Association of Teachers of Mathematics in Maine

Week 3: Laying the Groundwork: Setting Goals and Selecting Tasks

21 Jan 2014 10:59 AM | Anonymous member (Administrator)

Week Three:  Chapter 2:  Laying the Groundwork:  Setting Goals and Selecting Tasks


Read pp 13-20


The authors argue that what students learn depends on the nature of the task in which they engage.  Do you agree with this point of view?  Why or why not?


AND/OR


Look at the potential of a task to support students thinking and reasoning:


Use the task analysis guide (figure 2.3) to analyze a task(s) that you have used in one of your classes over the last few weeks.   Remember that we are all works in progress, so you may post an analysis of a task that you consider to be high level demand (according to your analysis) or maybe you will post what you consider to be a task with low level demand (according to your analysis) that has room for growth.


Comments

  • 21 Jan 2014 1:53 PM | Deleted user
    My remarks are about- The authors argue that what students learn depends on the nature of the task in which they engage. Do you agree with this point of view? Why or why not?

    I agree with the author that what students learn depends on the nature of the task they are engaged in. The key word here is engaged. I will use the Tiling a Patio as an example. This tasks requires a great deal of engagement and deeper levels of thinking. Students themselves need to develop a procedure to answer e. This takes time, but students are really then understanding the mathematics of the problem. Students working with a partner on this task are also engaged in discourse, asking questions, agreeing/disagreeing, attempting different strategies as they solve the problem. I also feel that once you have constructed and tested a"rule," that you are more likely to understand & remember it later. There are many rules I memorized during my school years that I never understood until I had to break a problem down in order to teach it.
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    • 26 Jan 2014 7:24 PM | Anonymous member (Administrator)
      I can appreciate Donna's last sentence about the amount of learning and relearning that we use in order to get ready to teach a lesson. I remember an elementary teacher commenting to me about how her level of understanding about the subtraction with regrouping algorithm had increased due to the fact that she was teaching it in her classroom. She'd had to spend time really analyzing her work for the problem in order to be prepared to address her students' potential misconceptions.
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  • 21 Jan 2014 6:04 PM | Anonymous member
    One key aspect is building students perseverance. Often students see in-depth tasks and they begin to shut down or believe they cannot do it. They do not have the perseverance to break it down in to small steps or have the motivation to rework or try again. Building perseverance is one of the biggest steps in middle school. I have seen with using the various iPad whiteboard apps students are willing to engage in reworking problems OR engage with various "hard" problems when presented through GeoGebra or in other electronic forms.
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  • 23 Jan 2014 7:06 AM | Deleted user
    I do agree with the authors, student's learning is dependent upon the tasks in which they engage. If we want students to learn to think and reason at high levels than the tasks we give them must also be at high levels of thinking and reasoning. Computation is essential for mathematics. At the least, it requires students to learn a procedure. Students can learn the procedure without knowing the mathematical concepts and be proficient. Many of them can not problem solve, reason mathematically and complete higher level tasks. Teachers need to incorporate high level thinking and reasoning tasks during instruction so students can become proficient at those skills also. Those are the skills that will be need in the adult world. Calculators are used by most adults for computation.
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  • 26 Jan 2014 7:21 PM | Anonymous member (Administrator)
    The authors argue that what students learn depends on the nature of the task in which they engage. Do you agree with this point of view? Why or why not?


    The authors start out in the very beginning of Chapter Two by stating that it's key to "specify a goal that clearly identifies what students are to know and understand about mathematics as a result of their engagement in a particular lesson." This indicates that the type of task is an important component of teachers' planning and provides teachers with what they are going to look for in order to determine if students learned what was expected from the lesson.
    The nature of the task can have an impact on student learning. If the task is not of interest to students they may not engage in the task with the same depth as if they were interested in the task. Also, tasks that cause students to engage in a way in which they have to work to solve it are rewarding. The level of thinking required in a task can impact the depth of learning and the class discussion about the various strategies used by the students to solve the task.
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  • 26 Jan 2014 8:12 PM | Deleted user
    I agree that what students learn depend on the nature of the task. One idea that stood out for me was the authors' comments on p. 18 regarding equity. A teacher must take into consideration various entry points available to students. Teachers need to frame the lesson and task in order for all students to be engaged. I wonder how often, especially when teachers feel constrained by a text book and/or curriculum demands, they just set up the task the way the books tells them to instead of manipulating it to work best for all their students.
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    • 27 Jan 2014 9:26 PM | Deleted user
      I like the various entry points that Tiling a Patio has. Some students might simply notice the pattern in the number of border tiles as the garden gets longer. They might have to extend the pattern to know how many tiles are needed. That's what's good about asking about garden #50--it's too big to be done by extending the pattern. Other students might see the relationship between garden length and the number of border tiles. Those students may see the relationship in various ways. Another thing I like about the task is that it can be extended by changing the width of the garden. This might keep more capable students engaged. I agree with you, Clare, that teachers often just go with what the book suggests. That means that they need books (like the NCTM Navigating series) that make good suggestions like this.
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  • 26 Jan 2014 8:22 PM | Anonymous member
    Yes I agree with the author's statement. I also agree with posts that address the word engagement. The tasks must be of interest to the students but, more importantly there must be a place for students to enter the problem. Setting clear and high expectations are important but we must be sure that the students have enough knowledge to be able to access the problem. I, myself, am guilty of choosing problems that I feel will be a perfect match to a desired outcome only to find out that 3/4 of the class does not even know where to begin. Of course, when this happens there is no engagement. One thing that I have found helpful is: when I decide on my learning goal for each lesson I also establish success criteria which tells me and the students what they will need to do in order to demonstrate that they have met the goal.
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  • 26 Jan 2014 8:50 PM | Deleted user
    As an edtech, I was asked to sub for an 8th grade math class. I had subbed for these students recently when they had been slogging through some early-algebra worksheets (x-8=-2). I thought I'd surprise them with a different sort of task.

    After we checked the answers from the previous day's worksheet and before they started in on the worksheets that had been left for them, I directed students to a KenKen riddle: Sally is 54 years old and her mother is 80. How many years ago was Sally's mother three times her age?

    I said that while answers are nice, its the strategy that we want to really think about. How will you go about solving this problem? What do you know? What do you want to know?

    Task Analysis:
    This could be considered to be a procedures with connections task.
    1) The problem does not name any procedures that can be used to solve the problem, so students' attentions are necessarily focused on finding those procedures. In searching, bad options are discarded (or not) and good options are held (or not), so students who are engaged with the problem are, with guidance, "developing deeper levels of understanding of mathematical concepts and ideas."
    2) I emphasized that students should catalog everything they know about the problem as they find it. All that is known are the women's current ages, and the problem has to do with comparing their ages. I would say that the problem "suggests implicitly" that paying attention to the age difference (number relationship) is key to finding a solution, only because that is all there is to work with. Several students realized that mom was 26 years older than Sally, then jumped with that data to various incorrect (and untested) conclusions.
    3) Singular initial representation (language) is a weakness here (I'm not counting the cartoon illustration that accompanies the text), but I thought that producing a table would have been soon coming. So, I TOLD students that a table would be a good way to represent this data, and I started it out for them (0, 26)...
    4) As for cognitive effort, some students were willing to look at the table I drew and then "hunt and peck" their way through, looking for a mother's age that was three times Sally's. I suppose I should not have GIVEN them the table outright. When I TOLD them that there is another procedure, one that involves using the idea of a variable, as I had seen them working with recently, there were audible groans. 

    There are ways this task could be improved. If students don't know how and why a table is useful, then that should be explicitly suggested as an organizing strategy. If students don't know how and why a variable is useful, then that should be explicitly suggested as a procedure for dealing with unknown quantities. Granted, students need to have some sense of both of those ideas before they are able to use them to solve problems. The class seemed utterly confused when I talked about using a variable here, and got even quiter when I showed how the algebraic procedure gave the same result as the table-inspection procedure. Too much too soon, but even if seeing the same variable on both sides of an equation was new to them (it was), I had expected that they could have expressed the relationship between the two ages. 

    Another way to improve this task would be for me to become a better practitioner of the Five Practices, especially ANTICIPATION. Had I better known the students, I could have correctly anticipated their reaction to this problem and I would have known what kinds of scaffolding would likely be needed.
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  • 27 Jan 2014 9:13 PM | Deleted user
    The authors are right that what students gain depends on the nature of the task. Learning a procedure may result in procedural competence. It's not likely to result in rich conversation. A task such as Tiling a Patio, which has a variety of entry points and solution paths is more likely to result in engagement, interest, discussion, and connections between methods. It's also more likely to result in students learning that they can be creative and effective problem solvers.

    I agree with Peggy's comments. If we expect students to reason, we need to give them tasks that require reasoning, and students can complete procedures without understanding the underlying mathematics.

    I also agree with Jenny when she writes about teacher planning for the type of learning that is expected. Planning that includes thinking about the mathematics and how a task will help to develop that mathematics is important. Unfortunately, there's often not enough of that type of thinking on teachers' part. I think often planning is about what students will do rather than what they'll think.
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    • 29 Jan 2014 9:36 PM | Deleted user
      One element of this chapter that struck me is how it drives teacher thinking. I do agree with the authors about the nature of the tasks effecting what students learn; but, it occurred to me that in creating problems that require deeper reasoning, structuring the problem to fit the learners where they are, and constructing a meaningful dialogue process - that brings students to a clear understanding of the mathematical concepts embedded - really pushes teachers to think deeply about how to pull those goals off. I would argue (or suggest) that if colleagues got together to discuss some of those lessons and planned collectively, the benefits would be significant for them and their kids. There were many great points made in this week's commentary and I appreciated being part of this discussion. My guess is, so would the authors.
      It was interesting to read about the Patio Tile problem and the 50th garden piece. At last springs ATOMIM conference, Steve Leinwand gave all of us a taste of such an extension problem in his key note address, and you could see the thinking going on in that room. I'm hopeful that some of you will try using the Task Analysis Guide because I would really like to hear about your experience with it.
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      • 30 Jan 2014 8:57 PM | Deleted user
        You're right, Wayne, that teachers need to consider where students are and structure problems so that they have entry points that students can access. Planning together and discussing problems along with likely solution paths may help. Teachers can also consider where students may have problems or misconceptions. Groups developing lessons and problems may be able to develop richer outcomes than individuals.
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  • 03 Feb 2014 6:53 AM | Deleted user
    I think the type of task is important. I also think that the way the task is presented is critical. Teachers are tempted to give too many hints and bring in too many opinions on the way that they would do it. This keeps the student from taking responsibility for solving it. It is all part of a class culture that needs to be developed for many tasks to be engaging.
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    • 05 Feb 2014 10:56 AM | Anonymous member
      Good point about giving hints. Often it is so hard to see kids struggling, but struggling and trying more than one strategy builds perseverance and the ability to stick with a task. So many of our kids see a hard task and give up within 10 seconds, but if they could just try something they would be well on their way to solving it.
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  • 03 Feb 2014 2:29 PM | Deleted user
    I agree with the author that what students do for a task will have an impact on their learning. I think that students need to feel a connection to what they are learning, a personal connection to the content and a reason for learning it. Through clear objectives (targets) and material that makes students think, not just do they will get a deeper meaning to the content that is being taught. I think it is important to remember that we are preparing students to enter into a high school that may still focus on the "skills or rules" of math without diving into content. Therefore, as teacher we have the responsibility to give our students the best of both worlds, deep content and old school computation. Face it, not every student will grow up and use algebra or geometry in their daily lives. Teaching them to think about a problem and come up with a logical solution is something that they will need to do and engaging in tasks in math class that bring that thinking out will help them as adults.
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  • 05 Feb 2014 7:17 PM | Anonymous member
    I recently did a task that I would consider to be a task that made connections to previous learning and connected to new learning that was coming up with the new lesson. It asked students to write an equation for a line parallel to another line, the slope of the original line was given and the intercept of the second line was given. It was a quick task but I feel that it gave students a place to consider y=mx + b. The issue for me as a teacher is letting kids struggle and the different rates at which kids learn and understand. With 27 students some get it quickly and some don't, How long do you wait for the strugglers?
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