John Selden
I received my Ph.D. in topological semigroups from the University of Georgia in 1963. I am a 5th generation Moore student [R. L. Moore, Gordon Whyburn, Alexander Donaphin Wallace, Robert Koch, Robert P. Hunter, John Selden]. I have taught at various universities in the U.S., Canada, Turkey, and Nigeria, and have directed 9 Ph.D.'s in mathematics. I served as Dean of Science, Bayero University in Nigeria, and upon returning to the U.S., took up research in undergraduate mathematics education. I was a member of the AMS/MAA Joint Committee on Research in Undergraduate Mathematics Education (CRUME) and various other MAA Committees. I was a Visiting Scholar at Peabody College; Vanderbilt, at the Division of Education in Mathematics, Science, and Technology, University of California at Berkeley; at the Center for Research in Mathematics and Science Education, San Diego State University; and the Department of Mathematics, Arizona State University. I am currently Adjunct Professor of Mathematics, New Mexico State University.
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Math Ed Papers by John Selden
continuous associative operations on S such that one (called multiplication) distributes across the other (called addition). That is, we insist that
x(y + z) = xy + xz and (x + y) z = xz + yz for all x,y and z in S. Note that, in contrast to the purely algebraic situation [1,2], we do not postulate the existence of an additive identity which is a multiplicative zero.
In this note we point out conditions under which the existence of such an
element is equivalent to the double simplicity of the semiring. We also discuss
maximal and minimal double ideals together with several examples.
with two continuous associative operations on S such that one (called
multiplication) distributes across the other (called addition). That
is, we insist that x(y+z)=xy+xz and (x+y)z = xz+yz for all x, y,
and z in S. Note that, in contrast to the purely algebraic situation,
we do not postulate the existence of an additive identity which is a
multiplicative zero.
In this note we point out a rather weak multiplicative condition
under which each additive subgroup of a compact semiring is totally
disconnected. We also give several corollaries and examples.
postulate the existence of an additive identity which is a multiplicative
zero.
In this note we characterize compact additively commutative semirings which are multiplicatively left zero simple. This is accomplished by first examining semirings which are multiplicatively groups with zero and then proceeding to the general situation. For other remarks on compact semirings the reader is referred to [3], [4].
The notation follows closely that of topological semigroups [5].
one-parameter inverse semigroups and then describe their congruences.
In the mathematics education literature, a variety of genres of mathematical writing have been considered. For example, in discussing mathematical writing at the school level, Marks and Mousley (1990, p. 119) distinguished narrative genre, procedural genre, description and report genres, exploratory genre, and expository genre. While all these have a place when examining mathematical writing more generally, in this short contribution, we restrict our considerations to the genre of proof, and in particular, to how mathematicians write proofs for publication. As Konior (1993) argued, studying the genre of mathematical proof is particularly important for mathematics educators, as this can inform how students should read and write proofs.
Mathematics educators, mathematicians, and philosophers have written about the genre of mathematical proof, emphasizing it special nature, its long evolution, and the impossibility of making it entirely explicit. For example, Ernest (1998, p. 169) stated, “Mathematical proof is a special form of text, which since the time of the ancient Greeks, has been presented in monological [rather than dialogical] form.” Jaffe (1990, p. 146) asserted that “The standards of what constitutes a proof have evolved over hundreds of years; there is no doubt in the minds of traditional mathematicians what a proof means.” Furthermore, according to Kitcher (1984, p. 163), in mathematical practice both tacit knowledge, or “know how,” and meta-mathematical views (including standards for proof) are important, and it is not possible for those standards to be made fully explicit. However, it may be possible to identify some significant features that generally occur in the genre of proofs.
We consider what we call a hierarchical structure based on subproofs, subconstructions, and parallel arguments (such as case arguments). This is an extension of our proof framework idea and is illustrated by a proof of the theorem: is continuous at a point provided and are, which appears below in the Appendix A.
We propose to study the effects of one curriculum project workshop and subsequent implementation attempts on its college mathematics teacher-participants. ["Learning Abstract Algebra: A Research Based Laboratory and Cooperative Learning Approach,"
funded through NSF Research in Learning and Teaching Program, Ed Dubinsky, Purdue University, PI.] Specifically, we propose to study the teaching practices and personal pedagogies of the participants, i.e., their pedagogical and mathematical beliefs, attitudes, priorities, and behaviors. We plan to observe the workshop and evaluate how its college teacher-participants' personal pedagogies and behaviors are changed by it. We will gather interviews and self-reported data and evaluate participants' subsequent teaching practices, as well as their teaching environment and backgrounds in mathematics.
The first third of this survey course will emphasize the
concepts, perspectives, and vocabulary arising from
research in undergraduate mathematics education.
These will then be used during the remainder of the
course to examine and compare various forms of teaching
(and assessment) including: lecture, whole-class discussion,
group and collaborative work, problem solving, distance
learning, discovery learning, and the Moore method.
The course will also examine the use of writing, computers
and calculators, various kinds of software (including CAS's
and aids to visualization), student projects and labs, and
programming as an aid to conceptual understanding.
The course will meet for the equivalent of a 15-week
3-credit hour course and involve discussions (equivalent
to three hours per week), conducted over the internet, partly
asynchronously and partly synchronously. Where practical,
participants will work in small groups and may be asked to
discuss their own and each other's teaching. The course will
require at least six hours' reading per week and at its conclusion
each participant will write a brief (individually produced) paper.