Manipulatives: The Missing
Link in High School Math
by Marilyn Curtain Phillips, M. Ed.
The thrill of achievement comes from overcoming adversity in the
accomplishment of an important goal ~ Unknown
Algebra, Geometry and Trigonometry can be challenging and complex subjects
for many high school students in the United States. These subjects are usually
taught using textbooks, workbooks and examinations. While these resources are
essential in developing learning skills in math, it doesn’t encourage
problemsolving skills and retention. It is not surprising to learn many
students view math as boring, difficult and irrelevant, rather than fun and
interesting. The scope of what it means to be successful in mathematics expands
beyond procedural knowledge and skillacquisition to include sensemaking and
conceptual understanding (Buckley 2004). An integrated approach is needed to
motivate students to learn math in a more relaxed environment, thus eliminating
persistent math anxiety, the lack of math application skills and poor
standardized test performance. The Third International Mathematics and Science
Study (TIMSS) tested the students of 41 nations. Children in the United States
were among the leaders in the fourth grade assessments, but by high school
graduation, they were almost last. High school math should be taught in a way in
which students can grasp and understand skills. Manipulatives are visual
objects that help illustrate mathematical relationships and applications. These
resources are used primarily in elementary schools and somewhat in middle
schools. Manipulatives are valuable resources for accelerating and deepening
students understanding of math, yet its use is almost nonexistent in high
school. Marilyn Burns, Creator of Math Solutions has "used manipulative
materials at all levels for 30 years." In every decade since 1940, the
National Council of Teachers of Mathematics (NCTM) has encouraged the use of
manipulatives at all grade levels (Bellonio), yet many high school teachers are
reluctant to use this type of resource.
Almost all mathematicsteaching activities take place at the abstract level
(Sharma, 1997). According to Sharma, students have a tendency to forget when
taught only at the abstract stage. Thus students become frustrated because
mastery was never fully attained. Students will begin to have difficulty in
learning mathematics. The results in failure will cause many students to develop
a fear of mathematics (Sharma, 1997). Students’ attitudes towards any
curriculum area can be related to their achievement in ways that reinforce
higher or lower performance (timss.org). Full 88% of Bill and Melinda Gates
Foundations’ survey respondents said they had passing grades in high school.
Asked to name the reasons they had left school, more respondents named boredom
than struggles with coursework. Over 1 million students drop out of school each
year. That includes nearly half of all African Americans, Hispanics, and Native
Americans who fail to graduate from public high school with their class. Leaving
many of them with a host of poor outcomes to follow, from low lifetime earnings
to high incarceration rates to a high likelihood that their children will drop
out of high school thus eliminating the cycle (Thornburg, 2006).
Sharma (1997) feels there are six levels of mastery of mathematical concepts:
intuitive, concrete, representation (pictorial), abstract, applications and
communication. According to Sharma, ideally, each mathematics concept should be
introduced beginning at the communication level. Manipulatives will teach
concrete understanding to the abstract math process, especially when the student
may not understand the concept behind the skill. When students develop a
concrete understanding of math skills, then they are more likely to perform that
math skill and understand math concepts at the abstract level. Manipulatives can
make math concepts come alive. According to Spikell (1993), most learners
whether adults or children, will master mathematical concepts and skills more
readily, if they are presented first in concrete, pictorial and symbols. By
using manipulatives, pictures and symbols to model or represent abstract ideas,
the stage is set for learners to understand the abstractions they represent (Spikell,
1993). Students will be able to practice and demonstrate mastery using concrete
objects. Manipulatives appeal to the learning style of kinesthetic learners
because they actually touch the objects. Pictures appeal visually for
visual/spatial learners. "Visualization is the natural way one begins to
think. Before words, images emerge" (Sharma, 1987). "Almost every
mathematics idea, except simple arithmetic facts, consists of three components:
linguistic, conceptual and skill/procedural" (Sharma, 1987). Therefore manipulatives offer benefits to a variety of learning styles. They
also provide a change from the textbook for mathematical/logical learners.
Games with manipulatives are also valuable with helping students to apply
what they learned to the real world, as well as provide a means to improve their
math skills interactively. Using board games and card games along with
cooperative learning are ways that students can become involved in a positive
mathematical environment. Games are highly motivational to students and can be
used effectively to practice specific skills. "Using games in the classroom
and at home will maximize students' problemsolving competence, ability to
communicate and reason mathematically, perception of the value of mathematics,
and selfconfidence in their ability to apply mathematical knowledge to new
situations." Cooperative groups provide students a chance to exchange
ideas, to ask questions freely, to explain to one another, to clarify ideas in
meaningful ways and to express feelings about their learning. These skills
acquired at an early age will be greatly beneficial throughout their adult
working life.
Many teachers feel as though they do not know how to teach using
manipulatives and therefore, hesitate to use them in the classroom. Many math
teachers, who attended college more than ten years ago, were usually taught on
the abstract level, textbook, pencil and paper, all through their school days.
There are classes and workshops for teachers to learn how to teach using
manipulatives. The companies that make the manipulatives also provide books and
pamphlets on ways the material can be used. There are articles on using
manipulatives in mathematics teaching journal such as the National Council of
Teachers of Mathematics’ Arithmetic Teacher magazine. Manipulatives help relieve boredom in children allowing them to explore and
use their imagination.
Many manipulatives are inexpensive and can be everyday objects. Money, 2 –color
counters, calculators, rulers, dominoes, playing cards, button and number cubes
are a few of the commonly available manipulatives that can successfully be used
in the classroom. These manipulatives can be used to teach such concepts such as
angles, area, decimals, factoring, estimation, fractions, measurement, counting,
percent, prime numbers, probability, geometry and whole numbers.
There are companies, which specialize in manipulatives that can be ordered
from a catalog or online. No matter where a school is located, materials can be
made available through the mail. These companies also make available to teachers
such manipulatives as tangrams, pattern blocks, fraction towers, geoboards,
algebra tiles, Cuisenaire rods, miras and polyhedral models.
When I initially introduced pattern blocks to my Geometry students, I was met
with opposition and disbelief. It was a consensus that these students had not
used pattern blocks since elementary school. However, many students actually
began to explore and build figures with these blocks. I gave them this little
bit of play time before taking on the task for that particular lesson, which was
angle measurement of polygons. We must remember, everyone likes to play games
even adults, look at our various hobbies, such as bowling, tennis, golf, art,
race cars, chess, playing cards, etc. This holds true for high school students,
they still enjoy playing games. Manipulatives require a great deal of prior
planning and organization. But considering the many benefits that manipulatives
offer, it is well worth the effort. Students will find math more attainable, fun
and exciting, hopefully returning them to the prior successes of their early
math experiences.
REFERENCES:
Buckley, L. A. (2004, Oct) Course Taking and Equity: The Efforts of One High
School Mathematics Department Paper presented at the annual meeting of the
North American Chapter of the International Group for the Psychology of
Mathematics Education, Delta Chelsea Hotel, Toronto, Ontario, Canada Online
<.PDF> Retrieved 20080112 from <http://www.allacademic.com/meta/p117666_index.html>
Burns, M. How to make the most of math manipulatives – A fresh look at
getting students’ heads — and hands! —
around math concepts. Retrieved 200801110 from <http://teacher.scholastic.com/lessonrepro/lessonplans/instructor/burns.htm>
Sharma, M. (1987). How to take a child from concrete to abstract. Math Note
book, 5, 810
Sharma, M. (1997, July). Improving mathematics instruction for all. Fourth
Lecture in Colloquium "Improving Schools from Within: Your Role. Pp. 212
Spikell, M. (1993). Teaching mathematics with manipulatives: A resource of
activities for the K12 teacher. New York: Allyn and Bacon.
Third International Mathematics and Science Study (TIMSS), Institute of
Educational Sciences, United States Department of Education
Thornburg, N. (2006), April 7), Dropout Nation, Time Magazine
TIMSS 1999 International Mathematics Report. Findings from IEA's Repeat of
the Third International Mathematics and Science Study at the Eighth Grade. Ina
V.S. Mullis, Michael O. Martin, Eugenio J. Gonzalez, Kelvin D. Gregory, Robert
A. Garden, Kathleen M. O'Connor, Steven J. Chrostowski, Teresa A. Smith.
Retrieved 20080112 from <http://timss.bc.edu/timss1999i/pdf/T99i_Math_TOC.pdf>
YaleNew Haven Teachers Institute Home MultiSensory Manipulatives in
Mathematics: Linking the Abstract to the Concrete by Judith L. Bellonio Contents
of Curriculum Unit 01.06.12: Narrative Math Anxiety Types of ... Retrieved
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