The word that best describes a constructivist mathematics classroom is energy. Young people bring tremendous energy. Rather than fight to contain it, teachers in constructivist classrooms direct this energy by engaging students actively in the learning process.
In these classrooms, students are more likely to participate in hands-on activities than to listen to lectures. They are more likely to discuss with other students their solution strategies than to ask the teacher to tell them the right one. They are more likely to work cooperatively in small groups as they shape and reformulate their conceptions than to practice mathematics rules silently at their desks. In constructivist classrooms, teachers establish interest, confidence, and a need for mathematics by capitalizing on students' energy.
Active engagement requires a classroom that looks different from a traditional mathematics classroom and contains such supplies as manipulatives, measuring devices for hands-on activities, and reference material for problem-solving activities and projects. Usually, desks are not lined up in rows. Arranging a classroom so that groups of students can work together signals an active learning environment, invites student interaction, and supports a learning community. All eyes are not focused on the teacher at the front of the room. With diligent work toward developing trust between teacher and student as well as among students, a teacher can create a culture and a climate of community.
In our years of teaching, supervising, and developing curriculums, we have observed outstanding teachers create these classroom environments. Even though many did not know the word, their classrooms were and are models of constructivism. Each of these teachers is unique, and each uses diverse methods. But we have observed five common attributes, which we call contextual teaching strategies: relating, experiencing, applying, cooperating, and transferring. These strategies focus on teaching and learning in context—a fundamental principle of constructivism.
Relating
Relating is the most powerful contextual teaching strategy and is at the heart of constructivism. We use the term relating to mean learning in the context of one's life experiences.
Ms. Herrera (all teachers' names are pseudonyms) is a 9th grade pre-algebra teacher. She uses relating when she links a new concept to something completely familiar, thus connecting what her students already know to the new information. When Ms. Herrera is successful, her students gain almost instant insight. Caine and Caine (1994) call this reaction "felt meaning" because of the "aha" sensation that often accompanies the insight.
Insight can be momentous. We have all experienced the relief and energy that occur when the many seemingly disparate pieces of a complicated problem fall into place. At that moment, we finally understand the problem in its entirety, and we can see the solution.
But felt meaning can also be subtle when these insights lead to a milder reaction: "Oh, that makes sense." Consider a lesson on ratio and proportion. A traditional approach typically begins with a definition, followed by an example: A ration is a comparison of two numbers by division. Suppose that a bag contains five marbles. Three of the five marbles are blue. The numbers 3 and 5 form a ratio.
Ms. Herrera begins by asking two questions that almost every student can answer from life experiences outside the classroom: "Have you ever made fruit punch from frozen concentrate? What did the instructions say?" She then reads from a real container: "Mix 3 cans water with 1 can concentrate." Now she can connect this familiar situation to the definition of ratio.
When they are presented with the fruit punch example first, most students feel that they already know about ratio because they are familiar with the experience of making fruit punch. They are also more likely to remember the definition of ratio because they can relate it to the fruit punch instructions.
Experiencing
Relating draws on the life experiences that students bring to the classroom. Teachers also help students construct new knowledge by orchestratrating hands-on experiences inside the classroom. We call this strategy experiencing. It is learning by doing—through exploration, discovery, and invention. Three general categories of hands-on experiences create meaning for all students.
Manipulatives. Students move these simple objects around to model abstract concepts concretely. For example, base-ten blocks model numeric representation in the decimal system. Fraction bars demonstrate the meaning of simple fractions and adding and multiplying fractions. Area tiles model the multiplication of polynomials.
Problem-solving activities. These hands-on activities engage students' creativity while teaching problem-solving skills, mathematical thinking, communication, and group interactions. In her fruit punch lesson on ratio and proportion, Ms. Herrera poses a follow-up question: "How many cans of concentrate and how many cans of water are needed to make punch for the whole class?" Several problem-solving approaches and solutions are possible because the answers depend on her students' assumptions: How much punch is needed? How can we make sure that we use the same 3 : 1 ratio of water to concentrate? At the end of the lesson, the students as a class decide on a single best solution and then make the fruit punch to "check their answer."
Laboratory activities. During laboratories, students collect data by making their own measurements, analyze the data, and then reflect on the mathematics concepts. In Mr. Anderson's first-year algebra class, groups of students measure their heights and arm spans. The class combines the groups' data, plots the data, and draws a line of best fit. Then students measure Mr. Anderson's arm span and use the fitted line to predict his height. This activity teaches ordered pairs, plotting ordered pairs on a coordinate plane, drawing a line of best fit, and the power and utility of a correlation. By using their own data, students are more likely to develop a sense of understanding, or felt meaning, for these concepts.
Teachers can orchestrate problem-solving and laboratory activities to show how students' assumptions and methods affect the final outcomes. Many of us are attracted to mathematics because it is a "pure" science—there is always a right answer to a problem and all others are wrong. But when individuals' perceptions are involved, assumptions, formulations, and interpretations of results can differ.
In Mr. Anderson's class, if two students independently use the same set of data points to draw a line of best fit "by eye," the two lines will not be identical and will lead to different predictions of Mr. Anderson's height. In a constructivist classroom, these differences are important. Through them, students learn that multiple perceptions exist and that even in mathematics, the "right" answer can be a matter of interpretation.
Applying
We define the strategy applying as learning by putting the concepts to use. Obviously, students apply mathematics concepts in hands-on, experiential, and problem-solving activities. Some teachers successfully use open-ended problems or projects as opportunities for applying mathematics. In addition, teachers can use realistic and relevant exercises to stimulate a need for mathematics.
These math-application exercises are similar to traditional textbook word problems, with two major differences: They pose a realistic situation and they demonstrate the utility of mathematics in a student's life, current or future. Both are important for a math application to be motivational. The following is a typical word problem from a lesson on the volume of solids: A hemispherical plastic dome covers an indoor swimming pool. If the diameter of the dome measures 150 feet, find the volume enclosed by the dome in cubic yards.
It may be real, but how would a teacher answer a student who asks, "So what?"
Ms. Hayes assigns a problem in her geometry class that also involves calculations with volumes of solids. In this problem, mathematics is crucial in a believable decision-making situation. The problem inherently answers "So what?" Montgomery is a compounding pharmacist at a pharmaceutical manufacturing plant. He is responsible for selecting the correct capsule sizes for specified dosages of the company's products. When a compound is prepared, the capsule size determines the dosage. The company uses eight sizes. The body length l<SUBSCRPT>B</SUBSCRPT>, cap length, 1<SUBSCRPT>C</SUBSCRPT>, diameter dof the capsules are shown in figure 1.Montgomery must select a capsule size for a 25–milligram dosage of an antidepressant. Each capsule must contain 650 ±10 mm3 of the compound. Which size should Montgomery select?
Figure
Strategies for Mathematics: Teaching in Context - table
Capsule Size | Body Length (mm) | Cap Length (mm) | Diameter (mm) |
|---|---|---|---|
| 0 | 22.96 | 13.44 | 9.52 |
| 0 | 20.5 | 12 | 8.5 |
| 0 | 18.86 | 11.04 | 7.82 |
| 1 | 16.51 | 9.65 | 6.86 |
| 2 | 15.35 | 9.1 | 6.25 |
| 3 | 13.6 | 8.13 | 5.47 |
| 4 | 12.3 | 7.2 | 5.1 |
| 5 | 9.84 | 5.76 | 4.08 |
All students will see the importance of the math concepts in solving this realistic problem. But because not all students aspire to become pharmacists, Ms. Hayes assigns problems that cover diverse situations. All her students find realistic scenarios that are applicable to their current or possible future lives outside the classroom, as consumers, family members, recreationists, sports competitors, workers, and citizens.
Relating and experiencing are strategies for developing felt meaning or understanding. Applying is a strategy for developing a deeper sense of meaning—a reason for learning. Relating and experiencing foster the attitude that "I can learn this." Applying fosters the attitude that "I need (or want) to learn this." Together, these attitudes are highly motivational.
Cooperating
Many problem-solving exercises, especially when they involve realistic situations, are complex. Students working individually sometimes cannot make significant progress in a class period and become frustrated unless the teacher provides step-by-step guidance. But students working in groups can often handle these complex problems with little outside help. When Ms. Herrera, Mr. Anderson, and Ms. Hayes use student-led groups to complete exercises or hands-on activities, they are using the strategy of cooperating—learning in the context of sharing, responding, and communicating with other learners.
Working with their peers in small groups, most students feel less self-consciousness and can ask questions without a threat of embarrassment. They also will more readily explain their understanding of concepts or recommend a problem-solving approach for the group. By listening to others, students re-evaluate and reformulate their own sense of understanding. They learn to value the opinions of others because sometimes a different strategy proves to be a better approach to the problem.
Hands-on activities and laboratories are best done, and sometimes must be done, in groups. Many teachers assign student roles for these activities, such as equipment custodian, timer, measurer, recorder, evaluator, and observer. Roles instill a sense of identity and responsibility and become important as students realize that successfully completing an activity depends on every group member doing his or her job. Success also depends on other group processes—communication, observation, suggestion, discussion, analysis, and reflection. These processes are themselves important learning experiences.
Cooperative learning places new demands on the teacher. The teacher must form effective groups, assign appropriate tasks, be keenly observant during group activities, diagnose problems quickly, and supply information or direction necessary to keep all groups moving forward. As with the other contextual teaching strategies, the teacher's role changes. He or she is sometimes lecturer, sometimes observer, and sometimes facilitator (Davidson, 1990).
Transferring
In a traditional classroom, the teacher's primary role is to convey knowledge to students. In a constructivist classroom, knowledge moves in three directions: from teacher to student, from student to student, and even from student to teacher (Brooks & Brooks, 1993). Contextual teaching adds another dimension to this person-to-person transfer. Transferring is a teaching strategy that we define as using knowledge in a new context or situation. Transferring is especially effective when students use newly acquired knowledge in unfamiliar situations.
Excellent teachers have the ability to introduce novel ideas that motivate students intrinsically by evoking curiosity or emotions. In his second-year algebra class, Mr. Whan distributes a magazine article whose author cites statistics to argue that young people should not be allowed to obtain a driver's license until they are 18. Predictably, Mr. Whan's 16- and 17-year-old students react emotionally to this argument.
Mr. Whan uses this source of energy to engage his students in a lively debate. Then he assigns the class, in groups, to evaluate the article. Their written critiques must include an analysis of the mathematics. Were statistics misused? Were facts or assumptions misrepresented or omitted? Was the argument logical? If the critiques are persuasive, Mr. Whan will encourage the students to submit them to the editor of the magazine as rebuttals.
Students also have a natural curiosity about unfamiliar situations. Mr. Whan capitalizes on this curiosity with the following exercise: A sheet of notebook paper is approximately 2 mils thick. (A mil is one-thousandth of an inch.) If you fold a sheet of notebook paper in half, the total thickness is 4 mils. If you fold it in half again, the thickness becomes 8 mils. Suppose that you could fold the paper 50 times. Which of the following best describes the total thickness?a. less than 10 feetb. more than 10 feet, but less than a 10–story buildingc. more than a 10–story building, but less than Mt. Everestd. more than the distance to the Moon
Although folding a sheet of paper is not novel, students cannot be familiar with 50 folds because it is impossible to fold the paper that many times. Mr. Whan asks his students to discuss the possible choices of thickness and to vote as a group for the one they predict to be true. A spokesperson for each group explains the rationale for its prediction. After the votes are tallied, students have bought in to the problem and are eager to know the right answer. At this point, Mr. Whan has each student group calculate the thickness. The mathematics involves sequences, patterns, conversion factors, powers, and scientific notation. As a wrap-up, Mr. Whan leads a class discussion about why most predictions were wrong.
Mr. Whan uses exercises like this to evoke curiosity and emotion as motivators in transferring mathematics ideas from one context to another. And conversely, felt meaning created by relating, experiencing, applying, cooperating, and transferring engages emotions. One of Caine and Caine's 12 principles of brain-based learning says that "emotions and cognition cannot be separated and the conjunction of the two is at the heart of learning" (1994, p. 104). Although they did not use the term constructivism, their ideas about felt meaning, emotions, and cognition clearly paved the way: The brain needs to create its own meanings. Meaningful learning is built on creativity and is the source of much of the joy that students could experience in education. (P.105)
Creativity and joy are two descriptors that we often associate with the classrooms of our best teachers. Others are laughter, motivation, engagement, attention, imagination, communication, and group processes. How much could mathematics learning improve if these described the classrooms of our average teachers?
