Association of Teachers of Mathematics in Maine
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Week 8: Call to Action (Option 2)

12 Jan 2018 4:52 PM | Anonymous member (Administrator)

Think about models you teach. Do your students currently see connections among them? What might you take from Becky's Example (191-193)? Try it, what did you learn?

Comments

  • 13 Jan 2018 1:16 PM | Anonymous member
    In order for true mathematics proficiency to be achieved, connections must be made. We need to connect new mathematical ideas to what we already know and build on that new connection to develop understanding. I am a visual learner; I need to see the math even though I wasn’t taught that way. I learned that I needed to draw, diagram, chart, etc. to develop understandings that I could retrieve later and apply to more complex mathematical concepts. These often self-developed models and representations allowed me to make connections and learn to love mathematics.

    My sixth graders have been working on Greatest Common Factor and Least Common Multiple. We used multiple models and representations: charts with lists of factors and multiples, Venn diagrams, area models, the Sieve of Eratosthenes, factor trees, factor ladders. By using multiple models and representations, I find that students are able to make sense and choose the model that works for them. With prime factorization, we discussed the merits of factor trees versus factor ladders that use repeated division. The group is divided almost equally in their preference in this area. By experiencing both methods and seeing the connections between them, they were able to make a reasonable choice.
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  • 15 Jan 2018 1:28 PM | Anonymous member
    For my students, operations with Integers is a new concept. For adding Integers we started with a number line and then moved to two-colored Integer chips. Both the number line and Integer chips required movement on the students' part, either using their fingers to "hop" along the number line to moving chips to make zero pairs and adding the remaining chips. From there, I introduced them to an Integer song I had written. We also used lecture style math notes. As Becky stated, all students have preferences and helping them bridge from one model to the next deepens their understanding. It is nice to have a conversation about how two different models can have the same result. It also lets students know that there is no one right way to approach a problem. Many students who lack confidence in math feel that if their approach is different, it must be wrong and that often is not the case. It is nice to diversity in math thinking.
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  • 17 Jan 2018 10:39 AM | Deleted user
    Math is all about taking what you know and applying the new strategies you learn and making connections with what works best for you. In Kindergarten we use a lot of visual manipulatives and physical movement to work out math strategies. We have been working on composing and decomposing numbers to 5. The students can easily take a stack of 5 linking cubes and break them in to two parts and come up with a number sentence like, 3 and 2 make 5, 1 and 4 make 5 and vice versa. We talked about how numbers can be reversed when adding and the learned a college term :)......Communitive Property of Addition, they felt really smart then. I asked the kids if they could break apart the cubes in a way they had not done so. Most students broke them in the reverse but I had a few students who broke them 1 and 1 and 1 and 2 make 5. I drew attention to thinking outside the box and asked the question again. Students then broke them apart in at least 3 ways and some did 4 and 5. For kindergarten learning is concrete and when attention is brought to different thinking, they are willing to attempt the new thought process. We moved from cubes to number bonds. I had hula hoops displayed as a large # bond and we used students to move from the whole to the parts and vice versa. We tried to make connections to our linking cubes and how we had whole and parts and how this strategy was the same but just a different way to show how we can make 5. We took the whole and put it into parts by having students move down a path to a party. This gave them a visual representation of how we seperated the whole. We then took the path back to make the whole again. We then moved on to paper number bonds and used beans to put into wholes and beans to seperate into parts. The students were all good while using manipulatives but the transfer over to using actual numbers was a bit confusing to some. They struggled to understand that the number stood for the amount of something. I went back and and had them write number sentences for the beans (addition only) so they could see that the numbers were representing the beans. Once they made this connection after some practice they then understood that the numbers they were placing in the number bonds were the whole and parts of the whole. After some practice they uderstood that the digits were representations of the pictures and could make the connections they needed to make the number bonds with digits and write the correct number sentences for the number bonds. Our next step is to now have missing digits and to be math detectives to find out what the missing part is to complete our sentences and number bonds in making 5.
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  • 17 Jan 2018 6:26 PM | Anonymous member
    I thought Becky’s example of making the connection between the equations generated from the rekenrek and the models generated from the number line were great. I am often astounded that connections that seem so obvious to adults are not obvious to students and how not making the connection obvious hinders their understanding. One model that gets used to introduce multiplication at our school is Circles and Stars. Another model that gets used is an array. Considering I saw the array model as a more efficient model to show the same idea, I was surprised to find that few students could explain where the circles (groups) and stars (objects in the group) were in the array model. I never would have realized that they didn’t see the two models as connected if I hadn’t blatantly asked them where those things were. This was powerful for me to learn as a teacher. We need to be very intentional with helping students make the connections between the models we use. John Tapper uses an analogy comparing our job to that of planting carrots or potatoes. Carrots grow straight down, we don’t want to cultivate carrots, we want to plant potatoes that spread out and produce more potatoes/ideas/connections. I try to keep that analogy in mind when I plan instruction.
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  • 26 Jan 2018 2:14 PM | Anonymous member
    n my practice models are used to make connections when developing and strengthening conceptual understanding, then we move toward using a procedure. If students are just given the procedural process as their first step to solve an operation and not allowed to explore and develop models of their own, then the goal of helping students develop strong mathematical practices is meaningless.

    Our students need to be flexible in their thinking about how to solve for themselves and the more opportunity we give for connecting with models, the more opportunity they'll have to think and do foe themselves and find answers in ways that are meaningful to them and not how we say it should be done.
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  • 01 Feb 2018 9:22 AM | Anonymous member
    I teach third graders and I don't think that they always see or look for the connections between each other's strategies naturally. I think that a teacher can ask questions like can you find the 2x4 in this problem of 8 x 6? To try and get students to see the connections. (2 x 4 = 8).

    I often have students share their strategies but don't always ask the follow up questions that will lead to deeper connections being made between the strategies. So, that is what I would like to focus on over the next few weeks. I think that if I do this, it will lead students to becoming more efficient in their strategy use.
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  • 28 Feb 2018 11:02 PM | Anonymous member
    Our math program provides learners with numerous models for learning new concepts and procedures. The goal is to give learners options for problem solving in a way that works best for them. However, the models are taught in quick succession without the time allowed for making connections between the models provided. Also, the models become confused, because students are trying them out without fully experiencing how they work, before moving on to another model, and trying to understand how it works. As a teacher, I have often provided a means of making these connections. I want students to learn the variety of ways to solve problems, as well as why each model works. Learners do land on a preferred method, but I also want them to value all of the methods, and understand how they are connected and related to doing the mathematical reasoning and problem solving. The experiential learning is not emphasized in our program. This is an area that teachers can address easily, though it certainly requires an allotment of time, and planning for efficient learning.
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  • 13 Mar 2018 10:35 PM | Anonymous member
    Our students do see them in the YouCubed Unit on Patterns and Functions. They get there pretty quickly and know that there are often a variety of ways to see patterns growing as well as how we might come up with general rules that are structured differently, but still represent the explicit and recursive patterns. Our students love to share their thinking about how they see a pattern growing, shrinking, changing. We usually throw them for a loop when we get into the staircase problem, but once they see the light, they generally understand what make this problem a bit different from many of the others that they are asked to evaluate. The solving method makes sense and can be brought to life for them via some interesting coloration and use of our SmartBoard technology.

    Oddly, not too long ago we use to see IEP's that had learning objective goals that would state that students should be shown only one way to do a math problem so that they wouldn't be confused with needing to see the problems done in other ways!
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