Thursday, March 3, 2011

The Burden of Proof

Math teachers do not care if you know how to get an answer. They care more that you can prove that you know how to get an answer. Here is an example to illustrate the difference:

I was helping an 11-year-old with his math homework. The assignment was to find the greatest common factor (gcf) for eleven pairs of numbers. He is really good at this. In fact, he figured out the answers to all eleven problems completely in his head, in less than one minute. This should be a good thing. This means he "knows" this concept. On to the next concept! Right? No, wait... what does that say on the assignment? Something about "showing your work..." Yes, that's right, even though this child did absolutely no work to get these answers, he has to write down some work. But not just some work. A lot of work.

So for the pair of numbers (60, 18), the gcf is 6. My little friend got this answer immediately upon looking at the numbers, as anyone who knows the concept would. But instead of writing down "6" and moving on, he has to write all this:

     18             60
      ^              ^
   6   3         6     10
  ^              ^      ^
2  3           2  3  2  5

18 = 2*3*3
60 = 2*2*3*5

Both numbers have one 2 and one 3 in common, so the gcf = 2*3 = 6

An assignment that was completed in less than one minute has turned into an hour-long session of unnecessary writing. He has to write this process over and over again, eleven times, to prove that he knows the steps. Even though he did not use these steps to get the answer in the first place! I don't know a better way to make a kid feel like math is boring and hard and requires tedious processes.

I ask: WHY? Why do teachers do this? It would take one minute of a teacher's time, sitting with this child, to realize he knows this concept. Instead, so much of his precious childhood is eaten up filling out paperwork like this, night after night.

This is a child who is brilliant in math. He often arrives at answers to complicated problems in his head, without writing down any work, and without being taught any steps. Sometimes it takes me ten minutes, after he has already told me the right answer, to explain the steps he needs to write down for his teacher's benefit.

On tests, he often loses for not showing work. I have already shared some of my feelings about tests in general (Part I and Part II). But here is another great example of TESTS (Teaching Everyone Some Terrible Stuff). Now this mathematically gifted child is bringing home test scores in the "C" range, even if most of his answers are correct. This is harmful to both his confidence and his interest in math. How can he ever believe he is good at math if he brings home such low grades? And this will only get worse as he gets older.

I can remember being upset about this when I was a student, but fortunately it did not completely turn me off from my favorite subject. And I have both a Bachelor's and a Master's degree in math to prove it.

This practice is ruining math for our children. Can we please make it stop?


  1. From my discussions with my daughter, she now talks about when the grade is measuring obedience instead of understanding. That helps with self-confidence (not that this is an issue for her).

    The pedagogical problem might be Fair means Everyone Does the Same. So some students get practice they don't need and others don't get the support they do need. Maybe you could ask your teacher to switch to asking for a certain length of time working on an idea... which would open up more constructive choices for your son.

  2. Commented over at Innovative Educator :)

  3. @John, That is an interesting observation, about the grade measuring obedience. I may share that with the students I tutor. Isn't that fact ridiculous? School should not be a place where our children are trained to obey.

    And you are right about that it seems like "fair" means "everyone does the same." But how crazy are the consequences of that! It ends up being not fair to anyone, since some are getting more than they want, and others getting less than they need.

    The child I am referring to above is a child I tutor,not my own, so I don't have much to say about it unfortunately!

    Thanks for your comment.

  4. Great post. At what point does a student throw his hands up and say, "Math could be cool, but you adults totally ruined it." On the other hand, if you're a teacher with 33 kids of varying abilities, most of which don't instinctively *see* the GCF off the bat, and you're teaching in a school that absolutely must meet AYP this year, etc., etc., what do you do? Ed school professors can talk all they want about "differentiated instruction," but it ends up being a really hard thing to do, if there's in fact anyone who's done/doing it well.

    I share your frustration. Instead of wondering why teachers teach "this way," though, what if we asked: Why are teachers put in a position where they have to teach this in the first place?

    It's not that GCF isn't important. It is, as it's fundamental to a good understanding of number theory. However, part of the problem is that we as teachers, curriculum developers, etc., haven't done a good enough job at distinguishing between what is pure math (abstract) and what is applied math (concrete). Thus, everything ends up looking the same, which is to say, random and nonsensical.

  5. @ John... "measuring obedience instead of understanding." I love this and may use it often as this is what much of school is about it seems.

  6. It sounds like that boy should be in an advanced class where he wouldn't be required to "show his work".

  7. @Voice, He's IN the advanced class! And they wanted to knock him down a level BECAUSE he has trouble showing his work! How messed up is that?

  8. I appreciate the sentiment here. I hated math in school, mainly because I was lazy and didn't want to spend my time "showing my work," and probably in part because I got the concepts and didn't feel like it was useful to keep practicing. I did better and had way more fun in high school physics. But, the principle behind "show your work" is a sound one (even if it's poorly taught). It's one of the basic principles behind all of science: you have to be able to show your work so that others can try to replicate your results. The bigger, and much more important, lesson that this kind of assignment should be teaching is how to reason and how to present an argument. When it comes right down to it, this is the difference between "trust me, it's right because I say it's right" and "here's how I know it's right and how you can confirm for yourself that it's right as well." It's the kind of skill that helps refute ridiculous ideas like "creationism is science too."

  9. @Scott, Thanks for commenting. I totally get what you are saying too. I think learning how to argue a point is a great thing, but I don't think everyone should have to learn how to argue these types of things in exactly the same way, and I don't think this kind of excessive "work" requirement is helpful at all. Let them show they understand a concept by consistently getting the right answers without showing the work (especially in math where there IS a well-accepted "right answer"), rather than boring them with learning the teacher's way. Let them be proud of their methods and their understanding. Let the kids make arguments about their own interests, about their own discoveries, not about the greatest common factor...

    Also, I bet lots of people who believe in creationism went through math classes where they had to show work as well, and look what good it did them... ;)

  10. Math is a very strong point in my family. My dad, my brother and I, and my youngest son all "get it" without going through all the steps.
    I attended public schools. My dad fought long and hard with each and every math teacher that my brother and I had. He usually ended up challenging them - "if my kid sits in front of you and answers 100 problems correctly, you stop making them show all the steps". It worked with almost every one of those teachers.
    I've chosen to stop the cycle and the fights with my son. The way things are taught, and the way kids are taught to be followers, in a traditional school setting sickens me. My boys have always been unschooled - and have consistently "got it" faster and at a more advanced level than their public school friends.

  11. I remember a particularly horrific moment in 5th grade when I was accused of cheating (using a calculator) on my math homework. In fact, my older sister had recently taught me how to do long division without writing it all short-hand. Embarrassed, I explained it to my teacher in the hallway and she said, "Well, if you're telling the truth, then you won't mind teaching that to the whole class." So, I did.

    She didn't say much to me after that.

    And to this day, I still feel like I have to prove myself to EVERYBODY.


  12. @Anonymous, Thanks for sharing your experience. It's great to hear about other moms who are loving life without school. The way math is taught in schools is especially bad, with the idea that there is one right way to do something, and all the drilling and busy work. It really ruins math for so many kids.

    @Andrea, That would have be awful for me too. It's amazing how one experience can stick with you or so long. That's reason enough for all of us to want to be nicer to children! :(

  13. My son has been "bitten" a few times by this, getting 1/2 off for not showing all his work, even though one part of the math program is "mental math," where they teach them how to do math in their heads. I know the intent is to ensure they know how to do it, but...

  14. The point of the question isn't to figure out the gcd of 60 and 18, after all, who cares? The answer doesn't matter.

    What matters is demonstrating that you have learned what they taught you, which was a specific technique of breaking down a larger problem into a smaller one, and then repeating this. Recursive thinking. Once you get this technique for finding a gcd, maybe you can write a computer program that does it. And maybe you can learn more complicated techniques, like the euclidean method, and then maybe someday you will invent ways of solving problems that we can't solve well today, and write a calculator program to do it, and one day your techniques will be taught in school to kids who say, "but can't I just write down the answer, if I know it, or use a calculator?".

    Maybe they should just make the problems harder and make only the answer matter, so that the problems clearly break down into ones you can solve in your head, and ones you need a pencil to solve. But speaking for myself, once I've done half a page of math, I've _always_ made a small error somewhere, so I'd always get the answer wrong if the problem was too large. A small problem, where I can easily prove I know the technique and not have to work for 20 minutes on it is better.