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?

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4 thoughts on “Math Begins With an Answer?

  1. Shawn: It is important to be clear that I was not stating that answer-first mathematics is the best way, but your reflections coupled with the rest of my thoughts seem to be on to something. Maybe our questions don’t spark learning, but rather a waiting game to get to the inevitable answer. I have since changed my wording to be less universal. The post now reads: “For many students, no matter their age, math begins with an answer.”
    I also restated your thoughts here. http://musingmathematically.blogspot.com/2011/07/messy-mathematics.html
    Teaching needs to direct thinking and learning toward the problems mathematics affords us, and not the wide selection of “clean” answers.

    • Hi Nat,

      This exchange of posts and comments highlights for me two aspects of learning and teaching, and not just of math, which I think need more attention.

      1. What does a lesson look like from a student’s perspective?

      It is relatively “easy” to anticipate student misconceptions and stumbles when designing and delivering a lesson. I have little trouble anticipating student reception of particular lessons (I use lessons here to refer to activities as well). So, I think that, for the most part, I understand how kids received my teaching or their learning. But I really don’t know what a student experiences / learns during a lesson. When I was a student, I viewed math as subject of puzzles and games. Though intellectually I understand that this is not how most people view math, deep down I don’t feel it. I have no idea what math is to other people, how students perceive what they are taught, if they learn whole-heartily, if they learn the way I think I teach, and if they expect and anticipate an answer, even if I do not supply any. This last admission is interesting; it has two meanings, only one of which until now I was actively aware of and actively guarded against. I knew most students often waited for an answer to be given to them and I tried my best not to supply any. But I never actively considered that they thought from an answer-first perspective; I was passively aware of the possibility and the obviousness of this perspective. I will now actively consider both. But I am left again with the question: what does a lesson look like from a student’s perspective, rather than, say, my perspective or my idea of what a student’s perspective is?

      This really goes to asking how am I teaching and how can I do better.

      2. Fri 29 Jul 13:36 @NatBanting: Do I have the wherewithal to know when I am teaching to an answer? or have I been blind to this the whole time?

      Fri 29 Jul 08:46 @NatBanting: I loved what you said on my blog. I hope you didn’t think I believe all math begins with an answer; my wording may have been confusing.

      Fri 29 Jul 13:26 @stefras: Hi Nat. Thank you for your post. It really got me thinking and I responded in a post of my own. I don’t think you believe all or even any math begins with an answer, but your example was undeniably honest. I have done the same myself, even as a teacher. It brings the question about how true the statement might actually be.

      I think this is a comment more on how we set up and deliver math ed rather than on math and mathematical thinking themselves. In a very real sense, we contrive the math we teach. So answers are predetermined. We therefore teach with answers in mind, or at least with the knowledge, rather than the faith, that an answer, or outcome lesson, exists — or not — and can be found.

      Even open ended questions, such as the one in my last math challenge, have some planning built into them, some outcome they are designed to “teach”, some Math for students to discover. (I love Dan Meyer‘s latest post in this regard.) This means the activity the students engage in is planned, within certain probability. We call this tailoring or individualized differentiation of instruction.

      What strikes me is that students consciously or subconsciously pick up on this and mimic it. Or perhaps, this is just a natural way of posing problems. But is does bring to light an aspect of our teaching that we should be aware and wary of. I like the way you put it: Do I have the wherewithal to know when I am teaching to an answer? or have I been blind to this the whole time?

      I believe I was blind this whole time and I’m not yet sure I passionately disagree with the predetermined-answer phenomenon. I think the big professional advance here is now I am aware of the “problem” and that gives me insight into student understanding and misconception.

      This is a good thing.

      Thank you for your post,
      Shawn

  2. Very interesting points made. I have a few questions to add to this discussion. Have we promoted the “answer first” mentality in our students by continuing to use multiple choice standardized assessments? Would it not be more beneficial to students if all questions were open-ended and asked students to simply show their work and thought process in attempting to answer the question posed? Definitely thought provoking. This perspective of teaching Math would go a long way to answering the question most students ask, “why do I need to learn this?”

    • Hi Mack,

      You may have a point. Perhaps our use of quick response assessments has contributed to student anticipation of answers in math lessons and tests. I think also our choice of clean-answer questions contributes to this phenomenon. Our questions are also “drill” questions for the most part. I think there should be a balance between open-ended, challenge problems and drill questions, since both contribute to student realization of the outcomes we wish them to master. Further, we should teach them to find and pose their own questions and reflect on how they solved the questions and problems they work on. (This is the premise of the Math labs I so vigorously promote.)

      But, we can also create quick response questions on tests which require reasoning and work, rather than formulaic response, to answer. In addition, I am unsure that student anticipation of answers does not begin before students are exposed to quick response assessments. It is easy to scapegoat standardized tests — and I believe they leave a lot to be admired — but they are part of a system, both symptom and driver.

      I think showing work, or at least a cleaned solution, is vital to mathematical communication. An answer is useless without some context and justification, or at least without some reference to the question. This is why I put more marks on solutions than I do on answers. Most teachers do this, so this is nothing new. But I think it would be beneficial if students at times also showed reflection on how they solved the question, what they learned, what Math processes and patterns they used and where their math connects to other Math concepts, processes and patterns.

      As for students asking “why do I need to learn this?”, I always learned that students ask this when they become disengaged from the math they are doing. They either mastered it and want to move on, or they stumbled somewhere and can not move on. When this happens, after failed attempts to “correct” the situation, they naturally begin to wonder what use the math they are learning is. In essence, frustration turns to demoralization. Adults experience this too and react the same way. Add to that the possibility that students may anticipate answers, rather than challenge and engagement, and “why do I need to learn this?” becomes even more potent.

      Thanks for the questions and the comment.
      Shawn

      (I make a distinction between solution and solving, proof and proving. The former are the cleaned works we communicate. The latter are the Math we do and the Math we learn while working on a question or problem. The solving and proving are the important activities where Math is done and learned, but what we grade are the solutions, proofs and answers. I think this also contributes to the answers-first anticipation of our modern culture.)

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