Open tasks can be powerful ways to engage students and can provide a teacher an opportunity to learn more about students' mathematical thinking. An open task will have a high level of cognitive demand for students because they cannot simply apply an algorithm or use memorized facts to provide a solution. Below are some of my favorite resources for finding ready-made open tasks and information about how we can create them from our existing materials in use in our schools. The related videos are clips taken from an online teacher professional development session.

## Why Open Tasks

NCTM Principles to Actions identifies eight effective mathematics teaching practices. Practice 2 states that “effective teaching of mathematics engages students in solving and discussing tasks that promote reasoning and problem solving and allow multiple entry points and varied solution strategies.” Researcher John Hattie did a meta-analysis of educational research and determined that problem-solving teaching has an effect size of 0.61, which indicates that this particular teaching strategy is in the “zone of desired effects.” It is clear that the incorporation of open tasks in mathematics instruction has the potential to offer students more opportunity to develop their mathematical thinking. “When students are assigned rich tasks, they use a variety of skills and ask themselves questions, make meaning of mathematics, and ultimately build a healthy and realistic relationship to mathematics as something that is engaging, interesting, and useful--and something that makes sense.” (Hattie, 83)

## "Open Up" a Task

In some curriculum materials, there is a built in design for problem-based teaching and the materials themselves have lots of possibilities and examples of open tasks. In other materials educators use, there may be more limited availability of rich, open tasks, so educators need to adapt existing tasks to “open them up” and ask for deeper thinking and engagement on the part of students. Additionally, there are many great online sources to go to for inspiration and new thinking about how to adapt tasks in one’s current materials, as well as being great sources of tasks. Some strategies are described below. These strategies can be combined and some of the examples include more than one strategy.

## 1. Problematize the Answer

Typically, students are provided with a problem to solve and then they find the answer to the problem. This strategy reverses that thinking: invite more thinking and more opportunity for students by giving an answer and asking students for the problem or question. Even a seemingly straightforward computation problem can be transformed into a task that may invite more discussion and debate as students defend their reasoning to peers. A few examples are below:

The answer is 42... what is the problem?

Can you make 42 a sum, a product, a quotient and a difference?

The sum of three numbers is 20. One of the numbers is a multiple of 4. What are possibilities for the three numbers?

## 2. Replacement?

You can open a task by replacing a number in a task with one or more blanks or question marks (?). In the elementary task below from Open Middle:Challenging math problems worth solving, students are asked to make the smallest (or largest) difference by filling in the blanks using the whole numbers 1-9 no more than one time each. There is a high degree of flexibility asked for in tasks like these.

______ ______ — ______ ______

## 3. Similarities and Differences

The task to the left is based on a similar task from Would you Rather Math. The task offers two situations and asks students to make a choice and offer mathematical justification for their decision. The Web site is filled with a variety of tasks that could be incorporated into the regular classroom routines and align to a wide range of mathematical topics and learning targets. These tasks present a choice (see below!), but more importantly they help students attend to similarities and differences between the choices. You could also present two or more approaches to a problem and ask students to explain their similarities ("Where do you see the original amount in each approach?") and differences ("How is this approach different from that one?").

## 4. Student Choices

This Kindergarten task is based on a similar task from Inside Mathematics Problems of the Month. It offers students the opportunity to make a choice. In this case, students will decide how many dinosaurs there are based on how many legs are below the surface of the lake. The student will need to convince their classmates why their answer for the number of dinosaurs is accurate.

The task:

You are swimming underwater in a lake and you see dinosaur legs!

You don’t want to look above the surface since they might not be friendly dinosaurs. This is what you see:

How many dinosaurs do you see standing in the lake? Explain how you know.

A related task can also be written about a trip to the dog park: "I brought my dog Moxie to the dog park. I counted 28 legs at the park. How many people and how many dogs could have been at the park?

A tasty example: "Tonya is planning a party and she’s making 12 cupcakes. She will use 3 colors of sprinkles: pink, purple, and blue. Each cupcake will have only 1 color of sprinkles. How can she make 12 cupcakes so at least 1 cupcake has each of these three colors?"

Another way to offer students choice is to provide more than one task and allow students to choose a task on which to work. The tasks could vary by one or more characteristics such as problem context, language, anticipated level of difficulty, and degree of openness.

## Resources

Additional resources for open tasks are listed below:

- Interactive STEM - Open Tasks (Educator STEM Learning Collection by EDC)
- Interactive STEM - What are Open Tasks? (EDC)
- 3 Act Tasks by Graham Fletcher
- NRICH (University of Cambridge)
- youcubed (Stanford University)
- Hattie, J., Fisher, D., & Frey, N. (2017). Visible learning for mathematics, grades K-12: What works best to optimize student learning. Thousand Oaks, CA: Corwin Mathematics.
- Small, M. (2009). Good questions: Great ways to differentiate mathematics instruction. New York: Teachers College Press.
- Sullivan, P., & Lilburn, P. (2012). Open-ended maths activities using good questions to enhance learning in mathematics. Victoria: Oxford Univ. Press.