Where do we draw the line with technology in math education?
There is a technology our students use that is destroying their ability to learn actual mathematics.
This technology is so advanced that you can use it to find any formula. In fact, almost all of mathematics we know is available via this technology. Some students, lacking the resources at home, are able to go to the library and use this technology and learn all of the mathematics we teach ahead of time, thus destroying our lessons during the year. Other students can even access this technology from home!
When using this technology, students don't need to think. It can record every step they do during a problem so they don't need to actually remember where they are. In fact, if they have done a similar problem before, they can just copy the steps from the previous time.
Students are using this technology to share ideas outside of class, and plagiarism and cheating is rampant because of the ease of sharing provided by this technology. This technology is inequitable because not all students have equal access to it at home. Some of our homeless students, for example, do not have access to this technology at all.
Further, the production of this technology is damaging our ecosystem, and we throw millions of tons of it away each year. It is an environmental catastrophe in our schools, and something must be done./p>
Yes folks, it's time. We need to stop using paper in our mathematics classes.
This sarcastic description of the problems with paper aside, there are issues with the use of computers and other technology in mathematics education. However, by focusing on the issues, we forget what we gain with each new technology; efficiencies and capabilities are not present in the older forms of technology. In our discussion of whether students should do mathematics by hand, or by computer, we forget that there have been many other technologies over the years that we have used in mathematics, and that each of these technologies had their advantages and disadvantages. Even our use of language is a technology! Who would seriously argue that language, paper, or slide rules in math education hinder our ability to learn mathematics?
Conrad Wolfram has a suggestion: Why don't we use the tools many real mathematicians, scientists and engineers use for their mathematical work in schools?
Whether you calculate using a by-hand method or a computer, both are mechanical operations; without understanding the algorithm, one cannot really be considered to be doing math. Paper, pencil and language itself are all forms of technology. If the technology changes, the way the algorithm is done changes. When we use a computer to do a calculation rather than doing it by hand, we are merely trading one algorithm that students could potentially understand or not understand for a different one.
Critically, pushing around symbols on paper is just a symbolic representation of the real math taking place within one's head. When one does a calculation, whether it is by hand or by machine, an important feature of whether or not one can be said to be doing the calculation is whether or not one can predict the potential output from the algorithm, or if one understands the process they are using.
By prediction, I mean have the ability to recognize nonsensical answers, and to have a feel as to the approximate size of your answer at least, if not always the exact value.
It is important to recognize that this is not a new perspective. Consider this statement from the Agenda for Action produced by the NCTM in the 1980s.
"It is recognized that a significant portion of instruction in the early grades must be devoted to the direct acquisition of number concepts and skills without the use of calculators. However, when the burden of lengthy computations outweighs the educational contribution of the process, the calculator should become readily available."
Obviously, we can easily substitute calculator for computer.
Control over what one does is a key aspect of "doing something" and is often the chief complaint against using a computer to do mathematics. "If you just enter it into the machine, you aren't doing mathematics, the machine is doing it for you." A story might be useful here, so you can understand my perspective on this.
One of my friends is an oceanographer, and at the end of the summer, he and I had a conversation at a party about what he does for a living. I asked him if he does any math as part of his job, since I am, of course, naturally interested in where mathematics is used outside of school. He replied, "No. My computer does all of the math for me."
He explained to me that he spends about half of his time creating mathematical models to describe ocean currents and climate on a small scale, and then uses the computer to crunch data and compare it to his model. For example, he recently proved that of three data-collecting stations a company he is working for deploys, one of them is unnecessary since the other two can predict the conditions at the third station with 88 percent accuracy.
So here is this person who is creating complex models involving differential equations, writing scripts to crunch data, comparing the output of the scripts to his models, then communicating his analysis to his employer, and he doesn't consider himself to be doing mathematics because the calculation step is done by his computer.
I think we probably agree that my friend has done a great deal of mathematics, and that what he does for a living models some of the mathematics we'd like our students to be able to do. His creation of a model, programming of that model into his computer, analysis and organization of the resulting data afterward is all highly mathematical, and is the kind of stuff that we could consider to be done "by hand."
What I also see from this story is that my friend is most definitely "in control" of what he is doing. He has both control over the process he is following, and over the machine that is helping with calculations he could not possibly do "by hand."
When you program the machine, you are in control of what it does. If you make a mistake in your program, the computer complains. If you program it correctly, it works what appears to be magic, unless you understand what the computer is doing.
So we require then an ability to predict and understand an algorithm, an ability to use it to model contextual situations, and an ability to use the output of an algorithm to reason and communicate mathematics. None of these is hindered by using a computer to do the algorithmic portion of your mathematical thinking.
We also require, as a system, much more flexibility in the mathematics taught at the K-to-12 level. I'd like to see a system where many different types of mathematics are taught besides just the standard hierarchy leading to calculus. Sol Garfunkel and David Mumford recommend focusing more on quantitative literacy. Other people have suggested over the years that more pure mathematics needs to be taught in schools. I'd like to see many paths to mathematical success rather than just our algebra-heavy route. A committed student can probably learn all of the algebra we teach in schools in a couple of years; so why not focus on other mathematics, which is more intrinsically interesting and will help develop mathematical reasoning?
Many mathematicians, scientists, and mathematics educators discussed these details in depth at a recent summit in London, England. I attended this conference, and was part of a debate on "Hand vs. Computer: Where do we draw the line?" We may not be able to make this switch yet from calculations done entirely by hand to a mixture of some by hand and with the computer, but it certainly is worth having the conversation.