Association of Teachers of Mathematics in Maine

Week 9: Discussion Question (Option 1)

17 Jan 2018 5:50 PM | Anonymous member (Administrator)

Discuss the opening passage about intution. What caught your ear? Any surprises?

Comments

  • 18 Jan 2018 8:00 AM | Anonymous member
    I really liked this opening passage because I am a strong supporter of intuition. I always told my teenage daughter and her friends as they ventured off onto new experiences, “Trust your gut. If a situation or person doesn’t feel safe, they probably aren’t.” This seemed simple and true enough. I never thought about it in terms of math. I have heard students say, “It feels like the right answer,” but I don’t think I ever probed their thinking enough, or had the language or understanding to know this was their mathematical intuition working for them. The concept of mathematicians using “contrasting pairs of ideas – logic and art, creativity and structure, truth and beauty, etc…” really struck a chord with me. Mathematics is both a creative and intellectual endeavor. Now I need to make sure I help my students know and understand this, unfortunately, I think most of them view math as an intellectual endeavor. If they can learn to use and trust their creative, intuitive self they may enjoy math ore and gain confidence.
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    • 21 Jan 2018 11:44 AM | Anonymous member
      We chose the same quote. I agree that intuition is what makes math enjoyable.
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    • 04 Feb 2018 9:50 PM | Anonymous member
      Math is beautiful art and language! Mindset is difficult to change for adults; not for kids. We can try to change ourselves for the benefit of students and the future.
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  • 18 Jan 2018 7:58 PM | Anonymous member
    I really liked this section of the chapter and I liked the way math intuition and number sense were stressed as far as going together and being vitally important in understanding math. I say to my class on a daily basis, "Does your answer make sense? If it doesn't then you need to go back and look at it again. Math should always make sense. I can clearly see which students don't have any math sense when I see that they don't pick up on an answer they have given that makes no sense at all. Then I know they have only memorized the algorithm without any thought as to why or who it works. I also like how in this chapter it talks about math intuition giving students something to then prove. When looking at a math challenge they need somewhere to start. Their math tuition gives them that and then they can proceed from their. It helps all those kids who don't know what do to or how to start a place to begin.
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    • 08 Mar 2018 9:41 AM | Anonymous member
      I find some agreement with what you are saying. I always asked my students if their answers made sense. However, the cue they heard was that they were wrong. I had to spend time teaching about the how and why of "making sense".
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  • 20 Jan 2018 5:42 PM | Anonymous member
    Being a High School Geometry teacher, the part that caught my attention was that intuition and proof complement each other. Having some mathematical intuition is very helpful when trying to prove something. Students need to have a hunch in order to complete a proof. Students need a proof in order to show that a hunch is a good hunch.

    Having intuition and number sense are important for learning math, but very hard to teach. Students need lots of experiences to develop that intuition, but have a limited amount of time.
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    • 26 Jan 2018 12:01 AM | Anonymous member
      Hi Renee,

      I also loved the connection to proof in the sense that we want them to "justify" their solutions or their step by step process in a proof. One question I often ask at this point in the trimester is "Can length be negative?" When their response is unsure, I ask them, "If I back-up in the car a distance of 7 ft, have I traveled negative 7 ft?" Having intuition of whether an answer is appropriate is a key to further learning. If they don't check their answers (after solving for 'x', for example), they cannot verify relationships.

      Limited time (with this winter's snow days) is definitely a factor in providing those exploratory experiences.

      Pam
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    • 04 Feb 2018 9:52 PM | Anonymous member
      Time and money. Forever a problem in education.
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  • 21 Jan 2018 11:43 AM | Anonymous member
    Mathematics consists of two essential components, intuition and proof, which are of equal importance. To quote from Stacy, “Contrasting pairs of ideas- logic and art, creativity and structure, truth and beauty, instinct and reason, passion and discipline- are the left and right hands of mathematicians.” Intuition is what breathes life into mathematics.

    Intuition develops when students are allowed to play and experiment with math, when they are encouraged to explore new math concepts through activity and the use of manipulatives. As students experiment, it is the teachers job to guide conversations to allow them not just to discover the concept you are working toward, but also to make connections to other concepts, and discover new ideas: Where have you seen this before? Tell me what you’re thinking. What else did you try? Will this always work?

    Estimation is a powerful strategy, which requires intuition. Students can be so focused on the right answer that they don’t recognize when they have made significant errors. Without intuition they forget that important question: Does this make sense?
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    • 25 Jan 2018 11:12 AM | Anonymous member
      I like your use of language "play and experiment with math" - so important yet something I know I don't do enough of.
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  • 21 Jan 2018 12:31 PM | Deleted user
    I teach a high school level geometry class to a small group of advanced 8th graders. I love to start off units with some sort of investigation in which the students develop conjectures. Students use inductive thinking as they play around with geometric shapes to recognize patterns. I like to think of this as a development of intuition. We follow up with proofs. I also coach our MathCounts team. Here, I tell the students to trust their intuition. If they think two triangles are similar, go for it. Don't worry about the proof.

    I guess I was surprised to read that intuition is an important aspect of mathematics. I loved the quote, "That's why a disciplined mathematician subjects her intuition to the skepticism and scrutiny of mathematical reasoning and proof."
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  • 21 Jan 2018 9:21 PM | Anonymous member
    I guess I never thought of Math related to intuition. It is, because a student needs to decide what step to take next, make a decision, and just try it. I do often say, "What do you think you should do next?" They will tell me, and often I will say try it! Sometimes when I know that solution will just lead them so far astray, that they will be so far off base, I may say what if you did this instead? I do not give the answer, but just put them on the right path, so they are not totally lost before they even get started.
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  • 22 Jan 2018 1:03 PM | Anonymous member
    This passage made me think about how kids are naturally mathematically intuitive, but could lose some, if not all of this intuitiveness if they are taught math in a systematic and orderly way with no regard for creativity, risk taking and making connections. Approaching math as “a creative art within a logical structure” is important so that kids can develop, or continue to develop their “gut instincts” in math.

    I like the quote “The gut leads to the mind, and the mind checks the gut.” I agree that the relationship between intuition and proof compliment each other well. It has been my experience in grade 2, that when kids “follow their gut” they automatically want to share their “proof.” I think it is this because…” “So, if you ….” “Well, I know that…” Sometimes their thinking correctly proves their “gut” and other times it does not, but either way students are beginning to develop rigorous thinking and hone their intuition. I do, however, have kids that do not seem to have that “gut instinct” for math. It is important to help these kids develop intuition by structuring exploration and experiences where they do their own thinking and reasoning. It is sometimes hard to sit back and let kids have that productive struggle. It is also crucial to ask questions and listen to kids’ thinking during problem solving, and provide lots of practice!
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    • 25 Jan 2018 11:13 AM | Anonymous member
      It is scary to me how many kids may be losing, or have lost their math intuition and playfulness.
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  • 22 Jan 2018 1:16 PM | Deleted user
    For students, learning to trust their gut means empowering them to take a chance. It’s a student centered approach that is often undermined by standard classroom practices. Students begin to look to the teacher to affirm what they are doing. Not only that, but some students need this support even within the process: “Am I doing this right?”, “What do I do next?”, and “ Is this right?”; are questions that indicate the students dependence on the teacher to work through a problem. One simple way to empower students is to simply answer those questions with a question. For example, Saying “How would you know if you are right?”Or to say “I don’t know, think about it and let me know what you find”. For students in our class we talk about trusting our math brain and checking our work with one another as well as other strategies to ensure we are on the right path.
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  • 22 Jan 2018 2:27 PM | Anonymous member
    As voiced by others, I am a strong supporter of the value of intuition. The line that really caught my ear - "It is only with a combination of both rigorous formalism and good intuition that one can tackle complex mathematical problems; one needs the former to correctly deal with the fine details, and the latter to correctly deal with the big picture." Seeing/grasping the "big picture" is often where the students I support struggle because their true, deep understanding of mathematics is not always solid enough to support the bridge to the "big picture". It is exciting to see a student make the connection to the "big picture".
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  • 22 Jan 2018 11:15 PM | Anonymous member
    Students always ask me how I can factor so quickly. They painfully go through a process which lead them to the answer while I just look at it an write it down. They think it is magic. I try to convince them that it takes practice. With lots of practice comes an intuition so that you just know what makes a great guess. I love the way they described this as experience leave a trace in the mind. I think it is a good visual way to think about this. Any time we can connect thinking about the way the brain works with practice and experience, I think we can motivate students to keep trying.

    I have a group of students who have been pulled out of the mainstream math class for years. They have not had the same experiences that the other students have had and they have no intuition. When problem solving, I often ask them why they did something. They often said something like "We were told when in doubt that we should divide." Because of these things they have no sense of whether an answer seems correct. If they do have a feeling that something might be wrong, they don't voice it.

    I think an implication for my teaching is that I need to explicitly teach them that intuition will grow.
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    • 26 Jan 2018 12:17 AM | Anonymous member
      Hi Ellen,

      I love the comment about "when in doubt divide". Haven't heard that one, but probably a challenge to "undo". I also co-teach a class of students that have an alternate setting for math. They have been amazingly resilient this year as we have definitely pushed them out of their comfort zone. The results have been positive when I take the time to ask them to work through the process of giving a meaning to their results. A week or so ago we were drawing diagrams for Pythagorean Theorem word problems. I tried to model a sentence for them based on their answer ("The students is 10 blocks from where they started."). With a few students, I also asked them if they knew what a city block was. Living in rural Maine does not lend itself to the idea of walking city blocks and other contexts.

      Pam
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  • 23 Jan 2018 11:55 AM | Anonymous member
    When reading this chapter the what really stuck out to me is how math is not just about following steps and procedures. Too often children develop the feeling that they are not good at math because they have a hard time following steps and procedures. You start hearing children say "I am bad at math." Reading this chapter helped me to take a step back and think about how to create opportunites to explore and practice to develop their math intuition. Often times as classroom teachers we worry about getting through our math unit and completing every worksheet, but without providing our students time to explore and experience math they are not about to build their matematical intuition. I really enjoyed reading the section about Jennifer Clerkin Muhammad and how she help to build mathematical intuition around new concepts. She took the time for her students to explore and manipulate in an meaningful way then used the curriculum material to reinforce what students had learned through exploration. I have noticed in my own classroom when my students have chances to manipulate and experience concepts before just presenting them with a worksheet they have a deeper understanding and confidence that helps them to move their way through problems presented in written form. I want to start using more exploration within my classroom so my students can develop their mathematical intuition.
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    • 16 Feb 2018 10:41 PM | Anonymous member
      In response to "Often times as classroom teachers we worry about getting through our math unit and completing every worksheet, but without providing our students time to explore and experience math..." One way that I try to give students space is to follow a philosophy of "less is more" - more time spent on fewer problems...there's always extras if needed....
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    • 08 Mar 2018 9:48 AM | Anonymous member
      I also felt inspired by the ideas around math as play and discovery, not just a series of steps and procedures. I feel that many of our students have developed this narrow view of math. I have also seen some "drill and kill" teaching of steps and procedures. Time is often mentioned, but I agree with the suggestions in the book that fewer and more authentic mathematical experiences, will lead to better math experiences and understandings.
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  • 23 Jan 2018 3:38 PM | Anonymous member
    The idea that intuition is gained through examples really stuck with me. Very much like "common sense" and "gut feelings". These things, like being good at math are not always natural, they are learned.
    I think I need to be really mindful of that concept and be sure to give my students more "hands-on" and "exploratory" experience when tackling a new topic as well ask more relevant live-rapport questions. Hopefully this will help the students build intuition and gain confidence in their problem solving.
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  • 23 Jan 2018 6:26 PM | Anonymous member
    I love thinking about math students using their intuition I am thinking of ways that I can encourage students to pay attention to their intuition and what feels reasonable or not. It makes sense that we want to encourage students to use their intuition because this shows they are really thinking about math and making connections. What caught my ear was the way we have destroyed their natural intuition by simply telling kids "this is how you solve the problem. These are the steps you need to do - because that's how it's done." As I read through this section (and other chapters in this book), I keep thinking that all math teachers would benefit from reading this book and thinking about how math is taught.
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  • 24 Jan 2018 4:25 PM | Anonymous member
    As a student math was not enjoyable to me because I was taught the formulas and rote memorization of facts. In reading this section I liked that it stated that mathematics is not always logical and certain or black and white and that mathematicians often need time when thinking about problems. In our current school day this is often a challenge because teachers are required to teach to a particular curriculum and learning targets. As a first grade teacher, I take advantage of the questions that my students ask and suggestions that they make so a problem makes sense to them. Making math fun is very important especially with our younger learners as well as reinforcing the skill part of what makes 10 or what makes 20.
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  • 25 Jan 2018 6:48 AM | Anonymous member
    I never really thought about intuition in the math setting. This piece is giving me a lot to think about while observing how my students approach math problems.
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  • 25 Jan 2018 4:05 PM | Anonymous member
    Zager writes “I have come to see mathematics as a creative art that operates within a logical structure.” New solutions to real world problems start with a creative response. The logical structure provides the parameters for the solution and within those parameters, the problem solver’s creativity and imagination is applied. The same applies to math problems and challenges posed in class.

    The discussion of intuition was thought-provoking. Whether working on word problems from a textbook, one of Andrew Stadel’s Estimation 180 challenges, or a local community issue, a “gut” feeling about how to approach the problem is essential. The discussion about emphasizing estimating was a reminder to not skip right to the calculating but to think about what is being asked and then set an expectation for a reasonable answer.

    I work with several high-level math groups who seem annoyed when asked to estimate. The intuition discussion and the list of questions referenced in the second prompt are great tools to use to prompt math discussion before the pencils hit the paper.
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  • 25 Jan 2018 11:45 PM | Anonymous member
    Hi all,

    A few times this trimester I have posed a question to one of my Geometry classes and had the students just sit and look at me with no response at all. Note that they are not asleep, but almost stumped by a question in which my purpose is to see what connections any of them might make to prior knowledge. After having a few “moments of uncomfortable silence” (which I might secretly enjoy briefly, hoping that they are considering the question in order to respond), I try to adding some additional pieces of information. When clearly no one is comfortable enough to even make a suggestion or I get the “raised eyebrow”, I start asking more simplistic questions that allow a few students to provide a response. Of course, if I had asked students to write a response to my question I might get a bit more feedback.

    At this point I am wondering what happened in their collective mathematical school experience that might have prevented so many of them from at least making an “intuitive response” or even asking a question about the question I am posing. Intuition - “a thing that one knows or considers likely from instinctive feeling rather than conscious reasoning” - is definitely something that my students use in their lives (but not always in math class).

    I definitely related to the quote from Terry Tao, saying that “It is only with a combination of both rigorous formalism and good intuition that one can tackle complex mathematical problems”. What I take from this is a need to create learning opportunities for my students to further develop their intuitive mathematical knowledge. At the same time, I want them to improve their ability to tackle more rigorous problems (so they are better prepared for their next math course).

    I also wish that my student had the opportunity to participate Jen’s angle exploration activity.

    Pam
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  • 26 Jan 2018 11:28 AM | Anonymous member
    I work primarily with young (k-2) struggling learners. Their intuition regarding numbers is pretty limited, and the experiences they have had thus far are not particularly affirming of any intuition that they do bring. This chapter has made me really think about how I can foster their intuition in regards to math. I feel that concrete experiences with quantities are imperative to helping them develop, and learn to trust, their intuition.
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  • 30 Jan 2018 10:02 AM | Anonymous member
    I have really never thought about there being a level of intuition in math. I found the opening passage very intriguing. It made me think of those kids who, when asked how they got their answer, often answer that they "just knew." It makes sense that with some problems one may recognize the relationship between the numbers. I think the more fluid the math facts become for the students, the quicker they recognize the relationships and perhaps build that intuition. After reading this passage, I intend to experiment with the idea of "gut feelings" and helping the students learn how to "prove" them.
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  • 30 Jan 2018 2:37 PM | Anonymous member
    I liked the quote by Burton on page 211,"My intuitions are based on my knowledge and my experience. The more I have, the more robust my intuitions are likely to be."

    I was excited to read about the connection between mathematics and intuition. I don't think this idea is given a lot of attention because math is known to be filled with proof and justification and reasoning, but I think we've all had those students who have had something a bit more "special" when working with math. They just knew where to go or what to do. Why is that? Maybe they had intuition in their favor, but it also seems to me from reading the opening passage that they probably had lots of experience and practice to explore and develop that intuition as well as opportunities to identify, reason and prove.

    As a teacher who works primarily with students who struggle with math concepts, it's encouraging to know that the opportunities I afford my students by exploring methods, using models and tools to develop deep understanding of concepts is not only solidifying their understanding but also aiding in strengthening their intuition and confidence as learners.
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  • 03 Feb 2018 10:27 AM | Anonymous member
    Chapter 9
    “Intuitive experiences must be acquired by the student through his/her own activities they cannot be learned through verbal instruction.”
    Manipulating mental images or manipulatives helps students to “see”
    “We acquire those representations not through memorizing formulas, but by repeated experiences. (Hersh 1997, 65)
    Yet, we continue to lecture, cover, and follow a pacing guide????
    All the while expecting different results. That reminds me of Einstein’s definition of insanity.
    We have math programs that are entrenched in language, the language limits access for struggling readers and Ell students? Should it be difficult for students to understand what the question is?
    How many times have you read a question in a math text book and had a difficult time understanding what is being asked of you?
    What do you do?.
    You go with your intuition.
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  • 04 Feb 2018 8:46 AM | Anonymous member
    This chapter resonated with me. Not only do I have students who have found math difficult but I have two daughters that have been in special education classes for math throughout their public schooling and these classes have scaffolded and taught algorithms but never taken the opportunity to really help develop number sense. It is obvious that I along with the teachers should have structured exploration and experience to help develop mathematical intuition. They still suffer from a lack of mathematical understanding and intuition as adults. SO, I am taking one day a week away from my math program for exploration, wondering, questioning and discussions.
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  • 04 Feb 2018 3:59 PM | Anonymous member
    I have never thought of the role intuition plays in math. The quote from Erich Wittmann, "Intuitive experiences must be acquired by the student through his/her own activities-they cannot be learned through verbal instructions." The quote makes realize how important "play" in Kindergarten math is for the students. These experiences set the foundation for intuition.
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    • 04 Feb 2018 9:54 PM | Anonymous member
      Yes. We set the tone. It is scary to teach Pre-K and Kindergarten with so, so, so much responsibility!
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  • 04 Feb 2018 10:14 PM | Anonymous member
    "Mathematicians experience the full range of feelings and use all their intellectual and creative faculties when they work." No wonder we're exhausted! It takes all of ourselves to do it best! We must listen to our feelings and trust they mean something. Very valuable chapter. I have not checked out all the suggested sites yet, but highlighted several quotes and sites to review. I love, "Before you calculate that, can you tell us why you'd want to?" and "What are you planning to do with that information, once you find it out?" We can't do much without number sense. Students have asked me many times, "Why do we have to do math?" I would say, "It teaches you how to think." Had I known, "developing an intuitive feel for numbers, shapes, quantities, operations and functions and how they relate to one another is the most important element of intuition we teach in school," I could have packed a deeper punch. Had I realized from the beginning of my teaching the importance of estimation, I would have done it every day and not just the chapter it was in. Oh, if I could do it over...
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  • 07 Feb 2018 10:19 AM | Anonymous member
    What I have noticed about intuition is that the students are better at using it before they have been formally taught the concept. For example, before I teach addition of fractions, I have found that students can answer questions such as 1/4 + 1/4 = 2/4 when we talk about it informally, but when I start to teach it, lots of kids want to add the numerators and denominators. I almost don't want to teach formal procedures at all because they lose that all- important intuition!
    Has anyone else noticed this?
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  • 07 Feb 2018 1:43 PM | Deleted user
    Steps and procedures is exactly how I learned math. I do not think I ever thought of intuition as something you use in math but it makes sense. I want my students to tackle their math learning not only from steps and procedures but by using their past experiences and learnings to solve new problems through their intuitions. As a Kindergarten teacher I am very careful to teach steps for mathematical thinking and problem solving as I do not want my students to feel that if they cannot remember the steps and procedures that they are then not good at math. I like what Terry Tao stated, "Once you are fully comfortable with Rigorous mathematical thinking, you should revisit your intuitions on the subject and use your new thinking skills to test and refine these intuitions." I often ask my students to tell me something about the problem or what they notice about the math we are about to do. Students of course state the obvious but a few students think outside the box and notice something not so obvious about the math. This in return helps the obvious students think deeper about what they originally thought. Intuitions, testing new thinking, and refining(reworking) your intuitions can be a very powerful tools to use in the math classroom. I will put these ideas to good use as I continue to instruct my students as they develop their minds into mathematical thinkers and problem solvers.
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  • 07 Feb 2018 3:05 PM | Anonymous member
    What caught my eye was the "instinct and reason" as listed in the second paragraph. It reminds me of one of my students. Good golly, this kid has math sense. This student could go on to be a really great math student. Where the student falls down is the ability to explain how they got the response. The student is very reluctant (like pulling teeth) to write down any notes or in learning the why and the process to find the answer. My fear, as the student continues on, is that the skills I am trying to teach now - the student will not have because of this reluctance to learn the process. So in this, the student has the instinct - but not the reason behind it.
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    • 16 Feb 2018 10:37 PM | Anonymous member
      Or that student may grow up to be the student I have now in my Honors Algebra 2 class, who can explain how he got the answer but prefers to do much of the work in his head, so there's a correct answer with little or no work on any given paper. What makes this student stand out is his mathematical curiosity...asking me questions that clearly show he thinking about the problem/process from different angles.
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  • 08 Feb 2018 4:50 PM | Anonymous member
    This opening passage was very interesting. I never thought that one could have intuition around math. Math has always been numbers and steps. But the more that I work with students and the relationship of math and numbers I can see how one could have intuition. I need to set up my math groups with this in mind I want to dig deeper and allow my students to find theirs.
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  • 09 Feb 2018 6:35 PM | Anonymous member
    I love the idea that the more we practice our intuition -- and we CAN -- the better it gets. Just like a police officer can eventually play that game where he (or she) predicts the car speed and matches the radar gun, the more we play with estimating or conjecturing, the better we get at it.

    Particularly in geometry classes, a lot of times I find myself saying, "right, so that's some great intuition, but WHY ..." and pressing students to be able to articulate (or at least explore) the math that supports the intuition. I don't, however, know how well I go BACK to the intuitive first answer to revisit how close it was, etc. I want to say I do, but on reflecting, I am not sure how well I do it. I definitely plan to keep this in mind as we move forward in Geometry. Students may be doing that on their own, but specifically and intentionally coming back to it can support their intuition and confidence in a meaningful way!
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  • 17 Feb 2018 10:16 PM | Anonymous member
    I guess I hadn't really thought about intuition as something to develop, or being "good" or "bad". The comment (p 211) "mathematical intuition--comes from from experience and practice" does makes sense to me--especially when I think of my own experiences as a student learning how to do proofs.
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  • 25 Feb 2018 6:05 PM | Anonymous member
    "The gut leads to the mind" and the "mind leads to the gut." I would have considered it obvious in terms of life and making every day decisions, however I never connected it to math class. Perhaps being taught math in a very systematic order caused more issues than I realized. The further I go in this book, the more I understand why the way I learned is not the best way to teach our students.
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  • 08 Mar 2018 9:56 AM | Anonymous member
    My ear was caught on the word creativity. I am a very visual and creative person. I see things readily in a system or pattern. This speaks to the ying/yang nature of mathematics. I feel both sides, the orderliness of math procedures, and the creative means to understanding a problem. I need to draw or make marks on a page in order to "think" about a math problem. I then find a satisfaction with using a procedure to solve a problem. It feels comfortable when both sides work in tandem. If one piece is missing, I can feel the imbalance.
    In my teaching, I have tried to recognize this in students. I have tried to find connections that are relevant and meaningful to their experience. Drawing pictures and using routines helps with procedures. Clapping rhythms or discovering relationships helps with creative thinking. I was surprised with the number of times I connected with the remarks about intuition. I have never talked about intuition in terms of mathematics, but I definitely rely on intuitive reasoning while doing and teaching math.
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  • 13 Mar 2018 10:48 PM | Anonymous member
    As stated in the reading,"The gut leads the mind and the mind checks the gut," sums it up nicely. We do use a variety of approaches in our mathematical studies and sometimes lean heavier in one direction vs. another depending on what we are pursuing. For example, maybe we need to use our intuition more when we are trying to analyze a situation and then we might check our intuition by using a "let's check our thoughts more scientifically" approach.

    I marveled at the conclusion that shows how rigor should be employed to assist with moving away from bad intuition towards intuition. How both rigor and good intuition are partners in assisting with tackling higher level mathematical problems. There is a lot of testing and refining of intuition to transport us to higher levels of thinking.
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