2015 Colloquia

Spring

Doctoral Dissertation Defense

An Investigation of the Impact of iPad Usage on Elementary Mathematical Skills and Attitudes

Grant Swicegood PhD Candidate, University of Montana

Wednesday, May 13, 2015 at 2:10 pm in Math 103

Currently, many schools are implementing one-to-one initiatives, where the goal is to give every student in a classroom a tablet or laptop.  However, there is a dearth of research backing up the assumption that they significantly improve student learning.  This study explored the effects of these new instructional devices by focusing on two second-grade classrooms implementing a one-to-one iPad program.  Specifically, it investigated how iPad usage affects student and teacher attitudes toward mathematics, student mathematics performance in and out of app environments, the instructional purposes for which iPads are used in the classroom, and implementation issues of the technology.  This primarily observational study employed a mixed methods approach to capture a picture of an active program to serve as a source for further questions that may be better answered by experimenting with different treatments.  Quantitative data was gathered on student performance in two apps, Addimal Adventure and Splash Math 2nd Grade, as well on the frequency and type of iPad usage.  Qualitative data came from interviews with six students and two teachers near the beginning and end of the four month research period. In this presentation, I will present these results and discuss their implications for the education community.

Colloquium

An Investigation of the Impact of iPad Usage on Elementary Mathematical Skills and Attitudes

Grant Swicegood PhD Candidate, University of Montana

Tuesday, May 12, 2015 at 3:10 p.m. in Math 103 4:00 p.m. Refreshments in Math Lounge 109

Currently, many schools are implementing one-to-one initiatives, where the goal is to give every student in a classroom a tablet or laptop.  However, there is a dearth of research backing up the assumption that they significantly improve student learning.  This study explored the effects of these new instructional devices by focusing on two second-grade classrooms implementing a one-to-one iPad program.  Specifically, it investigated how iPad usage affects student and teacher attitudes toward mathematics, student mathematics performance in and out of app environments, the instructional purposes for which iPads are used in the classroom, and implementation issues of the technology.  This primarily observational study employed a mixed methods approach to capture a picture of an active program to serve as a source for further questions that may be better answered by experimenting with different treatments.

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Special Event

Colloquium

Factors Considered by Elementary Teachers When Modifying Mathematical Tasks to Support or Extend Children’s Mathematical Thinking

Mike Fredenberg Mathematics Education Candidate San Diego State University & University of California SD

Mathematics educators and researchers have aligned themselves with John Dewey’s argument to concentrate on characterizing and organizing the knowledge and activities that enable teachers to bridge the gulf between theory and practice.  In this vein, I ask the question, what factors do exemplary elementary teachers consider when modifying a task for students during the enactment of a lesson?   In this presentation, I situate my dissertation study within the arena of Cognitively Guided Instruction (CGI), and the theoretical foundations of the professional noticing of children's mathematical thinking.  I describe the motivation for the study, the methodology and data analysis, and I present emerging results.  I conclude with a discussion of contributions to the field, and thoughts on future research projects.

Colloquium

Cross-disciplinary training: A vehicle for research collaboration and STEM education promotion

Lia Harrington PhD student, Department of Psychology

Monday, April 13, 2015 at 3:10 p.m. in Math 103 4:00 p.m. Refreshments in Math Lounge 109

Cross-disciplinary training and collaboration is becoming not only a fad, but a true trend of the future as the divisions between fields are crumbling due to the changing landscape of research.  Modern researchers often have to understand mathematical concepts when they implement new computational techniques to understand the underlying structure and relationships of the phenomena they are investigating. I will explore how a cross-disciplinary approach can aid and enrich research collaborations. I also will explore cross-disciplinary training as a possible motivating vehicle for promoting STEM education, especially with engaging women to pursue science careers.

Colloquium

Beyond rise over run! Learning slope in a cascade of artifacts

Frederick Peck University of Colorado- Boulder Mathematics Education Candidate

Monday, April 6, 2015 at 3:10 p.m. in Math 103 4:00 p.m. Refreshments in Math Lounge 109

In this talk I address two questions. The first is, how can we understand the role of culture in school mathematics? The second is, how do students in Algebra I learn slope? To explore these questions, I conducted a design experiment as a teacher-researcher in my own Algebra I classroom. In discussing the results, I’ll introduce the concept of a cascade of artifacts to describe learning from a cultural perspective, and I’ll present a local instructional theory for how students learn slope. I’ll conclude with a plan for how I will marshal my current body of work into a robust agenda of future research.

Colloquium

Equivalence criteria in the safety evaluation of a genetically modified crop

Christopher I. Vahl Kansas State University

Monday, March 16, 2015 at 3:10 p.m. in Math 103 4:00 p.m. Refreshments in Math Lounge 109

The safety evaluation of a genetically modified (GM) crop is accomplished by establishing its substantial equivalence to conventional non-GM food crops with a history of safe use.  Toward this end, equivalence testing rather than difference testing is the more appropriate statistical approach.  A pivotal step in this process is to specify a reasonable equivalence criterion that encompasses a measure of the discrepancy between the GM and reference crops as well as a regulatory threshold.   We explore several possible equivalence criteria and discuss their pros and cons.  Each criterion will be shown to address one of three ordered classes of equivalence.  Their implications will be examined over an array of parameter values estimated from a real-world dataset.  Furthermore, our literature search indicates that the linear mixed model proposed by the European Food Safety Authority is adequate for assessing substantial equivalence despite its lack of genotype-by-environment interaction terms.

Colloquium

Set membership with two bit probes

Jaikumar Radhakrishnan Professor, Tata Institute of Fundamental Research, Mumbai and Visiting Scientist, Simons Institute for the Theory of Computing, Berkeley

Monday, March 9, 2015 at 3:10 p.m. in Math 103 4:00 p.m. Refreshments in Math Lounge 109

We will consider the bit-probe complexity of the set membership problem, where a set $$S$$ of size at most $$n$$ from a universe of size $$m$$ is to be represented as a short bit vector in order to answer membership queries of the form "Is $$x$$ in $$S$$?" by adaptively probing the bit vector at $$t$$ places. Let $$s(m,n,t)$$ be the minimum number of bits of storage needed for such a scheme. Alon and Feige showed that for $$t=2$$ (two bit probes), such schemes can be obtained from dense graphs with large girth. In particular, they showed that for $$n < \log m$$,

$$s(m,n,2) = O(m n \log((\log m) / n) / \log m).$$

We improve their analysis and obtain a better upper bound and a corresponding lower bound.

Upper bound: There is a constant $$C>0,$$ such that for all large $$m$$,

$$s(m,n,2) \leq C \cdot m^{1-\frac{1}{(4n+1)}}.$$

Lower bound: There is a constant $$D>0,$$ such that for $$n\geq 4$$ and all large $$m$$, we have

$$s(m,n,2) \geq D \cdot m^{1-\frac{1}{\lfloor n/4 \rfloor}}.$$

(This is joint work with Mohit Garg.)

Colloquium

Using numerical optimization techniques for sampling in statistical inverse problems

Johnathan M. BardsleyDepartment of Mathematical Sciences

Monday, March 2, 2015 at 3:10 p.m. in Math 103 4:00 p.m. Refreshments in Math Lounge 109

Many solution methods for inverse problems compute the maximum a posteriori (MAP) estimator, or equivalently, the regularized solution, by solving an optimization problem. Uncertainty quantification (UQ), on the other hand, typically requires sampling from the Bayesian posterior density function. In this talk, we bring these two ideas together and present posterior sampling methods that make use of existing algorithms for computing regularized solutions/MAP estimators. Theoretically correct samplers for both linear and nonlinear inverse problems will be presented.

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Mobile Educational Technology and the Mathematics Classroom

Grant SwicegoodPhD Candidate, Department of Mathematical Sciences

Monday, February 23, 2015 at 3:10 p.m. in Math 103 4:00 p.m. Refreshments in Math Lounge 109

In this talk, I will discuss the current state of mobile educational technology initiatives involving tablets (e.g. iPads), laptops, and other mobile electronic devices.  What does this mean to the modern mathematics classroom at the elementary or high school level?  What does the implementation actually look like?  In addition to exploring these questions, I will demonstrate the functionality of several popular educational mathematics apps and discuss the possible impacts on student learning.  Against this background, I will describe the iPad initiative at Paxson School in Missoula and my current research there in second grade classrooms.​

Colloquium

Big Data Goes to Work: Liberating Latent Value in a Connected World

Peter Coffee VP for Strategic Research, Salesforce

Conversation around "big data" is too often focused on sheer volume, and the technical challenges of collection. More important are recent findings that roughly 7/8 of organizations look to data as history, not as news: that they use it to document past behavior, seek opportunities for cost reduction, and justify familiar and comfortable strategies and practices. The greater value lies in building predictive tools and recommendation engines that have the power to change behavior and create new value. Peter Coffee, the global VP of Strategic Research for cloud-computing leader Salesforce, will share observations on the engagement and transformation being enabled – and the ethical and regulatory challenges being created – by 24×7 connection, 'social graph' mathematics, and a sensor-rich "Internet of Things."

Colloquium

The Dynamics of Vector-Borne Relapsing Diseases.

Cody PalmerPhD Candidate, Department of Mathematical Sciences

Relapsing fever is a disease spread by lice and ticks among humans and other mammals. As the name suggests it is characterized by 3-4 relapsing periods of fever and muscle aches. Clinical descriptions of the disease date back to the ancient Greeks and it was a problem among troops during the World Wars. Tick-borne Relapsing Fever (TBRF) occurs in the Pacific Northwest and has recently been used to motivate models for vector-borne relapsing diseases.

In this talk we will be concerned with the effect that the number of relapses have on the dynamics of the disease. We quantify this by computing the fundamental reproductive number R0, the average number of new infections caused by a single infected individual.

We will be introducing and using the compartmental model of disease spread developed by van den Dreissche and Watmough to find a form for R0 as a function of the number of relapses.

Also of interest are the existence and stability to endemic equilibria (EE), fixed points of the system where only a portion of the population are infected. We will show the existence of EE for R0 sufficiently close to 1.

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A framework for teaching mathematical modeling in elementary grades.

Beth  Burroughs & Mary Alice CarlsonMontana State University

Modeling, a cyclic process by which mathematicians develop and use mathematical tools to represent, understand, and solve real-world problems (Lesh & Doerr, 2003), provides important learning opportunities for students. Two questions are critical for mathematics teacher educators interested in mathematical modeling in K-5 settings. (1) How should opportunities for modeling in K-5 settings be constructed and carried out? (2) What are the tasks of teaching when engaging students in mathematical modeling? We present a framework for teaching mathematical modeling in the elementary school and illustrations of its use by elementary grades teachers.

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Dimension reduction for Bayesian inference of large-scale systems

Tiangang Cui Massachusetts Institute of Technology

Algorithmic scalability to high dimensional parameters and computational efficiency of numerical solvers are two significant challenges in large-scale Bayesian inversion. Here we will explore the intrinsic dimensionality in both state space and parameter space of inverse problems by analyzing the interplay between noisy data, ill-posed forward model and smoothing prior. The resulting reduced subspaces naturally lead to a scalable and fast model reduction framework for solving large-scale inverse problems with high dimensional parameters.

Colloquium

Applications of Adjoint Based Assimilation

Jesse Johnson Department of Computer Science

A numerical model is a tool for reasoning about situations where direct observations or experiments are impossible or very expensive to conduct. For example, it would be interesting to know the impact of warmer oceans on climate, but expensive to heat and cool them. In this approach to doing science, the credibility of the numerical models can be called into question because they fail to reproduce even the most obvious features of observations. This is in spite of their fidelity to the physics of the situation. To address such criticism, many models now assimilate large observational data sets to achieve an initial state that is consistent with observation. The assimilation problem is formulated as an optimization problem where the mismatch between observation and model result, or objective function, is minimized. Optimization is done in the typical way, by determining gradients or search directions of the objective function with respect to unknown parameters. At this point, a difficulty arises: the number of parameters is very large, and determining the gradients in a naive way is computationally prohibitive. Specifically, a derivative of each degree of freedom must be computed with respect to each unknown parameter. Nowadays, the number of degrees of freedom is in the millions and the number of parameters in the hundreds of thousands. The adjoint equation is the solution to this problem. The solution to a single linear system which is the size of the number of degrees of freedom provides search directions. In this talk I will discuss a numerical model for ice sheet modeling and the solution to adjoint equations to drive data assimilation. I will show several examples where adjoint based assimilation has been useful, and outline our next task, which will be to explore time dependent problems. A critical component of data collection, low cost, low power custom data loggers, will also be discussed.

Colloquium

A Forum on Philanthropy at the College and Department

Bitty Balducci Assistant Director of Development, College of Humanities and Sciences University of Montana Foundation

Please join us for a discussion of existing efforts in the Math. Dept. for philanthropic development, and the incorporation of new strategies

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Spectral algorithms to find communities with special structure

Matthew Yancey IDA/CCS

A recent trend in data-mining is to find communities in a graph. Generally speaking, a community of a graph is a vertex set such that the number of edges contained entirely inside the set is "significantly more than expected." These communities are then used to describe families of proteins in protein-protein interaction networks, among other applications. Community detection is known to be NP-hard; there are several methods to find an approximate solution with rigorous bounds.

We present a new goal in community detection: to find good bipartite communities. A bipartite community is a pair of disjoint vertex sets S, S' such that the number of edges with one endpoint in S and the other endpoint in S' is "significantly more than expected." We claim that this additional structure is natural to some applications of community detection. In fact, using other terminology, they have already been used to study correlation networks, social networks, and two distinct biological networks. We will show how the spectral methods for classical community detection can be generalized to finding bipartite communities, and we will prove sharp rigorous bounds for their performance. Additionally, we will present how the algorithm performs on public-source data sets.

Colloquium

Teaching and learning of Mathematics using technology: opportunities and pitfalls

Gerrit Stols Head of the Department of Science, Mathematics, and Technology Education at the University of Pretoria

In this talk, I will give an overview of the different kinds of software that are used in mathematics classes. Teachers and lecturers technology use can be categorized as the use of technology outside the classroom (to improve the effectiveness and professionalism) and inside the classroom to enhance the conceptual development of their students. Both uses generate new opportunities, but if used incorrectly, can impede students’ conceptual development. I will therefore reflect on the importance and limitations of technology use. Lastly I will focus on the question: Why do teachers and lecturers, in general, not use technology in their classrooms?