Characterizing reasoning about fraction arithmetic of middle grades teachers in three latent classes

ABSTRACT

We analyzed item responses provided by a nationwide sample of 990 in-service U.S. middle grades mathematics teachers to a novel survey focused on fraction arithmetic. The survey targeted four components of reasoning in terms of measured quantities: Referent units, Reversibility, Partitioning and iterating, and Appropriateness. Using the mixture Rasch model, a psychometric model that combines unidimensional scaling with classification, we detected 3 latent classes. To understand what distinguished the performance of teachers in the three classes, we analyzed their responses to each item. One main result is that the performance of teachers across the three classes was more clearly distinguished on items targeting Reversibility and Partitioning and iterating than it was on items targeting Referent units and Appropriateness. A second main result is that the three classes formed a nested structure that aligned with good performance on more and more components of reasoning targeted by the survey items.

KEYWORDS:

• Fraction arithmetic
• latent classes
• mathematical knowledge of teachers

Acknowledgement

This study was part of the first author’s dissertation. We wish to thank committee members Allan S. Cohen and Sybilla Beckmann for advice they provided throughout the study.

Ethics

The authors confirmed that all original research procedures were consistent with the principles of the research ethics published by the American Psychological Association.

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Supplementary material

Supplemental data for this article can be accessed on the publisher’s website.

Correction Statement

This article has been republished with minor changes. These changes do not impact the academic content of the article.

Notes

1. Our use of the term measured quantity is similar to Thompson’s (2010).

2. Traits like knowledge of mathematics are termed latent, because they cannot be observed directly.

3. The red dot indicates Weighted Likelihood Estimation and the blue dot indicates Maximum Likelihood Estimation. These are the two estimation techniques that WINMIRA program provides and, as the figure shows, their estimations are nearly identical.

4. We used the probability of .90 or higher as the boundary for latent class membership because Izsák et al. (2012) reported 100% classification consistency for teachers who had a probability of .90 or higher. Specifically, Izsák et al. (2012) reported that, for teachers who had a probability of .90 or higher, classification was 100% consistent whether based on all items from the adapted LMT instrument, a subset of items used for a pretest, or a different subset of items used for a posttest. Therefore, we hypothesized that teachers with the probability of .90 or higher for being in a latent class will demonstrate the characteristics of that latent class.

5. Within each latent class, item difficulties sum to 0. This allows us to compare item difficulties within classes but not between classes.

6. As shown in Appendix B, trends in graphs for both the original 990 and the 649 teachers were almost identical across the three classes.

7. Item 18 measures a second sub-component, which is Partitioning in stages. One exception to this trend occurred in strongly classified Class-A teachers who had a higher percentage of responding correctly to item 18. This performance might be due to Class-A teachers’ performing well on an item that measures Partitioning in stages.

8. Item 22 measures a second sub-component, which is Partitioning using common denominators. One exception to this trend occurred in strongly classified Class-A teachers who had a higher percentage of responding correctly to item 22. This performance might be due to Class-A teachers’ performing well on an item that measures Partitioning using common denominators.

Additional information

Funding

The development of the Diagnosing Teacher’s Multiplicative Reasoning (DTMR) Fractions survey was supported by the National Science Foundation under Grant No. DRL-0903411. The opinions expressed are those of the authors and do not necessarily reflect the views of NSF.

Notes on contributors

İbrahim Burak Ölmez

İbrahim Burak Ӧlmez is interested in applications of psychometrics to mathematics education, including investigating students’ and teachers' reasoning about fraction arithmetic, proportional relationships, multiplication, and division.

Andrew Izsák

Andrew Izsák is interested in how students and teachers reason about topics related to multiplication (including fraction arithmetic, proportional relationships, and linear equations). He is especially interested in how they make use of inscriptions (drawings) and resources for reasoning about quantities.