One of my principals asked me once what I thought we should do about the math department and how we could improve math education at the school. My response was pretty quick and seemed to make him leery of taking me seriously, but I wasn’t joking. My two cents? “Get rid of math class.” He didn’t ask why I thought that, nor do most people when I say it. Sometimes I tell them anyway. Now I’m telling you.
There are many reasons for wondering whether or not math departments and math classes are doomed. Khan Academy comes to mind. It is a tool that schools are using to teach kids the content of mathematics while the teacher becomes the “guide on the side,” helping students with troubleshooting of the website and a math problem or two. Another piece of technology that I just saw from a friend on Facebook is called PhotoMath. It apparently takes a picture of a math problem in a textbook with your smartphone and then shows the student step-by-step solutions to the problem. Math teachers have gotten used to the problem presented by graphing calculators, that when a student inputs the parameters of a problem, the calculator helps them solve it and they do less of the “work” themselves. They have learned that these devices promote access to mathematical thinking and allow students to work on higher levels of mathematics than they previously would not have been able to with only pencil and paper. Of course there are graphing calculators that also solve problems symbolically. Just type in the integral and the calculator spits out the result. We didn’t want those so we had the calculator companies make two versions, CAS versions (Computer Algebra Systems), and the non-symbolic kind, the kind that is allowed in math class and on standardized tests such as the AP Calculus Exam and the SAT. But now kids can answer all homework questions with this new app and not even have to think about the problem. Does this kill math class? I say yes, and my response is a mock proof by exhaustion in just two simple cases, based on a given definition of the future math class.
Case 1: The traditional math class. There are many people who go with the argument that math has been taught the same way since Plato taught Aristotle and that it is the one subject that should not be tampered with. This hypothesis does not seem to be self-evident to me, but let us just assume that it is what everyone agrees on and go from there. Khan Academy teaches students in the traditional way, with procedures and algorithms and a linear cannon of topics that build on one another. They show some applications but are mostly concerned with math for the sake of math. It’s logical, orderly, and consistent. One can even imagine this technology going further. It is already a system that is correcting itself as it evolves. It takes criticism and uses it to improve. Since math taught as a step-by-step linear, algorithmic discipline is so universal and programable, its easy for me to imagine it to be the first subject to go. Where teachers give way to facilitators who don’t need math degrees or even bachelors degrees. They can be paid less as the robots do the cognitive work (I am imagining an R2D2 like robot who sends holographic projections of the mathematics to be studied in two and three dimensions to different stations of students, and who interacts with different groups simultaneously, differentiating according to their misconceptions, “Help me Obi-Wan Kenobi, you’re my only hope”). This math teacher robot will also have the interactive capability of PhotoMath programmed into it, along with all the capabilities of WolframAlpha, and other technologies that we haven’t even thought of yet, but you know are coming. So to me it is clear that the traditional math teacher teaching a class of students in rows with number two pencils and books will go the way of the slide-rule. In this sense, math class is already dead, its just a question of when.
Case 2: The progressive math class. There is a wide range of “progressiveness” when it comes to math education. The common core standards are actually an outgrowth of some of that thinking with its emphasis on deeper learning rather than the mile-wide, inch-deep model prevalent in our textbooks. There are advocates of problem-based learning, game-based learning, project-based learning, and maker classrooms to start with. For the sake of simplification, let’s assume that in order to avoid the fate of the math class discussed in case 1 above, we create a national math class of the future based on real-world problems that don’t have only one solution, or even any solution. Where kids use technology to do research and collaborate with each other on the best methods for solving the problem at hand. Where the teacher guides them through learning experiences that can help them solve these and other problems and takes them on a mathematical journey that helps them to not only use mathematics to solve the problem, but to see the kind of mathematics generated by proposed solutions and ideas related to the given problem. As an example, I am about to start a new job at a large public high school in California. It still has a math department and math classes. I am thinking of ideas for how to kick off the year and I thought of a problem that the students can help me work on. The school is an easily bike-able distance from my house and there is a nice trail that takes me almost to the school. When I get to that point, however, the trail veers off and I need to compete with rush hour traffic at a dangerous and busy intersection. Anyone who knows me or has read the first chapter of my book knows how much I hate it when cars hit me on my bike, so I naturally wonder how the city could better connect the bike trail to the school, especially since it is already so close. I checked and Khan Academy does not have a solution to the problem and neither does Google. It is also a problem that may seem to have nothing to do with math class, that maybe someone won’t need to use mathematics at all to solve it. But when I think of mathematical thinking and content that could be used, I realize that someone who understood that content could have some important insight into the problem. I immediately think of the Seven Bridges of Königsberg problem and how learning about it and some basics of graph theory, maybe even an introduction to transition matrices and basic arithmetic on them, could give students a more complete mathematical picture of what they are dealing with and a way to think about the problem. They would be taken on this mathematical journey for a while and then back to the task at hand. They would look at other critical junctures in town where the bike-friendly areas need to be connected and improved and use similar problem-solving strategies. Letters can be written to relevant political figures (after all it is election season, which comes with its own set of mathematical morsels worth biting into), models can be created, math class can really be a different place. A place where socrates might be proud to be.
However seen this way, math class becomes this place where math teachers not only need to be masters of their content, but also need to have a wealth of real-world problems for which this content can be illuminating. When teachers try to teach like this, the problems very often become oversimplified, non-compelling, and irrelevant. Though if I have to have a math class, I want to teach it like this, but I would much rather be paired up with several other teachers who have problems to be solved that mathematics could help them with. An art teacher, auto shop teacher, history teacher, government teacher, anyone who is dealing regularly in the business of actual real-world problem-solving. Math teachers would then act as consultants to these classes, pull groups of kids aside to teach them and guide them through the mathematical ways of thinking.
Pushing this even further though, one might ask, “Why even have subjects at all?” That seems to be what Finland is asking and it sounds pretty good to me. It is actually happening in some schools in the United States as well, I have had the pleasure of teaching at one of them in New Mexico. You can read all about that in my book.
In my mind, math class is dead, and not just “as we know it.” The only question left is, do we just let it go, or is there a better way to kill the beast?