By Chandra Hawley Orrill, Posted May 25, 2015  –

In the last post, we thought about the relationships among tasks, questions, and practices. When we parted, you were going to spend some time thinking about two similar tasks. In this post, I will give my thoughts on the two tasks as one opinion about the tasks and how they may play out in the classroom. I’m sure many of you had a lot of good ideas, too. Here are our two tasks:

Task 1: Lunchroom Problem

Ella would like to buy lunch in the lunchroom today. Meals cost $2.50. Dessert costs an extra 75 cents, and milk costs 75 cents. If Ella has $5.00, can she buy lunch in the lunchroom?

Task 2: Buying Lunch

You are going to the amusement park. You will buy lunch there. You don’t want to carry coins, only dollar bills, because coins can fall out of your pocket on the roller coaster. You can purchase any of the items on the menu, but you cannot buy more than one of the same item, and you cannot spend more than $10.00. Using the menu in the table below, determine how many different lunches you could buy without receiving any change.

Analysis of Tasks

Task 1 is of a lower cognitive demand than task 2. However, task 1 has some opportunity for young learners to reason mathematically. Specifically, task 1 can promote mathematical argument because Ella will be getting $1.00 back, which some students may find confusing because Ella has too much money. This provides an excellent real-world opportunity to think about how much money you need to purchase items and whether it is OK to have too much or too little money.

Task 2 is a higher cognitive demand task because it is open ended. This provides the basis for a rich discussion in class: A teacher can ask questions about how students know whether they have found all combinations. Extend the problem by changing the amount of money that students have available (e.g., “How many combinations do you think you could buy with $5.00?”) or by structuring what must be included with lunch (e.g., “You have to buy a drink as part of your lunch.”) All these variations require students to revisit the data they have been given and think about the effects of the changes on their solutions. Another important benefit of this kind of task is that it can offer lots of practice with basic arithmetic as students engage in meaningful problem solving.

Analysis of Questions

The questions asked as students work on a task can raise or lower the cognitive demand of the task. For example, in task 1, you could ask, “How did you decide she has enough money?” That is a pretty good question that gets the student working on the Common Core’s Standard for Mathematical Practice (SMP) 3. The question raises the cognitive demand of the task by requiring the student to not only add the numbers but also explain why we add the numbers. You could also use extension questions to further engage students, thus building their perseverance. For example, you could ask, “Can Ella buy more than one milk if she wants?” or “Is there any combination of these items Ella could buy that would cost exactly $5.00?” These challenges offer a playful way for students to engage in more mathematical reasoning and more construction of viable arguments.

Task 2 offers a chance to ask many great questions about different combinations, what counts as lunch, and how students know they have found all the answers. In contrast to the examples provided in task 1, we could also use questions to lower the cognitive demand in task 2. For example, we could ask, “Which one item could I buy along with a hot dog so I won’t receive any change?” reducing this rich task that is focused on combining amounts to a much simpler subtraction task.

Practices

As written, task 2 is much richer for engaging students in making sense of the problem and persevering in solving it. Like many tasks of this kind, it becomes a right site for rich mathematical practice in a way that does not overwhelm or bore students. This is one strength of this kind of task. Students are unlikely to start the problem-solving process with a strategy in mind for determining how many combinations they can make, but to start exploring, they will begin with the skills they need.

Because students have to invent a strategy to determine whether they have found all the combinations in task 2, the task inherently presents students with an opportunity to build an argument for why their way proves they have all the possible solutions. Task 1 has little room for argument other than for the natural minor variations students have in solving any task.

The Common Core’s SMPs highlight the need for teachers to help students develop the habit of attending to precision in terms of their communication about mathematics. Both of these tasks provide students with opportunities to talk to one another about real-world situations. Task 2 offers more opportunity for rich discussion. However, precision is not guaranteed by either task. Classroom norms and teacher’s’ expectations are what will bring precision to the discussion. For example, if a student answers task 1 by saying, “Yes, she can buy lunch because she has $5 and it’s only $4,” it is up to the teacher to either ask for clarification or set up the classroom culture so that other students will question, “What is it?” and “What do you mean by ‘only $4’?” In this way, students learn to communicate more clearly.

I hope this pair of blog posts has helped you think about the ways in which tasks, questions, and practices are connected. It takes paying attention all the way from task selection to task completion to ensure that students are getting the most out of their experiences.

Chandra Hawley Orrill, corrill@umassd.edu, is an associate professor and department chairperson in STEM Education and Teacher Development at the University of Massachusetts–Dartmouth. She teaches courses on mathematics content, like proportional reasoning and number sense, for teachers seeking their professional license as well as teaching a variety of courses in the Mathematics Education PhD program. Her interest is in how teachers understand the mathematics they teach and how we can better support teachers in understanding mathematics. She has conducted hundreds of hours of professional development focused on standards-based mathematics and on technology integration in mathematics for elementary school teachers.