Des-blog

The Desmos Teaching Faculty has been thinking hard about modeling.

For too long, many students’ experience with mathematical modeling has been limited to problems that begin “The relationship between this and that can be modeled by the equation….”

Our goals are more ambitious. We want our tools to help students model even when they’re not using our tools.

Our work is partially informed by these excerpts from the Standards for Mathematical Practice:

Mathematically proficient students are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later.

They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.

A new version of LEGO Prices represents our current best efforts here.

Students make a prediction about the price of a large LEGO set.

They sketch a relationship between price and number of pieces. The teacher can lead a discussion about those sketches—selecting and displaying ones that represent differing assumptions.

Then we impose a proportional relationship.

Students are free to argue back, and we hope that they will! In other modeling activities, we’ll ask them to select a model type or to build one from scratch. But at this early stage of modeling, we’re content to impose a structure in order to let students explore within these constraints.

We ask students to interpret the models they build, and to critique the constraints.

Finally, we build a little suspense by revealing the answer via video.

We’ll have a bunch of new and madeover modeling tasks in the near future. In the meantime, let us know how modeling is going in your classroom, and how we can strengthen our approach.

We just added support for multiple choice questions and video exhibits in our Activity Builder. These are the first features lots of companies add to their online activity platforms so we wanted to explain why we waited so long.

First, we wanted to get them right, to build them in alignment with our pedagogical principles. Multiple choice questions can offer meaningful assessment, but it’s also easy to answer them correctly without knowing much of anything and it’s easy to answer them incorrectly while knowing quite a lot.

So we took our time to integrate multiple choice with explanation. Students will select an option and they’ll also have to explain that selection. Students will also see explanations from three other students, consistent with our interest in connecting students and their thinking together.

These are our default options, too. We needed time to communicate our pedagogical preferences through design and code.

Second, we wanted to cultivate a teacher base and a house style. What interests Desmos more than many of our colleagues in this space is informal mathematical thinking. It’s harder for machines to automatically grade that kind of thinking but we know it’s the kind of thinking that often interests students in mathematics and it makes formal mathematics easier to learn. So we built a text box and we built sketch – two useful conduits for informal mathematical thinking – before we built multiple choice and video.

We also speculated that our earliest teacher base would come to define our company and challenge our ideas about mathematics education in essential ways. If multiple choice and video were the first features we built, we anticipated registering many teachers who were interested primarily in uploading video lectures and creating quizzes. We think Desmos is useful for those purposes and we welcome those teachers. But more than we need teachers who will challenge us to be an adequate Khan Academy, we need teachers who will challenge us to be the best Desmos.

We couldn’t be happier with the results of those efforts. Feature requests are a joy to receive more often than not because our teachers understand our project.

We’ve set a bar for ourselves. Our teachers push it higher and help us clear it.

Stay tuned.

We pose a conversation starter in the Desmos Fellows program every week. It helps us reflect on practice and grow as teachers and technologists. In a recent prompt, we asked the fellows to share do’s and don’t do’s when designing Desmos Activities. We followed this prompt by asking the fellows the same question about how they teach with Desmos Activities.

The fellows reflected on how whole-class instruction differs when your activity is on a computer rather than on paper. Some highlights are below.

• After giving students a chance to explore a topic, teachers can use the classroom conversation tools to bridge informal thinking to more formal ways of thinking about concepts.
• The dashboard enables teachers to collect and act on student thinking through whole class conversation around misconceptions, making connections, and summarizing learning.
  

Check out what the Desmos Fellows had to say below around exploring, practicing, and summarizing with Desmos Activities.

Exploration

Many of us use Desmos Activities to introduce a concept. An introduction may include a series of exploration screens followed by a class conversation around how the exploration connects to the topic that students will study. Whole class instruction during the exploration phase of an activity may be useful in the following ways:

• Paul Jorgens has used teacher pacing to take students through the initial exploration, sharing the overlay screen in order to help students make generalizations and explain their thinking.
  
• Jenn Vadnais uses a cycle here called Interact - Process - Connect - Repeat. She helps students make connections through questioning and by organizing their thinking on the whiteboard. Check out her blog post for examples.
  
• Lisa Bejarano has also used Desmos Activities to provide her students with a visual introduction that helps them build an informal understanding of the concept. She helps them think through how the concept works graphically, numerically and analytically, using the activity as a reference. Students then record these connections in a composition book. This is a strategy that can be used after the introduction or as part of the summary of the activity.
  

Practice

• Mark Alvaro has bridged the transition from introduction to practice by modeling how to do a problem together so that students know what the expectations are. This is especially important when completing a problem can depend both on understanding the mathematics and on interacting with the technology in the appropriate manner.
  
• Linda Saeta has used pause and teacher pacing to help students review specific screens. She noticed that after investing a lot of time on the first screens, the rest of the activity was more productive for students.
  

Summary

• Dave Sabol likes to find places where the class can stop and collect thoughts and students can catch up on a topic. This might involve addressing misconceptions, or summarizing learning and making connections.
  
• Anna Scholl likes to use card sorts to help students summarize their thinking and make connections.
  
• Linda Saeta has used a Desmos Activity that spanned multiple days. At the end of each day, she picked out some of the student responses to use to start the next day. This helped students see all the correct ways to think about the problems and how to make answers more complete.
  
• Bob Lochel aims for activities that build towards more open ended problems. He uses the dashboard of graphs as a gallery for students to discuss during the summary portion of the lesson.
• Sarah Vandivort has also used Activity Builder for whole class summary, choosing a key screen towards the end of the activity to launch the summary discussion.
  
• Nick Corley uses Polygraph: Lines at the beginning of his unit on Linear Functions in Algebra 1. In this activity students who had prior knowledge played with students who didn’t, and their vocabulary transferred from one student to another. Nick noticed that by the end of the activity most students were using words like slope and intercept. He used the teacher dashboard to foreshadow some topics and to start to introduce the vocabulary of the chapter.
  

What are some other ways in which we might use a Desmos Activity for whole-class instruction?