Inductive problem‐sequences for independent study of linear algebra

Three chapters of the syllabus of a basic mathematics course for engineering students at the Technion? ? Technion is the oldest and largest school of engineering in Israel. were adapted for the student's independent development of mathematical knowledge by means of solving a chain of inductive problem‐sequences: (i) vectors and scalars in the plane; (ii) vectors in the three‐dimensional space; (iii) planes and lines in the 3‐dimensional space. The main principles which guided the adaptation of course materials for this purpose are presented. In particular an inductive problem‐sequenceis defined as a sequence consisting of problems which lead the learner to investigate particular cases, discover a common regularity, make a conjecture about a unifying rule, and finally prove or refute the conjecture. The major part of the paper is an annotated analysis of an excerpt of the materials.     

A comparative analysis of British and Taiwanese students’ conceptual and procedural knowledge of fraction addition

This study examines students’ procedural and conceptual achievement in fraction addition in England and Taiwan. A total of 1209 participants (561 British students and 648 Taiwanese students) at ages 12 and 13 were recruited from England and Taiwan to take part in the study. A quantitative design by means of a self-designed written test is adopted as central to the methodological considerations. The test has two major parts: the concept part and the skill part. The former is concerned with students’ conceptual knowledge of fraction addition and the latter is interested in students’ procedural competence when adding fractions.

There were statistically significant differences both in concept and skill parts between the British and Taiwanese groups with the latter having a higher score. The analysis of the students’ responses to the skill section indicates that the superiority of Taiwanese students’ procedural achievements over those of their British peers is because most of the former are able to apply algorithms to adding fractions far more successfully than the latter. Earlier, Hart [1 Hart KM. Children's understanding of mathematics: 11–16. Oxford: Alden Press; 1981. [Google Scholar]] reported that around 30% of the British students in their study used an erroneous strategy (adding tops and bottoms, for example, 2/3 + 1/7 = 3/10) while adding fractions. This study also finds that nearly the same percentage of the British group remained using this erroneous strategy to add fractions as Hart found in 1981.

The study also provides evidence to show that students’ understanding of fractions is confused and incomplete, even those who are successfully able to perform operations. More research is needed to be done to help students make sense of the operations and eventually attain computational competence with meaningful grounding in the domain of fractions.     

A Survey of the Development of Operational Calculus

Although the name of Oliver Heaviside is usually associated with the foundations of operational calculus, he was in fact preceded in introducing operational methods by a number of mathematicians. Heaviside's purely operational approach was often intuitive rather than rigorous and fell into disfavour in mathematical circles, being replaced by methods involving the concept of an integral transformation. In particular, the Laplace transformation came to be adopted by applied mathematicians and engineers despite the need for the Schwartz theory of distributions to account satisfactorily for the Dirac and other impulse functions. Jan Mikusinski,? ? It is intended that an expository article on Jan Mikusinski's operational calculus by H. G. Flegg will appear in a later issue ‐ Editor. however, has returned to the earlier operational approach and, by placing operational calculus upon a rigorous algebraic basis, has provided a calculus which is more general than those based upon integral transformations, and one which includes generalized functions without recourse to any specialized external theory.

     

WHEN WILL THEY EVER MAKE UP THEIR MINDS? THE SOCIAL STRUCTURE OF UNSTABLE DECISION MAKING

French (1977) French, J. R. 1977. “A formal theory of social power”. In Social networks: A developing paradigm, Edited by: Leinhardt, S. pp. 35–48. New York: Academic Press.  [Google Scholar], Harary (1959) Harary, F. 1959. “A criterion for unanimity in French's theory of social power”. In Studies in social power, Edited by: Cartwright, D. pp. 168–182. Ann Arbor: Institute for Social Research.  [Google Scholar], and Abelson (1964) Abelson, R. P. 1964. “Mathematical models of the distribution of attitudes under controversy”. In Contributions to mathematical psychology, Edited by: Frederiksen, N. and Gulliksen, H. pp. 142–160. New York: Holt, Rinehart &; Winston.  [Google Scholar] initiated a prominent line of social influence models to explain social norms or collective decisions from the structure of influence networks. These models fail to generate unstable decision dynamics, a phenomenon that can be observed in collective decision-making. To capture instability, we assume that decision-makers raise their level of salience to reduce expected losses from decision-outcomes. Our model generates persistently unstable outcome patterns under conditions related to the social network and to intolerance for expected losses. A 6-actor example reveals stable outcomes for low intolerance, complex oscillations for intermediate levels of intolerance, and simple and regular oscillation for high intolerance. We discuss implications for the predictability of collective decision-making.     

Features of mathematical activities in interdisciplinary, project-based learning

Project-based learning in mathematics education leads to mathematical activities that are uncommon in regular lessons at school. Among these activities, the following are identified and examined more closely:

⊙ the elaboration and formulation of relevant mathematical problems, including necessary definitions.

⊙ the search for the mathematically feasible, and

⊙ the recognition of opportunities to apply mathematical methods.

Also, implications for the design of project-based learning environments are developed.     

An Investigation of Relationships between Seventh-Grade Students' Beliefs and Their Participation during Mathematics Discussions in Two Classrooms

As mathematics teachers attempt to promote classroom discourse that emphasizes reasoning about mathematical concepts and supports students' development of mathematical autonomy, not all students will participate similarly. For the purposes of this research report, I examined how 15 seventh-grade students participated during whole-class discussions in two mathematics classrooms. Additionally, I interpreted the nature of students' participation in relation to their beliefs about participating in whole-class discussions, extending results reported previously (Jansen, 2006 Jansen, A. 2006. Seventh graders' motivations for participating in two discussion-oriented mathematics classrooms. Elementary School Journal, 106: 409–428. [Crossref], [Web of Science ®] , [Google Scholar]) about a wider range of students' beliefs and goals in discussion-oriented mathematics classrooms. Students who believed mathematics discussions were threatening avoided talking about mathematics conceptually across both classrooms, yet these students participated by talking about mathematics procedurally. In addition, students' beliefs about appropriate behavior during mathematics class appeared to constrain whether they critiqued solutions of their classmates in both classrooms. Results suggest that coordinating analyses of students' beliefs and participation, particularly focusing on students who participate outside of typical interaction patterns in a classroom, can provide insights for engaging more students in mathematics classroom discussions.     

Global solutions in higher dimensions to a fourth-order parabolic equation modeling epitaxial thin-film growth

The initial-value problem for $$u_t=-\Delta^2 u - \mu\Delta u - \lambda \Delta |\nabla u|^2 + f(x)\qquad \qquad (\star)$$ is studied under the conditions ${{\frac{\partial}{\partial\nu}} u={\frac{\partial}{\partial\nu}} \Delta u=0}$ on the boundary of a bounded convex domain ${\Omega \subset {\mathbb{R}}^n}$ with smooth boundary. This problem arises in the modeling of the evolution of a thin surface when exposed to molecular beam epitaxy. Correspondingly the physically most relevant spatial setting is obtained when n?=?2, but previous mathematical results appear to concentrate on the case n?=?1. In this work, it is proved that when n??? 3,??? ?? 0, ???>?0 and ${f \in L^\infty(\Omega)}$ satisfies ${{\int_\Omega} f \ge 0}$ , for each prescribed initial distribution ${u_0 \in L^\infty(\Omega)}$ fulfilling ${{\int_\Omega} u_0 \ge 0}$ , there exists at least one global weak solution ${u \in L^2_{loc}([0,\infty); W^{1,2}(\Omega))}$ satisfying ${{\int_\Omega} u(\cdot,t) \ge 0}$ for a.e. t?>?0, and moreover, it is shown that this solution can be obtained through a Rothe-type approximation scheme. Furthermore, under an additional smallness condition on??? and ${\|f\|_{L^\infty(\Omega)}}$ , it is shown that there exists a bounded set ${S\subset L^1(\Omega)}$ which is absorbing for ${(\star)}$ in the sense that for any such solution, we can pick T?>?0 such that ${e^{2\lambda u(\cdot,t)}\in S}$ for all t?>?T, provided that ?? is a ball and u ₀ and f are radially symmetric with respect to x?=?0. This partially extends similar absorption results known in the spatially one-dimensional case. The techniques applied to derive appropriate compactness properties via a priori estimates include straightforward testing procedures which lead to integral inequalities involving, for instance, the functional ${{\int_\Omega} e^{2\lambda u}dx}$ , but also the use of a maximum principle for second-order elliptic equations.     

Three-Dimensional Alexandrov Spaces with Positive or Nonnegative Ricci Curvature

We study closed three-dimensional Alexandrov spaces with a lower Ricci curvature bound in the CD ^?(K,N) sense, focusing our attention on those with positive or nonnegative Ricci curvature. First, we show that a closed three-dimensional CD ^?(2,3)-Alexandrov space must be homeomorphic to a spherical space form or to the suspension of \(\mathbb {R}P^{2}\). We then classify closed three-dimensional CD ^?(0,3)-Alexandrov spaces.     

Lesson study in Asia Pacific classrooms: local responses to a global movement

Japanese Lesson Study is a model for teacher professional learning that has recently attracted world attention particularly within the mathematics education community. It is a highly structured process of teacher collaboration, observation, reflection and practice. The world focus has been mainly due to the work of American researchers such as Stigler and Hiebert (Am Educ Winter:1–10, 1998; The teaching gap: Best ideas from the world’s teachers for improving education in the classroom. Free Press, New York 1999), Lewis and Tsuchida (Am Educ Winter:14–17; 50–52, 1998) and Fernandez [J Teach Educ 53(5):395–405, 2002]. These researchers have documented Lesson Study from the perspective of their social, cultural and educational contexts. In order to develop a deeper understanding of Lesson Study in a post-modern global world, there is a need to seek views beyond those presented from an American perspective. This paper will provide further additional perspectives from an Australian state view and a Malaysian state district view and a university view. The aim is to develop an understanding of how the different contexts have influenced the structure and implementation of the Japanese Lesson Study model.     

Procrustes Problems and Parseval Quasi-Dual Frames

Parseval frames have particularly useful properties, and in some cases, they can be used to reconstruct signals which were analyzed by a non-Parseval frame. In this paper, we completely describe the degree to which such reconstruction is feasible. Indeed, notice that for fixed frames \({\mathcal{F}}\) and \({\mathcal{X}}\) with synthesis operators F and X, the operator norm of FX ^??I measures the (normalized) worst-case error in the reconstruction of vectors when analyzed with \({\mathcal{X}}\) and synthesized with \({\mathcal{F}}\) . Hence, for any given frame \({\mathcal{F}}\) , we compute explicitly the infimum of the operator norm of FX ^??I, where \({\mathcal{X}}\) is any Parseval frame. The \({\mathcal{X}}\) ’s that minimize this quantity are called Parseval quasi-dual frames of \({\mathcal{F}}\) . Our treatment considers both finite and infinite Parseval quasi-dual frames.     

Characterizing the Development of Specialized Mathematical Content Knowledge for Teaching in Algebraic Reasoning and Number Theory

This article characterizes the development of a deep and connected body of mathematical knowledge categorized by Ball and Bass' (2003b) Ball, D. L. and Bass, H. Toward a practice-based theory of mathematical knowledge for teaching. Proceedings of the 2002 Annual Meeting of the Canadian Mathematics Education Study Group. Edmonton, AB, Canada. Edited by: Davis, B. and Simmt, E. pp.3–14. CMESG/GCEDM.  [Google Scholar] model of Mathematical Knowledge for Teaching (MKT), as Specialized Content Knowledge for Teaching (SCK) in algebraic reasoning and number sense. The research employed multiple cases across three years from two content courses designed for elementary and middle-level mathematics specialists. Qualitative data were collected and a grounded theory approach to data analysis was employed. The resulting framework characterizes developmental levels of deep and connected mathematical content knowledge for teaching algebraic reasoning and number theory content. The framework consists of four intertwined components related to a teacher's ability to (1) solve problems and justify his/her reasoning, (2) use multiple representations, (3) recognize, use, and generalize conceptually similar tasks, and (4) pose problems. Implications for mathematics teacher education programs are discussed as well as directions for further research.     

A Multilevel Event History Model of Social Diffusion: Medical Innovation Revisited

This article presents a multilevel event history model of social diffusion and applies it to Coleman, Katz, and Menzel's (1966 Coleman , J. S. , Katz , E. , &; Menzel , H. ( 1966 ). Medical Innovation: A Diffusion Study . Indianapolis , IN : Bobbs-Merrill . [Google Scholar]) data on the adoption of tetracycline by physicians. The simplest form of a multilevel model allows a random intercept. In the present application of this simple model to the Medical Innovation data, structured for an event history analysis, the physicians are nested in city and time. Random intercepts capture effects of contextual conditions that are shared by event history cases with the same city–time status. The intercepts also reflect any baseline internal contagion effects, that is, the proportion of physicians in the city–time network who have adopted the drug at time t ? 1. Here, I show that Van den Bulte and Lilien's (2001 Van den Bulte , C. &; Lilien , G. L. ( 2001 ). Medical innovation revisited: social contagion versus marketing effort . American Journal of Sociology , 106 , 1409 – 1435 .[Crossref], [Web of Science ®] , [Google Scholar]) finding of an important contextual effect of drug firms' marketing effort is misleading. I also show that the social network in which physicians are situated significantly contributes to their adoptions, controlling for baseline internal contagion effects and individual-level characteristics of physicians, which have been emphasized in investigations of these data.     

Quasi-factors in ergodic theory

Motivated by the notion of quasi-factor in topological dynamics, we introduce an analogous notion in the context of ergodic theory. For two processes,X andY , we haveX?Y if and only ifY has a factor which is isomorphic to a quasi-factor ofX. On the other hand, weakly mixing processes can have nontrivial quasifactors which are not w.m. We characterize those ergodic processes which admit only trivial continuous ergodic quasi-factors, and use this characterization to conclude that a process with minimal selfjoinings is of this type. From this we derive the fact that for every suchX and any ergodicY eitherX ⊥Y orY extends some symmetric product ofX.     

How large dimension guarantees a given angle?

We study the following two problems: (1) Given $n\ge 2$ and $0\le \alpha \le 180^\circ $ , how large Hausdorff dimension can a compact set $A\subset \mathbb{R }^n$ have if $A$ does not contain three points that form an angle $\alpha $ ? (2) Given $\alpha $ and $\delta $ , how large Hausdorff dimension can a subset $A$ of a Euclidean space have if $A$ does not contain three points that form an angle in the $\delta $ -neighborhood of $\alpha $ ? An interesting phenomenon is that different angles show different behaviour in the above problems. Apart from the clearly special extreme angles $0$ and $180^\circ $ , the angles $60^\circ , 90^\circ $ and $120^\circ $ also play special role in problem (2): the maximal dimension is smaller for these angles than for the other angles. In problem (1) the angle $90^\circ $ seems to behave differently from other angles.     

The Isometry Group of an <Emphasis FontCategory="SansSerif">RCD</Emphasis><Superscript>∗</Superscript> Space is Lie

We give necessary and sufficient conditions that show that both the group of isometries and the group of measure-preserving isometries are Lie groups for a large class of metric measure spaces. In addition we study, among other examples, whether spaces having a generalized lower Ricci curvature bound fulfill these requirements. The conditions are satisfied by R C D ^?-spaces and, under extra assumptions, by C D-spaces, C D ^? P-spaces. However, we show that the M C C P-condition by itself is not enough to guarantee a smooth behavior of these automorphism groups.     

A Local Instruction Theory for the Development of Number Sense

Gravemeijer's (1999 Gravemeijer, K. 1999. How emergent models may foster the constitution of formal mathematics. Mathematical Thinking and Learning, 1: 155–177. [Taylor & Francis Online] , [Google Scholar], 2004 Gravemeijer, K. 2004. Local instruction theories as a means of support for teachers in reform mathematics education. Mathematical Thinking and Learning, 6: 105–128. [Taylor & Francis Online] , [Google Scholar]) construct of a local instruction theory suggests a means of offering teachers a framework of reference for designing and engaging students in a set of sequenced, exemplary instructional activities that support students' mathematical development for a focused concept. In this paper we offer a local instruction theory to guide the design of a set of instructional activities in support of the development of number sense. We make explicit the goals, assumptions, underlying rationale, and related instructional activities and provide examples from a mathematics content course for preservice elementary teachers. In this way, we contribute to an elaboration of the construct of local instruction theory.     

Semi-Heyting Algebras Term-equivalent to Gödel Algebras

In this paper we investigate those subvarieties of the variety $\mathcal {SH}$ of semi-Heyting algebras which are term-equivalent to the variety $\mathcal L_{\mathcal H}$ of Gödel algebras (linear Heyting algebras). We prove that the only other subvarieties with this property are the variety $\mathcal L^{\rm Com}$ of commutative semi-Heyting algebras and the variety $\mathcal L^{\vee}$ generated by the chains in which a?<?b implies a → b?=?b. We also study the variety $\mathcal C$ generated within $\mathcal{SH}$ by $\mathcal L_{\mathcal H}$ , $\mathcal L_\vee$ and $\mathcal L_{\rm Com}$ . In particular we prove that $\mathcal C$ is locally finite and we obtain a construction of the finitely generated free algebra in $\mathcal C$ .     

Completing Hermann Schubert's Work on the Enumerative Geometry of Cuspidal Cubics in ℙ3

Abstract

We outline the results of our revisiting Hermann Schubert's work on the enumerative geometry of cuspidal cubics in ?³(Sec. 23 of his Kalkül der abzählenden Geometrie, Teubner, [1879] Schubert, H. 1879. Kalkül der abzählenden Geometrie Teubner. Rep. in 1979 by Springer-Verlag [Google Scholar]. Rep. in 1979 by Springer-Verlag). There are three main aspects that we would like to point to. First, we describe the spaces parameterizing cuspidal cubics in ?³, as well as several different degenerations, using modern algebraic geometry language and techniques. Then we get formulas, by means of today's intersection theory, for the relevant relations among conditions and degenerations, and for allthe intersection numbers in which Schubert was in principle interested. And finally there is the computational aspect, which has been an adventure on its own: the computations have been performed by means of the mathematical computation system OmegaMath, together with the WITmodule. They are discussed briefly in the final Section, with references to detailed information, and here we would just like to say that one of our motivations has been to test that system in what has turned out to be an interesting computational project. Our final table for the cuspidal cubics, which has the 19778 nonzero numbers involving the nine first-order conditions considered by Schubert, fully confirms the fraction of numbers computed by Schubert, as listed on pages 140–143 of the Kalkül. (For those interested in getting our table, please see the indications in the last section of the full paper.) For an assessment of whether or not the numbers computed by Schubert are fully representative of the problems involved in computing all of them, see the Remark at the end of Sec. 3.     

THREE-GENERATOR INDECOMPOSABLE RA LOOPS

ABSTRACT

In [2] Orin, Chein and Edgar G., Goodaire. Minimally Nonassociative Nilpotent Moufang Loops Preprint [Google Scholar], we showed that the minimally nonassociative RA loops (those which are not themselves associative but for which every proper subloop is associative) are precisely the RA loops which are indecomposable (that is, not nontrivial direct products) and which can be generated by three elements. Here, we investigate which RA loops have these two properties.     

On the Analytic Structure of Commutative Nilmanifolds

In the classification theorems of Vinberg and Yakimova for commutative nilmanifolds, the relevant nilpotent groups have a very surprising analytic property. The manifolds are of the form \(G/K = N\rtimes K/K\) where, in all but three cases, the nilpotent group \(N\) has irreducible unitary representations whose coefficients are square integrable modulo the center \(Z\) of \(N\). Here we show that, in those three “exceptional” cases, the group \(N\) is a semidirect product \(N_{1}\rtimes \mathbb {R}\) or \(N_{1}\rtimes \mathbb {C}\) where the normal subgroup \(N_{1}\) contains the center \(Z\) of \(N\) and has irreducible unitary representations whose coefficients are square integrable modulo \(Z\). This leads directly to explicit harmonic analysis and Fourier inversion formulae for commutative nilmanifolds.