Description:

Co-Principal Investigator: Jennifer Kaminski (Wright State University)

Purpose: Mathematical knowledge is crucial for science, engineering, economics, and many other fields of study, as well as many aspects of everyday life; however, even when students successfully learn mathematical concepts, they often have difficulty transferring their knowledge to new situations and at later points in time. One approach to teaching mathematics to facilitate both learning and transfer has been the use of concrete instantiations, including visual displays, as well as real world contextualized examples. The goal of this project is to understand how the different ways in which concepts are instantiated throughout learning affect transfer across domains, contexts, and time.

Project Activities: In order to address how the instantiation of concepts affects mathematics learning and transfer, the research team will conduct three studies. Study 1 will investigate elementary school students’ acquisition of basic fraction arithmetic and Study 2 will investigate middle school students’ acquisition of basic probability. Studies 1 and 2 will each include four experiments. In addition to varying the instructional format (i.e., the extent to which visual representations are included in instruction), these experiments will also test transfer by varying the differences between learning and testing. In some experiments, the contexts of learning differ (i.e., one-on-one or group); in some the locations of learning and testing differ (i.e., in the student’s school or at the university campus); and in some the domains of testing will differ (i.e., numbers, everyday word problems, or scientific topics). Study 3 will consist of aggregated analyses of data collected in Studies 1 and 2. In addition, for Study 3, researchers will compare the relative contributions of instructional format and student ability, including mathematics achievement and components of executive function (working memory and inhibition).

Products: The products of this project will be preliminary evidence of potentially promising instructional practices and peer reviewed publications.

Structured Abstract

Setting: The majority of the data collection will take place at public and private elementary and middle schools in an urban area in Ohio. Some data collection will take place at the research team’s laboratory, located on a university campus in Ohio.

Sample: For each experiment, about 40 students will be recruited per condition. This will result in approximately 1440 students for the entire project. Half of the participants will be third grade students and the other half will be sixth grade students. The sample will include schools with a wide range of ethnic and racial composition, ranging from schools with over 90% African American students to schools with mostly White students. The sample will have approximately equal numbers of female and male students.

Intervention: Because this is an Exploration project, no intervention will be developed. Instead, this project will seek to understand how the different ways in which concepts are instantiated throughout learning affect inter-domain transfer as well as transfer across contexts and across time.

Research Design and Methods: This project consists of three studies. Studies 1 and 2 will be conducted in parallel across the first three years of the project. Study 1 will investigate elementary school students’ acquisition of basic fraction arithmetic and Study 2 will investigate middle school students’ acquisition of basic probability. Each of these studies includes four experiments, and in each experiment, participants will be randomly assigned to one of three between-subjects conditions that vary the level to which instruction is grounded in visual representations (no grounding, minimal grounding, and protracted grounding). Each experiment will have four sessions. Session 1 includes a pre-test and initial instruction, session 2 (one week after session 1) includes a test of what students retained from session 1 as well as additional instruction, session 3 (2 months after session 2) includes a post-test, and session 4 includes a series of assessments to measure individual factors (mathematics achievement and components of executive function). Within both Study 1 and Study 2, the four experiments will address 1) initial learning and transfer across time; 2) transfer across time, domain, and location; 3) the effects of group instruction; and 4) allocation of attention (through eye-tracking). After completing Studies 1 and 2, the research team will conduct Study 3, which will consist of aggregated analyses of data collected in Studies 1 and 2. The goal of Study 3 is to establish which instructional formats are more effective than others. An additional goal of Study 3 is to compare the relative contributions of instructional format and student ability, including mathematics achievement and components of executive function (working memory and inhibition).

Control Condition: There is no formal control condition. Instead, conditions will vary the level to which instruction is grounded in visual representations (no grounding, minimal grounding, and protracted grounding).

Key Measures: For Studies 1 and 2, pre-tests and post-tests will be developed by the research team. Within these studies, the tests for experiments 1 and 4 will include open-ended questions and contextualized real-world word problems in a paper and pencil format. For experiments 2 and 3, the post-tests will consist of a set of questions that are set in the context of climate change (Study 1) or environmental pollution (Study 2). Experiment 4 will use eye-tracking measures, specifically looking time and scan path data, to explore the allocation of participants’ attention. Mathematics achievement will be measured with various math-based subtests of the Woodcock Johnson III Test of Achievement. Working memory will be measured using a numeric Stroop task, in which strings of numbers of symbols are presented on a computer screen and participants must name the quantity of items in each string as quickly as possible. Inhibition will be measured using the Counting Span task. For this task, participants will be shown cards that display between one and nine green spots and between one and nine red spots on a computer screen and will be told that either green or red is their target color. Participants will be instructed to count the number of items of the target color on the card.

Data Analytic Strategy: For Studies 1 and 2, the research team will use general linear modeling to analyze the pre-test and post-test scores across conditions to determine effects of instruction on learning and transfer. For the eye-tracking data, researchers will use analysis of variance to consider differences between conditions in the mean looking times to particular information as well as scan paths. Machine learning classifiers will determine the subset of eye-tracking variables that best predict the participants’ training condition. For Study 3, the research team will use structural equation modeling (SEM). The SEM model will capture effects of the three conditions on measures of transfer as well as effects of the covariate variables, mathematics achievement, working memory, and inhibition.

Products and Publications

Journal article, monograph, or newsletter

Deng, W., and Sloutsky, V. M. (2016). Selective attention, diffused attention, and the development of categorization. Cognitive Psychology, 91: 24–62.

O'Leary, A. P. and Sloutsky, V. M. (2017). Carving Metacognition at Its Joints: Protracted Development of Component Processes. Child Development, 88(3): 1015–1032.

Plebanek, D., J., and Sloutsky, V. M. (2017). Costs of selective attention: When children notice what adults miss. Psychological Science, 28: 723–732.

Proceeding

Kaminski, J. A. (2017). A transfer advantage of learning diagrammatic representations of mathematics. In Proceedings of the 39th Annual Meeting of the Cognitive Science Society.

Lee, S. and Sloutsky, V.M. (2015). Independent Recognition of Numerosity Requires Attention. In Proceedings of the 37th Annual Meeting of the Cognitive Science Society (pp. 1279–1284). Pasadena, CA: Cognitive Science Society.

Smalley, E., and Kaminski, J. A. (2017). Gender or Community: What Drives STEM Interest Among Middle School Students?. In Proceedings of the 39th Annual Meeting of the Cognitive Science Society.