Inquiry-Based Learning and the Art of Mathematical Discourse

Abstract

Our particular flavor of inquiry-based learning (IBL) uses mathematical discourse, conversations, and discussions to empower students to deepen their mathematical thinking, building on strengths of students in the humanities. We present an organized catalog of powerful questions, discussion prompts, and talk moves that can help faculty facilitate a classroom focused on mathematical discourse. The paper brings this discourse alive through classroom vignettes and explores various teacher moves and their impacts. The mathematical theme of the classroom investigations, Maypole dance patterns, stems from the learning guide “Discovering the Art of Mathematics: Dance.” Both authors are part of the NSF-funded project “Discovering the Art of Mathematics,” which provides IBL materials for mathematics for liberal arts courses, see www.artofmathematics.org.     

Need-solution pair recognition by household sector individuals: Evidence,and a cognitive mechanism explanation

Problem-solving by everyday individuals is thought to occur as a two-step process. First, an individual identifies or formulates a problem, followed by entering into a subsequent search to find the best solution. Here, however, we consider an alternative process that everyday individuals may use for solution finding first theorized by von Hippel and von Krogh (2016). Specifically, von Hippel and von Krogh proposed that everyday individuals may sometimes discover a solution and the need it satisfies simultaneously without the need for apriori problem formation, a cognitive process they called “need-solution pair recognition”. Utilizing a rich literature from psychology and neuroscience, we propose that seemingly spontaneous discoveries found by need-solution pair recognition are natural products of the object recognition system and its underlying mechanisms. This view asserts that on encountering an object and reasoning how it might be used (i.e. functional object understanding), an individual's perception of an object may culminate in recognizing the object as a solution, and in some cases, as a solution to a problem previously unknown to him or her, thus bypassing formal problem-formulation and active solution searching entirely. To empirically test this view, we manipulated the ability of everyday individuals to functionally reason about objects while we examined the spontaneous occurrence of solutions found by either need-solution pair recognition or traditional problem-first problem-solving. Consistent with our hypothesized mechanism, our results indicate that need-solution pair recognition occurs more frequently when constraints on functional object understanding are reduced. That is, we found that needsolution pair discoveries outpaced solutions found from traditional problem solving, in environments with unfamiliar objects, where participants were not directed to solve specific problems. Our results provide clear evidence that everyday individuals in the household sector do not always innovate through traditional problem-solving processes, but instead may arrive at solutions as they recognize and reason about objects. Implications for research and practice in household innovation, and for innovation more generally are considered.     

Local Observed‐Score Kernel Equating

Three local observed‐score kernel equating methods that integrate methods from the local equating and kernel equating frameworks are proposed. The new methods were compared with their earlier counterparts with respect to such measures as bias—as defined by Lord's criterion of equity—and percent relative error. The local kernel item response theory observed‐score equating method, which can be used for any of the common equating designs, had a small amount of bias, a low percent relative error, and a relatively low kernel standard error of equating, even when the accuracy of the test was reduced. The local kernel equating methods for the nonequivalent groups with anchor test generally had low bias and were quite stable against changes in the accuracy or length of the anchor test. Although all proposed methods showed small percent relative errors, the local kernel equating methods for the nonequivalent groups with anchor test design had somewhat larger standard error of equating than their kernel method counterparts.     

Improving the Bandwidth Selection in Kernel Equating

We investigate the current bandwidth selection methods in kernel equating and propose a method based on Silverman's rule of thumb for selecting the bandwidth parameters. In kernel equating, the bandwidth parameters have previously been obtained by minimizing a penalty function. This minimization process has been criticized by practitioners for being too complex and that it does not offer sufficient smoothing in certain cases. In addition, the bandwidth parameters have been treated as constants in the derivation of the standard error of equating even when they were selected by considering the observed data. Here, the bandwidth selection is simplified, and modified standard errors of equating (SEEs) that reflect the bandwidth selection method are derived. The method is illustrated with real data examples and simulated data.     

Movement planning reflects skill level and age changes in toddlers

Kinematic measures of children's reaching were found to reflect stable differences in skill level for planning for future actions. Thirty-five toddlers (18-21 months) were engaged in building block towers (precise task) and in placing blocks into an open container (imprecise task). Sixteen children were retested on the same tasks a year later. Longer deceleration as the hand approached the block for pickup was found in the tower task compared with the imprecise task, indicating planning for the second movement. More skillful toddlers who could build high towers had a longer deceleration phase when placing blocks on the tower than toddlers who built low towers. Kinematic differences between the groups remained a year later when all children could build high towers.     

Practical Application of a Synthetic Linking Function on Small-Sample Equating

The synthetic function is a weighted average of the identity (the linking function for forms that are known to be completely parallel) and a traditional equating method. The purpose of the present study was to investigate the benefits of the synthetic function on small-sample equating using various real data sets gathered from different administrations of tests from a licensure testing program. We investigated the chained linear, Tucker, Levine, and mean equating methods, along with the identity and the synthetic functions with small samples (N = 19 to 70). The synthetic function did not perform as well as did other linear equating methods because test forms differed markedly in difficulty; thus, the use of the identity function produced substantial bias. The effectiveness of the synthetic function depended on the forms' similarity in difficulty.     

Systematic reduction of leg muscle activity throughout a standard assessment of running footwear

Purpose:This study aimed to investigate whether there is a systematic change of leg muscle activity,as quantified by surface electromyography(EMG),throughout a standard running footwear assessment protocol at a predetermined running speed.Methods:Thirty-one physically active adults(15 females and 16 males) completed 5 testing rounds consisting of overground running trials at a speed of 3.5 m/s.The level of muscle activity from 6 major leg muscles was recorded using surface EMG.The variables asse...     

A dynamic analytic approach to national innovation systems: The IC industry in Taiwan

The paper attempts to develop a mathematical model of the national innovation system (NIS) of Taiwan, particularly with regard to its integrated circuit (IC) industry. The method utilised is that of system dynamics (SD). Drawing on both questionnaire/in-depth interviews and computer simulation, the paper aims to increase insights into the dynamic processes of the Taiwanese IC industry system of innovation and understand the interdependence and interaction among capital flows, human resource flows, knowledge and technology flows, and product flows in the NIS. The paper concludes with some simulations of policy alternatives confronting the IC industry and the Taiwanese government.     

Combined Approach to Multi-Informant Data Using Latent Factors and Latent Classes: Trifactor Mixture Model

Although collecting data from multiple informants is highly recommended, methods to model the congruence and incongruence between informants are limited. Bauer and colleagues suggested the trifactor model that decomposes the variances into common factor, informant perspective factors, and item-specific factors. This study extends their work to the trifactor mixture model that combines the trifactor model and the mixture model. This combined approach allows researchers to investigate the common and unique perspectives of multiple informants on targets using latent factors and simultaneously take into account potential heterogeneity of targets using latent classes. We demonstrate this model using student self-rated and teacher-rated academic behaviors (N = 24,094). Model specification and testing procedures are explicated in detail. Methodological and practical issues in conducting the trifactor mixture analysis are discussed.