Mathematics teacher educators’/researchers’ collaboration with teachers as a context for professional learning

This paper describes our joint activity as mathematics teacher educators and academic researchers in collaborating with both experienced and novice teachers in two contexts: an emergent community of inquiry into mathematics teaching and its development; and a research methods course, offered as part of a mathematics education Master’s program, aspiring to initiate participating teachers into research practice through inquiry. Adopting an Activity Theory (AT) perspective, we analyse our activity, identifying its nature and transformations that frame our professional learning. The results indicate that our professional learning is the outcome of a continuous process of becoming aware of our own activity and its transformation in relation to that of the teachers.     

Statistical modeling to promote students’ aggregate reasoning with sample and sampling

While aggregate reasoning is a core aspect of statistical reasoning, its development is a key challenge in statistics education. In this study we examine how students’ aggregate reasoning with samples and sampling (ARWSS) can emerge in the context of statistical modeling activities of real phenomena. We present a case study on the emergent ARWSS of two pairs of sixth graders (age 11–12) involved in statistical data analysis and informal inference utilizing TinkerPlots. The students’ growing understandings of various statistical concepts is described and five perceptions the students expressed are identified. We discuss the contribution of modeling to these progressions followed by conclusions and limitations of these results. While idiosyncratic, the insights contribute to the understanding of students’ aggregate reasoning with data and models, with regards to samples and sampling.     

The role of model comparison in young learners’ reasoning with statistical models and modeling

The goal of this study is to explore the role of model comparison, which is a key activity of young learners’ informal reasoning, with statistical models and modeling in the context of informal statistical inference. We suggest a framework to describe this reasoning (the RISM framework), and offer an illustrative case study of two-sixth graders showcasing its utility. In particular, we illustrate the benefit of untangling the informal modeling process into three separate, though not independent, modeling processes: modeling a conjecture, modeling data, and comparing them by means of a comparison model. This case study shows the possible progression of a comparison model, and its potential role as a catalyst for the development of the other two modeling processes. Finally, an expansion of our initial framework is discussed, highlighting the centrality of model comparisons.     

Risk preference modeling with conditional average: an application to portfolio optimization

The paper introduces a new risk measure called Conditional Average (CAVG), which was designed to cover typical attitudes towards risk for any type of distribution. It can be viewed as a generalization of Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR), two commonly used risk measures. The preference structure induced by CAVG has the interpretation in Yaari’s dual theory of choice under risk and relates to Tversky and Kahneman’s cumulative prospect theory. The measure is based on the new stochastic ordering called dual prospect stochastic dominance, which can be considered as a dual stochastic ordering to recently developed prospect stochastic dominance. In general, CAVG translates into a nonconvex quadratic programming problem, but in the case of a finite probability space it can also be expressed as a mixed-integer program. The paper also presents the results of computational studies designed to assess the preference modeling capabilities of the measure. The experimental analysis was performed on the asset allocation problem built on historical values of S&P 500 sub-industry indexes. The research was supported by the grant PBZ-KBN-016/P03/99 from the State Committee for Scientific Research.     

Evolution equations with Gel’fond-Leont’ev generalized differentiation operators: II

We establish the solvability of the Cauchy problem for evolution equations with Gel’fond-Leont’ev generalized differentiation operators in spaces of the type W as well as in spaces of generalized functions (analytic functionals) of the type W′.     

Chaplygin’s ball with a rotor: Non-degeneracy of singular points

A problem of a dynamically balanced asymmetric ball with a rotor rolling over a rough horizontal plane is considered. Earlier, A. Y. Moskvin constructed bifurcation diagrams of the momentum map and bifurcation complexes in order to study the dynamics of the system and to obtain singular solutions. A natural development of this research is a fine Liouville analysis of the system. The first step in this direction is presented in the paper, namely, we verify the non-degeneracy of singularities and describe the Liouville foliation in a neighborhood of singular points of the momentum map.     

Coping with the Renewable Portfolio Standards: A utility’s perspective

Renewable Portfolio Standards require utilities to supply a percentage of their energy from renewables. Utilities demonstrate their compliance by purchasing renewable energy or renewable energy certificates. Certificates can be traded in a secondary market. This flexibility is intended to foster an efficient environment. Utilities, however, have not been reacting strategically. We formulate the utility’s problem as a stochastic dynamic program and present optimal policies for the utility to trade in the REC market and to purchase from the energy market.     

The unit commitment model with concave emissions costs: a hybrid Benders’ Decomposition with nonconvex master problems

We present a unit commitment model which determines generator schedules, associated production and storage quantities, and spinning reserve requirements. Our model minimizes fixed costs, fuel costs, shortage costs, and emissions costs. A constraint set balances the load, imposes requirements on the way in which generators and storage devices operate, and tracks reserve requirements. We capture cost functions with piecewise-linear and (concave) nonlinear constructs. We strengthen the formulation via cut addition. We then describe an underestimation approach to obtain an initial feasible solution to our model. Finally, we constitute a Benders’ master problem from the scheduling variables and a subset of those variables associated with the nonlinear constructs; the subproblem contains the storage and reserve requirement quantities, and power from generators with convex (linear) emissions curves. We demonstrate that our strengthening techniques and Benders’ Decomposition approach solve our mixed integer, nonlinear version of the unit commitment model more quickly than standard global optimization algorithms. We present numerical results based on a subset of the Colorado power system that provide insights regarding storage, renewable generators, and emissions.     

Students’ experiences with mathematics teaching and learning: listening to unheard voices

This study documents students’ views about the nature of mathematics, the mathematics learning process and factors within the classroom that are perceived to impact upon the learning of mathematics. The participants were senior secondary school students. Qualitative and quantitative methods were used to understand the students’ views about their experiences with mathematics learning and mathematics classroom environment. Interviews of students and mathematics lesson observations were analysed to understand how students view their mathematics classes. A questionnaire was used to solicit students’ views with regards to teaching approaches in mathematics classes. The results suggest that students consider learning and understanding mathematics to mean being successful in getting the correct answers. Students reported that in the majority of cases, the teaching of mathematics was lecture-oriented. Mathematics language was considered a barrier in learning some topics in mathematics. The use of informal language was also evident during mathematics class lessons.     

Students’ geometric thinking with cube representations: Assessment framework and empirical evidence

While representations of 3D shapes are used in the teaching of geometry in lower secondary school, it is known that such representations can provide difficulties for students. In order to assess students’ thinking about 3D shapes, we constructed an assessment framework based on existing research studies and data from G7-9 students (aged 12–15). We then applied our framework to assess students’ geometric thinking in lessons. We report two cases of qualitative findings from a classroom experiment in which Grade 7 students (aged 12–13) tackled a problem in 3D geometry that was, for them, quite challenging. We found that students who failed to answer given problems did not mentally manipulate representations effectively, while others could mentally manipulate representations and reason about them in order to reach correct solutions. We conclude with the proposition that this finding shows the framework can be used by teachers in instruction to assess their students’ 3D geometric thinking.     

Synchronization of Chua’s circuits with multi-scroll attractors: Application to communication

In this paper, the synchronization problem of coupled Chua’s circuits generator of n-scroll chaotic attractors in master–slave configuration is numerically studied. In particular, we consider a modified Chua’s circuit generator of 5-scroll chaotic attractors by using Hamiltonian systems and state observer approach. A potential application to transmit encrypted audio and image information is also given.     

Quasi-classical asymptotics for pseudodifferential operators with discontinuous symbols: Widom’s conjecture

In 1982 H. Widom conjectured a multi-dimensional generalization of a well-known two-term quasi-classical asymptotic formula for the trace of the function f(A) of a Wiener—Hopf-type operator A in dimension 1 for a pseudodifferential operator A with symbol a(x, ξ) having jump discontinuities in both variables. In 1990 he proved the conjecture for the special case when the jump in any of the two variables occurs in a hyperplane.     

Sequences of 0’s and 1’s: Special sequence spaces with the separable Hahn property

As pointed out in [4] the paper [2], authored by G. Bennett, J. Boos and T. Leiger, contains a nontrivial gap in the argumentation of the proof of Theorem 5.2 which is one of main results of that paper and has been applied three times. Till now neither the gap is closed nor a counterexample has been stated. That is why the authors have examined in [4] the situation around the ‘gap’ aiming to a better understanding for the gap. The aim of this paper is to prove the mentioned applications of the theorem in doubt by using gliding hump arguments (quite similar to the classical proofs of the Theorems of Schur and Hahn in the first case (cf. [3]) and a very technical and artful construction, being of independent mathematical interest, in the second case). Research of T. Leiger supported by Estonian Science Foundation Grant 5376.     

Optimal stopping with a capacity constraint: Generalizing Shepp’s urn scheme

We formulate an optimal stopping problem for a variant of Shepp’s urn model in which it is possible to sample more than one item at each stage. Using a Markov decision process model, we establish monotonicity of the optimal value function and show that the optimal policy is a monotone threshold policy that prescribes either not sampling, or sampling the maximum number of items permitted. A special case exhibits convexity and submodularity, but these properties do not hold in general.     

A degenerate Newton’s map in two complex variables: Linking with currents

Little is known about the global structure of the basins of attraction of Newton’s method in two or more complex variables. We make the first steps by focusing on the specific Newton mapping to solve for the common roots of P(x, y) = x(1−x) and Q(x, y) = y² + Bxy − y. There are invariant circles S₀ and S₁ within the lines x = 0 and x = 1 which are superattracting in the x-direction and hyperbolically repelling within the vertical line. We show that S₀ and S₁ have local super-stable manifolds, which when pulled back under iterates of N form global super-stable spaces W₀ and W₁. By blowing-up the points of indeterminacy p and q of N and all of their inverse images under N we prove that W₀ and W₁ are real-analytic varieties. We define linking between closed 1-cycles in W_i (i = 0, 1) and an appropriate closed 2 current providing a homomorphism lk: H₁ (W_i, ℤ) → ℚ. If W_i intersects the critical value locus of N, this homomorphism has dense image, proving that H₁ (W_i, ℤ) is infinitely generated. Using the Mayer-Vietoris exact sequence and an algebraic trick, we show that the same is true for the closures of the basins of the roots      

Motion of a Swinging Atwood’s Machine: Simulation and Analysis with Mathematica

A generalized model of the Atwood machine when one body is constrained to move along a vertical axis while the other one can swing in a plane is considered. Combining symbolic and numerical calculations, we have obtained equations of motion of the system and analyzed their solutions. We have shown that oscillation can completely modify a motion of the system while the simple Atwood machine demonstrates only the uniformly accelerated motion of the bodies. The validity of the results obtained is demonstrated by means of the simulation of motion of swinging Atwood’s machine with the computer algebra system Wolfram Mathematica.     

Correction to: On a variant of Pillai’s problem with transcendental numbers

Acta Mathematica Hungarica -     

Solving equations with fractions: An analysis of prospective teachers’ solution pathways and errors

We analyzed the solution pathways and errors found in the written responses of 469 prospective teachers solving an equation containing fractions. The majority (332, or 70%) used an algebraic method; 141 of the 332 (42%) were correct, and 22% of the algebraic methods were abandoned before a solution was obtained. We identified the steps in the written solutions, determined which solution pathways led to the correct solution, and identified common errors in the solution pathways of respondents who incorrectly solved the equation. Respondents initially attempted different methods. The most common method was solving by using fractions, but the majority of respondents who solved by using mixed numbers were able to correctly solve the problem. Common errors related to fraction arithmetic and the distributive property. Nearly all of the abandoned pathways contained no errors, but ended with a step that likely would precede an operation with fractions. Our findings suggest that the ability to solve an arithmetic equation with no fractions was necessary, but not sufficient, to solve an arithmetic equation involving fractions, and that the problem of solving equations with fractions was more closely tied to one's difficulties with rational number arithmetic and less with one's understanding of algebra.