VISIBLE THINKING IN MATH: ASSESSING STUDENT LEARNING THROUGH CLASSROOM ARTIFACTS
In terms of misconceptions to be anticipated for this lesson, there are several that the fourth and fifth graders might demonstrate. The students may not understand the following: The value of the slices does not have to be whole numbers (can be decimals); same-sized slices must have the same fractional value and dollar value; half a cake should cost half of the $10 total and one-fourth of a cake should cost one-fourth of the $10 total; and information from the previous day’s cake may help in determining the fractions and dollars values for the next day’s cake.
Research indicates that there are five levels of progression regarding partitioning:
We propose that the acquisition of the partitioning process is the culmination of a gradual progression through five levels. Each level is distinguished by certain conceptual characteristics, procedural behaviors, and partitioning capabilities. In particular, the partitioning process is mastered through the successive attainment of five subsets of the unit fractions: the fraction 1/2, fractions whose denominators are powers of 2, fractions with even denominators, fractions with odd denominators, and fractions with composite denominators. (Pothier & Sewada, 1983, p. 316)
If teachers who include this lesson in their instruction keep these levels in mind, it may inform their assessment of student understanding and can guide the follow-up questions they pose to students in both individual and whole-group discussions.
Students who demonstrate misconceptions should be provided multiple opportunities for remediation in order to correct said misconceptions. At the same time, teacher must be cognizant of the role their language and word choice play in student understanding. For example, Siebert and Gaskin (2006) discuss partitioning and iterating. An example of partitioning is when “5/8 is 5 one-eighths, where 1/8 is the amount we get by taking a whole, dividing it into 8 equal parts, and taking 1 of those parts,” while an example of iterating is when “5/8 is 5 one-eighths, where 1/8 is the amount such that 8 copies of that amount, put together, make a whole.” The authors point out that “these two different ways of conceptualizing 5/8 can lead to a variety of ways of actually creating and justifying the quantity 5/8” (p. 397).
Figure 7.1 illustrates some examples of some of the misconceptions students demonstrated. The first one displayed showed that a student did not think that the triangular fractional part was equal to the square part of the other cake. This misconception is not surprising as students are used to congruent pieces to represent fractions (i.e., fraction circles and fraction bars). This model of having congruent shapes and sizes represent the region model. However, what they see here are two pieces that have the same area. As mentioned before, this is a more complex model related to fraction. The parts must be equal in area but not necessarily congruent.
The next common error displayed by many was when they gave dollar value to each of the fractional pieces. They were able to understand the problem and attack the first day where the baker was dealing with fairly concrete halves. As they moved along, however, they relied heavily on the use of the dollar bills and guess and check. They would divide the dollars evenly into piles based on the number of pieces the
Figure 7.1 Incorrect and correct responses from Student A and Student B.
Table 7.1 Mapping out learning progression and ways to differentiate a task across grade bands
Grades 3-4 |
Grades 5-6 |
Grades 7-8 |
VA Math SOL 3.3a: The student will name and write fractions (including mixed numbers) represented by a model. |
VA Math SOL 6.2 b: The student will identify a given fraction, decimal, or percent from a representation. |
VA Math SOL 7.4: The student will solve singlestep and multistep practical problems, using proportional reasoning. |
For the third grade, the total cost of the cake could be a multiple of 2, 4, and 8 and the fractional pieces could be limited in value. For example, the cake could cost $16 and the only fraction parts shown could be halves, fourths, and eighths. [This is a simplified version of the original task involving.] |
For the sixth grade, the baker could start out with more than one cake. For example, the baker has three cakes that cost $10 total. [The whole has changed, therefore the value of the fractions has changed—what looks like one-half of a cake is actually one-sixth of the whole.] |
For the seventh grade, the baker could start out with more than one cake and have to provide certain sized portions. For example, the baker starts with four- and-a-half cakes and must provide servings that are one-third of a cake. [This involves fractions divided by fractions.] |
baker sliced the cake into. At times, they would divide the dollars with disregard to the fractional amounts.
Meanwhile, some of the students were able to determine the correct cost of each slice of cake for all three days; they were stuck on labeling the fractional parts of each cake.
Teachers used the Exemplar Rubric called the Thermometer Student Rubric for their assessment. While this student’s work is not complete in the picture, he was able to complete the problem and was determined to be at a “Practitioner” level in all categories. Student B was able to correctly solve the three days’ cakes cost per slice as well as the fractional amount. This student’s work was assessed as “Practitioner” in understanding and strategies but “Expert” in communication. The student was able to solve all three days and began working on the extension with great enthusiasm. He was able to identify the price per slice of cake as well as define the fractional piece for each cake as well. Table 7.1 above maps out the conversations teachers had about Fractional Cake problem and the learning progressions in fraction related to this problem. Having a vertical team of teachers also offered opportunities to think about the Fractional Cake problem and consider ways they could differentiate this problem to extend the conceptual understanding of partitioning a whole.