> Math >
Role of manipulatives in middle and secondary math instruction
The role of manipulatives in introducing and developing
mathematical concepts in elementary and middle grades
by Susan Jones: analysis of research about using manipulatives to teach mathematical
(Warning: This is long. )
Manipulatives are defined as materials that are physically
handled by students in order to help them see actual examples
of mathematical principles at work. Manipluatives are designed,
marketed, purchased, researched and used throughout educational
systems, but primarily in elementary schools. Unfortunately,
the exact manner in which people do these things is often
based more on the definition of manipulatives than on their
ultimate purpose. Manipulatives should be used as an effector
of understanding and applying mathematical principles, not
In conducting research and making decisions about manipulative
use, the differences in these instructional goals must be
considered. Manipulatives are designed with the idea of illustrating
a certain mathematical principle. The manipulative may be
successfully marketed on the basis of its price, simplicity
and clarity of illustration of the principle. However, attributes
important to its effective demonstration of universal mathematical
concepts are sometimes ignored because the nature and relative
importance of these attributes have not been ascertained.
In turn, instructional goals are mistakenly based on using
the manipulative to illustrate a mathematical pattern, such
as place value or addition, then translating the pattern into
mathematical symbols; students are then drilled in following
that pattern of symbols. At the point that the student masters
the translated pattern (that is, can get the right answer),
the teacher proceeds to the next task. This is often how mathematics
is taught throughout grade school, with or without manipulatives.
There is a monstrous assumption behind this progression:
that learning mathematics is fully achieved by learning symbolic
patterns. It has been clearly demonstrated that learning these
patterns produces skill in computation, but analysis of current
research demonstrates that without intervention many students
learn these patterns and do not progress beyond this into
understanding how to use these patterns to solve problems.
If manipulatives are wisely incorporated into the curriculum,
they can be used at the middle and secondary level to induce
understanding of the concepts underlying the computation.
I. MATHEMATICS: UNDERSTOOD, MISUNDERSTOOD AND NOT UNDERSTOOD
In school systems around the western world, researchers and
educators express dissatisfaction with results of mathematics
education: There are far too many children who dislike mathematics,
more so as they get older, and many who find great difficulty
with what is very simple. Let us face it: the majority of
children never succeed in understanding the real meanings
of mathematical concepts. At best they become deft technicians
in the art of manipulating complicated sets of symbols, at
worst they are baffled by the impossible situations into which
the present mathematical requirements in schools tend to place
them. An all too common attitude is 'get the examination over,'
after which no further thought is given to mathematics."
Research reveals that the problem is a real one, following
a consistent pattern: students are unable to apply well-learned
computation skills to problem solving. Several examples of
this follow: The National Assessment of Educational Progress
II determined that while performance of routine skills and
algorithms was at an acceptable level, students in junior
high and high schools lacked understanding of certain underlying
concepts, especially those concerned with fractions (Wiebe,
1983). Bell, Swan and Taylor (1982) concurred with
this in disclosing that when students were presented with
problems identical save different numbers, they did the problems
differently and did not know why one was wrong. Twice as many
students got the problem "1.25 x 15" right as were
correct on "15 x 1.25." A basic understanding of
the commutative property would have precluded this occurrence.
Blume and Mitchell (1983) found that without reviewing,
fewer than 10% of 83 eighth graders could "state the
distributive property of multiplication and addition;"
they noted that the primary emphasis in texts at the 6-8th
grade level was on completing the pattern of sums of products,
and considered it to be symptomatic of the "common classroom
malady known as 'symbol pushing.'" Upper elementary students
often use correct or incorrect procedures for the purpose
of getting an answer, not solving the problem (Barody,
1985); if a student is aware of and confident in using
what they considered a "good" strategy, they would
use it inappropriately and forego seeing if their final answer
makes sense (Garofalo, 1985). More advanced strategies
are being used without being understood; there is also a prevalence
of simple strategies being overextended to higher level problems.
In O'Brien and Casey's (3/83) study, students proficient
at computing multiplication failed dismally in attempts to
construct a multiplication problem. 37% of fourth and, surprisingly,
44% of fifth graders constructed additive problems, e.g.,
"Johnny had five apples and Mary had 3 apples. How many
would they have together?" Booth (1981) tested
10,000 high school students in England on their level of understanding
of concepts typically taught in secondary schools. He found
that many problems involving secondary school skills were
correctly answered by fewer than half of the students. He
also found that students tended to choose simpler, "intuitive"
methods over those taught in school. Even students who were
successful in solving higher level problems got their answers
by adapting the simple methods, as opposed to using higher
level skills. The students demonstrated both an inefficiency
and an inability to formally express what they had done in
mathematical terms. The "knowledge" students are
comfortable with using is not what they are being taught.
"It was as if two completely different types of mathematics
were involved; one where children used common sense, the other
where they had to remember rules." Wearne-Hiebert
& Hiebert (1985) tested 6th and 8th graders in nine
fraction problems with accompanying drawings, exemplifying
concepts of varying difficulty from "find the whole number,
given that ½ of it is 8" (which 96% of both sixth
and eighth graders answered correctly) to "find 1½
of 12" (which fewer than 15% of students from either
group could answer. They found a strong tendency for students
to double or halve in the right direction (if the answer should
be larger, they doubled; if the answer should be smaller,
they halved) - using the correct process, but unable to apply
it to thirds and fourths. More remarkable was the small change
from sixth to eighth grade in the number of correct answers
for many of the problems. Clearly, while students were developing
some mathematical understanding of the fraction concepts,
the understanding was not keeping pace with educational expectations.
Wiebe offered a succinct and cogent interpretation of this
prevailing tendency: "Relationships and meanings we attempt
to develop are understood by only the brightest few, and for
the rest arithmetic becomes a process of memorizing algorithms.
Eventually, for certain children/adults, understanding may
come after further cognitive development and after prolonged
use of the memorized algorithms. "
Apparently, mathematical processes are mastered without the
level of understanding necessary to know when to use them,
or even to be able to tell whether a problem has been solved
or not. Numbers are moved through learned mathematical patterns
via algorithms - but mathematics has not been applied.
II. The Role of Manipulatives in Concept Formation
The aforementioned research indicates strongly that a special
effort should be made in cultivating concept understanding.
When and how should manipulatives be used to help teach these
concepts? Most thought and words on the use of manipulatives
are based on Piaget's theories. However, evidence strongly
suggests that "learning" as Piaget defines it does
not happen in much of elementary mathematics. This has led
some people to conclude that Piaget's theory was not really
designed to be applied there (Halford, 1978). Indeed,
Barody (1985) maintains that "according to current
theories, efficient production of number combinations is exclusively
a reproductive process," and therefore the role of reasoning
and analysis skills is limited. A vital and vast portion of
the computation skills taught in elementary school is no more
than "number combinations," and in that aspect of
school mathematics, concentration should be trained on the
reproductive process. But this should not be all there is
to learning mathematics. Kuchemann (1981) grouped 83
eighth graders by individual interview and test into Piaget's
"Concrete" or "formal operational" stages.
He found that while they did not differ in how they performed
the mechanics of problem-solving, the "formal-operational"
children were far more advanced in deductive reasoning, approximation,
and evaluation processes. This difference was even more pronounced
in complex problems than in simpler ones. Given this background,
it is not surprising that manipulatives have been found to
be of dubious value in teaching computation skills. Friedman
(1978) considered manipulatives to be "the latest
Pied Piper," citing four studies concerning them. Of
the four, three tested the effectiveness of manipulatives
in teaching children below the fourth grade efficient multiplication
computation, and showed the manipulatives to be of little
or no benefit. From this, Friedman inferred that manipulatives
were useful only at the first grade level, teaching the most
basic properties of numbers. However, other information shows
that it is more likely that manipulatives would increase in
value in later grades, in teaching more complicated skills,
as children mature and become mentally able to develop understanding
of operations such as multiplication. Since it seems the ability
to understand mathematics is not needed to successfully compute
operations, then manipulatives might indeed confuse matters
if used to try to explain things that a child isn't developmentally
ready to understand. Simple models the child is able to understand
would interfere with the complex algorithm. Nonetheless, this
is often how manipulatives are used. Manipulatives are brought
in to relate physical experience to written symbols when a
new algorithm and corresponding concept is introduced; then
the manipulatives are put away and further instruction is
based on the symbols, not on further concrete experience.
Attempts to increase the "physical experience" aspect
of the concept quickly get bogged down because the students
are not ready to relate the algorithms to concrete experience.
The computation skills taught in mathematics far surpass in
formality and complexity what school children can be expected
to grasp; Kuchemann (1981) also found a wide mismatch
between cognitive demand of what elementary and secondary
students are taught and what age-related constraints determine
that they can understand. There are two conceivable ways of
bridging this gap: First, to slow down the pace of teaching
computation so that children could understand what they are
doing. This would entail a complete and probably unwarranted
upheaval of most curricula. It is certainly a less than useful
idea for someone trying to figure out how to incorporate manipulatives
into their teaching. The second is to attempt to bridge that
gap between the symbolic routine and the "simple intuitive"
logic as the children mature, to ensure that they are of that
population of people who eventually understand the material.
Manipulatives could be useful in this effort.
III. THE ROLE OF MANIPULATIVES IN PROMOTING UNDERSTANDING
Manipulatives should be incorporated into the curriculum
with the goal of helping the student understand mathematics,
rather than increasing efficiency in calculation. To help
a student understand, we need to have some specific knowledge
of how mathematics comes to be understood. Hans Freudenthal
claims that "all agree and textbook writers witness that...arithmetic
cannot be learned in any way other than by insight, whether
it is taught that way or not." Most others will maintain
that while textbook writers may "witness" that insight
is necessary for understanding, they are geared very strongly
toward learning algorithms and imitating patterns (such as
the distributive property). Freudenthal, too, cites the ominous
gap between what can, in fact, be understood through insight
and what is learned through drill and practice. He warns against
pitfalls that caused the confusion and lack of success of
the "new math:" "The wrong perspective of the
so-called New Math was that of replacing the learner's insight
by the adult mathematician's insight." Zoltan P. Dienes
has written several books based on using manipulatives to
enhance understanding of mathematics. Since then, Zbigniew
Semdadeni (1984) has written a concise condensation and
evaluation of the principles behind Dienes work, and some
improvements and additional applications. A most important
aspect of Dienes' approach is that he emphasizes using manipulatives
to provide a concrete referent for a concept, usually at more
than one level, instead of a referent for a given abstract
idea or procedure. In keeping with Piaget's thoughts, Dienes
and Semdadeni view new knowledge as the extension of old knowledge
into new areas. For example, learning the nature of positive
and negative integers is an extension of the nature of natural
numbers. Dienes established several principles on which to
base teaching mathematics. The ones that apply to use of manipulatives
- Permanence Principle: When extending knowledge
into a new area, pick an extension of properties that is
most like the rules the students are already comfortable
with; explore familiar, manipulative examples and extend
the concept to generic numbers, then explore examples of
the new domain. (For example, subtraction of fractions would
be introduced through subtraction of whole numbers, and
then extended through questions.)
- Mathematical variability principle: To enhance
full understanding of a concept, you need to vary all the
variables possible so students can understand which properties
are constant; in other words, you proceed from the specific
concrete examples demonstrated with the manipulatives, to
the abstract, generic properties of the concept.
- Multiple Embodiment Principle: In order to abstract
a mathematical concept, you should demonstrate it concretely
with as many different situations as possible, again changing
variables, so that the student can abstract what its "purely
structural" properties are. These principles and many
of Dienes' suggested techniques all suffer somewhat from
the "new math" syndrome: they rely on fairly sophisticated
abstracting ability, and generally demand that too many
insights be garnered from a given situation. When kept within
the constraints of a child's development, the principles
do hold up. Semdadeni emphasizes that the Permanence Principle
be carried out with extensive manipulatives, because the
extension of known rules could foster the habit of applying
old rules to new situations without checking to see whether
it is reasonable or not.
The principle that has engendered the most controversy is
the "Multiple Embodiment Principle." Many educators
do not think it applies to the use of manipulatives. Vest
(1985) maintains that students do not need to experience different
types of concrete developments for each complex algorithm;
one should be enough. Jackson (1979) concurred with this,
on the grounds that using more than one manifestation of a
concept required advanced, formal skills including reflective
thinking, generalizing and transferring knowledge. These tasks
would render the "concrete" experience more distracting
and confusing than useful. Others disagree with this thinking,
and would try to incorporate reflection on problems as early
as possible. Indeed, reflection is not that sophisticated
a skill - so long as the concept is not too sophisticated
(Freudenthal, 1981 and Skemp, 1972). And certainly, in the
middle and secondary grades, such reflection should be insisted
upon. Skemp felt this was especially important in learning
algebra, embracing Semdadeni's "concretization schema"
(a foundation of varied concrete experiences for abstract
concepts) as the most effective way to learn mathematics.
In this way, mathematics beyond algebra becomes an extension
of algebra instead of a new set of algorithms.
Wearne-Hiebert and Hiebert recommend using several concrete
examples for the fraction concept, such as weight, length
and area; they suggested that students could learn the logistics
of "1½ times" something by starting with
"3 x" and descending to "2 x," "1
x," "O times," back to "1 x," and
then "1 and a half times." Burton and Knifong (1983)
also felt that many concrete referents should be used with
division, to avoid the arbitrary "15 divided by 3 is
5 because 5 x 3 is 15." Fischbein, Deri, Nello and Marino
(1985) based their support for using more than one example
of a concept on their conclusion that the concrete models
children did use were so limited. They felt that in order
to avoid the type of oversimplification (as in where children
could "halve" but not split into thirds), the simple
models the children already had needed to be worked with and
expanded so that they would not interfere with problem-solving.
He agrees that division involves many concrete manifestations,
but thinks children are not advanced enough to understand
many examples, such as velocity, area, and volume. Kulm(1982)
summed up the usefulness of different models for early adolescents:
"Solution of related problems helps subjects to focus
on relevant strategies, but distantly related structures or
different contexts appear to interfere with transfer in solving
some problems." Jackson (1979) stresses that manipulatives
need to be objects of "relevant characteristics,"
and removes from consideration toys and games such as dice,
spinners, number lines, puzzles, and cards that present or
generate abstract problems; these are merely more interesting
representations of the symbolic, not the concrete.
Juraschek (1983) explored the application of Piaget's "learning
cycles" to mathematics in the middle schools. A "learning
cycle" has three stages: exploration, concept introduction,
and application. Concrete exploration should be designed so
learners confront information slightly beyond their understanding.
Ideally, the student should feel this lack of understanding
(a manifestation of "reflection") so the teacher
can initiate the second, "concept introduction"
phase. The teacher then introduces the concept that will allay
that lack of understanding and explains it; the student, finally,
applies and practices the use of the skill, proving and re-proving
to himself the validity of the concept.
By keeping the "application" stage concrete, the
connection between the learner's "simple intuitive"
understanding and the mathematics he is expected to learn
has a much better chance of being established. Here, the child's
intuitive models can change and advance. However, If the "application"
is abstract, then the student must not only jump from being
introduced to the concept to applying it, but must simultaneously
leap from concrete to symbolic. When confronted with this,
most students rely on learning the algorithm for correct calculation.
Their "common sense" model may be even more confused
by the "lack of understanding" imposed upon it,
despite their having mastered a new computation algorithm.
Likewise, if the information is too distant from the child's
intuitive model, that simple model will not change, creating
what Fischbein, et. al., hypothesizes: that children's models
actually limit their ability to expand their mathematical
Children need to learn that their knowledge can be expanded,
and their intuitive models changed - not by giving them "how
to change your intuitive model" lessons, but by practice
in doing it. Learning that while truth does not change, we
can change what we do with it is called "self-regulation"
by Garafalo and Lester (1985). They break down cognition and
metacognition into existing knowledge/models. Only five of
the 17 steps within the four stages are based on imitating
a known pattern; the others all involve some sort of evaluation
on a "how does this solve my problem" level, as
opposed to "what do I know how to do with these numbers"
approach. They noted that when students were trained in problem
solving strategies, they were most successful when the training
included "self-regulation" skills that tried to
fit the problem into the student's known universe.
Hawkins (1985) considered the process of learning a "basic
skill," warning against overemphasis of the products
of knowledge over the processes. Manning (1984) also focuses
on processes, and recommends that students be taught to notice
when they don't understand instructions, when they don't have
enough material to solve a problem, when they should slow
down or recheck algorithms, and the relative ease/difficulty
of different tasks. However, "problem-solving training"
has an inherent potential for further bastardizing insight
by inflicting upon a student an algorithm for inducing insight
- a memorized algorithm for the purpose of avoiding learning
by memorizing algorithms. The teacher does need to see that
learning is not a straight path to reproduction, but the child
hardly needs to be able to reproduce the steps he takes toward
The study that most harmoniously married the principles of
Piaget and Dienes and Semdadeni was that of Wiebe (1983).
He concentrated on building a structured framework so that
different models would be used to progress gradually from
the concrete to the abstract. Not every example conceivable
to the adult mind would be feasible, but by starting with
manipulatives that correspond closest to the simplest concrete
model that the child already owned, and then extending that
to slightly more abstract representation (but still a manipulative
representation), and progressing deeper and deeper into abstractness
the connection is made between concrete experience and the
general concept, as well as its verbal explanation. It is
quite apparent that this approach is ideally suited to the
student in the middle or secondary schools who is just learning
to think formally, and has gained linguistic skills that can
offer insights into mathematics that were completely impossible
when the computations were first introduced.
IV. EXAMPLES OF MANIPULATIVES: USABLE, UNUSABLE AND MISUSABLE
While manipulatives to explain the basic nature of numbers
and the meaning of numerals have been fairly well explored
and refined, manipulatives used to demonstrate higher level
concepts are not so well researched, especially with regard
to their use as has been outlined here. I will explain and
analyze the use of some manipulatives that can be used in
the latter circumstances. Some manipulatives are very broad-based
and can be used to exemplify many concepts. One of these is
the balance. Dienes and others (Burns, 1975 and Moses and
Speer, 1982), advocate using the balance extensively to explain
basic operations (addition, etc.; see Fig. 1).. FIGURE ONE
The teacher places a weight on the "six" on one
side, and weights on the "two" and "four"
hooks on the other side, demonstrating that "2 + 4=6;"
three weights placed on the "two" hook on one side
will balance one weight placed on the "six " hook
on the other side. Subtraction and division are explained
as the inverses of addition and multiplication. (The child
figures out what "8-2" is by asking "2 + what=8?")
This is the scenario Dienes outlines in Building up Mathematics.
However, he falls heavily into the "adult insight"
trap several times over. Firstly, the nature of the balance
demands that the student understand levers, or at least seesaws.
To many students, the display of "2 + 4=6" could
just as easily be interpreted as "2=1." After all,
two equal weights on one side balance one -also equal - weight
on the other. Secondly, explaining subtraction and division
in terms of their inverses requires a quantum leap into interpreting
symbols. A "double pan" version of the balance,
however, is an elegant vehicle for explaining concepts from
equality to addition to algebraic unknowns (see Fig. 2).
FIGURE TWO Handmade (from Civil War cannon shot embedded
in styrofoam for weights), or lifted from the science lab,
the balance can be used as a manipulative or for demonstration.
The following lesson plan outlines how addition can be introduced
and progressed to mathematical symbols.
MATERIALS NEEDED: scale weights (pref. in values of 1,
10 and 100), pennies and dimes (may be substituted with
different color chips, but these don't have familiar values)
- Introduce the scale, explaining how it works and the=and
not equal signs.
- Draw a diagram of it on the board.
- At the same time, have a student hand out worksheets
- . Explain that equality means having the same value
or amount of something; demonstrate or have students demonstrate
examples of equality (3=3).
- Put 5 "ones" on one side of the scale. Ask
what number of weights you have put on the scale and write
it on the board.
- Have students put five pennies on the same side of
the scale on their worksheet, and write "5"
on the first line under it.
- Put 3 more weights on that same side. Ask how you would
express that action without erasing anything. Write +3
next to the five (on board and papers.)
- Ask how many weights you have now; ask how many you'd
put on the other side to balance it out. Do so, writing
the number on the board. Have students mimic with coins.
- Ask for other ways of saying "8" and show
and write this.
- You may want to do several examples like this.
- Erase everything but the scale; write examples of simple
equations under the scale, perhaps some like "3+3=2
+ 4" and have students prove their equality or inequality
on the scale.
The balance can be used for many other kinds of equations.
It is especially valuable in introducing the concept of
the "unknown" in basic algebra. The following
is a lesson plan for demonstrating "n + 1=7."
(It assumes prerequisite understanding of equality and addition.)
- Put some weights in a (lightweight) bag marked "a"
and place it on one side of the scale. Write "a"
under the drawing on the board; have students put "a"
on their worksheets under the scale.
- Have a student come forth and figure out how many are
in the bag by balancing the scale. Write the represented
number under the other side on the board; have students
do the same on paper.
- Have a student put more weights on the "a"
side (not in the bag). Write the corresponding addition
operation underneath the drawing. See how many weights
must be added to the other side to balance; figure out
how to express that in symbols.
- Figure out what number is represented on the bagless
side of the scale. Explain that you can interchange equal
Review or progress in concept development as reception
dictates; there is much room in either direction for adaptability.
The balance can also be used to exemplify multiplication
(division gets difficult, but with the right engineer to
design units which could be attached like Legos into blocks,
it might be feasible). Semdadeni cites several other viable
illustrations of multiplication that prove useful in generalizing
the multiplication concept to include 1 and 0:
- Start with the concept concretized: 3 boxes, each with
four cookies, as 3 x 4=12.
- Vary one factor (either cookies or boxes); keep the
other constant. Decrease that factor all the way to zero.
- If there is trouble with "no boxes, 4 cookies
in each," explain: Ms. Jones bought 3 boxes of cookies
with 4 in each box: how many cookies did she buy? Ms.
Jones bought 2 boxes... how many cookies? Ms. Jones bought
1 box... how many cookies? Ms. Jones bought 0 boxes..
how many cookies did she buy?
Other multiplication scenarios include, in order of level
of abstraction: arrays (rows x columns, as in a parking
lot or case of soda), area, number lines (or distances),
and counting forward by numbers (repeated addition). Many
of these scenarios can be adapted to division as well.
Multiplication and division are two operations that a child
can excel in without ever understanding; fractions involve
much more complex algorithms and are much harder to apply
correctly in problem solving. Much time and energy and research
has been devoted to developing strategies for teaching them.
Sets and areas of cookies and cakes are perhaps the most
common, but it becomes difficult to demonstrate equivalence
and operations with these. Burns (1979 suggests the following
game, once the basic concept of the fraction has been explored:
MATERIALS: dice marked with fractions: "1/2, 1/4/,
1/4, 1/8, 1/8, 1/16" construction paper and rulers,
whole strips of construction paper of one color, and fractions
thereof in another.
GAME ONE: OBJECT- to cover your strip exactly. Roll the
die and put that fraction of the strip on it; you have to
completely and exactly cover it.
GAME TWO; Start with two halves covering your whole strip;
you want to uncover your whole step. Roll to see what you
can uncover. Of course, if you're allowed to uncover "1/4"
then you have to trade in your "1/2" for two of
the "1/4" pieces. Burns recommends that written
equivalent fraction exercises be illustrated and performed
with the paper strips. Depending on the age level, it may
or may not be possible to generalize the concept of fractions
so that the relationship of part to whole regardless of
the size of the object in question. In keeping with the
idea of multiple-embodiment, fractions can also be referenced
to sets of objects, volume measurements, hours (1/2 hour,
1/4 hour), and distance on number lines.
Multiplication of fractions is challenging to represent,
but can be done, especially if the child's understanding
of the "half" concept is used, e.g. half of a
half of a pie. Areas of rectangles can be used, too (what
is the area of a 6 x 1/2 in. rectangle?). The concept can
also be approached from whole numbers: If you have 6 kg
of sugar, and empty 2 kg bags, how many hags can you fill?
If you have 6 kg, and 1 kg bags... if you have 6 kg. of
sugar and 1/2 kg. bags... Some research has focused on using
manipulatives to illustrate even higher skills through geometry,
algebra and calculus (but they tend to turn into physics
at some point). Most of these involve a good deal of abstractness
- but if someone can understand the prerequisite concepts,
they have the abstracting ability to understand it. Many
students with agile memories are rudely awakened when they
find themselves expected to apply concepts and solve problems
in more advanced courses. Many teachers are frustrated with
students in secondary mathematics courses who have relied
exclusively on algorithms and struggle with concepts.
V. OTHER TRANSITIONS FROM CONCRETE TO ABSTRACT
There are several other channels through which a connection
can be made between a child's intuitive models and more
abstract concepts. One is to develop a child's ability to
visualize mathematics. For the most part, visualization
is associated with high-level mathematics conceptualization,
but is considered a hindrance in elementary and middle school
and most secondary mathematics; linguistic processing requires
far less short-term memory space (Lea and Clements, 1981).
However, while visualization may be a poor way to solve
many mathematics problems, it can be an efficient and effective
way of learning mathematics (Kend and Hedger, 1980). (Few
people if any claim that manipulatives should be used to
solve problems once a concept is understood and abstracted.)
Kend and Hedger found visualization a useful skill in analyzing
problems, especially when the skills were practiced and
refined. They incorporated a transitional stage from the
manipulative to imagined manipulatives, the practice of
which could be highly useful at advanced levels where manipulatives
are less likely to be found. Kulm and Days (1979) and Bell
and Bell (1985) emphasize using linguistic and verbal skills.
Kulm and Days offer the following suggestions for bridging
intuitive and formal processes:
- making drawings or sketches
- lists or tables of data
- graphing relationships
- classifying objects/concepts into similar categories
- breaking problems into parts
Bell and Bell concentrated on using written synthesis and
explanations, with carefully structured and practiced written
exercises incorporated into the mathematics curriculum.
Manipulatives are not the be-all to end-all in teaching
mathematics. They can be a waste of time and effort. Concept
development cannot replace learning algorithms for computation;
children must have a strong command of computation to apply
the concepts. However, all indications are that they can
be very useful in middle and secondary education, if they
are wisely planned and executed to build a firm, concrete
model for abstract mathematical concepts. They do not always
succeed; when they fail, it is generally because of one
of the following: 1) the child is not developmentally ready
for the concept, 2) the child has not mastered prerequisite
concepts, 3) the model used is too abstract for the student,
4) the instruction shifts to symbolic before the child has
developed the cognitive concrete model to embrace the new
concept, or 5) the gap between the model and its symbolic
representation is too large (Wiebe, 1983). The manipulative
must be a model from which the child can gather meaning
from his actions; gathering 125 popsicle sticks doesn't
have much to do with what "125" means; "borrowing"
from piles to subtract does. Without a firm grasp of what
a child can understand, it is easy to slip abstraction into
teaching, especially if making the lesson manipulative is
expected te compensate for any abstractions. It is the responsibility
of educators to detect the developmental level of the children
they are teaching, and work to build the child's insight
to accommodate new horizons..
BIBLIOGRAPHY ******VITALLY IMPORTANT OR STIMULATING******
Behr, Merlyn J., Ipke Wachsmuth and Thomas R. Post. Construct
a Sum: A measure of Children's Understanding of fraction
size. Journal for Research in Mathematics Education v. 16
no. 2 3/85
Bell, Eliz. S. and Ronald N. Bell. Writing and Mathematical
Problem Solving: Arguments in Favor of Synthesis. School
Science and Mathematics v. 85 n. 3 3/85
Blume, Glendon W. and Charles E. Mitchell. Distributivity:
a useful model or an abstract entity? School Science and
Mathematics v. 83 n. 3 3/83
Booth, Lesley R. Child-methods in Secondary Mathematics.
Education in Science and Mathematics v. 12 n. 1 2/81
Burns, Marilyn. Computation Activities that Make Connections.
Learning, v. 7 n. 5 5/79
Days, Harold C., Grayson H. Wheatley and Gerald Kulm. Problem
Structure, Cognitive Level, and Problem Solving Performance.
Journal for Research in Mathematics Education v. 14 n. 1
Dienes, Zoltan Paul. Building Up Mathematics. Hutchinson
Ed. LTD London, 1961.
Fischbein, Efraim, Maria Deri, Maria Sainati Nello, and
Maria Sciolis Marino. The Role of Implicit Models in Solving
Verbal Problems In Multiplication and Division. Journal
for Research in Mathematics Education v. 16 n. 1 1/85
Friedman, Morton. The Manipulative Materials Strategy:
The Latest Pied Piper? Journal for Research in Mathematics
Education v.9 n. 1 1/78
Freudenthal, Hans. Major Problems of Mathematics Education.
Education in Science and Mathematics v. 12 n. 2 5/81
Halford, G.S. An Approach to the Definition of Cognitive
Developmental Stages in School Mathematics. British Journal
of Educ. Psych. v. 48 n. 3 11/78
Jackson, Robert L. Hands-on Math: Misconceptions and Abuses.
Learning, vol. 7 n. 5 1/79
Juraschek, William. Piaget and Middle School Mathematics.
School Science and Mathematics v. 83 n. 1 1/83
Kent, David and Keith Hedger. Growing Tall. Education in
Science and Mathematics v.11 n. 2 5/80
Kuchemann Deitmar. Cognitive Demand of Secondary School
Mathematics Items. Education in Science and Mathematics
v. 12 n. 3 8/81
Kulm, Gerald. The Development of Mathematics Problem Solving
Ability in Early Adolescence. School Science and Mathematics.
v.82 n. 8 12/82
Kulm, Gerald and Harold Days. Information transfer in solving
problems. Journal for Research in Mathematics Education
v.10 n. 2 3/79
O'Brien, Thomas C. and Shirley A. Casey. Children Learning
Multiplication Part I. School Science and Mathematics v.
83 n. 3 3/83
O'Brien, Thomas C. and Shirley A. Casey. Children Learning
Multiplication Part II. School Science and Mathematics v.
83 n. 5 5/83
Semadeni, Zbigniew. A Principle of Concretization Permanence
for the Formation of Arithmetical Concepts. Education in
Science and Mathematics v. 15 n. 4 11/84
Skemp, Richard. Schematic Learning: The Process of Learning
Mathematics. ed. L.R. Chapman Pergamon Press, N.Y. 1972.
Souviney, Randall J. Cognitive Competence and Mathematical
Development. Journal for Research in Mathematics Education
V.11 n. 3 5.80
Vest, Floyd. Using Physical Models to Explain a Division
Algorithm. Scholl Science and Mathematics v. 85 n. 3 3/85.
Wearne-Hiebert, Diana C. and James Hiebert. Jr. High School
Students Understanding of Fractions. School Science and
Mathematics v. 83 n. 2 2/83
Webb, Norman L. Processes, Conceptual Knowledge, and Mathematical
Problem-Solving Ability. Journal for Research in Mathematics
Education v. 10 n. 2 3/79.
Wiebe, James H. Physical Models for Symbolic Representations
in Arithmetic. School Science and Mathematics v. 83 n. 6
BIBLIOGRAPHY **of secondary importance**
Barr, David C. A comparison of three methods of introducing
2-digit numeration. Journal of Research in Mathematics Education
v. 9 n. 1 Jan 1978 p. 33-43.
Barody, Arthur J. Pitfalls in Equating Informal Arithmetic
Procedures with Specific Mathematical Concepts. Journal
of Research in Mathematics Education. v. 16 n. 3 5/85 p.
Barody, Arthur J. Mastery of Basic Number Combinations:
Internalization of relationships or facts? Journal for Research
in Mathematics Education v. 16 n. 2 3/85 p. 83-98.
Bender, Peter and Alfred Schrieber. The Principle of Operative
Concept Formation in Geometry Teaching. Education in Science
and Mathematics v. 11 n. 1 2/80 p. 59-90.
Burton, Grace M. and J. Dan Knifong. What does division
mean? School Science and Mathematics v. 83 n. 5 10/83
Cooney, Thomas J., Edward J. Davis and K.B. Henderson.
Dynamics of teaching secondary school mathematics. Houghton
Mifflin Co., Boston, 1975.
Day, Mary Carol and C. Addison Stone. Developmental and
individual differences in the use of the Control-of-Variables
strategy. Journal of Educational Psychology v. 74 n. 5 10/82.
Garofalo, Joe and Frank K. Lester. Metacognition, cognitive
monitoring and mathematical performance. Journal for Research
in Mathematics Education. v. 16 n. 3 5/85.
Hawkins, Vincent J. A model of the 8 basic trigonometric
identities - and others. School Science and Mathematics
v. 85 n. 2 2/85 Hiatt, Arthur A. Basic Skills: What are
they? Mathematics Teacher v. 72 n. 2 2/79
Hiebert, James and Lowell H. Tonnesen. Development of the
fraction concept in two physical contexts: an exploratory
investigation. Journal for research in Mathematics Education
v. 9 n. 5 11/78.
Hoyles, Celia. The pupil's view of mathematics learning.
Educational Studies in Mathematics v. 13 n. 4 11/82.
Kohn, Judith B. A physical model for operations with integers.
Mathematics Teacher v. 71 n. 9 12/78.
LaBar, Martin. A practical use of exponents: eliminating
an unnecessary search for a compound interest rate. School
Science and Mathematics v. 82 n. 5 5-6/82.
Leah, Glen and M.A. (Ken) Clements. Spatial ability, visual
imagery and Mathematical performance. Education Studies
in Mathematics v. 12 n. 3 8/81.
Manning, Brenda H. A self-communication structure for learning
mathematics. School Science and Mathematics v. 84 n. 1 1/84.
McLeod, Terry M. and Stephen W. Armstrong. Learning disabilities
in mathematics - skill deficits and remedial approaches
at the intermediate and secondary level. Learning Disabilities
Quarterly v. 5 n. 3 summer 82.
Moses, Barbara and William R. Speer. Keep Your Balance.
School Science and Mathematics. v.82 n.5 5-6/82.
Moyer, John C., Larry Bowder, Judith Threadgill-Sowder,
and Margaret B. Moyer. Story problem formats: drawn vs.
verbal vs. telegraphic. Journal for Research in Mathematics
Education v. 15 n. 5 11/84
Pratton, Jerry and Lloyd W. Hales. The effects of active
participation on student learning. Journal of Educational
Research v. 79 n. 4 4/86.
Prigge, Glenn R. The differential effectors of the use
of manipulative aids on the learning of geometric concepts
by elementary school children. Journal for Research in Mathematics
Education v.9 n. 5 1//78.
Swafford, Jane O. and Henry S. Kepner, Jr. The evaluation
of an application-oriented first-year algebra program. Journal
for Research in Mathematics Education v.11 n. 3 5/80.
Tatsuoka, Kikumi K. Changes in error types over learning
stages. Journal of Educational Psychology v.76 n. 1 2/84
Thompson, Alba Gonzalez. The relationship of teachers conceptions
of mathematics and mathematics teaching to instructional
practice. Education Studies in Mathematics v.15 n.2 5/84.
Threadgill-Sowder, Judith A. and Patricia A. Julifs. Manipulative
vs. Symbolic Approaches to teaching logical connectives
in junior high school: an aptitude x treatment study. Journal
for Research in Mathematics Education v.11 n.5 11/80.
copyright © 1998-2000
Susan Jones, Resource Room. All Rights Reserved.