# Technical Reports

## Technical Reports 2005

### The Oval Experiment, Measuring Distance

Rudy Gideon
University of Montana
Missoula, MT 59812

Abstract

An actual experiment in an applied statistics class is beneficial in helping students understand the many complexities of estimation. In a senior- graduate level applied statistics class the following independent two-sample experiment was carried out by the students. An Oval sits in the heart of the University of Montana campus with only grass and sidewalks on the edge and interior of this Oval. Bordering on this Oval is the mathematics building in which the classes are held. The distance around this Oval is taken as an unknown parameter and many people walk it every day. There are two classes of students and each class was assigned a different way to estimate the distance. The questions of interest are; (1) are the two methods equally valid (an independent two-sample question), and (2) what statistic best estimates the distance around; a robust estimation procedure or a classical one. The main emphasis is the introduction to robust estimation through rank based correlation coefficients, in particular the Greatest Deviation correlation. A general method of estimation with any correlation coefficient is demonstrated with a rank based correlation. Location, scale, and simple linear regression parameters are estimated and compared to the classical estimators. The simplicity and usefulness of the new method is apparent.

Keywords: two-sample problem, scale estimation, robust estimation, Greatest Deviation Correlation, robust location estimation

AMS Subject Classification: 62

### The Utility and General Definition of Correlation CoefficientsRudy Gideon University of Montana Missoula, MT 59812

Abstract

Previous attempts at defining other correlation measures mostly tried to generalize the inner product definition used in Pearson's correlation coefficient. This does not allow for certain useful correlation's, like the Greatest Deviation, or Gini's. In this work the idea in Gideon and Hollister (1987) of seeing correlation, as the difference between distance from perfect negative and perfect positive correlation will be used to bring together a general setting. Pearson, Spearman, and Kendall correlation coefficients are then seen as special cases where a linear restriction holds. It will also be seen how to define a wide variety of correlation coefficients. Simple linear regression with these correlations will be discussed in order to illustrate an introduction to statistical estimation with correlation coefficients. The general focus of this paper is simply to outline notation and concepts necessary for using correlation coefficients as estimating functions.

Keywords: correlation coefficients, Greatest Deviation Correlation, Kendall Correlation, regression, scale estimation, minimization

AMS Subject Classification: 62

### Issues and Tendencies in German Mathematics- Didactics

Günter Törner
Duisburg-Essen Universität (Germany)

and

Bharath Sriraman
The University of Montana (USA)

Abstract

It is a positive sign that an international discussion on theories of mathematics education is taking place especially in the wake of TIMMS and PISA. It is laudable of PME to take the initiative to closely examine specific geographic trends in mathematics education research in comparison with trends that are concurrently occurring (or occurred) elsewhere (as reported in English et al., 2002; Schoenfeld, 1999, 2002). In doing so we can reflect and hypothesize on why certain trends seem to re-occur, sometimes invariantly across time and geographic location. Numerous reviews about the state of German mathematics didactics are available in German (see [1], Hefendehl et al., 2004; Vollrath et al., 2004). However there are no extant attempts to trace and analyze the last hundred years of "mathematics didactic" trends in Germany in comparison to what is happening internationally. This is our modest attempt to fill this void.

Keywords: German Mathematics Didatics; History of Mathematics Education

AMS Subject Classification: 97

Pre-print of:
Sriraman, Bharath & Günter Törner. (2005). Issues and tendencies in German Mathematics Didactics: An epochal perspective. In H. Chick & J.L. Vincent (Eds). Proceedings of the 29th Annual meeting of the International Group of Psychology of Mathematics Education: Melbourne, Australia, vol.1, pp. 197-202.

### Structural stability and dynamic geometry: Some ideas on situated proofs

Luis Moreno Armella
CINVESTAV-IPN, Matematica Educativa (Mexico)

and

Bharath Sriraman
The University of Montana (USA)

Abstract

In this paper we survey the historical and contemporary connections in mathematics between classical "conceptual" tools versus modern computing tools. In this process we highlight the interplay between the inductive and deductive, experimental and theoretical, and propose the notion of situated proofs as a didactic tool for the teaching of geometry in the 21st century.

Keywords: Dynamic Geomtery; Mathematics History; Napoleon's Theorem; Situated Proofs

AMS Subject Classification: 97

Pre-Print Of:
Sriraman, Bharath (with Luis Moreno-Armella) (2005). Structural Stability and Dynamic Geometry: Some Ideas on Situated Proof. International Reviews on Mathematical Education. Vol. 37, no.3, pp.130-139.

### Playing with Powers

Bharath Sriraman
The University of Montana (USA)

and

Pawel Strzelecki
Institute of Mathematics
Warsaw University (Poland)

Abstract

This paper explores the wide range of pure mathematics that becomes accessible through the use of problems involving powers. In particular we stress the need to balance an applied and context based pedagogical and curricular approach to mathematics with the powerful pure mathematics beneath the simplicity of easily stated and understandable questions in pure mathematics. In doing so, pupils realize the limitations of computational tools as well as gain an appreciation for the aesthetic beauty and power of mathematics in addition to its far-reaching applicability in the real world.

Keywords: Elementary Ergodic Theory; Recreational mathematics; Technology Limitations

AMS Subject Classification: 97

Pre-print Of:
Sriraman, Bharath & Strzelecki, Pawel. (2004). Playing with powers. The International Journal for Technology in Mathematics Education, Vol 11, no.1, pp.29-34.

### An Empirical Taxonomy of Problem Posing Processes

Constantinos Christou
The University of Cyprus

Nicholas Mousoulides

The University of Cyprus

Marios Pittalis

The University of Cyprus

Demetra Pitta-Pantazi

The University of Cyprus

and

Bharath Sriraman
University of Montana

Abstract

This article focuses on the construction, description and testing of a theoretical model of problem posing. We operationalize procesess that are frequently described in problem solving and problem posing literature inorder to generate a model. We name these processes editing quantitative information, their meanings or relationships, selecting quantitative information, comprehending and organizing quantitative information by giving it meaning or creating relations between provided information, and translating quantitative information from one form to another. The validity and the applicability of the model is empirically tested using five problem-posing tests with 143 6th grade students in Cyprus. The analysis shows that three different categories of students can be identified. Category 1 students are able to respond only to the comprehension tasks. Category 2 students are able to respond to both the comprehension and translation tasks, while Category 3 students are able to respond to all types of tasks. The results of the study also show that students are more successful in first posing problems that involve comprehending processes, then translation processes and finally editing and selecting processes.

Keywords: Problem Posing; Problem Solving; Quantitative Experimental Designs; Structural Equation Modeling

AMS Subject Classification: 97

Pre-Print Of
Sriraman, Bharath [2005] (with C. Christou, N. Mousoulides, M. Pittalis & D. Pitta-Pantazi). An empirical taxonomy of problem posing processes. International Reviews on Mathematical Education (ZDM) , vol 37, no.3, pp.149-158.

### Collapsing Heat Waves

William R. Derrick and Leonid V. Kalachev
Department of Mathematics, University of Montana
Missoula, MT 59812, USA

and

Joseph A. Cima
Department of Mathematics, University of North Carolina
Chapel Hill, NC 27599, USA

Abstract

In certain combustion models, an initial temperature profile will develop into a combustion wave that will travel at a specific wave speed. Other initial profiles do not develop into such waves, but die out to the ambient temperature. There exists a clear demarcation between those initial conditions that evolve into combustion waves and those that do not. This is sometimes called a watershed initial condition. In this paper we will show there may be numerous exact watershed conditions to the initial-Neumann-boundary value problem $$u_{t}=Du_{xx}+e^{-1/u}-\delta(u-\alpha)$$, with $$u_{x}(0,t)=u_{x}(1,t)=0$$, on $$I=[0,1]$$. They are composed from the positive non-constant solutions of $$Dv_{xx}+e^{-1/v}-\delta(v-\alpha)=0$$, with $$v_{x}(0)=v_{x}(1)=0$$, for small values of $$D$$. We will give easily verifiable conditions for when combustion waves arise and when they do not.

Keywords: reaction-diffusion equation, combustion, domain of attraction

AMS Subject Classification: 35K57, 35K55