Math Problem-based Learning

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Introduction

The target audience for this document is preservice and inservice K-12 teachers who teach math, and teachers of such teachers. This document provides an introduction to uses of Problem-based Learning (PBL) in math education.

The website Math Project-based Learning is a companion IAE-pedia document that covers Project-based learning (PBL). As the Venn diagram given below indicates, Problem-based Learning and Project-based Learning overlap. Since they are both abbreviated PBL, and they overlap, it is easy to confuse the two.

PBL Venn.jpeg


K-12 teachers of math need to know about both project-based learning and problem-based learning.

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 receiving.

The main focus in this document is on problem-based learning, where 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 a homework assignment.

Definition

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.

To help you understand the difference between math problem-based learning and math project-based learning, two examples are given below.

Example of Math 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, one also can 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 4/30/2014 Google search of four fours math problem produced over 28,000 hits. Click here if you want to see solutions to a very broad generalization of the problem.

Example of Math Project-based Learning

Here is an example of math project-based learning. In this example, 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 Problem-based Learning Is a Large Topic

A 4/30/2014 Google search of math problem-based learning produced over 51 million hits. A Google search of math project-based learning produced over 47 million hits. There is a substantial overlap between the two lists of hits.

For the most part, the publishers of math texts used at the K-12 levels place considerable emphasis on PBL, but do not attempt to make a clear distinction between Problem-based and Project-based Learning.

Math Problems and Math Exercises

Precollege math textbooks typically contain sets of exercises. A section in a chapter discusses a quite specific problem-solving topic and gives examples of how to solve that type of problem.Then students are asked to practice solving this type of problem over and over again.

For example, the problem may be, "Given the coordinates for two different points on a straight line in a plane, find the equation of the line." The text "teaches" how to solve this problem. Students are then given a set of "exercises" in which they are asked to find the equations for various sets of points. While students and their teacher may call these "problems to be solved," they are more appropriately called exercises. The assignment to do a bunch of these exercises is not math problem-solving. (Look back at the definition given earlier.)

Here are four examples of math problems related to the equation of a line problem given above.

1. Given the coordinates of three different points in a plane, how can one determine if they all lie on one line or if they form the vertices of a triangle?

2. Given the coordinates for the end points of segment AB and of segment CD, how can one tell if both segments lie on the same line? How can one tell if the line segments intersect but do not lie on the same line? How can one tell if the line containing the segment AB intersects with the line containing the segment CD?

3. Given the coordinates of the vertices of two different triangles in a plane, how can one tell if the triangles intersect each other of if one is completely inside the other?

4. Suppose that A and B are two distinct points on a sphere. Is there more than one line segment on the sphere that connects the two points? Of course, this problem is not a "plane geometry" problem. What do students who are studying plane geometry know about other types of geometry? Is it useful to "stretch their brains" with questions such as this one concerning lines on a sphere?

Think about what math knowledge and skills a student might need to use in attacking these problems. What math might a student learn by working on these problems?

Some Examples of Geoboard Problem-based Learning

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 three useful references:

Math Learning Center (n.d.). Free Math Apps. Retrieved 4/30/2014 from http://www.mathlearningcenter.org/blog/topic/free-math-apps.
CT4ME (n.d.). Math Manipulatives. Computer Technology for Math Excellence. Retrieved 4/30/2014 from http://www.ct4me.net/math_manipulatives.htm.
NCTM (n.d.). Interactive Virtual Geoboard. Retrieved 4/30/2014 from http://standards.nctm.org/document/eexamples/chap4/4.2/index.htm#applet.

Problem: 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?

Problem: Make an equilateral triangle on a geoboard. 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?

People like myself, who sometimes become quite nitpicking, will note that the problem is not carefully stated. It does not state, "Use only the pegs and rubber bands." So, I can solve the problem by taking a marking pen and drawing an equilateral triangle on the geoboard. The intent, of course, is to try to make an equilateral triangle that has as vertices any three pegs on the geoboard.

A somewhat more challenging problem. Can one make two triangles that each use three geoboard pegs as vertices, placed so that the two triangles overlap in an equilateral triangle?

An Example of a Problem-based Learning Assignment

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?

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 methods may be more efficient or "elegant" than others.
  • A student working on the problem can recognize 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 students' backgrounds.

Here are three possible problems:

1. Identify some "so-called" math problems in the math textbook. Analyze their strengths and weaknesses in satisfying the list of characteristics given above.

2. Make up one or more "good" math problems that draw on the math topics that have been covered in the class in the past week. What math from the previous week does a student learn by solving this problem? What math from earlier in the course is used in solving the problem? Is it possible to solve the problem by looking it up in the textbook or on the Web?

3. Analyze some of the word problems ("story problems") in the text. What distinguishes a word problem from an exercise? Give examples from the text that you consider to be good "word problem" problems. Find some that seem to be relatively routine exercises. Do math journaling on this problem/task. (See Math Journal Writing in the the next section.)

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. Does this mean that when computers are available and students are skilled in math information retrieval, then more and more of the problems they encounter in their assignments can be considered to be merely information retrieval problems—where the retrieval can be done from their brains, from their math book, or from the Web?

Math Journal Writing

We want students to learn math communication skills. This includes oral communication as well as reading and writing.

After a student solves (or fails to solve) a challenging math problem, the student can write about what worked, what didn't work, what was learned, and so on.

Note that this activity is also useful in math project-based learning. This can tie together project-based and problem-based math learning. See http://math.about.com/library/weekly/aa123001a.htm. Quoting from this 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 some snippets 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.
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!

Writing Computer Programs to Solve Math Problems

If you are faced by a simple math task of determining 3,897 times 374, you can do the exercise by use of a memorized paper and pencil algorithm, by use of a calculator, or by use of a computer. In using a paper and pencil algorithm, the challenge is to remember the algorithm and to apply it without making any errors. With a calculator or computer, the challenge is to avoid making key press errors. Regardless of the approach, you are doing an exercise—you are completing a routine math task that you know you can do.

Now, suppose that you are faced by a somewhat more complex task. For each teacher in a (hypothetical) school, you are given information that can be used to determine what "step" each teacher is placed at on the pay scale and the amount of their current salary. You are given information that will allow you to determine the medical and retirement benefits for each teacher for the next year. You also have information about each teacher as to whether they will be moving up a step at the start of the next school year, and the amount of a step increase. Finally, you know that the current contract calls for a 2% cost of living increase for all teachers. What will be the total cost to the school district for these same teachers for the next year?

This problem can be solved using arithmetic that students "are supposed to have" mastered by the end of the sixth grade or so. Thus, the challenges in the problem lie in figuring out what to calculate, and doing the calculations without error. But, most people no longer do such calculations by hand or with a calculator. Instead they use computer programs that someone (perhaps, occasionally, themself) has written. Writing such a program requires communicating a set of directions to a computer. The thinking of what needs to be done and how to communicate with the computer is left to the human—the actual "doing it" of the calculations is left to the computer.

Think again about what I have just said. There is a "thinking" part and a "doing" part. This is true whether the "doing" is done by a human working with paper and pencil, or by a computer. Since one of the major goals in math education is to improve a student's math-oriented thinking, learning and doing computer programming is a useful vehicle in teaching math.

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.

Resources: Problem-based Learning

Gardner, H. (2002). Howard Gardner Interview [with] Steen Nepper Larson on January 30, 2002. Retrieved 4/30 2014 from http://howardgardner.net/Papers/documents/Interview%20with%20Steen%20Nepper%20Larsen.pdf. See the discussion about an education for interdisciplinary problem solving.

IMP (n.d.). The Interactive Mathematics Program. Retrieved 4/30/2014 from http://mathimp.org. Quoting from this site:

The Interactive Mathematics Program (IMP) is a growing collaboration of mathematicians, teacher-educators, and teachers who have been working together since 1989 on both curriculum development and professional development for teachers
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.

For more information about IMP see http://mathforum.org/library/ed_topics/sub_sequence/. Quoting from this site:

The Interactive Mathematics Program (IMP) - Chicago, Univ. of Illinois.
A Model for Mathematics Education Reform. A four-year, problem based mathematics curriculum for high schools, to meet the needs of both college-bound and non-college-bound students. 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.

PBLNetwork (n.d.). 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 4/30/2014 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.

Roh, Kyeong Ha (2003). Problem-Based Learning in Mathematics. ERIC Digest. Retrieved 4/30/2014 from http://www.ericdigests.org/2004-3/math.html. Quoting from the short article:

The effectiveness of PBL depends on student characteristics and classroom culture as well as the problem tasks. Proponents of PBL believe that when students develop methods for constructing their own procedures, they are integrating their conceptual knowledge with their procedural skill.
Limitations of traditional ways of teaching mathematics are associated with teacher-oriented instruction and the "ready-made" mathematical knowledge presented to students who are not receptive to the ideas (Shoenfeld, 1988). In these circumstances, students are likely to imitate the procedures without deep conceptual understanding. When mathematical knowledge or procedural skills are taught before students have conceptualized their meaning, students' creative thinking skills are likely to be stifled by instruction. As an example, the standard addition algorithm has been taught without being considered detrimental to understanding arithmetic because it has been considered useful and important enough for students to ultimately enhance profound understanding of mathematics. Kamii and Dominick(1998), and Baek (1998) have shown, though, that the standard arithmetic algorithms would not benefit elementary students learning arithmetic. Rather, students who had learned the standard addition algorithm seemed to make more computational errors than students who never learned the standard addition algorithm, but instead created their own algorithm.

Author or Authors

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