One of the initiatives identified by students—opening a school store—became one of the overarching mathematical modeling tasks Alice posed to her students. The process of opening and running their school store provided the students with many rich opportunities for engaging in 21st century skills while learning and applying mathematics. Within the context the school store task, we highlight the challenges encountered by Alice’s students and Alice’s description of how solving these mathematical modeling tasks elicited student’s 21st century skills.
4.2.1. Critical Thinking and Problem Solving
In order to provide examples of student engagement in critical thinking and problem solving in, we include five segments and describe the way in which they prompted the use of 21st century skills. The segments are: (1) Defining the task; (2) Processing survey data; (3) Predicting sales; (4) Determining pricing; and (5) Moving dead stock.
(1) Defining the task—Once the overarching task was defined, student brainstormed as a whole class to identify what they needed to know in order to open their store. This process engaged them in looking for a logical structure to address their problem. As they grappled with where to begin, students identified the following questions through a whole class discussion:
What items should we sell at our store?
How should we price our items?
What will our profit be? What should we do with the profit?
What hours should the store be open?
Where should the store be located?
How will we staff the store?
Where will we store our inventory and keep it secure?
At first glance, not all of these questions appeared to lead to mathematical thinking but through familiarity with the curriculum, Alice was able to guide her students to focus on questions that would access meaningful mathematics and lead to critical thinking and problem solving. Alice commented, “with the realization that there will be a lot of data collection involved in setting up and running our school store, we sidetracked to see how many types of graphs we could name, and brainstormed for how each type might reflect school store data”.
(2) Processing survey data—After listing ideas of items they would like to sell at their school store, students realized they needed to know what their “customers” would be interested in buying. They worked together to develop and send out a school-wide survey to identify the most desirable items for their store. Over 600 surveys were returned and students were overwhelmed by the amount of data they had collected. Alice noted, “We are still working on it. The marketing group has their hands full with 600 kids’ worth of survey data... They still aren’t open to ‘outsourcing’ data crunchers, and, we haven’t had much time for them to get a lot done.” In order to address the large amount of data and move the project forward, Alice noted she:
[P]lanned a status-of-the-school-store conversation with the entire ‘company’, with the goal of firming up a timeline for provisioning, advertising, and opening. The data crunching challenges will come up, so I will challenge the entire class to think about how we could expedite the process. My hope is that one option that’s thrown out is narrowing the sample space by taking a random sampling from each grade level. But, I want that to come from them (knowing that I will need to teach them the correct terminology). And, if the class thinks that using a random sampling is prudent, I’m even wondering if validity will come up, with someone suggesting that we take several random samplings, crunch the data, then compare results.
The way Alice’s students met this challenge demonstrated perseverance in problem solving. Importantly, the students also began developing the statistical skills needed to understand large amounts of data. They organically discovered through mathematical modeling that big data is messy and without the means to analyze it, it is useless.
(3) Predicting sales—While waiting for the survey results, the teacher asked her students to “predict the first week sales.” In order to make predictions, students made assumptions about how they would stock their store, researched wholesale costs, determined unit costs, estimated retail prices and evaluated various profit scenarios. Students were able to identify and ask significant questions about mathematics as they moved between defining the problem to making assumptions and mathematizing a solution. Through this mini-task, students were able to make sense of applied mathematics and explain and justify the mathematical choices they made—an important component of critical thinking and problem-solving skills.
(4) Determining Prices—Another episode highlighting critical thinking and problem solving was related to students pricing their inventory. After setting the store prices, students received negative feedback from the other sixth graders about how expensive things were. The feedback prompted a discussion amongst the store “departments” about the balance between profit and affordability. Students decided to revise and reprint their price lists, order forms, and marketing signage resulting in an initial model for opening the store.
(5) Moving dead stock—The initial opening of the store was a great success. After being open for a week, the store sold out of many items and closed to reorder stock and to assess the store model. As a class, students discussed what worked well and what needed improvement. Two particular dilemmas, that developed into mini-mathematical modeling tasks were identified: (a) how to reduce the customer line rate in order to expedite traffic flow issues and students being late to class; and (b) how to move dead stock. Solving both of these dilemmas required the use of meaningful mathematics.
This dialogue shows how Alice’s students engaged in the problem posing phase of the mathematical modeling cycle and demonstrated critical thinking and problem solving and creativity to address the issue of moving dead stock:
- “They don’t like these pencils. The mechanical ones are way cooler, so they don’t even consider buying the graphite ones.”
- “I think they’re too expensive for a boring old graphite pencil!”
- “Talk to us about your thinking, Dan”
Dan continued by elaborating on his comparison of the cost of the mechanical pencil and graphite one to show that they were charging too much for something much less sophisticated.
- “Yea, maybe they do cost a little too much!”
- “But it’s more than that—customers just aren’t interested in them. We need to really push them.”
- “Yea, I’m thinking a combination of dropping the price and making people WANT to buy them is what we need to do.”
Throughout this rich discourse, students were making assumptions, realizing constraints, and defining the variables. Figure 4
lists the student-generated thoughts during class discussion for each of these processes. After this discussion, students were ready to develop plans to move the dead stock.
Students worked in triads to develop solutions and create profit projections based on their ideas for solving the dilemma of the dead stock. They discovered that their profit projection tables were tables of values that could be represented with linear functions. They realized that graphing their predictions would allow them to select which triad’s plan was optimal for moving the dead stock. Students were able to defend their assumptions and choices, analyze each other’s proposals, and agree on the best approach to moving their dead stock. Figure 5
below shows the graphs students created to present their triad’s plan to the rest of the class. In addition to creating a graph of a linear function from their table of values, students wrote equations for their functions, and discussed the significance of the y-intercept for their specific situation.
4.2.2. Creativity and Innovation
The open-ended nature of mathematical modeling tasks offered plenty of opportunity for students to be creative. On multiple occasions, Alice referred to the MM process as a creative endeavor involving iterations and sometimes failure, that was part of the creative mess. She emphasized that the MM cycle provided an organizational structure—we refer to it as “order in chaos: structure in the creative mess in MM”—for her students to systematically solve a relevant problem.
You got to define your problem, you have to narrow it down...so what are we really asking? What’s our goal here? What are the constraints? And so it’s a natural problem solving process so by putting it up there. It’s just a good anchor for the kids, like today it was a wonderful way to reiterate the point I was making that would take them through college to really figure out that it is, you don’t just come up with a solution and your problem is solved. It’s messy.
And another one and another one, so by having that visual it’s continuing to help them be ok with the messiness of multiple iterations of going back to the drawing board and failure even, we’ve had failure in some of the things that we try but that’s all ok because if we know that if it just swings back and forth.
We saw this in Alice’s classroom as students determined that a school-wide survey would be the best way to choose how to stock their store, and as they developed final survey questions. This was also evident when, after students realized the scope of the task they had set out to do, they resolved to create different departments in order to efficiently proceed with opening their school store. They created the following departments: Marketing, Merchandise, Accounting, Sales/Personnel, Security/Storage, and Customer Service. As they made these decisions, they listened to and evaluated others’ reasoning.
Creativity and innovation in mathematics involves comparing different ways of approaching mathematical problems and finding innovative solutions ([10
], p. 5). After operating their store for one week, the class evaluated what was working well and where there was opportunity for improvement, coming up with a list of problems they needed to fix. We highlight two problems identified by students that demonstrated creativity and innovation in mathematics: (a) Long wait-time; and (b) How to handle dead stock.
For the “long wait-time” problem, students considered how to improve wait-time in the store line by listening to and evaluating their classmates’ ideas. After exchanging various views, they decided that they would post prices at the end of the line and have students preselect items to purchase before arriving at the cashier. Mathematically, they had to test how this would impact the wait-time. Students positioned themselves with timers and used a sample group of clients to determine how long it would take to get through the line with the new method. Once they determined rate/flow of customers under the new system and compared it to the original rate, then decided to make the change. Figure 6
shows the constraints discussed by students in addressing this dilemma and their initial recording of the line rate under their new system. This segment illustrated creativity and innovation in mathematics as students looked for patterns that suggested creative shortcuts and made generalizations from patterns they observed in repeated calculations.
For the problem of “How to handle dead stock”, we also observed creativity and innovation in mathematics as students pondered different ideas for dealing with their dead stock dilemma. As mentioned above, the dead stock problem was an issue identified by the students. Students came up with a variety of ideas including a price drop, a buy-one-get-one free (BOGO), and a give-away-with-purchase plan. As discussed above, they presented their proposals to each other by graphically representing their predicted sales. After listening to and evaluating each other’s mathematical reasoning, students selected the BOGO option. By discovering fresh insights and communicating them to others in this process, students had the opportunity to understand that mathematics is a creative endeavor that builds on previous knowledge.
Alice used the word “messy” six times during one of her interviews to refer to the open-ended nature of the MM tasks and the creative process that is necessary to progress through the phases of MM. In her use of the word “messy”, Alice referred to the disorderly nature of the authentic, real-world MM problems that differentiates them from the straightforward, procedural problems often found in textbooks. When asked, how did you make sure the students focused on the mathematics? Alice responded,
That is really by giving them something they are accountable for...that they all had to think about what they would put in a survey. They had to have a poster and they had to include... I’m just asking for a deliverable, so just asking for something so ok you guys are all going to be thinking about how would you survey to find out and so they took thought to that or another day, the day that you were here, I asked them just to project for a week and I knew they would all be talking about.
She continued on and said that she required “some accountability, some recording, some deliverable.” To focus her students on using meaningful mathematics, Alice gave her students a MM process tool as shown in Figure 7
. While allowing the MM process to develop organically in the classroom, at times, Alice used these MM process tool to take advantage of a rich mathematics opportunity. The following tool provided students opportunity to use creativity and maintain the openness of the task, while allowing to focus on building a model to predict sales. Students were asked to develop three business move to address the deadstock issue—while focusing on using mathematics to justify and explain their choices. We note that for elementary students, teachers’ support and guidance provided a rich opportunity for students to discover how mathematics can be used to make decisions among the creative choices faced in a real-life situation or phenomena.
4.2.3. Communication and Collaboration
In developing a final survey, students made assumptions about what items students might be interested in, what would sell, and possible profit margins. While waiting for the survey results, the teacher asked students to, “predict the first week sales”. Students gathered 600 survey responses in order to determine what goods they should stock in the store. Students used the cost of the store’s inventory to set initial prices of their stock and estimate profit. Students listened to each other’s ideas as they brainstormed ideas for tasks and came to a consensus. They worked in small groups to develop marketing surveys for their school store. The groups compared their drafts and crafted a final survey that was administered to all students and teachers in the elementary school.
Students created a method to measure the customer line rate and worked together to gather data from a sample group of clients to determine the average rate for clients to complete a transaction. They used this rate/flow information to revise their system and provide customers with order forms and price lists before reaching the cashier in order to expedite the transaction time. As we observed students working together during the Opening a Store project, we noted that they were respectful of each other, asking thoughtful questions of their classmates and affirming the ideas of others.