Investigating Reversibility (InReV) Project

Ibrahim Burak Olmez & Michael Lawson

June 1, 2020

Regarding persistent difficulties that teachers and their students experience in reasoning about fractions, there is an urgent need to understand such difficulties. Investigating Reversibility (InReV) is a research study that investigates elementary/middle grades (K-8) mathematics teachers’ understanding of reversibility, a central component of reasoning about fractions (e.g., Izsák, Jacobson, & Bradshaw, 2019). Reversibility can be defined as reconstructing the whole from either a proper or an improper fraction. The purpose of the present study is to identify distinct latent classes of U.S. mathematics teachers on reasoning about reversibility. Administering a measure that targets only reasoning about reversibility to U.S. in-service mathematics teachers and analyzing the national sample data using a mixture item response theory model will provide finer-grained information about particular strengths and weaknesses of teachers in terms of reasoning about reversibility. Furthermore, this study will examine the relationships between latent class membership and aspects of teachers’ professional preparation and histories teacher such as years of teaching experience. This, in turn, will inform about teachers with which particular aspects of professional preparation and histories are more likely to reason about reversibility. This study will administer a survey that consists of demographic information questions about teachers’ professional preparation and histories, and items that target reasoning about fractions from the literature (e.g., Izsák et al., 2019, Wilkins, Norton, and Boyce, 2013). Currently, we are collecting data from a national sample of 250 mathematics teachers across the United States.

Mathematics Teachers’ Gender-Specific Beliefs About Mathematical Aptitude

Yasemin Copur-Gencturk, David Quinn, & Ian Thacker

June 16, 2019

Persisting stereotypes and beliefs that certain disciplines require innate ability and individuals from different ethnic backgrounds or genders have different ability levels seriously impede students’ academic career paths. In this study, we examined mathematics teachers’ beliefs about mathematical aptitude and identified teacher characteristics associated with these beliefs. An analysis of data from 434 K-8 teachers indicated that teachers did not see mathematical aptitude and girls’ mathematical ability as two different constructs. Those who believed mathematics requires brilliance also thought girls have less on average than boys. Although most teachers did not believe that mathematics requires innate ability, teachers who were more experienced and who worked with students at risk seem to hold beliefs that could impede their students’ career paths. This research study was presented at the American Education Research Association (AERA) in Toronto and a manuscript is under review.

The Role of Warm Constructs in Fraction Learning

Ian Thacker, Jessica Rodrigues, & Gale Sinatra

May 20, 2019

Mathematics teaching and learning does not occur in an emotional vacuum. The extent to which people engage in and learn difficult content is influenced by motivational and emotional factors, also called warm constructs. However, there is limited research that investigates the extent to which warm constructs predict mathematics learning outcomes. The purpose of this study was to investigate the role of motivational and emotional factors during teachers’ mathematics learning about fraction arithmetic. Ian Thacker and Jessica Rodriguez found that teachers’ math-specific beliefs about their ability to succeed (self-efficacy) was associated with greater learning about fraction arithmetic while their mathematics-anxiety was not. This initial exploration provides promising evidence that emotional and motivational constructs are indeed important predictors of mathematical conceptual change. Teacher educators might therefore target improving teachers’ mathematics self-efficacy in order to improve learning outcomes.

Research Update from Yasemin Copur-Gencturk

Proficiency in Teaching Mathematics Study

Yasemin Copur-Gencturk, Jessica Rodrigues, and Shauna Campbell

Proficiency in teaching mathematics has several important components: content knowledge, mathematical knowledge for teaching, pedagogical content knowledge, and the ability to notice important aspects of the mathematics instruction. Although scholars have focused on some of these components, no research to date has investigated how all these different constructs are related to one another. In this study, we aimed to fill this gap in our field. We investigated the relationships among these constructs by collecting data from the same teachers on these key aspects of mathematics teaching proficiency. Furthermore, we refined what we meant by content knowledge. Rather than focusing only on what teachers know, we also captured how teachers reason mathematically and how teachers solve mathematical problems. We are extremely excited about this project because we mainly used open-ended items to capture these constructs!


This project had two phases. In the first phase, we developed items to capture teachers’ mathematical reasoning, conceptual understanding, and problem-solving skills. We also identified video clips that captured teachers’ noticing skills as well as their knowledge of students’ mathematical thinking and knowledge of instructional practices. During this phase, several prominent scholars (Andrew Izsak, Chandra Orrill, Randy Phillip, Erik Jacobson, and Sarah Lubienski), many of whom attended our first Initiative meeting, provided feedback on the items we developed. We then interviewed 20 teachers to test our items and partnered with the Qualtrics panel to recruit teachers from across the United States. We collected data from hundreds of teachers and are now analyzing our data.

So far, we have been able to analyze the data on teachers’ noticing. We wish to acknowledge Miriam Sherin, Beth Van Es, Randy Phillips, and Lisa Lamp, who provided feedback on how to conceptualize the key aspects of teachers’ noticing. We would also like to thank Nicole Kersting for letting us use her videos in this study. Our paper on teacher noticing, written by Copur-Gencturk, Rodrigues, and Campbell, will be the first large-scale study on teacher noticing and will be submitted to a journal soon. Here’s a sneak peak of our findings!

April 31, 2019

We are excited to share a cluster of Math Initiative research studies led by Yasemin Copur-Gencturk that have been making rapid progress. 

What are Teachers Noticing?

More than 500 elementary teachers across the U.S. watched brief videos of classroom math instruction. We asked them to list the most significant things they noticed about the math content of the lessons. 

Noticing Topic

Noticing Depth

Problem Solving Findings

Yasemin Copur-Gencturk and Tenzin Doleck

We also coded the data detailed above for teachers’ problem-solving strategies. We identified common strategies elementary school teachers used to solve word problems. This paper, which was written by Yasemin Copur-Gencturk and Tenzin Doleck, has been submitted for publication. We believe that it will inform teacher educators about how teachers deal with mathematics problems, which could reveal insights into their mathematical thinking. Here’s a sneak peak of our findings!

Teachers' Problem Solving Strategies

More than 300 fourth and fifth grade teachers across the U.S. solved four open-ended fraction problems. We examined the strategies that teachers use correctly to solve the problems.

For the problem below, we identified the following common strategies.

Suppose that Max, the Wonder Dog, ate 1/4 of a bag of treats on Sunday. Each night after that, Max ate 3/16 of the bag. How many nights (including Sunday) would it take to finish the whole bag?

Problem Solving



Examples of solutions using the most common strategies for the Max the Wonder Dog Problem.

We (Yasemin Copur-Gencturk and Tejvir Grewall) are also writing a paper in which we use two of the conceptual understanding items to examine the types of explanations teachers provided to explain the meaning behind algorithms for adding two fractions with unlike denominators and dividing two fractions. We have coded their explanations for these problems (we thank Sybilla Beckman for providing us with feedback) and are presently analyzing the data. 

Learning From Teachers Study

Yasemin Copur-Gencturk & Paul Lepe

Teachers gain invaluable experience over time regarding the struggles their students are having and strategies that are effective in helping them overcome such struggles. In this study, we aimed to update the present knowledge base regarding students’ struggles and effective teaching strategies. Specifically, we asked teachers to list the struggles they noticed their students having when learning fraction concepts and operations and the strategies they found useful in helping their students overcome these struggles. We collected data from almost 200 teachers. Findings will be presented at forthcoming education conferences and prepared for publication in the near future.

Mathematics Teachers' Implicit Race- and Gender -Biases

Yasemin Copur-Gencturk, Ian Thacker, Joe Cimpian, & Sarah Lubienski

April 14, 2019

Mathematics teachers can significantly affect students’ perceptions of their mathematical ability and future career choices. Hence, it is important for teachers’ assessments of students’ mathematical abilities to be free from bias. Still, most research conducted on teacher bias has failed to discern whether teachers are biased or if their assessments of their own students’ abilities are based on valid evidence not captured by researchers. In this experiment, 390 mathematics teachers evaluated 18 mathematical solutions to which gender- and race-specific names were randomly assigned. Teachers displayed no detectable bias when assessing the correctness of students’ solutions; however, they perceived White students’ ability to be higher, especially relative to Black and Latina girls. Surprisingly, non-White teachers displayed greater bias favoring White-sounding names. This research study was presented at the American Education Research Association (AERA) in Toronto, April 2019. 

An Instructional Approach for Remediating a Fraction Misconception

Jessica Rodrigues, Ian Thacker, & Gale Sinatra

March 14, 2019  

Our study explores a brief instructional approach for helping teachers overcome a mathematics misconception. We know from prior research that some teachers incorrectly overgeneralize the whole number rule of “multiplication always makes bigger” to multiplication with fractions. To our knowledge, our study is the first to evaluate an intervention for addressing this misconception. A sample of 100 in-service and pre-service elementary teachers completed a fractions task that assessed the presence of the misconception. Teachers were then randomly assigned to read either a refutation text that we designed to directly refute the misconception or a control text. Finally, teachers completed the fractions task again to assess for changes in their understanding. Results showed that (a) the misunderstanding was even present among items with familiar denominators such as halves and fourths, demonstrating the pervasiveness of the misconception and (b) the refutation text supported teachers in overcoming the misconception. Findings suggest that teachers have misconceptions that may be limiting their readiness to support their students’ mathematics understanding, and reading a short refutation text may be a powerful approach for remediating these misconceptions.

Epistemic Cognition and Mathematics Teaching and Learning

Ian Thacker, Gale Sinatra, & Richard Rasiej

March 14, 2019

Over the last three decades, there has been increased attention to the problem of how students should convince themselves and others that mathematical statements are true. For example, the Common Core State Standards considers the construction and evaluation of formal arguments to be an essential mathematical skill that cuts across all topics in mathematics. Cognitive processes involved in constructing and evaluating arguments — called epistemic cognition — has been well studied in the educational psychology literature. However, while much attention has been devoted to understanding the role of epistemic cognition in disciplines such as science and history, there is little research that explores its role in mathematics learning. The purpose of this literature review is to synthesize the existing work on epistemic cognition in mathematics teaching and learning and to highlight future directions for theoretical and empirical work. This research synthesis is under preparation and will be ready for submission by summer 2019.



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