Unit 2: What is a Problematic Task?
Home Page: Problem-Based Learning in a Mathematics Classroom
Unit 1: What is PBL and Why is it Important?
Unit Objectives:
1. The learner will list in writing the key factors to consider when creating problematic mathematics contexts.
2. Given a variety of mathematical tasks, the learner will discriminate between problematic tasks and non-problematic tasks by listing the characteristics of the given task, and determining which tasks fit into each category.
What is the Difference Between an Exercise and a Problem?
To put it simply, you have a problem when you are required to act, but don't know what to do (Wiggins, 2010).
♦ An exercise is a question that tests the student's mastery of a narrowly focused technique, usually one that was recently 'covered.' Exercises may be hard or easy, but they are never puzzling, the path toward the solution is always apparent.
♦ In contrast, a problem is a question that cannot be answered immediately. Problems are often open-ended, paradoxical, and sometimes unsolvable, and require investigation before one can come close to a solution.
A real problem is often open-ended, paradoxical, and requires investigation.
Differing Characteristics: Problems vs. Exercises
| Problems | Exercises |
|---|---|
| A puzzle - not all needed information is provided explicitly. Some investigation, inference, and logic is required. Lots of prior knowledge is needed and tested. | May be a hard challenge to memory, but never puzzling. Once the right algorithm is found in memory, it is simple to stay on track to the answer. "Plug and Chug." |
| Paradoxical - on the surface it seems unsolvable, self-contradictory. "Non-routine" problem. | Straight-Forward Request - understand the prompt, not confused or puzzled, looks routine. |
| Solution path not apparent - multiple approaches may work, different approaches may have different strengths and weaknesses. | Solution path is apparent. |
| Open-ended - work may require to re-frame problem statement, consider what counts as an answer, and consider that the answer may depend. | Single and unequivocal right answer. |
| Investigation is a must - need to experiment with numbers before fully knowing how to proceed. | Little or no investigation - know how to proceed. |
Mathematical Tasks: Problematic or Traditional?
Now that you know the characteristics of a problematic task, look back at the exercises in the beginning of Unit 1 and the following mathematical tasks. 1) List the qualities of the task and 2) using those qualities, define if the task is problematic in nature or a traditional task.
1) What is the relationship of perimeter to area in any square?
2) What number, when added to itself four times equals 55?
3) Which uses more water: the average bath or the average shower?
4) Which is the bigger area: A 3-foot square figure or an equilateral triangle in which each side is 4 feet long?
Reflection
Which examples from above, did you classify as traditional?
List at least two reasons why these tasks are traditional?
How can these traditional tasks be changed to problematic tasks using the same mathematical concepts?
Hints: A problematic task has more than one answer. How can you re-word the above traditional tasks, to make them open-ended?
Inside a Problem-Based Learning Classroom
♦ Now that you have a better idea of what problematic tasks look like, let's see them in action. Take a look at the following 7th grade mathematics lesson, What's the Savings? to see problem-based learning in action. You will need to click on Clip 2 to see the lesson itself, Clip 1 is a pre-lesson talk, and Clip 3 is a follow up to the lesson.
While watching the video reflect on the following questions:
• How does this mathematics lesson differ from a traditional mathematics lesson? Is the mathematics lesson problematic in nature?
• What is the role of the teacher?
• What are the roles of the students?
• Why do you think this lesson is considered problematic in nature? (Hint: What are the key qualities of the lesson that differentiate it from a traditional mathematics lesson?)
• How do you think the students are benefiting from this collaboration and inquiry?
After reflecting on the video, it is time for Unit 3: How do I Generate Problematic Tasks?
Resources
Wiggins, Grant. "The Problems of Non-Problems in Math Programs." Big Ideas: An Authentic Education E-Journal. 31, Aug. 2010. Web. Sept. 2012. <http://www.authenticeducation.org/ae_bigideas/article.lasso?artId=122>.
7th Grade: What's the Savings? Inside Mathematics. Web. Sept 2012. <http://insidemathematics.org/index.php/7th-grade-whats-the-savings>.