Hi Catherine,

Thank you for your questions. We developed many apps specifically for this project that are authored in GeoGebra, so they work on most devices and they are free for everyone. The apps are on the GeoGebra Web site as a book/collection at https://go.edc.org/vam-apps, and you can also find these and related individual apps by searching for 'edc vam maine' (they are posted under the user "EDC in Maine") in the classroom resources search bar on the GeoGebra site.

The apps support understanding and working with visual representations of ratios and proportional relationships. Below are three examples of how they are used in the course (note that depending on their use, most of the examples given could be listed under more than one category): 

• as representations of rational numbers (e.g., fraction locations on a number line and comparing rational numbers with fraction strips and number lines)
• as tools for problem solving and reasoning (e.g., comparing paint mixtures and modeling percent change on a double number line)
• as tools to deepen conceptual understanding (e.g., exploring slope as a ratio using similar rectangles on a coordinate graph).

The apps are integrated throughout the professional development in both face-to-face workshops and online sessions. Some of the apps are associated with a course activity such as a specific math task, but almost all the apps can be used with a wide variety of contexts. Some of the apps are included in the course as optional resources. Although the primary audience for the apps is educators/course participants, they can certainly be used by students and many participants report using them with students or planning to use them with students in the future.

An example way that an app is used in the course is to connect participants' knowledge of ratio and their knowledge of slope. Participants work on a task that asks them to answer questions about two given visual representations—a double number line and a coordinate graph—for the same problem context. Afterward, they watch a screencast (video) that models working with an app on the same problem context, and then they explore the app themselves. In the Double Number Line to Coordinate Graph app, dragging a blue handle on the right side of the top number line transforms a double number line to a coordinate graph.

Multiple participants have reported that using this app has been important for their own learning. Three example responses from participants are below:

"When the double number line unfolded into a quadrant of a graph with points already plotted I was struck by what a powerful visualization it was for me. It was like I was thrown into the deep-end of the algebra pool and learned a new appreciation for the graphic representation of a problem."

"When we used the applet that asked us to pull up the top line to make a set of coordinate axes I was blown away. When I taught slope to students, I didn't teach it with the current understanding of its connection to proportional relationships and similar triangles. This course has added a lot to my understanding of the connections between these representations. Thanks for pushing me outside my comfort zone."

"When I saw the applet go from the double number line to the x and y axis it was an unbelievable experience. I understood the connection of the double number line and the importance of teaching it."

We are excited to share the apps and we hope they are helpful for educators and students.

Susan Jo,

I appreciate the focus of your question—integrating learning language in context with using visual representations. Including the use of visual representations is such an important feature of our work. Although most of the language strategies teachers learn about and implement can be used effectively in many content areas, within the VAM course these strategies are regularly and deeply integrated with the use of structural visual representations (such as tape diagrams and double number lines), which can themselves be used to support students who are English learners as they learn language while doing mathematics.

For example, teachers design a task with a word problem that uses a partial (or ‘starter’) visual representation and sentence starters. During this process, teachers consider questions such as, “How will the partial visual representation support students’ understanding of the information in the problem?” and “What sentence starter can draw students’ attention to details in the visual representation or task?” One possible reason to use a partial visual is that it can act as a language access strategy, since its structure and features can support student understanding of the language and the meaning of the associated word problem. The use of a sentence starter can explicitly prompt this use of a partial visual.

Another example is when teachers plan a lesson that incorporates their learning about differentiating teacher questions based on the English proficiency level(s) of their focus students. Teachers wrote questions that refer to visual representations, such as, “What does the 10 [on the visual] represent in the problem?” and “[pointing to answer] show me the answer here [pointing to double number line]”.

Teachers also used visual representations as a starting point for whole-class discussions. One teacher reflected on how her practice changed during the VAM course, writing that, “I now [at the end of the VAM course] plan lessons to have time to share visuals, particularly on the document camera…. By having students look at and examine the visual representations that their peers have made, it gives them new options for their own work, and it increases math conversation and discourse between students. Visual representations also help me to help students, because they might be able to show their reasoning with a visual even though the calculations are hard or explaining their logic might be a challenge.”

Over time, as teachers implement lessons with combinations of language strategies and visual representations, and as they attend to their students’ math and language understandings during these lessons, the potential of visual representations as a support for learning language while doing math may become more salient for them. I feel the teacher reflections below capture this well:

“Previously, I hadn't considered how visual representations could help support students in language access and language production. I used visual representations in lessons because I knew they were valuable tools for solving and understanding problems, but I hadn't considered their other benefits. I now consider the other structures I can add to the use of visuals to support EL students. The videos [in the course] we watched of students really helped to solidify this understanding for me. I saw how students could use their double number line or other visual representation as a place of reference when describing their work and that the [visual] model helped ground the language of the problem in a visual structure.”

"My most important learning in the course so far is is how you can see a students thinking by using visual representations. Before this course I just thought of visual representations as a strategy to help students solve a problem but not as a way I could understand more about their mathematical thinking. I have also learned that they can use visual representations to decode the meaning of text."

"I have also adjusted how I run class-wide discussions. I now ground so many of these discussions in student work, pushing students to be specific and talk explicitly about a model. I show a student's work on the Doc Cam and have another student go up and point to what they see represented. It has made our discussions much richer and I think made my students' understanding deeper. It has also helped with discourse, as students are a tool right in front of them to comment on."

"Providing them with a visual representation of the relationship between two quantities in a double number line and in a tape diagram has helped provide something tangible for my students and I to ground our conversations in and to literally point to relationships that exist within a word problem."

Thanks for this thorough response--very helpful!  We are also using visual representations as a central tool in helping teachers incorporate mathematical argument into core math content at the elementary level.  By its very nature, mathematical argument involves articulation of math ideas, and, for young students, representations are the tools for argument, so this is a natural connection.  I appreciate hearing about how this work supports EL students.

I am very interested in learning any insights or findings you can share from your work around argumentation with elementary educators and students. There are so many exciting aspects to pursue and consider related to visual representations in mathematics; using visual representations can support language learning, can be a tool for argumentation and communication more broadly, can be a tool for problem-solving and mathematical modeling, can be a tool for investigating mathematical concepts, can support engagement with mathematical practices, etc. I am especially interested in continuing to learn about how dynamic visual representations, like our GeoGebra apps, are related to these purposes, especially student language learning and argumentation. Thanks again for your question.

Kathe, great questions.

We are working on additional analyses of the study educators' notebook responses, surveys, and other instruments that will help understand this work and its potential impacts more fully.

As we begin to disseminate more broadly, we are very interested in learning what information would be most useful for various audiences to know about this project and its associated research. Do you have ideas about this? We have multiple papers and opportunities for dissemination planned related to educator self-efficacy and other topics, but we are interested in what others would like to learn.

We are hoping to not be one of those programs you describe ("end with funding even with documentation of success"). We do consider our course and elements of it scalable, and we are actively planning additional proposals and working to determine if other methods of offering the course or related professional development are possible in the near future. We have received some feedback that collections of materials may be of interest, for example for higher education faculty for use in mathematics methods courses or other courses and programs. We have also received requests to run this course or related professional development with other mathematics topics (beyond our focus on ratio and proportional reasoning) and other grade spans, including upper elementary grades. Any suggestions or ideas related to expanding and scaling this type of work are appreciated!

Thanks

Beth,

Great questions!

The VAM course is blended professional development for educators and not a curriculum for use with middle grades students—in fact, it's designed so that an educator can take their learning about the strategies and representations and integrate them into any curriculum or program, or even in small group work and intervention work with students. We have also had participants who are coaches of other educators and integrated their learning in their coaching work. As previously enacted, the course includes an expectation that a participating educator picks 2 focus students to follow through the course and for whom they plan most consciously; over time they implement a variety of strategies and tasks with at least their focus students (although often educators report using the strategies with all their students, and yes, many participants have reported to us that they find these strategies help all their students). There is a fair amount of latitude and choice in these implementation cycles, including some choice related to tasks and strategies and whether and how to adapt the materials for their unique focus students, context, enacted curriculum and program, etc. Part of what we work on is related to attending to student strengths and understandings and using the information to inform instructional planning. However, we do have particular foci and parameters around each implementation "assignment".

For example, at the end of the initial VAM summer institute, educators write a plan for an activity or lesson to implement within the first six weeks of school. Their planning process includes writing at least 1 mathematics [non-language] objective and 1 language development objective, choosing a mathematics task that includes having students create, use, or interpret a visual representation, and anticipating what students at different English language development levels might do, say, and find challenging during the activity. In addition, educators must also plan to use the 3 Reads strategy (you can also read about this and other language strategies—please note this was written for a different project) in the first 6 weeks of school, either independently or integrated with the previously mentioned plan. With both expectations, educators can choose the mathematics task (from their own curriculum or from the many we've already introduced in the VAM summer institute), the timing (within the 6 weeks), the setting, and students who will do the activity, among other things.

I also wonder what a shift in student understanding could look like if students engaged in a regular and cohesive way with visual activities throughout middle school (especially if it followed directly on similar prior engagement!). In this vein, we have had course participants ask us to offer the course again so their colleagues can participate and work toward this vision, as well as ask if/when we will develop a course for elementary grades for their colleagues who teach younger students.... we are excited and motivated based on their interest, but we have no current funding to do so.

As far as sharing content as an open resource, we are currently exploring options and considering best approaches within the parameters of our current funding, which does not include extensive development work to share everything in that way with the necessary supporting materials. Also, as part of our learning, we are finding at least anecdotally through participant (educator) writing and interviews that it's important for educators to engage with the course in a sustained way with knowledgeable facilitators, so we are considering this feedback when we think about how best to share in a thoughtful and impactful way. In the mean time, we are sharing many aspects of the course and many of our materials when we have the opportunity, including during this showcase, and we are very interested in what you and others find most interesting and would recommend sharing for various audiences. I appreciate your question and please note that sharing widely is a high priority for us, and part of the reason why we developed the interactive apps for the course using the free open source application GeoGebra under their non-commercial license, which allows worldwide access and the ability to remix/adapt the apps for different languages and in any other way.

Thank you, Bill, and I appreciate your shameless plug. I am excited by your work on conceptually-focused mathematical discussions and appreciative of your support and feedback for the VAM project. The book you mention, Mathematical Thinking and Communication: Access for English Learners, is a resource we use in the VAM course and many educators report it has been helpful, including the sections on differentiating teacher questions and the design principles for mathematics tasks.

We are excited by the feedback we have received about the VAM course, and by the preliminary analyses of our research on the cluster randomized trial. We are actively looking into ways we could offer the VAM course—or a portion or a version of the course!—beyond this funding, as well as ways to make the material available more broadly. Suggestions related to what aspects of the project would be most valuable to different audiences are appreciated as we prioritize our efforts during this phase of our work.

Thanks, Peter - these are really interesting. The teachers really were specific about new connections they made - cool.

Sarah,

I think one of the nicest things is the delight and, at times, humor some VAM course participants express when writing about shifts in their own thinking and/or teaching practice. I find it very inspiring. For example:

"This course has shown me the importance of the world beyond the algorithm! I have come to see just how limiting the algorithm method is in solving many math tasks and that using VRs [visual representations] effectively expands student possibility for entry into the problem and provides much opportunity for extended thinking. At the start of the course I was hesitant at using VRs. I had not had a lot of practice using them myself so I would tend to avoid making VRs part of my lessons if possible. That has changed for me."

"This has been a big brain shift for me...my math mind has been equation centered for quite some time. I have learned that by asking my students to use VRs, mathematical language production and sharing becomes the focal point of most lessons."

"Before this course, I was relied on finding the algebraic form of any task.... As I began to shift my thinking, I began to observe how my teaching began to shift.... I noticed this change several weeks ago. We were working on our algebra unit when we encountered a problem. I read the problem and I solved it using a visual representation. I took a step back and surprised myself that I did not quickly solve it using my algebraic thinking. In that moment, I was very proud of myself... It was a very important teachable moment because we are not always going to remember rules/formulas but if we understand the problem and use a visual to help us we can then make the connections."