Tuesday, November 19, 2013

Is Your Own Math Work Shareable?

(Lots to disagree with here, kids. Get excited!)

Students don't like to write about their reasoning. They don't present their work in a way that allows anyone else to comprehend their path to a solution. But we want kids to write about their reasoning. Conflict! Drama!

Why do kids hate writing about math? Here are some possibilities:
  • Kids care more about answers than about the underlying reasoning
  • Kids are lazy
  • Kids don't know how to write clearly about math
Without a doubt, these factors all play their part. But I think that there's something else going on.

Here are a few pages from one of the notebooks that I do math in:

There's a ton of scribbling, some diagrams, some arrows, all of which make a ton of sense to me right now but would take a significant amount of work for anyone to decipher.

I tend not to do math in a way that would make sense to anyone else. The question is, should I? What would I gain by presenting my notes in a cleaner, more legible way?

I can see two reasons for writing this up in a cleaner way. 
  1. For sharing my thoughts with others.
  2. For clarifying my own thoughts.
Let's immediately dismiss the first possibility as unrealistic. These aren't new discoveries, and they aren't even new ways to think about old problems. This is me trying to understand some ridiculously well-trodden math. Why would I share this?

The second possibility is a more serious one. And, look, I'm numero uno in line for the "Writing Helps Me Think More Clearly About Things" rally. (I'm getting clearer on these ideas as I'm writing it right now which is fairly meta.) But writing a clear statement about math is only occasionally what I do when I want to get clearer on a mathematical idea. When I want to test my learning, sometimes I rederive my result. Sometimes I try to tackle a new, but related, problem. Sometimes I take a walk and think about it.

What's behind my own tendency to skip the writing-up process when I'm doing my own math? As before, it might be laziness, it might be a lack of skill. Upon reflection, I think that it mostly has to do with my preference for problem solving. I typically figure that if I don't understand something, then eventually it'll get in the way of my ability to solve problems. Sure, I could prevent that by trying to sort everything out now, but that wouldn't be nearly as much fun as trying to solve another problem. "I'll understand everything with the depth that it currently needs," seems to be the principle by which I usually operate in my own math work.

So, here are my (somewhat loaded) questions:

  • What does your own math work look like? How often do you create something that could be shared with others as part of your learning process?
  • Should I be changing the way that I do math in my notebooks?
  • Should we be asking students to practice math in a way that differs from our own?
  • Does writing about reasoning typically solve a problem for the teacher or the student? 
  • What exactly is the value of writing one's reasoning in math, as opposed to articulating it in speech or in thought?
Lots, lots to disagree with here. Please do! Let's get closer to the truth together.


  1. When I was taking graduate classes in math, I would re-write my homework 4 or 5 times. I always scribble at first. I needed to re-write it clearly to help me think it out carefully.

    These days I write things up now and then to explain them to students. Doing that helps me to understand better, actually. I find that I was glossing over a point that I was fuzzy on. Typing it all up, I figure out every detail.

    I think asking students to explain their reasoning will help them to understand better, but shouldn't be used too often. I am right now reading my students' write-ups for the murder mystery, in which they are to play the prosecutor explaining the math to the jury - how we knew the time of death (which led us to knowing the murderer). I wouldn't want to do this often, but I think some of them improved their understanding substantially.

    1. Interesting!

      If you find that writing something helps you think more clearly about it, then why not have students do it all the time?

  2. It's a lot of work, and would burn them out. I only do it once in a while myself. I wouldn't want to ask more of them than I do of myself.

  3. My math notebook: https://www.dropbox.com/s/8g4udra7nyxmhir/131119_0001.jpg

    A big question for me right now is how to assess this type of written work. What are my benchmarks? Are they consistent from assignment to assignment, and how does being consistent (or inconsistent) help or hurt my students?

  4. I think that asking ourselves or our students to change the way we solve problems and do math would be detrimental to the problem solving process. The simple act of forcing myself to organize, plan and explain in a clear way poses a certain level of cognitive demand. This then reduces the amount of mental energy one can expend on analyzing a problem, planning a solution path, and solving the problem. Being able to clearly articulate ourselves and our ideas in writing, in a legible manner, with others in mind is a refinement. Just as I do not ask students to solve a problem in the most efficient way the first time around without additional work, I would not ask them to perform such a refinement of their solution (writing clearly, articulately, and in an organized fashion) while they are problem solving.

    Having said this, I do believe there is value in refining one's work after solving a problem. It is similar to teaching others, which has significant benefits for students. Writing a solution clearly with a quality explanation is teaching essentially (just in a written form). As a result, I think it is reasonable to, on occasion, have students refine their work. This is no different than asking students to find the most efficient method to solving a problem (or at least a more efficient method). Such activities are useful and they have their place in education, but I do not think they have to be done for every problem solved.

    As for your last question regarding the difference between articulating a solution method orally vs. in writing, I would defer to an English teacher on some of the benefits here. However, I do feel that when we communicate things orally, we have a chance to clarify while explaining based on questions from our audience and/or based on non-verbal cues we receive. Such second chances do not exist when we are writing as the audience can not ask questions nor can we read their faces. In writing, we have to articulate clearly and concisely the first time around. This requires a greater degree of refinement on our own part (something I find myself doing right now). As a result, I believe there is room for both, with articulation orally being used more frequently and also as a way to scaffold to more refined, written explanations.

    1. I agree completely with your description of problem solving -- keeping clear notes on what you are doing is often an additional cognitive load on top of actually solving the problem.

      While I do agree that it is worthwhile to (occasionally) ask students to present refined solutions (this is the problem, this is how to solve it, etc.), I think it is also worthwhile to (occasionally) ask students to go through the problem-solving process. This can often be more helpful than presenting novel techniques for solving certain problems. (First I tried X, because the problem was Y, but that didn't work, although the failure hinted at alternative method Z that did work.)

    2. I agree with the idea that it is beneficial to explain the problem solving process sometimes also. I have done that in the past, and while it can be lengthy, it can also lead to awesome reflection on ideas, rationale, and problem solving in general.

  5. Okay - so the conversation about the second possibility will take a while for me to process. I do want to get an idea out quickly while it is (hopefully) cogent regarding the first possibility. You dismiss - seemingly out of hand - the process of sharing YOUR ideas about well-known facts. Isn't this essentially at the heart of what we do with our students on a day to day basis? I blogged recently about this powerful quote regarding enquiry and the heart of it is this "But it is difficult to enquire genuinely about the answer to problems or tasks which have well-known answers and have been used every year. However, it is possible to be genuinely interested in how students are thinking, in what they are attending to, in what they are stressing (and consequently ignoring). Thus it is almost always possible to ask genuine questions of students, to engage with them, and to display intelligent directed enquiry."
    If it is an essential part of our job to behave this way - and I think that it is - then isn't it a pattern of behavior that we can hope our students adapt? I know that at the AP level - especially in Stats - we spend a good deal more time and energy explaining our process and decisions than we do on actual calculations of those conclusions. I don't feel bad about asking students to present their work in a way that might not be the most natural or in a fashion that is consistent with how they would work if they were working just on their own.

    1. Tend to agree here (I think). I found it quite jarring, even shocking to see the casual dismissal of sharing thoughts with others as unrealistic.
      I'm reminded of a math contest question that a student wasn't sure they got right. Neither of us knew the actual answer. So I went through the process myself, when I had time, literally writing things out as they occurred to me, with a few arrows so the student could follow my process when they got the sheet later. Was it a well formulated solution? No, I wanted the student to follow my reasoning. Could I have drawn up a formal solution? Sure! But why? And just because the guy who created the math contest had a solution written up somewhere doesn't make my explanation less worthwhile.
      Mr Dardy's mention of statistics seems key too... just because one guy has some numbers on global warming, doesn't mean it's now an "old problem" so we'll just run to his explanation whenever the subject comes up. What if someone doesn't buy his logic? What if someone has a BETTER explanation?
      Also, if you can't explain how you reason through problems we DO have solutions to, how can we hope to understand you when you create solutions to problems we DON'T have solutions to? That's the whole problem with last year's proof of the "ABC Conjecture"! No one has any idea if it's actually valid because no one knows what the heck Mochizuki was talking about.
      I think there's definitely something to clarifying your own thoughts. But if you can't articulate them later in a way that makes sense, we all miss out.

    2. I'm not saying that mathematical communication isn't important. But I am saying that it's hard to feel motivated to clearly communicate routine results or common knowledge.

      I solved a problem today. I'm proud of my solution. I want to share it with friends of mine. But I'm completely unmotivated to write it up clearly and share it with others. The thing that I "found" has been known for centuries. What would be the point of me writing it up and sharing it?

    3. Michael, you've just intrigued me. What did you solve? I am totally motivated to write up what I've done when I solve something myself (regions in a circle, Pythagorean triples, and Spot It - ask if you want links). But that's only in the past 5 years. I don't care if I'm not the first. I want to describe my thinking. Until recently, too much math writing has been above most people's heads. I like trying to explain in an accessible way.

    4. I was working on pi approximations for class, and I accidentally found myself looking at this.

      You write "I like trying to explain in an accessible way."

      Great! If you think that you have something special to share, then I understand why you'd share it. But there are more-or-less clear explanations of the pi approximations out there, and I'm not sure that I could do much better.

      More to the point: would our students be motivated to write their explanations by the drive to explain something in an accessible way for others?

    5. I think it could be motivating for students. For instance, why would I sit through a lecture by some guy I haven't heard of, when I could hear your explanation instead? The point to writing something up isn't always the message, it may be partly the messenger. "Better" is subjective.

      I dare say I gamble on that all the time with my web serial. There's no new maths in there. Just a reframing in terms of personification. I'm hoping people would prefer to read about uses of Versine from me than some other source.

  6. Well, I am very curious about your journey to "that". I had no desire to read the wikipedia article, but I'll bet I'd want to read your take on this problem. And that's what our writing offers others - a personal take on a problem.

    The one time an explanation didn't seem to be "out there" (I couldn't find anything on centroids), I delighted in writing it up. But I wasn't solving a problem that time, just explaining some Calc II content.

    The regions in a circle problem is too fun to think about. I didn't even want to put that solution on my blog for more people to find. Same with Spot It. Pythagorean triples are written up all over, and my solution was not the usual one, so I had fun writing about that. Although I think I never did write it all up. I like leaving questions for others to ponder.

    One reason students could get interested in writing up their explanations is to put together their own textbook. Someone has mentioned doing that, I believe. Sorry I can't remember who right now.

  7. I just re-read your original question ("Why do kids hate writing about math?"), and realized I hadn't gone there. As a student I hated that what I wrote for class would be read by exactly one person, who I now know dreaded reading the 20 or so papers they needed to grade. Whatever subject it was, I hated that aspect of it.

    If I had been asked to write about my mathematical thinking, I'd have been puzzled about why the teacher wanted to see that.

    I think audience is a huge part of writing. If our students will read each other's writing, and might learn from it, then maybe they wouldn't hate it.