USE OF STUDENTS' DIVERSITY Of STRATEGIES AS PEDAGOGICAL CONTENT TOOLS
Text Box 8.4 A Math Happening 8d: The Mango Problem
One night, the King couldn’t sleep, so he went down into the Royal kitchen, where he found a bowl full of mangoes. Being hungry, he took 1/6 of the mangoes. Later that same night, the Queen was hungry and couldn’t sleep. She, too, found the mangoes and took 1/5 of what the King had left. Still later, the Prince awoke, went to the kitchen, and ate 1/4 of the remaining mangoes. Even later, his sister, the Princess, ate 1/3 of what was then left. Finally, the royal dog woke up hungry and ate 1/2 of what was left, leaving only 3 mangoes for the kitchen staff. How many mangoes were originally in the bowl?
One of the research lessons we designed with a group of fourth- through eighth- grade teachers and two math specialists was called the Mango Problem. The problem was rich in that it provided lots of learning opportunity for the students and teachers. Acting out the problem was important to be sure the students understood and could visualize the problem. Students were given independent time so that all students could “own” the problem by making sense of the problem first. In addition, we wanted to assess how individuals were making sense or experiencing misconceptions before the group conversation.
Some of the students encountered this problem with the mental model of thinking of fractional parts of the mango as a “region model” and ran into problems when they arrived at the last part of the problem where “1/2 of what was left, leaving only 3 mangoes.” This problem requires students to have a more sophisticated understanding of fraction including the “set model” and unitizing.
A seventh-grade teacher who retaught this lesson after the first cycle, reflected on how for an older group of students, the students did not have the same misconceptions as the fourth graders who were beginning to understand the multiple meanings of fractions. This illustrated that the task had an important mathematical agenda embedded in the problem, which was the understanding of unitizing fractions.
I believe many of the 4th graders viewed fractional pieces as 1 whole as opposed to a
unitized whole. The 7th and 8th graders had understood the notion of a unitized whole.
(Sonny, 7th and 8th grade teacher)
Second, teachers became more intentional about monitoring students’ thinking so that they could use student strategies or representations as pedagogical tools for classroom discourse. In the professional development, we focused on how teachers can use tools like student artifacts and representations (diagrams, manipulative models, small group discussions, and numeric notations) to discuss important mathematical ideas. Some nameable strategies included:
Text Box 8.5 Nameable Discourse Moves to Highlight Student Thinking
- 1. Connecting: making connections among representations;
- 2. Marking: marking critical features, which the students should pay attention to;
- 3. Directing: keeping students on task and encouraged to persist;
- 4. Extending: Pressing on for justification;
- 5. Clarifying: clarifying to work through misconceptions/partial understanding
- 6. Zooming in and zooming out: making generalizations.
While planning and anticipating student strategies, teachers noted in their lesson plan that one of the important ideas for marking was to look for students making 6 equal groups with 3 mangoes in each group. In addition, teachers would mark for students who thought about multiples of 3s or 6s since 3 was the remaining mangos in each group and 6 was the number of groups. Finally, they noted that students would need to realize that the amount that was taken was the same portion each time.
The important idea was to see if students were able to unitize by constructing a reference unit.
Teachers noticed that tools can help or hinder the interpretation during the problemsolving process. The tool was only helpful if the learner could attach meaning or match their mental interpretation with the physical manipulatives. From the first cycle, teachers noticed how offering students’ access to multiple manipulatives actually hindered their thinking. In the previous excerpt, a student had success with connecting cubes, using it to “chunk” the portion taken by each of the characters. However, there were some students who chose fraction circles and had a difficult time because they could not take their mental picture of 3 remaining mangoes and see they fit into the fraction circles.
They wanted the fraction circles to represent the 3 mangoes remaining in the bowl and the fractional parts that the characters took out of the bowl but did not see how 3 discrete mangos could be represented by a region of the fraction circle. After observing the first research lesson, one of the observers, another fourth-grade teacher decided to experiment by limiting the number of manipulatives to see how students would represent their thinking with the available tools, but wondered whether it would limit some students. Another teacher who taught the lesson to their own classroom decided to allow students to draw their own pictures to represent their own mental images of the problem and found that students were more successful. In their diagrams, teachers noticed how students showed their understanding of unitizing.
One of the observers of the research lesson who retaught the lesson in her own classroom decided to introduce a related problem called the People on the Bus problem, where a fractional portion of the riders got off the bus. This is a classic problem where drawing a picture is more elegant than a formula
At some stop, 2/5 of the people got off the bus and 3/5 of the original number got on. At the second stop, 1/2 of the people got off and 1/3 of the number that were left on the bus got on. At the last stop % of the people got off, leaving 5 people on the bus. How many people were on the bus before the bus reached the first stop?
Teachers’ ability to experiment through multiple research lessons allowed for us to conclude that good problem solvers who have a strong grasp will have versatile mental models to represent the problem whether it is through their own drawings or manipula- tives. That is, while some struggled to model the problem with fraction circles, others who had an understanding of unitizing fraction were able to represent each fractional piece as a unit of 3 mangoes and could model their thinking. It wasn’t so much the choice of manipulative, rather the types of mental models students had of fractions that allowed them to be effective with manipulatives and other tools for thinking.
One teacher noted that students, who were correct in their thinking and modeling of the mango problem, whether abstractly or concretely, were very confident that they were right. Their models matched, or proved, their thinking. However, students who were incorrect in their solutions were typically dissatisfied with their models as if something was missing that they were unable to explain. And these latter students felt their results were unreasonable or the results lacked adequate proof, so they were confident that they were wrong. However, these students were unable to explain why
Figure 8.11 Worked examples of the "People on the Bus" problem.
their solution was wrong. To extend students’ thinking, the teacher asked students to share their responses within a group and then each group explained their approach to the problem to the class.
The teacher sequenced their work for display and discussion starting with concrete models, followed by the logical backward design model, then to the guess and check which could be more abstract. The class discussed the connections and differences between each of the three strategies as well as their efficiency and effectiveness. Since the clear and explicit learning goal was to get students to unitize the mangoes so that 1 unit = 3 mangoes, students who engaged in the backward design and the concrete strategy of drawing out models that displayed the unit seemed to arrive at the correct conclusion with greater efficiency and understanding than the guess-and-check method. The effective use of questioning and mathematics discourse in the classroom allowed students’ strategic competence to take center stage and moved more students along understanding the efficient and advanced strategies for the problem using multiple models.