Week Four: Chapter 3: Investigating the 5 practices in Action

28 Jan 2014 7:05 AM | Anonymous member (Administrator)

Week Four: Chapter 3:  Investigating the 5 practices in Action

Read pp 21-30

What, if anything, would you like to see Darcy Dunn do differently?  

How do you think the changes that you propose would have affected student learning?

Have you used a problem like this?  

What did you learn about effective questioning?


  • 30 Jan 2014 10:58 AM | Deleted user
    First, I like how Ms. Dunn gave students five minutes of “private think time” at the beginning of the task or working with others, and also had concrete materials to support their understanding of the concept!!

    During the first discussion with Beth, no student asked questions. Ms. Dunn could have utilized one of the 8 Mathematical Practices, Construct viable arguments & critique the reasoning of others. I would have pointed to the poster in my room, and encouraged students to participate. So instead of Ms. Dunn doing all of the effective questioning of Beth, the students would have been.

    Also during the dialog between Ms. Dunn & Devon, Ms. Dunn commented, “Ooh-I like that.” This invalidates the other students’ ideas when they stood up in front of the classroom to present, as well as the students who did not share. To me it also states that this was the best representation so far and the method students should use.

    Another method I might have used in order to get all representations presented (ex. Tamika’s) would have been to ask certain students to record their ideas on the board at one time, then ask one student at a time to explain their reasoning. This may have saved some time so that Tamika’s table could have been explained. I have done this before, and it is interesting watching & listening to the rest of the class as the student’s write their representations on the board. Quite a few side bar discussions occurred, and some students who were struggling had an opportunity to discuss someone’s idea with another student. Many times I heard, “I get it now!”
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    • 02 Feb 2014 10:38 AM | Deleted user
      I share Donna's concerns about Ms. Dunn's "Ooh…" remark [line 16], but I wonder if a remark like that could have been salvaged or do such remarks need to be barred outright? I think that registering delight should be welcome in any classroom's discussion of any topic, as for when an unusual or unanticipated perspective sheds light on a situation. It's those sparks of insight, not always correct, that measure the aliveness of classroom engagement. In this case, an expression that involved subtraction was not unanticipated by Ms. Dunn, and arriving at it does suggest a creative leap, some kind of think-different disposition. Is there a way that Ms. Dunn could have registered delight without downplaying other student's efforts and without marking it as the "best way" or the "favored way?" Could she have said: "Ooh, that's a different way of looking at these patios. I like the way you stuck to your vision and found a rule that that explains it." I might then throw it to the class to identify in what way(s) this approach was different or similar to the other approaches. I'd be tempted to add "Very creative work indeed," but that would be going too far again. As teachers, we DO SEE real creative leaps from individual students from time to time, but if we acknowledge that publicly, then we are telling students that we have a scale against which we can measure and rate creativity; that this effort IS creative but those efforts are NOT or are LESS creative. It would be better, I think, to congratulate a hard-working class on the creative thinking they had just shown so much evidence of having done. Students can not hear too many times that doing mathematics is a creative process.
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    • 02 Feb 2014 9:23 PM | Deleted user
      She could have saved herself from invalidating other students with the "Ooh-I like that" comment by following it with the idea that what is great about learning is that we can often "see things differently" yet come to the same conclusions. Myabe she should have added an "ooh-I like that" to the chart idea too.
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      • 04 Feb 2014 11:03 AM | Deleted user
        Clair, Ooh I like your thought regarding this!!
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    • 03 Feb 2014 7:31 PM | Deleted user
      Yes, replying with added enthusiasm can project unintended negatives to other responses. That's a good point. One thing Darcy could have done is to ask students what they thought of Devon's strategy. This has the plus of having students respond directly to Devon, state their own understanding, and it comes with the possibility that a student might say, "Ooh, I like that."

      Individual think time is great.

      I agree that attempting to have students ask questions of Beth and generate student-to-student conversation would have been an improvement.
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  • 02 Feb 2014 11:03 AM | Deleted user
    [line 67] After Beth’s presentation, Ms. Dunn pressed students to express Beth’s way of
    viewing the pattern symbolically as w = 2b + 6, where w is the number of white
    tiles in the patio and b is the number of black tiles.

    We don't know Ms. Dunn's class's (prior) experience with variables here, so it's not clear what understanding her students had that could support their efforts while "being pressed" to "express Beth’s way of viewing the pattern symbolically." One thing I might have done differently is to ask all students to write the (linguistic) sentence of the pattern rule that Beth had just reported: "The number of white tiles is two times the number of black tiles and six more." Depending on student's prior knowledge and their love/hate of writing words, by the second or third pattern rule report, some especially irritated student might lament; "C'mon! Isn't there a shorter way to write this?" Good question! Is there?
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    • 02 Feb 2014 9:21 PM | Deleted user
      I, like Barry, would have asked the students to write in words what the pattern was. I think this would have helped both the students who were presenting have some clarity before they explained it and those who were watching/listening to be able to connect the presenter's thoughts with their own. For some students, the written description might have more easily led to the numerical equation.
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    • 03 Feb 2014 7:35 PM | Deleted user
      Writing the relationship between white and black tiles in words is a good idea. Students have a harder time with the symbolic. Students could also state the patterns they notice. That could lead to some understanding of which two white tiles are added each time.
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  • 02 Feb 2014 3:46 PM | Anonymous member
    Like Donna, I noticed that no students asked questions during the first discussion with Beth. In fact, there seemed to be very little discussion between the students at all during the whole group portion of this lesson. If felt like it was very teacher driven. The only time that the students interacted without the teacher was when she pointed out that the class seemed to be confused and she directly asked if anyone had any questions.(82-85) I would like to see a classroom expectation that the students engage in direct dialogue with each other even in the whole group setting. It appears that this is in place in this classroom with the small group setting but not for the whole group.

    I also felt that the teacher's questions were too leading. It seemed to me that the teacher did most of the work in making the connections from one strategy to the others. I like to provide more opportunity for the students to find the similarities and differences between the strategies.
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    • 03 Feb 2014 7:39 PM | Deleted user
      I'm thinking of using this chapter with teachers in a PD session. I've been using the book, Classroom Discussions, from Math Solutions, and this case study would make a good addition. The comments in this week's discussion here are very helpful! I agree that there are ways that Darcy could encourage more student discussion.
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      • 07 Feb 2014 10:09 AM | Anonymous member
        I agree with Jim Cook and recommend Math Solution's Classroom Discussions. Added classroom scenarios always help.
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  • 02 Feb 2014 7:21 PM | Anonymous member (Administrator)
    I was pleased to see that Ms. Dunn provided individual think time before students had a chance to work together. I was also pleased to read that she used the students' discussion time to think about her summary portion of the lesson. She had notes about what she observed and used them to plan which students would share and in what order. I am sometimes frustrated when I see that teachers are not taking advantage of this "discussion time" to learn more about their students by listening in on the discussion. I do not see it as a time to be at my desk.
    In line # 82 and 83 Ms. Dunn says she doesn't think everyone understood... I wonder what made her say this? I might have asked the class about Faith's explanation rather than stating that students didn't understand. I think Pedro might have still asked his question. I wonder if she made the statement to make students feel comfortable asking clarifying questions about Faith's explanation.

    I have used a problem, The Border Pool Problem, to elicit student understanding and have a class discussion about the different approaches to solving the problem. The problem is very similar to the "Tiling a Patio" problem. The difference with the Border Pool problem is that the shape of the pool remains a square. I like the Border Pool problem because there are many different approaches to the problem and ways to represent the patterns seen. I have use the Border Pool problem as a way for students to recognize that there are different strategies for solving a problem. I've also used the students' equations that they found for the problem as a follow up discussion. I've presented the class with the list of equations from the last class and then asked them to show how they represent the tiles in the problem by providing them with a drawing that then has to get labeled to match a particular equation.
    Effective questioning should bring all the students into the discussion. I also like to have questions that cause students to continue thinking about the problem; whether it's to extend their thinking or to help clarify another student's thinking. I also like my questions to spark a discussion among the students; my job then is to facilitate the discussion and keep it moving from student to student.
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    • 03 Feb 2014 7:49 PM | Deleted user
      I like your ideas about providing students with the various equations and then asking them to represent with tiles, Jenny. That should help students make connections between symbolic and actual representations. It would be interesting to match equations with gardens that are 1, 2, 3, etc. tiles wide.
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  • 03 Feb 2014 6:49 AM | Deleted user
    I noticed that tiles were provided for students to use. I would like to have seen the problem introduced as a concrete tile problem. Handling manipulatives helps provide entry for every student. It is tempting not to value this at grade 8 ( and I do not know the composition of the class) but think most students would value touching them(tiles) as a way of learning. I understand that tiles can also be distracting but think it would be important to consider doing this. All children benefit from the concrete. Some stay there longer as other move to paper and symbols but it benefits all.
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  • 03 Feb 2014 8:14 PM | Deleted user
    I wonder if Darcy had selected students first who might have only recognized the growing +2 pattern of white tiles. This might have validated their thinking and contribution, and it might lead them to make a connection to the slope when they get to graphing. Students who only see the +2 pattern will probably have trouble getting the 50th element. They might simply multiply #5 by 10. Darcy's questioning then could be directed toward the relationship between black and white tiles.

    I've used similar tiles patterns using drawings with fifth and sixth graders. I'm going to alter the task and have students build with tiles. I agree with Mary that actually building with objects can help. My patterns are "buildings" made of squares on grid paper. Each building is the same width, and they grow in height, one extra square each time. I ask students what they notice is the same and different between successive buildings before I ask them to draw the next few buildings. Then I ask them to predict the number of squares in the 100th building and write a rule in words before trying to write an equation. I learned that having students recognize and state the relationship between building number and squares needed to build it helps them write meaningful equations.
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    • 05 Feb 2014 7:00 AM | Deleted user
      This chapter was a good illustration of what the authors are wanting to get across in the structuring of problem solving discussion. I liked the analysis provided at the end of the chapter, so one can get a pretty clear picture of how this looks in a classroom. The authors note that Darcy engaged 8 different students in the discussion. In fact, she engaged nine of them. When you think back over many of your experiences in math class, there's a pretty good chance that you experienced lessons where the teacher did nearly all of the talking, demonstrating a solution, and assigning homework. The beauty of these models of discussion lies in a teacher's access to his/her kids thinking. On page 27, the authors point out that Darcy's "monitoring of her student's work provided the information that she needed about their mathematical thinking to modify her lesson"; and that led to engaging nine different students, providing several different ways to solve the problem; and, more importantly, to facilitate everyone's grasp of her 3 lesson goals. Her utilization of the five practices reflect Tom Carpenter's description (from his book, Children's Mathematics, Cognitively Guided Instruction), of a Level 3 teacher - "their classrooms are strongly influenced by their understanding of children's thinking, ...posing appropriate problems, ...with questions that elicit a variety of solutions", pg. 108. I do agree that having students actually use tiles to construct with, would enhance some of that thinking.
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      • 07 Feb 2014 10:12 AM | Anonymous member
        Also providing students time to work in small groups allows student to communicate with each other in addition to the teacher.
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  • 05 Feb 2014 10:51 AM | Anonymous member
    I have used a similar problem called Flower Beds.

    First, I think you have to check your own problem-solving abilities. My mathematical instruction focused on computation after computation after computation worksheet. Our students have the best of both worlds: time to learn how to do the computation and time to apply the math to real life situations.

    Second, as a teacher anticipating and monitoring become valuable and important so I can predict both incorrect and correct models students may use. Monitoring allows me to successfully choose students to get the mathematical focus across to my students. I think this became most evident for me as I read this section: choosing students so they are clearly teaching my math message. As students tackle more than just computation, they are solving complex tasks which ask them to use a variety of strategies and models.

    When I used the Flower Beds (it was during the first part of year as a part of our unit Variables and Patterns), my questioning involved selecting the popular strategy, the strategy I wanted students to think about/use, and a popular wrong strategy. I know who has each because of my incessant walking around the classroom. Once up on the AppleTV, I ask students to prove why one works or why one doesn't work allowing students to dictate the sequence. I often use Turn and Talk to your Seatmate before I will take a proposal whole class.

    So I am still thinking about do I still use the disprove one of the strategies/models or does this take away from students connecting to a clear math message?
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