Proof validation in real analysis: Inferring and checking warrants

In the study reported here, we investigate the skills needed to validate a proof in real analysis, i.e., to determine whether a proof is valid. We first argue that when one is validating a proof, it is not sufficient to make certain that each statement in the argument is true. One must also check that there is good reason to believe that each statement follows from the preceding statements or from other accepted knowledge, i.e., that there is a valid warrant for making that statement in the context of this argument. We then report an exploratory study in which we investigated the behavior of 13 undergraduates when they were asked to determine whether or not a particular flawed mathematical argument is a valid mathematical proof. The last line of this purported proof was true, but did not follow legitimately from the earlier assertions in the proof. Our findings were that only six of these undergraduates recognized that this argument was invalid and only two did so for legitimate mathematical reasons. On a more positive note, when asked to consider whether the last line of the proof followed from previous assertions, a total of 10 students concluded that the statement did not and rejected the proof as invalid.     

Traditional instruction in advanced mathematics courses: a case study of one professor’s lectures and proofs in an introductory real analysis course

It is widely accepted by mathematics educators and mathematicians that most proof-oriented university mathematics courses are taught in a “definition-theorem-proof” format. However, there are relatively few empirical studies on what takes place during this instruction, why this instruction is used, and how it affects students’ learning. In this paper, I investigate these issues by examining a case study of one professor using this type of instruction in an introductory real analysis course. I first describe the professor’s actions in the classroom and argue that these actions are the result of the professor’s beliefs about mathematics, students, and education, as well as his knowledge of the material being covered. I then illustrate how the professor’s teaching style influenced the way that his students attempted to learn the material. Finally, I discuss the implications that the reported data have on mathematics education research.     

Mathematical tasks,study approaches,and course grades in undergraduate mathematics: a year-by-year analysis

Students approach learning in different ways, depending on the experienced learning situation. A deep approach is geared toward long-term retention and conceptual change while a surface approach focuses on quickly acquiring knowledge for immediate use. These approaches ultimately affect the students’ academic outcomes. This study takes a cross-sectional look at the approaches to learning used by students from courses across all four years of undergraduate mathematics and analyses how these relate to the students’ grades. We find that deep learning correlates with grade in the first year and not in the upper years. Surficial learning has no correlation with grades in the first year and a strong negative correlation with grades in the upper years. Using Bloom's taxonomy, we argue that the nature of the tasks given to students is fundamentally different in lower and upper year courses. We find that first-year courses emphasize tasks that require only low-level cognitive processes. Upper year courses require higher level processes but, surprisingly, have a simultaneous greater emphasis on recall and understanding. These observations explain the differences in correlations between approaches to learning and course grades. We conclude with some concerns about the disconnect between first year and upper year mathematics courses and the effect this may have on students.     

The same tasks,different learning opportunities: An analysis of two exemplary lessons in China and the U.S. from a perspective of variation

This study examined the learning opportunities afforded in two exemplary lessons based on a theory of variation. Implemented in China and the U.S., the two lessons focused on the same topic of patterns in a calendar and were carefully developed through a lesson study approach. Both lessons set similar learning goals but enacted these goals differently. When compared with the U.S. lesson, the Chinese lesson provided more learning opportunities through high cognitively demanding tasks focusing on different identities within patterns. However, the U.S. lesson, which featured fewer tasks and focused on a single pattern identity, may have better supported students in discerning the critical features within the objects of learning. The implications for task design and implementation for effective mathematics teaching are discussed.     

我国居民收入、人口、教育、财政政策和货币政策与居民消费模型的实证分析

本文从影响消费的各个因素:居民收入、人口、教育、国家宏观政策等着手,对各因素进行了相关分析,运用多元统计中的岭回归估计法建立消费模型。从定量和定性分析的角度,分析了我国居民消费水平、居民收入、人口增长率、各层次教育、国家财政支出和银行利率等相关因素之间相互影响的数量变动关系和内在规律,就如何提高居民消费水平促进经济协调发展提出若干对策。     

A variant of radial measure capable of dealing with negative inputs and outputs in data envelopment analysis

Data envelopment analysis (DEA) is a linear programming methodology to evaluate the relative technical efficiency for each member of a set of peer decision making units (DMUs) with multiple inputs and multiple outputs. It has been widely used to measure performance in many areas. A weakness of the traditional DEA model is that it cannot deal with negative input or output values. There have been many studies exploring this issue, and various approaches have been proposed.     

The likelihood ratio and its applications in sequential analysis

The likelihood ratio is treated as a generalized Radon-Nikodym derivative for measures which need not be absolutely continuous. It is shown that a wide variety of identities and inequalities in sequential analysis follow directly from simple properties of likelihood ratios.     

Robust tests in group sequential analysis: One- and two-sided hypotheses in the linear model

Consider the linear modelY=X+E in the usual matrix notation where the errors are independent and identically distributed. We develop robust tests for a large class of one- and two-sided hypotheses about when the data are obtained and tests are carried out according to a group sequential design. To illustrate the nature of the main results, let and be anM- and the least squares estimator of respectively which are asymptotically normal about with covariance matrices ²(X ^t X)^–1 and ²(X ^t X)^–1 respectively. Let the Wald-type statistics based on and be denoted byRW andW respectively. It is shown thatRW andW have the same asymptotic null distributions; here the limit is taken with the number of groups fixed but the numbers of observations in the groups increase proportionately. Our main result is that the asymptotic Pitman efficiency ofRW relative toW is (²/²). Thus, the asymptotic efficiency-robustness properties of relative to translate to asymptotic power-robustness ofRW relative toW. Clearly, this is an attractive result since we already have a large literature which shows that is efficiency-robust compared to . The results of a simulation study show that with realistic sample sizes,RW is likely to have almost as much power asW for normal errors, and substantially more power if the errors have long tails. The simulation results also illustrate the advantages of group sequential designs compared to a fixed sample design, in terms of sample size requirements to achieve a specified power.     

Early work of N.G. (Dick) de Bruijn in analysis and some of my own

There are parallels between de Bruijn’s early work in analysis and that of the author. However, Dick’s work soon became much broader and deeper. While the present paper reviews several topics of common interest, its main content is a short version of Dick’s important article related to Riemann’s Hypothesis, entitled ‘The roots of trigonometric integrals’ [N.G. de Bruijn, The roots of trigonometric integrals, Duke Math. J. 17 (1950) 197–226]. The associated ‘de Bruijn–Newman constant’ is also discussed.     

Selection of variables in two-group discriminant analysis by error rate and Akaike's information criteria

This paper deals with two criteria for selection of variables for the discriminant analysis in the case of two multivariate normal populations with different means and a common covariance matrix. One is based on the estimated error rate of misclassification. The other uses Akaike's information criterion. The asymptotic distributions and error rate risks of the criteria are obtained. The result will prove that the two criteria are asymptotically equivalent in the sense of their asymptotic distributions and error rate risks being identical.