Plotting Uses of Technology for Learning

How can technology be best used as a tool for learning mathematics? Calculators can assist with computations when learning other mathematics. iPads can help students communicate their learning. I’m asking about something else. I’m asking about the use of technology to develop new mathematical understandings.

Last week, Marc and I explored this question with about twenty math department heads. First, teachers were given time to explore several dynagraphs. In this version, the equation of each function was hidden. This became the problem to solve.

Following this activity, we wanted to discuss the question above. One approach would be to present several different examples and evaluate each, sharing our criteria. Not very effective. It’s our evaluation, our criteria.

Inspired by this

we came up with the following:

Technology for Learning 1

After generating a list of possible uses, teachers were asked to plot them in the plane. For example:

Technology for Learning 2Quadrant I: The dynagraphs investigation was placed in the first quadrant (active-understanding). The NCTM Illuminations Pan Balance applet can also be placed here. In this quadrant, learners build depth of conceptual understanding, be it of function relationships or algebraic thinking, through problem solving. Learners encounter many, if not all, of the seven mathematical processes identified in the curriculum. They communicate mathematical ideas, make connections among mathematical concepts and to past experiences, reason and justify their mathematical thinking, and use visualization to make sense of mathematics.

Quadrant II: An alternate version of dynagraphs was placed in the second quadrant (passive-understanding). The equation is no longer hidden, thereby replacing problem solving with observation. My GeoGebra material also fits here. You know the type: drag a slider (or, worse, watch the teacher drag the slider); what do you notice?

Quadrant III: Ah, yes, Khan Academy. Enough said? Probably not. Activity is limited to pressing pause and rewind. The “intuition” video comes later, if at all. In the third quadrant (passive-knowledge), learners consume content.

Quadrant IV: In the fourth quadrant (active-knowledge), you will find Math Blaster, an iPad app in which students practice math facts (+, −, ×, ÷) through gameplay.

Many interesting comments were made by the group. Some highlights:

“It’s about teaching, not technology.”

Activities can slide from the first quadrant. Who’s doing the math? If it’s the teacher, then we’ve moved to the left. Is the focus on “how-to’s” or essential understandings? If it’s the former, we’ve shifted down. The same holds true for uses of manipulatives.

“We can’t always be in the first quadrant.”

Fair enough. This activity provides one answer to the opening question. Quadrant I is the ideal. Is there value in quadrants two through four? I think so. A demonstration can be helpful. For example, this applet can help learners make sense of A = πr². So, too, can this low-tech activity. Is there a place for grapefruit? There may be. But this can’t be where we live. FWIW, it’s not just that KA occupies this space. It’s that it goes about it so badly. If you must have a grapefruit…

“Your axes are wrong.”

At least one teacher suggested that the x-axis be labelled “active learner.” In Math Blaster, children are active in the sense that they are blasting through razor sharp blockades and speeding past the stars on their HyperCycles. A bit of a stretch to call this active learning. Other possibilities for each axis were suggested: student-centred/teacher-centred, conceptual/procedural, process/content, etc. We fully expected this. The intent of this activity was to generate discussion. The imperfection of our labelling of the axes only added to the conversation. The question “How can technology be best used as a tool for learning mathematics?” became “How do students best learn mathematics?”

Revisiting GeoGebra

Four years ago I learned about GeoGebra and made some applets to be used in my classroom. I started by creating applets that demonstrated the effect changing slider values had on the graphs of trigonometric functions. I’d change a value and then ask the class to describe what happened to the graph. These constructions made excellent demonstrations. But that was the problem. The spectator experience was improved, but students remained spectators. (SMART Board fans take note: having one student at a time come to the front of the class does not change this.)

I also posted these applets on my class website. I thought students would try them at home to reinforce learning and check for understanding. They didn’t.

I wanted to move more towards having students themselves do the investigating. I constructed dynamic worksheets to explore slope and circle geometry in Math 10 and 11. Twice, I threw in the towel halfway through the period because of technical difficulties. The 15 laptops had to remain plugged in because their batteries no longer held a charge. The wireless network couldn’t handle having 15 laptops on it. The files were copied from my flash drive to desktops but only worked on some of the computers.

So, we went back to pencil and paper. Each student drew and then measured his or her own angles. Some students immediately observed the relationship. Others observed it after seeing the results of each group member. They asked “What if we move the inscribed angle off to the side more?” and “What if the central angle is larger?” Then, they set off to find the answers. Listening to these conversations, I wondered what this would have looked like had I been able to carry out my lesson plan.

In the four years since then, I’ve seen several GeoGebra/Sketchpad constructions created by other math teachers but very little that really excites me. A new tool to use while I stand and deliver? An e-version of an investigation that my students do using pencil and paper? Okay. I guess. Just don’t try to sell it to me as being more than what it is.

I want to incorporate technology into my teaching in meaningful ways. Here’s something from David Cox that could get me back on the GeoGebra bandwagon. It starts with a great problem that is enhanced because it is posed using GeoGebra. Students continue to interact with the applet as they attempt to solve the problem.

I uploaded some of my dynamic worksheets to GeoGebraTube and was very pleased to see that they worked on my iPad. I’d love some feedback on them. Was my assessment of them correct or are they salvageable? Also, I’m willing to give GeoGebra another try. Can you point me to exemplars?