About the Author:  Dr. Jennifer Bay is a mathematics educator at Kansas State University.  She teaches preservice and practicing elementary teachers.  Her research interests include curricular reform and problem solving.  Her email address is jbay@ksu.edu.

Problem solving has been an emphasis in school mathematics for longer than the current “standards-based” reform in mathematics education.  It is considered to have begun with the release of the National Council of Teachers of Mathematics (NCTM) Curriculum and Evaluation Standards for School Mathematics in 1989.  The NCTM issued An Agenda for Action in 1980, which argued that problem solving should be the focus of school mathematics.  This resulted in a significant increase in problem solving in curriculum and classrooms, in particular the direct instruction of problem solving strategies such as “make a list,” “draw a picture,” or “make a simpler problem.”  In 2000, the vision for problem solving in mathematics teaching and learning is still expanding.  Principles & Standards for School Mathematics (NCTM, 2000, p. 52) states that instructional programs from prekindergarten through grade 12 should enable all students to:

• Build new mathematical knowledge through problem solving.
• Solve problems that arise in mathematics and in other contexts.
• Apply and adapt a variety of appropriate strategies to solve problems.
• Monitor and reflect on the process of mathematical problem solving.

This stance reinforces the need for students to develop problem solving strategies, but also asserts the need for students to learn content through problem solving.

The NCTM recommendations appear to be impacting teacher practices.  State standards and textbooks have reflected, to varying degrees, the increased emphasis on problem solving (Nibbelink, Stockdale, Hoover, & Mangru, 1987).  The 1996 National Assessment of Educational Progress (NAEP) found that 52% of teachers in fourth and eighth grades reported placing “a lot” of emphasis on “developing reasoning and analytical ability to solve unique problems” (Shaughnessy, Nelson, & Norris, 1998; U. S. Department of Education, 1999).

Studying the link between problem solving and student achievement requires defining what type of problem solving instruction improves what kind of student achievement.  Problem solving instruction can be conceptualized in three different ways (Schroeder & Lester, 1989) and there are different types of student achievement to assess.  In the following sections the types of problem solving instruction and student achievement are briefly discussed.  This is followed by a discussion of the research in each of the different areas of problem solving instruction.

One enigma in linking assessment with problem solving is identifying what type of problem solving is occurring in the classroom.  Schroeder and Lester (1989) describe three ways in which problem solving is interpreted in the classroom: teaching for problem solving, teaching about problem solving, and teaching via problem solving.  In teaching for problem solving, the goal is to first teach the concepts so students will later apply that knowledge to problem solving situations.  This approach has been commonly seen in textbooks where a page of skill practice is followed by story problems that apply the same concept. 

The second approach, teaching about problem solving, is the teaching of strategies, or heuristics, in order to solve problems.  One way to teach students to problem solve is to teaching the four step processes developed by Polya (1971): understand the problem, devise a plan, carry out the plan, and look back.  Common plans for solving problems include guess and check, making an organized list, and drawing a picture.  A classic example is the Pigs and Chickens problem where students are given the number of legs and number of total animals and must determine how many pigs and how many chickens there are.  This can be solved using the plans (strategies) listed above or in other ways.  The instructional purpose of this task is for students to learn to apply problem solving strategies, not necessarily to teach the mathematics content that is in a particular curriculum guide.

Teaching via problem solving is teaching mathematics content in a problem-solving environment.  Learning in this approach involves learning through a concrete problem and eventually moving to abstraction.  For example, students might be asked to build all possible rectangles for figures numbered from 1 to 20 and note any patterns.  Once completed, students might note that 1, 4, 9, 16 are the only numbers in which they could make a square and that 11, 17, and 19 are examples of numbers that can only be built one way.  Imbedded in this problem solving activity are mathematics concepts, in particular factors, primes, and square numbers.  This final definition is the one most closely aligned with the NCTM Principles &Standards (2000). 

These different types of problem solving are critical to the discussion of problem solving and student achievement because of the potentially different impact on student achievement.  A teacher teaching for problem solving and a teacher teaching via problem solving are doing very different activities with their students.  For example, Erickson (1993) found that a teacher teaching for problem solving was using much direct instruction in her classroom while a teacher teaching via problem solving was facilitating student explorations.  In the first case there were no gains in student achievement in computation, application, or problem solving. In the latter case, student achievement significantly increased in each category.  Though this is only a comparison of two teachers, it indicates that the type of problem solving used in classrooms and the instructional approach may be a factor in raising student achievement.  Schools that choose problem solving as a school improvement goal have to be very clear about the kind of problem solving they want to impact.

In addition to defining problem solving, there is the issue of determining what type of student achievement to assess.  One focus, for example, is to study whether increased problem solving in the curriculum increases student achievement in problem solving.  Few instruments exist that are developed to measure student problem solving ability solely.  Another focus is to study the impact of problem solving on student skills and/or conceptual understandings, for example those assessed by national standardized tests like the Iowa Test of Basic Skills (ITBS) and many state level student achievement tests.

A third issue in assessing the impact of problem solving on student achievement is determining the actual problem solving that is occurring in the classroom.  It is commonly accepted that the written curriculum can vary from the taught curriculum.  Teachers' interpretations of how to teach problem solving and how much time to devote to problem solving vary (Grouws, 1996).  Even when teachers think they have aligned their practices with the NCTM Standards, there is much variance in the type of instruction that is occurring in the classroom (National Research Center, 1996; Bay, 1999).  There is some evidence that problem solving experiences have become “normal” to students in mathematics classrooms (Gay, 1999).  Research on problem solving, however, usually involves testing student achievement without considering the extent and the way in which problem solving is being implemented in the classroom.  Teachers' interpretations of how to include problem solving in the curriculum impacts the students' achievement (Erickson, 1993).  Hence, results linking problem solving to student achievement can vary greatly across classrooms.  These issues (defining problem solving, defining student achievement, and intended versus implemented curriculum) are presented in order to give some insights into why results on the effectiveness of problem solving might vary.  It also frames the following discussion of research on problem solving.

Despite the difficulties in assessing the impact of problem solving on student achievement, there is a significant body of research that shows gains in student achievement.  Research has been conducted on the impact of teaching problem solving and on teaching via problem solving, though recent work has primarily focused on the latter of the two categories.

Research on teaching problem solving has considered the impact on students' problem solving abilities as well as their mathematical understanding and skill.  Butkowski, Corrigan, Nemeth, and Spencer (1994) found that third graders who were explicitly taught problem solving strategies became better at using each strategy.  In a study of fifth and seventh graders, Charles and Lester (1984) found that students learning a process-oriented approach to problem solving scored better than their peers in problem solving.  Similarly, students using a structure-plus-writing model were more successful at problem solving over other students and the difference was more notable 10 weeks after learning the strategy (Rudnitsky, Etheredge, Freeman, & Gilbert,1995). Gains have also been found in student skill and concept development.  A longitudinal study of textbooks from the 1950s through the 1980s indicated that the years when problem solving was emphasized in textbooks, student achievement on the ITBS showed parallel gains (Nibbelink, et al. 1987).  Hoffer and Gamoran (1993) studied the impact of various instructional approaches on student achievement.  They found that one of the three main determinants was emphasis on problem solving, and this was particularly effective with low- and middle- ability groups.  Research findings support that teaching problem solving does positively impact student achievement in problem solving and in skill and conceptual development.

Textbooks play an important role in what is taught in mathematics classrooms (Cohen & Hickman, 1998; Edwards, 1995; Willoughby, 1990).  Much recent research in linking problem solving to student achievement is related to standards-based curricula (textbooks).  Standards-based curricula were developed by mathematics educators, mathematicians, and teachers with National Science Foundation (NSF) funding.  The curricula are “problem-based,” meaning mathematics content is presented in problem solving situations (teaching via problem solving):

There are two needs in developing . . .curriculum.  One is to find appropriate, engaging problems . . . .The other is to develop a pedagogy in which the emphasis is on the development of a mathematical frame of mind.  The focus for young children, as in later mathematics, must be on thinking and reasoning mathematically. (Russell, 1993, p. 187)

Two studies report gains in student achievement in proportional reasoning.  Flowers and Kline (1998) found that fourth graders in a standards-based curriculum (Investigations in Data, Number, and Space) improved in skills, concepts, and problem solving.  In a second standards-based curriculum (Everyday Mathematics) and in a traditional curriculum there were gains in skills and problem solving, but not in concepts.  Flowers and Kline attribute the gains in conceptual understanding to the fact that Investigations encourages invented strategies, which involves more reasoning and problem solving on the part of students.  In a study focused on addition and subtraction skills of second graders, students using Investigations were significantly more accurate than students in a traditional curriculum in addition and subtraction problems (Mokros, Berle-Carman, Rubin, O'Neil, 1996).  In seventh and eighth grade, students using Connected Mathematics Project (CMP) and traditional textbooks were tested in several areas related to proportional reasoning (Ben-Chaim, Fey, Fitgerald, Benedetto, & Miller, 1997).  CMP students scored significantly higher on a proportional reasoning test in both their willingness to explain their thinking and getting the answers right.  This sampling of studies provides evidence that problem-based curriculum improves not only students' problem solving abilities but also their conceptual understanding and skills.

In 1994, the U.S. Congress mandated a study of curriculum that resulted in improved student achievement, emphasizing the need for research-based programs.  The expert panel encouraged all curriculum developers to submit their textbooks for review.  The result was a document identifying “exemplary” and “promising” programs (U.S. Department of Education, 1999).  The criteria for being an “exemplary” or “promising” program was based on (a) the quantity and quality of research that demonstrated gains in student achievement and (b) the inclusion of students of diverse economic and cultural backgrounds.  For example, the Core-Plus Mathematics Project was rated an exemplary secondary mathematics project.  This curriculum is based on the rationale that mathematics concepts should be developed in the context of modeling real-world problems.  One study showed increases in students' ITBS scores after one year of Core-Plus, and in subsequent years Core-Plus students gained 10 points per year, whereas control group gains were 4-6 points per year.  Core-Plus twelfth-graders scored higher on 1990 and 1992 NAEP items in all content areas.  Other studies showed increased achievement for girls, gains in confidence, and more positive attitudes regarding the value of mathematics.  Each of the exemplary and promising programs describes similar studies in which students are performing better than control groups (this report is available on the web: http://www.enc.org/).  In all, this provides a significant amount of research supporting the implementation of curriculum that employs problem solving as a vehicle through which mathematics is learned.  These data indicate that teaching via problem solving results in gains in student achievement in mathematical problem solving, concepts, and skills.

National trends in student achievement in mathematics support the continued emphasis on problem solving.  In recent years, when problem solving has been emphasized in mathematics instruction, the National Assessment of Education Progress (NAEP) has shown gains in student achievement in the United States (NAEP, 1992, 1994, 1996).  The Third International Mathematics and Science Study (TIMSS), however, shows that students in the United States still score well below the international average in eighth grade mathematics (U.S. National Research Center, 1996).  A closer look at Japan, a high-achieving country, finds students learning mathematics via problem solving.  “It seems clear that Japanese teachers come closer to implementing the spirit of current ideas advanced by American reformers than do American teachers” (NRC, 1996, p. 7).  NAEP and TIMSS indicate that a curriculum emphasizing problem solving does have a positive impact on student achievement and that the U.S. needs to continue its efforts to integrate problem solving into instruction.

Problem solving is, and will continue to be, an important consideration in the teaching of mathematics.  Research indicates that teaching problem solving and teaching via problem solving improve students' problem solving, skills, and concepts.  As school districts continue to consider the role of problem solving in their curriculum, it is important to consider the type of problem solving that is to be developed and to provide support to teachers to implement those strategies.  The process of increasing problem solving in the classroom parallels Polya's (1971) four-step process to solving problems.  The first step, understanding the problem, includes being familiar with the research and with what is already happening in classrooms.  Then, a focus for increasing problem solving must be determined.  For example, if problem solving has not been emphasized, one focus might be a district-wide emphasis on the four-step approach to problem solving or on the use of one explicit approach to teaching problem solving, such as those referred to in the research on teaching problem solving.  If teachers are already doing this type of problem solving, the focus might be on teaching via problem solving.  Once the focus is determined, implementing the plan begins. If the focus is on teaching via problem solving, then reviewing, piloting, and adopting NSF-sponsored mathematics curriculum are viable considerations.  Textbooks have a significant impact on what teachers do in the classroom.  Each curriculum development site offers professional development in a variety of formats.  More information on these curricula is available from the following web sites:

http://www.edc.org/mcc/ (K-12 Implementation Center)
http://www.arccenter.comap.com/ (elementary)
http://showmecenter.missouri.edu/ (middle school)
http://www.ithaca.edu/compass/ (secondary)

At this stage, professional development and ongoing support is critical (Bay, Reys, Reys, 1999).  Implementing a problem-based curriculum is a slow and challenging process.  Sufficient time must be given to teachers to become familiar with the new curriculum and learn new instructional approaches.  Students also need time to become comfortable with a new approach to learning mathematics.  Step four of Polya's process is reflection.  As teachers implement problem solving in the classroom, regardless of what the specific approach is, reflecting on teaching practice and on student achievement locally is essential.  Selecting, implementing, and refining strategies for increasing problem solving in mathematics classrooms is a major endeavor, however, research indicates that this process (implementing new practice and a problem-based approach to teaching mathematics) offers much promise in improving student achievement in mathematics.

References

Bay, J. M. (1999). Middle school mathematics curriculum implementation: The dynamics of change as teachers introduce and use standards-based curricula. Unpublished doctoral dissertation.

Bay, J. M., Reys, B. J., & Reys, R. E. (1999).  The top 10 elements that must be in place to implement standards-based mathematics curricula.  Phi Delta Kappan, 80(7), 503 - 506.

Ben-Chaim, D., Fey, J. T., Fitgerald, W. M., Benedetto, C., Miller, J. (1997).  A study of proportional reasoning among seventh and eighth grade students—a short report.

Butkowski, J., Corrigan, C., Nemeth, T., and Spencer L. (1994). Improving student higher-order thinking skills in mathematics. Unpublished Master's thesis. Wheeling, IL.

Charles, R. I. & Lester, F. K. (1984). An evaluation of a process-oriented instructional program in mathematical problem solving in grades 5 and 7. Journal for Research in Mathematics Education, 15(1), 15-34.

Cohen, S. B. & Hickman, P. (1998, April). Paper presented at the annual meeting of NARST, San Diego, CA.

Edwards, T. G. (1995, October). Cooperative learning in response to an innovative curriculum as a manifestation of change in teaching practice. Paper presented at the annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, Columbus, OH.

Erickson, D. K. (1993, April). Middle school mathematics teachers' views of mathematics and mathematics education, their planning and classroom instruction, and student beliefs and achievement.  Paper presented at the annual meeting of the American Educational Research Association, Atlanta, GA.

Flowers, J. & Kline, K. (1998, April). A comparison of fourth graders' proportional reasoning in reform and traditional classrooms. Paper presented at the annual meeting of the American Educational Research Association, San Diego, CA.

Gay, A. S. (1999). Is problem solving in middle school mathematics “normal”? Middle School Journal, 31(1), 41- 47.

Grouws, D. A. (1993). Critical issues in problem solving instruction in mathematics. In D. Zhang, T. Sawada, & J. P. Becker (Eds.), Proceedings of the China-Japan-U.S. seminar on mathematical education (pp. 70-93).  Carbondale, IL: Board of Trustees of Southern Illinois University.

Hoffer, T. B., & Gamoran, A. (1993, April). Effects of instructional differences among ability groups on student achievement in middle-school science and mathematics.  Report Center on Organization and restructuring of Schools. University of Wisconsin, Madison, WI.

Mokros, J., Berle-Carman, M., Rubin, A., O'Neil, K. (1996, April). Learning operations: Invented strategies that work. Paper presented at the annual meeting of the American Educational Research Association, New York, NY.

National Council of Teachers of Mathematics (1980). An agenda for action: Recommendations for school mathematics of the 1980's. Reston, VA: author.

National Council of Teachers of Mathematics (1989). Curriculum and evaluation standards for school mathematics. (1st ed.). Reston, VA: author.

National Council of Teachers of Mathematics (2000). Principles and standards for school mathematics. Reston, VA: author.

Nibbelink, W. H., Stockdale, S.R., Hoover, H. D. & Mangru, M. (1987). Problem solving in the elementary grades: Textbook practices and achievement trends over the past 30 years. Arithmetic Teacher, 35(1), p. 34 – 37.

Polya, G. (1971). How to solve it. Princeton, N. J.: Princeton University Press.

Rudnitsky, A., Etheredge, S., Freeman, S. J. M., & Gilbert, T. (1995). Learning to solve addition and subtraction word problems through a structure-plus-writing approach. Journal for Research in Mathematics Education, 26(5), 467-86.

Russell, S. J. (1993). Changing the elementary mathematics curriculum: Obstacles and challenges. In D. Zhang, T. Sawada, & J. P. Becker (Eds.), Proceedings of the China-Japan-U.S. seminar on mathematical education (pp. 174-189).  Carbondale, IL: Board of Trustees of Southern Illinois University.

Shaughnessy, C. A. Nelson, J. E., & Norris, N. A. (1998).  NAEP 1996 mathematics cross-state data compendium for the grade 4 and grade 8 assessment. Washington, D.C: National Center for Educational Statistics.

Schroeder, T. L. & Lester, F. K. (1989). Developing understanding in mathematics via problem solving. In Trafton, P. R., & Shulte, A. P. (Eds.) New directions for elementary school mathematics. 1989 yearbook. Reston, VA: NCTM.

U.S. Department of Education Office of Educational Research and Involvement (1999). Student work and teacher practices in mathematics. Washington, D.C: National Center for Educational Statistics.

U.S. Department of Education (1999). Exemplary & promising mathematics programs. Washington, D. C: author.

U.S. National Research Center (1996, December). Third International Mathematics and Science Study. Report #7. Washington, D.C: author.

Willoughby, S. S. (1990). Mathematics education for a changing world. Alexandria: Association for Supervision and Curriculum Development.