The Roles of Teacher and Technology: Mathematical Modeling Using Desmos
We have many tools in our math teacher toolkits. Need to demonstrate place value? Base ten blocks. Explore area and perimeter? Geoboards. Probability, you say? Out come my cards and dice.
The thing is, we don’t expect students to understand place value simply because we handed them the base ten blocks. We don’t expect them to derive formulas for area and perimeter without some guidance with the geoboards. We don’t expect them to grasp probability because we let them play dice and card games for the week.
Yet, we often expect students to learn math simply because we hand them a computer with a particular site or app. Digital technology is just another tool in our kit, albeit a dynamic and potentially powerful one. Still, left unsupported, it is equally unlikely to lead to real learning gains. Instead, the teacher and the technology have to work hand in hand. I think Matthew Lynch said it best in his article on using technology in the classroom:
“We must remember that technology is a tool to enhance instruction. It's not meant to be a replacement for teachers or for the quality instruction they provide in the classroom. When used to supplement and enhance lessons, technology becomes a powerful resource that promotes student achievement, especially when students are engaged in higher-order thinking skills.”
So what can we do to support technology during modeling activities? The same things we do to support the blocks, the geoboards, the cards, and the dice! Before we ever bring out the tool, we make sure our classroom supports productive interactions by strategically arranging furniture and by fostering productive classroom expectations. We identify concrete learning goals and then design a learning experience around them while making sure each part of the activity reinforces them. We identify what students already know about the topic so we can anchor what they will experience to what already makes sense to them. We pause for small group and whole group discussion so students can process what they see and do. We offer individualized feedback when frustration becomes unproductive. We give students time to work on meaningful activities.
That all sounds well and good in theory, but let’s see how it plays out in a practical example. Enter mathematical modeling. We can unleash the transformative power of technology when students are engaged in higher-order thinking like mathematical modeling. For example, take Charge!, a lesson from Desmos’s modeling bundle. In it, students build, use, and evaluate models for predicting how long it will take for a smartphone to fully charge based on real data.
A preview of activities students engage in during Charge!
How can we support the technology? First, identify the concrete learning goals.
- CCSS.Math.Content.8.SP.A.1 Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line.
- CCSS.Math.Content.8.SP.A.2 Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept.
- CCSS.Math.Content.HSF.LE.B.5 Interpret the parameters in a linear or exponential function in terms of a context.
Desmos makes calculating and graphing different lines of fit incredibly easy so students can focus on the intended objective--creating, comparing, interpreting, and assessing a suitable linear model. Since the technology does the task more efficiently than pencil and paper, it’s an appropriate time to use it.
Next, how can we anchor the experience to prior knowledge? I would strategically place this lesson after students are familiar with linear equations, slope, and y-intercept. There are many places to build on that knowledge in this lesson.
Now, how can we support the learning goals? Look for places to insert small or whole group discussion. I would want to share student responses after everyone builds a model (step 4), so I would customize the lesson to pause there. In other words, I would prevent any student from seeing the model in step 5 until we had a class discussion. That way, students have the opportunity to think through and refine their reasoning about their models before they use them to interpret the slope and intercept as shown in screens 5 through 7. I would probably pause again so everyone sees the big reveal of screen 8 at the same time. (This screen is a great example of reconciling models with real life!) Then, I would probably do screen 9 as a group to reflect together and discuss appropriate domains followed by screen 10 extending to new areas as my exit slip or homework.
Next, I would anticipate what mistakes students might make. For example, someone will assume that the table increases by the same time interval when they are completing the table as shown on screen 3. When interpreting the graph on screen 5, someone will accidentally start at 100 minutes and read the corresponding charge instead of starting with 100% charge and looking for the corresponding number of minutes. Then, I would prepare a few prompts to redirect their thinking. “Tell me about your response. How do you know? What does each axis on the graph represent?”
Those are some of the ways teachers can support the technology, so how does the technology support the teacher? In addition to what I already mentioned, Desmos allows teachers to see student responses in real time providing helpful formative data so you can determine the next best move for the class as a whole, for a group of students, or even an individual student who may need some extra help. It also allows teachers to share student responses (anonymously) with the class so you can initiate productive conversations. Furthermore, Desmos gives us reminders to support the technology in the teacher’s guide accompanying the lesson. I really appreciate people who design thoughtful educational resources. Such a breath of fresh air!
I hope you have seen how invaluable we are to the success of technology in our classrooms. I like how Tools for Learning: Technology and Teaching Strategies sums it up:
“With technology now being part of our everyday lives, it is time to rethink the concept of integrating technology into the curriculum and instead aim to embed technology into pedagogy, to support the learning process. This means that technology becomes an integral part of the learning experience and an important consideration for teachers, from the onset of preparing learning experiences through to teaching and learning with students.”