Examining Teaching-Learning Associations: Theory and Measurement for Conceptual Understanding in Grades 4 Through 8 Mathematics

• In Event: Connections Between Inquiry-Based Teacher Practices and Student Outcomes

Abstract

Purpose
We replicate analyses to further support our theory and large-scale measures of teaching in mathematics for conceptual understanding (Stein et al, 2017). We examine teaching-learning associations in a volunteer sample of 4th through 8th grade teachers.
Theoretical Framework

We examine a quadrant theory for mathematics teaching defined by a 2-by-2 matrix for the interaction of high and low levels of two constructs that past research has shown to influence students’ development of conceptual understanding - explicit attention to concepts (EAC) and students’ opportunity to struggle (SOS). Our validity argument seeks to confirm that teaching practices on self-reported surveys fit patterns defined by each of the quadrants, especially the off-diagonals (high conceptual instruction with low struggle, and low conceptual instruction with high struggle), which represent different types of partial implementation of teaching practices known to influence students’ conceptual understanding. Furthermore, our validity argument examines a priori conjectures for associated patterns of student learning.

Data Sources/Methods
We administered surveys (n = 811) and constructed response assessments (hereafter referred to as CRAs; n = 203) to 4th through 8th grade teachers in Tennessee and linked these data to state administrative files with standardized test scores (TCAP). We examined a confirmatory factor analysis utilizing a 2nd order factor structure using MPlus (v7.2) for our two main constructs – EAC (4 sub-constructs) and SOS (3 sub-constructs). We also generated classroom value-added scores from two level covariate-adjusted models for each learning outcome – the TCAP (n = 656 teachers) and CRAs (n = 201 teachers). Using a set of unique survey items, including vignettes of teaching and only items not used in the prior CFA, we examined a latent profile analysis to place teachers into one of four quadrants. Using these quadrant placements, we examined three different ANOVAs to understand: 1) patterns of EAC and SOS within each quadrant; 2) patterns of achievement on TCAP; and 3) patterns of achievement on CRAs.
Results
The CFA revealed desirable psychometric properties for our survey measures of EAC and SOS (χ^2(df)=1127.88 (397), CFI=.913, TLI=.905, RMSEA=.045, SPMR=.046). There were significant differences between quadrants in patterns of EAC [F(3,803)=58.53, p=.000] and SOS [F(3,803)=107.81,p=.000]. Finally, there were significant differences between quadrants on the TCAP as an outcome [F(3,652)=3.03, p=.029], and marginally significant differences (p<.10) between quadrants on the CRA as an outcome [F(3,197)=2.26, p=.083]. Post-hoc analyses reveal some significant differences between diagonal and off-diagonal quadrants that support the overall theory, demonstrating the relative strength of high conceptual instruction on TCAP versus CRA; and vice-versa for high struggle, student-centered, instruction.
Significance
In general, our findings replicate our validity argument. We discuss implications including the potential for measures at scale that could be obtained at a reasonable cost; as well as constructs that are interpretable and embedded in a theory of teaching and learning that could inform professional learning. Finally, we discuss limitations, including the relative difficulty distinguishing between two different types of partial implementation when teaching for students’ conceptual understanding in mathematics (off-diagonals).

Authors

• Richard J. Correnti, University of Pittsburgh

• Mary Kay Stein, University of Pittsburgh

• Bilge Yurekli, University of Pittsburgh