- Teachers are
always interested in looking for ways to improve their
teaching and to help students understand mathematics. Research in
England, Japan, China, and the United States supports the idea that
mathematics instruction and student mathematics understanding will be
more effective if manipulative materials are used (Canny, 1984; Clements
& Battista, 1990; Dienes, 1960; Driscoll, 1981; Fennema, 1972,1973;
Skemp, 1987; Sugiyama, 1987; Suydam, 1984). Mathematics
manipulative is defined as any material or object from the real world that
children move around to show a mathematics concept.
Manipulative
materials in teaching mathematics to students hold the
promise that manipulatives will help students understand
mathematics. At
the same time as with any "cure", manipulatives hold
potential for harm if
they are used poorly. Manipulatives that are improperly
used will
convince students that two mathematical worlds exist
- manipulative and
symbolic. All mathematics comes from the real world.
Then the real
situation must be translated into the symbolism of mathematics
for
calculating. For example, putting three goats with
five goats to get eight
goats is the real world situation but on the mathematics
level we say 3+5 =
8 (Read three add five equals eight). These are
not two different worlds
but they are in the same world expressing the concepts
in different ways.
What are manipulative
materials? Manipulative materials are concrete
models that involve mathematics concepts, appealing to
several senses,
that can be touched and moved around by the students
(not demonstrations
of materials by the teacher). The manipulative
materials should relate to
the students' real world. For example, the use
of an abacus is not
something that is used in Malawian daily life.
Instead stones, eating
utensils, tins, beans, apples, peanuts, sticks, etc.
would be more
appropriate.
Each student
needs material to manipulate independently.
Demonstrations by the teacher or by one student are not
sufficient. With
students actively involved in manipulating materials,
interest in
mathematics will be aroused. Manipulative materials
must be selected that
are appropriate for the concept being developed and appropriate
for the
developmental level of the students. For example,
one stick may be placed
on a place value chart in the ones place.
However one stick should not be
placed in the tens place. Instead a package of
ten sticks bundled together
with string or an elastic should be placed in the tens
place. Students need
to realize and conceptualize the idea of tenness.
The same is true for the
concept of the hundreds place; a bundle of 100 identical
things should be
used. As the students' concept of place value develops,
then single sticks
can be used for place value of numbers with greater value.
Good mathematics
manipulative materials are durable, simplistic
(easily manipulated), attractive (to create interest),
and manageable. A
systematic method should be developed for storage and
distribution of
materials. Baskets or boxes are convenient for
storage and distribution
purposes.
Using manipulative
materials in teaching mathematics will help
students learn:
1. to relate real world
situations to mathematics symbolism.
2. to work together cooperatively
in solving problems.
3. to discuss mathematical
ideas and concepts.
4. to verbalize their mathematics
thinking.
5. to make presentations
in front of a large group.
6. that there are many
different ways to solve problems.
7. that mathematics problems
can be symbolized in many different
ways.
8. that they can solve
mathematics problems without just following
teachers' directions.
If mathematics
is taught using manipulative materials, then the methods
of evaluating mathematical achievement must also change.
Just calculating
correct solutions to mathematics problems is not sufficient.
Concept
development and understandings should be valued more
highly. Effective
use of mathematics manipulatives contributes to conceptualization
and
understanding. Evaluation of students' mathematics
is changing from tests
and testing to assessment. Assessment is much broader
than testing or
evaluation. For teachers to assess students' understanding
of concepts,
different techniques of evaluation will be needed.
Teachers will receive
more insight into students' mathematics understanding
by:
1. listening to students'
talk about their mathematics thinking.
2. observing students working
individually and in cooperative groups.
3. asking why and how questions
rather than asking:
a.
yes or no questions.
b.
for results of calculating activities.
c.
for answers.
4. having students write
a solution to a problem rather than by only
responding with correct or incorrect values.
Paper-and-pencil
method of assessment limits the scope of student
evaluation. Requiring students to defend their
mathematical reasoning
provides insight into the development of the students'
thinking skills.
Observation of students' functioning within a group will
provide data for
assessment. The teacher will move around the classroom
observing how
students are working and interacting.
To facilitate
collecting assessment data, different types of questions
will need to be asked by the teacher. The traditional
questions which
focus on calculating and correct answers will change
to:
1. how and why questions.
2. probing questions to
stimulate the thinking process of the students.
3. having students write
responses to mathematics problems.
- a. integrates writing with mathematics
b. numerical values are not sufficient for answers to mathematics
problems.
c. presents an opportunity for reflection, which is conducive to
students' cognitive development.
d. helps identify students having mathematical difficulties.
e. helps identify the conceptual level of development of the
students.
might be:
1. How do you know that ?
2. What would happen if ... ?
3. Why do you suppose ... ?
4. What makes you think your answer is correct ?
5. How could you prove your answer is correct ?
6. Could you express your answer in a different way ?
7. What is another way to solve the problem ?
8. How many different ways can you find to solve problems ?
9. How can you convince the other members of your group that your
way is the best method to solve the problem ?
Conclusions
Mathematics teachers are learning to direct their attention
toward the
facilitation of students' understanding and conceptualization
rather than
drill and practice of rote procedures. The use
of manipulative materials in
mathematics classrooms supports this attempt. Incorporating
the use of
manipulative materials with an emphasis upon the thought
process of
students provides an opportunity for the teacher to assess
and meet the
needs of primary school students as they construct personal
mathematical
knowledge.
Bibliography
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achievement in three areas of Fourth-Grade mathematics: computation,
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