Association of Teachers of Mathematics in Maine
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Week 10: Discussion Question (Option 1)

25 Jan 2018 4:30 AM | Anonymous member (Administrator)

What do you make of the argument that counterintuitive or paradoxical math is a motivator for proof?

Comments

  • 25 Jan 2018 11:10 AM | Anonymous member
    I definitely agree with the argument that counterintuitive math is a motivator for proof If it is handled right. It must be part of a culture of embracing challenge, asking questions, persevering - not as a "gotcha," followed by "prove it to me." I love the idea that mathematicians must, "question our assumptions." I intend to incorporate and teach this idea across all subject areas. I predict it will stretch students thinking about concepts, themselves and others. I also like the thought of teaching students to embrace the conflict arising from cognitive dissonance. Again, doing so can have benefits across all content areas and life situations.
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    • 01 Feb 2018 11:53 PM | Deleted user
      Hi Julie,

      For those of us that have been around for a while, I think back to the introduction and chapter 1. Our childhood math experience was based on the rule that we should "not question teachers or adults in general". The "because I said so" was often frustrating.

      We now have such a great opportunity to embrace and share students' questions. I also find myself writing notes and explaining processes in greater detail, so I don't assume prior knowledge. It is easy to take for granted that students "should understand and make connections", when they truly don't.

      Pam
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  • 26 Jan 2018 6:09 PM | Anonymous member
    I think that counterintuitive or paradoxical math is a great motivator for proof in math. Any thing that makes, you think, use mathematical thinking to create a conjecture and then investigating to prove your thinking is going to get kids thinking deeply about the math they are using. This type of thinking is what we remember. When students simply memorize an algorithm with no thought behind it, they don't remember it. Students need to learn how to prove what they think. Using counterintuitive or paradoxical math is a way to get that thinking going. It makes the math more real and alive for students. I can see students using this way of thinking in all subjects in school and even in their life outside of school.
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  • 30 Jan 2018 1:11 PM | Anonymous member
    I definitely agree with Kathleen - that counterintuitive math is a motivator for proof If it is handled right. I especially agree with her "If". If it is not handled "right", it can turn students off, making them feel like they don't have what it takes to make that big "proof leap", which may not be that big at all. I think we can learn a lot about how a student thinks, what motivates them, their learning strategies and their Habits of Learning while watching them work on proving the counterintuitive. I agree with the comment that this can extend to all subjects and to life beyond academics.

    I would like to know how others handle "moving on" when proofs have not been figured out. I deal with a small group of students who receive math support, and our time together is limited. I do encourage them to keep working on a proof on their own and let them know they can share their findings any time with me/ their peers. If we have a few extra minutes at the end of support time, I will ask if there is any more to share. I had a group of students who were trying to figure out proof for why the product of a negative number times a negative number is positive. That was one we kept coming back to and after a couple of months, one of the students shared a multiplication pattern that satisfied most of the students but not all.
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  • 01 Feb 2018 11:28 PM | Deleted user
    Hi,
    What stood out for me in reading the list of the brainstormed counterintuitive concepts, was negative numbers. Not so much with the negative numbers themselves (since we had plenty of below zero temperatures during our holiday vacation), but the concept that multiplying two negative numbers gives you a positive result.

    Many years ago, I taught an Algebra class at USM. A couple of women (around the age of 40) were coming back to school and needed to refresh their math skills. When I stood in front of that class and “reminded” the students that two negatives made a positive, I could clearly tell that the two women were completely lost. I spent some time in the Library after each class trying to help them explore and review some of the math concepts that had escaped them over time. The negative number multiplication rules were definitely counterintuitive.

    Calculating with signed numbers is a particular challenge for some of my (high school) students as well. In Geometry class this week, the students have been learning to write linear equations that are parallel or perpendicular to a given equation. When solving for the y-intercept, subtracting 6 from 2 or 6 from - 2 (for example), I often draw a number line and an arc-shape arrow to help students understand what the calculation is asking. With that visual, they have a much better sense of the solution. Using the calculator gives them an “answer”, but no concept of the process that they are performing. Even entering the process in the calculator can confuse them. “Do I subtract a positive or add a negative?” is a common question. So I have them explore both to see what they get. Some have never had a chance to try it and don’t have the number sense to know when the result should be negative and when it should be positive.

    I believe that counterintuitive math is a motivator, if and only if students are given the time to explore and make sense of concepts that stump them. I frequently get the “I never learned how to do that” comment. I’m not always sure how to interpret that comment from a student. It is somehow implied as an excuse for explaining to me that I should have lower expectations for them. That I couldn’t possibly expect them to do something so hard. Or that they expect themselves to fail as a result. How defeating that must feel to them. I can’t always take a long detour, but addressing gaps or misconceptions is critical to their success.

    Pam
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  • 04 Feb 2018 1:57 PM | Deleted user
    I really liked the list of counter-intuitive ideas in math listed on pages 244-5. I coach our math team, and one of the areas that we've been exploring lately is exponents. How can we make sense of numbers to negative or fractional powers? Why is it that sometimes we add or subtract the powers and sometimes we multiply? When is a number to a certain power equivalent to a different number to another power? It doesn't work to just memorize rules. Students need to explore the bases and powers using numbers they can easily manipulate, and formulate their own justification for what they need to do. They need to see 2 to the power of 3 as 2x2x2. They need to see the pattern of 8,4,2, 1, 0, 1/2, 1/4, 1/8 as powers of 2 reduce from +3 to -3. I find that if the students are given time to explore exponents in varied contexts, they can begin to make sense of the different situations and justify the operations.
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  • 07 Feb 2018 3:11 PM | Anonymous member
    I often tell my students that if it was easy - where would the challenge be? I often link it to sports - you want to be a better athlete - then you have to practice, practice, practice. Same as learning anything new. I want them to question why something works and not just take my word for it. I want to encourage them to challenge themselves and not give up. Even though we are discussing this in relation to a math book - this is a life skill that I am encouraging them to learn. If something doesn't make sense - how can you find the answer?
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  • 08 Feb 2018 5:33 PM | Anonymous member
    I agree with this statement. I find that students want to make sense of math concepts and struggle when it doesn't make sense to them. I loved the ideas in this chapter to help students question, explore, and come up with their own conjectures. This is when true learning happens and is so much more powerful than simply "telling" students what is happening and why. I know I struggle to let students critique their own conjectures. I'm tempted to confirm or deny their claims in some way, so I'll need to train myself to ask questions that allow students to reason for themselves and test out their ideas.
    The other area where I find my students need work is communicating their claims and reasoning in writing. I can usually help students to clearly express ideas orally, but when they are asked to put it in writing, it lacks clear, precise language. This is a skill I hope to work on this year as well.
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  • 17 Feb 2018 11:00 PM | Anonymous member
    I agree with the statement...and believe that the exploration for proof a necessity for learning. In a recent unit, we were working with exponents, powers, and roots (item from list on p. 245)... Many (high school) students do not have a good sense of operations with fractions. They may remember the "rules" or parts of the "rules" and apply them incorrectly (2 * 3/5=6/10 or 2/3+5/6=7/9 but 1/4+1/4=1/2 not 1/8!). In an activity we did early in our trimester, I asked students how is it that 10/(1/2)=20, could they explain their answer (without the rule)...most found that difficult to do.
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  • 20 Feb 2018 9:24 AM | Anonymous member
    I agree with the argument that counterintuitive math is a motivator for proof. I teach high school Geometry and it is difficult to get students to understand the purpose of a proof. It now makes sense, since we are usually proving something that our intuition is already telling us that it is true. Counterintuitive math, if given enough time to explore and prove, will develop that deep understanding that we are constantly trying to get at. To be honest, there are times that I struggle with it, because I learned it as a rule and never really knew or explored why it was true until I had to teach it.
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  • 14 Mar 2018 8:44 PM | Anonymous member
    I make lots of heads and tails of it! It is critical to test our knowledge, reasoning, and intuition so that we can be better equipped for engaging in the proof process. We need to teach students to test their ideas so that they can feel like they have walked all around their conjectures, several times, in order to let them have confident ownership of their claims. Along the way we should stress that we all need to learn to accept positive criticism and come to welcome differing opinions that could possibly help us further our claims. Flexible thinking, coupled with intuition, exploration, and willingness to work hard to arrive at their evolving conclusions.
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  • 25 Mar 2018 8:00 PM | Anonymous member
    I like the "Prove it to me...." prompt. It is much like what I have used with students when I ask, "How do you know?". When they hear that question, they always think that I am telling them they are wrong. So, I have to explain that I just want to know more about what their thinking is. Students are not used to having to dig deeper and explain their thinking. It is an important way to strengthen thinking, organizing, and communication skills. I am trying to push learners to go beyond the answer and check themselves by asking, "Why?" or "How?" as a follow up measure to double check their work. This strategy, when presented as a positive challenge can be motivating. If the student feels it as a negative, you may not get very far.
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