In a nutshell, we LOVE learning and thinking about how people learn and think about math (very meta, we know!). How does mathematics knowledge develop over time? Why are so many people so afraid of math while others are so in love with it? Why do some people view math like a puzzle or game while others view them as confusing procedural rules and algorithms? Where do people who are "bad" at math get stuck? What are mathematical and contextual predictors of mathematics learning and achievement? What does good mathematics teaching look like? Why is teaching math so hard? What skills (other than math knowledge) are required to successfully teach and understand mathematics? These are just a few of the questions I try to answer in my research. 

My research aims to develop and evaluate classroom interventions that improve mathematics teaching and learning. My research is highly interdisciplinary and focuses on the intersections of educational, cognitive, and developmental psychology. My work is focused on the development, design, and testing of innovative classroom interventions and technologies that embed cognitive and educational principles of learning into everyday practice. I also use classroom observations, longitudinal data, and multi-level modeling to examine how mathematics and social-emotional learning (SEL) interventions in schools can enhance students’ opportunities to learn. Ultimately, I am interested in understanding how cognitive and non-cognitive pathways combine to produce learning and growth for all children in K-12 classrooms (and beyond!)

Over the past five years, my work has been supported by over $7.5 million in research funding from the Institute of Education Sciences, National Science Foundation, Spencer Foundation, Hewlett Foundation, and the American Educational Research Association.

My work explores the ways that instructional interventions that integrate perceptual learning, gesture, and embodied cognition can improve student learning and engagement in mathematics. Many leading curricular programs in mathematics and science have often used more traditional algorithms and static representations to teach algebraic notation; however, there is some reason to think that students may better recognize the construction of algebraic notation if they dynamically transform expressions using manipulation of physical objects. In theory, perceptual training may help build students understanding of algebraic concepts by internalizing the appropriate way of visualizing and gesturing patterns in algebraic structure. Symbols are presented as literal (sometimes virtual) objects, which children can touch and move, and which respond in natural, object like ways (constrained by mathematical law). This novel, dynamic program allows students to explore patterns and properties of mathematics by rearranging, splitting, and manipulating numbers and expressions. To date, much of my work has focused on building and designing several instantiations of the approach and coordinating across multiple representations for flexible use in classrooms by both teachers and students.  Working with an iterative design cycle has allowed me to evaluate what components of the intervention work best in applied classroom settings and has provided me great feedback about how to adapt the program to better fit the needs of teachers and students. My quantitative work provides evidence of the program’s efficacy and offers new and innovative insight into thinking about how we learn abstract concepts.

NSF ECR CORE GRANT

• Examining the Effects of Perceptual Cues on Middle School Students? Online Mathematical Reasoning and Learning

• Effective Years: 2023-2026

• Many students lack the necessary procedural and conceptual knowledge to progress beyond Algebra I. This limits their ability to pursue STEM interests, training, and careers. In this promising approach to giving support to learners, investigators will study the use various perceptual cues embedded in online math materials for middle school students in an effort to evaluate whether manipulating the visual perceptual cues in the materials, such as coloring and spacing, influence learning and performance by directing students’ attention towards key pieces of information during problem solving. The outcomes will provide an understanding of whether and how different types of perceptual cues work in order to provide low-cost, and effective instructional support in online math learning environments that will better prepare students for algebra and higher-level mathematics. It is proposed that viewing cues that highlight relevant mathematical structures will support immediate performance and learning, but that viewing cues that break the pattern or structure of the notations, in combination with training, will support greater retention.

• The purpose of this research is to implement two experimental studies that investigate whether and how congruent and incongruent perceptual cues within math notation influence middle school students’ order-of-operations performance, learning, and retention, and to understand how perceptual cues create desirable difficulties, or productive struggles, in mathematics practice. Study 1 will experimentally manipulate and test the individual effects of perceptual cues in math notation on middle school students’ performance on order-of-operations problems. Study 2 will use a 3×3 factorial design to systematically test the individual and synergistic effects of congruent and incongruent perceptual cues on students’ order-of-operations performance, learning, and retention. Additionally, both studies will investigate whether students’ prior knowledge, math anxiety, and perceptual processing skills moderate the effects of perceptual cues. This project will produce evidence of the roles of perceptual cues on students’ math performance, learning, and retention, including preliminary evidence of any differences in the effects of congruent vs. incongruent perceptual cues, and whether these variations in perceptual effects influence learning synergistically.

• This project is funded by the EDU Core Research (ECR) program, which supports fundamental research on STEM learning and learning environments, broadening participation in STEM, and STEM workforce development.

NSF BCSER GRANT 

• From lab to math classroom: Utilizing eye gaze and cognitive control tasks to examine the effects of perceptual cues and structure on mathematical performance

• Effective Years: 2023-2026

• Using perceptual cues in mathematical problem solving has been shown to improve students' performance. Attention and executive function skills are two potential mechanisms that explain how and for whom perceptual supports have the greatest impact. However, measuring executive function, attention, and cognitive control with students in educational settings is challenging. Using more scalable and affordable web-based tools such as open-source executive function and eye gaze measures provides a means to understand individual differences in cognitive processing related to math. This project aims to build capacity and expertise in the principal investigator's research on perceptual learning in mathematics by examining how open source tools such as eye gaze/trackers and executive function/cognitive control measures can be embedded into online technology platforms to conduct ecologically valid classroom-based research.
  
  To move research from the lab to classrooms with broader populations and to test mechanisms, two studies will be conducted. The first study will establish the usability and feasibility of using open-source eye tracking (Webgazer) and executive function measures (ACE-X) online with college students. The second study will test these relations in classroom contexts with middle school students. By using translational research to investigate key potential moderators and mediators, this work can contribute to research on how attention and executive function impact student use of perceptual scaffolds during mathematics problem solving. The outcomes of this project will also guide methods to conduct translational research in education and may also inform the future design of educational interventions or technology supports.
  
  This project is supported through a partnership with the Bill & Melinda Gates Foundation, Schmidt Futures, and the Walton Family Foundation. This project is also supported by NSF's EDU Core Research Building Capacity in STEM Education Research (ECR: BCSER) program, which is designed to build investigators' capacity to carry out high-quality STEM education research in the core areas of STEM learning and learning environments, broadening participation in STEM fields, and STEM workforce development.

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NSF CAREER GRANT (RETTL)

• Grasping Understandings of Students Mathematical and Perceptual Strategies Using Real-Time Teacher Orchestration Tools

• Effective Years: 2022-2027

• Many middle and high school students in the United States do not reach proficiency in algebra. When solving algebraic expressions and equations, students not only need to perform procedures, but also identify mathematical structure, attend to important perceptual cues, and make decisions about which steps are most appropriate or productive in a particular problem context. Math teachers are critical to supporting and improving students’ math achievement by providing high-quality feedback, instruction, discourse, and opportunities to their students. However, many teachers struggle to find algebra-based teaching tools that efficiently provide a means to challenge students to think conceptually, keep their students engaged, review student work efficiently in real-time, and better support their instruction. This project focuses on the design, development, and use of new algebra-focused teacher tools that use artificial intelligence (AI) to efficiently provide teachers with detailed information about their students’ math problem solving steps, behaviors, errors, and learning in real-time. The underlying hypothesis is that if teachers are given detailed information and feedback about their students’ perceptual and mathematical processes using real-time analytics, teachers will better notice and interpret student struggles. In turn, teachers will be able to make better decisions and differentiate their instruction for a broader range of students.
  
  The main research question is to determine whether teachers are better able to detect, attend to, interpret, and make actionable decisions when using the AI-supported tool. Researchers will conduct a sequence of activities during this five-year project. First, to determine what behaviors best predict learning, a database of log files generated from students solving problems will be analyzed using statistical and learning analytics methods. Next, researchers will utilize machine learning approaches to create automated detectors that capture the use of effective math strategies, errors, and focus that has led to improved learning. Third, the project will use design-based research alongside teachers to co-design, develop, and prototype AI-supported teacher tools. The tools provide critical information about students’ mathematical and perceptual processes and help teachers quickly identify what gaps students have in their math knowledge. The researchers will conduct classroom-based observations and interviews to examine how teachers’ instruction and students’ understanding might be altered with the real-time tools and feedback. The outcome of the project will advance theories and foundational research in the fields of learning science, computational data science, human-computer interaction, and math education, as well as offer new insights into automatic detection of mathematical strategies and classroom orchestration. The technical and educational agendas also provide opportunities for interdisciplinary research and practical training and collaboration between graduate students, postdocs, teachers, and students. This award is funded in whole or in part under the American Rescue Plan Act of 2021 (Public Law 117-2).

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About 12  years ago, my colleague David Landy and I designed an initial prototype for a dynamic math notation that was based on theory and research in cognitive science and perceptual learning. What started as an initial idea has turned into a widely used dynamic algebra notation tool called Graspable Math. With the support by iES, we have developed several different initiations of this tool and my research has explored its impact on learning and the ways that we can use the data from students problem solving to understanding perceptual and mathematical mechanisms of learning.

• Graspable Math (IES) -I co-developed Graspable Math , an interactive web-based tool that allows you to manipulate and solve mathematical expressions and equations. My empirical work uses data collected from the GM technology to answer important theoretical questions about student interactions, problem solving, and mathematics teaching and learning. 

• Graspable Math Activities (IES/SBIR) -  With Graspable Math Activities, teachers can assign algebra tasks to your students and see live feedback of their step-by-step work. It is powered by Graspable Math, a digital algebra notation that provides a powerful and flexible way to work with algebra in the digital space. The  phase I and II SBIR grants were focused on developing the product and examining its initial promise in classrooms. My lab and WPI lead the initial user-centered design research. 

• From Here to There! (IES-CASL)- Algebra can be a fun and engaging game! (really, it can!). FH2T is a game-based application and intervention that I co-developed (and have tested) that embeds perceptual learning and cognitive science principles into mathematical puzzles. From Here to There (FH2T)  allows students to explore patterns and properties of arithmetic and symbolic algebra by rearranging, splitting, and manipulating numbers and expressions to reach a specified goal. We know that playing it relates to improved mathematical understanding but we don't yet know WHY or HOW....A large scale RCT study with 4,000 7th grade students was conducted in Fall 2020 to compare the efficacy of From Here to There! to other math technologies. While a team of external evaluators are conducting the impact analyses, my lab is currently analyzing the data from this study to understand mechanisms of learning. 

• Wearable Learning Cloud Platform (NSF-Cyberlearning) - I have worked with my colleague Ivon Arroyo on the Wearables for Learning Project. The Wearables for Learning project is investigating mathematics learning in elementary and middle school via authentic games that children play in the classroom and the playground, which are full of rich mathematical properties. We integrate technology with relay races, and scavenger hunts that target the development of number sense, measurement and geometry, and overlap technology to authentic children’s games that capitalize on this infrastructure to learn mathematics and excite kids about learning math. With an EAGER award and a recent NSF award from the National Science Foundation, we will investigate middle schooler’s development of computational thinking as children create math games themselves, defining the behavior of wearables as finite-state-machines. This research explores how learning with  physical-technology compares to learning with classic/passive-technology and how problem-posing adds inquiry-based approaches and ownership into the game, as well as motivational/affective impact on perceptions of mathematics and the drive to learn. 

• You can learn more about this project and tool by visiting wearablelearning.org 

• Teachers Are the Learners: (NSF- Cyberlearning)-  I am Co-PI on a project exploring how machine learning and computer vision can be harnessed to characterize automatically the interpersonal dynamics between students and teachers from videos of school classrooms, and how machine perception can deliver new training experiences for both pre- and in-service teachers. The project is an interdisciplinary collaboration that spans multi-modal machine learning, data visualization, classroom observation, and teacher training. This project brings me back to my roots in classroom observation and social and emotional learning.

• Towards Computer-Assisted Coding of Classroom Observations: A Computer Vision Approach to Measuring Positive Climate (Spencer Foundation)-  We explored how machine learning and computer vision algorithms can be used to characterize automatically the teacher-student and student-student interactions from classroom observation videos. Partially automating the process of classroom observation could facilitate finer grained and more efficient feedback on teachers' classroom interactions, which could provide teachers with improved professional development feedback and researchers with a more powerful lens with which to measure the impact of educational interventions. As first steps, the researchers are investigating how automatically extractable features such as the facial expression of each participant (teacher, student), as well as auditory features of speech, can be analyzed by recurrent neural networks to predict positive climate and negative climate of the Classroom Assessment Scoring System (CLASS). Later work will consider how to identify which participants are interacting with whom at each moment in time based on their audiovisual profiles, as well as how to combine machine-coded with human-coded CLASS scores to improve accuracy. This study harnesses a dataset of hundreds of CLASS-coded videos of toddler classrooms collected and annotated by the Center for the Advanced Study of Teaching and Learning at the University of Virginia.,