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To make algebraic work experiential, a group of us - math educators, psychologists, mathematicians, and computer scientists - have developed Graspable Math (GM). GM is a dynamic algebra notation tool that allows users to manipulate algebraic expressions and observe an immediate response to their actions by dragging and dropping mouse, or using touch gestures. Thanks to GM, solving algebraic transformations no longer needs to be an arbitrary, rule-driven process, but instead the demonstration of one’s structural intuition.

Using the GM as a foundation, we are currently developing Graspable Math Activities (GMA) - a student activity app with novel kinds of algebra practice and assessment tasks. With the goal of supporting both students and teachers in the journey of learning and teaching algebra, the activities in GMA promotes conceptual understanding and procedural fluency while allowing students to focus on strategy and the connection between concepts and procedures.

With support from IES, we are also conducting a large scale efficacy trial of From Here To There - a dynamic game-based version of GM designed to promote algebraic thinking. In this game, students use gesture-actions in GM to solve engaging mathematical equivalence puzzles.

We have conducted several studies that demonstrates gains in mathematics achievement and engagement after using GM. We are now using the rich clickstream data logged from the technology to analyze student behaviors and the strategies students use while solving mathematical problems to reveal variation and flexibility in problem solving and expand our understanding of mathematical cognition and development.

## Erin Ottmar

Lead PresenterAssistant Professor of Learning Sciences and Technology

Hello everyone! In this video we present an overview of the Graspable Math and From Here to There Projects that have been supported through IES research and development awards over the last several years!

The goal of our projects are to develop new ways for students and teachers to engage with math notation both in and outside of the math classroom!

Please post your questions or comments below! We look forward to an exciting exchange this week!

## Stacy Shaw

I wish I had this when I was learning algebra! Have you noticed if students who engage with this compared to more traditional forms of algebra teaching show a shift in their attitude toward? I can imagine many students find this more engaging, interesting, and helpful with the feedback, and this might make them like math more than they previously had.

## Jenny Yun-Chen Chan

Co-PresenterHi Stacy,

We have anecdotal evidence on changes in students' attitude towards math after using Graspable Math. This summer, we will be able to empirically test your question using our RCT data collected last fall :D We are all super excited for you to join this effort in understanding how GM experience influence math thinking and learning!

## Mia Dubosarsky

Congrats Erin & team!

## Jenny Yun-Chen Chan

Co-PresenterTHANK YOU, Mia :-) Hope you are doing well! I would love to catch up with you some time soon!!!

## Rebecca Vieyra

Dear Erin and Graspable Math Team,

Congratulations on such an awesome tool!

Based on the video, I am most struck by your last statement, that this project is focused on figuring out "how algebra notation bridges to other representations." I see from some of the video snippets that the tool includes the ability to dynamically move around pieces of an expression, and also to sometimes have a graphical representation. Are these the kinds of "other" representations you are speaking of? (Or are you mostly focused on multiple algebraic-only representations, such as ratio expressions, exponential numbers, functions versus expressions, etc.).

I am particularly curious about this because of my own work in Modeling Instruction in physics, where we do a lot of coordination of representations among algebraic expressions, graphs, vector diagrams, verbal descriptions, physical models, pie charts, histograms, etc., and, most recently, computer programs (https://aapt.org/K12/Computational-Modeling-in-Physics-First.cfm).

Another point I note is the implication that the software makes it easier for students to approach problems from different ways. If I am accurate in my assumption, what is it about the software that facilitates this kind of "try it again" attitude? And, do you think that this pedagogy is replicable without the software?

Thank you so much!

Sincerely,

Rebecca

## Erin Ottmar

Lead PresenterAssistant Professor of Learning Sciences and Technology

Hi Rebecca. Thanks for the comments. We have integrations with Geogebra which allows users to dynamically link graphs and equations. Users can also use the GM canvas as a white board which some teachers and students drag images onto the screen for modeling other representations. While there is not the functionality to embed dynamic pie charts, histograms, etc, they can create those in other systems, take a screenshot, and drag that image onto the screen. There is also a draw feature where students and teachers can draw and submit their work to the teacher.

Since the system allows students to attempt any mathematical action, but only commits to valid transformations, students can approach problems in any way that they wish. In the data, we definitely see enormous variability in problem solving approaches and increased persistence and engagement. In our empirical RCTs we have found shifts in student conceptual understanding and flexibility on assessments and transfer tasks (not using GM) as well as decreased math anxiety. Teachers also have shared that students begin to mimic the GM actions, both as gestures when explaining their thinking as well as drawing arrows indicating motion on their paper-based work. This is an area of research that we are interested in exploring in the future.

## Rebecca Vieyra

Dear Erin,

Thanks for the response! The integration with Geogebra makes sense, and that expands the power of your tool even more! I think the core of my question is really about what students actually do with the GM canvas -- are they spending time using multiple representations, or is it more about the manipulation of formulas on both sides of the equation? (Both are important; I'm just curious about the points of emphasis, because in the video I see a lot of manipulation of the expressions).

One thing I'd love to know about is what student's initial tendencies are before and after exposure to CM.

I'm super glad to hear about decreased math anxiety -- given the tendency for females to have higher performance anxiety on academic work, this tool could be a real game-changer for supporting equity!

## Teruni Lamberg

Would this tool be publically available?

## Erin Ottmar

Lead PresenterAssistant Professor of Learning Sciences and Technology

Yes, they are both publicly available now at graspablemath.com and activities.graspablemath.com

Lorna Quandt

## Michael Haney

This is a wonderful teaching, learning and research tool with an excellent student interface that makes explicit how mathematical notations are related to various representations. It supports a more flexible form of problem solving that would likely lead to a deeper understanding of mathematics. Its sophisticated yet appears deceptively simple to the user. This is very promising work and an excellent platform for examining the strategies that were used.

If I understand the tool, it could help teachers know what student understand and maybe even what misconceptions they are carrying forward. Your system can track the steps the student uses and somehow characterize his or her work, information critical in any teaching/learning system. I am curious about the how that is done...do you use technology to do some sort of analysis. How is it reported?

## Erin Ottmar

Lead PresenterAssistant Professor of Learning Sciences and Technology

Thank you Michael! We are currently deep in the process of identifying ways to analyze student strategies as they solve problems. The GM system logs all user actions, clicks, errors, mouse trajectories, and behaviors. We then aggregate that information by problem, student, content, and overall. Using the raw data and timing information, we can create detailed visualizations about their strategies using data analytics (we are currently using sankey diagrams and tree diagrams). We can also conduct cluster analyses and machine learning techniques on the data to understand patterns in students problem solving strategy. The data files are extremely large and we are just beginning to dig into new ways to extract patterns of student mathematical thinking and behaviors. Its an area that has endless possibilities to apply computer science, data science, and machine learning approaches to educational intervention data.

## Erin Ottmar

Lead PresenterAssistant Professor of Learning Sciences and Technology

I also want to add that a huge interest of mine is thinking about how to make the data recorded in this system actionable for teachers. We are currently designing dashboard systems that could provide teachers with different information and visualizations about individual and overall students problem solving and behaviors. While this requires extensive software engineering to make this happen in real time and we are some time away from making this available in the wild, we are making great strides on the research side to create visualizations of student problem solving and behaviors and determine which kinds of information is best for teachers to understand where their students thinking is.

Sarah Hampton

Lorna Quandt

## Michael Haney

This is all very encouraging. One real benefit of technology is that it could help us understand what type of student behavior (thinking) leads to better learning. Whether that uses experts models, more pragmatic means (X pattern tends to learn more, Y do not etc.) or more nuanced analysis, to date teachers have had little more than delayed testing and intuition to guide them. Your data and interests are filled with possibilities.

Sarah Hampton

## Susan Kowalski

I love how students can use touch screens to quickly manipulate algebraic expressions. I'm curious about the learning curve - do students struggle to understand what their manipulations mean? Or, by manipulating expressions, do they develop better understanding of those expressions? I'd love to hear more about what you know about student thinking as they manipulate expressions on screen.

## Erin Ottmar

Lead PresenterAssistant Professor of Learning Sciences and Technology

Susan- we find that students pick up the gesture-actions fairly quickly. We have found that gif videos that show one action at a time, mapping it onto a property, and the following it up with a few practice problems is enough for most students.

We do find that students with higher initial prior knowledge pick up the system much faster and progress through activities much faster than lower knowledge students. However, lower knowledge students are more likely to reset and reattempt problems multiple times, even after completing them. We are currently writing a manuscript that examines the relations of these clusters of profiles to learning gains for conceptual understanding and flexibility. We find that both of these clusters of students demonstrate improved gains in their understanding and there are no interactions by prior knowledge.

We also have found positive impact findings in both elementary and middle school students of the From Here to There game, compared to both business as usual controls, as well as comparisons to other technology programs that have demonstrated efficacy.

One study that we were planning on running in the fall was conducting structured interviews to ask students.to explain their thinking as they solved problems and then map their explanations onto their strategies and data that is logged in the GM system. These in person will likely have to be delayed due to COVID but it is definitely a direction that we will go as we explore the variation in problem solving strategies in more detail.

## Rebecca Vieyra

Could those interviews be conducted virtually via webcam and screen share? I wonder how outcomes from remote interviews might be different than in-person ones. I actually think kids might be more comfortable with a remote interviewer...!

## Meeko Oishi

Really interesting! I look forward to reading about some of the outcomes of this work.

## Mitchell Nathan

This is such a great project and has developed into a mature technology that can really transform how students engage in algebra and arithmetic symbol manipulation. The emphasis on developing multiple ways of understanding is so important for an area like algebra. I am curious if you see students' gestures carry over from one solution method to another when they explain their thinking or during peer collaboration, suggesting that the gestures they develop from the interface serve to ground their understanding of symbol manipulation. I would especially expect to see this as a way to reveal how they think about equality of expressions, and when they perform inverse operations, either with addition/subtraction or multiplication/division. Thank you for sharing this and for making such a well crafted video!

## Erin Ottmar

Lead PresenterAssistant Professor of Learning Sciences and Technology

Thanks Mitch! That is definitely something that we have observed when working with students in our research and teachers tell us they notice the same when kids are working with GM and explaining their thinking. Id love to get some video tapes and do some analysis of gestures with students explanations to unpack this. Definitely an area that my students are I are excited to dig into soon! (when we can get back into classrooms!)

## Kathryn Kozak

What level of students is this appropriate for? I teach at a community college. Has your research looked at how higher ed students do with this program? I am especially interested in community college student gains.

## Erin Ottmar

Lead PresenterAssistant Professor of Learning Sciences and Technology

We have not conducted studies with FH2T or Graspable Math in community college students but we anticipate that this would be especially useful for community college students. We have conducted studies with high school interrupted education students and teachers have been extremely pleased with how engaged and persistence the students were. if you would like to collaborate, we could arrange to conduct a study to explore that. We have conducted studies with college students at WPI (a STEM focused university) to examine attention to structure and have found that there is enormous variability in strategy and attention to structure and performance even among students who achieved a high level of high school and college math.

## Jianke Zhang

Enjoyed the presentation!

Great research from Drs. Ottmar and Chan and team

## Jianke Zhang

Enjoyed the presentation!

Great research from Drs. Ottmar and Chan and team

## Erin Ottmar

Lead PresenterAssistant Professor of Learning Sciences and Technology

Thank you!

## Jonathan Margolin

Dear GM team, I can't wait to share this resource with my son, who is just starting to learn algebra! I was wondering what sort of role you envision for teachers scaffolding or facilitation of this tool. Furthermore, do you have a vision for how this resource will be integrated into the classroom?

Thanks in advance.

Sarah Hampton

## Erin Ottmar

Lead PresenterAssistant Professor of Learning Sciences and Technology

Teachers use this tool in different ways. Much of our earlier work has been focused on students using the tool in a 1:1 technology setting in class. We are working with a few teachers now who are using the tool for whole group instruction, 1 on 1 tutoring, and also student practice. We feel that the best use of the tool is by having teachers embed it in their whole group instruction and discussion and then have students also use the tool for practice both in class and for homework. We feel strongly that it should complement what teachers are doing, not replace. The GM canvas tool provides an open ended white board for teachers to use for any kind of instruction, and the GMA teacher focuses system provides this learning environment to support instruction, practice, and reflection.

## Erin Ottmar

Lead PresenterAssistant Professor of Learning Sciences and Technology

Also- let us know how your son enjoys it! You can make your own activities for him too and see his progress. If you would like to learn how, visit https://sites.google.com/view/gmactivities for some resources!

Sarah Hampton

## Traci Higgins

What an interesting project! I'm fascinated with the dynamic modeling of algebraic equations and transformations and the ability for kids to try things out and see what happens. My current project (Data Clubs) engages students in creating graphs to explore data and having good technology makes all the difference! But getting back to algebra-- Often I've found that some of the biggest stumbling blocks for students in algebra is a disconnect with the action of the operation. Students move from work with numbers in the elementary grades to work with variables in the upper grades--what is left for them to focus their reasoning on is the operation, yet many have fragile understanding of the behavior of the operations. I am wondering if you have had a chance to begin developing supports for students based on what you see them doing when they use the graspable math technology. My earlier work involved the transition from elementary to middle school and I'm really curious about how the difficulties we saw with that transition play out as students continue with algebra. I'm also nearby and would love to connect at some point learn more about this project!

## Erin Ottmar

Lead PresenterAssistant Professor of Learning Sciences and Technology

Thanks Traci! We are starting to visualize and code student strategies to see if we can develop algorithms to reveal those difficulties or patterns using data analytics and and educational data mining techniques. There isnt much work to build our approach off with data like this of but our efforts so far are creating really exciting and innovating insights!

We would love to connect. Ive been in touch with a few others at TERC about the ways that our work intersects.

## Sarah Hampton

I had a chance to see your project at CL19 and loved it! Your video is great, but I would really encourage the viewers to try out From Here to There for themselves so they can understand why it's such a powerful tool. As a math educator with 10+ years of experience, I can say it's one of the best I've seen for conveying the usefulness of algebraic properties. Thank you for sharing your work!

## Erin Ottmar

Lead PresenterAssistant Professor of Learning Sciences and Technology

Thank you Sarah for the shoutout! We feel that we have really developed a game that is theoretically, conceptually, and pedagogically sound and also integrates the best of what we know from research in the science of learning and math cognition. I feel strongly that the iterative development and slow evaluation approach, interdisciplinary approach that we took (crossing boundaries between perceptual learning, cognitive science, math education, and developmental psychology) has made this work stronger! We have exciting results that suggests that this does lead to conceptual gains in students understanding of math equivalence and other concepts and is transferable for contexts outside of the fh2t technology.

Sarah Hampton

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