# Technical Reports

## Technical Reports 2007

### #18/2007: Johnathan M. Bardsley

### Stopping Rules for a Nonnegatively Constrained Iterative Method for Ill-Posed Poisson Imaging Problems

**Johnathan M. Bardsley**

Department of Mathematical Sciences

University of Montana, Missoula, MT 59812-0864, USA

email:bardsleyj@mso.umt.edu

**Abstract**

Image data is often collected by a charge coupled device (CCD) camera. CCD camera noise is known to be well-modeled by a Poisson distribution. If this is taken into account, the negative-log of the Poisson likelihood is the resulting data-fidelity function. We derive, via a Taylor series argument, a weighted least squares approximation of the negative-log of the Poisson likelihood function. The image deblurring algorithm of interest is then applied to the problem of minimizing this weighted least squares function subject to a nonnegativity constraint. Our objective in this paper is the development of stopping rules for this algorithm. We present three stopping rules and then test them on data generated using two different true images and an accurate CCD camera noise model. The results indicate that each of the three stopping rules is effective.

**Keywords:** iterative methods, image reconstruction, regularization, statistical methods.

**AMS Subject Classification:** 65F20, 65F30.

**Download Technical Report:** Pdf (320 KB)

### #17/2007: Johnathan M. Bardsley

### An Efficient Computational Method for Total Variation-Penalized Poisson Likelihood Estimation

An Efficient Computational Method for Total Variation-Penalized Poisson Likelihood Estimation

**Johnathan M. Bardsley**

Department of Mathematical Sciences

University of Montana, Missoula, MT 59812-0864, USA

email:bardsleyj@mso.umt.edu

**Abstract**

Approximating non-Gaussian noise processes with Gaussian models is standard in data analysis. This is due in large part to the fact that Gaussian models yield parameter estimation problems of least squares form, which have been extensively studied both from the theoretical and computational points of view. In image processing applications, for example, data is often collected by a CCD camera, in which case the noise is a Guassian/Poisson mixture with the Poisson noise dominating for a sufficiently strong signal. Even so, the standard approach in such cases is to use a Gaussian approximation that leads to a negative-log likelihood function of weighted least squares type.

In the Bayesian point-of-view taken in this paper, a negative-log prior (or regularization) function is added to the negative-log likelihood function, and the resulting function is minimized. We focus on the case where the negative-log prior is the well-known total variation function and give a statistical interpretation. Regardless of whether the least squares or Poisson negative-log likelihood is used, the total variation term yields a minimization problem that is computationally challenging. The primary result of this work is the efficient computational method that is presented for the solution of such problems, together with its convergence analysis. With the computational method in hand, we then perform experiments that indicate that the Poisson negative-log likelihood yields a more computationally efficient method than does the use of the least squares function. We also present results that indicate that this may even be the case when the data noise is i.i.d. Gaussian, suggesting that irregardless of noise statistics, using the Poisson negative-log likelihood can yield a more computationally tractable problem when total variation regularization is used.

**Keywords:**total variation, nonnegatively constrained optimization, image recon- struction, Bayesian statistical methods.

**AMS Subject Classification:**65J22, 65K10, 65F22.

**Download Technical Report:** Pdf (438 KB)

### #16/2007: P. Mark Kayll, Jennifer McNulty, James Mihalisin

### Magic squares and antimagic graphs

**P. Mark Kayll**

Department of Mathematical Sciences

University of Montana

Missoula MT 59812-0864, USA

mark.kayll@umontana.edu

**Jennifer McNulty**

Department of Mathematical Sciences

University of Montana

Missoula MT 59812-0864, USA

jenny.mcnulty@umontana.edu

**James Mihalisin
** Department of Mathematical Sciences

University of Montana

Missoula MT 59812-0864, USA

mihalisi@mso.umt.edu

**Abstract**

An *antimagic labeling* of a graph with *m* edges is a bijection \(\lambda\) from its edge set to \(\left\{1,2,\ldots,m\right\}\) such that the vertex sums are distinct, a vertex sum being the sum of the \(\lambda\)-values on the edges incident with the vertex. An *antimagic* graph is one that admits such a labeling. It was conjectured in [N. Hartsfield and G. Ringel, Supermagic and antimagic graphs, *J. Recreational Math.* **21** (1989), 107–115] that every connected graph other than \(K_{2}\) is antimagic. We verify this conjecture constructively for a class of graphs derived from the complete graphs \(K_{n} = (V,E)\) using a variant of magic squares. In support of our main result, we establish that \(K_{n}\), with \(n\geq3\), possesses a property stronger than being antimagic: for every function \(\omega : V\rightarrow \mathbb{N}\), there exists a bijection \(\lambda:E\rightarrow\{1,2,\ldots,\binom{n}{2}\}\) such that the sums \(\omega(\nu)+\sum_{e\ni\nu}\lambda(e)\), for \(\nu \in V\), are all different. This ‘robust’ property of \(K_{n}\) proves useful in establishing that graphs of the form \(K_{n}-F\), for certain \(F\subset E\), are antimagic.

**Keywords:** antimagic, latin square, magic square, RC-magic, transversal system

**AMS Subject Classification:** Primary 05C78 Secondary 05C50, 05B15

**Download Technical Report:** Pdf (144 KB)

### #15/2007: Johnathan M. Bardsley

### An Analysis of Methods for Wavefront Reconstruction from Gradient Measurements in Adaptive Optics

**Johnathan M. Bardsley**

Department of Mathematical Sciences

University of Montana, Missoula, MT 59812-0864, USA

email:bardsleyj@mso.umt.edu

**Abstract**

The use of adaptive optics (AO) in ground-based astronomy is becoming increasingly mainstream. While classical methods, such as de- convolution, remove the blur in an image only after it has been collected, AO systems seek to remove phase error in incoming wavefronts prior to image formation, resulting in higher resolution images. If the phase error is known, it can be removed via the creation of a counter wavefront using, e.g., a deformable mirror. In the AO systems used on ground-based tele- scopes, an estimate of the phase error is typically obtained by solving an inverse problem involving measurements of the wavefront gradient. The standard approach for obtaining phase estimates from measurements of its gradient is least squares. However, a more robust solution can be ob- tain if a minimum variance, or penalized least squares, approach is taken instead. In this paper, we will perform a theoretical analysis of these approaches in a continuous, i.e. function space, setting.

**Keywords:**adaptive optics, wavefront reconstruction, partial di®erential equations, variational methods.

**AMS Subject Classification:**35A15

**Download Technical Report:** Pdf (366 KB)

### #14/2007: Arne C. Bathke, Solomon W. Harrar, Laurence V. Madden

### How to Compare Small Multivariate Samples Using Nonparametric Tests

**Arne C. Bathke ^{1}, Solomon W. Harrar^{2}, Laurence V. Madden^{3}**

Department of Statistics, University of Kentucky

^{1}

Department of Mathematical Sciences, University of Montana

^{2}

Department of Plant Pathology, Ohio State University

^{3}

**Abstract**

In plant pathology, in particular, and plant science, in general, experiments are often conducted to determine disease and related responses of plants to various treatments. Typically, such data are multivariate, where different variables may be measured on different scales that can be quantitative, ordinal, or mixed. To analyze these data, we propose different nonparametric (rank-based) tests for multivariate observations in balanced and unbalanced one-way layouts. Previous work has led to the development of tests based on asymptotic theory, either for large numbers of samples or groups; however, most experiments comprise only small or moderate numbers of groups and samples. Here, we investigate several tests based on small-sample approximations, and compare their performance in terms of levels and power for different simulated situations, with continuous and discrete observations. For positively correlated responses, an approximation based on Brunner et al. (1997) ANOVA-Type statistic performed best; for responses with negative correlations, in general, an approximation based on the Lawley-Hotelling type test performed best. We demonstrate the use of the tests based on the approximations for a plant pathology experiment.

**Keywords:** Rank Test, Small Samples, ANOVA-Type Test, Lawley-Hotelling Test, Bartlett-Nanda- Pillai Test.

**Download Technical Report:** Pdf (232 KB)

### #13/2007: Solomon W. Harrar

### Asymptotics for Tests on Mean Profiles, Additional Information and Dimensionality under Non-normality

**Solomon W. Harrar ***

*Department of Mathematical Sciences, University of Montana, Missoula, MT 59812, Email:harrar@mso.umt.edu

**Abstract**

We consider the comparison of mean vectors for k groups when k is large and sample size per group is fixed. The asymptotic null and non-null distributions of the normal theory Likelihood Ratio, Lawley-Hotelling and Bartlett-Nanda-Pillai statistics are derived under general conditions. We extend the results to tests on the profiles of the mean vectors, tests for additional information (provided by a sub-vector of the responses over and beyond the remaining sub-vector of responses in separating the groups) and tests on the dimension of the hyperplane formed by the mean vectors. Our techniques are based on perturbation expansions and limit theorems applied to independent but nonidentically distributed sequences of quadratic forms in random matrices. In all these four MANOVA problems, the asymptotic null and non-null distributions are normal. Both the null and non-null distributions are asymptotically invariant to non-normality when the group sample sizes are equal. In the unbalanced case, a slight modification of the test statistics will lead to asymptotically robust tests. Based on the robustness results, some approaches for finite approximation are introduced. The numerical results provide strong support for the asymptotic results and finiteness approximations.

**AMS Subject Classificatios:** Primary 62H10; Secondary 62H15

**Keywords:** Asymptotics, Dimensionality, Distribution of Eigenvalues, MANOVA, Perturbation

Expansion, Tests for Additional Information, Robustness

**Download Technical Report:** Pdf (299 KB)

### #12/2007: Solomon W. Harrar and Arne C. Bathke

### Nonparametric Methods for Unbalanced Multivariate Data and Many Factor Levels

**Solomon W. Harrar * and Arne C. Bathke ****

*Solomon W. Harrar is Assistant Professor, Department of Mathematical Sciences, University of Montana, 32 Campus Drive, Missoula, MT 59812, USA; Email: harrar@mso.umt.edu

**Arne C. Bathke is Assistant Professor, Department of Statistics, University of Kentucky, 875 Patterson Office Tower, Lexington, KY 40506-0027, USA; Email: arne@ms.uky.edu

**Abstract**

We propose different nonparametric tests for multivariate data and derive their asymptotic distribution for unbalanced designs in which the number of factor levels tends to infinity (large *a*, small *n _{i}*case). Quasi gratis, some new parametric multivariate tests suitable for the large

*a*asymptotic case are also obtained. Finite sample performances are investigated and compared in a simulation study. The nonparametric tests are based on separate rankings for the different variables. In the presence of outliers, the proposed nonparametric methods have better power than their parametric counterparts. Application of the new tests is demonstrated using data from plant pathology.

**AMS Subject Classificatios:** 62G10, 62G20, 62H10, 62H15, 62J10.

**Keywords:**Multivariate Analysis of Variance, Nonnormality, Nonparametric Model, Ordinal Data, Rank statistic, Unbalanced Design.

**Download Technical Report:** Pdf (215 KB)

### #11/2007: Solomon W. Harrar and Jin Xu

### A note on the null distributions of some test statistics for profile analysis under general conditions

**Solomon W. Harrar* and Jin Xu†**

*Department of Mathematical Sciences, University of Montana, Missoula, MT 59812, USA, Email:harrar@mso.umt.edu

†Department of Statistics, East China Normal University, 500 Dongchuan Road, Shanghai 200241, China, Email:jxu@stat.ecnu.edu.cn

**Abstract**

In this note, we present the asymptotic expansions of the null distributions of some test statistics for k-sample profile analysis under general conditions. It extentsMaruyama (2007, Asmptotic expansions of the null distributions of some test statistics for profile analysis in general conditions. J. Statist. Plann. Inference, 137, 506-526)’s result of two-sample case to *k*-sample (*k* = 2) situations. Further, two new test statistics are devised for testing flatness based on the normal theory likelihood ratio criterion under two different specifications for the parameter space. The asymptotic expansions for these statistics are obtained. Our derivations are much simpler and elegant in that they are based on a transformation on some known results for one-way MANOVA and Hotelling’s *T*^{2} statistics, etc. The accuracy of the asymptotic expansions in approximating the exact null distributions of the test statistics is examined via simulation studies. An application to a real data set is illustrated.

**AMS Subject Classificatios:** Primary 62H10; Secondary 62E20

**Keywords:** Asymptotic expansion, Multivariate skewness, Multivariate kurtosis, MANOVA, Robustness

**Download Technical Report:** Pdf (187 KB)

### #10/2007: Heikki Haario, Leonid Kalachev, and Marko Laine

### Reduced models of algae growth

**Heikki Haario*, Leonid Kalachev**, and Marko Laine*****

*Lappeenranta University of Technology, Lappeenranta, Finland

**University of Montana, Missoula, MT, USA

***Finnish Meteorological institute, Helsinki, Finland

**Abstract**

The simulation of biological systems often is plagued with a high noise level in the data as well as models loaded with a large number of correlated parameters. As a results, the parameters are poorly identified by the data and the reliability of the model predictions may remain questionable. Recently, the advance of Bayesian sampling methods has provided new methods for proper statistical analysis in such situations. Nevertheless, simulations should employ models that, on the one hand, are reduced as much as possible, and, on the other hand, are still able to capture the essential features of the phenomena studied. Here, in the case of algae growth modeling, we show how a systematic model reduction may be done. The simplified model is analyzed from both, theoretical and statistical, points of view.

**Keywords:** Algae growth modeling, asymptotic methods, model reduction, MCMC, Adaptive Markov chain Monte Carlo.

**Download Technical Report:** Pdf (931 KB)

### #9/2007: Adam Nyman, S. Paul Smith

### Watts Theorem for Schemes

**Adam Nyman**

Department of Mathematical Sciences

University of Montana

**S. Paul Smith**

Department of Mathematics

University of Washington

**Abstract**

We describe obstructions to a direct-limit preserving right-exact functor between categories of quasi-coherent sheaves on schemes being isomorphic to tensoring with a bimodule. When the domain scheme is affine, all obstructions vanish and we recover Watts Theorem. We use our description of these obstructions to prove that if a direct-limit preserving right-exact functor F from a smooth curve is exact on vector bundles, then it is isomorphic to tensoring with a bimodule. This result is used to prove that the noncommutative Hirzebruch surfaces constructed by Ingalls and Patrick are noncommutative P^1-bundles in the sense of Van den Bergh. We conclude by giving necessary and sufficient conditions under which a direct-limit and coherence preserving right-exact functor from P^1 to P^0 is an extension

of tensoring with a bimodule by a sum of cohomologies.

**Keywords:** Watts theorem, Morita theory for schemes, non-commutative Hirzebruch surface

**AMS Subject Classification:** 18F99, 14A22, 16D90, 18A25.

**Download Technical Report:**Pdf (393 KB)

### #8/2007: Adelaida B. Vasil'eva, Leonid V. Kalachev, Alexander A. Plotnikov

### On Certain Control Problems for a Class of Singularly Perturbed Parabolic Equations

**Adelaida B. Vasil'eva**

Department of Physics,

Moscow State University,

Moscow, 119899 Russia

E-mail: abvas@mathabv.phys.msu.su

**Leonid Kalachev**

Leonid V. Kalachev Department of Mathematical Sciences,

University of Montana,

Missoula, MT 59812, USA

E-mail: kalachev@mso.umt.edu

and

**Alexander A. Plotnikov**

Department of Physics,

Moscow State University,

Moscow, 119899 Russia

**Abstract**

We consider singularly perturbed parabolic equations that have alternating boundary layer type solutions: two boundary layer type solutions (upper and lower) may exist, and at certain instants of time the switching between these two solutions is observed. The problem formulations for these parabolic equations contain parameters that can be chosen to regulate the duration of periods corresponding to upper solution and to lower solution stages. We present a biological model for which it is possible to choose the values of control parameters in such a way that only one of the mentioned above boundary layer type solutions persists.

**Keywords:** singular perturbations, parabolic equations, boundary function method, Dirichlet boundary conditions, control

**AMS Subject Classification:**34E10, 35B05, 35B25

**Download Technical Report:** Pdf (391 KB)

### #7/2007: Johnathan M. Bardsley

### Wavefront Reconstruction Methods for Adaptive Optics Systems on Ground-Based Telescopes

**Johnathan M. Bardsley**

Department of Mathematical Sciences

The University of Montana (USA)

**Abstract**

The earth's atmosphere is not a perfect media through which to view objects in outer-space; turbulence in the atmospheric temperature distribution results in refractive index variations that interfere with the propagation of light. As a result, wavefronts are non-planar when they reach the ground. The deviation from planarity of a wavefront is known as phase error, and it is phase error that causes the refractive blurring of images. Adaptive optics systems seek to remove phase error from incoming wavefronts. In ground-based astronomy, an estimate of the phase error in a wavefront is typically obtained from wavefront gradient measurements collected by a Shack-Hartmann sensor. The estimate is then used to create a counter wavefront, e.g. using a deformable mirror, that (approximately) removes the phase error from the incoming wavefronts. The problem of reconstructing the phase error from Shack-Hartmann gradient measurements requires the solution of a large linear system whose form is defined by the configuration of the sensor. We derive this system and present both the regular least squares and minimum variance approaches to its solution. The most effective existing approaches are then presented alongside new computational methods, and comparisons are made.

**Keywords:** adaptive optics, wavefront reconstruction, minimum variance estimation.

**AMS Subject Classification:**

**Download Technical Report:** Pdf (297 KB)

### #6/2007: Bharath Sriraman, Libby Knott, Harry Adrian

### The Mathematics of Estimation: Possibilities for Interdisciplinary Pedagogy and Social Consciousness

**Bharath Sriraman**

The University of Montana

**Libby Knott**

The University of Montana

**Harry Adrian**

Ottawa, Illinois

**Abstract**

The purpose of this article is to report on the importance of providing pre-service and in-service teachers with experience and specific training in critical thinking skills. The essential concepts in elementary mathematics curricula can be augmented to include and cultivate critical thinking skills that have tremendous ramifications for future leaders and for those who move on to more technical training. A sample problem, along with pre-service teacher responses, is used here to show the necessity of and importance of this kind of training because the responses show clear evidence of a certain naiveté on the part of these college level students. The responses do show evidence of budding social conscience in these students, but the level of expertise in critical thinking is not at all sophisticated. We discuss and explore the implications of our approach.

**Keywords:** critical thinking; Fermi estimates; interdisciplinarity; pre-service teachers; social consciousness

**AMS Subject Classification:** 97

**Download Technical Report:**Pdf (86 KB)

### #5/2007: Nicholas Mousilides, Bharath Sriraman, Marios Pittalis, Constantinos Christou

### Tracing students' modelling processes in elementary and secondary school

**Nicholas Mousoulides**

Department of Education, University of Cyprus

**Bharath Sriraman**

Department of Mathematical Sciences, The University of Montana

**Marios Pittalis, Constantinos Christou**

Department of Education, University of Cyprus

**Abstract**

This study examines 6th and 8th grade students’ mathematization processes as they worked a mathematical modelling problem. Students participated in four 40 minute sessions for deciding on the best city to live in among a number of different cities. In the present study we report on an analysis of the mathematization processes and developments of two groups of students, one 6th and one 8th grade, as they worked the problem, with special emphasis on the similarities and differences between the two groups. Results provide evidence that students developed the necessary mathematical constructs and processes to actively engage and solve the problem through meaningful problem solving. Both groups of students set hypotheses, evaluate, modify and refine their models. Among the differences between the two groups, 8th grade students were involved in higher level of mathematical communication, projected and effectively employed higher order mathematical concepts and processes and reached better and more refined solutions.

**Keywords:** cognitive processes; mathematical modelling ; teaching and learning

Pre-print of: Mousoulides, N., Sriaman, B., et al (2007). Tracing students’ Modelling processes in elementary and secondary school. *Proceedings of the 5th European Congress on Mathematics Education* (CERME5), Larnaca, Cyprus, Feb.21-27, 2007.

**AMS Subject Classification:** 97

**Download Technical Report:**Pdf (48 KB)

### #4/2007: Bharath Sriraman, Bettina Dahl Søndergaard

### On Bringing Interdisciplinary Ideas to Gifted Education

**Bharath Sriraman**

The University of Montana, USA

**Bettina Dahl Søndergaard
** University of Aarhus, Denmark

**Abstract**

This chapter is based on the premise that the utopian goal of education is to unify various strands of knowledge as opposed to dividing it. Ideally education should nurture talent in the classroom and create well-rounded individuals akin to the great thinkers of the Renaissance. That is, individuals who are able to pursue multiple fields of research and appreciate both the aesthetic and structural/ scientific connections between mathematics, arts and the sciences. We will explore an under addressed aspect of giftedness, namely the role of interdisciplinary activities and problems to foster talent in and across the disciplines of mathematics, science and humanities, increasingly important for emerging professions in the 21st century. Examples from the history of mathematics, science and arts will be used to argue for the value of such activities to foster polymathic traits in gifted individuals, particularly the questioning of paradigms. Recent findings from classroom studies will be used to illustrate the value of such an approach to gifted education.

**Keywords:** domain general creativity; domain general giftedness; history of science; interdisciplinarity; interdisciplinary pedagogy; paradigm shifts; polymathy; Renaissance; talent development

Pre-print of: Sriraman, Bharath & Dahl, Bettina. (2007). On Bringing Interdisciplinary Ideas to Gifted Education. To appear in L.V. Shavinina (Editor). ** The International Handbook of Giftedness.** Springer Science

**AMS Subject Classification:**97

**Download Technical Report:**Pdf (420 KB)

### #3/2007: Bharath Sriraman, Günter Törner

### Political Union/ Mathematics Education Disunion: Building Bridges in European Didactic Traditions.

**Bharath Sriraman**

The University of Montana, USA

**Günter Törner
** University of Duisburg-Essen, Germany

**Abstract**

In this chapter, the historical development of mathematics didactic traditions of Germany, France and Italy are carefully traced and analyzed since the dawn of the Renaissance. Particular attention is paid to analyzing original historical sources for the purpose of delineating similarities and differences between these three traditions. The chapter argues that despite the proximity of these three countries, the mathematics didactic traditions are extremely diverse. This diversity is attributed to political structures, the culture of mathematics inherited from academic institutions and culture. The concluding sections of this chapter outline particular domains of inquiry within which bridges can be built between the mathematics didactic traditions of Germany, France and Italy.

**Keywords:** History of Didactics, History of mathematics; French mathematics didactics, German mathematics didactics, Humanism, Italian mathematics didactics; Political structures in education

Pre-print of:

Sriraman, B & Törner, G. (2007) Political Union/ Mathematics Education Disunion: Building Bridges in European Didactic Traditions. To appear in L. English (Editor)

Handbook of International Research in Mathematics Education (2nd Edition).

Lawrence Erlbaum & Associates

**AMS Subject Classification:** 97

**Download Technical Report:**Pdf (209 KB)

### #2/2007: Olof Steinthorsdottir, Bharath Sriraman

### Gender and Strategy Use in Proportional Situations: An Icelandic Study

**Olof Steinthorsdottir
** University of North Carolina - Chapel Hill, USA

**Bharath Sriraman**

The University of Montana, USA

**Abstract**

This study was conducted to investigate the influence of semantic type and number structure on individuals’ use of strategies in solving missing value proportion problems, and to examine gender differences in strategy use. Fifty-three eighth graders in one school in Reykjavik, Iceland, participated in this study. Twenty-seven females and twenty-six males, were individually interviewed as they solved sixteen missing value proportion problems. The problems represented four semantic structures: Well chunked (W-C), part-part whole (P-P-W), associated sets (A-S), and symbolic (S-P). For each semantic structure there were four problems, each representing a distinct number structures: Integer-integer with an integer answer (I-I-I), integer-noninteger with an integer answer (I-N-I or N-I-I), noninteger-noninteger with an integer answer (N-N-I), or noninteger-noninteger with a noninteger answer (N-N-N). The findings in this study indicate that number structure influenced strategy use and success to a greater extent than semantic type. The easiest problems for students to solve were I-I-I tasks and the most difficult problems were the N-N-N tasks. Most students used multiplicative strategies on the I-I-I problems and I-N-I problems, but resorted to less sophisticated reasoning on the N-N-I and N-N-N problems. No gender differences were identified in the overall success rate. Girls were more successful than boys in associated sets and symbolic problems, and boys were more successful than girls in part-part-whole problems. Girls used less mature strategies for all semantic types except the case of the symbolic type. The data suggest that the semantic type influences females’ choice of strategy more than that of males.

**Keywords:** Gender differences, Iceland, proportion, ratio, proportional reasoning, problem sequencing, semantic types, teaching and learning proportions.

**AMS Subject Classification:** 97

**Download Technical Report:**Pdf (141 KB)

### #1/2007: Olof Steinthorsdottir, Bharath Sriraman

### Icelandic 5th grade girls developmental trajectories in proportional reasoning

**Olof Steinthorsdottir
** University of North Carolina - Chapel Hill, USA

**Bharath Sriraman**

The University of Montana, USA

**Abstract**

Understanding of ratio and proportion are critical to the development of higher level mathematical skills. Following the proposal by Carpenter et al (1999) of a four-level trajectory in the development of proportional reasoning, a twelve week investigation was undertaken of the developmental trajectory of proportional reasoning of girls in two fifth-grade classes in Iceland. Students in these classes were used to instructional practices that encouraged them to devise and explain their own solutions to mathematical problems. Results of the study confirm the learning trajectory with the addition of a further distinct level of development between Level 2 and Level 3. Results showed that girls moved easily, with minimum scaffolding, from Level 1 to 2 and from Level 2 to 3. The transition to Level 4, which involves explicit awareness of ‘within’ and ‘between’ multiplicative relationships, took greater time and effort. Teacher awareness of the four-level learning strategy, with the new emerging Level 3, will assist in the design of appropriate problems, class structure and teaching strategies.

**Keywords:** Iceland, Proportion, ratio, proportional reasoning, teaching proportion, learning trajectory, multiplicative thinking, mathematical learning, mathematical understanding.

**AMS Subject Classification:** 97

**Download Technical Report:** Pdf (192 KB)