Association of Teachers of Mathematics in Maine
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Chapter 5: Response Choice 1

04 Feb 2019 9:43 AM | Anonymous member

1. How are manipulatives used in your mathematics instruction?


2. Mathematical practice 5 calls for students to use appropriate tools strategically. How do you allow students to make decisions about the tools they use in their work?


3. How are these tools used to move learning from surface learning to deep learning?

Comments

  • 05 Feb 2019 5:42 PM | Anonymous member
    1. Math manipulatives are usually used in my classroom when I am introducing a new concept. I use them to make sure my students are solid in their number sense and mathematical thinking. For example, when I began my unit on fractions I had my students show the fractions they were working with with fraction circle pieces. Then when we started adding and subtracting fractions we used them to show why the denominator did not change. We used them to show equal fractions, etc.

    2. When we are first learning about a topic we do use the manipulatives whole class. At that point students do not have a choice. They have to use them. Then the manipulatives are places in a bin on a shelf that has all of our math manipulative and they stay there in a bin students can access whenever they like. They can use them at this point or not. I leave it up to them.

    3. These tools are used to move learning from surface to deep learning because once students know what they can be used for and how they can use them, they will go to get them to help them solve more difficult problems. So they are using them to build on what they already know and try something new. The tools give the the security to be willing to try something new.
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    • 15 Feb 2019 10:34 AM | Anonymous member
      Julie, using manipulatives when starting a new concept is a great way for students to develop conceptual understanding before moving to abstract understanding. Gradual release of having all students use the manipulatives to then allowing students to use when they need is a good way to scaffold the use of manipulatives.
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  • 09 Feb 2019 6:47 PM | Anonymous member
    As mentioned in my Chapter 4 response, I wish I was able to incorporate manipulatives more in my instruction. I have used chips with integer operations, measuring circles for circumference and mobiles for balancing equations. I’m always looking for great suggestions!

    I encourage the use of models almost to the point of requiring it. I think that using models helps students show their thinking - which 7th graders can have trouble doing. I get a lot of “I just know” or “I did it in my head”. I usually reply with “show me”! Which is how they can move from surface learning to deep learning.

    That being said, I do try to give students choice in the tools that they use to show their thinking. We use Notability on the iPad a lot, but they can also use graph paper or white board or however they can express themselves best.
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    • 15 Feb 2019 10:37 AM | Anonymous member
      Julie, models are such a great way for students to show their understanding of a concept. It really shows their conceptual understanding and helps teachers determine where there may be misconceptions or misunderstandings. I would love to hear more about notability in the classroom and how students are using it.
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      • 24 Feb 2019 2:21 PM | Anonymous member
        We use notability as our 'math notebook". I have made electronic version of each of our CMP problems (such as: https://drive.google.com/file/d/196QhRhoo8HWprmWPqKTZfvh0XBdvFOdR/view) and the students import it into Notability. They then can hand write or type answers and can make graphs or other visuals. And of course they love picking their own colors! :-)
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        • 25 Feb 2019 9:39 AM | Anonymous member
          Julie that is really cool!
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  • 11 Feb 2019 4:35 PM | Anonymous member
    1. I use manipulatives in my instruction when I believe they can help students visualize what I am trying to teach. For example, when teaching fractions, it makes sense to show them either using online tools or physical fraction circles or fraction bars, how is 1/2 = 2/4. I usually talk about pizza and tell them stories about having parties with certain students, etc.

    I also use manipulatives if I have a group of students who are struggling with concepts. Sometimes manipulatives can help students understand concepts in a different way. Manipulatives can help make an abstract concept more concrete.

    2. When students ask to use base ten blocks or fraction circles, I ask them why. How do you think that will help you? I often find that some students play with the tools instead of using them as tools. By asking them to explain how they are going to use them, it helps them to think about what they are going to do.
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    • 15 Feb 2019 10:43 AM | Anonymous member
      Penny, manipulatives are a great first step when introducing new content to help students gain conceptual understanding. Often times if students are just shown an algorithm they may be able to memorize it but will have no idea of what they are actually doing and why it works. Manipulatives can be a distraction for some students but asking them how they will use those tools is definitely a good way to hinder playing with them.
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  • 11 Feb 2019 4:38 PM | Kristan Curtis
    As many have said, manipulatives are often used when new concepts are being introduced. Manipulatives give students the ability to figure things out on their own rather than being led by me. They allow students to see for themselves what works and what doesn't and have an opportunity to try and fail and make discoveries as they work.

    I leave my manipulatives out in open containers where they can be seen. I will often remind my students that "we have a tool somewhere in the classroom that can help you with that" or simply "use your tools!" I also make sure that students know when there are different choices for use in solving problems - fraction circles or fraction rods or strips or stax, for example.

    Because students are accustomed to using manipulatives when trying to figure things out, as concepts become deeper and more complex, it has become second nature for students to use tools to figure out the answers to questions.
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    • 15 Feb 2019 10:49 AM | Anonymous member
      Kristan, leaving manipulatives out in open containers is a great way to invite students to use the tools around the room. They are readily available for them to grab when needed. This invites problem solving into the classroom where students can really use what they know to figure out problems. I am curious how long it took your students to really make the shift to grabbing the tools when they need and it to become second nature to them? I have seen classrooms where materials are out and available but students do not necessarily seek them out and use them. How did you get them to make that shift to becoming problem solvers?
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  • 13 Feb 2019 1:54 PM | Anonymous member (Administrator)
    As I go into classrooms I've been trying really hard to highlight manipulatives that can be used with a lesson, small group activities, or stations.

    When I was in the classroom I used to leave my manipulatives open for use as needed. When students would get stuck on questions, I would always have students draw a picture to have a visual representation. This really helped with problem solving and tackling an unfamiliar problem.

    I like to challenge students to find a visual representation to prove their answer is correct. Students have been able to transfer what they know and apply that in later units.
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    • 15 Feb 2019 10:50 AM | Anonymous member
      Jennifer, I have been trying to do the same when I notice that a student is struggling on a concept and that a certain manipulative may help. Making manipulatives available and making sure students know that they are there for their use (as a tool) when they need them, really is key.
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  • 18 Feb 2019 6:51 AM | Anonymous member
    I pretty much do the same as the two Julie's above with manipulatives. I teach Geometry. The manipulatives that I use most are patty paper, ruler & protractor, and Geogebra. My students typically won't go get the manipulatives later when they are struggling, but I can usually reference the activity to refresh their memory to the concept that we were doing. "Remember that Geogebra activity that had the polygon with all the exterior angles a different color, what happened to the angles when we changed the size of the polygon? What did that show us about the exterior angle?" If they don't remember, then they would go look at that activity again, but they usually remember.
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    • 25 Feb 2019 9:44 AM | Anonymous member
      Renee, it is interesting that your students won't typically go get manipulatives when they are struggling. Why do you think that may be? The activities must have been engaging for students to reflect back and remember an activity they completed when learning the concept.
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  • 19 Feb 2019 2:17 PM | Anonymous member
    1. How are manipulatives used in your mathematics instruction?

    Previously I used manipulatives daily with my students, primarily base 10 blocks, unifix cubes, or one to one counters. Now that I am teaching at the upper level, I feel that the manipulatives have shifted a bit. WE use whiteboards and markers daily, and students are well aware of the use of "pictures, numbers, or words" to demonstrate their thinking.


    2. Mathematical practice 5 calls for students to use appropriate tools strategically. How do you allow students to make decisions about the tools they use in their work?

    I try my best to give them a variety of options to "show their work" but they have to be able to move beyond the "I did it in my head" or "I used (multiplication, division, etc.)". I allow students to use white boards or their desks to demonstrate their understanding of problems. Whenever we are approaching a problem solving activity, I give examples and non-examples of what I am looking for for responses, and then reinforce the "pictures, numbers and/or words" for their responses. From the beginning of the year until now, they appear to have made significant progress. Having exemplars for them to use had made the learning even more meaningful, so now they see how other people use the same strategies and tools to demonstrate understanding of the task.

    3. How are these tools used to move learning from surface learning to deep learning?

    They use the tools to elicit discussions, demonstrate concrete understanding, while giving them an opportunity to demonstrate understanding of the "why", not just the "what". They are using the tools to help them solve the problem, and through the tools they are learning to explain their reasoning and understanding of what they are learning. The tools are used to demonstrate the deeper understanding of the initial "surface" learning.
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    • 25 Feb 2019 9:48 AM | Anonymous member
      Holly, I love the idea of using examples and non-examples for students. Showing students an exemplar really shows them what they should be trying to do without making them guess. What grade level are you currently teaching?
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  • 24 Feb 2019 10:54 AM | Anonymous member
    When reading about manipulatives in this chapter, as always, I immediately related it to my own classroom and my instruction. Reflecting honestly, I know that there are concepts where I use manipulatives more often, and more effectively. Examples of this would be geometry, fractions, and measurement. Areas of improvement for incorporating manipulatives would be place value and the operations. I plan on incorporating more in the future. Also, the reading reacquainted me with virtual manipulative sites that I used more frequently in the past that I want to use more in the future. I assumed students would be familiar with place value manipulatives when they got to fourth grade, but they often tell me they have not used them for 2-3 years and they do not know how, which adds to the amount of time needed to reteach how to use them effectively. Now I see the value of this and plan on doing that more in the future, and perhaps speak to colleagues in the younger grades about using them as well.
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    • 25 Feb 2019 9:54 AM | Anonymous member
      Kristin, some units or concepts are definitely more conducive to using manipulatives. It is always good to try and have some sort of manipulative that would be helpful for those students still working on conceptual understanding. It is interesting that students have told you they have not used place value manipulatives in the a couple of years. Students could definitely benefit from a refresher and then be able to use them when doing place value at your grade level.
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  • 28 Feb 2019 8:11 AM | Anonymous member
    With my younger students, I try to model the use of different tools to answer questions. I try to use both tangible tools like tiles and blocks, as well as apps when appropriate, and organizational tools like making a table, picture, or diagram to solve a problem. I have not been as good this year at using apps due to lack of technology and time. As my groups get older, I try to to work towards use the most efficient tool for the question or task. When solving open tasks, I have students use the tool that they think will work best for coming to a solution or solutions, but sometimes I help to guide them when they struggle with how that tool is used in the context of solving the problem. With tangible objects or picture/diagrams, students can visualize better what the question is asking. I struggle with getting my older students to "draw the picture" for geometry questions, when many times if they do, then they can see how to make the next move in solving the question.
    The biggest struggle I have with my students is they want to solve everything in their head and quickly when deeper learning generally requires one or more representation to get to the end result.
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    • 28 Feb 2019 10:08 AM | Anonymous member
      Carolyn, this happens so often. Students want to just hurry up and get an answer when they really need to slow down and think about what the question is asking and what strategies or tools they can use to help solve the problem. I find it interesting that the older students are not really interested in drawing a picture for geometry problems. I wonder if they feel there is some sort of stigma attached to using different manipulatives or tools to try and help solve a problem? I know in the past I had students embarrassed when I asked if they wanted to use a number line to work on adding and subtracting integers but they did not want to because other students were not necessarily using it.
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  • 04 Mar 2019 6:24 AM | Anonymous member
    My students in grades 4 and 5 use a variety of manipulatives including fraction strips, place value materials, and cubes, especially for volume. For my geometry students we use tools for constructions such as rulers, protractors, compasses. I use Desmos and Geogebra with the geometry students. I am starting to use Desmos and Geogebra also with the younger students to demonstrate concepts.

    For all grade levels, I ask the students to represent their thinking through drawings. This is something I have been emphasizing at all three grade levels. As I read through others’ comments I noted that my students are not the only ones who want to jump right to the answer without showing their thinking. Some students are responding while others are anxious to answer the question without thinking too deeply. When a student can visually represent their thinking I see the learning move from surface learning to deep learning.
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    • 11 Mar 2019 2:37 PM | Anonymous member
      Mary, I agree with you. "When a student can visually represent their thinking I see the learning move from surface learning to deep learning." I think without seeing their thinking it is really hard to determine the level of learning and understanding that has taken place. I am interested to hear how you are using Desmos and Geogebra with your grade 4 and 5 students.
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  • 04 Mar 2019 8:41 AM | Anonymous member
    How are manipulatives used in your mathematics instruction?
    I use manipulative in two ways-
    1. When I am introducing a topic, I demonstrate how the manipulative represents the symbols in the concept, place value for example.
    Secondly I make the manipulative available for use as students are working independently. At times I will prompts students to access the materials by reminding them how we used them when learning a concept. I find they are more likely to use manipulative when they are taking a concept to a different level. For example- if we used hands on shapes to measure and calculate perimeter, the students may return to the shapes when we begin applying the concept of perimeter to complex shapes.

    2. Mathematical practice 5 calls for students to use appropriate tools strategically. How do you allow students to make decisions about the tools they use in their work?
    I try to be sure that tools are available and accessible. I may also make a comment like: "Oh cool idea, using place value blocks to help solve that problem"

    3. How are these tools used to move learning from surface learning to deep learning?
    When students can transfer a concept from physical to theoretical or from theoretical to physical they are demonstrating their understanding of relationships. They are also able to discover and understand the mathematical principals.
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    • 11 Mar 2019 2:42 PM | Anonymous member
      Ann, I like that you brought up the idea of understanding relationships between physical and theoretical. It shows flexibility in their thinking and multiple representations of the same problem. I also appreciate the positive reinforcement to students that are using manipulatives.
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  • 05 Mar 2019 11:13 PM | Anonymous member
    I used fraction pieces to introduce the topic. I then got bold and had a 'station' be exploring with the pieces. They didn't know what to do with them other than be goofy. However, once we got into the unit, those pieces became a resource for those who need something to look at and use when calculating. Although they are out to be used, I forget to use them when trying to explain other parts of fractions--like why 1/3 + 1/3 isn't 2/6--manipulatives would be great and I didn't think to use them. For a real-world problem they are working on, I meant to bring in measuring cups (I will tomorrow!) to see and 'feel' along with something like sand or rice to play with. This would take students from the surface learning of following instructions (same denominator means leave it and only add the numerator) to applying knowledge.
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    • 11 Mar 2019 3:10 PM | Anonymous member
      Rachel, I like how when you were responding to the post it seems to have sparked more ideas for you and you showed those ideas. I think bringing in measuring cups is a great idea and will really help students develop understanding conceptually. I hope it went well!
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