Math Problem-based Learning

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Work in Progress

Currently, this is merely a place holder for an initial collection of materials related to math problem-based learning.

Eventually this will be a companion to the Math Project-based Learning page in the IAE-pedia.

The target audience for this document is preservice and inservice K-12 teachers who teach math, and teachers of such teachers.

PBL Venn.jpeg

Introduction

This document provides an introduction to uses of Problem-based Learning (PBL) in math education. It includes arguments supporting use of PBL and it includes a number of examples that can be adapted for use in a wide range of math courses at the precollege level and in teacher education.

Keep in mind that the overriding goal or purpose of this document is to improve math education. As a teacher, you should consider making use of problem-based learning when you believe it will help to improve the quality of the math education your students are getting.

PBL is used to represent both problem-based learning and project-based learning. The two topics overlap, but are not the same. The main focus in this document is on problem-based learning. There, students in a class are assigned a challenging math problem. They may work individually or perhaps in teams. While the assignment sometimes continues over several class periods, more typically it is done in one class period and/or as homework.

A Math Problem for Problem-based Learning

Here is an example of math problem-based learning.

Perhaps you have encountered the four 4s math problem. The goal is to combine four 4s in various ways in order to make as many different integers as possible. The "combine" rules are that one can use addition, subtraction, multiplication, division, and parentheses.
Thus, (4 + 4 + 4)/4 = 3. For a more complex version of the problem, also allow concatenation (thus, 444/4 = 111), exponentiation, or other types of operation.

This math problem and its variations is widely used in math education. It illustrates that a math problem may have more than one solution. It illustrates the need for very careful definition of a problem. It is a problem that can engage individual students or a team of students over an extended period of time. Thus, you can see it has some of the characteristics of a project. However, typically all students are required to work on the exact same problem. Some students are likely to produce more answers than others. In any case, there is the added challenge of trying to prove that one has found all possible answers.

There are many variations of the problem, both in the base number (for example, how about using four 3s) and the allowable operations. A Google search of four fours math problem produces many thousands of hits.

Quoting from an ERIC Digest article on Problem-based Learning in Mathematics:

Problem-Based Learning (PBL) describes a learning environment where problems drive the learning. That is, learning begins with a problem to be solved, and the problem is posed is such a way that students need to gain new knowledge before they can solve the problem. Rather than seeking a single correct answer, students interpret the problem, gather needed information, identify possible solutions, evaluate options, and present conclusions. Proponents of mathematical problem solving insist that students become good problem solvers by learning mathematical knowledge heuristically.

A Math Project for Project-based Learning

Here is an example of math project-based learning. Students are given a general project area, and they are asked to work on it individually or in teams.

Select an academic discipline other than math. Investigate roles of math in helping to represent and solve the problems in this discipline. Pay special attention to identifying specific areas of math that are important in the academic discipline you select.

This "assignment" might include the requirement that students write a paper and make a presentation to the whole class. The assignment could well extend over a number of weeks, with much of the required work being completed outside of class. If this assignment is given in a specific math class (such as geometry or algebra) then the assignment might include the requirement that the roles of the math being studied in the class be emphasized in the project.

The project leads to both a written paper and an oral presentation to the class. As a teacher in this setting, you might want to emphasize the idea of developing a paper that will be useful to other students in the class, students in other similar classes, and future students in the class. The intended audience is much larger than just the teacher!

Math PBL is a Large Topic

PBL in math education is a large and challenging topic. A recent Google search of math "problem-based learning" produced 86,400 hits. A Google search of math "project-based learning" produced 90,100 hits. Many of these hits are documents covering both project-based and problem-based learning.

At the Northwest Mathematics Conference held October 9-11, 2008, in Portland, Oregon, each of the exhibitors was asked about their availability of project-based learning materials. Many responded by pointing to certain sections in their textbook series. For the most part, the materials identified were better classified as problem-based learning rather than as project-based learning.

This little bit of evidence suggests that problem-based learning is considered by many publishers to be a major topic in the current K-12 math curriculum.

Some Examples

A geoboard is a useful and widely used math manipulative. The board consists of rows and columns of parallel, equally spaced nails (or pegs, or equivalent) that are firmly embedded in the board and that stick up far enough to allow rubber bands to easily be stretched around them. The figure given below is a 5 X 5 geoboard.

Blank Geoboard.jpeg

Geoboards can be of different sizes, and there are variations in the design. These variations increase the versatility of the manipulative. The 5 X 5 geoboard is widely available through outlets that sell math manipulatives. One can also use virtual geoboards—computer versions of geoboards. Here are two useful references:

CT4ME (n.d.). Math Manipulatives. Computer Technology for Math Excellence. Retrieved 10/22/08 from http://www.ct4me.net/math_manipulatives.htm.
NCTM (n.d.). Interactive Virtual Geoboard. Retrieved 10/22/08 from http://standards.nctm.org/document/eexamples/chap4/4.2/index.htm#applet.

With a standard 5 X 5 geoboard such as the one pictured above, one can make a variety of triangles. It is easy to see how to make right triangles. How many right triangles can one make on a 5 X 5 geoboard? Answer the same question for a 1 X 1 geoboard, a 2 X 2 geoboard, and so on. Can you find a pattern or formula for your answers? If so, can you give arguments that your pattern or formula is correct?

Equilateral triangle. This is an example of a math problem that has no solution within the constraints of a geoboard with parallel and equally spaced horizontal and vertical rows of pegs. How might you become convinced of this result? Can you give a good argument (a proof) supporting the result?

Some Math/Computer Problems

The Project Euler Website contains a number of problems that require use of both math and computer programming to solve. Quoting from the website:

Project Euler is a series of challenging mathematical/computer programming problems that will require more than just mathematical insights to solve. Although mathematics will help you arrive at elegant and efficient methods, the use of a computer and programming skills will be required to solve most problems.
The motivation for starting Project Euler, and its continuation, is to provide a platform for the inquiring mind to delve into unfamiliar areas and learn new concepts in a fun and recreational context.
The intended audience includes students for whom the basic curriculum is not feeding their hunger to learn, adults whose background was not primarily mathematics but had an interest in things mathematical, and professionals who want to keep their problem solving and mathematics on the edge.

References: Problem-based Learning

Gardner, Howard (2002). An interview. Retrieved 4/6/09 from http://www.pz.harvard.edu/PIs/SteenNepperLarsenInterviewJan2002.pdf. See the discussion about an education for interdisciplinary problem solving .

Roh, Kyeong Ha (2003). Problem-Based Learning in Mathematics. ERIC Digest. Retrieved 10/15/08 from http://www.ericdigests.org/2004-3/math.html. Quoting from the short article:

Since Problem-based Learning starts with a problem to be solved, students working in a PBL environment must become skilled in problem solving, creative thinking, and critical thinking. Unfortunately, young children's problem-solving abilities seem to have been seriously underestimated. Even kindergarten children can solve basic multiplication problems (Thomas et al., 1993) and children can solve a reasonably broad range of word problems by directly modeling the actions and relationships in the problem, just as children usually solve addition and subtraction problems through direct modeling.
Those results are in contrast to previous research assumptions that the structures of multiplication and division problems are more complex than those of addition and subtraction problems. However, this study shows that even kindergarten children may be able to figure out more complex mathematical problems than most mathematics curricula suggest. PBL in mathematics classes would provide young students more opportunities to think critically, represent their own creative ideas, and communicate with their peers mathematically.
...
Problem-Based Learning (PBL) describes a learning environment where problems drive the learning. That is, learning begins with a problem to be solved, and the problem is posed is such a way that students need to gain new knowledge before they can solve the problem. Rather than seeking a single correct answer, students interpret the problem, gather needed information, identify possible solutions, evaluate options, and present conclusions. Proponents of mathematical problem solving insist that students become good problem solvers by learning mathematical knowledge heuristically.
Students' successful experiences in managing their own knowledge also helps them solve mathematical problems well (Shoenfeld, 1985; Boaler, 1998). Problem-based learning is a classroom strategy that organizes mathematics instruction around problem solving activities and affords students more opportunities to think critically, present their own creative ideas, and communicate with peers mathematically (Krulik & Rudnick, 1999; Lewellen & Mikusa, 1999; Erickson, 1999; Carpenter et al., 1993; Hiebert et al., 1996; Hiebert et al., 1997).

PBLNetwork. The Illinois Mathematics and Science Academy (IMSA) established the PBLNetwork in 1992. The network engages in PBL professional development, research, information exchange, curriculum development and networking in K-16 educational settings. Retrieved 10/6/08 from http://pbln.imsa.edu/. The goals of IMSA's PBLNetwork are:

To mentor educators in all disciplines as they design and develop effective problem-based learning (PBL) materials and become skillful coaches in K-16 classrooms and other educational settings.
To explore problem-based learning (PBL) strategies as the context in which knowledge is acquired, ethical decision-making is nurtured, and problem-solving skills are developed with learners of all abilities.
To connect problem-based learning (PBL) educators through numerous networking options designed to meet a variety of needs.

Problem-based Learning in Mathematics: Researching PBL. Retrieved 10/6/08 from http://www.polaris.edu/iltli/math_pbl.htm

Chicago IMP (n.d.). The Interactive Mathematics Program.

A Model for Mathematics Education Reform. Retrieved 10/6/08 from http://www.math.uic.edu/~cpmp/. Quoting from the site:

The Interactive Mathematics Program (IMP) is a four-year, problem based mathematics curriculum for high schools. IMP is designed to meet the needs of both college-bound and non-college-bound students.
The fundamental principles of IMP reflect major changes currently taking place in mathematics education. National reports issued by experts and professional organizations recognize that existing ways of teaching mathematics are inadequate in meeting the needs of today's public school students.
A Problem-Based Mathematics Curriculum 
IMP is designed to replace the four-course sequence typical of most high school mathematics programs with an integrated course. While the traditional Algebra I-Geometry-Algebra II/Trigonometry-Precalculus structure emphasizes rote learning of mathematical skills and concepts, the IMP curriculum is problem-based, consisting of four-to-eight-week units that are each organized around a central problem or theme. Motivated by this central focus, students solve a variety of smaller problems, both routine and non-routine, that develop the underlying skills and concepts needed to solve the central problems in that unit. 


Especially for Teachers of Teachers

This section contains some math project-based learning activities that can be used with students in preservice and inservice math education classes or workshops. Each activity begins with whole class instruction and discussion. Then a project-based learning assignment is given to teams of one or more students. Each teams finds and/or develops materials and resources that are designed to be used with specified categories of students and that are shared with the whole class. For example, teams might be based on grade level, with teams for grades K, 1, 2, and so on. Teams might be based on categories of students within a grade level—for example, based on the math maturity or math learning needs of categories of students.

Math Puzzles for Problem-based Learning

This assignment begins as a whole class activity in which there is general discussion on what constitutes a good math puzzle to be used in teaching and learning math. Some possible characteristics include:

  • The rules are relatively easy to learn. Thus, it is easy to get started working on the puzzle.
  • Doing a puzzle helps one gain in math maturity, knowledge, and skills.
  • The puzzle comes in a number of variations and difficulty levels.
  • The person doing the puzzle can provide feedback to himself or herself on the possible correctness or usefulness of a particular step (move) or sequence of steps (moves).
  • There are one or more relatively obvious and important transfer of learning opportunities.

Math Problems for Problem-based Learning

This assignment begins as a whole class activity in which there is general discussion about what constitutes a good math problem to be used in math problem-based learning. Some possible characteristics include:

  • The problem is relatively easy to understand.
  • The problem can be understood by a wide range of students—that is, by students with diverse math backgrounds.
  • The problem is solvable by a variety of methods. Some may be more efficient or "elegant" than others.
  • A student working on the problem can see when progress is occurring.
  • It is not easy to look up a solution from readily available resources such as books and the Web.
  • Work on solving the problem contributes to increasing math maturity and draws upon a wide range of a students' backgrounds.

What constitutes a good problem for problem-based learning? What does one learn through working on such problems versus working with the "exercises" in a typical K-12 math textbook? What is the difference between a problem and an exercise?

Here is a major challenge in math education. In a math assignment, most students think that "the goal" is to get "the answers." Actually, that is not the case. "The" goal is to learn math.

Consider, for example, the four 4s problem given earlier in this document. A Google search of four fours math problem produced thousands of hits. When this search was done on 10/18/08, the first hit was an article, Solutions from 0 to 116, not including 113 which is probably not findable by the rules given.

More generally, the accumulated math literature can be thought of as an accumulation of solutions to math problems (including, or course, proofs of math theorems). As more and more of this accumulated knowledge is made available on the Web or in other searchable databases, it becomes easier to solve a math problem by merely retrieving a solution.

This is a little like the situation of human chess playing versus computer chess playing. Computers can now readily defeat the best of human chess players. However, chess remains a very popular game. Variations of this game now include human versus human, human plus computer versus human plus computer, human versus computer, and computer versus computer.

See videos at http://www.metacafe.com/watch/798368/extraordinary_way_to_solve_math_problem/. Solving a problem and making a proof are closely related ideas. In this case, a method for solving a problem (doing a computation) is given. Prove that it works.

Math Journal Writing

The journaling is done after one does a math problem, exercise, or activity. Thus the journal writing can be thought of as a continuing project. This ties together project-based and problem-based math learning. See http://math.about.com/library/weekly/aa123001a.htm. Quoting from the article:

Journal writing can be a valuable technique to further develop and enhance your mathematical thinking and communication skills in mathematics. Journal entries in mathematics provide opportunities for individuals to self-assess what they've learned. When one makes an entry into a math journal, it becomes a record of the experience received from the specific math exercise or problem solving activity. The individual has to think about what he/she did in order to communicate it in writing; in so doing, one gains some valuable insight and feedback about the mathematical problem solving process. The math no longer becomes a task where by the individual simply follows the steps or rules of thumb. When a math journal entry is required as a follow up to the specific learning goal, one actually has to think about what was done and what was required to solve the specific math activity or problem.
Math instructors will also find that math journaling can be quite effective. When reading through the journal entries, a decision can be made to determine if further review is required. When an individual writes a math journal, they must reflect on what they have learned which becomes a great assessment technique for individuals and instructors.

See also: http://math.about.com/aa123001a.htm. Quoting from this article:

A journal should be written at the end of a math exercise [and] should contain specific details about the areas of difficulties and areas of success.

Math journals can be done with children and adults. Younger children will draw pictures of the concrete math problem they have explored.

Math journals should not be done daily; it's more important to do math journals with new concepts in areas specifically related to growth in mathematical problem solving. The entries should be in a separate book, one used specifically for mathematical thinking, and should take no more than 5-7 minutes.

Be patient, math journaling takes time to learn. It is critical to understand that math journaling is an entry of the mathematical thinking processes. There's no right or wrong way of thinking!

Moody's Mega Math Challenge

The Moody's Mega Math Challenge problem is a open-ended, realistic, applied math modeling problem focused on a real-world issue. Quoting from the website:

The purpose of the Moody's Mega Math Challenge, in addition to being a contest for the best, brightest and most creative minds, is to elevate high school students' enthusiasm and excitement about using mathematics to solve real-world problems and to increase students' interests in pursuing math-related studies and careers in college and beyond. Moody's and SIAM are interested in improving the pipeline of young people into studies and careers in applied mathematics, and encourage students to participate in this contest as an educational process.

The website contains sample challenge problems from previous years. Here is some information provided on 3/3/09:

Instead of sleeping in this weekend like most American teenagers, 2086 high school juniors and seniors will get up before 7:00 a.m. on Saturday and Sunday—to do math. Why? How does winning a share of $80,000 in scholarship prizes sound?
Four hundred sixty-six teams, each consisting of three to five students, will gather in kitchens, libraries, cafés, and classrooms—actually anywhere they choose—to solve an applied math-modeling problem based on a real-world issue. The topic is entirely unknown until they download the problem at 7:00 a.m. on March 7 or 8, whichever day they selected at registration. Using any free, publicly available, and inanimate sources of information to help them, teams will have until 9:00 p.m. that same night to research the problem, formulate assumptions, develop and test a model, analyze their findings, and summarize their response in a solution paper, which they will upload to the Challenge website. The goal of this entirely Internet-based Challenge is to increase interest in and encourage high school students to pursue math-related studies and careers.

It is important to note the bolded section above. The contest is shaped like a real-world, but time limited problem-solving challenge.

Author or Authors

The original version of this document was developed by David Moursund.