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### 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 concepts.

(Warning: This is long. )

### I. Introduction

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 efficient computation.

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

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 follow:

• 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 knowledge.

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 learning.

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) and worksheets

1. Introduce the scale, explaining how it works and the=and not equal signs.
2. Draw a diagram of it on the board.
3. At the same time, have a student hand out worksheets (appendix B).
4. . Explain that equality means having the same value or amount of something; demonstrate or have students demonstrate examples of equality (3=3).
5. 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.
6. Have students put five pennies on the same side of the scale on their worksheet, and write "5" on the first line under it.
7. 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.)
8. 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.
9. Ask for other ways of saying "8" and show and write this.
10. You may want to do several examples like this.
11. 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.)

1. 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.
2. 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.
3. 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.
4. Figure out what number is represented on the bagless side of the scale. Explain that you can interchange equal expressions.

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:

2. Vary one factor (either cookies or boxes); keep the other constant. Decrease that factor all the way to zero.
3. 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:

1. making drawings or sketches
2. lists or tables of data
3. graphing relationships
4. classifying objects/concepts into similar categories
5. 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.

### VI. CONCLUSIONS

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..

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