The American Computer Museum opened its doors in Bozeman in 1990. It presently has the most comprehensive collection of the history of communications and computing on public display. Drawing on this collection, the presentation will highlight the mileposts in the evolution of mathematics, calculation, data processing, telecommunications and the Internet. Using scores of photographs and a video, George Keremedjiev, the director of the museum will offer a tour of the information highway, the personalities behind its inventions and the social impact to come, through the use of computers and the Internet, in the unraveling of nature's most complex code, DNA . The presentation will appeal to both novices and experts in the fields of mathematics, computing and the history of science and technology.

Thursday, 1 February 2001
4:10 p.m. in Math 109
Coffee/treats at 3:30 p.m. Math 104 (Lounge)

Presenting a multi-level investigation of how people understand division of fractions, this research considered middle school students and their teachers, pre-service elementary students, university graduate students, and mathematics faculty.  The study responds to research conclusions of Ma (1999) in Knowing and Teaching Elementary Mathematics in which Chinese teachers were, on average, better prepared than their US counterparts to teach mathematics.

Thursday, 8 February 2001
4:10 p.m. in Math 109
Coffee/treats at 3:30 p.m. Math 104 (Lounge)

Postponed until March 29

In four years of teaching mathematics at Salish Kootenai College, these questions have haunted me.  My students routinely span each spectrum:  age, academic preparation, natural ability, math anxiety, and learning style.  Leveling the playing field enough to cover essential content is an overwhelming task.  There have been many worthy attempts to adapt course content, but my greatest successes have utilized student group interaction.  I will share a few of my cooperative learning experiments, ranging from open-ended calculus projects to base groups and readiness testing in statistics.  In addition, I hope to entertain a short discussion of the advantages/disadvantages of using these techniques in a university setting.

Thursday, 29 March 2001
4:10 p.m. in Math 109
Coffee/treats at 3:30 p.m. Math 104 (Lounge)

Do today's mathematics students learn enough geometry to succeed in later courses?

What is the role of geometry in today's mathematics curriculum?

What do we lose if geometry is gone?

Some history, some data, some opinions, and some suggestions.

Thursday, 22 February 2001
4:10 p.m. in Math 109
Coffee/treats at 3:30 p.m. Math 104 (Lounge)

Tourism in rural states is starting to show signs of wear on the residents.  A mail-back random survey of Montana residents was used to examine residents' opinions about tourism and about the dispersion of a statewide bed tax.  Residents ranked tourism fifth as an opportunity for future economic development.  While residents support tourism development, most feel no direct connection with their lives to the tourism industry. Residents placed a high priority on spending the bed-tax on the environmental product of state parks, fish and wildlife resources, and purchasing land for open space in contrast to the current policy of spending the majority of tax dollars for promotion. Three years of data on bed tax dispersion shows consistent findings.  The future for resident backlash toward the industry is becoming evident.  A commission designed to examine tourism policy is recommended.

Thursday, 8 March 2001
4:10 p.m. in Math 109
Coffee/treats at 3:30 p.m. Math 104 (Lounge)

A number of authors have addressed the idea that mathematics or the way we understand and think about mathematical ideas is a cultural artifact.  This work represents partial results from a case study of the enculturation experiences of a new mathematician.  That ethnography documents and describes the effects of mathematics and involvement in doing mathematics on socio-cultural behavior within the newcomer's department.  Three behaviors--sassing students, despairing the new professional evaluation system, and regaling others with tales of public innumeracy--are discussed in terms of their importance to group self-definition and self-esteem.  These behaviors make up “the burden of mathematics” and represent the primary coping system generated in response to a public prejudice in favor of innumeracy.  Implications for future research in mathematics education both in terms of the emerging role of larger units of analysis (the school instead of the classroom) and in terms of the blame- and solution-free focus of the approach will be discussed.

Wednesday, 14 March 2001
4:10 p.m. in Math 109
Coffee/treats at 3:30 p.m. Math 104 (Lounge)

Although the numerical solution of linear differential equations via the Hermite collocation discretization has been widely studied, there have been (until now) no results giving analytical formulas of the solution of the linear systems that arise from this discretization.

This talk will present recent results in this area, including comparison of the discrete Hermite collocation solution with the corresponding continuous solution.

Friday, 16 March 2001
4:10 p.m. in Math 109
Coffee/treats at 3:30 p.m. Math 104 (Lounge)

Dr. Luebeck will describe ongoing qualitative and descriptive research efforts to identify the content and pedagogical needs of isolated rural K-12 mathematics teachers and to measure the effectiveness of intervention models that offer mentoring and reform-based renewal to this population. Issues of methodology and design will be discussed, along with new lines of inquiry linking the current research to cognition and communication issues.  Finally, Dr. Luebeck will summarize her current work with doctoral students on research in undergraduate mathematics education.

Monday, 26 March 2001
4:10 p.m. in Math 109
Coffee/treats at 3:30 p.m. Math 104 (Lounge)

In four years of teaching mathematics at Salish Kootenai College, these questions have haunted me.  My students routinely span each spectrum:  age, academic preparation, natural ability, math anxiety, and learning style.  Leveling the playing field enough to cover essential content is an overwhelming task.  There have been many worthy attempts to adapt course content, but my greatest successes have utilized student group interaction.  I will share a few of my cooperative learning experiments, ranging from open-ended calculus projects to base groups and readiness testing in statistics.  In addition, I hope to entertain a short discussion of the advantages/disadvantages of using these techniques in a university setting.

Thursday, 29 March 2001
4:10 p.m. in Math 109
Coffee/treats at 3:30 p.m. Math 104 (Lounge)

As a at Juneau Douglas High School (Alaska) teacher, I linked mathematics to a modern day art form using a group-based program called "Phoenix".  The link gave students the potential either to create a unique colorful image or to learn programming skills. As an additional feature, it captured the attention of less attentive students.

Content leading to the development of each student's fractal included a modulus operator, base conversion, binary code, logical operators, group theory, the concept of a mapping, matrix addition and multiplication, eigenvalues, probability, and writing C++ source code.  Though this sounds impossible, it was possible to get all this into student projects on fractals and to have them produce very colorful results.

Thursday, 5 April 2001
4:10 p.m. in Math 109
Coffee/treats at 3:30 p.m. Math 104 (Lounge)

Math Awareness Month Event

Earth's climate is changing, with both natural cycles and human-induced change. These are not uniform warmings and coolings that

happen everywhere, but instead come in patterns.  Natural oscillations of the oceans and atmosphere can make the Labrador Sea chill down while Canada and northern Asia warm.

In this talk we describe how expeditions to the sub-Arctic on ships lead to mathematical models of ocean circulation and its variability. By taking math and physics 'out-doors' we find a world of inviting problems, whose solutions affect our future water supplies, weather and the health of all our ecosystems. And, we also find some adventure in this final wilderness.

Thursday, 12 April 2001
4:10 p.m., North Underground Lecture Hall
Coffee/treats at 3:30 p.m. Math 104 (Lounge)

Determining sources of item difficulty and using them for selection or development of test items is a bridging task of psychometrics and cognitive psychology. A key problem in this task is the validation of hypothesized cognitive operations required for the correct solution of test items. This issue is addressed in the presentation with examples from testing students' proficiency on algebra tests of simple linear equations.

Friday, 13 April 2001
4:10 p.m. in Skaggs 114
Coffee/treats at 3:30 p.m. Math 104 (Lounge)

In  his famous paper of 1935, Phillip Hall answered the following question:

What are necessary and sufficient conditions on a collection of finite sets S1, S2, . . . , Sn such that there exists an n-tuple (x1, x2, . . . , xn) of n distinct elements such that xi is in Si?

We call such an n-tuple a system of distinct representatives.  Since this result, there have been many related results, applications and generalizations of Hall’s Theorem. 

In this presentation we will state a more general question to that above and consider how Hall’s Theorem may apply.  We will discuss how previous work may be viewed through this generalization and present a few applications of this theory to universal algebra and graph coloring problems.

Thursday, 19 April 2001
4:10 p.m. in Math 109
Coffee/treats at 3:30 p.m. Math 104 (Lounge)

In partial fulfillment of the requirements for a doctoral degree.

A class of models with only a few easily identifiable parameters are introduced to allow the long-term consequences of disturbances in a natural forest to be qualitatively described. Formulated in terms of the von-Foerster partial differential equation, these models can be reduced to an integro-differential equation for the seedlings' density as a function of time. This seedlings equation contains a small parameter, the ratio of seedlings' re-establishment time and the average life span of a tree. The re-establishment time, typically 2-5 years, measures the time for the number of seedlings to adapt to a change in available resources.

The problem is solved using numerical and asymptotic methods. By means of Banach's fixed point theorem in a suitable function space, it can be shown that these methods converge even when the number of seedlings intermittently vanishes and the solution is not continuously differentiable. In this case, for a certain range of parameters, periodic solutions occur. 

Thursday, 26 April 2001
4:10 p.m. in Math 109
Coffee/treats at 3:30 p.m. Math 104 (Lounge)

This presentation will examine funding possibilities and proposal preparation at The National Science Foundation. While concentrating on the programs in both fundamental and interdisciplinary mathematicians, there will be content of utility to other scientists as well. Emphasis will be placed on the potential within the context of the forthcoming Mathematical Sciences Investments.

Thursday, 3 May 2001
4:10 p.m. in Math 109
Coffee/treats at 3:30 p.m. Math 104 (Lounge)

We show that there is an (amazing!) integer valued function f(n) (n=0,1,2,...) such that 

(a) the sequence f(0)/f(1), f(1)/f(2), f(2)/f(3), ... consists of all of the positive rational numbers, each occurring once and     only once, and 

(b) f(n) and f(n+1) are always relatively prime, so each rational occurs in part (a) in reduced form, and 

(c) the function f(n) actually counts something of combinatorial interest.

Thursday, 6 September 2001
4:10 p.m. in James E Todd Building CE 203-204
Reception at 3:30 p.m. CE 204

This talk is part of The Big Sky Conference, sponsored by the National Science Foundation and the Department of Mathematical Sciences.

In the town of Königsberg there were seven bridges connecting a number of islands. The question was whether a citizen could take a single walk that would cross every bridge exactly once. Euler solved this problem and a whole family of problems like this one, and we will discuss the solution that he found. We will also describe some of the  applications of this subject to contemporary DNA sequencing, so these questions are of practical importance as well as being very pretty. You might enjoy having a look at the web pages The Beginnings of Topology, and "New path lays DNA puzzles bare", to get a little background material. I'll also say a few words about how computers can now do some things that  used to be thought of as do-able only by human mathematicians. The list of such things is growing steadily.

Thursday, 6 September 2001
8:00 p.m. in Music Recital Hall

This lecture is intended for a general audience.

This talk is part of The Big Sky Conference, sponsored by the National Science Foundation and the Department of Mathematical Sciences.

Tutte discovered an interesting connection between coloring planar graphs and nowhere-zero flows.  For planar dual graphs G and G*, he showed that k-colorings of G dualize no nowhere-zero k-flows of G*.  Based on this duality, Tutte made three fascinating conjectures concerning nowhere-zero flows.  All three of these conjectures are still open.

After exploring this duality and discussing some known properties of nowhere-zero flows, we move on to a more general perspective:

choosability in vector spaces.  In this realm we discuss what is known to be true and what is conjectured to be true.   In particular, we give two choosability theorems about matrices over fields of characteristic two which generalize Jaeger's 4-flow and 8-flow theorems, and we suggest a new conjecture about matrices over fields of odd characteristic which would imply Tutte's 5-flow conjecture.

Friday, 7 September 2001
4:10 p.m. in James E Todd Building CE 203-204
Reception at 3:30 p.m. CE 204

This talk is part of The Big Sky Conference, sponsored by the National Science Foundation and the Department of Mathematical Sciences.

A random walk on a graph is a walk where at each vertex visited, an edge incident to the vertex is chosen at random, and the walk proceeds along the edge. 

A random walk on a graph represents a reversible Markov chain, whose transition probabilities depend on the degrees of the vertics. A graph is said to be recurrent on transient, according to wheather the corresponding Markov chain is recurrent or transient. We shall discuss how random walks on graphs can be used to classify Rievann surfaces, as to their hyperbolicity or parabolicity.

Thursday, 27 September 2001
4:10 p.m. in Math 109
Coffee/treats at 3:30 p.m. Math 104 (Lounge)

According to Hadamard, a problem defined by the operator equation 

Az = u(1)

(where z and u are elements of metric spaces Z and U,  respectively) is correctly (or  well)  posed  problem  if  the  following  three conditions are satisfied: (a) Eq. (1) is solvable for any u; (b) the solution of Eq.(1) is unique; (c) the solution of Eq.(1) is stable with respect  to  perturbations in the right-hand side u; i.e., that inverse  operator  exists, is defined throughout the space U, and is continuous.

If one of the conditions (a)-(c) does not hold, the problem is called ill-posed. A lot of mathematical problems are ill-posed. Among them there are the following very well known examples:  the Fredholm integral equation of the first kind; an operator equation (1) with a completely continuous operator in infinite-dimensional spaces, etc. 

A.N. Tikhonov proposed a special approach for solving ill-posed problems: for searching for stable solutions of unstable ill-posed problems it is necessary to use special regularizing operators  (algorithms), if they exist, which give stable approximations to exact solution of unstable problems. Tikhonov has proposed also concrete regularizing operators for linear ill-posed operator equations in the Hilbert spaces, for minimization of functionals, for unstable problems of linear algebra, etc.

At present, the theory of ill-posed problems is developed and is widely used to solve inverse problems in optics, spectroscopy, electrodynamics, plasma diagnostics, geophysics, astrophysics, image processing, etc. Regularizing algorithms, when being applied to process experimental data, significantly improve the accuracy with which the parameters of the physical objects are determined. The resolution of an experimental device can thus be greatly increased simply by using computer data processing without any expensive modification of the device itself. There is no doubt that the most effective systems with software packages for experimental work include programs based on regularizing algorithms. The methods for solving ill-posed problems now available can be successfully used in various branches of natural sciences.

In my lecture I would like to introduce some fundamental results of the theory of linear and nonlinear ill-posed problems and its applications.

Thursday, 4 October 2001
4:10 p.m. in Math 109
Coffee/treats at 3:30 p.m. Math 104 (Lounge)

Over the course of last summer, I developed a strong interest in hierarchical linear modeling in a collaborative work with Dr. Wes Snyder (Assistant Vice President for Research & Director of the Office of International Programs) as part of the Gates project to enhance technology for educational leaders. The project is funded by the Bill and Melinda Gates Foundation.

Hierarchical linear models (HLM) provide a conceptual framework and a flexible set of analytic tools to study a variety of social, political, and developmental (Biological) processes. HLM incorporate data from multiple levels in an attempt to determine the impact of individual and grouping factors upon some individual level outcomes. For example, student achievement may be a function of student level characteristics (e.g., IQ, study habits), classroom level factors (e.g., instruction style, textbook), school level factors (e.g., wealth), and so on. HLMs, or multilevel models, can incorporate such factors in a manner better than ordinary least squares since HLMs take into account error structures at each level. In Biology, animal and human studies of inheritance deal with a natural hierarchy where offspring are grouped within families. Hierarchy is usually referred to the fact that these problems consist of units grouped at different levels. Thus offspring may be the level 1 units in a 2-level structure where the level 2 units are the families: students may be level 1 units clustered within schools that are the level 2 units.

In this talk I will give an overview of HLM, consider the formulation of statistical models in educational applications, give several examples of the 2-level and 3-level structures, and explain the procedure of solving them using one of the available software.

Thursday, 11 October 2001
4:10 p.m. in Math 109
Coffee/treats at 3:30 p.m. Math 104 (Lounge)

Forensic DNA profiling has become a powerful and highly discriminating tool for identifying sources of biological material deposited during the commission of crimes. The degree of discrimination has reached a point to where the F.B.I. will identify, to a scientific certainty, the source of the biological material. This talk will review the history of forensic statistics, the formula commonly used in single source material analysis, and other situations where statistics are used in forensic DNA analysis.

Thursday, 18 October 2001
4:10 p.m. in Math 109
Coffee/treats at 3:30 p.m. Math 104 (Lounge)

Mr. Michael Allan Andrus
Adjunct Instructor
Department of Mathematics
The University of Montana

Thursday, 25 October 2001
4:10 p.m. in Math 109
Coffee/treats at 3:30 p.m. Math 104 (Lounge)

In the spring of 2001 large class sizes became commonplace in the Mathematics Department at the University of Montana. There are now classes of 125 to 250 students taught in the North Underground Lecture Hall, (and even less desirable rooms) in Mathematics courses 107, 117, 121, 150, and 241. It is common knowledge that it is difficult to successfully teach large lecture classes, especially for teachers who have been used to teaching class sizes of 30 or less. Six of the instructors of these classes will present their experiences and concerns associated with this assignment, and will discuss new issues that arise, such as, necessary changes in teaching styles, quality of instruction, use of group activities, logistics of administering tests, how to "reach the students in the back row", and other issues.

Mark S. Cracolice, Professor of Chemistry, and Director of the Center for Teaching Excellence at the University of Montana, will attend the discussion and present a follow up discussion next week at this same time. Libby Krussel will be the moderator and the following faculty will be the panel: Greg St. George, Lily Eidswick, Mark Heaphy, Carol Ulsafer, Regina Souza, and Karel Stroethoff. At a time when football teams have coaches for very specialized units, mathematics is asked to do the impossible and teach large classes of students, many of whom must work 20 or more hours a week and have little time to study mathematics.

Thursday, 1 November 2001
(Follow up on Thursday, 8 November 2001)
4:10 p.m. in Math 109
Coffee/treats at 3:30 p.m. Math 104 (Lounge)

The main idea of the twistor theory, created by R. Penrose to solve problems in Mathematical Physics, is that the geometry of a conformal manifold M can be "encoded" in holomorphic terms of the so-called twistor space associated to M. The Penrose ideas have been developed in the context of the Riemannian geometry by Atiyah, Hitchin and Singer in the case of manifolds of dimension four. In particular, they have defined an almost-complex structure, say J₁, on the twistor space Z of such a manifold M which is invariant under conformal changes of the metric of M and found the integrability condition for this almost-complex structure. J. Eells and S. Salamon have introduced another almost-complex structure on Z, say J₂, which, although is not conformally invariant and is never integrable, plays an important role in the harmonic maps theory. The twistor space Z admits a natural one-parameter family h_t, t > 0, of Riemannian metrics compatible with the Atiyah-Hitchin-Singer and Eells- Salamon almost-complex structures. Thus we have two almost-Hermitian manifolds (Z, J₁, h_t) and (Z, J₂, h_t) and the main purpose of this talk is to discuss some geometric properties of these manifolds. More precisely the following topics will be considered:

1. The Atiyah-Hitchin-Singer theorem for integrability of the almost-complex structure J₁.
2. The Penrose transform. Applications.
3. Twistor spaces with Hermitian Ricci tensor.
4. KähIer curvature identies on twistor spaces.

Friday, 9 November 2001
4:10 p.m. in Math 109
Coffee/treats at 3:30 p.m. Math 104 (Lounge)

In 1803 the Italian mathematician Giovanni Malfatti posed the following problem: Given a triangle, find three nonintersecting circles inside of it such that the sum of their areas is maximal. Malfatti and many other mathematicians have thought that the solution of this problem is given by the three circles each of which is tangent to the other two and also to two sides of the triangle. Malfatti has computed the radii of these circles and they are now known as Malfatti's circles. Later on it became clear that the conjecture of Malfatti is not true. Moreover Goldberg proved in 1969 that the Malfatti circles never give a solution of the Malfatti problem, i.e. for any triangle there are three nonintersecting circles inside of it whose area is bigger than the area of the Malfatti circles. As far as the author knows, the Malfatti problem has not been solved yet in the general case although it seems reasonable to conjecture that the solution is given by the greedy algorithm: We first inscribe a circle in the given triangle; then we inscribe a circle in the smallest angle of the triangle which is tangent to the first circle. The third circle is inscribed either in the same angle or in the middle angle of the triangle, depending on which of them has bigger area. In the first part of this lecture we shall discuss the Malfatti problem for two circles in a triangle or in a square. Then we shall consider some problems which, in some sense, are dual to the problems above. Our main purpose is to show how one can solve the Malfatti problem for an equilateral triangle.

*For five years the speaker has been the coach of the Bulgarian Mathematics Olympic Team, which constantly has been successful at the International Mathematical Olympiads for high school students. He will share his experience on working and interaction with gifted students.

Friday, 13 November 2001
4:10 p.m. in Skaggs 117
Coffee/treats at 3:30 p.m. Math 104 (Lounge)

This talk was presented at the 100th "birthday party" for the School Science and Mathematics Association in the Chicago Area a month ago. The research started as a comparison of past geometry books, which I collect, but soon led to the quest for answers to:

• Why change?
• Who were suggesting the changes? (You will be surprised.)
• When were the changes suggested?
• What were the changes?
• Were the changes accepted?
• Are the changes in the textbooks?

Handouts will be provided, and a few old books will be available for inspection with your TLC.

The speaker's background includes teaching at the high school and college levels, past president of the Metropolitan Mathematics Club of Chicago, past president of School Science and Mathematics Association, co-chaired the ISMAA Geometry Committee, General Electric Fellow at Purdue University, and the Burlington Northern award at North Central College.

He also has authored a geometry text and a CD-text titled "Mathematical Bridges For Your Future".

Thursday, 29 November 2001
4:10 p.m. in Math 109
Coffee/treats at 3:30 p.m. Math 104 (Lounge)

Four speakers will share a few thoughts on aspects on computer use in mathematics in the 21st Century. Each will present 10-12 minutes. This is the time allotment that many large scientific meetings give to speakers with contributed papers. The order of presentations:

1. Karel Stroethoff: The World Wide Web as a Teaching Tool (4:10 - 4:22)
2. Scott Stevens: A Couple of Maple Projects and "Style Points" (4:22 - 4:34)
3. Brian Steele: The use of Acrobat Reader and PDF in the class room (4:34 - 4:46)
4. Dick Lane: Maple - Animated Plots and Documents with Inline Mathematics (4:46 - 4:58)

This is the last scheduled colloquium of Fall Semester.

Thursday, 6 December 2001
4:10 p.m. in Math 109
Coffee/treats at 3:30 p.m. Math 104 (Lounge)

This presentation will be an exposition of mathematics of native peoples of North America related to the Western mathematics traditionally studied at the elementary through college level. This ethnomathematical review was made not only to allow instructors of Native American students to include relevant mathematics developed by Indian people in the school curriculum, but also to offer all students a fuller understanding of the universal nature and power of mathematics. Primary and secondary sources of Indian and Western mathematics were surveyed, summarized, analyzed and synthesized, and the results will be presented here. Sources of curriculum materials for inclusion of Native American approaches to various mathematical topics are offered throughout. This review concludes with a discussion of the implications for teaching and learning mathematics.

Tuesday, 18 December 2001
4:10 p.m. in Math 109
Coffee/treats at 3:30 p.m. Math 104 (Lounge)

The process of generalization is an important component of mathematical ability, and to develop this ability is an objective of mathematics teaching and learning (NCTM, 2000). The research study documented how high school freshmen in an accelerated algebra class developed generalization strategies in combinatorial problem-solving situations. Students were asked to solve five non-routine combinatorial problems in their journals, assigned at increasing level of complexity. The generality that characterized the solutions of the five problems was the pigeonhole (Dirichlet) principle.

The researcher documented the evolving strategies of the students. Data gathered through journal writings, open-ended interviews and classroom observations was analyzed using techniques from grounded theory. In particular the constant comparative method of Glaser & Strauss (1977) was used. The researcher expected that student strategies would evolve with the complexity of the problem and with time. Four students were successful in discovering, verbalizing, and in one case successfully applying the generality that characterized the solutions of the five problems, whereas five students were unable to discover the hidden generality.

The research categorized and described student behaviors that led to successful generalizations and those that led to unsuccessful generalizations, as well as identified the variables necessary for students to successfully arrive at mathematical generalizations. The research study resulted in a modification of Lester's (1985) problem-solving model, for the purpose of understanding the generalization process in problem-solving situations. The modified model was an adaptation and extension of Lester's (1985) model and elucidated the properties of the categories in terms of student behaviors. It included an explicit task component and an affective component. The modified model could serve as a pedagogical tool in a mathematics classroom.

Thursday, 20 December 2001
4:10 p.m. in Math 109
Coffee/treats at 3:30 p.m. Math 104 (Lounge)