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CTL jurnal

Strategies for mathematics: Teaching in context

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Active learning in motivating contexts is the foundation on which constructivist teachers build their teaching strategies and classroom environments. Within the context of mathematics education, Crawford and Witte explain how to craft learning experiences that invite interaction and help students apply knowledge.

Headnote

Active learning in motivating contexts is the foundation on which contructivist teachers build their teaching strategies and classroom environments.

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 ratio 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 I 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 feet 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, baseten 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 problemsolving skills, mathematical thinking, communication, and group interactions. In her fruit punch lesson on ratio and proportion, Ms. Herrera poses a followup 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 firstyear 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 problemsolving 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 openended 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^subb^, cap length l^sub C^, and diameter d of 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 mm^sup 3^ of the compound. Which size should Montgomery select?

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 selfconsciousness 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-yearold 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 feet

b. more than 10 feet, but less than a 10-story building

c. more than a 10-story building, but less than Mt. Everest

d. 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?

jurnal active learning

Active learning of mathematics

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Smith discusses the importance of active learning techniques in the mathematics classroom. A teacher working with this view would have to provide students with opportunities to create their own theories and to engage their mental model-making processes.

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SINCE THE COCKCROFT REPORT IN 1982, THERE HAS BEEN AN increasing emphasis on the use of active learning in school mathematics, with a typical and influential view being propounded in Better Mathematics: "Mathematics can be effectively learned only by involving pupils in experimenting, questioning, reflecting, discovering, inventing and discussing. Mathematics should be a kind of learning which requires a minimum of factual knowledge and a great deal of experience in dealing with situations using particular kinds of thinking skills" (Ahmed 1987, 24).

During the same period of time, there has been an increased explicit and implicit use of a constructivist epistemology, for example in an implicit way: "The teachers job is to organize and provide the sorts of experience which enable pupils to construct and develop their own understanding of mathematics, rather than simply communicate the ways in which they themselves understand the subject" (NCC 1989, para. 2.2).

And in an explicit way: "Many writers embed their view of active learning in a framework concerned with the nature of the intellectual activity taking place, most often located within a constructivist model of mathematical learning" (Kyriacou 1992, 312).

We can, with some consistency, summarize this particular constructivist framework as the hypothesis that human knowledge is personally constructed and consists of conjecture, unfalsified theories, modified theories and expectations.

A teacher working with this view of how we come to know would have to provide pupils with appropriate chances to create their own theories, to engage their mental model-making processes and to allow them the opportunity to develop expectations in order to subject their theories to the test of reality. This [approach] strongly suggests that pupils must be actively engaged in constructing their understanding, and that the activities themselves must be judged mainly by their contribution in assisting pupils to construct their own understanding of concepts selected by the teacher.

Within this particular constructivist framework, learning tasks should be chosen with the specific intention of actively involving learners in seeking to understand their external world by creating and testing their own models of what is going on "out there" in reality. The teacher has a clear initial role in selecting activities that are expected to focus the attention of the learner on constructing the intended learning outcomes of the session. The mathematical activities should therefore be selected or designed to encourage the learner to link between external world and internal thought. This involves a consideration of presentation, pupil activity, reflection and socialization. Activity by itself is not enough.

Presentation

WE MUST AIM TO PRESENT ANY MATHEMATICAL ACTIVITY in a way that invites pupils to fully engage their higher mental capacities. This can be by the use of a game, a puzzle, a surprise or some other intriguing challenge. In creating a challenge for learners, we must be aware of the need to choose an appropriate level of challenge; one that learners can perceive as offering them a realistic, but not certain, chance of meeting. The challenge can come from the teacher or from the pupil themselves. It can be in the form of a puzzle, a target to reach, a goal or a conflict to resolve.

An example of a presentation that involves all four of these aspects is to motivate pupils' work on geometrical construction. One way in which I have done this is to show pupils a hexaflexagon, on the front of which is drawn a pattern and on the reverse, a different pattern. Pupils are asked to memorize the patterns, and whilst they attempt to do this, I flex the hexaflexagon whilst maintaining the same pattern on the front, then ask, "Who can remember the pattern on the back?" and surprise the class by showing that the pattern has "disappeared." The goal is then set for the class members to make their own hexaflexagon, resolving the puzzling disappearance of the lost pattern. I have found this to be a simple yet highly motivating presentation, which sometimes generates a round of applause-I wish I could say that more often!

In creating the initial challenge, surprise and cognitive conflict can be useful to generate interest. For example, by asking pupils to work in pairs and giving one of each pair a basic calculator and the other a scientific calculator, we can create cognitive conflict and surprise if we ask pupils to carry out calculations (like 2 + 3 x 5) that give different answers on the two calculators. The role of the presentation is to indicate to the learner the need for new or revised theories; the role of the challenge, surprise or cognitive conflict is to engage the learners' full attention.

Pupil Activity

TO SOME EXTENT THE ROLE OF PUPIL ACTIVITY IS clear; it is for learners to undertake in order to meet the challenge set by themselves, the teacher or the text. They should be trying to make sense of the challenge, the activity and their findings. It can therefore be to move toward the goal, or to attempt to resolve a cognitive conflict, or even to explore the extent of the cognitive conflict. For example, to continue with the calculator challenge, we might ask pupils to find out as many sums as they can that lead to different answers on the different calculators.

The key aspect of any learning activity is that it must be constructed or chosen to demand mental involvement; perhaps this is best achieved by requiring pupils to deal with new, unfamiliar and nonroutine activity. Familiar mathematics can often be packaged in an unfamiliar way; for example, I recently worked with a Y10 top set class studying graphical inequalities. To do this, I began by placing the pupils in a rectangular array and systematically giving each person a set of coordinates. By asking pupils to stand up if they satisfied inequalities like 2x + 3y > 5, I involved all the pupils in thinking about the familiar coordinates in an unfamiliar way. The class teacher reported that "the activity maintained interest, was fun and entertaining and got across what I usually find pupils have difficulty with very clearly. The physical nature of the task helps keep interest and minds awake!"

The task should not be a passive routine, such as factorizing fifty similar-looking equations. Practice may be important to develop skill, but from this perspective the less routine there is in it, the more learning is likely to be achieved. In writing of the importance of a firm conceptual understanding, HMI state that ". . . progress in pupils' mathematical understanding is more important than progress in the performance of skills. In fact, when the early stages of learning are firmly established subsequent progress can take place more quickly and confidently" (HMI 1985, 36).

Reflection

ON COMPLETION OF AN ACTIVITY, A REVIEW OF THE learning achieved during the activity can be most helpful in assisting the learners to integrate their new or revised theories and expectations with their other mental systems.

"Activity per se is not a guarantee of mathematical learning" (Goodchild 1992, 24). Goodchild goes on to consider the importance of reflection and "interpretation," which he sees as a mechanism for making sense of the learning activity and for locating it in a wider framework of meaning and purpose. Alternatively, it may be seen as a process in which new personal theories are created or existing personal theories modified and in which new expectations may be created. The teacher's role in this process is to ensure that there is time for such reflection, and to provide a mechanism to ensure that reflection occurs. The classroom organization of reflection may involve pupils in writing, or it may involve structured discussion with other pupils or with the teacher.

"When asked for the connections between practical work and the symbolic statement of rule, the children's best reply was that one was a quicker route to the answer than the other. Nobody mentioned that the practical experience provided the data on which the formula was built. The teachers did not stress why this procedure was being followed, nor emphasize the generalizability of the rule and thus the advantage of accepting it" (Hart 1989, 139).

Within the theoretical framework outlined above, it may be seen as appropriate to ask pupils to be explicit about some new expectations. This obliges pupils to create both theories and expectations. To continue the calculator example: once pupils have amassed a number of calculations that give different answers on two calculators, they can be asked to predict the two answers for other calculations, to predict when the calculators will give the same answers and to predict when they will give different answers.

Socialization

"KNOWLEDGE, FROM THE CONSTRUCTIVIST POINT OF view, is always contextual and never separated from the subject. . . to know also implies understanding in such a way that the knowledge can be shared with others and a community thus formed. A fundamental role is played by the negotiation of meaning in this interaction, which is of a social nature" (Moreno-Armella and Waldegg 1993, 657).

I do not believe that many pupils achieve this socialization of knowledge whilst working from individualized schemes. Even the pronunciation of key mathematical words (e.g., "sin" instead of sine) is missing, let alone the opportunity to discuss mathematical problem solving in depth. Socialization cannot be delegated to textbooks.

Having developed an understanding of a mathematical concept, there is a need for pupils to be able to communicate effectively about it. For this [requirement], there is often a need to know, understand and use the appropriate language and terminology and to adopt the standard conventions. In other words, there is a need to socialize the personal and private understanding. The teacher has a role here in ensuring that the need for communication is apparent, that activities incorporate discussion work to facilitate such communication and that pupils are helped to become aware of conventional language and notations. "The use of discussion as a technique for teaching centres on the fact that it is above all else a means of escaping from our own individual perceptions of the world, with all their circumstances and boundaries into which we would otherwise be locked. It adds to the richness of understanding and enables us to make contact with the minds of others in the most direct way possible" (Van Ments 1990, 17). During the socialization process, the individual is being expected to negotiate a shared meaning with the teacher, peers, external examiners and textbooks largely by coming to grips with conventions and conventional language. If all pupils in a group are encouraged to communicate about their mathematical activity, this [approach] can provide a richer learning environment for each individual, as well as begin the process of negotiating shared meanings and socializing the new knowledge.

Conclusion

IN THIS ARTICLE, I HAVE CONSIDERED A CONSTRUCTIVIST framework for the analysis of mathematical learning activities. This analysis suggests that mathematical activities are not enough to achieve learning by themselves; they need to carried out with a consideration of aspects of presentation, the nature of the pupils' mental activity, the need to ensure pupil reflection and the achievement of socialization of the learning.