When I am having students work on something, whether it’s homework or something done in class, I’ll get a stream of questions that are variations on:
- Is this right?
- Am I on the right track?
- Can you tell me if I am doing this correctly?
And so on. They want verification. This is perfectly natural and, to some extent, conducive to learning. But I think that we math teachers acquiesce to these kinds of requests far too often, and we continue to verify when we ought to be teaching students how to self-verify.
In the early stages of learning a concept, students need what the machine learning people call training data. They need inputs paired with correct outputs. When asked to calculate the derivative of \(5x^4\), students need to know, having done what they feel is correct work, that the answer is \(20x^3\). This heads off any major misconception in the formation of the concept being studied. The more complicated the concept, the more training data is useful in forming it.
But this is in the early stages of learning. Life, on the other hand, does not consist of training data. In real life, students are going to be presented with ambiguous, ill-posed problems that may not even have a single correct answer. Even if there is one, there is no authoritative voice that says definitively that their answer is right or wrong. At least, you’d have to stop and ask how you know that the authority itself is right or wrong.
So as a college professor, working with young men and women who most of them are one step away from being done with formal education, it serves no purpose — and certainly does not help students — to pretend that training, the early stage, goes on forever. At some point I must resist the urge to answer their verifying questions, despite the fact that students take great comfort in having their work verified for them by an external authority and the fact that teachers usually are perceived as being better by students the more frequently they verify.
I’ve started making the training stage and the self-verification stage explicitly distinct in my classroom teaching. In a 50-minute class, I’ll usually break down the time as follows:
I’ll spend the first 20 minutes of class focusing in on one or two main ideas for the class along with some simple exercises, a few of which I’ll do (to help students get the flow of working the exercises and to provide training data not only on the math but also on the notation and explication) and more of which they will do, providing full answers to the “Is this right?” questions along the way. Then five minutes for further Q&A or to wrap up the work.
But then the training phase is over, and students will get more complicated problems (not just exercises) and are told: I will now answer any question you have that involves clarifying the terms of the problem. But I will not answer any question of the form “Is this right?” or provide any guidance on technology use. What I will do instead, if students persist in asking “Is this right?”, is answer their questions with more questions of my own:
- Are the units working out correctly? Are you getting cubic feet for volume, meters per second for velocity, etc.?
- Did you graph the function to see if the roots are really where you say they are?
- Have you seen a problem like this before in the book, your notes, or your homework?
- Does that answer make sense in the context of the problem? Did you get a negative derivative value for a function that is visibly decreasing?
- What did Wolfram|Alpha (or Maple or MATLAB, etc.) say? *
- What do your group-mates think?
And so on. Many of these are merely ripped from the pages of Polya’s How to Solve It, which ought to be required reading of, well, everybody. In other words, in this post-training phase of the class, students must simulate life in the sense that they are relying only on their wits, their tools, their experiences, and their colleagues, and not the back-of-the-book oracle.
Also, by telling students up-front that this is how the classes are going to be structured, they get the idea that there is a time for getting verification and another time for learning how to self-verify, and hopefully they learn that the act (or at least the urge) to self-verify is something like a goal of the course.
My hope here is to provide training data of a different sort — training on how to be independent of training data. This is the only kind of preparation that makes sense for young adults heading for a world without backs of books.
* You could make a good argument that Wolfram|Alpha used in this way is just a very sophisticated “back of the book” — an oracle that students use as an authority. I think there are at least a couple of reasons why W|A is more than that, and I’ll try to address those later. But you can certainly comment about it.