DescriptionBeginning Ordering and Comparing (Clip1) -- In this clip the researcher begins by posing the question of, "which fraction is bigger?" She puts 1/2 , 1/3,1/4, and 1/5 on the board and asks the students to be able to prove which is bigger. Notice how David, without the rods, draws rods to illustrate which is bigger. Then in an attempt to move the students from a rod model to a number line, the researcher puts up a number line segment from 0 to 1 and then asks the students to place those same fractions on this line. This clip ends with the researcher asking the students to think about where the number 1/10 would go on this line. Activity Continued (Clip 2) -- This clip is later on in activity 1. A student has volunteered to place 1/10 on the number line. A short discussion about the proper placement of 1/10 then commenced. Notice that the students are still looking at these fractions as operators and the line segment from 0 to 1 is their whole. At the end of the session students were asked to create their own number line from 0 to 1 and were asked to place all of the fractions discussed (including 1/10) on the line. In the next clip you will see the results of that assignment. Placing fractions on a Number Line (Clip 3) -- Students are asked to share their homework. In this clip, the question is raised about the fraction 1/3. Some students had 1/3 in different (even multiple) places. Alan then offers the explanation that 1/3 can go in any of 3 places. Alan continues to use fractions as operators, and not as numbers. Notice how comfortable he is placing multiple numbers in the same positions. Alan is concerned with each individual segment where the alternative explanation is that Andrew was concerned with additivity and starting at 0 and giving a number that covers the whole segment. Andrew finds it much easier to read and more logical if the numbers go in ascending order. The clip ends with the discussion of the placement of 3rds on a number line. Clip 4 The Ruler Saves the Day -- The researcher then makes an analogy that a ruler with each mark labeled one inch would be of no use to anyone and students agree. Clip 5 The Big Number Line -- In the front of the class there sits a large number line. All the previous activities have led up to this point. The number line runs from -3 to 3 with just the whole numbers pre-labeled. The first question posed to the class was where to place the number 1/2 on the number line. Notice where Audra placed the 1/2 and the discussion that stems from that placement. For more information about this analytic please refer to Schmeelk, Alston (2010), From Equivalence to Rational Numbers: The Case Study of Meredith.
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