Unit III: Planning & Activities
Problem-Based Learning in Math Home Page
Planning
Tasks do not just appear
- To select appropriate tasks, the teacher must draw on two resources: knowledge of the subject to select tasks that encourage students to wrestle with key ideas and knowledge of students’ thinking to select tasks that link with students’ experience and for which students can see the relevance of the ideas and skills they already possess (Hiebert et al., 1996).
Planning Principles
- In her article, Erickson (1999) outlines the planning principles for a PBL lesson to keep in mind. It is as follows:
Time
- In planning lessons, teachers must allow appropriate amounts of time for students to work on problem-solving tasks. If too little time is allowed, students often attempt to apply a known routine without connection to concepts, meaning, and understanding. If too much time is spent on a single activity when students are not making progress in constructing and understanding key mathematical ideas, students’ work degenerates into unsystematic exploration or declines into no mathematical activity at all (Henningsen and Stein, 1997; Erickson, 1999). The length of time for the activity should be proportional to the size of the problem.
Organization
- Problem-based lessons require instructional flexibility that responds to students’ interaction with the problem. Know what your goals are for the lesson and the essential elements that should be taking place during the lesson.
Implementation of problem-based instruction
- Teachers must establish the community and its rules. They should also clarify their expectation for the type of activity that is going to take place (i.e. large-group work, or small-group work). Teachers face two challenges in a problem-based lesson, keeping the character of the problem-solving tasks from changing after the students begin working and keeping the cognitive demand of high-level tasks from declining (Stein, Grover, and Henningsen, 1997; Erickson, 1999). They must continually assess the students’ progress and provide sufficient scaffolding or guidance to keep students interested and on-task.
Activities
Types
"Post-Hole" Problems
- Post-holes are short problems that can be used when teachers don’t want to design their entire course around problems but do want to introduce one occasionally (Stepien & Gallagher, 1993).
Open-Ended Problems
- These are problems that have no clearly defined question asked, therefore allowing many possible solutions. These types of problems should suit every single student by using familiar and interesting subjects, and should be suitable for mathematical thinking and should also be able to be generalized into new problems (Kwon, Park, & Park, 2006). Open-ended problems can give students a sense of achievement and fulfillment because it is possible even for students with less mathematical ability to set forth their own solutions within their own ability (Kwon, Park, & Park, 2006).
- Teachers can also develop more open-ended problems from existing problems in their own collection or textbooks by changing the wording of the problems, or suggested procedures or methods of solution could be deleted from the statement of the problem (Erickson, 1999). Erickson also suggests asking the students solve the problem in an alternative way, explain their thinking either orally or in writing, or propose an extension or generalization.
- Here is an example of an open-ended problem form the article by Kwon, Park, & Park (2006).
| Among the following numbers, choose a number which is different from the others. If possible, try to find many possible cases or answers 1, 2, 4, 6, 8, 12 |
|---|
Example Problems
Example 1
- From Erickson (1999):
Draw a square 1 unit on a side. Divide the square region into quarters, and shade the upper-left quarter. In the lower-right quarter of the original square region, do the same. Continue until you have a diagram that looks like figure 1. Find or estimate the sum of the areas of the shaded squares in as many ways as you can.
- Students can use multiple solution strategies, multiple representations of the concept, and mathematical communication that includes explanation and justification.
Example 2
- From Erickson (1999):
To what extent are today’s seventeen year-olds like their ancestors?
- Small groups of students could tackle this problem in a unit on statistics that includes averages, measures of dispersion, lines of best fit, and chi-square tests or t-tests. Students could generate more specific questions, then collect data and use statistical methods to produce possible models and reasonable conjectures or answers to their posed questions.
Example 3
- From Alper, Fendel, Fraser, & Resek (1996):
Picture this: You’re on a planning team, consulting to the city manager. Your task is to come up with a reasonable plan for the use of 550 acres of land recently obtained by the city. The acreage includes a recently closed army base, a 300-acre farm, and abandoned mining land.
Two conflicting parties are interested in the property. The business community is pushing for development schemes, while environmental groups are advocating for recreational space. The two factions have arrived at a partial compromise, which is what you have to work with. They have agreed that
- a maximum of 200 acres from the army base and the mining land will be used for recreation, and
- the amount of army land used for recreation plus the amount of farm land used for development will together total 100 acres.
Not only are you dealing with opposing factions, but with improvement costs ranging from $50 to $2000 per acre, depending on which parcel of land is involved and how it will be used. You have to satisfy everyone while minimizing the total cost for improvements. To arrive at a reasonable allocation plan will demand careful analysis and attention to detail.
- This problem is called Meadows or Malls?, and it is a six to eight week unit. In this unit, students use algebra, geometry, matrix operations, and written explanations to support their solution.
Example 4
From Kwon, Park, & Park (2006):
Make an expression to obtain 30 after calculating by the following rules.
Rules: 1. Use all or some of the following numbers.
2. You are allowed to use any kind of mathematical symbols.
3. Use the given numbers only once in one expression.
- e.g. 10+20 (o), 10+10+10 (x)
Given Numbers: 18 2 10 12 40 48 3 20 90 15
- This problem has students use their knowledge to explore basic operations, involutions, and order of operations .
Think About It
- What are the main concerns when planning a PBL lesson or activity?
- How do they shape the lesson or activity?
- Do you see some ways that you could create problems to use in your classroom?
Continue on to Additional Resources
Resources
- Alper, L., D. Fendel, S. Fraser, and D. Resek. (1996). Problem-Based Mathematics- Not Just for the College-Bound. Educational Leadership, 53(8), 18-21.
- Erickson, D. K.(1999). A Problem-Based Approach to Mathematics Instruction. Mathematics Teacher, 92(6), 516-521.
- Hiebert, J., et al.(1996). Problem Solving as a Basis for Reform in Curriculum and Instruction: The Case of Mathematics. Educational Researcher, 25(4), 12-21.
- Kwon, O. N., J. S. Park, & J. H. Park. (2006). Cultivating Divergent Thinking in Mathematics through an Open-Ended Approach. Asia Pacific Education Review, 7(1), 51-61.
- Stepien, W. & S. Gallagher.(1993). Problem-Based Learing: As Authentic as It Gets. Educational Leadership, 50(7), 25-28.
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