heterogeneous. Thus, an average curve, such as the PDP, can obfuscate the complexity of the modeled relationship. Accordingly, ICE plots refine the PDP by graphing the functional relationship between the predicted response and the feature for individual observations. Specifically, ICE plots highlight the variation in the fitted values across the range of a covariate, suggesting where and to what extent heterogeneities might exist. In addition to providing a plotting suite for exploratory analysis, we include a visual test for additive structure in the data-generating model. Through simulated examples and real datasets, we demonstrate how ICE plots can shed light on estimated models in ways PDPs cannot. Procedures outlined are available in the R package ICEbox.     

Generalized minimum χ2 test for the extreme values

There are some difficulties in applying the Pearson’s Chi-Square Test for the continuous distribution. The problems include how to form class intervals for the test of fit, how to employ in the test when the estimators of parameters are obtained from the ungrouped sample, so on. In order to solve these problems, we use the generalized minimum Chi-Square technique which is a test free of the complications associated with the Peason’s Chi-Square test. This paper show how to apply the goodness of fit tests based on generalized minimum Chi-Square technique to the extreme values.     

Causal geometry of Einstein-Vacuum spacetimes with finite curvature flux

One of the central difficulties of settling the L²-bounded curvature conjecture for the Einstein-Vacuum equations is to be able to control the causal structure of spacetimes with such limited regularity. In this paper we show how to circumvent this difficulty by showing that the geometry of null hypersurfaces of Enstein-Vacuum spacetimes can be controlled in terms of initial data and the total curvature flux through the hypersurface. Mathematics Subject Classification (1991) 35J10     

Trust and relational capital

Trust can be viewed at the same time as an instrument both for an agent selecting the right partners in order to achieve its own goals, and for an agent of being selected from other potential partners in order to establish with them a cooperation/collaboration and to take advantage from the accumulated trust. In this paper we will analyze trust as the agents’ relational capital. Starting from the classical dependence network with potential partners, we introduce the analysis of what it means for an agent to be trusted and how this condition could be strategically used from it for achieving its own goals, that is, why it represents a form of power. The idea of taking another agent’s point of view is especially important if we consider the amount of studies in social science that connect trust with social capital related issues. Although there is a big interest in literature about ‘social capital’ and its powerful effects on the wellbeing of both societies and individuals, often it is not clear enough what is it the object under analysis. Individual trust capital (relational capital) and collective trust capital not only should be disentangled, but their relations are quite complicated and even conflicting. To overcome this gap, we propose a study that first attempts to understand what trust is as capital of individuals. In which sense “trust” is a capital. How this capital is built, managed and saved. In particular, how this capital is the result of the others’ beliefs and goals. Then we aim to analytically study the cognitive dynamics of this object.     

Trust and relational capital

Trust can be viewed at the same time as an instrument both for an agent selecting the right partners in order to achieve its own goals, and for an agent of being selected from other potential partners in order to establish with them a cooperation/collaboration and to take advantage from the accumulated trust. In this paper we will analyze trust as the agents’ relational capital. Starting from the classical dependence network with potential partners, we introduce the analysis of what it means for an agent to be trusted and how this condition could be strategically used from it for achieving its own goals, that is, why it represents a form of power. The idea of taking another agent’s point of view is especially important if we consider the amount of studies in social science that connect trust with social capital related issues. Although there is a big interest in literature about ‘social capital’ and its powerful effects on the wellbeing of both societies and individuals, often it is not clear enough what is it the object under analysis. Individual trust capital (relational capital) and collective trust capital not only should be disentangled, but their relations are quite complicated and even conflicting. To overcome this gap, we propose a study that first attempts to understand what trust is as capital of individuals. In which sense “trust” is a capital. How this capital is built, managed and saved. In particular, how this capital is the result of the others’ beliefs and goals. Then we aim to analytically study the cognitive dynamics of this object.     

Displaying Variation in Large Datasets: Plotting a Visual Summary of Effect Sizes

Displaying the component-wise between-group differences high-dimensional datasets is problematic because widely used plots such as Bland–Altman and Volcano plots do not show what they are colloquially believed to show. Thus, it is difficult for the experimentalist to grasp why the between-group difference of one component is “significant” while that of another component is not. Here, we propose a type of “Effect Plot” that displays between-group differences in relation to respective underlying variability for every component of a high-dimensional dataset. We use synthetic data to show that such a plot captures the essence of what determines “significance” for between-group differences in each component, and provide guidance in the interpretation of the plot. Supplementary online materials contain the code and data for this article and include simple R functions to produce an effect plot from suitable datasets.     

A point set puzzle revisited

Two discrete point sets in Rⁿ are said to be homometric if their difference sets coincide. Homometric point sets were first studied in the 1930s in connection with the interpretation of x-ray diffraction patterns; today they appear in many contexts. Open questions still abound, even for point sets on the line. Under what conditions does a difference set S−S characterize S uniquely? If it does not, how can we find all the sets S_i,i=1,…, that give rise to it, and how are these sets related?     

A tabu search tutorial based on a real-world scheduling problem

We apply a tabu search method to a scheduling problem of a company producing cables for cars: the task is to determine on what machines and in which order the cable jobs should be produced in order to save production costs. First, the problem is modeled as a combinatorial optimization problem. We then employ a tabu search algorithm as an approach to solve the specific problem of the company, adapt various intensification as well as diversification strategies within the algorithm, and demonstrate how these different implementations improve the results. Moreover, we show how the computational cost in each iteration of the algorithm can be reduced drastically from O(n ³) (naive implementation) to O(n) (smart implementation) by exploiting the specific structure of the problem (n refers to the number of cable orders).