DescriptionThis analytic focuses on James, a seventh-grade student, working on a problem solving activity involving linear functions to help him recognize an isomorphic relationship between problems. The goal of recognizing this isomorphism is to help him draw connections between the problems, gain a deeper understanding of the concept, and generalize his findings.
Before this series of video clips, James worked on the ladder problem which asks to find the number of blocks needed to build a ladder with 10 steps. Some follow-up problems he worked on include finding how many blocks are needed to build a ladder with 100 steps, an odd number of steps, and an even number of steps. After that activity, James began working on the museum problem. In the museum problem, students are asked what the total cost would be to buy 10 craft pieces given that each craft piece costs $3 and there is a $2 entrance fee. Some follow up problems include considering what the total cost of buying 30 craft pieces, 100 craft pieces, and any number of craft pieces would be. As a result of this activity, James created the equation, “CPx3 + 2 = how much money,” (where CP is the number of craft pieces purchased) to represent the total cost of any number of craft pieces.
This analytic picks up with James continuing his work on the museum problem. Markus Hahkioniemi, a visiting researcher from Finland, points to James’ equation for the museum problem, and asks him to represent his past findings for the ladder problem in a similar way. James writes an equation, “Steps x3 + 2 = how many blocks,” following the same form of his equation for the museum problem. He comments, “there’s two different ways to explain it but it’s the same method.” This statement shows that he has identified a similarity in the structure of these two problems and recognized that they use the same method. Thus, he has begun to identify the isomorphic relationship between the two problems.
The researcher then prompts James to identify relationships between the variables of the two problems. James says that the blocks in the ladder problem are similar to money in the museum problem, and the steps in the ladder problem are similar to the craft pieces in the museum problem. From there, the researcher asks James what the meaning of the coefficient of 3 and the constant of 2 are in each problem. James responds, “In that [the museum problem], it’s money, because it depends on how much it costs for each craft piece, and that means how many [blocks] to make one step [in the ladder problem].” James then demonstrates his construction of a one-step ladder and transformation to a two-step ladder to show where the 3 and the 2 come into play. He concludes that the 2 in the museum problem is the two-dollar “entry fee,” and the two in the ladder problem is the two blocks required to complete a “full ladder.” Through his responses to the researcher’s questions and his demonstration of building ladders, he has successfully identified the relationship between the coefficient of 3 and the constant of 2 in both problems.
After making connections between all the structural elements of the problems, the researcher moves on to try to encourage James to use the blocks to represent the museum problem. This would be beneficial to transfer the concepts to a different context, and ultimately discover a way to generalize these ideas about linear functions. At first, James sets aside two blocks to each represent one dollar of the entrance fee. He identifies the remaining blocks on the table as craft pieces. In his initial representation, he is using the blocks to signify both money and craft pieces.
The researcher then asks James if he could consider all the blocks as one dollar. James replies, “You can’t. Every block can’t be one dollar […] because if each block was a dollar, then how would you find the craft piece… craft pieces? You wouldn’t.” James then marks all the craft piece blocks with a black marker to create a physical difference between the two different types of blocks. The researcher flips the marked pieces over, and makes several more attempts to ask James to try to consider them all as one dollar like how he considers the entrance fee as two blocks representing one dollar each. After thinking for some time, James announces, “Now, now, I see it. In a way, these are money.” He explains that the craft piece blocks could actually be considered as three dollars each. James now believes that all the blocks represent money, by using one-dollar blocks for the entrance fee and three-dollar blocks for the craft pieces.
Through this linear function problem strand, James is able to discover an isomorphism between two problems: the ladder problem and the museum problem. In this analytic, we see James recognize the same solution method for the ladder and museum problem, and draw the connections between their structure, including independent variables (number of steps and number of craft pieces) and dependent variables (total number of blocks and total cost), as well as their coefficients (3) and constants (2). We then watch as he tries to represent the museum problem with blocks similar to how he did with the ladder problem. He begins by having two blocks represent one-dollar each of the two-dollar entrance fee, and the rest of the blocks represent craft pieces. Despite the researcher’s attempts to get James to see each block as representing one dollar, James remains firm in his solution. By the end of the analytic, James has all the blocks representing money, with two different block representations: one-dollar blocks and three-dollar blocks.
It is important to be able to recognize isomorphisms because they help students make connections between problems, understand math concepts in different contexts, and generalize their findings. This skill will help them problem solve in the real world by finding connections to seemingly unrelated situations. According to Lave, “learning transfer is assumed to be the central mechanism for bringing school-taught knowledge to bear in life after school” (Lave, 1988, p. 23).
The Museum Problem: A museum gift shop is having a craft sale. The entrance fee is $2. Once inside, there is a special discount table where each craft piece costs $3. How could you represent the total amount that you would spend if you were to buy any number of craft pieces at the discount price?
The Ladder Problem: A company makes ladders of different heights, from very short ones to very tall ones. The shortest ladder has only one rung, and looks like this (we could build a model of it with 5 light green Cuisenaire rods.) A two-rung ladder could be modeled using 8 light green rods, and looks like this. Build a rod model to represent a 3-rung ladder. How many rods did you use? How could you represent the number of rods needed if you were to build a ladder with any number of rungs?
References:
Early algebra, investigating linear functions, series 6 of 7, Museum problem, Clip 4 of 6:
Comparing the Ladder problem with the Museum problem [video]. Retrieved
from https://doi.org/doi:10.7282/T3NG4NKT
Early algebra, investigating linear functions, series 6 of 7, Museum problem, Clip 5 of 6:
James modeling the Museum problem with rods [video]. Retrieved
from https://doi.org/doi:10.7282/T3HQ3WWJ
Greer, B., & Harel, G. (1998). The Role of Isomorphisms in Mathematical Cognition. Journal of
Mathematical Behavior, 17(1), 5-24.
Lave, J. (1988). Cognition in practice: Mind, mathematics, and culture in everyday life.
Cambridge: Cambridge University Press.