Mathematically Sane

In the Trenches: Three Teacher’s Perspectives on Moving Beyond the Math Wars

By Susan Brown, Antoinette Seidelmann, and Gwendolyn Zimmermann

Why can’t they remember how to graph a line? It’s January, and we are beginning the unit on systems of equations. Graphing is one way my students are expected to solve a system of equations, but they can’t even graph a line. We spent how many weeks on graphing linear equations back in November? So why does it seem as if they’ve never graphed a line before?

Does this vignette sound familiar? It describes what we experienced as beginning high school teachers who used a traditional approach. Although we had drilled and drilled on graphing lines, students could not apply the skill in a new situation. In fact, many lost the ability to graph an equation altogether. The students were frustrated and so were we. But since that time our teaching has changed. We now teach less through “teacher talk” and more through “student solve;” that is, we base our instruction on the students’ thinking and ability to solve problems. And we feel that the result is that our students learn more. They have a better understanding, and they are more able to apply their knowledge and build on it.

In today’s debate about how mathematics should be taught, it seems that the voice of the teacher is often unheard. We would like to add our voices to the discussion. But what we offer goes beyond the viewpoint of classroom teachers because we are also researchers. Thus, we bring a unique perspective that links our experiences in the classroom with our knowledge of current research on mathematics education.

For us, the key issue in the whole debate about how mathematics should be taught centers on the balancing and sequencing of conceptual knowledge and procedural knowledge. We think it is a key issue because it impacts teachers’ decisions in the classrooms, thus affecting what students learn. This is also a divisive issue in the current debate about how mathematics should be taught and we want to offer our views on the subject.

Two Approaches to Teaching

Teaching for procedural knowledge means teaching definitions, symbols, and isolated skills in an expository manner without first focusing on building deep, connected meaning to support those concepts (Skemp, 1987). Teaching for conceptual understanding, on the other hand, begins with posing problems that require students to reason flexibly. Through the solution process, students make connections to what they already know, thus allowing them to extend their prior knowledge and transfer it to new situations (National Council of Teachers of Mathematics, 2000).

Teaching for Procedural Knowledge: A Direct Instruction Approach

Hung-Hsi Wu (1999), a university mathematics professor at Berkley who is often critical of reform mathematics education, believes that “the subject of mathematics is a logical unfolding of ideas starting with clear and precise definitions and assumptions” (p. 16). He also says instruction should begin with “the proper infusion of precise definitions, clear explanations, and symbolic computations” (p. 17). His statements are congruent with our definition of teaching for procedural knowledge. Wu implies that definitions and logical deduction obviously lead to understanding. Another mathematician, David Ross (2001), writes “The best way to advance students’ conceptual thinking about mathematics is to have them master the traditional algorithms.” He believes that students who have learned a computational process will then go on to understand the mathematical principles that underlie it.

Direct instruction is the predominant teaching method that critics of reform espouse. Another critic of reform, Frank Allen (1996), defines a secondary school teacher as “one who explains.” According to Wu, “Children always respond to reason when it is carefully explained to them” (1999, p. 19). Students learn by hearing, and it is the teacher’s job to transmit that knowledge. With this approach, teaching students to graph a linear equation, such as y = 2x + 15, is primarily a matter of laying out a careful step-by-step procedure, demonstrating the procedure to the class, having students practice the procedure in class, and then assigning more practice for homework. Assessing students’ understanding is a matter of checking to see if their lines are in the right place. In the past, we have had some students who have been able to draw meaning from this way of learning. However, most stop at the stage of gaining a minimum level of proficiency with the routine skill, which is all too often a temporary attainment. Skemp (1987) would refer to this kind of approach as learning “rules without reasons” (p. 86).

Direct instruction is not our preferred teaching strategy. In fact, we agree with James Fey (1999), who says “to suggest that students get only correct and complete conceptions from teacher exposition is to deny extensive research evidence of student misconceptions that have developed from traditional teaching experiences.” Fey’s comment resonates with us because it reinforces both our teaching and research experiences. Algorithms themselves do not necessarily connect in students’ minds with larger ideas. Perfecting and polishing those procedures may never address underlying mathematical misconceptions and gaps.

Teaching for Conceptual Knowledge

Teaching for conceptual knowledge means that students must make sense of mathematics. Hiebert et al. (1997) say we understand something if we see how it is related to other things we know. For example, students who have a conceptual understanding of graphs of linear equations will be able to connect an equation to specific ordered pairs that satisfy it. They will also be able to draw on experiences with equations that describe real world situations. The equation y = 2x + 15 might represent the cost of renting videos, where y represents the cost of renting x videos from a club with a $15 membership fee. By providing students with a variety of problems, ideas such as slope as a rate of change become more accessible. Students can relate the varying quantities to their personal experiences. How does this differ from the traditional way of ending chapters with application problems? The difference is fundamental. For conceptual learning the applications are often the beginning point, and the mathematics is drawn from them. They are not merely a place for applying previously mastered skills, time permitting, as is the case in a more traditional setting.

Proponents of direct instruction (Wu, Ross, and Allen) claim that students’ experiences with new mathematical ideas should begin with definitions and theorems. But while these may be good starting points for mathematicians, they do not provide the examples and experiences that students need before working with abstractions. Definitions themselves are exact and precise but convey limited meaning. Tall (1992) differentiates between the mathematical definition of a concept and the concept image, which is the entire cognitive structure that a person has formed related to the concept. This concept image is made up of pictures, examples and non-examples, processes, and properties. A strong concept image is a rich, integrated, mental representation that allows the student to flexibly move between multiple formulations and representations of an idea. A student who has connected mathematical ideas in this way can create and use a model to analyze a situation, uncover patterns and synthesize them to form an integrated picture. They can also use symbols meaningfully to describe generalizations. On the other hand, a student with a narrow concept image has trouble applying it and is blocked when a situation does not exactly fit prior experience. Ideas are separate and disjointed and flexibility is lacking (Dreyfus, 1991). Our teaching experience shows us that textbook definitions are barren and a poor starting point. In contrast to direct instruction, teaching for conceptual knowledge begins with problems and asks students to uncover patterns and relationships. The generalizations, definitions, and symbolic descriptions follow. This process produces rich, interconnected structures that students can use for thinking.

Conceptual understanding does not “just happen.” Lessons have to be carefully designed so that students have experiences that will help them make connections. Tasks must be chosen with a particular mathematical idea as a focus and they must connect closely to students’ prior knowledge. A fruitful task allows the development of multiple strategies and the use of more than one representation. This way, students are asked to explain their thinking, compare their methods, and justify their results. The focus should not only center on the correct answer, but on the valid reasoning and vital connections elicited. The teacher’s role is not to tell; rather it is to raise questions that help students bridge the gap between what they already know and the new ideas they are developing. In essence, teachers need to lead summative class discussions, making sure that meaningful connections are made, while continually pushing students toward explanation and meaning (Hiebert & Carpenter, 1992; Hiebert et al., 1997; Stein, Grover, & Henningsen, 1996).

Often critics will accuse reform-minded teachers of sacrificing procedures for conceptual understanding, or even more strongly, of saying practice and procedures are unnecessary. As reform-minded teachers, we value both procedural and conceptual knowledge. Wu, a critic of reform, agrees both are important but goes beyond that to say “the standard algorithms embody conceptual understanding” (1999, p. 19). In other words, he believes as one is learning procedures, he or she is building conceptual understanding. The research on achievement data (Kouba, Zawojewski, & Strutchens, 1997; U.S. Department of Education, 1996) and our experiences in the classroom demonstrate conclusively that this is not the case.

Procedural or Conceptual: Which Comes First?

Several studies have shown that learning procedures actually interferes with the development of meaningful knowledge if a solid conceptual foundation has not been laid (Carpenter, Franke, Jacobs, Fennema, & Empson, 1998; Mack, 2001; Pesek & Kirshner, 2000). In the study by Pesek and Kirshner, students were divided into two groups for a unit on area and perimeter. One group (PF) studied procedures first, learning to use and compute with formulas. Next, this group of students received conceptually oriented instruction, with an emphasis on developing relationships. No formulas were given. Instead, students were assisted in developing their own methods to solve problems. The second group of students (CO) received only the conceptually oriented instruction, spending less than half the time that the PF group did studying area and perimeter.

The study found that the CO group, whose instruction focused exclusively on relationships and concepts, demonstrated better understanding of area and perimeter than the PF students, who were taught formulas first and then the concepts. The PF students could not articulate the difference between area and perimeter in the context of a real life situation. When asked which measurement, area or perimeter, would be needed to determine the amount of wallpaper needed to decorate a room, most of the PF group chose perimeter because, as one student explained, “Walls don’t have area because they go around” (p. 535). However, all students in the CO group understood that area was needed to solve the wallpaper problem. When students were asked to explain how area and perimeter formulas work, the PF students spoke about the processes in terms of computations. They could not explain why the formulas worked, but many in the CO group were able to make sense of the formulas even though they had not received explicit instruction on them.

Compared to their weaker classmates, stronger students in the PF group were more easily able to overcome the negative effects of learning procedures first. Thus, the evidence shows that the direct instruction method that traditionalists think will help weaker students is actually handicapping them! This finding has important implications for the classroom as we strive to reach more students with mathematics.

The Pesek and Kirshner study clearly illustrates the potentially detrimental effect of teaching for procedural knowledge prior to teaching for conceptual understanding. Procedural knowledge does not easily transfer to conceptual knowledge. In contrast, by working with meaning and concepts first, students form a much richer knowledge base. Moreover, they are able to apply that knowledge and give meaning to the formulas and computations.

What does this mean for us as teachers? It tells us how to approach instruction. On the one hand, if we want our students to have a conceptual understanding of mathematics, we need to teach conceptually. On the other hand, a rapid push for procedural skill will actually do more harm than good. In other words, if skills are taught before the concepts are developed and connected to the understanding students bring to the topic, students are likely to struggle to develop conceptual knowledge.

As teachers we know that teaching for conceptual understanding can be challenging. However, we believe it is imperative for all teachers to examine their teaching practices and assess the extent to which they are helping students make sense of mathematics.

Conclusion

It may appear that this paper is addressing two issues, what mathematics students should learn (procedural skills versus conceptual knowledge) and how mathematics should be taught (direct instruction versus reform-based approaches.) We would argue that there is no real debate about what students need to learn; they need the skills and they need the concepts. The issue is about how to teach to help students learn.

We know telling does not work. It did not work for the many high school students we have seen struggle with ill-memorized rules for integer operations or trigonometric values. It did not work for the college students who came to us able to define slope without having a clue about what it meant. On the other hand, we have also seen evidence of what does work. For us, student learning occurs when teachers develop conceptual knowledge before procedural knowledge. In addition to our own compelling teaching experiences with a conceptual-procedural sequence, we have observed second and third grade students solve storybook problems that involved division; they were even able to explain what the remainder meant! We have also seen “low level” algebra students use negative exponents in the context of a problem without ever having been exposed to formal definitions.

We agree that both procedural and conceptual knowledge are important. Furthermore, to debate this issue is to argue a moot point. The key issue, as we see it, is in the manner and order in which procedures and concepts are taught. Our experiences and current research bear this out. The good news is both are attainable. Teaching first for conceptual knowledge leads to the acquisition of procedural knowledge, but the converse is not true.

References

• Allen, F. (1996). A program for raising the level of student achievement in secondary school mathematics. Retrieved June 27, 2001, from Mathematically Correct Web site. http://www.mathematicallycorrect.com
• Carpenter, T. P., Franke, M. L., Jacobs, V. R., Fennema, E., & Empson, S. B. (1998). A longitudinal study of invention and understanding in children’s multidigit addition and subtraction. Journal for Research in Mathematics Education, 29(1), 3-20.
• Dreyfus, T. (1991). Advanced mathematical thinking processes. In D. Tall (Ed.), Advanced Mathematical Thinking (pp. 25-41). Dordrecht: Kluwer.
• Fey, J. (1999). Standards under fire: Issues and options in the math wars. Retrieved June 27, 2001, from the University of Missouri, Show-Me Center Web site. http://www.showmecenter.missouri.edu/showme/perspectives/keynote1.html
• Hiebert, J., & Carpenter, T. P. (1992). Learning and teaching with understanding. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 65-97). New York: Macmillan.
• Hiebert, J., Carpenter, T. P., Fennema, E., Fuson, K., Wearne, D., Murray, H., Oliver, A., & Human, P. (1997). Making sense: Teaching and learning mathematics with understanding. Portsmouth, NH: Heinemann.
• Kouba, V. L., Zawojewski, J. S., & Strutchens, M. E. (1997). What do students know about numbers and operations? In P. A. Kenney & E. Silver (Eds.), Results from the sixth mathematics assessment of the National Assessment of Educational Progress (pp. 87-140). Reston, VA: National Council of Teachers of Mathematics.
• Mack, N. (2001). Building on informal knowledge through instruction in a complex content domain: Partitioning, units, and understanding multiplication of fractions. Journal for Research in Mathematics Education, 32(3), 267-295.
• National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics.
• Pesek, D. D., & Kirshner, D. (2000). Interference of instrumental instruction in subsequent relational learning. Journal for Research in Mathematics Education, 31(5), 524-540.
• Ross, D. (2001). The math wars [Electronic version]. Navigator, 4(5).
• Skemp, R. (1987). The psychology of learning mathematics. Hillsdale, NJ: Erlbaum.
• Stein, M. K., Grover, B. W., & Henningsen, M. (1996). Building student capacity for mathematical thinking and reasoning: An analysis of mathematical tasks used in reform classrooms. American Educational Research Journal, 33(2), 455-488.
• Tall, D. (1992). The transition to advanced mathematical thinking: Functions, limits, infinity, and proof. In D. A. Grouws (Ed.), Handbook of Research on Mathematics Thinking and Learning (pp. 495-511). New York: Macmillan.
• U.S. Department of Education. (1996). Pursuing excellence: A study of U.S. eighth-grade mathematics and science teaching, learning, curriculum, and achievement in an international context. Washington, DC: U.S. Government Printing Office.
• Wu, H. (1999). Basic skills versus conceptual understanding: A bogus dichotomy in mathematics education. American Educator, 23, 14-19, 50-52.

Note about the authors: Susan Brown, Antoinette Seidelmann, and Gwendolyn Zimmermann currently teach high school mathematics in the Chicago area. Gwendolyn Zimmermann has a Ph.D. in Mathematics Education. Susan Brown and Antoinette Seidelmann are Ph.D. candidates in Mathematics Education at Illinois State University. They collectively have 56 years of teaching experience spanning grades three through college.