In this post I’d like to specifically lay out how pedagogy, technology, and content will be balanced in the context of my current classroom while utilizing this technology (tablets for every student). You can read the basic outline of my plan here, so I’ll get right into the details. My ultimate goal is for students to leave my classroom as deeper mathematical thinkers and I didn’t take the decision to write a grant for tablets lightly. I considered what is required for students to understand concepts deeply and settled on a few necessities: writing, exploration and play, and feedback. There are countless ways to achieve these in the context of a math classroom, the problem lies in accomplishing them on a regular basis. Tablets provide a means by which we can accomplish these aspects of learning as frequently as possible (or as frequently as pedagogically sound).
When I first settled on these three aspects of learning as a focus for this grant I began searching for what other math teachers do to increase these things in their classroom. The first thing I noticed was the emphasis on providing lot’s of collaborative space (which essentially meant whiteboards) as well as having a student centered environment. I currently have my classroom organized into pods and each pod has a collaborative whiteboard on it. In addition I have large whiteboards on three of my walls. I currently have a fairly student centered, collaboration focused classroom. In addition, I teach higher level math where much of the content cannot be accessed by simple “hands-on” manipulatives that are common in lower mathematics. This is to say that I considered “low-tech” options but have either covered them (collaborative space) or don’t need them (hands-on manipulatives).
I then looked to “high-tech” options. I decided that most of the technology I looked at would be most effective in the hands of my students. For instance, if I only had one tablet in my classroom I might be able to show a demonstration with it, but students wouldn’t be able to explore the mathematics on their own. Once I decided on devices, I needed to choose between tablets or laptops. I decided to go with tablets for two main reasons. First, they are better for creating things that require a camera (video projects, interesting image projects, etc.). Second, they are much easier to draw or write on.This comes up frequently in mathematics and is especially handy with some formative assessment tools (Nearpod for example). This often much more intuitive for students than using an equation editor. As outlined below, the positive changes these devices can bring about are many.
I currently have my students maintain blogs to increase writing in my class. On days in which there is a blog post assignment I must reserve a computer lab. This usually is not a problem but it requires breaking up whatever we are doing in class to go to the lab. Students usually don’t finish the writing at the same time so students that finish early are frequently left with not much to do. I want writing to be more of a “regular” thing in class and less of a field trip. I think it would send the message that writing is simply an important part of the learning process. Having a class in which each student has a device means that the flow of class doesn’t need to be interrupted for writing and students can do the assignments as soon as they reach that point in the lesson.
Tablets also provide students with powerful tools for exploring higher level mathematics. Software like Desmos, Geogebra, and Wolfram Alpha demonstrations allow students to easily play with math and make difficult concepts more accessible. Pedagogically speaking it makes it easier for a teacher to allow students time to engage with concepts prior to formal instruction. This gives students a chance to actively engage their prior knowledge with the new ideas (Bransford, 1999). They can begin to construct understanding of concepts and then the teacher can follow up with a more formal lesson to help the student make the final connections. The more frequently this happens, the more the student owns the learning.
In addition to exploration and play, historically many higher math concepts (slope fields, solids of revolutions, complex graphs, etc.) were only accessible through sketching graphs (by hand) or through graphing calculators. Both have severe limitations in their capabilities compared to the aforementioned utilities. (For instance, see the activities on composition and operations on functions located here. This is a concept that’s very difficult to see purpose in, but in creatively using Desmos it becomes one of the more engaging topics in the unit.) With these new tools students can access content that historically would’ve seemed mysterious. Granted, I could take students to the computer lab whenever I wanted them to use these tools, but the situation where these tools are more beneficial than drawing or graphing calculators comes up so frequently in higher math that it’s not reasonable to do so. Not only does this technology make these tools consistently available to students, but it also can help make my pedagogical decisions more in line with the constructivist philosophy of learning.
In order to decide how to teach a concept (or especially re-teach a concept) it’s helpful to have an understanding of your students’ understanding of a given concept (or group of concepts). To do this teachers are frequently formatively assessing their students. The more frequently we formative assess the better idea we have what students know. One problem with it is that much of the time it’s based on our “feel” for their understanding. For instance, we might use questioning frequently to get an idea of their preconceptions or misconceptions. The problems with this are that it doesn’t give us specific data about our entire class, we often overestimate our students’ understanding of prerequisite skills, and it’s difficult to narrow general understandings down to specific concepts. There are countless tools designed for tablets that give very specific formative assessments to students and provide valuable insights into their learning in real time. These assessments can happen frequently and drive instruction almost on a minute to minute basis is every student has a device.
When trying to design tasks in which students construct knowledge we must do our best to gain insight into how far they’re coming in their understanding. The better our understanding of their understanding, the more effective each class period is for students. In addition, this gives the students an understanding of their weaknesses and in preparation for their summative assessments they will know which concepts to focus on. Lastly, this allows students to develop a good study habit. Instead of just reading about concepts students should use self quizzing to reveal misunderstandings and force themselves to try to recall information or concepts. This forces them to honestly evaluate which ideas are stored in long term memory and which ones they merely thought were stored there.
In the context of my current classroom and teaching assignment, tablets can make the three main areas of teaching (technology, pedagogy, and content) blend together well. We can dig into content more easily and deeper than before. We can create more situations in which students play with mathematics as an entry point to lessons. We can assess in low stakes situations more frequently and effectively. Students can write on a regular basis. Tablets, implemented effectively, give my students countless more opportunities to understand mathematical ideas more deeply.
Bransford, J. (1999). How people learn brain, mind, experience, and school. Washington, D.C.: National Academy Press.