JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.. The Johns Hopkins University Press is collaborating with JSTOR to digitize, preserve and extend access to American Journal of Mathematics. 1. Introduction. Let Mn be a compact oriented Riemannian n-manifold with Riemannian metric < , > and Riemannian connection V. We use APM for the space of p-forms, Tq,PM for the space of tensors of type (q,p), x(M) for the space of vector fields on M, d for exterior derivative, 8 for codifferential and A = d + 8d: APM--APM for the Laplace-Beltrami operator on p-forms. (As a general reference, see [4].) Let SpecPM={XERj There is 0 # co &APM for which Aco=Xco} be the spectrum of the Laplace-Beltrami operator. If X & SpecPM, then the multiplicity of X is the dimension of {co e APM IAo = Xco}. There has been a great deal of interest in the spectrum of a Riemannian manifold in terms of Minakshisundaram-Pleijel type expansions on both functions ([4]) and forms ([11]). The spectra of a Riemannian manifold has been used as the main tool in an analytic proof of the Atiyah-Singer Index theoreom ([1], [6], [7]). There is also interest, coming from physics, in expansions of functions in terms of eigenfunctions of the Laplacian ([2]) and expansions of forms in terms of eigenforms of the Laplacian ([3]). Unfortunately, SpecP(M) has actually been computed only in isolated cases. Speco(M) is known forflat tori, the Klein bottle, Sn, CPn and RP' as well as coverings, products and submersions with totally geodesic fibers [4] of the above. We know of no computations of SpecP (M) with p >0 except for the result of Lichnerowicz quoted in section 1, the fact that SpecP(M)=Spec'-P(M) and the fact, immediate from Hodge theory, that 0 & SpecP (M) if and only if 8, (M) 0 (and 0 has multiplicity /k (M)).In this note we announce that we have reduced the problem of computing SpecP (G) for an arbitrary compact semisimple Lie group (with the Killing form metric) to a completely algebraic problem. The solution to this algebraic problem is known in general and can be readily computed in any specific case although no general formula is known for an arbitrary group. This method also enables us to actually compute both the value of the eigenvalue, X, and its Manuscript

This article provides an overview of mathematical habits of the mind and discusses how the concept relates to Polya's problem solving principles as well as exemplification. Specific problems are discussed as a means to assist preservice elementary school teachers' in their development of mathematical habits of the mind. Without a technique to begin solving these rich problems, preservice teachers may have difficulty getting started. The process of preservice teachers outlining their thinking as they progress through Polya's process is discussed. Students' reflections from this technique are discussed to explore the outcomes that may be expected from establishing an environment where students are encouraged to develop mathematical habits of the mind.

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The present study aims to explore the use of assessment in mathematics content courses for future elementary school teachers. Analysis of self assessment data on mathematical understanding and peer assessment data on oral mathematical presentation showed that pre-service teachers had a balanced understanding of procedural knowledge and problem solving. Conceptual understanding was not in the structure of pre-service teachers' mathematical knowledge. Understandings of conceptual knowledge, procedural knowledge, and problem solving had no meaningful effects on gains in mathematics performance. Aspects of oral mathematical presentation were associated with improved understanding of procedural knowledge and in particular conceptual knowledge. The result of the study calls for a conceptual approach to mathematical knowledge and sufficient mathematical problem solving in college-level mathematics content courses and in particular the infusion of assessment into college-level mathematics education for pre-service teachers.Keywords Pre-service teacher education · Structure of mathematical knowledge · Infusion of assessment into mathematics content courses We explored in the present analysis the use of assessment in mathematics content courses for future elementary school teachers. 1 At most institutions in the United States, these courses are taught by mathematicians whose emphasis is on learning or relearning the mathematics rather than pedagogical implications of mathematical knowledge, usually as a direct result of the disconnection between mathematicians and mathematics educators. These courses are typically difficult to a large percentage of students majoring in elementary education because their mathematical competency is significantly lower than the established norms of the general population of university students (e.g., Rech et al. 1993;Southwell and Penglase 2005).1 Because 90% of the participants in these mathematics content courses are pre-service K-5 (kindergarten to grade 5) teachers at the university where the present study was conducted, we used the phrase "pre-service elementary teachers" for this group throughout this article. The others (10%) are pre-service middle school teachers (grade 6 to grade 8).