DescriptionThis analytic illustrates how third-graders, Stephanie and Dana, justify their solutions to counting problems, displaying different representations to justify the number of different outfits that could be created with a certain number of shirts and pants available.

In this analytic, notice that the girls use words, pictures, and diagrams to justify their solutions. Notice also, that they predict a solution with more articles of clothing, suggesting recognition of a pattern.

Stephanie and Dana work independently, speaking aloud and monitoring each other’s approach to solve problems requiring the number of outfits that can be made from a given number of different shirts and pants to make all possible outfits.

The students began with the following task: “Find the number of outfits that Stephen can make if he has three shirts, a white shirt, a blue shirt, and a yellow shirt, and two pairs of pants, a pair of white jean pants, and blue jean pants.”

In events 1 and 2, Stephanie and Dana’s initial attempt to solve the problem was to draw pictures of shirts and pants, each labeled with a letter such as “y”, “b”, and “w” to represent each different color and used lines to connect the shirts and pants to account for each different outfit. This representation led students to account for a total of 6 different outfits.

In event 3, Stephanie explained the purpose of using lines to connect the shirts and pants to Researcher Martino. This event demonstrates an awareness to keep track of the outfits, and not to produce any duplicates.

In event 4, Dana wrote a statement to explain the representations and the procedure they used to reach the answer to the problem. Dana indicated that each line was drawn to connect a shirt and pants, to illustrate an outfit and accounted for 6 distinct outfits that she called combinations.

In event 5, The students were given an altered version of the original task to include another pair of pants. They were asked to figure out how many outfits Stephen could make if he had a new pair of black pants. Before attempting the problem, Dana said “It would be twelve.” Dana elaborated “Like everything goes with black. Cause six plus six is twelve.” Stephanie started naming the colors of the shirts and pants as she counted 9 outfits. When Dana said “Okay.” Stephanie told Dana “No Dana first I want you to figure it out, you may get different answer.” Students elaborated on and solved this extended problem with the additional colored pants by again drawing lines connecting shirts with pants, using the same approach they used in the earlier problem.

In event 6, Stephanie and Dana Justified their solution of getting 9 outfits from 3 different shirts and 3 different jeans. Dana said, “We just drew lines again.” And Stephanie explained: “See we drew shirts, and since each one of them can go to three pairs of jeans, three, six, nine.” as she was pointing to the pictures and lines on her paper Researcher Martino asked, “What do you think if you had four pairs of jeans? What would happen? Think about it, you don’t have to do that one.”. Dana replied, “it would be 3, 6, 9, 12. It would be 12.” Dana responded: “You would have to do this one like four. Like three, six, nine, twelve. It would be twelve. Three, six, nine, twelve.” Dana solved the problem, using words to support her solution as she was making counting hand gestures.

Stephanie and Dana found solutions to the multiple variations of the shirts and pants problem first, by making drawings of shirts and pants, then connecting shirts to pants to count outfits, to later conjecturing solutions to more complex problems, making use of patterns from simpler versions of the problem.

By examining student problem-solving, we can see that when young students, as early as grade 3, are challenged to justify their solutions, they can recognize patterns and make conjectures for the solutions to more challenging problems. (Maher & Martino, 1996).

In this analytic, notice that the girls use words, pictures, and diagrams to justify their solutions. Notice also, that they predict a solution with more articles of clothing, suggesting recognition of a pattern.

Stephanie and Dana work independently, speaking aloud and monitoring each other’s approach to solve problems requiring the number of outfits that can be made from a given number of different shirts and pants to make all possible outfits.

The students began with the following task: “Find the number of outfits that Stephen can make if he has three shirts, a white shirt, a blue shirt, and a yellow shirt, and two pairs of pants, a pair of white jean pants, and blue jean pants.”

In events 1 and 2, Stephanie and Dana’s initial attempt to solve the problem was to draw pictures of shirts and pants, each labeled with a letter such as “y”, “b”, and “w” to represent each different color and used lines to connect the shirts and pants to account for each different outfit. This representation led students to account for a total of 6 different outfits.

In event 3, Stephanie explained the purpose of using lines to connect the shirts and pants to Researcher Martino. This event demonstrates an awareness to keep track of the outfits, and not to produce any duplicates.

In event 4, Dana wrote a statement to explain the representations and the procedure they used to reach the answer to the problem. Dana indicated that each line was drawn to connect a shirt and pants, to illustrate an outfit and accounted for 6 distinct outfits that she called combinations.

In event 5, The students were given an altered version of the original task to include another pair of pants. They were asked to figure out how many outfits Stephen could make if he had a new pair of black pants. Before attempting the problem, Dana said “It would be twelve.” Dana elaborated “Like everything goes with black. Cause six plus six is twelve.” Stephanie started naming the colors of the shirts and pants as she counted 9 outfits. When Dana said “Okay.” Stephanie told Dana “No Dana first I want you to figure it out, you may get different answer.” Students elaborated on and solved this extended problem with the additional colored pants by again drawing lines connecting shirts with pants, using the same approach they used in the earlier problem.

In event 6, Stephanie and Dana Justified their solution of getting 9 outfits from 3 different shirts and 3 different jeans. Dana said, “We just drew lines again.” And Stephanie explained: “See we drew shirts, and since each one of them can go to three pairs of jeans, three, six, nine.” as she was pointing to the pictures and lines on her paper Researcher Martino asked, “What do you think if you had four pairs of jeans? What would happen? Think about it, you don’t have to do that one.”. Dana replied, “it would be 3, 6, 9, 12. It would be 12.” Dana responded: “You would have to do this one like four. Like three, six, nine, twelve. It would be twelve. Three, six, nine, twelve.” Dana solved the problem, using words to support her solution as she was making counting hand gestures.

Stephanie and Dana found solutions to the multiple variations of the shirts and pants problem first, by making drawings of shirts and pants, then connecting shirts to pants to count outfits, to later conjecturing solutions to more complex problems, making use of patterns from simpler versions of the problem.

By examining student problem-solving, we can see that when young students, as early as grade 3, are challenged to justify their solutions, they can recognize patterns and make conjectures for the solutions to more challenging problems. (Maher & Martino, 1996).

Created on2021-04-09T13:18:50-0400

Published on2023-08-04T09:31:35-0400

Persistent URLhttps://doi.org/doi:10.7282/t3-hmf8-cw58