Week 11: Discussion Question (Option 1)

01 Feb 2018 4:34 PM | Anonymous member (Administrator)

Consider the architect's work versus the draftsmen's work from the Halmos quote (pp 303-304). What thinking does it spark in you? Are you pushing your students to be more like the architect or more like the draftsman?


  • 03 Feb 2018 11:32 AM | Anonymous member
    I hope I am pushing my students to be architects. It takes time to "nurture and develop skeptical, logical, rigorous, insightful mathematical thinkers." pg 304 . Pacing guides do not give enough time for students to develop a conceptual understanding. Districts keep pushing us to "cover" this material. What good does covering do? We all know the answer to that question.

    This year, I have been giving students credit on test questions that involve higher level thinking, even if they do not get the answer correct and they are not even in the ballpark. When they attempt the problem and they know their answer doesn't make sense, and they can explain to me why it doesn't make sense. They are thinking and they are using their intuition.
    I love that they are thinking and trying, and I do not want to squelch this behavior. It is so much more than "getting the right answer", and to show them that the process is valued, I give them credit and accolades.
    When I first started doing this the kids thought I was crazy, but now more of them are attempting the exceeds questions on our grade level tests.

    Having a class where there is discourse, and deliberation makes the class fun and interesting. It is not quiet, it is not me up in front modeling how I solve problems. The document camera is a great way for kids to show how they got their answer. I have middle schoolers passionate about their thinking. It is not from something I said or did, it comes from them.

    As Henderson said, the why questions students ask, are not our why questions. And questions lead to more questions, and that leads to intrinsic learning. The more ways we look at a problem, the more ways we see to understand. The mind is a magnificent organ, and when the neurons are firing the energy in a classroom is exilarating.
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  • 03 Feb 2018 5:34 PM | Deleted user
    I am a first year teacher in the classroom with first graders. I have found that in first grade it is important to learn computational skills (being the draftsman) and being able to explain how you use strategies to solve a problem (architect). Examples of some of the strategies that first graders can articulate involve using the number line, drawing a picture to explain their thinking, using manipulatives to group items together, and using a number grid. In first grade, I feel having a combination of the draftsman and the architect develops future mathematical thinking.
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  • 03 Feb 2018 8:56 PM | Anonymous member
    I found this section interesting. We have had conversations at school about proofs and how students struggle. I have read many article where teachers believe they are the best thing about Geometry and others who think they should be removed. I think there are several reason why students react the way they do. Forcing students over the years to keep a certain pace in the classroom has lead to poor understanding. Teachers get caught in the frustration of trying to complete a syllabus without students doing more math practice at home. Students see math as just a series of worksheets. Suddenly in Geometry comes the proof. We want them to think and reason on their own, but this has not been their experience in most of their math classes before this. We have given them so many days of I Do, You Do that they don't know any other experience. I used to think we were being kind to students with this type of teaching. They were comfortable with it. Now I see that we are limiting their ability to think on their own. Hard to change these practices quickly, but we really need to work on it.
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    • 19 Feb 2018 1:50 PM | Anonymous member
      You explained exactly why geometry was my first C (and only C) grade in math! Having ever had only A's, it was a turning point for me. Thank goodness I had an exceptional teacher!
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  • 07 Feb 2018 3:20 PM | Anonymous member
    I would hope I am encouraging architects. These are the thinkers outside the box for me. When they are thinking about their math - what questions are they coming up with? How do we go about and find the answers? We wouldn't want to hire an architect that just does cookie cutter designs - we want a design that speaks to us - and this is the same way with students. We want the math to speak to them - think about it - and then challenge themselves to find the answer. I don't want my students to just regurgitate what I say - I want them to question what I say. I love giving them a problem to think about - and they have come to know that we're just thinking and coming up with ideas - on how to solve it. I have some students that enjoy being the teacher - demonstrating in their words - how to solve a problem.
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    • 07 Feb 2018 9:57 PM | Anonymous member
      I found this section very interesting. I love how the investigative behind the scenes work of math is described. That is what I try to teach my class. I try to have the take risks, see if there is more than one way of solving a problem, find out if it works. I give my class problems that have more than one solution on purpose to see what they come up with and to have them have the experience of sharing out with the class and having them learn from each other. I find my students love to explore and try new things. That is the fun part of math. I do also find that my student need fundamental skills like long division and adding and subtracting fractions. Those parts of the class are not as much fun, but essential so that they can tackle the more challenging problem.
      I do agree with some of the other comments about following a pacing guide doesn't fit with "architect" type teaching. It is something we have to deal with and it is not easy.
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  • 07 Feb 2018 10:31 PM | Deleted user
    I definitely think that for many of my students, the “architectural” foundation is a bit wobbly. When students have gaps in their mathematical past, that have resulted in lowering their confidence, they do not trust that their knowledge is correct. They are often second-guessing themselves. I try very hard to encourage questions, but I need to work harder on providing an atmosphere where students are more comfortable asking questions in front of their peers. I also want to encourage students to build friendly rivalries within the classroom. Some days that occurs naturally, but not as often as I would like. Open-ended questions with time to process seem to work the best.

    I have also found that “find the error problems” encourages students to look at the reasoning of others (thus proving why something is incorrect, why it doesn’t make sense). It helps my less confident students, see common mistakes that are made and how to analyze a “process” or “proof”. It could also be called “math errors anonymous”. It provides a great launching pad for follow up questions (“Why didn’t this work?”, “What alternative strategy might have worked?”, “Does the answer make sense in the context of the problem?”).

    At the high school, we live in the world of variables and formulas. In Geometry, we are often trying to prove the “general” case. We can split 180 degrees into many pairs of supplementary angles or 90 degrees into many pairs of complementary angles. When looking at similar triangles we explore why we couldn’t multiply the angle measures by the scale factor and still have a valid triangle.

    I find myself in a balancing act between encouraging exploration and meeting timelines for completing units. I really wish that we had a fluid 4-year math program. One that would allow students to work at a pace that they are comfortable with, based on the skills they come into 9th grade with. Writing down their thinking does not seem to be a skill that most are equipped with. I want my students to gain a curiosity about the math concepts we cover. I want them to “craft” their mathematical futures (instead of praying to survive them).

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  • 09 Feb 2018 9:23 AM | Anonymous member
    My husband is an engineer (draftman) who often works with architects. It is interesting for me to look at working with my students in this way. All good/great architects have a solid foundation in structural design, and how they work with this knowledge is varied and limitless visually and creatively. It is exciting to view what we do with our students in this way.
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  • 11 Feb 2018 12:00 PM | Deleted user
    I love the idea of promoting our students to explore mathematics, in such a way that they become insightful mathematical thinkers. Halmos describes the mathematician going from conjecture and intuition to that final proof as a arranging and rearranging of ideas, conclusion making, failures, and insights - an exciting process that would engage our young math students. The list of verbs (pg 305) would be practiced over and over.

    Unfortunately, I think that although my heart is there, my practice is not. I hear time knocking on my door, and with just 15 minutes left in a period, or 3-months until the end of the year, I move the students forward - giving them answers to their questions and paths to a conclusion. There is just not enough time in a public school day to keep the exploration going long enough for every concept to be discovered through exploration. Hopefully, I give some balance. Students are involved in inquiry. I do allow them to travel down some blind alleys. But then I end up saying, 5 minutes left until clean up.

    Here's an example. On Friday, a group of seven 4th graders met with me for math club (30 min., 1x/wk). In first problem, they were given a cube's net, with numbers 1 to 6 written on the six faces. They had to determine the product of the 4 numbers adjacent to the 1 when the net was folded into a cube. The word "net" was new to them, and so after they had solved the given problem, I asked if the net in the problem was the only possible net for a cube. They spent the remaining time drawing different nets and folding them into cubes. There were successes and failures. Unfortunately, there was not enough time to develop a rule for a cube's net; and since I won't see them for another week, we'll move on.
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  • 11 Feb 2018 12:14 PM | Anonymous member
    I am really trying to have my kids be more like draftsmen, and one way I am doing this is by being sure they don't erase mistakes. For example, when the kids get back their end-of-unit tests, I have marked which ones are wrong, and they use a colored pencil to go back through and find/fix their mistakes. I don't want them erasing because I want them to see their errors and to see if they can understand why they made them. I have been doing my best this year to celebrate mistakes and to especially celebrate when they realize why they happened, and seeing where they have been not only helps them, but also helps me to know their misconceptions.
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  • 18 Feb 2018 11:06 AM | Anonymous member
    I am trying to teach my students the skills of the architect - the foundations of mathematical thinking and reasoning. It does take time to build this culture of questioning and intuition. I, too, feel the pressure of covering all of the standards in a given year, or before the standardized testing, and find myself not taking the time to go into depth of thinking and exploring that I know is best for students. I am constantly evaluating and prioritizing so that I can have students explore their own conjectures and strive to support or refute their claims. I need more work in this area so that students can gain confidence to take risks and feel successful even if they have disproved their own claims.
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    • 25 Feb 2018 11:59 AM | Anonymous member
      I agree with you, Carrie! It is such a balance to cover the curriculum AND give students the time they need to understand and have confidence with the concepts we teach. We know their understanding would be deeper, and their confidence greater, if we could go at the pace they need.I think it is a struggle for us all!
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  • 23 Feb 2018 6:43 PM | Anonymous member
    By the nature of the course that I teach, I am working at getting my students to do the work of an architect. It is a challenge to get the kids to think on their own and to notice,generalize, test, claim, revise, explain and prove. They want me to just tell them or at least tell them if they are right. Our department as a whole is working on their thinking and reasoning skills by giving them more open problems to do.
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  • 16 Mar 2018 8:11 AM | Anonymous member
    As of the last 4 years, I have developed a teaching tone that is pitched on building more full-time architects, yet also I allocate some time having my students practice as draftsmen (with support as needed). I generally co-teach in Algebra classes, yet am available to assist students in our schoolwide TASC block (Teacher Assistance Student Centered) in any math curriculum. Some of our Geometry teachers tend to focus on the draftsmen building as they try to "cover" the curriculum. This is unfortunate as it perpetuates the traditional old-school nature of way too much mystery and frustration in the process. Students often express feeling ill-equipped to handle the proving part of proofs. I am going to copy this section of the book and share it with our math department with the goal of making the time to discuss the nurturing and development of our students to be more "skeptical, logical, rigorous, insightful mathematical thinkers"!
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  • 27 Mar 2018 8:06 PM | Anonymous member
    I would like my students to "feel" confident and capable while doing math. I want them to feel like they know how to solve problems in math. I want them to know what tools to choose from when solving a word problem, versus what tools to use when calculating perimeter. In this, I feel like I am more alongside the architect's position. I want students to know the underpinnings of the choices they make while actively involved with math.

    My students have often tried to erase or hide any marks they've made on the paper that are not "correct". Over and over, I ask them to show all of their work. "Leave it there so I can see how you worked through the problem." But they do not want anyone to think they didn't know the answer from the get go......
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