Variation of student numerical and figural reasoning approaches by pattern generalization type,strategy use and grade level

This paper explored variation of student numerical and figural reasoning approaches across different pattern generalization types and across grade level. An instrument was designed for this purpose. The instrument was given to a sample of 1232 students from grades 4 to 11 from five schools in Lebanon. Analysis of data showed that the numerical reasoning approach seems to be more dominant than the figural reasoning approach for the near and far pattern generalization types but not for the immediate generalization type. The findings showed that for the recursive strategy, the numerical reasoning approach seems to be more dominant than the figural reasoning approach for each of the three pattern generalization types. However, the figural reasoning approach seems to be more dominant than the numerical reasoning approach for the functional strategy, for each generalization type. The findings also showed that the numerical reasoning was more dominant than the figural reasoning in lower grade levels (grades 4 and 5) for each generalization type. In contrast, the figural reasoning became more dominant than the numerical reasoning in the upper grade levels (grades 10 and 11).     

Middle school children’s cognitive perceptions of constructive and deconstructive generalizations involving linear figural patterns

This paper discusses the content and structure of generalization involving figural patterns of middle school students, focusing on the extent to which they are capable of establishing and justifying complicated generalizations that entail possible overlap of aspects of the figures. Findings from an ongoing 3-year longitudinal study of middle school students are used to extend the knowledge base in this area. Using pre-and post-interviews and videos of intervening teaching experiments, we specify three forms of generalization involving such figural linear patterns: constructive standard; constructive nonstandard; and deconstructive; and we classify these forms of generalization according to complexity based on student work. We document students’ cognitive tendency to shift from a figural to a numerical strategy in determining their figural-based patterns, and we observe the not always salutary consequences of such a shift in their representational fluency and inductive justifications.     

Identifying Kinds of Reasoning in Collective Argumentation

We combine Peirce’s rule, case, and result with Toulmin’s data, claim, and warrant to differentiate between deductive, inductive, abductive, and analogical reasoning within collective argumentation. In this theoretical article, we illustrate these kinds of reasoning in episodes of collective argumentation using examples from one teacher’s practice. Examining different kinds of reasoning in collective argumentation can inform how students engage in generating and examining hypotheses using inductive and abductive reasoning and move toward the deductive reasoning required for proof. Mathematics educators can build on their understanding of these kinds of reasoning to support students in reasoning in productive ways.     

Reflective abstraction, uniframes and the formulation of generalizations

In mathematics, generalizations are the end result of an inductive zigzag path of trial and error, that begin with the construction of examples, within which plausible patterns are detected and lead to the formulation of theorems. This paper examines whether it is possible for high school students to discover and formulate generalizations similar to ways professional mathematicians do. What are the experiences that allow students to become adept at generalization? In this paper, the mathematical experiences of a ninth grade student, which lead to the discovery and the formulation of a mathematical generalization are described, qualitatively analyzed and interpreted using the notion of uniframes. It is found that reflecting on the solutions of a class of seemingly different problem-situations over a prolonged time period facilitates the abstraction of structural similarities in the problems and results in the formulation of mathematical generalizations.     

A Conversation With Zoltan P. Dienes

In this article, we analyze the visual and symbolic strategies developed by students to express generalizations of number patterns and the connections they make between them. By analysis of a series of case studies, we compare the approaches adopted by students working through parallel task sequences, which integrate different computer tools in different ways. Finally, we make suggestions as to how students might be encouraged to exploit visual reasoning alongside the symbolic and draw out implications for curriculum design.     

Visual and Symbolic Reasoning in Mathematics: Making Connections With Computers?

In this article, we analyze the visual and symbolic strategies developed by students to express generalizations of number patterns and the connections they make between them. By analysis of a series of case studies, we compare the approaches adopted by students working through parallel task sequences, which integrate different computer tools in different ways. Finally, we make suggestions as to how students might be encouraged to exploit visual reasoning alongside the symbolic and draw out implications for curriculum design.     

“Rising to the challenge”: using generalization in pattern problems to unearth the algebraic skills of talented pre-algebra students

This study focuses on the generalization methods used by talented pre-algebra students in solving linear and non-linear pattern problems. A qualitative analysis of the solutions of three problems revealed two approaches to generalization: recursive–local and functional–global. The students showed mental flexibility, shifting smoothly between pictorial, verbal and numerical representations and abandoning additive solution approaches in favor of more effective multiplicative strategies. Three forms of reflection aided generalization: reflection on commonalities in the pattern sequence’s structure, reflection on the generalization method, and reflection on the “tentative generalization” through verification of the pattern sequence. The latter indicates an intuitive grasp of the mathematical power of generalization. The students’ generalizations evinced algebraic thinking in the discovery of variables, constants and their mutual relations, and in the communication of these discoveries using invented algebraic notation. This study confirms the centrality of generalizations in mathematics and their potential as gateways to the world of algebra.     

On a Generalized Chu–Vandermonde Identity

In the present paper we introduce a generalization of the well–known Chu–Vandermonde identity. In particular, by inductive reasoning, the identity is extended to a multivariate setup in terms of the fourth Lauricella function. The main interest in such generalizations derives from the species diversity estimation and, in particular, prediction problems in Genomics and Ecology within a Bayesian nonparametric framework.     

Recursive and explicit rules: How can we build student algebraic understanding?

Differing perspectives have been offered about student use of recursive and explicit rules. These include: (a) promoting the use of explicit rules over the use of recursive rules, and (b) encouraging student use of both recursive and explicit rules. This study sought to explore students’ use of recursive and explicit rules by examining the reasoning of 25 sixth-grade students, including a focus on four target students, as they approached tasks in which they were required to develop generalizations while using computer spreadsheets as an instructional tool. The results demonstrate the difficulty that students had moving from the successful use of recursive rules toward explicit rules. In particular, two students abandoned general reasoning, instead focusing on particular values in an attempt to construct explicit rules. It is recommended that students be encouraged to connect recursive and explicit rules as a potential means for constructing successful generalizations.     

A Generalization of Commutativity Degree of Finite Groups

The commutativity degree of a finite group is the probability that two arbitrarily chosen group elements commute. This notion has been generalized in a number of ways. The object of this article is to study yet another generalization of the same notion, which further extends some of the existing generalizations.     

Generalizations of the centroid with an application in stochastic geometry

The centroid of a subset of with positive volume is a well‐known characteristic. An interesting task is to generalize its definition to at least some sets of zero volume. In the presented paper we propose two possible ways how to do that. The first is based on the Hausdorff measure of an appropriate dimension. The second is given by the limit of centroids of ε‐neighbourhoods of the particular set when ε goes to 0. For both generalizations we discuss their existence and basic properties. Then we focus on sufficient conditions of existence of the second generalization and on conditions when both generalizations coincide. It turns out that they can be formulated with the help of the Minkowski content, rectifiability, and self‐similarity. Since the centroid is often used in stochastic geometry as a centre function for certain particle processes, we present properties that are needed for both generalizations to be valid centre functions. Finally, we also show their continuity on compact convex m‐sets with respect to the Hausdorff metric topology.     

Is it a function? Generalizing from single- to multivariable settings

Generalizing is a hallmark of mathematical thinking. The term ‘generalization’ is used to mean both the process of generalizing and the product of that process. This paper reports on five calculus students’ generalizing activity and what they generalized about multivariable functions. The study makes two contributions. The first is a fine-grained, actor-oriented characterization of the ways undergraduates generalized. This adds to knowledge in two areas: the use of the actor-oriented perspective and generalization in advanced mathematics. The second contribution is the products of students’ generalizing: what they generalized about what it means for a multivariable relation to represent a function). This adds to the literature about student reasoning regarding multivariable topics by characterizing the powerful ways of reasoning students possess pre-instruction.     

Reinventing solutions to systems of linear differential equations: A case of emergent models involving analytic expressions

An enduring challenge in mathematics education is to create learning environments in which students generate, refine, and extend their intuitive and informal ways of reasoning to more sophisticated and formal ways of reasoning. Pressing concerns for research, therefore, are to detail students’ progressively sophisticated ways of reasoning and instructional design heuristics that can facilitate this process. In this article we analyze the case of student reasoning with analytic expressions as they reinvent solutions to systems of two differential equations. The significance of this work is twofold: it includes an elaboration of the Realistic Mathematics Education instructional design heuristic of emergent models to the undergraduate setting in which symbolic expressions play a prominent role, and it offers teachers insight into student thinking by highlighting qualitatively different ways that students reason proportionally in relation to this instructional design heuristic.     

Relations between Types of Reasoning and Computational Representations

This paper examines the idea that particular representations differentially support and enhance different cognitive processes, in particular different types of reasoning. Five case studies were conducted consisting of detailed observations of pairs of middle-school students interacting with a computer-based learning environment. The software environment, called NumberSpeed, deals with kinematics concepts by having students construct various motion scenarios by adjusting numerical motion parameters: position, velocity and acceleration. NumberSpeed provides feedback about the student-specified motion using two representations: the motion representation and the number-lists representation. Two distinct types of reasoning were recognized in students’ learning while interacting with NumberSpeed: (1) model-based reasoning and (2) constraint-based reasoning. These two types of reasoning are characterized in detail and their roles in problem-solving are analyzed. A cross-analysis between the types of reasoning and the use of particular NumberSpeed representations reveals a correlation between type of reasoning and representational choice. These findings are explained by analyzing the representations’ characteristics and the ways they may differentially support and enhance particular types of reasoning.     

Generalization and Justification: The Challenge of Introducing Algebraic Reasoning Through Patterning Activities

The expectation that students be introduced to algebraic ideas at earlier grade levels places an increased burden on the classroom teacher to help students construct and justify generalizations. This study provides insight into the reasoning of 25 sixth-grade students as they approached patterning tasks in which they were required to develop and justify generalizations while using computer spreadsheets as an instructional tool. The students demonstrated both the potential and pitfalls of such activities. During whole-class discussions, students were generally able to provide appropriate generalizations and justify using generic examples. Students who used geometric schemes were more successful in providing general arguments and valid justifications. However, during small-group discussions, the students rarely justified their generalizations, with some students focusing more on particular values than on general relations. It is recommended that the various student strategies and justifications be brought to the forefront of classroom discussions so that students can examine the mathematical power and validity of the various strategies and justifications typically introduced by students.     

Requiem for the Miller–Tucker–Zemlin subtour elimination constraints?

The Miller–Tucker–Zemlin (MTZ) Subtour Elimination Constraints (SECs) and the improved version by Desrochers and Laporte (DL) have been and are still in regular use to model a variety of routing problems. This paper presents a systematic way of deriving inequalities that are more complicated than the MTZ and DL inequalities and that, in a certain way, “generalize” the underlying idea of the original inequalities. We present a polyhedral approach that studies and analyses the convex hull of feasible sets for small dimensions. This approach allows us to generate generalizations of the MTZ and DL inequalities, which are “good” in the sense that they define facets of these small polyhedra. It is well known that DL inequalities imply a subset of Dantzig–Fulkerson–Johnson (DFJ) SECs for two-node subsets. Through the approach presented, we describe a generalization of these inequalities which imply DFJ SECs for three-node subsets and show that generalizations for larger subsets are unlikely to exist. Our study presents a similar analysis with generalizations of MTZ inequalities and their relation with the lifted circuit inequalities for three node subsets.     

Commentary: Linking epistemology and semio-cognitive modeling in visualization

To situate the contributions of these research articles on visualization as an epistemological learning tool, we have employed mathematical, cognitive and functional criteria. Mathematical criteria refer to mathematical content, or more precisely the areas to which they belong: whole numbers (numeracy), algebra, calculus and geometry. They lead us to characterize the “tools” of visualization according to the number of dimensions of the diagrams used in experiments. From a cognitive point of view, visualization should not be confused with a visualization “tool,” which is often called “diagram” and is in fact a semiotic production. To understand how visualization springs from any diagram, we must resort to the notion of figural unity. It results methodologically in the two following criteria and questions: (1) In a given diagram, what are the figural units recognized by the students? (2) What are the mathematically relevant figural units that pupils should recognize? The analysis of difficulties of visualization in mathematical learning and the value of “tools” of visualization depend on the gap between the observations for these two questions. Visualization meets functions that can be quite different from both a cognitive and epistemological point of view. It can fulfill a help function by materializing mathematical relations or transformations in pictures or movements. This function is essential in the early numerical activities in which case the used diagrams are specifically iconic representations. Visualization can also fulfill a heuristic function for solving problems in which case the used diagrams such as graphs and geometrical figures are intrinsically mathematical and are used for the modeling of real problems. Most of the papers in this special issue concern the tools of visualization for whole numbers, their properties, and calculation algorithms. They show the semiotic complexity of classical diagrams assumed as obvious to students. In teaching experiments or case studies they explore new ways to introduce them and make use by students. But they lie within frameworks of a conceptual construction of numbers and meaning of calculation algorithms, which lead to underestimating the importance of the cognitive process specific to mathematical activity. The other papers concern the tools of mathematical visualization at higher levels of teaching. They are based on very simple tasks that develop the ability to see 3D objects by touch of 2D objects or use visual data to reason. They remain short of the crucial problem of graphs and geometrical figures as tools of visualization, or they go beyond that with their presupposition of students' ability to coordinate them with another register of semiotic representation, verbal or algebraic.     

Models and numerical schemes for generalized van der Pol equations

This paper presents three generalizations of the van der Pol equation (VDPE) using newly proposed three new generalized K-, A- and B-operators. These operators allow kernel to be arbitrary. As a result, these operators provide a greater generalization of the VDPE than the fractional integral and differential operators do. Like the original VDPE, the generalized van der Pol equations (GVDPEs) are also nonlinear equations, and in most cases, they can not be solved analytically. Numerical algorithms are presented and used to solve the GVDPEs. Results for several examples are presented to demonstrate the effectiveness of the numerical algorithms, and to examine the behavior of the GVDPEs and the limit cycles associated with them. Although the numerical algorithms have been used to solve the GVDPEs only, they can also be used to solve many other generalized oscillators and generalized differential equations. This will be considered in the future.     

A generalized Malfatti problem

Malfatti?s problem, first published in 1803, is commonly understood to ask fitting three circles into a given triangle such that they are tangent to each other, externally, and such that each circle is tangent to a pair of the triangle?s sides. There are many solutions based on geometric constructions, as well as generalizations in which the triangle sides are assumed to be circle arcs. A generalization that asks to fit six circles into the triangle, tangent to each other and to the triangle sides, has been considered a good example of a problem that requires sophisticated numerical iteration to solve by computer. We analyze this problem and show how to solve it quickly.     

Diffusion maps tailored to arbitrary non-degenerate Itô processes

We present two generalizations of the popular diffusion maps algorithm. The first generalization replaces the drift term in diffusion maps, which is the gradient of the sampling density, with the gradient of an arbitrary density of interest which is known up to a normalization constant. The second generalization allows for a diffusion map type approximation of the forward and backward generators of general Itô diffusions with given drift and diffusion coefficients. We use the local kernels introduced by Berry and Sauer, but allow for arbitrary sampling densities. We provide numerical illustrations to demonstrate that this opens up many new applications for diffusion maps as a tool to organize point cloud data, including biased or corrupted samples, dimension reduction for dynamical systems, detection of almost invariant regions in flow fields, and importance sampling.