# Unit 2: Designing Problematic Tasks

## Introduction:

Now that we know problem-based learning is beneficial to the classroom how do we create an environment and appropriate tasks that correlate with such inquiry, in a math classroom? Well, this is the question we will be answering throughout this particular unit.

## Unit Objectives:

• At the end of this unit, you will be able to do the following things:
1. Given some examples, you will identify and describe ways to create problematic tasks while acquiring the ability to design authentic tasks in your individual classrooms.

## Analyze the Examples Below:

While you are analyzing the below tasks and lessons think about them in comparison to lessons you have observed, implemented, and designed. What are the differences and similarities? Then, think about the definition of problem-based learning and how the tasks may relate to such inquiry?

Analyze the lesson and problematic task called Hexagons, Hexagons, & More Hexagons. Note: The task is an example of an activity that could be used in an Algebra classroom that aligns with the components of problem-based learning.

Problem-based learning can be utilized in Geometry just like it can be used in Algebra; it simply comes down to the creation of an appropriate task. Take a look at the following lesson/task Determining Surface Area

Online Video

Watch/observe at least one lesson from the following website: Inside Mathematics. Note: In order to get back to this page you will have to push the back button on your browser).

1. Go to the website and click on classroom video visits.
2. Click on Videos under the heading: Public Lessons.
3. Once there, choose a lesson to view.
• While you are watching the video think about the nature of the task, what is the role of the teacher? Do the students have individual roles? What is different about the tasks viewed in this online lesson as opposed to those seen in other math classrooms? Do the lessons represent problematic mathematics? Why or Why not?

## Research:

After reviewing and analyzing the lessons above, read the following article: A Problem-Based Approach to Mathematics Instruction. While reading, think about the components of the tasks you have just analyzed. What did they all have in common and what specifically makes a task problematic?

Problematic Mathematics:

There are three main components needed to ensure that a task is problematic in nature and fits into problem-based learning:

1. Tasks must be created in a problematic manner, or in other words tasks must give students something to think about rather than rules or a prescription they need to follow.
2. What is problematic about the task must specifically pertain to mathematics and not other aspects of the task or situation.
3. In order for students to take the task seriously and work diligently, the task must meet students where they are. Or in other words the task must give students a chance to use the skills and knowledge they already possess, i.e. their prior knowledge (Hiebert et al., 1997, p. 18).

Tasks should encourage reflection and communication:

• In the end, tasks should encourage students to reflect on their work and the work of others, which is explained nicely by Hiebert et al, (1997). "Reflection means turning something over in your head, thinking again about it, trying to relate it to something else you know. If a task encourages you to reflect on something, you do not rush through it as quickly as you can. Tasks that encourage reflection take time" (p. 18). However, how can we encourage students to take their time, think, and reflect on their work? Well the answer to this question correlates with the student's motivation to succeed. Therefore, the task must be worthwhile for the student, something that is intriguing and a problem that needs to be solved. The students need to see that the task is difficult (challenging) and they must want to know the answer. Thus, the students will work to achieve a goal if and only if the goal is worth the effort. (Hiebert et al., 1997, p. 19).

• Tasks must also encourage communication between students. However, before this can occur students must explore the problem, try out different solutions, expand on their ideas, and choose a method. Once this transpires students are more likely to engage in conversation and discussion in order to explain, expand, clarify, and defend their methods. As a result, students learn from one another. Different students see different relationships and different ways of solving the problem, which in turn creates new understandings that may have otherwise gone unnoticed (Hiebert et al., 1997, p. 45). "Communication increases the likelihood that students will think again about their own method, and hear about other methods that would work just as well or better. It is not hard to see that understanding would be a natural outcome of this kind of task" (Hiebert et al., 1997, p. 20).

• The tasks should meet students where they are. Or in other words the tasks created should not be too easy or too difficult but should allow students to use their prior knowledge to begin solving the problem. Students must be able to use their pre-existing ideas, knowledge, and understandings to start the problem-solving process. Thus, tasks should challenge students, but link up to where they are. If the tasks do not follow this rule, the students will either become frustrated with the task or will breeze through it without learning how to critically think.

Tasks should leave behind "mathematical residue:"

• William Brownell (1946) explained that it is better to think of understanding as something that comes naturally when solving mathematical problems rather than something teachers teach directly (Hiebert et al., 1997, p. 22). Thus, tasks should be created as problems that students develop methods for solving; since what students take away from that experience is what they have learned. This type of learning, that students take with them after solving a problem is called, according to Davis (1992), mathematical residue (Hiebert et al.,1997, p. 22). When students have tasks that encourage them to construct their own methods, they learn the general approaches for inventing specific procedures and adapting the ones they have already created to new situations. This type of residue allows students to solve a variety of different problems without having to memorize formulas or specific procedures or mathematical recipes. So the tasks created by the teacher must encourage the students to develop their own procedures, which further students problem solving skills.
• "It is appropriate to engage students in solving problems because it is only through solving problems that their concepts and procedures develop together and remain connected in a natural and productive way" (Hiebert et al., 1997, p. 25).

## Reflection

1. How can tasks be designed so they are problematic in nature? What are the common components of an effective lesson that follows PBL?
2. How can you create such tasks to implement in your existing classroom?
3. Create another authentic and problematic task for an Algebra course and evaluate it using the 'rules' described above.

After you are finished with the reflection, you may proceed to... Unit 3: Assessment & Feedback

Note: To learn about the roles of both the teacher and the student in a problem-based learning environment, please visit the following wiki space.Teacher & Student Roles

## References and Resources

Boaler, J., & Humphreys, C. (2005). Connecting Mathematical Ideas: Middle School Video Cases to Support Teaching and Learning. Portsmouth, NH: Heinemann.