Nicole

Back to Psych 204B Projects 2012

Introduction

Broad Context

Instructional methods influence the likelihood that an individual will learn either a symbolic rule, its relation to the referent, or both. If students are provided with the rule and later shown example situations in which it applies, they may not fully understand the problem space. In contrast, inventing a symbolic representation based on the context provides students with a better sense of the relationships among variables (Schwartz, Chase, Oppezzo, & Chin, 2011; Star & Rittle-Johnson, 2007). Alternatively, students may not know why operations are used in the ways that they are. To teach formulas, many typical instructional methods rely on clear lectures. These can effectively guide students toward procedural fluency. However, this type of instruction does not always emphasize the relationships between the symbolic notation and the referenced context. Consequentially, students may perform poorly in new contexts, failing to transfer or showing evidence of negative transfer, emphasizing their inability to understand the limits of problem space. Additionally, they may exhibit only a brittle understanding of the rules and their meanings (Lehrer, Schauble, Carpenter, & Penner, 2000). Even more, people may attempt to develop connections to the referent but fail to do so if they already have the formula to fall back on (Schwartz, Chase, Oppezzo, & Chin, 2011). With my colleagues, I hypothesize that with only the referent or only symbolic understanding, knowledge would not transfer to new contexts.

Prior behavioral piloting has shown that learners who search for a formula are less likely to overgeneralize (negatively transfer) the formula to new contexts than their counterparts who are told the formula directly.

This study explores the role of the mental model, comparing different ways to integrate knowledge: using two representations versus only one. Individuals who receive direct instruction about the formula are compared to those who learn both the formula and a spatial strategy that ties it to the referent. With this design, this study compares purely symbolic knowledge to an integrated understanding of multiple aspects of the relationship. It is predicted that people who have both an algebraic and spatial representation of the problem will be more flexible in applying either strategy and will show neurological overlap between the two types of problem solving. Additionally, individual differences in neurological signatures among individuals who successfully generalize the formula and those who find this task difficult may provide insight into the effects of understanding the spatial referent of the formula.

Materials

All participants solved the Polygon Problem, a growth pattern problem in which the solver’s task is to determine the perimeter of a row of regular polygons arranged in a singe line, as in Figure 1. These figures consisted of shapes ranging from 3 sides (triangles) to 6 sides (hexagons) in rows of 3 to 10 shapes. The relationship between the total perimeter and these two variables can be expressed in various formulae which simplify to this canonical form: Perimeter = (s-2)n + 2 , where s represents the number of sides per polygon and n represents the number of polygons in a row. The Polygon Problem has been used in professional development programs with teachers as an example of growth pattern problem that allows for a general abstract solution to be built from a range of possible contexts (Koellner et al., 2007).

Trial Paradigm

The trial procedure is shown in the figure below. To ensure that potential differences in brain activations to the different conditions would not simply be an effect of different visual stimuli, the presentation of each problem was the same for all trials. As shown below, this screen consisted of the graphic representation of the problem, the formula, and the values of the variables ‘s’ and ‘n’ such that it provided all of the necessary information to use the formula or use the spatial strategy. In each trial, participants clicked a button to indicate that they had solved the problem. On the next screen, participants used the trackball to scroll to their answer.

The study timing was self-paced, a method that has been used in other studies of cognitive processing (e.g. Kalbfleish, VanMeter, & Zeffiro, 2006). The interstimulus interval (ISI) was jittered between each trial.

Blocks consisted of 24 trials with varying lengths and types of shapes presented. Each block lasted approximately six to eight minutes depending on the participants’ efficiency in working through the problems.

Study Design

In this study, participants received training about the Polygon Problem prior to entering the scanner, solved several blocks of problems in the scanner, and were tested on transfer questions after scanning. The study included a between subjects manipulation of instruction. In the Formula + Spatial condition, participants built up the formula from the referent, learning both the formula and an analogous spatial strategy involving skip-counting along the figure to geometrically represent the formula. The Formula Only condition simply learned the formula to be able to do the task but did not learn about its relationship to the spatial referent. This design is outlined below.

In the present investigation, data from seven participants in the Formula Only condition are examined. In particular, I considered data from the first two scanner blocks, during which the participants are using the formula to solve the problems.

The present investigation: Subject Motion

(Example of subject motion graph(s).)

Methods

Subjects

Sixteen right-handed participants were recruited from the Stanford Psychology Participant Pool to participate in a functional MRI study of the Polygon Problem. The age range of participants was constrained to 18 to 40 years of age as we observed that some older participants had difficulty with simple math facts in the pilot study. Nine of the participants were male and the average age of participants was 23.2 years of age.

The dataset selected for the present study is comprised of seven healthy volunteers who participated in the Formula condition of the overall study. Their mean age was 23.8 years old and four of these participants were male.

MR acquisition

Data were obtained on a 3T GE scanner at Stanford's Center for Cognitive and Neurobiological Imaging (CNI)

MR Analysis

The MR data was analyzed using SPM software tools in MATLAB: [1]. To identify and correct motion outliers, the Artifact Repair toolbox ([| ArtRepair]) was used in conjunction with SPM.

Pre-processing

All data were slice-time corrected, motion corrected, and repeated scans were averaged together to create a single average scan for each subject.

MNI space

MNI is an abbreviation for Montreal Neurological Institute.

After a pRF model was solved for each subject, the model was trasnformed into MNI template space. This was done by first aligning the high resolution t1-weighted anatomical scan from each subject to an MNI template. Since the pRF model was coregistered to the t1-anatomical scan, the same alignment matrix could then be applied to the pRF model.
Once each pRF model was aligned to MNI space, 4 model parameters - x, y, sigma, and r^2 - were averaged across each of the 6 subjects in each voxel.

Group Level Analyses

No Motion Correction

Some text. Some analysis. Some figures.

Using Motion Correction

Some text. Some analysis. Some figures.

Dropping A Subject With Too Much Motion

Some text. Some analysis. Some figures. Maybe some equations.

Conclusions

Here is where you say what your results mean.

References - Resources and related work

References

Kalbfleisch, M.L., VanMeter, J.W., & Zeffiro, T.A. (2007). The influences of task difficulty and response correctness on neural systems supporting fluid reasoning. Cognitive Neurodynamics, 1: 71-84.

Lehrer, R., Schauble, L., Carpenter, S., & Penner, D. (2000). The interrelated development of inscriptions and conceptual understanding. In P. Cobb, E. Yackel., & K. McClain (Eds.) , Symbolizing and communication in mathematics classrooms (pp. 325 – 360). Mahwah, NJ: Lawrence Erlabaum Associates, Inc.

Schwartz, D.L., Chase, C.C., Oppezzo, M.A., & Chin, D.B. (2011). Practicing versus inventing with contrasting cases: The effects of telling first on learning and transfer. Journal of Educational Psycholgy, 103(4), 759-775.

Star, J.R., & Rittle-Johnson, B. (2008). Flexibility in problem solving: The case of equation solving. Learning and Instruction 18(6), 565-579.

Software

Art Repair

For more information about the SPM plugin ArtRepair, see: Art Repair