How have you used technology to create an opportunity for mathematical reasoning in your classroom?

Question

Shaffer and Kaput (1998) wrote that technology has the power to transform the mathematical thinking that is possible in classrooms, giving students new ways of experiencing mathematics as a sense-making activity:

We suggest here that the ability to externalize the manipulation of formal systems changes the very nature of cognitive activity. These changes will have important consequences for mathematics education in coming decades. In particular, we argue that mathematics education in a virtual culture should strive to give students generative fluency to learn varieties of representational systems, provide opportunities to create and modify representational forms, develop skill in making and exploring virtual environments, and emphasize mathematics as a fundamental way of making sense of the world, reserving most exact computation and formal proof for those who will need those specialized skills. (p. 97)

My question is twofold:

• How have you used technology in your classroom in a way that created opportunities for students to reason mathematically?*
• What result, if any, did you note? (For example, did students express insights related to mathematical relationships represented within the technology? Or did you have a memorable interaction related to mathematical reasoning?)

[* Kaput (1992) describes four classes of mathematical activities in school mathematics:

1. Syntactically constrained transformations within a particular notation system, with or without reference to any external meanings,
2. Translations between notation systems, including the coordination of actions across notation systems,
3. Construction and testing of mathematical models, which amount to translation between aspects of situations and sets of notations,
4. The consolidation or crystallization of relationships and/or processes into conceptual objects or "cognitive entities" that can then be used in relationships or processes at a higher level of organization. (pp. 524-525)

Technology could play a role in any of these types of activities. I provide them in case they help people to consider what role technology might play with respect to the mathematics in their classrooms.]

Clarification: I'm primarily looking for people to share their personal experiences and impressions. Research you may have been involved with is, of course, also of interest.

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Cited

Kaput, J. J. (1992). Technology and mathematics education. In D. Gurows (Ed.), Handbook of research on mathematics teaching and learning (pp. 515–556). New York, NY: Macmillan.

Shaffer, D. W., & Kaput, J. J. (1998). Mathematics and virtual culture: An evolutionary perspective on technology and mathematics education. Educational Studies in Mathematics, 37(2), 97–119.

Answer 1

I have many ways I have used technology in my math class to promote mathematical thinking. Here are five examples.

1. Share a dynamic Geogebra applet at the front of the room, and ask students to share things they notice, and share things they wonder, and ask the person at the front to change the sliders. As students notice things, other students notice other things, and pretty soon all of the important features of the graph fall out of their collaborative thinking. Imagine trying to do this activity (noticing and wondering about different forms of a linear function) without the technology!

2. Instead of giving students tutorial videos, or asking them to create tutorial videos, ask students to create video word problems, ones which involve the current mathematics they are learning, but with a challenging problem solving narrative. Here are some examples of the final product.

3. With kindergarten students, I showed them briefly how to use Google's Blockly. They then used the simple programming language to draw shapes, and I asked students to attempt challenges like "draw a square" or "draw a stair case".

4. I created a "Graph Game" for students to use. The idea is that students are attempting to recreate the distance-time graph by sliding the little stick-man across the screen. It turns out that this forces students to confront some of the most common misconceptions related to what graphs actually mean.

5. We did a project once with my 9th grade class, which was to look at the number of digits that repeat in any given fraction (or the number of digits before the decimal expansion terminates) and find a relationship between the numerator and the denominator. It turns out that a calculator display is too limiting to do this project, and in fact, most programming languages are not up to the task using their default operations, so I created this calculator that just does division, up to an arbitrary length. In this case the technology enables the problem solving activity since actually doing the division would be much too time-consuming.

Answer 2

I am most happy about technology when it comes to dynamic processes.

There is just nothing like seeing something actually move, whether you rotate a transparency or whether you move a point in Geogebra/Cinderella/etc.

Another big subject class where visualization is extremely enlightening is asymptotics. I have let my students plot the prime counting function (already implemented) and then "zoom out" and see how the graph gets smoother and smoother. (I let them do this prior to telling them anything about the result in the lecture.)