Association of Teachers of Mathematics in Maine

Week 7: Discussion Question (Option 1)

04 Jan 2018 3:39 PM | Anonymous member (Administrator)

In the section on standards (169), Debbie never wrote an objective on the board, yet her students engaged in rich exploration of the standards. In your teaching context, how might you give students opportunities to uncover the standards through inquiry?

Comments

  • 04 Jan 2018 5:12 PM | Anonymous member
    I loved reading about how her students addressed the majority of the standards through their own questions, with some skillfully planned instruction intended to give the students the background information necessary to explore those questions for themselves. A unit on shapes seems to lend itself nicely to that approach.

    I am struggling to figure out a way to get students to wonder about addition and subtraction strategies. Shapes is a much broader topic than addition or computation strategies, is that where I am falling short? Am I thinking too narrowly? I want students to wonder about efficient strategies and start using related facts instead of fingers. I’m at a loss as to how to address it with an inquiry approach. I’m sure there is a way to do it, but I’m not coming up with one at the moment. My initial thought is to launch the inquiry with a number talk where multiple strategies can be used to get the correct answer. Our focus during sharing often centers around which strategy is most efficient, but the conversation seems lost on the students I most want it to rub off on! I realize that developmentally maybe they just aren’t ready to use those derived facts or more efficient strategies, but at some point I feel that we just skip to trying to get them to memorize for the sake of recall…but at the expense of flexibility and deeper understanding. In retrospect this response doesn’t seem to address the prompt all that well, but it is a question that I struggled with throughout the chapter.
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  • 04 Jan 2018 6:41 PM | Anonymous member
    If the context in which I am teaching allows my students to learn through inquiry, the rich exploration they will be involved in will allow them to learn in a way that is much more interesting to my students. I loved reading about how into the lesson her students were. When I have taught lessons where students were interested in what they are learning about they are so into the topic it is hard to get them to stop as opposed to when they are learning about something they they have not interested in and they keep asking if the class is over yet. I would think that at the end of the class time, if students were shown the math standards of their grade they would be able to pick out which ones they were just working on. I love the idea of the students sorting through what they had done and creating the bulletin board to show what they had learned on page 163. I teach 5th grade and I am sure my students could do something similar.

    I would present them with a inquiry problem and then do as Debbie did on page 156, #2. Think about what the standards are I need to teach and then think about learning experiences I could provide to get my students there.
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  • 05 Jan 2018 11:40 AM | Anonymous member
    I too enjoyed reading about the unit on shapes. I, like Kimberly, am struggling to figure out a way to get students to wonder about addition and subtraction strategies. I did not do a timed sprint this week. After reading about math anxiety, I chose not to. Instead students used snap cubes to look at adding and subtracting. They were trying to get them to reach the ceiling of the classroom. It was a great cooperative lesson. Somehow I can use that enthusiasm to build to help with the standard adding and subtracting up to 100.
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  • 05 Jan 2018 12:22 PM | Anonymous member
    We use a strategy that is called, Recharged. The students are presented with a problem, that follows the standard of what we are working on. They work in groups of 4 discussing the problem and the strategies they would use to solve them. As I was reading the section on standards I realized I needed to find more open ended problems for these groups to work on to create a more inquiry based opportunity for these teams.
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  • 06 Jan 2018 10:01 AM | Anonymous member
    In thinking about how I might give students opportunities to uncover the standards through inquiry my initial thought was that some concepts, such as geometry, lend themselves to inquiry more than others. The more I thought about it, the more I realized any concept is appropriate for inquiry. Connection to standards can easily be made after inquiry. Although I understand and support the importance of letting kids know the learning target recently I have felt the practice has become a bit “rote” and without heart. Inquiry invigorates the heart! I plan to do this on Monday with ordering fractions – curious to see how the kids solve this without direct instruction on finding the least common multiple!
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  • 06 Jan 2018 10:50 PM | Anonymous member
    We have many standards from similar topics. As a group the teachers put them in an order that made sense to us. I have not been asking students if they have any other questions about the topics or if they are wondering about anything else. I think it would be interesting to follow students questions and maybe do some math in a different order or at least take a peak at different topics in order to answer student questions. I worry that it might lead to a bit of chaos. However, it would be worth it to have more engaged students.
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    • 17 Jan 2018 7:25 PM | Anonymous member
      Ellen, I am thinking that we could do this with congruence maybe at the beginning of the year and build from it as we need to based on their questions. Maybe use a Notice - Wonder strategy to start the unit.
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  • 07 Jan 2018 8:53 AM | Deleted user
    In my geometry class, I often use inquiry or exploration as a way for students to uncover the standards. For example, in the unit on special parallelograms, as an introduction, I created geogebra applets so the students could explore sizes of angles, edges, and diagonals in parallelograms, rectangles, squares, and rhombuses. I based my exploration on a java applet published by Kendall Hunt (which our computers can no longer open and use: http://math.kendallhunt.com/x19432.html) I hoped that through this exploration, the students would not memorize special parallelogram attributes, but experience and understand them.

    Standard CCSS.MATH.CONTENT.HSG.CO.C.11: Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.
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  • 07 Jan 2018 1:31 PM | Anonymous member
    I am getting ready to introduce a new unit tomorrow on place value using the base ten manipulatives to figure out two digit numbers. Part of the lesson involves counting how many cubes make up a ten and then counting up how many tens are there. Then the students are going to play a game to figure out which two digit number is larger. I have noticed that my first graders respond to having a hands on activity using manipulatives to get them excited about learning new math content. This lesson will tie into two of the common core math standards.

    SMP2: GMP2.2: Make sense of the representations you and others see
    SMP3: GMP2.3: Make connections between representations.
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  • 07 Jan 2018 2:04 PM | Anonymous member
    Lately, I have been really trying to take a step back and give my students more opportunities to uncover the standards through inquiry to help them engaged in what they are learning. When I started my unit on measurement I wanted students exploring how to compare lengths of objects. Before introducing the standard I was working on I introduced my students to a game called straw wars. Each set of students got a group of straws at different lengths and a token that had longest written on one side and shortest written on the other side. Each player picked a straw out and compared the lengths of their straws. They then filpped the token to see who would win the war. If the token said shortest the player with the shortest straw one the round and kept the straws. Once the students had had 20 wars they lined their straws up to see who got the longest straw out of the straws they had compiled. This activity got my students learning how to effectively measure by lining their straws up side by side and also covered comparing the lengths of objects. The students were way more engaged and interested in using correct measurement skills because it lead them to winning the game. When children have time to explore they become invested in what they are learning and the students can learn so much more from eachother.
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    • 28 Feb 2018 9:36 PM | Anonymous member
      Nice idea Amanda! The "work" of playing the game, is the thinking you don't have to coerce out of the learners. Comparing and measuring became meaningful to your learners and no pencils or workbook pages were needed! The game format is more engaging, and fun. It allows for social interaction, argumentation, providing evidence, and critical thinking. Development of these skills goes beyond the learning standards.
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  • 07 Jan 2018 6:27 PM | Anonymous member
    One way I have used inquiry is to have the students use fraction strips and circles and explore with them and talk with each other about their discoveries. The standard is for the students to understand equivalent fractions. The students start realizing that some fraction pieces "match up" with others. Then I have the students extend that learning by asking them to find patterns. For example, if 1/4=2/8, how many eighths would be equal to 2/4? The students continue the pattern and talk with others about their discoveries. It's a lot of fun to see how excited they get when they notice the patterns!
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  • 08 Jan 2018 1:17 PM | Deleted user
    One of the ways we get students to find the standard they are working on through their problem solving is to have students find the error and write about the mistake they found. We recently did this for the adding with regrouping standard and understandable place value standard. We gave them 32+9 with the 9 in the tens place. It had a sum of 122. Students were tasked with solving the problem correctly and explaining the error that they found. Math writing has been very valuable for us in getting students to uncover the standard. Students were able to recognize that the problem did not have 9 tens but rather 9 ones. They were also able to see that they could build a ten and one out of their sum of 11units and move the ten to the tens place. In this way they solidified their thinking in place value which is the basis of addition with regrouping.
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  • 08 Jan 2018 1:58 PM | Anonymous member
    We have known for a long time how math writing can strengthen a students understanding of a math concept. We begin each year modeling the great parts of students answers and how to improve math writing. It is one of out tried and true strategies for extending a students thinking without constantly posting objectives. Generally, by the end of the lesson we can ask students what they learned and through this discussion and I write their thoughts up on the board and end with writing the standard it relates to. Its amazing how close they come! In working to understand place value and adding with regrouping we ask students to find the error in a problem, solve it correctly, and write about the error that they discovered.
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    • 10 Jan 2018 11:30 PM | Anonymous member
      Hi Sheila,

      One of our teacher expectations is posting daily objectives. In doing so, I often feel like I am giving students the answers before I ask the questions. I think that posting some "essential questions" and student-generated questions is a much better place to start from (and still get to some of the same end-results).

      Pam
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  • 13 Jan 2018 11:36 AM | Anonymous member
    This is an area that I have been exploring this year. I have been putting up questions as the target and asking students to be thinking about the question as they work. The answer to the question is the standard or at least part of it.

    I have also been trying to get my third graders to reflect more in writing and will often have them write their answers to our questions as well as work to prove them. My class seems motivated and excited to share their thinking with the class as well as their "proofs" to try and change the thinking of their peers.

    It has been fascinating to watch them grow and learn as mathematicians.
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  • 15 Jan 2018 9:15 AM | Anonymous member
    It was amazing to me to read about how first- and second-grade students were able to construct such diverse and deep questions! I think I have been generally giving students the standards about which we are going to be learning, without giving them the opportunity to work out the standards on their own. I wonder how many of my students understand that there is a point to the activities that I provide them every day-- that there is a standard that we are supposed to be learning about? I think they sometimes do not see the connection between a standard and the activity. I think that time is definitely a constraint in my classroom-- a half hour lesson period three days a week does not give me much time to allow for open inquiry-- but that is not a good enough reason! I think I need to be more proactive about looking at the standards and then designing activities that will allow for students to question their thinking while working on the activities.
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  • 15 Jan 2018 11:06 AM | Anonymous member
    Our next lesson is on “Understand solving an equation or inequality as a process of answering a question: which values, from a specified set, if any, make the equation or inequality true?” This is 6.EE.B.5. Every day I write our objective on the board and we talk about it before heading off into exploring it and the objectives for this lesson are 1) identify equations and variable and 2) use substitution to find solutions to equations. If I wanted this lesson to be an inquiry lesson - I could just put a problem on the board - with a set of numbers or maybe without - and have them work in a group or individually without any discussion first. But then I’m thinking they might get a little frustrated. I am very confident my students can identify an equation - the challenge will be substitution. I see with Debbie that this was an ongoing lesson on shapes. I am wondering how others would approach this - it seems relatively simple in Debbies example for 1st and 2nd grade. I would’ve liked to have seen this at the older levels.
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  • 17 Jan 2018 11:09 AM | Anonymous member
    I am getting ready to do a unit of volume of cylinders, cones and spheres. I am thinking about ways that I can introduce the unit to get students excited about exploring the concept of volume, and review properties of solids from previous years. I love the ideas presented here about having students ask questions and encouraging their curiosity. In 8th grades, students have often lost that natural curiosity about math and simply want to be told how to solve the problems. I plan to start the unit by a "notice and wonder" activity to generate some creative thinking and then look at answering some real life questions about volume.
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  • 17 Jan 2018 12:59 PM | Anonymous member
    In Kindergarten we have just started to talk about 2-D and 3-D shape, standard-K.G.B.4. I would like to use a format of engaging students in exploration of shapes. I would have the students use pattern blocks and attribute blocks and do some exploration by sorting the shapes without prompts from me. After the students explore and sort we can have a discussion about how they sorted the shapes. During this time vocabulary would be introduced so enhance our discussion. I would then have the students try sorting the shapes in a different way. Again a discussion would take place. Students would be engaged in a discussion describing similarities, difference, number of sides, and corners and other attributes.
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  • 18 Jan 2018 3:11 PM | Anonymous member
    I really enjoyed reading the various ideas/scenarios that I could use to get students to begin asking their own questions related to math, instead of always being the "poser of questions". I think just making a shift from asking "What did you notice?'' to "What do YOU wonder?" will really open up some higher thinking questions that will lead to some very rich discussion and engagement.
    Sheryl's work in Problem Posing (pg 145- 147) really resonated with me. How often do we really know what our students are thinking about in a problem or if our students really even know what they're solving for because they are so busy trying to answer the question other's have asked them to answer that they just do the adding or subtracting to get the answer because that's what they see math as.
    I think using some of the ideas like 101questions and Notice and Wonder will really help all students access a whole other level of opportunity in mathematics.
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  • 21 Jan 2018 9:01 PM | Anonymous member
    My students know that we have standards that we have to learn, or a LIST of things they need to learn, before they go onto the next grade. On page 169, the standards that Debbie was working on were Geometry standards. In fact, these standards are our grade level. Our class has already worked with our Geometry unit. When I first began, I did not tell them what the standards or things were, that we needed to learn. I gave the students various shapes, and had them work with partners, just to tell me various things that they noticed. So, I gave them time to explore the shapes, before learning the standards. From here, I began with the vocabulary, and things we would be learning with these shapes. Many of the vocabulary words were what the kids discussed, they just used their own terminology. I think it was good to have the students time to just "play" with the pieces before they learned each standard.
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  • 25 Jan 2018 1:48 PM | Anonymous member
    We are starting a unit on Area: After reading about the geometry exploration , I've decided to change how we approach this topic. I can start by having the children create questions about Area; what it is, how it is used, how to find it. The students can then break into small groups to work on different questions using graph paper, pattern blocks, geoboards, pictures. I know that many students express their dislike for math. I believe that this approach, questioning and exploration will be motivational, active and profitable.
    I know that I'll have to write a new objective on the board to please my district but it could be something like. I will use exploration to answer my questions about Area.
    I'm excited about trying this approach.
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  • 01 Feb 2018 5:48 AM | Anonymous member
    I like to begin my units with giving my fifth grade students an opportunity to play with manipulatives, numbers, or games. Sometimes this works and sometimes it doesn't. I find about half of my students (boys) don't want to play with materials or numbers, they want to just have the handout or problems to complete and then to move on. They are afraid to play and afraid to make mistakes, so to give them lots of time to explore or wonder, some students get frustrated.
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  • 03 Feb 2018 10:08 AM | Anonymous member
    Chapter 7
    Writing the objective on the board and explaining the objective is not what I do. I begin each class with the Essential Question that we will exploring. Sometimes we have the same question for a week.
    As we develop an understanding, and students are able to answer the question, we go onto another question.
    When I assess their understanding, I am assessing how they are able to answer the essential questions. The essential questions are derived from the learning targets, but they are real life examples that students can and do relate to.
    Big Ideas, our text book, lists essential questions for each chapter, and I add to this list.
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  • 08 Feb 2018 4:28 PM | Anonymous member
    For me I try to give students a chance to discover the standards by giving them time to explore the new concept. I may use shapes for geometry, base ten blocks for place value, or coins for money. I want to give my students a chance to explore the new tools we would be using. I hope that it allows my students time to question learning ideas.
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  • 09 Feb 2018 5:41 PM | Anonymous member
    I agree with Sarah (above me? Below me?!) about how easily Geometry lends itself to explorations where students in turn uncover the standards.

    Generally speaking, I find that my books and resources -- even once I love -- tend to lay out a rule or standard, and then give examples and activities that demonstrate why it's true. It's easy, sure, to present a rule and show why it works, but Dan Meyer's big idea about "finding the headache" for particular "aspirin" has me trying to get my students to figure out why something MUST be a rule.

    For example, when we got to triangle congruence proofs, I asked a version of "are these two triangles congruent?" and that forced students to explore and defend their decisions. I provided guidance in terms of how manipulatives might be helpful, and this led very nicely to the SSS, SAS, etc. instead of listing the postulates/theorems and simply showing how (or that) they "work." When we get to quadrilaterals, I will be able to scaffold even less, I hope, because students will have the experience of using manipulative to help demonstrate -- prove! -- their conjectures.

    Side note: make no mistake: we're going much more slowly by doing explorations as much as possible than we otherwise would, and while I'm fortunate to be in a school where I'm not worried about racing to a state test, I do worry about pacing and how much "traditional geometry curriculum" we will complete. It's a tricky balance to cover it all in a way where I am confident kids are learning in a rich way, but still covering as much of the broad content to really feel like we've done "Geometry" (and our students!) justice.
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  • 16 Feb 2018 9:54 PM | Anonymous member
    Sometimes I post an SAT question, or one I've seen in an SAT practice book, related to the current topic on the board, for students to explore. Recently it was - "What is the sum of the first 100 positive integers?" This problem generates conversation around the language (what do we mean when we say positive integer"). It is always interesting to see the varied approaches taken by students to "solve" the problem; invariably one or more groups will simply add the numbers (1+2+3+4 ... +100). Occasionally a student might figure out the formula for the sum of an arithmetic series.
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  • 25 Feb 2018 11:00 AM | Anonymous member
    I am really struggling with this question of how to get students to uncover standards through inquiry. I love how Debbie worked with her students and found a natural way to lead into the geometry unit. I have not asked students what they wonder about in regards to fractions, the unit I am currently still on with one of my fifth grade classes; I honestly never thought to ask. Like everyone else, I learned math through memorization and rote practice. Questioning and wondering was never a part of it and therefore doesn't feel natural. My fifth grade class is having a very difficult time with adding fractions with different denominators. I have incorporated manipulatives such as fraction bars and fraction strips, yet the students are still not making sense of the problems enough to apply to situations where they are unable to use the manipulatives to solve. I feel the pressure to move forward even though they don't understand and yet I know that can't be the real answer.
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  • 28 Feb 2018 9:45 PM | Anonymous member
    One way that I have worked on equations and inquiry learning was through the use of pattern blocks and equation strips. I used sentence strips with a variety of equations, leaving the values blank. Some examples: ________ + ____________ = ___________ + ___________

    ________ - ____________ = ___________ - ___________

    __________ = ___________ + _____________ + __________

    and so on......

    Learners used pattern blocks in place of numbers to show equalities, and learn more about what "equal" means in mathematics.
    Often there is confusion and frustration at not being able to "just use numbers". Yet, when one student makes an exciting discovery, well, that inspires others to persevere and keep going.
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  • 13 Mar 2018 10:07 PM | Anonymous member
    Know your standards, set up opportunities for kids to explore and ask questions to then pursue various paths to get their answers, let them work in groups and if this exploration leaves out any part of a standard then guide them there. How neat to see how each class does this exploratory work and how your role changes.

    At the high school level we get so caught up in the UBD's and put the essential questions right out there every time. Maybe we could do it in reverse some of the time to let kids think differently about how to take what they've learned and develop the essential question! Again, if they don't get there, guide them there. I'm thinking that for the most part they are capable of getting there is our work is sound!
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