Math Begins With an Answer?

We all hear it; we all say it. Students want the answers; teachers want the learning. But now, it seems students might also begin with, and not just want, the answer, or maybe they ignore everything but the answer, despite our effort to encourage and facilitate learning.

I am in the middle of reading Nat Banting‘s post Measuring Roots. He makes one statement in this post that I wish to address (indeed, my response below is cross-posted as a comment on his post).

I found this statement so profound, I had to stop reading the post and respond to it.

Nat’s statement is here:

For students, no matter how young, math begins with an answer.

Here is my response:

I stopped reading this post as soon as I read the point that, for students, math begins with an answer. (Don’t worry, I plan to read the rest of the post. But I needed to respond to this; it is so mind-blowing.)

I learned under Dr. David Pimm of Open University (UK) and the University of Alberta (CAN) during my Diploma in Math Education studies. He argues that math begins with a question; in fact, it does not exist until a question is asked. All the demonstrating and lecturing about math in the world does not involve math until a mathematical question is asked.

This discrepancy is very revealing. It tells us what math is and what math education is. Most students learn to expect math questions and problems to be short, quick, to the point, solvable and structured around “clean” answers (often related in some way to integer components). They anticipate the answers before they anticipate the questions. I am not sure if they even consider the math, and if they consider the questions mathematical, or mathematically.

I wonder what they are really learning? Is it math? What to them is math? Is this why so many students are so disconnected with math and why they are proud to have failed it and ashamed to have aced it? After all, from their perspective, if answers they anticipate before math, what have they aced?

I think we have done students a great disservice if they ace math in elementary, secondary and even tertiary school without ever actually learning that math is all about the question, the quest and struggle to tackle it and the discovery of pattern that possibly limits to (an) answer(s). They completely miss the point and the empowering strength of math process and pattern. And in the end they really have nothing to use in their lives beyond the “math” lesson.

So, why do they need to learn this? That question makes so much sense now.

I shall now return to your post.

Further Reflection

It is easy to lose sight of our students’ understanding of what we teach. Sure, we anticipate their individual problems and our scaffolding of these problems. We tailor our lessons to help each student. We ask leading questions or offer leading hints to open the door for them to learn. But then we learn that they have a completely different fundamental take on what we are teaching (and how) and what they are learning (and how). Sometimes these takes are so fundamental in fact that we can not even conceive them, never mind address them.

And this is where professionally developing with our PLNs really helps us to grow, to learn, to better ourselves.

Sometimes what we think makes sense only makes sense because we think about it within a certain frame and from a certain premise. David Hewitt would consider this generated, rather than necessary, knowledge. Until the moment I read Nat’s statement, I thought that students shared the same fundamental sense I did; that is, I thought this sense was necessary and common. They might want the answer and to skip the learning, but they start learning when a question is considered and asked, regardless of who asks it and whether it is internal or external. Thinking, in short, starts with a question.

All it takes is one statement, simple to others, even the author, to change our view of what we are doing forever.

Are we teaching our kids that thinking starts with an answer? That an answer even always exists? Do our kids think backward from us — speculating an answer then working through, rather than on, the question and its “solution” until they match their guess or verify no match? Are we teaching them the wrong way, literally?

The way we teach (think) now, we solve problems. Do our students anticipate solutions and then test them, like a computer batch tests scenarios or a player navigates a game? These are very different ways of learning, teaching and thinking. Much skill and strategy is lost by exchanging deliberate problem consideration and solving or playing for rapid testing of many outcome and solution (scenario) possibilities. In a very real sense, problem solving is deductive, while answer testing is inductive. New skills and strategies are needed to solve problems this way. And new methods of teaching need to be added to our portfolios and lessons.

What do you think? Do students start learning by considering an answer or a question first? Are we teaching them the “right” way? Should we teach both directions, or are their still more directions, more senses, we need to consider?

Any thoughts on Nat’s claim and its implications?

I am seriously considering that there is some truth in his statement. Perhaps kids approach learning inductively and deductively. Perhaps some kids approach learning one way under certain circumstances, the other under others, and different kids apply different strategies or strategy mixes differently.

Either way, is there a premise change here?