Critical thinking vs education: Teaching kids math without "correct" answers

Brooke Powers assigned her middle-school math class a probability exercise with no single correct answer and was monumentally frustrated by her kids' inability to accept the idea of a problem without a canonical solution. After a long and productive wrangle with her kids about how critical thinking works and why divergent problem-solving is much more important than mechanically calculating an answer that you could just get out of a computer, she salvaged the exercise and made something genuinely wonderful out of it.

When did we brainwash kids into thinking that math was about getting an answer? My students truly believe for some reason that math is about combining whatever numbers you can in whatever method that seems about right to get one “answer” and then call it a day. They rarely think about what they are doing as long as at the end of the day their answer is “correct”. Today they were given a task with no real correct answer and they lost it. It did however lead us to have a very productive discussion about that fact that they are lucky, after all they live in the 21 century where they can solve any computation problem with technology with no issue. The problem I told them lies in the fact that they have no idea how to interpret that answer. We talked about the need for them to stop worrying about if I think their answer is right and to start worrying about whether or not they thought their answer was right. I told them I was sorry someone (maybe me) broke their desire to think about math and instead taught them that math was a means to an end where there was always one right and one wrong answer and then I told them to try their assignment again.

Who or What Broke My Kids? [Brooke Powers] (via Hacker News)

Notable Replies

  1. LDoBe says:

    I'm not sure what's meant by a math problem with no one answer. I mean, quadratic roots can be one, or two real or complex numbers, but the roots definitely can be found.

    I fell in love with math after getting on effective treatment for ADHD which allowed me the ability to finally focus two seconds on something that isn't immediately riveting.

    In high school, while the teacher would go through their preferred method for solving a given probem ten times, I would come up with multiple equivalent processes for doing the same thing. I'd prove them, and re-teach the methods I came up with when half the class came to me not getting the standard method the teacher failed to get across.

    I probably should have been in more challenging math classes, but I enjoyed teaching, and had the determination to make sure other students I helped actually understood the math instead of just letting them copy my answers (which always made me feel dirty. Sticking to my guns, I passed up making a lot of potential friends because I wouldn't give them a freebie on my own work.)

  2. Math class is bollocks and could easily be incorporated into other subjects. Modern education is hamstrung by the compartmentization of its core "subjects". This fundamentally deprives traditional "core subjects" of the rich interconnectedness that actually permeates the world. We should be teaching math in science, history, music, and art classes. Numeracy and history is amazing. Music theory is math.

    Skills practiced in isolation lack meaning. I mean, if you are studying a new language, what is more effective: reading a dictionary or sitting down with someone and having a conversation? Math, like language, is a tool for understanding; but in isolation it is just dull.

  3. Looking at the actual work, it looked to me the confusion was the set of instructions. The second run looked like to me, the students finally understood what the teacher was trying to get--the reasoning behind the answer.

    The answer could be different and was not the desired result, rather the reasoning for the result was the true answer. Dang, I barely understand what I just typed. I'll try to be clear, I think the student's didn't understand what the teacher wanted. What I think the teacher wanted was the reasoning for the answer, where the answer could be anything as long as you can make a good reason for it.

  4. There are many problems with no one answer.

    For example, I remember a professor in college throwing this out: "how many ping pong balls would it take to fill this lecture hall?"

    Some people shouted out things like "a million", "a billion", but he was trying to get us to think abou the problem, not simply guess.

    "How big is a ping pong ball? About an inch in diameter. ABOUT how many in a cubic foot, then? 12 is close to 10 so 1,000 to a fair approximation. Don't worry too much about how they're packed."

    "I'm about 6 feet tall, the ceiling is about three times my height, so how high is the ceiling? About 18 feet, call it 20. The floor tiles are about a foot square, so how wide and long is the room?" About 60 by 100 feet.

    So 20 x 60 is about 1,000 (we over-estimated the height and under-estimated the width). Multiply that by 100 and we get 100,000 cubic feet. Each cubic foot holds about 1,000 ping pong balls, so we estimate about 100 million balls.

    That's certainly not the exact answer but thinking about a problem like that helps one figure out when something just doesn't make any sense. I've see many "cool inventions" where a simple estimate blows holes in the claims... "Let's put spring loaded generators in roadways in front of turnpike toll booths. The cars driving over them will provide all the power for them..."
    Or "solar freakin' roadways...", which seem to have recently been popularized on the web. Being able to estimate, not getting any one particular "correct" answer, but within an order of magnitude or so is a very useful skill.

    In the article, being able to rank "unlikely" "50% chance" and "will I go to the beach?" is a good skill to have so one can visualize and discuss a problem.

    Beyond all that, Kurt Gödel's Incompleteness Theorems mathematically demonstrated that some problems can provably have an answer that can't be determined, some statements are provably both true and false, and other mind blowing concepts. Douglas Hofstadter's excellent book Gödel's, Escher, Bach; an Eternal Golden Braid provides a wonderful explanation and contemplation of number theory and it's limitation.

  5. Maybe our daughter's kindergarten teacher is an example of bucking the trend. I was at a school party. All the kids had gathered around the teacher as she prepared to draw a name out of a box for a raffle.
    She asked them, "Won't it be cool to win?"
    "Yes!", they all replied.
    "Are you probably going to win?"
    "No!", with the same amount of enthusiasm.

    There was certainly a correct numerical answer to the probability that any one of them was going to win, but they didn't need to know it to prepare emotionally for the outcome.

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