In 1977, when I taught 4th grade, I had a PET computer in the classroom. This was the first home computer, pre-dating even the first Apple. I created assorted math games for my students, which I incorporated in the daily routine. This was fun and educational for me and for them.
A few years later, I switched to teaching high school. I remember a time, early in my Urban School career when the purchase of a $200 printer was a big decision ("a printer? are you sure you need a printer"). Now our school has printers in every hallway, LCD projectors and interactive whiteboards in every classroom and laptops in the hands of every teacher and student.
In those 30 years, I saw what had started as an eccentric preoccupation of mine become a mainstream concern. Along the way, I had to let go of the rosy expectation that computers would automatically revolutionize education, but I also learned that they do have the potential to help us reflect upon our teaching, and make changes for the better. For example, thanks to graphing technology, instead of "here's a formula -- draw the graph", we can ask "here's a graph -- what is the formula?", which is a much higher level question. [Make These Designs]
In the computer age, speed and accuracy in computation are no longer legitimate goals of math instruction. Students cannot compete with machines in this arena. Instead, this is a time to recommit ourselves to teaching for understanding. Our students live in a world where people in almost every walk of life use the computer as an essential tool. It is absurd to shield our students from this potentially powerful aid to their learning.
Whether we like it or not, today's calculators and computers (or the free WolframAlpha Web site) can solve equations, factor expressions, etc. at the push of a button. It will take much time, classroom experimentation, and teacher discussion before we fully grasp the educational implications of this state of affairs, but we already know that we can make use of computers to make our math classes more visual, more interactive, and more creative.
Ideas such as the Pythagorean theorem or completing the square can be visualized with the help of animated computer graphics/. This sort of demonstration is helpful, but in isolation it does little to convey concepts. It is most effective as a trigger for conversation and reflection, and in combination with traditional pencil / scissors / paper approaches.
Interaction is where the computer shines. There are many ways for students to manipulate mathematical concepts, using applications that provide "objects to think with", in the words of early educational computing pioneer Seymour Papert. The interactive whiteboard and accompanying software allow teachers (and students) to use virtual manipulatives to complement and support work with hands-on materials. [SmartBoard galleries: Lab Gear; pattern blocks.]
Statistical software such as Fathom makes it possible to interact with data, and (for example) bring the real world of sports into the math classroom. Learning to think quantitatively about politics, finances, and the environment is more important to most people than (say) the quadratic formula, but it is laborious without technological tools. [Fathom file: Dream Team]
Interactive geometry software, such as Cabri or Geometer's Sketchpad, opens up a whole world of exploration and conjecture. Cabri 3D takes this to the third dimension, a dimension sorely missing from most of our classes. [Cabri 3D unit on volume of a pyramid]
In a traditional math class, problems are narrowly stated, and typically we expect a single correct answer. True, it is possible and important to encourage different pathways to the solution, but the computer allows us to go further, and really open up the field to student creativity. One way to do that is to start with a blank document in programs such as the ones mentioned above, and working "from scratch". Students can devise and illustrate a lesson based on a manipulative model. They can collect their own data, or find data of interest to them on the Internet, and analyze it in Fathom. They can be challenged to make various constructions in two or three dimensions in Cabri.
Programming can be the most creative endeavor on the computer. Computer science is not mathematics, but the two fields are intimately related. Programming is omnipresent in our lives, though it is not always in plain sight. Every educated adult should have some understanding of it. "I don't need to understand electricity to turn on the lights", as the saying goes, but that does not stop us from teaching students about electricity! At Urban, we introduce all our students to programming as a part of a required math class.
Technology complements other approaches in math education -- it does not replace them. Used well, it can yield more motivation, more access, more challenge, and more understanding for our students. For us as educators, to be honest, it's a lot of work. Given the relentless pace of innovation in hardware and software, using technology requires us to constantly learn new things. An unexpected benefit is that as we learn to use this or that application in new ways, we deepen our understanding of math and pedagogy, we design new lessons, and our curriculum evolves. This requires a lot of time, patience, and support, but it is worth it!