Assessing learning progression in the domain of fractions

Author: Bakir Haryanto

Haryanto, Bakir, 2019 Assessing learning progression in the domain of fractions, Flinders University, College of Education, Psychology and Social Work

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Rational numbers are an important domain of mathematics learning and one which many students find difficult to learn. The present research has developed and validated an assessment instrument to assess the progression of student learning in the domain of rational numbers, with an emphasis on fractions. The assessment was based on a cognitive model of fraction learning and was innovative in that it distinguished two essential dimensions of fraction knowledge, namely conceptual and procedural.

The research has developed a hypothetical model of the two-dimensional fraction learning progression based on existing research. The hypothetical learning progression was first validated in a qualitative study, carried out through cognitive interviews. The results from the interviews were used to evaluate and revise the learning progression, which was subsequently tested in a second study with 516 students from grades 7 to 9 in a junior high school in Bogor, Indonesia. The fraction learning progression was validated using Bayesian Network Analysis. Two Bayesian Networks models were developed. Model 1 was a single latent variable model, while Model 2 was a multiple hierarchical latent variables model. Model 2 was found to have a better fit with the students’ responses than Model 1 and had a number of innovative characteristics, such as incorporating the assumption of the hierarchical dependencies between the levels in the learning progression into a formal statistical model, measuring students’ competency for each level and performing pseudo-guessing item analysis.

A confirmatory analysis was developed through Bayesian Network item level analysis and student level analysis to validate the hypothesized fraction learning progression empirically. The analysis has resulted in a learning progression with 7 validated levels of conceptual and 7 validated levels of procedural knowledge. About 48% of the students were grouped at very low levels of conceptual knowledge, indicating that the Indonesian curriculum is ineffective in developing a conceptual understanding of fractions beyond part-whole. In the procedural knowledge dimension, about 50% of the students reached the goals of the Indonesian curriculum at grade 7. However, the remaining students had difficulties with both additive and multiplicative fraction operations and often misapplied the algorithms for addition to multiplication. There were substantial individual differences in the relationship between students’ conceptual and procedural knowledge but some important dependencies between conceptual and procedural knowledge were also identified.

The present research is innovative in the area of fraction assessment research, because it has developed the first two-dimensional fraction learning progression based on conceptual and procedural knowledge and also because it has included aspects of fraction knowledge that were missing from previous assessments. The two-dimensional learning progression provided more accurate profiles of students’ progression levels compared with previous research, thus making a significant contribution to research into fraction education. Finally, the development of Bayesian Networks Models has made a contribution to educational measurement research both in that it has validated a learning progression using item and student level analysis, and in that it has developed Bayesian Networks analyses of item difficulty, item discrimination, and pseudo-guessing.

Keywords: Learning Progression, Fractions, Bayesian Networks, Item Analysis

Subject: Education thesis

Thesis type: Doctor of Philosophy
Completed: 2019
School: College of Education, Psychology and Social Work
Supervisor: Professor Stella Vosniadou