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6 Making Thinking Visible: Modeling and Representation
Pages 109-126

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From page 109...
... The students used "Air Puppies" as a model to represent air mol ecules. They depicted Air Puppies as dots in some scenarios and as numbers in others (see Figures 6-1 and 6-2)
From page 110...
... For example, in modeling air molecules with Air Puppies, certain characteristics of molecules are represented, such as the fact that they move constantly without intention, and other characteristics are not, such as their being composed of hydrogen and oxygen atoms. Students need guidance in recognizing what characteristics are included in a model and how this helps fur ther their understanding of how a system works.
From page 111...
... This similarity in patterns would not have been noticeable without the mathematical representation afforded by the graph. Given the importance of mathematics in understanding science, elementary school mathematics needs to go beyond arithmetic to include ideas regarding space and geometry, measurement, and data and uncertainty.
From page 112...
... In contrast, a scatterplot of children's height by children's age would yield a linear relationship between height and age. An important goal for students -- one that extends over several years -- is to come to understand the conventions and proper ties of different kinds of data displays.
From page 113...
... However, with good instruction, middle and upper elementary students can learn to simultaneously consider the center and the spread of the data. Students also can generate various mathematical descriptions of error.
From page 114...
... , and mathematical descriptions of space, such as polar coordinate representations. Modeling and Learning Progressions In a study involving biological growth, Richard Lehrer and Leona Schauble observed characteristic shifts in the understanding of modeling over the span of the elementary school grades.1 They developed a learning progression that emphasized different and increasingly complex ideas in different grade bands.
From page 115...
... . Reasoning about changes in the height differences of the FIGURE 6-4 strips, students identified times when their plants grew "faster" and "slower." Displays of plant height depicted in Their study of the plant heights was firmly grounded in prior discussions about bar graphs.
From page 116...
... Why would the growth of two different FIGURE 6-5 A display of plant height over time depicted in an S-shaped curve. 116 Ready, Set, SCIENCE!
From page 117...
... As students developed and used new mathematical means for characterizing growth, they understood biological change in increasingly dynamic ways. For example, once students understood the mathematics of changing ratios, they began to conceive of growth not as a simple linear increase but as a patterned rate of change.
From page 118...
... Let's take a closer look at how children develop scientific represen tations. In the following case, also taken from the work of Lehrer and Schauble, we examine a group of fifth graders working on an investigation of plant growth.
From page 119...
... Rather than assigning children particular data displays to use in capturing data, he asked them to invent displays. He introduced additional uncertainty into the assignment by asking students to identify typical values.
From page 120...
... " A data display representing individual specimen height with a vertical line. Will: "Yes, then you can tell how many of those there are." difficult to read, especially from the back of the room.
From page 121...
... " A student replied, "One value per bin." Another student asked, referring to Figure 6-8, "Why did you select bins of 10? " Tanner and Erica, the authors of the graph, explained their reasoning: FIGURE 6-7 A display featuring ordered values of plant heights.
From page 122...
... Katie and Greg, the present ers, noted that the authors had writ ten their proposed typical value on the lower right of the display and that they had also marked out the 160s in their FIGURE 6-9 display, presumably to indicate that these were the A data display with rows of ascending values and repeated values selected as typical. However, Katie and Greg values stacked.
From page 123...
... Although few of the high value is 555. Which graph would help us original displays met this criterion, all of the displays see that it's more spread out?
From page 124...
... It took two full days of discussion before students finally surrendered their focus on novelty of design and gravitated instead toward criteria favoring clarity of the mathematical ideas. FIGURE 6-10 A data display on a two dimensional coordinate grid.
From page 125...
... As they analyzed and discussed the data displays, they practiced scientific norms by critically appraising each other's displays and explicitly reasoning about how well the displays accomplished the intended communicative goals (Strand 4)
From page 126...
... Teachers can help students reflect on the features and purposes of rep resentations by asking them to generate and critique their own representational solutions to problems, by encouraging them to interpret the representations developed by other students, and by asking them to consider what a representa tion shows and hides so that they come to understand representational choices as trade-offs. Although working with representations poses challenges for learners, it also can help bridge between the knowledge and skills they bring to the classroom and more sophisticated scientific practices.


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