# PBL and Math Education

This page was originally authored by Stacy Chirico (2009). This page was edited by Anita Aytona (2011)

## Contents

## Overview

Problem Based learning (PBL) is an instructional method of hands-on, active learning centered on the investigation and resolution of messy, real world problems.

Students want to make connections between what they learn and what they experience in their lives and a problem based learning (PBL) approach is a perfect way of marrying theory and pracice in mathematics. The goal of PBL is to prepare students to be ready for true-to-life settings by “[increasing] higher-level thinking skills … by requiring them to think about problems critically and analyzing data to find solution” (Sungur et al., 2006).

Mathematics is a human endeavor - there exists a rich history of mathematics applied to human civilization and its uses in our modern society. PBL approach prepares students to think critically and analytically so that the knowledge learned through school can be better applied to the real world.

## History

Problem Based learning started in the late 1960s at the medical school at the
McMaster University in Canada. Shortly thereafter, three other medical schools — the University of Maastricht in the Netherlands, the
University of Newcastle (Australia), and the University of New Mexico (United States) took on the McMaster model of problem-based learning. Soon after, it found its way in the study of engineering, business, education and others.

PBL is based on the educational theories of Vygotsky, Dewey, and others, and is related to social-cultural and constructivist theories of learning and instructional design. PBL is inherently social and collaborative in methodology and teaches students essential "soft skills" as well as domain specific content and skills.

## Difference between problem based learning and traditional learning

Many schools use the traditional approach to learning where a teacher ‘transmits’ information to students. Theories are learned first then the problems are given. In PBL, the students are presented with a problem prior to the instruction. The problem is the focal point for knowledge acquisition and application.

In traditional learning, the teacher is the dominant part of the learning process. In PBL, it is learner centered. The students play an active role in learning. Roles of the teacher and student are therefore shifted, as a teacher should facilitate and collaborate with his students in order to develop meaning construction in students. Learning therefore becomes a reciprocal experience for the students and teacher.

## Benefits of PBL in Math education

- Allows the learners more responsibility and independence.
- Gives students realistic dilemma because the problems or cases are context specific.
- Shows students that there is more than one way to solve a problem and there may also be more than one answer to the problem.
- Promotes group work and collaboration in math.
- Increases self-motivation and critical thinking.
- Enables students to demonstrate their understanding and knowledge in a non-traditional way.
- Encourages life-long learning.

## Challenges

**PBL should not be confused with the traditional math word problems**

A traditional word problem in math simply translates equations into mathematical sentences and it requires students to focus on the arithmetic operation in the end. Also, these problems usually have only one right solution and teachers usually teach them in a way that there is only one approach to solving them. The reality that PBL students will be facing is too dynamic and uncertain, however, the conventional instructional approaches are “too straight-line cause-effect” and persist on too general models (Polanco, 2004). A PBL math approach deals more with situational problems in real life and such situational problems rarely give students all the necessary information in one tidy package in a convenient form. Hence, students may not be sure if the solution/direction they choose is right or wrong, just like real life problems, students have no immediate answer but require trial and error. In this way students will learn from their missteps along the way to a solution.

**Significance of teachers' roles in Math PBL**

Roh (2003) indicates that teachers skills should not be limited to just math but they need to develop broader range of pedagogical skills. In other words, they not only need to provide mathematical knowledge to students but also need to know how to engage students in the problem solving process and applying knowledge when they face novel situations.

Roh also notes that teachers' deep understanding of math curriculum is essential to guide the learners to apply the knowledge in a variety of situational problems. Without an in-depth knowledge, teachers will not be able to choose appropriate tasks that best fits the students at the right age level (Roh, 2003).

Also, some teachers may find it difficult at first to "relinquish control" and become a facilitator. He should encourage the students to ask the right questions and explore the needed facts rather than handing them the solutions.

**Challenge of Time Management**

A common criticism is that the PBL approach may not be able to cover as much material as a conventional lecture based course. PBL entails a lot of planning, creativity and resourcefulness from the teacher.

**Resource problems**

An initial problem that teachers will often site is that there is a lack of rich problems that will be ideal for PBL. In his book Look to the Mountain Dr. Gregory Cajete shows that problems are everywhere in our own local environments (1994). An example of one such problem (sturgeon math) for the environment in Chilliwack or along the Fraser River is shown below.

## Steps in Problem Based learning

The PBL approach begins with a problem that is contextualized and ill structured. The problem becomes more defined as students separate the known facts from unknown issues. This is the main difference between between a PBL classroom and a traditional one where the instruction of theories come before the problem. In PBL, the problem becomes the focal point for knowledge acquisition and application. The students research, investigate and evaluate to arrive at the solution. There is no one correct answer. The students are not judged on how well their answers match the teacher's but on the validity of the solution. After several cycles of data collection and analysis, possible solutions to the problem are formulated. The potential solutions are examined in the light of all the evidence collected and the most viable solution is then selected. The PBL experience culminates with the public sharing of the solution and some type of evaluation. This evaluation may be formal or informal; self-, peer-, or instructor-assessed, written or oral.

## Examples of Math PBL

**1. Sample Math Case Study**

*One More Minute Between Classes*

The student council has asked the principal to add one more minute between classes, to cut down on the number of students tardy for class. The principal has asked your class to study the matter and make a recommendation. Here are guidelines provided by the principal:

- Bus schedules cannot be changed; because of after school activities and bus schedules.
- Classes must meet at minimum of 210 minutes a week or 420 minutes every two weeks.
- Homeroom time is required.
- A multimedia presentation is expected in two weeks.
- Be prepared to defend any plan with facts, figures, and charts

**2. Sturgeon Math**

Many people come to Chilliwack to fish for sturgeon. Sturgeon were also a resource to indigenous peoples in the area. For this reason basing an activity in math class in Chilliwack around sturgeon should interest the students.

Mathematical problems can be designed so that students can:

- Use graphs and statistics to analyze data
- Critique information presented in a variety of media
- Perform arithmetic operation on irrational numbers, using appropriate decimal approximations (Golden Ratio in the tail fin of sturgeon).
- Perform operations on irrational numbers of monomial and binomial form, using exact values (Golden Ratio same as above).
- Use the quadratic formula to solve problems.

**Extention to Cross Curricula**

- Biology - Anatomy, Physiology, Evolution
- Geography - Location, Ecology of Sturgeon, Habitat study
- History/First Nation Studies - Relations to Indigenous peoples and European settlers
- Art

**3. Inheritance Case**

*What to do with Mrs. HugandKiss' inheritance.*

Mrs. HugandKiss is a well respected community volunteer worker and school board member of Selma Town. She passed away and left a huge inheritance for the town.

These are her conditions:

- The money should be used to build a school, library, or recreational center.

- The town will donate the lot for the structure.

- The structure should be named after her late husband, Mr Hugsandkiss.

Form a committee composed of teachers, architect, business owners, parents and children to decide how to spend her gift.

**Stop Motion Video**

Video provides a brief overview of Problem-Based Learning in Math.

https://www.youtube.com/watch?v=WqlbvKHUNTE

## Useful Links

- History of PBL

http://www.learning-theories.com/problem-based-learning-pbl.html

- PBL in Universities
- McMaster University http://www.mcmaster.ca/welcome/aboutmac.cfm
- University of Windsor http://www.uwindsor.ca/vabe/problem-based-learning
- University of California, Irvine http://www.pbl.uci.edu/whatispbl.html

- Lev Vygotsky and John Dewey

http://www.learning-theories.com/vygotskys-social-learning-theory.html

http://en.wikipedia.org/wiki/John_Dewey

- Steps in PBL

http://www.ncsu.edu/pbl/design.html

- Sample PBL Math Problems

http://www.suite101.com/content/math-problem-solving-stories-and-process-skills-a66091

http://www.ncsu.edu/pbl/pdf/hugandkiss.pdf

## Stop Motion Video

Sabrina Holats' Submission for ETEC510 65D Problem-Based Learning and Math Education https://www.youtube.com/watch?v=zX4_R_VK32M&t=9s

## References

Boaler, J. (1998). Open and Closed Mathematics: Student Experiences and Understandings. Journal for Research in Mathematics Education, 29 (1), 41 – 62.

Cajete, Gregory (1994). Look To The Mountain: An Ecology of Indigenous Education. Skyland, NC: Kivaki Press.

Overbaugh, R.C. & Lin, S.Y.(2005). Problem-based Learning and Fourth Grade: Who Really Benefits? The Constructivist, 16(1), 1 - 21.

Polanco, R., Calderon, P., & Delgado, F. (2004). Effect of a problem-based learning program on engineering students’ academic achievements in a Mexican University. Innovations in Education and Teaching International, 41 (2), 145 – 155.

Rodd, M. (2006). Commentary: Mathematics, Emotion and Special Needs. Educational Studies in Mathematics, 63, 227 – 234.

Roh, K. H. (2003). Problem-Based Learning in Mathematics. ERIC Digest.

Shore, M., & Shore, J. (2004). Allied Health Applications Integrated into Developmental Mathematics Using Problem Based Learning. Math Comput Educ., 38 (2), 183 – 189.

Sungur, S., Tekkaya, C., & Geban, O. (2006). Improving achievement through problem-based learning. Journal of Biological Education, 40 (4), 155 – 160.

Wee, K.N., Kek, Y.C., & Sim, H.C. (2001). Crafting Effective Problems for Problem-based Learning. PBL Conference