Strategies for mathematics: Teaching in context
Abstract (summary)
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?
