PRINCIPLES FOR DESIGNING INVENTION TASKS FOR UNDERGRADUATE MATHEMATICS Ben Davies Supervised by Caroline Yoon Traditional undergraduate instruction rarely produces meaningful first interactions with abstract mathematical concepts. In this talk I present five principles for designing tasks that address this absence, not only promoting meaningful interactions for students through the process of personal mathematical invention (Schwarz, 2004), but also helping prepare students for understanding the expository lectures that constitute the majority of undergraduate mathematics education. These design principles are influenced by the six principles for designing Model Eliciting Activities (Lesh, Hoover, Hole, Kelly and Post, 2000), but are modified in order to design tasks intended specifically for mathematical invention in undergraduate mathematics settings. These principles will be illustrated by a discussion of the successes and failures of two tasks designed to prepare students for more formal exploration of mathematical induction. One such task exploits the dominoes metaphor for induction, while the other explores the recursive pattern in the maximum number of pieces of pie yielded by n cuts. Lesh, R., Hoover, M., Hole, B., Kelly, A., and Post, T. (2000). Principles for developing thought revealing activities for students and teachers. In Research Design in Mathematics and Science Education, chapter 21, pages 591–646. Lawrence Erlbaum Associates, Mah- wah, NJ. Schwartz, D. L. and Martin, T. (2004). Inventing to prepare for future learning: The hidden efficiency of encouraging original student production in statistics instruction. Cognition and Instruction, 22:129–184. THE DOMINOES PROBLEM Find an arrangement of 9 dominoes is such that, depending on the chosen ‘starting domino’, it is possible to have 1,2,3,4,6,7,8 or 9 dominoes fall but it is always impossible to have 5 dominoes fall. Rules: • You may choose a different starting domino to topple each time • Every domino must fall in the same direction Is it always possible to set up n dominoes such that you can fell any number of dominoes except i with just one touch? THE PIE CUTTING PROBLEM Let " be the maximum number of pieces of pie produced by n cuts. So # = %, % = ’, ’ = (, etc. Find a formula for in terms of n.
Transcript
PRINCIPLES FOR DESIGNING INVENTION TASKS FOR UNDERGRADUATE MATHEMATICS
Ben Davies
Supervised by Caroline Yoon
Traditional undergraduate instruction rarely produces meaningful first interactions with abstract mathematical concepts. In this talk I present five principles for designing tasks that address this absence, not only promoting meaningful interactions for students through the process of personal mathematical invention (Schwarz, 2004), but also helping prepare students for understanding the expository lectures that constitute the majority of undergraduate mathematics education. These design principles are influenced by the six principles for designing Model Eliciting Activities (Lesh, Hoover, Hole, Kelly and Post, 2000), but are modified in order to design tasks intended specifically for mathematical invention in undergraduate mathematics settings.
These principles will be illustrated by a discussion of the successes and failures of two tasks designed to prepare students for more formal exploration of mathematical induction. One such task exploits the dominoes metaphor for induction, while the other explores the recursive pattern in the maximum number of pieces of pie yielded by n cuts.
Lesh, R., Hoover, M., Hole, B., Kelly, A., and Post, T. (2000). Principles for developing thought revealing activities for students and teachers. In Research Design in Mathematics and Science Education, chapter 21, pages 591–646. Lawrence Erlbaum Associates, Mah- wah, NJ. Schwartz, D. L. and Martin, T. (2004). Inventing to prepare for future learning: The hidden efficiency of encouraging original student production in statistics instruction. Cognition and Instruction, 22:129–184.
THE DOMINOES PROBLEM
Find an arrangement of 9 dominoes is such that, depending on the chosen ‘starting domino’, it is possible to have 1,2,3,4,6,7,8 or 9 dominoes fall but it is always impossible to have 5 dominoes fall. Rules: • You may choose a different starting domino to topple each time • Every domino must fall in the same direction Is it always possible to set up n dominoes such that you can fell any number of dominoes except i with just
one touch?
THE PIE CUTTING PROBLEM Let 𝑃" be the maximum number of pieces of pie produced by n cuts.
So 𝑃# = %, 𝑃% = ', 𝑃' = (,etc. Find a formula for 𝑷𝒏in terms of n.
Name: Mareli de Lange Supervisor: Hinke Osinga
TheMathematicsofKeepingPacewithMorrisLecar In this talk I will present a numerical analysis of oscillatory behaviour of biological systems. My interest is in the effects of applying a given external stimulus at different phases of the oscillation. I will use the Morris-Lecar model [1] as an example, which is given by a two-dimensional system of differential equations. I will present the numerical tools that I implemented in Matlab to approximate solutions to initial value problems and to determine the phase of the eventual oscillation relative to a given point on the periodic orbit. The initial values are obtained by perturbations that resulted in a horizontal shift of different lengths for 20 points along the periodic orbit. I will present an analysis of the phase portraits together with the numerically obtained phase-resetting-curves (PRC) [2]. The PRC illustrates the deviation of the new phase of the system (after it was perturbed) in comparison to the initial phase of the system (before it was perturbed). The effects of the perturbation on the new phase of the system is importance as (1) it illustrates how the ongoing oscillation activity resets, and (2) clearly highlights whether that phase reset is (a) always in a particular direction (Type I), or (b) in alternating directions (Type II). References:
[1] Catherine Morris and Harold Lecar (1981), "Voltage oscillations in the barnacle giant muscle fiber," Biophysical Journal 35(1): 193-213.
[2] Carmen C. Canavier (2006), "Phase response curve," Scholarpedia, doi:10.4249/scholarpedia.1332.
[3] David Hansel, German Mato, and Claude Meunier (1995), "Synchrony in excitatory neural networks," Neural Computation 7(2): 307-337.
Computational and Algebraic Methods for
Bayesian Inverse Problems
Owen DillonSupervised by Jari Kaipio
May 14, 2016
Abstract
Inverse problems arise in a wide variety of applications, from astro-physics to medicine. It is often the case that the models developed togive “sufficiently accurate” estimates of parameters of interest from data,are too complex to give results in the required timeframe. For such prob-lems, the Bayesian Approximation Error (BAE) approach was developed,which allows inferences to be made using just a reduced model alongside aconditional corrector term. The Bayesian framework for inverse problemsalso allows us to produce spread estimates on these parameters, which isoften useful in practice.
The implementation of BAE requires both a large amount of “offline”(prior to application) computations, and also an often nontrivial amountof “online” (in final application) computation. This talk will outline someof the techniques that have been created to reduce these hurdles in boththe offline and online stages.
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Reverse Engineering Biological Models
Author: Shintaro Fushida-HardySupervisors: Vivien Kirk and James Sneyd
May 18, 2016
Mathematical models of biological processes are typically constructed by ini-tially developing an understanding of the biology, then including terms in themodel that match the biological understanding. The model is then analysed, topredict further behaviour of the system. A problem with this approach is thatit inherently requires an understanding of the biology. Hence it can be difficultto create a model with the intention of learning more about the system withoutalready understanding a great deal about the system.
In recent years, a novel approach to mathematical modelling of phenomenain physics and engineering has been proposed: the use of symbolic regressionto reverse engineer models. This approach has been shown to work successfullyin a number of physical systems, for example, the motion of a pendulum1. Itenables models to be generated which accurately predict the behaviour of aphysical system, even when no theory about the behaviour is available.
The aim of this research was to assess the extent to which symbolic regression isuseful for the modelling of biological phenomena, particularly the dynamics ofintracellular calcium concentration. In this talk, I will describe one particularsymbolic regression package (Eureqa, Nutonian Inc.). Some of the main obsta-cles to the generation of useful models will be discussed, as well as progress andthe potential for future work.
1R. Riolo et al. (eds), Genetic Programming Theory and Practice VII, Springer US (2010)
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Movement in Mathematical Problem Solving
Robyn Gandell
Supervisor: Caroline Yoon
From Plato through Descartes to the present it is widely believed that thinking occurs only in the mind
(Hall & Nemirovsky, 2012; Radford, 2014). Within this paradigm the body merely provides sensory
information and carries out motor actions for the mind. However, the last three decades of research, in
the fields of cognitive psychology, linguistics, neuroscience and more recently mathematics
education, has increasingly demonstrated that both the brain and the sensorimotor functions of the
body are inextricably linked to cognitive function (Cook, 2011; Edwards, 2003; Radford, 2014).
Mathematics education studies, supporting these findings in the field of embodied cognition, have
primarily focused on the use of gestures and their contribution to narrative and problem solving
activities (Cook, 2011). As a teacher of mathematics and dance I observed that students’ movements
during problem solving activities involved more of the body than just gesture. The focus of my
research is, therefore, whether whole body movements are linked to cognition.
More specifically this research investigates what kinds of whole body movements are used, and when
and where these movements occur during a mathematical problem solving activity. Preliminary
results show that not only is more of the body used during mathematical problem solving but that the
space used also changes. In sitting, gesture space in front of the body is used (McNeill, 1992), while
standing, however, movement space that surrounds the body is used. This suggests that embodied
cognition does encompass whole body movements.
References Cook, S. W. (2011). Abstract thinking in space and time: Using gesture to learn math. Cognitie,
Edwards, L. D. (2003). A natural history of mathematical gesture. American Educational Research
Association Annual Conference, Chicago,
Hall, R., & Nemirovsky, R. (2012). Introduction to the special issue: Modalities of body engagement
in mathematical activity and learning. Journal of the Learning Sciences, 21(2), 207-215.
McNeill, D. (1992). Hand and mind: What gestures reveal about thought University of Chicago press.
Radford, L. (2014). Towards an embodied, cultural, and material conception of mathematics
cognition. Zdm, 46(3), 349-361.
Name: Anton Gulley Supervisors: Prof Jari Kaipio and Dr Jennifer Eccles (Environment)
Inverse approaches to Fault Zone Guided Waves
A mature fault often has a zone of deformed rock with lower seismic velocities than the surrounding intact rock. If a seismic source is located near a fault zone then this low velocity zone can trap some of the seismic energy through internal reflections. This trapped energy is called a Fault Zone Guided Wave (FZGW) and they were first observed in the San Andreas Fault in 1985 [1]. FZGW’s are recorded on seismometers that are located near the fault and have a slow moving, high amplitude waveform. They are also dispersive which means that different trapped frequencies arrive at different times.
The amplitude and the dispersion of FZGW’s is highly dependent on the elastic and geometric properties of the fault zone and surrounding rock. This means that FZGW’s can be used to estimate these properties of the fault zone [2]. The analysis of the FZGW’s is carried out to gain further understanding in the physics of earthquakes and earthquake hazards. The estimation of the fault zone properties is, however, a challenging task since the problem is mathematically characterized as an ill-posed inverse problem [3,4]. In this talk the development of a 2-D and 3-D Bayesian inversion approach for FZGW data will be discussed. These algorithms rely on a simple 1-D wave propagation code coupled with a full elastic waveform solver using the Bayesian Approximation Error method. Synthetic examples will be demonstrated along with examples from the Alpine Fault, New Zealand. References: [1] Leary, P. C., Li, Y. G., & Aki, K., 1985. Borehole observations of fault
zone trapped waves, Oroville, CA, EosTrans. AGU, 66, 976.
[2] Li, Y.-G., 2012. Imaging, Modeling and Assimilation in Seismology, vol.
V1, chap. Fault-Zone Trapped Waves: High-Resolution Characterization of the
Damage Zone of the Parkfield San Andreas Fault at Depth, pp. 107– 150,
Beijing in China and Boston in US: China High-Education Press with De
Gruyter.
[3] Ben-Zion, Y., 1998. Properties of seismic fault zone waves and their utility
for imaging low-velocity structures, J. Geophys. Res.-Solid Earth, 103, 12,567–
12,585.
[4] Tarantola, A., 2004. Inverse Problem Theory and Methods for Model
Parameter Estimation, Society for Industrial and Applied Mathematics.
The Pursuit of SynchronisationJames Hannam
Supervisors: Hinke Osinga and Bernd Krauskopf
Many physical systems feature stable oscillations. One practical approach to studying suchsystems is to apply a perturbation and record how the state returns to oscillation. In applicationsthe relative phase difference between the original and the perturbed system may be of interest.When modelling such a system with ordinary differential equations the stable oscillation isrepresented by an attracting periodic orbit. We can then use the concept of isochrons to assigna notion of phase associated with that periodic orbit to all points in its basin of attraction. Thusthe relative phase difference between the original and perturbed system can be understoodgeometrically by considering these isochrons. An isochron defines a connected set of pointsin a periodic orbit’s basin of attraction such that all trajectories starting at these points willsynchronise with the same phase at the periodic orbit.
There are very few systems for which isochrons can be computed analytically; as soon as thedynamics become interesting one has to make use of numerical methods. I will present one suchmethod for computing isochrons in the plane, based on the technique of numerical continuationwhich accurately computes isochrons of a periodic orbit and visualises them as smooth curves.As a test example, I will present a system modified from the Hopf normal form, showcasing someof the intriguing geometric properties that isochrons display in planar systems. As an extensionof this planar example, I will then consider the isochrons of a saddle-type periodic orbit inthree-dimensional phase space, which will lie on that periodic orbit’s two-dimensional invariantmanifolds. This constitutes a first step towards computing isochrons in higher dimensions, andshowcases the intricacies in their visualisation.
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MASS OF FUNCTIONS IN THE CLASS L+(V)
JESSE HART
SUPERVISOR: SIONE MA’U
In pluripotential theory a common class of functions to study is the class
L+(CN ) = u ∈ PSH(CN ) : log+ |z|+ α ≤ u(z) ≤ log+ |z|+ β, for some α, β ∈ R
where PSH(CN ) 3 u : CM → [−∞,∞) if u is upper semicontinuous and satisfiesthe sub-mean inequality on any complex line ([3]). The Monge-Ampere mass of afunction u ∈ PSH(CN ) is defined to be the quantity∫
CN
(ddcu)N ,
where ddcu = 2i∂∂u is interpreted in the sense of currents ([3]). It is a remarkablefact of classical pluripotential theory that if u ∈ L(CN ) then the mass of u is (2π)N
([5]).Our research concerns generalising certain aspects of pluripotential theory in
CN to pluripotential theory on an algebraic variety V. Conspicuously absent fromthe literature is the mass of functions in L+(V) (which is defined analogously toL+(CM )). Using a refinement of a geometric critereon for algebraic varieties dueto Rudin ([4]), we constructed coordinates z = (x, y) which satisfy the growthproperty |y| ≤ (1 + |x|) and for which V is a finite branched holomorphic coveringover x ∈ CM ([2],[1]). This key technical result allowed us to compare the behaviorof u ∈ L+(V) to the behavior associated functions ui ∈ L(CM ) and hence calculatethe mass of u to be d(2π)M where d is the number of branches of V.
This talk will show how the special coordinates (x, y) ∈ CM × CN−M can beconstructed and highlight the key steps of the proof.
References
[1] D. Cox, J. Little, D. O’Shea, Ideals, Varieties and Algorithms: An Introduction to Computa-
tional Algebraic Geometry, 4th ed., Undergraduate Texts in Mathematics, Springer, 2015.[2] R. C. Gunning, Introduction to Holomorphic Functions of Several Complex Variables II: Local
[3] M. Klimek, Pluripotential Theory, London Mathematical Society Monographs New Series 6,Oxford University Press, 1991.
[4] W. Rudin, A geometric criterion for algebraic varieties, J. Math. Mech., 17 (1967/1968), pp.671–683.
[5] B. A. Taylor, An estimate for an extremal plurisubharmonic function on CN , In Seminaire
d’analyse, annees 1982/1983, ed. P. Lelong, P. Dolbeault, and H. Skoda, Lectures Notes in
Mathematics 1028, Springer, Berlin, pp. 318–328.
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Slow-fast Dynamics in Systems of Chemical
Reactions
Cris Hasan
Supervisors:Hinke Osinga and Bernd Krauskopf
Abstract:
In our daily life, we encounter many phenomena that are character-ized by dynamics with slow and fast episodes. For example, spiking andbursting oscillations are observed widely in many applications, such assemiconductor lasers, neuronal models and chemical reactions. Whatis the nature behind such dynamics? Can we model this mathemati-cally? We concentrate here on mixed-mode oscillations (MMOs) whichare complex waveforms with patterns of alternating small- and large-amplitude oscillations, as found in a model of an autocatalytic chemicalreaction. We explore the geometry of this model, which features a mix-ture of slow and fast components, to explain the mechanisms behindthe underlying MMOs.
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Clustered Ventilation Defects in AsthmaticsAustin Ibarra
Supervisor: Graham Donovan
Asthma is a chronic lung disease of reversible airway constriction, and during an asthma at-tack we see heterogeneous ventilation, or specifically, clustered ventilation defects. These effectsare seen in imaging studies of lungs in asthmatics during asthma attacks in the form of the imagesshowing clustered regions of very low ventilation, which is expected from a person struggling todraw breath, but interestingly, the images also show hyper-ventilated regions [6]. These clustersvary from event to event, even in the same subject, and thus, it is believed that the causes aredynamic rather than structural.
We want to understand the dynamic mechanisms behind the formation of these ventilationclusters, both in terms of the underlying mathematics and the physiological implications. Answersto the idea of spatial clustering have been suggested by Anafi and Wilson [2], and Venegas etal. [5], and we use these concepts to help us formulate and analyse models representative of theformation of clustered ventilation defects.
In particular we decompose the problem into so-called Type I, and Type II coupling, with muchof the work in analysing the Type I coupling model done by Donovan and Kritter [1]. This talkwill look at the Type I coupling model and the results we get from that, and how we use that inthe construction of a Type II model by also taking into account other airway models [3, 4]. Finally,we answer the question of whether it is possible to obtain clustered solutions in our Type II modeland how we can extend it to be more sophisticated.
References
[1] Donovan, G. M. and Kritter, T. (2015) Spatial Pattern Formation in the Lung. Journal ofMathematical Biology, volume 70, pages 1119-1149
[2] Anafi, R. C. and Wilson, T. A. (2001) Airway Stability and Heterogeneity in the ConstrictedLung. Journal of Applied Physiology, volume 91, pages 1185-1192
[3] Lambert, R. K.; Wilson, T. A. and Hyatt, R. E. (1982) A Computational Model for ExpiratoryFlow. Journal of Applied Physiology, volume 52(1), pages 44-56
[4] Lambert, R. K. (1989) A New Computational Model for Expiratory Flow from NonhomogeneousHuman Lungs. Journal of Biomechanical Engineering, volume 111(3) pages 200-205
[5] Venegas, J. G.;Winkler, T.;Musch, G.;Vidal Melo, M. F.;Layfield, D.;Tgavalekos, N.;Fischman,A. J.;Callahan, R. J.;Bellani, G. and Harris, R. S. (2005) Self-organised Patchiness in Asthmaas a Prelude to Catastrophic Shifts. Letters to Nature, volume 434, pages 777-782
[6] Svenningsen, S. et al. (2013) Hyperpolarised 3He and 129Xe MRI: Differences in Asthma beforeBronchodilation. Journal of Magnetic Resonance Imaging, volume 38, pages 1521-1530
The Significant Fourier Transform
Joel Laity
May 20, 2016
Some functions f : Zn → C can be well approximated by taking theirdiscrete Fourier transforms and discarding the terms that have small Fouriercoefficients.
The sparse Fourier transform is an algorithm that computes such an ap-proximation more efficiently than computing the entire Fourier transform.
The sparse Fourier transform has many applications to problems in math-ematics and engineering. For example, in mathematics the sparse Fouriertransform can be used to solve the chosen multiplier hidden number problem[1]. In engineering, the sparse Fourier transform can be used to compressaudio or video data [2].
The engineering community is mainly interested in computing Fouriertransforms over Zn where n is an integer power of two. Considerable efforthas been invested in implementing and optimising algorithms for this case.
In this talk I will describe how the sparse Fourier tranform works, andhow the problem of calculating the sparse Fourier transform over Zn can bereduced to calculating it over Z2k where k is the smallest integer such thatn ≤ 2k.
References
[1] S. D. Galbraith and B. Shani, “The multivariate hidden number prob-lem,” Information Theoretic Security, 2015.
[2] H. Hassanieh, P. Indyk, D. Katabi, and E. Price, “Simple and PracticalAlgorithm for Sparse Fourier Transform,” in Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1183–1194, Philadelphia, PA: Society for Industrial and Applied Mathematics,Dec. 2013.
The symmetric genus of a finite group G, denoted by σ(G), is defined as the smallest non-negative integer g such that there exists a compact Riemann surface of genus g on which the groupG has a faithful action as a group of automorphisms, some of which may reverse the surface’sorientation. If we restrict G to contain only orientation-preserving elements, then instead weget the parameter called the strong symmetric genus, denoted by σo(G). These concepts weredefined formally in the early 1980s by Tucker [3].
An interesting question that has been considered by many people over the last 20 yearsor so but which remains open is this: Is the symmetric genus function surjective? In otherwords, for each non-negative integer g, does there exist a finite group G such that σ(G) = g?The corresponding question for the strong symmetric genus σo(G) has been answered in theaffirmative by May and Zimmerman [2], but the above question is more difficult.
Conder and Tucker have proved in [1] the existence of a finite group G with symmetric genusg for all g 6≡ 8 or 14 (mod 18). Further, Conder and Tucker made the stronger conjecture thatsome of these gaps in the spectrum (range of values) of the symmetric genus can be filled byconsidering finite metabelian groups. A group G is said to be metabelian if it contains an abeliannormal subgroup N such that the quotient G/N is also abelian. A subset of this is metacyclicgroups, which are groups having a cyclic normal subgroup whose quotient is also cyclic.
Our research aims to determine the symmetric genus of various families of finite metabeliangroups, starting with metacyclic groups. In this talk, we will describe the method of determiningthe symmetric genus of finite groups using the Riemann-Hurwitz equation, which allows us totreat the problem as a purely algebraic one. We will also present results for some families ofmetabelian groups that we have considered, including split and non-split extensions of the formCm.C2, and split extensions of the form Cm : Cp where p is an odd prime.
References
[1] M. D. E. Conder and T. W. Tucker. The symmetric genus spectrum of finite groups. ArsMath. Contemp., 4(2):271–289, 2011.
[2] C. L. May and J. Zimmerman. There is a group of every strong symmetric genus. Bull.London Math. Soc., 35(4):433–439, 2003.
[3] T. W. Tucker. Finite groups acting on surfaces and the genus of a group. J. Combin. TheorySer. B, 34(1):82–98, 1983.
Name: John Griffith Moala (PhD Student)
Supervisor: A/Prof. Caroline Yoon
Title: On Overcoming Cognitive Obstacles: Towards a Structural Characterization of Overcoming Cognitive Obstacles in the Context of a Mathematical Task
Abstract: Adapting existing ways of thinking during a complex and non-routine task is not easy for many students, especially when these ways of thinking were useful in the past. The success and productivity of a way of thinking establishes it as a useful tool for organizing other current ways of thinking, interpreting new situations and acquiring new ways of thinking (De Bono, 1967; Skemp, 1971). But, the predominance of this way of may be what will impede the student’s progress in a non-routine task. These ways of thinking are known as cognitive obstacles. The inability to revise these particular ways of thinking (cognitive obstacles) can be a major hindrance to the student’s performance on a task.
The process of overcoming a cognitive obstacle during a task is the primary focus of my PhD thesis. This process, I propose, involves not only revising one’s current way of thinking about the problem, but more importantly strengthening what one knows about this particular way of thinking. More specifically, I hypothesize that the process of overcoming a cognitive obstacle during a task involves shifts in the structure of attention (Mason, 2003). The overall goal of the thesis is to investigate the nature of the shifts in the structure of attention that occur when students attempt to overcome cognitive obstacles during non-routine tasks. The primary research question is: How are students’ structures of attention in instances of overcoming obstacles different from, or similar to, their structures of attention in instances where obstacles are encountered but not overcome?
To provide some answers to this question, I will first coordinate and combine (Prediger et al., 2008) several different theories in mathematics education to conceptualize: (i) a cognitive obstacle in the context of a task, and (ii) overcoming a cognitive obstacle in the context of a task. I will then design a set of non-routine and challenging mathematical tasks, which will be implemented on students from three different educational sectors: secondary, pre-university bridging programs, and undergraduate. The students’ mathematical performance/thinking during these tasks will be conceptualized as both the ‘mathematical structures’ attended to, in the sense of the structuralist (Shapiro, 1997) philosophical stance, and how these structures are attended to, in the sense of Mason’s (2003) structure of attention framework, as manifested in the semiotic resources (Arzarello et al., 2009) that students create and use during the tasks. I will also create SPOT (Structures Perceived Over Time) diagrams (Yoon, 2012) of the students’ work and analyze them using, primarily, the structure of attention framework.
Name: Rachel Passmore
Supervisor: Maxine Pfannkuch
The Impact of Curriculum Change on the Teaching and Learning of Time Series
The secondary school statistics curriculum in New Zealand has experienced substantial change since 2010. The catalyst for these changes was a desire to improve students’ statistical reasoning and to narrow the gap between the statistics taught in secondary school classrooms and the practices and thinking of professional statisticians. Anecdotal evidence suggested the quality of Year 13 student work in time series had improved. My research, therefore, sought to provide a robust analysis of changes in learning outcomes of Year 13 students in time series, in order to determine whether these claims could be supported. Furthermore a literature review produced no evidence of any prior research in the area of student reasoning with time series at this level.
Thirty five exemplars of student work distributed by the New Zealand Qualifications Authority (NZQA) were analysed. In order to obtain a more holistic perspective of the curriculum change, 18 teachers were surveyed and five were interviewed. These secondary data sources provided data about teachers’ perceptions of the curriculum change. Since a framework for assessing student learning outcomes for time series did not exist, a framework was developed based on the student data and a synthesis of established frameworks concerned with levels and development of mathematical reasoning, dimensions of statistical reasoning and interpretation of data and data displays.
Analysis of student exemplars against the framework provided strong evidence that after the curriculum change higher levels of reasoning were observed. The shift towards higher levels of reasoning was observed at all levels of achievement – Achieved, Merit and Excellence. The style of student exemplars changed to include a more complete report style response, integrating other research findings that either confirmed or refuted the student’s own findings. One of the major facilitators of this change was the availability of free data visualisation software, which liberated teaching and assessment time from a focus on procedures to one of data interpretation and interrogation. The implications of these findings suggest that the framework developed for this study could be used by teachers in order to scaffold their students’ reasoning to higher levels. Other NCEA Achievement Standards could be similarly scrutinised in order to evaluate the effect of curriculum change.
References:
Friel, S., Curcio, F., & Bright, G. (2001). Making sense of graphs: Critical factors influencing comprehension and instructional implications. Journal for Research in Mathematics Education, 32(2), 124-158.
Pirie, S., & Kieren, T. (1994). Growth in mathematical understanding: How can we characterise it and how can we represent it? Educational Studies in Mathematics, 26(2-3), 165-190. Shaughnessy, M. (2007). Research on statistics learning and reasoning. In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning (Vol. 2, pp. 957-1009). Charlotte, NC: Information Age Publishers.
Verhage, H., & De Lange, J. (1997). Mathematics education and assessment. Pythagoras, 14-20.
Wild, C., & Pfannkuch, M. (1999). Statistical thinking in empirical enquiry. International Statistical Review, 67(3), 223-265.
The arc-type of a vertex-transitive graph is a partition of the graph’svalency as a sum of the lengths of the orbits of a vertex-stabilizer on theneighbourhood of that vertex. Parentheses are used in the partition to denotepaired orbits.
In this talk we will outline some basic concepts from algebraic graphtheory. We will then discuss how vertex-transitive graphs may be classifiedby their arc-types. It has been shown by Conder, Pisanski and Zitnik thatevery marked integer partition except for 2 = 1+1 and 2 = (1+1) is the arc-type of some vertex-transitive graph [1]. We extend this result by showingthat every marked integer partition except for 1, 1+1 and (1+1) is the arc-type of infinitely many Cayley graphs.
References
[1] Conder, M., Pisanski, T., Zitnik, A. (2015). Vertex-transitive graphsand their arc-types. arXiv preprint arXiv:1505.02029.
Hard-core interactions in one-dimensional velocity jump models
Tertius Ralph
Supervisor: Dr Steve Taylor
Excluded-volume effects can play an important role in determining transport propertiesin diffusion of particles through crowded environments. Here, the diffusion of finite-sizedhard-core inter-acting particles is considered systematically using the method of matchedasymptotic expansions. We will use the Langevin approach to diffusion where stochasticincrements are applied to the velocity rather than to the space variable. The result is anon-linear PDE for the one-particle distribution function taking into account crowdingeffects. Stochastic simulations will be used for a comparison with the analytic solutionsderived.
References
[1] M. S. Bartlett, A Note on Random Walks at Constant Speed. Advances in AppliedProbability, 10(4) (1978), pp. 704-707.
[2] M. Bruna, Excluded-volume effects in stochastic models of diffusion, PhD thesis, StAnne’s College, University of Oxford, England (2012).
[3] K. P. Hadeler, T. Hillen and F. Lutscher, The Langevin or Kramers approach tobiological modelling. Mathematical Models and Methods in Applied Sciences, 14(10)(2004), pp. 1561-1583.
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Supporting Teachers Developing Mathematical Tasks
with Digital Technology
Iresha Ratnayake
Supervisors: Prof. Mike Thomas & Dr. Greg Oates
The use of digital technology (DT) in the classroom can help students to understand
mathematical concepts meaningfully, although the effectiveness of using DT in teaching
depends on the DT tasks used (Thomas & Lin, 2013). In addition, the teacher plays the
primary role in implementing DT tasks in the classroom (Sullivan et al., 2015), and so their
relationship to the tasks requires careful consideration. This suggests that if DT tasks are to
be effective in the development of students’ mathematical conceptual knowledge, it is crucial
we involve the teachers in the task development. Hence, this study seeks to identify ways in
which teachers may be assisted with DT task development and implementation. To achieve
this we worked with four groups of three teachers as they designed and implemented DT
tasks.
In this talk I would like to discuss some results from our preliminary analysis, which
examines the richness of the two tasks produced by one of the groups and seeks to explain the
difference between the two tasks. The results suggest that the intervention, focused on
features of rich task design, led to improved pedagogical technology knowledge for the
teachers, and hence a richer task. This indicates that the format and delivery of the
intervention could be of assistance in focusing professional development programmes to
facilitate better the training of teachers in the use of digital technology in teaching
mathematics (Ratnayake, I., Oates, G., & Thomas, M. O. J., 2016, in print).
References
Ratnayake, I., Oates, G., & Thomas, M. O. J., (2016, in print). Supporting teachers
developing mathematical tasks with digital technology. In Opening up mathematics
education research (Proceedings of the 39th annual conference of the Mathematics
Education Research Group of Australasia). Adelaide: MERGA.
Sullivan, P., Knott, L., Yang, Y., Askew, M., Brown, L., Bussi, M. G. B., . . . Gimenez, J.
(2015). The relationship between task design, anticipated pedagogies, and student
learning. In A. Watson & M. Ohtani (Eds.), Task design in mathematics education
(pp. 83-114). London: Springer.
Thomas, M. O. J., & Lin, C. (2013). Designing tasks for use with digital technology. In C.
Margolinas, J. Ainley, J. B. Frant, M. Doorman, C. Kieran, A. Leung, M. Ohtani, P.
Sullivan, D. Thompson, A. Watson & Y. Yang (Eds.), Task design in mathematics
education, Proceedings of ICMI study 22 (pp. 109-118). United Kingdom: Oxford.
Quantifier Elimination And The RealNullstellensatz
Given a set of polynomials p1, . . . , pm ⊂ R[x1, . . . , xn] with the setof common zeroes over real numbers being empty, the Real Nullstellensatzstates
1 +k∑
n=1
(hi(x1, . . . , xn))2 +m∑i=1
pi(x1, . . . , xn)gi(x1, . . . , xn) = 0,
for some k ∈ N and hi, gi ∈ R[x1, . . . , xn]. Now the main question is findingan algorithm that produces hi and gi and in particular to find bounds ontheir degrees.In this talk we discuss the bounds found for the equation above and presentsome of the methods used to attack the question.
Hardness of Computing the 5/6 MSB’s of theElliptic Curve Diffie-Hellman Keys
Barak ShaniSupervised by Prof. Steven D. Galbraith
Abstract
The elliptic curve Diffie-Hellman protocol was adopted by Googlein late 2011 and is used today by most internet sites to establish asecure communication channel. This includes: Amazon, Dropbox,Facebook, Instagram, Snapchat, Twitter, Yahoo and many more. Itwas recently adopted by WhatsApp for communication using mobiledevices.
Due to size limitations of mobile’s CPUs it is desirable to minimizethe computations and the amount of information being transferredduring the process of creating the secure communication channel. Nor-mally, to achieve some desired security level, the entire Diffie-Hellmankey is used to establish the channel.
In my talk I will present my recent result, that proves that onecan get the same level of security by taking only 5/6 of all the secretkey’s bits. That is, if an attacker can compute the 5/6 MSB’s of theDiffie-Hellman key, then she can compute the entire key. Achievingsecurity of bits for elliptic curve Diffie-Hellman keys has been an openproblem for more than 15 years. The result can be used to minimizethe amount of computation done on mobile devices, without harmingthe security level.
1
Conformal Geometry and Conserved Quantities
Daniel Snell
Supervisor: A. Rod Gover
Let (Mn, g) be a Riemannian manifold. Suppose that γ is a geodesic on Mwith tangent vector u, and that k is a Killing field for the metric g. A well-knownclassical result [1] then tells that the quantity u · k will be conserved along γ.Conserved quantities play an important role in mathematics and physics. Forexample, the presence of conserved quantities sometimes allows one to solvedifferential equations on the manifold [2].
Recall that a conformal manifold is a pair (Mn, c), where c is the equivalenceclass of metrics on Mn defined by the relation g ∼ g if, and only if, g =Ω2g for some positive real-valued function Ω. This is clearly a generalisationof Riemannian geometry, and so it is natural to ask what kinds of conservedquantities one might have. This programme presents several problems. Forone, since there is no longer a single distinguished metric on the manifold, onedoes not have a canonical Levi-Civita connection on the tangent bundle. Foranother, we must find conformal analogues of the geodesic and Killing field fromthe statement of the original theorem.
It turns out that all of these problems are solvable, and the resulting objectscan be used to manufacture conserved quantities. In this talk, I will present abrief overview of tractor calculus, the natural framework for studying conformalmanifolds. I will also show how we can form appropriate conformal versionsof geodesics and Killing fields, and that these can be used to make conservedquantities. Finally, we shall see that in the presence of some additional confor-mal geometric structure (such as the existence of a parallel standard tractor),we get several other quantities which are conserved along conformal geodesics.This is joint work with A. Rod Gover.
References
[1] R. M. Wald, General relativity. University of Chicago Press, Chicago, IL,1984.
[2] L. Andersson and P. Blue, “Hidden symmetries and decay forthe wave equation on the Kerr spacetime,” 2015. Available athttp://www.arxiv.org/abs/0908.2265.
1
On James’ Characterisationof weak Compactness
Samuel J. WhiteSupervisor: Warren Moors
If a set K is compact, then it is well know that every continuous linear functional on K attains itssupremum. More interesting is whether the converse is true: if every continuous linear functionalattains its supremum on a set, is this set necessarily compact in the weak topology? This questionwas famously answered in the affirmative in [1]. However, James’ proof was notoriously difficult,and many attempts were made to simplify it.
The first of these was in [3], which, like the original, used only elementary techniques. Subsequently,Moors developed a proof (in [2]), which was much shorter and simpler, but which required a lotmore knowledge of the theory of the geometry of Banach spaces. In particular, Moors’ proof reliedupon the Krein-Milman theorem, Milman’s theorem and the Bishop-Phelps theorem. The maintheorem of that paper was the following.
Given K, a weak∗ compact convex subset of the dual of a Banach space X, a subset B of K is calleda boundary of K if for every x ∈ X there exists an x∗ ∈ B such that x∗(x) = supy∗(x) : y∗ ∈ K.We shall say B, (I)-generates K, if for every countable cover Cn : n ∈ N of B by weak∗ compactconvex subsets of K, the convex hull of
⋃n∈NCn is norm dense in K.
Theorem 1 Let K be a weak∗ compact convex subset of the dual of a Banach space X and let Bbe a boundary of K. Then B, (I)-generates K.
In this talk, we present a new proof of this theorem. The proof we give is completely elementary,and in fact is based upon the techniques of Pryce’s paper, thus unifying the two approaches. kReferences
[1] R. C. James, Weakly compact sets, Trans. Amer. Math. Soc. 113 (1964), 129–140.
[2] W. B. Moors, An elementary proof of James’ characterization of weak compactness, Bull.Austr. Math. Soc. 84 (2011), 98-102.
[3] J. D. Pryce, Weak compactness in locally convex spaces, Proc. Amer. Math. Soc. 17 (1966),148-155.
