(Unit 1) What is problem-based learning?
Design Project: Problem-based Learning in Math through Technology
Objectives
- State what problem-based learning is and why it is beneficial for mathematics students
- State examples of problem-based learning and what it looks like in a mathematics classroom
- Examine strategies for problem-based learning in mathematics
- Learn how to create problem-based activities that could be utilized in a mathematics classroom
Introduction
This unit is to introduce and expand instructors' knowledge and practice with problem-based learning. The goal of this unit is to encourage instructors to use problem-based learning in their classrooms as it promotes an engaging and interactive environment. Problem-based learning can also serve as a motivator to students as it makes material and lessons relate-able an captures their interest.
Launch Activity
The launch activity for this unit is to introduce instructors on what problem-based learning is and get them thinking about problem-based learning in math. Instructors will have to think about three questions about incorporating problem-based learning when creating lessons: What problem-based learning lesson are not? What can problem-based learning lessons be? What must problem-based learning lesson be?
After reflecting on these three questions, instructors will be directed to the link: http://www.amphi.com/teachers/pgreenleaf/edtech/pbllesson.html with possible answers.
Assessing Prior Knowledge
Problem-based learning encourages problem solving and critical thinking skills which are the most desirable and beneficial skills to teach our students for their future as citizens of the 21st century.
Some questions to consider when thinking about what prior knowledge you (as an instructor) may already have:
When my students are actively engaged, what methods and strategies are in effect?
How can I relate real-life scenarios and issues to lessons, particularly in math?
Strategies for instructors to assess their students' prior knowledge during a problem-based learning activity:
- analyzing videos of classroom discussion
- discussions about animated vignettes
- create real-world activity that students can participate in and then reflect on
Assessing student prior knowledge will allow you (the instructor) create a framework of instruction for lessons
Introduction of New Concepts
Problem-based learning in a mathematics classroom describes an environment where an actual problem drives learning. Specifically, learning will begin with a "real-world" connection made to a problem that students will begin to find a solution. Additional information and a thorough description of what problem based learning is, why we should use it, and benefits of PBL can be found with the following links:
What is problem-based learning? Why should we use problem-based learning?
As well as examining, researching, and learning about what PBL is; instructors are also encouraged to explore samples of PBL math activities which have been previously used in classroom. The following are links with current PBL math lessons, activities, and examples:
http://www.ct4me.net/math_projects.htm#DesignProjects
http://www.amphi.com/teachers/pgreenleaf/edtech/pblactivities.html
Instructors are encouraged to discusses their questions, comments, ideas, critiques, suggestions, etc.
Connecting "Real-Life" to Mathematics
One of the main reasons for incorporating PBL in a mathematics classroom is to make the material relate-able to students. Often in mathematics classes, teachers are asked the questions "why do we need to learn this?" and "when are we ever going to use this?". If instructors can make students see the connections between the importance of math and the opportunities it can create for their future then students are more likely to become interested and engage in the process of learning. The best strategy for students to become interested in class is to connect mathematics to real-life scenarios and issues that students can understand and relate to. When instructors establish this connection they can facilitate their students to learn through understanding instead of a traditional skills and drills approach. The most significant attribute of problem-based learning in mathematics is to relate the material to the real-world. This acts as a motivator for students to learn math as they see why they need to learn it and how it can be of use in their future.
Activities
Planning
It is essential to think and plan each PBL activity as they require proper timing and organization. PBL activities should be flexible as many activities encourage discussions and interactions with other students. It is important to incorporate all objectives of the lesson, but to realize that the organization and structure may change during the lesson so being able to adapt is very important. Also, the timing for PBL activities can range because of diverse student learners and it may be hard to predict how long it takes students to work through various problems. Too little time can make students feel rushed and not spend sufficient time relating to the material through real-life experiences to understand concepts, ideas, and mathematical applications. Too much time may allow for students to be unfocused and end up off task and not exploring, examining, and analyzing the problem at hand.
Implementing
Once a topic and objectives are established for a particular lesson as well as the timing and organization considered, teachers can begin implementing a PBL activity by first planning the project. It's important to be creative in choosing a project that would interest and engage the interests of your students. In addition to providing project guidelines and directions its also essential to include expectation for students or a sample completed project that can be viewed by students. Group work and collaborative learning tasks are encouraged for problem solving problem-based activities.
Examples
Example 1:
(Pohlman 2010)
Elections provide students with a relevant example of math's practical applications. Tell your students that they are going to have ice cream for a class party but can only buy one flavor. Give students a choice between strawberry, vanilla or chocolate as their favorite flavor of ice cream. Ask students what method they want to use to choose the flavor. If students suggest voting, tell them to write their favorite flavor on an index card. Count the votes and tape the index cards to the whiteboard in columns according to the type of ice cream.
Depending on what level math class you have (perhaps a statistics unit in pre-algebra) student can construct a graph with the voting outcomes, analyze the differences between difference flavor choices, find standard deviations, etc. Perhaps if an AP stats class, students can create multiple graphs, research national average of ice-cream flavor choice, look at other variables for ice cream choices (males vs females, age, region, etc).
Example 2: March Madness
(Wetzel 2010)
Students:
Explore the many different areas of math found in the NCAA basketball tournament.
Use the NCAA bracket to determine which team or teams they want to follow in the tournament.
Find fractions, decimals, and percents; probability statistics; make predictions; and look for patterns within the basketball statistics of the team(s) they selected.
Make connections with and between the math contained within the NCAA tournament.
Explain the importance of mathematics in basketball.
Example 3:
(Ethier, Mounds Park Academy)
Using Graphing TI-83 calculator:
The Big Chill
Exponential functions appear in many situations in ordinary life. For this activity, you will gather some data about how long it takes a cup of cold water to warm up.
1. Get a cup of refrigerated water. Use a thermometer to measure the temperature. Allow some time for the thermometer to reach the temperature of the water. Then start timing and recording data as follows:
2. As you record your data, enter it into your graphing calculator. Enter the temperature as degrees below room temperature. If room temperature is 70° and the water temperature is 40°, enter 30° as your temperature.
3. Once you have at least 10 - 15 minutes of data, plot your data and find the best-fit exponential function:
y = a * bx
4. Write up your data, your graph, and your best-fit function neatly on a separate sheet of paper.
Example 4:
(Ethier, Mounds Park Academy) Applications of Linear Equations
1. A 50 ft long bungee will stretch a certain amount, depending on how much the person doing the jump weighs. The following table tells how much a bungee cord will stretch for certain weights. The stretch is in addition to the original 50 ft length of the bungee cord.
| Wt (lbs) Stretch (ft) |
| 100 ......... 80.9 |
| 110 ......... 86.7 |
| 120 ......... 92.4 |
| 130 ......... 98.1 |
| 140 ......... 103.7 |
Write an equation that will predict the stretch distance for a given weight:
stretch = ______ x weight + ______
Use your equation to find the stretch distance for your weight.
If the concrete is 200 ft below the point where the bungee cord is attached, what is the heaviest "safe" weight for a 6 ft tall jumper? Assume the bungee cord is attached to the jumper's ankles.
2. A gas has a volume of 500cc at 27°C and a volume of 605cc at 90°C. The volume and temperature of any gas vary in a linear fashion. Use these two data points to write an equation relating the volume and temperature of this gas (y=volume, x=temperature).
At what temperature will the volume be 0? What is the significance of this temperature? (Hint: Ask a science teacher!).
3. The world record for the mile was 4:10 in 1923 and 3:49 in 1975. Assuming that the record improves in a linear fashion (which it has so far!), write an equation to predict the mile world record (in seconds) for a given year (y=record in seconds, x=year).
What should the world record be this year?
What should the world record be in 2001?
The world record was broken in 1993 by Noredine Morceli. What is the current world record in the mile? How does this compare to your prediction above?
4. The figure below represents four puzzle pieces that can be arranged in an 8x8 square and as shown. The area seems to magically increase by 1. The truth is that there is a slight gap along the diagonal that adds the extra unit. Prove that the pieces do not fit together. (Hint: if they did fit, points A, B, C, and D would all be co-linear).
Assume that the lower left corner of the rectangle is at (0, 0). Thus B is at (5, 3) etc.
Resources
Ethier. "My Algebra Activities." Mounds Park Academy, Math Teacher. http://panther.moundsparkacademy.org/~dethier/activities.htm
Pohlman, Bruce. "Problem-based Activities for Math." June 2010. http://www.ehow.com/list_6662183_problem_based-activities-learning-math.html
Wetzel, David R. "Problem-based Learning-Math Activities. February 2010. http://www.teachscienceandmath.com/2010/02/24/project-based-learning-math-activities/
Assessment
In problem-based learning for math classrooms, assessments and evaluations can take place in a variety of different ways and methods. There can be a culminating evaluation of the progress, final product, and work along the way during the activity. Students can also be given a quiz or test on the subject matter of the PBL activity or lesson. No matter which assessment strategy used, its important to assess and evaluate the objectives of the lesson/PBL activity (not the activity itself). Feedback and thoughts about the PBL activity should be welcome, but that can not be in place of an assessment measure on what the students learned during the lesson.
Reflection
Not only is it helpful to provide students with the opportunity to have feedback on a specific lesson or unit, but it also important to provide an opportunity of reflection for yourself (the instructor). Ask yourself the key questions: Was this lesson successful? What can I do to improve this lesson? What would I chance in the future? Were the students actively engaged? Were the objectives met for this lesson? Did student understanding occur during this lesson? Were students able to establish a connection to the material?
References
http://www.amphi.com/teachers/pgreenleaf/edtech/pbllesson.html
http://www.ct4me.net/math_projects.htm#DesignProjects
http://iae-pedia.org/Good_PBL_Lesson_Plans