Saturday, November 12, 2011

Show your work!

One of my colleagues recently said to me "You know, nobody does proofs anymore." Seriously? I think that would be a catastrophic move in the wrong direction, and it reminds me of a nagging problem that I have avoided.

Since the beginning of my teaching career, there are a handful in every class: those students who, despite numerous pleas to "show work," refuse to write anything but the answer on the page. I give what I think is a thoughtful (and lengthy) practice set as homework, only to find that student who has successfully reduced it to a smattering of digits that fills up only a quarter of a notebook page!
"You need to show your work," I'll say.
"I did it in my head," he replies.
"You did what in your head?" I'm not letting him off, I don't care if he's a genius or not. "You need to write down what you did."
At this point, blank stares or shrugs are about all I ever get, but depending upon my mood, I might launch into one of the following attacks:
  • "You need to know what you've done, so you can correct yourself if mistakes are made." (You're not as clever as you think.)
  • "A good mathematician must show diligence in precision, and exercise good habits of mind." (You'll never amount to anything.)
  • "It may seem easy now, but it won't always be and showing steps now will make the steps more natural in the future." (Pleeeease... I need to validate my existence.)
  • "You only get partial (or no) credit. You need to show your work to get full credit." (You will do it because I SAID so!)
The truth is, I don't really understand this student. It shouldn't surprise you to learn that I never was this student. I was the neat girl, with the 5mm mechanical pencil, gum paste eraser, and tiny, Times New Roman handwriting. My geometry problem sets looked like architectural journals and my calculus tests got photocopied and passed around as examples. The aesthetic appeal alone of those neatly written problems in nicely aligned columns with balancing symbols and digits was enough motivation for me. Those other students have always frustrated me to no end!

So then comes the blow: the thing I've been forcing myself into these past months. I ask myself "WHY?" And I don't mean "Why do these kinds of students frustrate me?" I know the answer to that question. The real question is "Why is it important to show your work?"

That particular question has evaded me many times, probably because of a lingering fear that the answer might be "It isn't." Thankfully, I don't think I'll need to go there.

At the heart of the issue is the realization that there is a quintessential difference between an answer and a solution to a problem. A solution illustrates, generalizes, communicates, and verifies the results. An answer is just a number. In mathematics it is important to find the correct answer, sure, but more importantly I want my students to gain skills in writing accurate and convincing solutions. It's not enough to just know the answer, you need to be able to prove it beyond a doubt to yourself and others. Take this example:

Student A has successfully shown that 4 is AN answer to this problem (I took the liberty of writing what I believe is going on in Student A's head, because usually this version of the solution is simply a scrawled 4 on the page) Student B, on the other hand has successfully shown (and verified) that 4 is THE answer to this problem. The difference may seem subtle and unnecessary for a typical beginning algebra student, but it quickly becomes essential, like in the case where more than one solution exists or when extraneous solutions occur.

The result of treating 'work' as 'proof' could very well be an improvement in reasoning and logic in students across the board. Certainly it would make a difference to my geometry students when they are first confronted with conditional statements, laws of reasoning, and proofs. It seems that more and more, students (and people in general) are forgiven for their inability to write a logical and convincing argument.

So, is my colleague right? ARE we moving away from proof? Are YOU? Seriously, I want to know.

13 comments:

  1. I like what you have said here! I tutor middle school students and all I hear is that I don't have to show my work. It's drives me crazy! So I spend a lot of time getting them to show me the steps as they are thinking so I can correct and help along the way. Well said!
    Adrianne at Kidscogworks.com

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  2. Thanks Adrianne. I think we can all benefit from teaching our students some improved metacognition skills.

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  3. I was that student. Now I'm an engineer. I showed work once the problems got hard enough to need it. I'd say if they're getting it right, let them be.

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    1. Exactly 1:56 a.m. This is time consuming busy work. And I hope you'll consider this, Emily. We moved from homeschool using Singapore Math, which rewards metal calculation, to a typical American suburban middle school - it's looking like a HUGE mistake. BTW, since 2008, Singapore has ranked first in the world in maths. For GT kds, that writing out of EVERY problem is nothing more than frustrating busywork. If the student is getting answer wrong - then by all means, write it out to show where the mistake was made. Otherwise, leave the kid alone.

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  4. Before these comments digress too far into "showing work is for dummies," I just want to restate that I believe that there is merit in showing work BEYOND finding the answer. Yes, I see that you can do this problem in your head. But can you trust that I purposely have chosen problems that you CAN do in your head so that the focus can remain solidly planted on a higher goal: writing a convincing mathematical argument?

    I have nothing against mental calculations, and certainly have high regard for Singapore Math. AND, I agree that numeracy is a critical skill that tends to get left behind in traditional American curricula.

    The answer is great. The work is even better. When I assign a page full of two-step equations to a room of 13 year-olds I really don't think that none of them can tell me what number will make 10 when I multiply it by 2 and add 4? In fact, I'm counting on the fact that they can.

    I want them to be able to explain to me (over and over if necessary) that the reason why they know the answer is 3 and only 3 is because of the properties of equality and the order of operations and the concept of inverses. And I want them to be able to communicate the steps in the right order and show me how these steps lead to an unarguable conclusion about the truth of that answer. Now that's a valuable skill that I think we sometimes miss.

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    1. The thing is I can do work In my head also and well I actually couldn't write it out, and still can't. That's the problem with most students they can do it but can't write it out. Its something that only some people understand and can relate with.

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    2. I don't know if this is true that "only some people understand". I think its more correct to say that some people have not developed the skill of showing work. The skill is similar to elaborating on ideas. When I was a kid I might have just answered "yes" if someone were to ask me if I liked Disneyland. Today after learning how to express my thoughts in a fuller way would go on to explain why I liked it and what things I liked about it. The "yes" is the answer and the rest of the stuff is showing work. If you haven't practiced and learned how to do it its going to be hard.

      I was one of those kids who didnt want to show work. All through middle school and high school I just put the answer -no work. Eventually after I grew up and matured I realized that I thought I knew more that my teachers. Well I didn't! In my mid-20s when I went back to school, I listened and trusted that the math teacher probably knew better than I did and followed their directions. It worked. Once I started to do what they asked of me it was much more comprehensible.

      The teachers were always saying the same thing. The difference was in me; I chose to listen. .

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  5. I also was that type of student and now teach math. Something to consider: I was proud of doing math with little work and felt I should get recognition if anything, not a lecture. I felt these teachers were judgemental and had control issues, which may or may not have been true.

    Recognizing extraneous solutions typically involves remembering to plug values back into the original equation. This has nothing to do with showing the work used to get the candidates for a solution. In fact, those that don't show work are probably more likely to recognize an extraneous solution as they are more focused on the original equation. Maybe the most common "forgotten" solution is the negative square root. Again, this has nothing to do with showing work. I just don't follow your reasoning on this.

    I am willing to bet money that very few of your 13 year old students can clearly explain why the work shown implies only one solution in your example. On the other hand, a student with good mental math abilities will be able to give a convincing reason for only one solution - if you pick a number more than 4, the left side will be more than 5, etc.

    You don't understand this student, so maybe you should be a little more accepting of them. If you want them to explain their thinking, give them a question that truly requires some explanation. If you want to see if they can show steps for an equation, give them something that is too complicated to do in their head.

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  6. I can relate and enjoyed your article. Also, I found this article (http://www.byrdseed.com/to-show-or-not-to-show-work/) encouraging. Maybe you will too. I'm still learning as a math teacher and tutor, and didn't go higher than college algebra with an intro. to trig. and calc., as a student, so I am still learning from people more experienced in math, to help me prepare my elementary and middle school students.

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  7. look. I am a kid at school and I think that you shouldn't waste time getting us to show their workings out all of the time. you should definitely show us how to do it and make us practice a bit, but all of the time is not needed. I think that if we write our answers and we are right, than great, their way of doing it is working. if we are wrong, then you can go over with us and show us another method. I know the main reason is to see where they are going wrong but the main thing is that they get the answer right and as said before, just show us another method if we are wrong.

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  8. Found this essay while I was trying to find a convincing argument for a student who makes mistakes about 50% of the time, largely because he is convinced that if he is doing math on the computer, he "is supposed to do it in [my] head," regardless of what parents and teachers tell him. (He's doing it on the computer because a book "is too much and I'll never be able to get through all of that" and he (for unknown reasons) believes it has to be fully, 100% completed before he'll be able to do anything ever again. Yes, he has some issues, but he's a great young man, for all that.

    In his case, it would really benefit him to put it on paper. I think that's true for a large number of students. It's the convincing them of that that is the difficult bit for many of us.

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