Problematizing mathematics and its pedagogy through teacher engagement with history-focused and classroom situation-specific tasks

We explore the conjecture that engaging teachers with activities which feature mathematical practices from the past (history-focused tasks) and in today’s mathematics classrooms (mathtasks) can promote teachers’ problematizing of mathematics and its pedagogy. Here, we sample evidence of discursive shifts observed as twelve mathematics teachers engage with a set of problematizing activities (PA) – three rounds of history-focused and mathtask combinations – during a four–month postgraduate course. We trace how the commognitive conflicts orchestrated in the PA triggered changes in the teachers’ narratives about: mathematical objects (such as what a function is); how mathematical objects come to be (such as what led to the emergence of the function object); and, pedagogy (such as what value may lie in listening to students or in trialing innovative assessment practices). Our study explores a hitherto under-researched capacity of the commognitive framework to steer the design, evidence identification and impact evaluation of pedagogical interventions.     

The discursive footprint of learning across mathematical domains: The case of the tangent line

This paper employs the commognitive frame (Sfard, 2008) to investigate how experiences with tangents across mathematical domains leave their marks on students’ subsequent work with tangents. To this aim, I introduce the notion of the discursive footprint of tangents and its characteristics by reviewing how tangents are used across mathematical domains in school textbooks. Manifestations of this footprint were sought in 182 undergraduate mathematics students’ responses to a questionnaire about tangents by labelling their responses and by identifying patterns in the endorsed narratives. Manifestations include the identification of characteristics of sole (and combination of) discourses (geometry, algebra, calculus, mathematical analysis) in student responses. Five themes emerged from the analysis: apparent replication of word use in different narratives; geometry-local hybrid discourse; endorsement of conflicting narratives; enrichment of familiar narratives with new words; and, mathematical analysis as a subsuming discourse. Finally, I discuss the potency of the discursive footprint in research and teaching.     

Standing on each other’s shoulders: A case of coalescence between geometric discourses in peer interaction

In this paper we draw on the commognitive theory to examine novice students’ transition from familiar mathematics meta-rules to less familiar ones during peer interaction. To pursue this goal, we focused on a relatively symmetric interaction between two middle-school students given a geometric task. During their dyadic problem-solving, the students transitioned from configural procedures to deductive ones. We found that this transition included an interactive coalescence pattern in which one student “borrowed” her partner’s configural sub-procedures and built on them to develop a new deductive procedure. Furthermore, we found that during their peer interaction, the students oscillated between configural, coalesced and deductive procedures. Several patterns in the students’ interpretation of the task-situation contributed to these oscillations. We discuss the contribution of our findings to commognitive research, to geometry learning research and to peer learning research.     

Running to keep up with the lecturer or gradual de-ritualization? Biology students’ engagement with construction and data interpretation graphing routines in mathematical modelling tasks

Through a commognitive lens, we examine twelve first-semester biology students’. engagement with graphing routines as they work in groups, during four sessions of Mathematical Modelling (MM). We trace the students’ meta-level learning, particularly as they fluctuate between deploying graphs for mere illustration of data and as sense-making tools. We account for student activity in relation to precedent events in their experiences of graphing and as fluid, if not always productive, interplay between ritualised and exploratory engagement with graph construction and interpretation routines. The students’ construal of the task situations is marked by efforts to keep up with lecturer expectations which allow for changing degrees of student agency but do not factor in the influence of precedent events. Our analysis has pedagogical implications for the way MM problems are formulated and also foregrounds the capacity of the commognitive framework to trace de-ritualization and meta-level learning in students’ MM activity.     

Pre-university students square-root from squared things: A commognitive account of apparent conflicts within learners' mathematical discourses

This study explores the grasp of square roots among eleven students in a remedial mathematics course, with a special focus on where the same student generated apparently conflicting responses. Building on the commognitive framework, the analysis distinguished between routines that individual students consistently implemented in situations where roots “stood alone” and where they were incorporated in more compound exercises, where roots were extracted from square numbers and from squared radicands, where roots were applied to monomials and binomials, and where parameters named with different letters were involved. Differences were found in routines’ degree of objectification, procedures, and tasks. These differences are explained with a theoretical account, suggesting that what may seem as a conflict within a student’s discourse could be a sensible difference of actions taken in situations that this student construed as different. The contribution of this study to the body of knowledge on teaching and learning of roots and to commognitive research is discussed.     

An analysis of interorganizational resource sharing decisions in collaborative knowledge creation

Collaborative relationships between organizations for knowledge creation have received considerable attention in recent literature. For any collaborative endeavor to succeed, adequate allocation and sharing of resources is important. We explored the conditions of resource sharing in which organizations are motivated to collaborate for knowledge creation, as well as the conditions required for the collaboration to continue. Using the game theoretic framework, we modeled the collaboration for knowledge creation as a Stackelberg leader–follower game. Specifically, we modeled two scenarios, the first in which organizations expend current efforts only, and the second in which organizations have made relevant prior efforts. The equilibrium values of current efforts in knowledge creation, the leader organization’s resource participation rate, and the total expected system gain in both of these scenarios were determined. Both within scenario and cross-scenario analyses were conducted. It is important to maintain an optimal ratio between the leader’s and follower’s marginal gains for the formation and continuation of the collaboration.     

Optimal approximations with Rough Sets and similarities in measure spaces

When arbitrary sets are approximated by more structured sets, it may not be possible to obtain an exact approximation that is equivalent to a given set. Presented here, is a new proposal for a ‘metric’ approach to Rough Sets. We assume some finite measure space is defined on a given universe, and then use it to define various similarity indexes. A set of axioms and the concept of consistency for similarity indexes are also proposed. The core of the paper is a definition of the ‘optimal’ or ‘best’ approximation with respect to any particular similarity index, and an algorithm to find this optimal approximation by using the Marczewski–Steinhaus Index. This algorithm is also shown to hold for a class of similarity indexes that are consistent with the Marczewski–Steinhaus Index.     

Forecasting accuracy of behavioural models for participation in the arts

This paper assesses the forecasting performance of count data models applied to arts attendance. We estimate participation models for two artistic activities that differ in their degree of popularity – museums and jazz concerts – with data derived from the 2002 release of the Survey of Public Participation in the Arts for the United States. We estimate a finite mixture model – a zero-inflated negative binomial model – that allows us to distinguish between “true” non-attendants and “goers” and their respective behaviour regarding participation in the arts. We evaluate the predictive (in-sample) and forecasting (out-of-sample) accuracy of the estimated model using bootstrapping techniques to compute the Brier score. Overall, the results indicate the model performs well in terms of forecasting. Finally, we draw certain policy implications from the model’s forecasting capacity, thereby allowing the identification of target populations.     

Teaching practices aimed at promoting meta-level learning: The case of complex numbers

In this paper we seek to promote a conceptualization of “teaching toward meta-level learning” based on theoretical and empirical aspects. We adopt the commognitive distinction between object- and meta-level learning, and relate to meta-level learning as involving changes in the metarules that govern the discourse. Specifically, we refer to changes in the discourse on numbers emerging in the shift in discourse from real to complex numbers. We applied implications from the commognitive theory about meta-level learning to the planning and teaching of a lesson about complex numbers. Then, we analyzed the lesson to identify teaching practices that could promote meta-level learning. We found that these teaching practices can be clustered into three theory driven sub-sets: those referring to students’ current discourse on numbers, those referring to their new discourse and those referring to the transition between the two.     

The functions of function discourse – university mathematics teaching from a commognitive standpoint

This paper addresses a topic within university mathematics education which has been somewhat underexplored: the teaching practices actually used by university mathematics teachers when giving lectures. The study investigates the teaching practices of seven Swedish university teachers on the topic of functions using a discursive approach, the commognitive framework of Sfard. In the paper a categorization of the construction and substantiation routines used by the teachers is presented, for instance various routines for constructing definitions and examples, and for verifying whether an example satisfies a given definition. The findings show that although the overall form of the lectures is similar, with teachers using ‘chalk talk’, and overt student participation limited to asking and answering questions, there are in fact significant differences in the way the teachers present and do mathematics in their lectures. These differences present themselves both on the level of discursive routines and on a more general level in how the process of doing mathematics is made visible in the teachers’ teaching practices. Moreover, I believe that many of the results of the study could be relevant for investigating the teaching of other mathematical topics.     

Existence and uniqueness of similarity solutions of imbibition equation

The existence and uniqueness of the similarity profiles for the imbibition phenomenon which may arise due to the difference in the wetting abilities of the two immiscible fluids involved in the displacement process through porous media is discussed. By assuming the validity of the Darcy’s law, a mathematical model has been described and it is found that the investigated flow system is governed by a nonlinear diffusivity type equation. The existence and uniqueness of its similarity solutions have been proved by considering the bounds on the saturation coefficient,N(S _w) which is regarded as positive and piecewise continuously differentiable.     

On the Unitary Similarity of Matrix Families

The classical Specht criterion for the unitary similarity between two complex n × n matrices is extended to the unitary similarity between two normal matrix sets of cardinality m. This property means that the algebra generated by a set is closed with respect to the conjugate transpose operation. Similar to the well-known result of Pearcy that supplements Specht's theorem, the proposed extension can be made a finite criterion. The complexity of this criterion depends on n as well as the length l of the algebras under analysis. For a pair of matrices, this complexity can be significantly lower than that of the Specht--Pearcy criterion.     

Monotone convergence of Newton-like methods for M-matrix algebraic Riccati equations

For the algebraic Riccati equation whose four coefficient matrices form a nonsingular M-matrix or an irreducible singular M-matrix K, the minimal nonnegative solution can be found by Newton’s method and the doubling algorithm. When the two diagonal blocks of the matrix K have both large and small diagonal entries, the doubling algorithm often requires many more iterations than Newton’s method. In those cases, Newton’s method may be more efficient than the doubling algorithm. This has motivated us to study Newton-like methods that have higher-order convergence and are not much more expensive each iteration. We find that the Chebyshev method of order three and a two-step modified Chebyshev method of order four can be more efficient than Newton’s method. For the Riccati equation, these two Newton-like methods are actually special cases of the Newton–Shamanskii method. We show that, starting with zero initial guess or some other suitable initial guess, the sequence generated by the Newton–Shamanskii method converges monotonically to the minimal nonnegative solution.We also explain that the Newton-like methods can be used to great advantage when solving some Riccati equations involving a parameter.     

Analytical and numerical solution of a generalized Stefan problem exhibiting two moving boundaries with application to ocean delta formation

A model associated with the formation of sedimentary ocean deltas is presented. This model is a generalized one-dimensional Stefan problem bounded by two moving boundaries, the shoreline and the alluvial-bedrock transition. The sediment transport is a non-linear diffusive process; the diffusivity modeled as a power law of the fluvial slope. Dimensional analysis shows that the first order behavior of the moving boundaries is determined by the dimensionless parameter 0?Rab?1—the ratio of the fluvial slope to bedrock slope at the alluvial-bedrock transition. A similarity form of the governing equations is derived and a solution that tracks the boundaries obtained via the use of a numerical ODE solver; in the cases where the exponent θ in the diffusivity model is zero (linear diffusion) or infinite, closed from solutions are found. For the full range of the diffusivity exponents, 0?θ→∞, the similarity solution shows that when Rab<0.4 there is no distinction in the predicted speeds of the moving boundaries. Further, within the range of physically meaningful values of the diffusivity exponent, i.e., 0?θ∼2, reasonable agreement in predictions extents up to Rab∼0.7. In addition to the similarity solution a fixed grid enthalpy like solution is also proposed; predictions obtained with this solution closely match those obtained with the similarity solution.     

Where is Difference? Processes of Mathematical Remediation through a Constructivist Lens

In this study, we challenge the deficit perspective on mathematical knowing and learning for children labeled as LD, focusing on their struggles not as a within student attribute, but rather as within teacher-learner interactions. We present two cases of fifth-grade students labeled LD as they interacted with a researcher-teacher during two constructivist-oriented teaching experiments designed to foster a concept of unit fraction. Data analysis revealed three main types of interactions, and how they changed over time, which seemed to support the students’ learning: Assess, Cause and Effect Reflection, and Comparison/Prediction Reflection. We thus argue for an intervention in interaction that occurs in the instructional process for students with LD, which should replace attempts to “fix” ‘deficiencies’ that we claim to contribute to disabling such students.     

Leveraging interdiscursivity to support elementary students in bridging the empirical-deductive gap: the case of parity

Research has recognized deductive reasoning as challenging but not impossible for young mathematics learners. In this paper, we present a learning environment developed to assist elementary-school students to bridge the empirical-deductive gap in the context of parity of numbers. Using the commognitive framework, we construe the empirical-deductive gap as part of a broader divide between two discourses that abide by different rules of a “mathematical game”: a discourse on specific numbers and a discourse on numeric patterns. Interdiscursivity is leveraged as a mechanism for instructional design, where students’ familiar routines with specific numbers are teased out and advanced to make sense in the new discourse. We mobilize this mechanism to create opportunities for students to play an active role in recognizing issues with empirical reasoning and generating deductive arguments to establish the validity of universal statements. The environment is illustrated with a small group of 8-year-olds who learned to justify deductively that “odd + odd = even”.     

Elliptic curves and a new construction of integrable systems

A class of elliptic curves with associated Lax matrices is considered. A family of dynamical systems on e(3) parametrized by polynomial a with the above Lax matrices are constructed. Five cases from the family are selected by the condition of preserving the standard measure. Three of them are Hamiltonian. It is proved that two other cases are not Hamiltonian in the standard Poisson structure on e(3). Integrability of all five cases is proven. Integration procedures are performed in all five cases. Separation of variables in Sklyanin sense is also given. A connection with Hess-Appel’rot system is established. A sort of separation of variables is suggested for the Hess-Appel’rot system.     

On the Lipschitz continuity of certain quasiregular mappings between smooth Jordan domains

We first investigate the Lipschitz continuity of (K,K’)-quasiregular C ² mappings between two Jordan domains with smooth boundaries, satisfying certain partial differential inequalities concerning Laplacian. Then two applications of the obtained result are given: As a direct consequence, we get the Lipschitz continuity of ρ-harmonic (K,K’)-quasiregular mappings, and as the other application, we study the Lipschitz continuity of (K,K’)- quasiconformal self-mappings of the unit disk, which are the solutions of the Poisson equation Δw = g. These results generalize and extend several recently obtained results by Kalaj, Mateljevi? and Pavlovi?.     

客户企业参与下服务外包激励机制设计

服务外包下,由于客户企业参与程度、服务提供商的努力水平以及专用性资产投入水平都难以让合作双方有效验证,这三种不可验证性风险因素可能引发道德风险,从而降低服务外包的绩效,甚至导致服务外包的失败。为此,本文建立了服务外包下的委托—代理框架,设计了客户企业参与下的服务外包激励机制模型,运用最优化理论分析了客户企业的参与程度以及服务提供商的专用性资产投入水平等因素对激励机制契约设计的影响,并运用数值仿真验证客户企业参与程度和服务提供商专用性资产投入水平对激励系数和服务提供商的努力水平的影响。研究分析表明:在服务外包活动中客户企业更多的投入到合作生产中时,对服务提供商的激励也会增加但是增加的幅度较慢,同时服务提供商付出的努力将会加速的提高;服务提供商投入的专用性资产投入水平越高时,其自身的努力水平也会加速增长,客户企业对其的激励系数也会增长,当专用性资产投入水平达到某一程度时,激励系数增长速度达到最大。