Association of Teachers of Mathematics in Maine
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Prompt 1 - Option A

24 Oct 2014 6:00 AM | Anonymous member (Administrator)

In figure 2 (pp. 14-15), Mrs. Burke says that she wants students to "better understand these different types of word problems and be able to solve them." Find solutions for each of the three problems in the figure. What equations could be written to solve each problem? Which equations match the story situation? Discuss how these three problems offer different ways to think about subtraction.

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  • 02 Nov 2014 3:20 PM | Anonymous member
    In this vignette I like the way the Math Coach moves the teachers to focus on the mathematics that the students are learning (most teachers focus on what students need to be doing) and what other prompts students may need. From her questions - What indicators should the teachers consider during the planning so they can establish if the students are learning - she prompts the teachers to get at the learning - understanding subtraction and making a connection between addition and subtraction. The equations could include: 15 + ? = 22; 36 - ? = 9 and 43 - 27 = ?. The first one 15 + ? =22 - if the student records or explains one gets the result from 22 - 15 - shows the thinking connected to inverse operations. Getting at the thinking posed in the various problems - align with the CCSSM table on page 88 of the Common Core Standards: Add to / Change Unknown; Take from/ Change Unknown and Compare/Difference Unknown. Using examples such as these gives students the opportunity to discuss the variety of ways to think about subtraction and connecting to inverse operations versus just practicing take away problems to experience subtraction.
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    • 13 Nov 2014 10:14 AM | Anonymous member
      I love how both you and Angie point out the fact that the math coach is focusing the teacher on what the students are learning, not the procedure for solving the problem! When it is all said and done, these teaching practices are best practices for supporting student learning and understanding of mathematics!
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  • 04 Nov 2014 9:14 AM | Deleted user
    I like how the coach asks questions to lead the teachers through the thought process on what they really want the students to learn; not just get the correct answer for each word problem. It is good that each problem is about a different topic like books, balloons, & running; trying to connect with students interests. I think it is important to illustrate all the different acceptable equations that may be given by the students: { 15 + ☐ = 22, 36 = ☐ + 9 or
    36 – ☐ = 9, and 43 – 27 = ☐ or 43 = 27 + ☐ }. Students see that one way to think about subtraction is figuring out what has to be added to get the total. Students also see that subtraction is finding out how much is left or needed in a situation. And, subtraction can be used to decide how much bigger a number is compared to another number.
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  • 05 Nov 2014 5:19 PM | Deleted user
    The first example reminds me of how we try to teach "fact families" in Saco at the elementary and middle grades. You could solve it 22 - 15 = x , but it is almost framed as
    15 + x = 22. If students understand fact families they can better understand how to approach that problem and they will have more strategies.

    These are all clearly different ways of looking at subtraction. As a middle school teacher I would be interested in presenting these to my students and see if they could turn them into algebraic equations.
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  • 05 Nov 2014 6:33 PM | Anonymous member (Administrator)
    These comments remind me of Christopher Danielson's keynote at our last Spring Conference, when he talked about Cognitively Guided Instruction (CGI). That there are different ways to think about these types of problems: join, separate, part-part-whole, and compare. It would be interesting to hear what elementary students do with these prompts. I also thought of fact families. It seems as if we tend to tear away at that fact family number sense as students begin to "learn" algebra. Forcing procedure, for the sake of procedure, over sense-making. What I mean is, given a simple problem like x + 4 = 12, algebra teachers have a tendency to emphasize the procedure of "subtract 4 from both sides" to reach the solution of x = 8, rather than honoring the fact that x + 4 = 12 is equivalent to x = 12 - 4 because they belong to the same fact family. This, in turn, forces students to focus on procedure, rather than sense making. I know this is probably straying from the topic, but, as a high school teacher, I would love for a kid to say that x = 8 (in the previous example) because, well, it's just obvious.

    Sorry for the distraction, but thanks for indulging me.
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    • 08 Dec 2014 10:23 AM | Deleted user
      Pam,

      Loved your last line, "I would love for a kid to say that x = 8 (in the previous example) because, well, it's just obvious" because many elementary students do say just that. However, these students have had an opportunity to explore numbers/fact families in multiple situations and have had many opportunities to discuss their ideas. Students work with subtraction word problems look different depending on the student. Some will add on, others add back, while others will create a subtraction problem and try to solve it. Some use a number line model. Providing students with these types of problems enables them to see the connections to addition and subtraction, and become more fluent in solving them. It is especially important to let the students share their strategies with their classmates, and the teacher should help students connect these various strategies.
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  • 15 Nov 2014 1:03 PM | Anonymous member
    The coach encourages the teaching team to keep the learning targets in mind as they design activities for students to engage in. Using three different situations that require knowledge of addition and subtraction should result in a deeper mathematical understanding.
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  • 20 Nov 2014 8:19 PM | Anonymous member (Administrator)
    In solving the three problems I found that I would expect to see kids using a number line, an open number sentence or a subtraction problem. Equations that I wrote were 22-15= 7 and and open model of 15 + __ = 22. For the second problem I wrote 36-9=27 and 9 + __ = 36 and for the 3rd problem I wrote 443-27=16 and 27 + __ = 43. Students might think about these problems as counting on problems. In the first situation kids might count up from 15 and keep track on their fingers. I think the open number sentence is least likely to be used by younger students - these models don't seem to help with the answer - something more needs to be done to figure out the number that belongs in the blank space.
    I also smiled when I read Michele's Nov. 13th entry. I agree that the Math Practices are what students should be doing and that teachers need to be providing opportunities for students and encouraging them to "do" the practices. The Math Teaching Practices on pg.10 are a great connection to what teachers should be doing.
    I appreciate the fact family connection mentioned by several people. I've always been amazed at how hard it was for some second grade students to understand fact families. When I was working at the elementary school, a second grade teacher and I worked on the fact family lesson over several years. It's been good to see fact families being used at the middle school to teach operations with fractions and integers.
    Longwinded entry but it is my first and was great to read through the other entries as I was writing and thinking about the reading.
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  • 01 Dec 2014 8:29 PM | Deleted user
    Apologies to all for this late entry! Trying to get many things caught up. I am teaching a new grade level this year.....Grade 2! Previously I taught both first grade and third grade. Monkey in the middle! I wanted to join this book study group in order to hone my practice and pay more attention to learning outcomes. I feel that I teach process very well, but when I have tried to involve students more in conversations about math, they are not confident and often shut down. One difficulty is having the time to converse in a thoughtful way...not trying to beat the clock!

    So, on to the prompt: 15 + ? = 22 / 36 - ? =9 / 43-27 = ? . These are the first equations I would expect from students. The first problem does relate to inverse operations, but my students would simply count up on their fingers from 15 to get to 22. Some would get the correct answer. Many would make errors because they don't count carefully. The next problem requires more work. A number chart is what my students have readily available to them, so they might start at 36 and count back 9 to find the answer. Others would start at nine and try to count up to 36, but they wouldn't make it without an interruption or a mistake. The last problem asks students to subtract to find the difference between two numbers. They can find the answer, but they don't have a confident sense of the word difference and what it means. They know one number is greater than the other, but asking how different is that number, i.e.. a lot different or a little different, they should know, but don't. Not yet....
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