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Lesh, R., Post, T., & Behr, M. (1987). Dienes revisited: Multiple embodiments in computer environments. In I. Wirsup & R. Streit (Eds.), Development in School Mathematics Education Around the World (pp. 647-680). Reston, VA: National Council of Teachers of Mathematics.

 

Dienes Revisited: Multiple Embodiments in Computer Environments

Richard Lesh
WICAT and Northwestern University

Thomas Post
University of Minnesota

Merlyn Behr
Northern Illinois University

 

In the early 1960s, the names Cuisenaire, Dienes, Gottegno, Montessori, and Piaget had become fashionable in mathematics education theory development. Each was associated with a "mathematics laboratory" approach to instruction, based on activities for children using concrete materials. However, these theoretical perspectives had little impact on actual classroom practice.

This paper will describe significant ways in which computer-based instruction can encourage teachers and students to make greater use of activities with concrete materials, while at the same time providing a useful context in which to implement some of the best instructional strategies associated with mathematics laboratories - including some strategies which have never worked well using concrete materials.

A Closer Look at Dienes’ Instructional Principles

Among the "mathematics laboratory" theorists, Dienes (1960) was the most specific in his recommendations for mathematics teachers. Dienes’ perspective was based on the following four principles:

1. The constructive principle: Dienes claimed that mathematical ideas must be constructed. He treated mathematical ideas as abstract structural metaphors; that is, he believed the structure of a given mathematical idea cannot be abstracted from concrete objects, but instead must be abstracted from relational/ operational/organizational systems that humans impose on sets of objects. For example, Dienes claimed that when his famous arithmetic blocks (see Figure 1) are used to teach the "regrouping structure" of our base-ten numeration system, children must first organize the blocks using an appropriate system of relations and operations. Only after these organizational systems have been constructed can children use the materials as a model that embodies the underlying structure.

 

Figure 1

 

Part of what Dienes meant when he claimed that mathematical ideas must be constructed is that when arithmetic blocks, for example, are used to teach the structure of our numeration system, the relevant systems must first be "read into" the set of materials in the form of concrete activities. Only after this "reading in" has taken place can the structure be "read out" as abstract relational/ operational networks.

Let’s consider an example other than arithmetic blocks. Figure 2 shows a diagram that Dienes used to describe a network of relations that his students constructed when working with material designed to help them learn about an algebraic finite group. Notice in Figure 2 that the mathematical structure is pictured not simply as a network of relations and operations; it is also a network which has a pattern (or structure) in which the whole is more than the sum of its parts. According to Dienes, it is the pattern itself (and properties of the pattern) that must be abstracted. So, in activities with blocks, it is not so much the blocks that the child must organize as it is his or her own activities on the blocks; and it is the pattern of the activities that forms the basis for the abstraction. When individual activities cease to be treated as isolated actions and start to be treated as part of a systematic pattern of activities, the student begins to shift from playing with blocks to playing with mathematical structures.

 

Figure 2

 

Piaget furnished some of the best evidence to show that the first relational/ operational/organizational systems that children use are often based on overt actions (e.g., ordering, combining, separating) performed on concrete objects. Yet, even at an early age, children’s organizing and ordering activities are often carried out "in their heads" (often accompanied by spoken language) rather than through overt actions (Vygotsky, 1962) - and often are applied to pictures or diagrams rather than to concrete objects. In fact, when youngsters become too involved in the details of object manipulation, they often "lose sight of the forest because attention is focused on the trees" and fail to notice the patterns or structures that underly their own behaviors.

Using concrete materials does not guarantee the use of "concrete activities." In fact, using concrete materials may actually hinder conceptual development at the stages of learning where attention must shift from isolated actions toward systems of actions. Yet, when concrete materials have been used in instruction, more concern is often given to the "concreteness" of the materials than to the "activeness" of the activity - as though the abstraction were from the materials rather than from the structure that must be imposed on the materials.

The fact that whole mathematical structures have properties that go beyond those of constituent elements means that each individual child must construct these systems on a lower plane (e.g., using concrete objects) before they can be abstracted on a higher plane. Nonetheless, the role of the materials is simply to serve as a support for the student’s mental activities, not to serve as the direct basis for abstraction.

Dienes’ constructive principle implies two distinct corollaries: (1) mathematical relations and operations are considered to be abstracted from activities rather than from the materials, and (2) systems of activities must be constructed before they attain the status of mathematical structures.

2. The multiple embodiment principle: According to Dienes, mathematical ideas cannot be abstracted from single isolated patterns (or models which "embody" these patterns) any more than simpler abstractions can occur from single instances. Instead, mathematical abstractions occur when students recognize structural similarities shared by several related models. For example, when base-ten blocks are used to teach arithmetic regrouping operations, Dienes claimed that it is not enough for students to work with a single model; they must also investigate "mappings" to other models, such as bundling sticks or an abacus. So, in laboratory forms of instruction, a primary goal is to help students recognize how patterns of relationships in one model correspond to patterns of relationships in another model, as illustrated in Figure 3.

 

Figure 3

 

3. The dynamic principle: According to Dienes, the systems that must be abstracted from structurally related models are not simply "static patterns"; the are dynamic, that is, the most important features to be recognized have to with the way transformations within one model correspond to relate transformations in a second model. To reflect this dynamic aspect of mathematical structures, Figure 3 can be modified to include not only translation between models but also transformations within models, as shown in Figure 4.

 

Figure 4

In Figure 4, a transformation followed by a translation should produce the same result as a translation followed by a transformation; that is, the mathematical property of homomorphism should be true:

T(A*B) = T(A)*T(B)

4. The perceptual variability principle: Regardless of whether the material used to embody a given model is a set of concrete objects, graphics, written symbols, spoken language, or some other representational system, models always have some characteristics that the modeled system does not have - or conversely, they fail to have some characteristics that the modeled system does have. Consequently, to select a small number of especially appropriate models to embody a given system, the following characteristics should be used:

  • Irrelevant characteristics should vary from one model to another so that these characteristics will be "washed out" of the resulting abstraction.
  • Collectively, models should illustrate all of the most important structural characteristics of the modeled system.

The Rare Use of Concrete Mathematical Laboratories Principles

Why have Dienes’ mathematical laboratory principles made so little impact on actual classroom construction? Even when concrete materials like arithmetic blocks are used in mathematics instruction,

  • They tend to be used only in classroom demonstrations in which students are passive observers rather than constructing participants (a violation of the constructive principle).
  • They tend to be the only concrete model used, with attention being focused on the objects themselves rather than "mappings" (or translations) from one model to another (a violation of the multiple embodiment principle).
  • They can be cumbersome. To satisfy Dienes’ dynamic principles students must not only investigate transformations within a single model, they must also investigate corresponding transformations within other models. Yet with concrete materials, such correspondences are time-consuming to act out and difficult to keep track of, even in well-rehearsed teacher demonstrations. It is virtually impossible for students to create these demonstrations on their own and to benefit from them without guidance.

Often more concern is given to the "concreteness" of the materials than to the "activeness" of the activity - as though the abstraction were from the materials rather than the structure that must be imposed on the materials. The purpose of the materials is seldom clear.

Beyond the preceding kinds of difficulties, perhaps the main reason why Dienes’ instructional principles are so rarely implemented in classroom instruction is that teachers seldom understand or accept Dienes’ perspective about the nature of mathematics because they tend to view mathematics as simply a collection of isolated rules for manipulating symbols. Therefore, work with concrete objects is often viewed as "too baby-ish," when in fact the depth of understanding needed to model dynamic mathematical systems using concrete materials often far exceeds that needed to treat mathematics simply as a set of computational skills (Bell, Fuson, and Lesh 1978).

Another fact that discourages teachers from using concrete materials is that when such materials are used, students’ misconceptions tend to become clearly visible. Therefore, because many such misconceptions were formerly hidden behind superficial computational proficiency, the misconceptions that emerge actually seem to be caused by the use of concrete materials (rather than simply being revealed by them). Finally, the misconceptions that emerge when concrete activities are used tend to be unusually difficult to correct because they are so basic.

It does teachers little good to identify students’ misunderstandings if they are unable to correct them. So, if a class of errors is easy to ignore by focusing exclusively on low-level skills, and if higher-order errors are viewed as impossible to correct, then a seemingly reasonable teaching strategy is to devote valuable classroom time to objectives that can indeed be attained. Using concrete materials to correct misconceptions or teach conceptions presupposes

  • A firm belief in the ultimate payoff that will result.
  • A clear understanding about the higher-level misunderstandings that may be avoided.
  • Confidence that misunderstandings that do in fact emerge can be corrected.

This paper will briefly describe some ways computers can be used to address the preceding issues. Comments will be based on results from three recent or current NSF-funded projects on Rational Numbers Concepts (RN), Proportional Reasoning (PR), and Applied Mathematical Problem Solving (AMPS), as well as on recent research conducted at the World Institute for Computer Assisted Teaching (WICAT). 1

Dienes’ Multiple Embodiment Principle:

This section will describe some important factors to take into account in computer-based implementation of Dienes’ multiple embodiment principle.

Past RN, PR, and AMPS publications (e.g., Lesh 1981; Lesh, Landau and Hamilton 1980; and Behr, Lesh, Post, and Silver 1983) have identified five distinct types of representation systems that occur in mathematics learning and problem solving (see Figure 5). These are (a) "scripts" in which knowledge is organized around "real world" events that serve as models for interpreting and solving other kinds of problem situations; (b) manipulative models (such as Cuisenaire rods, arithmetic blocks, fraction bars, number lines, etc.) in which the "elements" in the system have little meaning per se, but the "built-in" relationships and operations fit many everyday situations; (c) pictures which, like manipulative models, can be internalized as "images"; (d) spoken languages, including specialized sub-languages (e.g., logic); and (e) written symbols which, like spoken languages, can involve specialized sentences and phrases, such as (x + 3 = 7, A’ È B’ = (A Ç B)’) , as well as normal English sentences and phrases.

 

Figure 5

 

To illustrate why translations between models in different representational systems are important, consider the following examples.

Figure 6 shows a simple type of question that space considerations seldom permit in textbooks and pencil-and-paper tests, but which is quite easy to generate in computer-based instructional materials. The question, taken from a written test on "rational number relations and proportions" used in our RN and PR projects, illustrates a "written symbol to picture" translation. The aim is to require students to establish a relationship from one representational system to another while preserving certain structural characteristics and meaning.

Figure 6

 

Figure 7 shows another item from the same "relations and proportions" test as Figure 6, but it was adapted from a recent "National Assessment" examination (Carpenter, et al. 1981). To answer the question in Figure 7 correctly, the student’s primary task is to perform a computational transformation.

 

The ratio of boys to girls in a class is 3 to 8. How many girls are in the class if there are 9 boys?

A.17    B. 14    C. 24    D. not given    E. I don’t know

Figure 7

 

Educators familiar with results from recent "National Assessments" (Carpenter, et al. 1981) may not be surprised that our students’ success rates for Figure 7 were only 11% for fourth graders, 13% for fifth graders, 30% for sixth graders, 29% for seventh graders, and 51% for eighth graders. Such performances by American students led to "Nation at Risk" reports from a number of federal agencies and professional organizations. However, success rates on the seemingly simpler question in Figure 6 were even lower: 4% for fourth graders, 8% for fifth graders, 19% for sixth graders, 21% for seventh graders, and 24% for eighth graders. On the translation in Figure 6, only one in four students answered correctly! Forty-three percent selected answer choice a; 4% selected b; 15% selected c; 34% selected d; 3% selected e; and 2% did not give a response.

One major conclusion from our RN and PR research is apparent from the preceding examples: not only do most fourth- through eighth-grade students have seriously deficient understandings with "word problems" and "pencil and paper computations," but many have equally deficient understandings about the models and languages needed to represent and manipulate these ideas. Furthermore, we have found that the ability to do these translations is a significant factor influencing both mathematical learning and problem-solving performance (Behr, Lesh, Post, and Wachsmuth 1985; Post 1986). For example, when we say that students "understand" an idea such as "1/3," we mean that they can (a) recognize the idea embedded in a variety of qualitatively different representational systems, (b) flexibly manipulate the idea within given representational systems, and (c) accurately translate the idea from one system to another. Translation processes are implicit in a variety of common techniques used to investigate whether a student "understands" a given textbook word problem (e.g., "Restate it in your own words." "Draw a diagram to illustrate what it’s about." "Act it out with real objects." "Describe a similar problem in a familiar situation."). Similarly, techniques for improving performance on word problems include (1) using several concrete materials to "act out" a given problem situation, (2) describing several everyday problem situations that are similar to a given concrete model, or (3) writing equations to describe a series of word problems, delaying the actual solutions until the student becomes proficient at this descriptive phase.

To diagnose a student’s learning difficulties or to identify instructional opportunities, teachers and computers can generate a variety of useful questions by presenting an idea in one representational mode and asking the student to render the same idea in another mode. Then, if diagnostic questions indicate unusual difficulties with one of the processes in Figure 5, other processes in the diagram can be used to strengthen or bypass it. For example, a student who has difficulty translating from situations to written symbols may find it helpful to begin by translating real situations to spoken words, and then translate spoken words to written symbols; or it may be useful to practice the inverse of the troublesome translation (i.e., identifying familiar situations that fit given equations).

Not only are the translation processes in Figure 5 important components of understanding a given idea, they also correspond to some of the most important "modeling" processes needed to use this idea in everyday situations. Essential features of modeling include (1) simplifying the original situation by ignoring irrelevant characteristics in order to focus on more relevant factors, (2) establishing a mapping between the original situation and the "model," (3) investigating the properties of the model in order to generate predictions about the original situation, (4) translating (or mapping) the predictions back into the original situation, and (5) checking to see whether the translated prediction is useful.

Here is an example where the preceding steps are used to solve a standard algebra word problem.

Al has an after-school job. He earns $6 per hour if he works 15 hours per week. If he works more than 15 hours, he gets paid "time and a half" for overtime. How many hours must Al work to earn $135 during one week?

To solve this problem, students may begin by paraphrasing the given "English sentence" into their own words, perhaps accompanied by a diagram or picture of the situation. Next, the description of the problem may be translated into an "algebraic sentence":

(6 x 15) + 9(x - 15) = 135

Then, a series of algebraic transformations may be used to convert this algebraic model into an arithmetic sentence that is sufficient to find the answer. The final transformed description is:

 

Finally, by using a series of arithmetic simplifications, this arithmetic sentence can be reduced to:

x = 20

So, beyond paraphrasing and diagramming, the entire solution process involves three significant translations: (1) from an English sentence to an algebraic sentence, (2) from an algebraic sentence to an arithmetic sentence, and (3) from an arithmetic sentence back into the original problem situation.

Notice that the algebraic sentence that most naturally describes the preceding problem situation does not immediately fit an arithmetic computation procedure. This possibility of "first describing and then calculating" is one of the key features that makes algebra different from arithmetic.

As the preceding problem illustrates, problem solving often occurs by (1) translating from the "given situation" to a mathematical model, (2) transforming the model so that desired results are apparent, and (3) translating the model-based result back into the original problem situation to see if it is useful. However, the modeling process usually is not this simple. Instead, modeling students frequently use several representation systems (or models) in series or in parallel, with each depicting only a portion of the given problem situation (see Figure 8).

 

Figure 8

 

In RN and PR research involving realistic textbook word problems, we found that students seldom work through solutions in a single representational mode (Lesh, Landau, and Hamilton 1983). In fact, many realistic problem-solving situations are inherently anti-modal from the outset. The following two pizza problems illustrate this point.

Show a sixth grader one-fourth of a real pizza, and then ask, "If I eat this much pizza, and then one-third of another pizza, how much will I have eaten all together?"

Show a sixth grader one-third of a real pizza, and then ask, "If I already ate one-fourth of a pizza, and now eat this much, how much will I have eaten all together?"

Neither of these pizza problems is a "symbol-symbol" or "word-word" problem. Instead, the "givens" in both problems include (1) a real object (a piece of pizza) and (2) a spoken word (to represent a past or future situation). Like many realistic problems in which mathematics is used, the situation in these two pizza problems is inherently multi-modal. Each of the problems is a "pizza-word" problem in which one of the students’ difficulties is translating the two givens into a homogeneous representational mode so that combining is possible.

Figure 8 suggests that solutions are often characterized by several mappings from parts of the given situation to parts of several (often partly incompatible) representational systems rather than by one mapping from the whole given situation to only one representational system.

Not only may problems like the above occur in a multi-modal form, but solution paths also may weave back and forth among several representational systems, each of which is typically well-suited for representing some parts of the situation, but ill-suited for representing others. For example, in the above two problems, a student may think about the static quantities (e.g., the two pieces of pizza) in a concrete way (perhaps using pictures), but may switch to spoken language (or to written symbols) to carry out the dynamic "combining" actions (Lesh, Landau, and Hamilton 1983); that is, the student may begin a solution by translating to one representational system and may then map from this system to yet another, as illustrated in Figure 9.

 

Figure 9

 

We found that for realistic textbook word problems, the actual solution students use often combine features depicted in both Figures 7 and 8, and good problem solvers are flexible in their use of various relevant representation systems - they instinctively switch to the most efficient representation at an given point in the solution process.

The main points of this section are:

  • Not only is it important for youngsters to work with concrete models that illustrate mathematical concepts, it is also important to focus on translations from one representational system to another.
  • We have extensive evidence from the RN, PR, and AMPS projects and from field tests of computer-based materials developed by WICAT that focusing on translation abilities can significantly influence learning and problem-solving capabilities.

Dienes’ Constructive Principle and Perceptual Variability

Concerning Dienes’ constructive principle and perceptual variability principle, this section will begin with a series of examples from our own RN and PR research showing that students do not necessarily "see" relationships that seem to be "built into" certain fraction diagrams. Instead, relevant systems of relationships must be organized (constructed) and imposed on the models before their attributes will be noticed. We will then give a series of examples of computer-based lessons designed to help youngsters construct the kind of relational/operational systems needed to correctly interpret fraction and ratio diagrams.

A major goal of our RN and PR projects has been to describe the relational/ organizational systems that youngsters use to make judgments involving rational number concepts or proportional reasoning (Behr, Lesh, Post, and Silver 1983; Lesh, Landau, and Hamilton 1983). The examples that follow will show how such judgments often inherently involve networks of part-whole and part-part comparisons as well as organizational schemes in which composite units are made, up of "lower-level" units.

Figure 10 is a fraction situation which involves composite units. The unit of measure (i.e., the "whole") is "a half dozen eggs"; one-third is "two eggs"; and two thirds is "four eggs."

 

Figure 10

 

Figure 11 illustrates how distinct but related types of relationships and unit comparisons often lie at the heart of children’s confusion between "fraction judgments" and "ratio judgments.’’ Answer choices A and B involve units made up of discrete pieces, whereas answer choice C involves a continuous quantity. Answer choices B and C both involve a part-part ratio of three-to-four, whereas only answer choice A shows three-fourths shaded. The success rate for eight graders was only 61.5%. ‘Me percentage of students selecting each answer choice is shown in Figure 11.

 

Figure 11

 

Figure 12 presents a question that was answered incorrectly by more than 50% of the eighth graders. If the two pictures in Figure 12 are compared by the total amount shaded, rather than by the relationships between parts and wholes, then answer choice b would be picked. Or, if the pictures were compared by part-part comparisons, then the ratio of shaded-to-unshaded parts in picture A would be only one-to-two, whereas the ratio for picture B would be two-to-three - again making choice b the answer chosen.

 

Figure 12

 

If a picture or diagram draws a student’s attention to "perceptual distractors" that increase the possibility of basing rational number judgments inappropriate cues, then the difficulty of the preceding kinds of questions can be increased significantly (Behr, Lesh, Post, and Silver 1983; Lesh, Landau, and Hamilton 1983; Behr, Post, Lesh, and Wachsmuth 1983). For example, a particularly impressive "perceptual distractor" occurred in the clinical interview phase of our RN and PR testing programs. Students were given a Hershey chocolate bar and were asked to give the interviewer one-third of the chocolate bar. The results showed that this question was significantly more difficult when a plain Hershey bar was used instead of a Hershey bar with nuts. Why? The apparent answer was that the ten subdivisions in the plain bar created a perceptual distractor that made it more difficult for youngsters to divide the bar into thirds. On the other hand, the bar "with nuts" did not have these distracting subdivisions, and so it was easier to divide into thirds.

The influence of perceptual distractors makes it clear why Dienes’ perceptual variability principle involving the use of more than a single concrete model, is such an important feature of his instructional approach. When students have not yet constructed the relevant system of relationships (i.e., part-whole, part-part, etc.) needed to make rational number judgments, they are more likely to become victims of perceptually compelling (but misleading) cues.

Next, we will give examples of computer-based activities that have helped youngsters gradually use appropriate systems of rational number relations in organized and flexible ways. In Figure 13, the child is asked to fill in a specified fraction of a whole using the computer’s arrow keys. If help is requested, then the whole is divided into an appropriate number of parts, and the child’s task is simply to fill in the correct number of parts.

Figure 14 is similar to Figure 13 except that it deals with discrete quantities rather than with continuous quantities. The particular question illustrated in Figure 14 is relatively difficult because the "whole" consists of three rows with eighth squares in each row, rather than simply four squares. (Note: For any of the examples given in this section, the difficulty of the items can vary considerably depending on the particular numbers and pictures used; computer programs can adjust difficulty levels to fit the capabilities of individual students.)

Figure 15 is similar to Figures 13 and 14 except that rational numbers are portrayed as single "things" (as points along a number line) more than as comparisons between pairs of "things" (parts and wholes). Number lines also emphasize position and order relationship (e.g., 3/5 is "between" 2/5 and 4/5, "after" 2/5, and "before" 4/5) more than quantitative relationships (e.g., "more than" and "less than").

Figure 16 involves comparisons of pairs of fractions (or fraction pictures). If help is requested, two pictures are partitioned into an appropriate number of subsections so that the student can more easily compare the relevant attributes. In answer feedbacks, the appropriate number of parts is colored in each picture so that perceptual comparisons are easier to make.

 

Figure 13

Figure 14

Figure 15

Figure 16

 

In Figure 17, rather than giving the picture of a "whole" and asking the student to shade in a specified part, the process is reversed; the part is shown and the student is asked to use arrow keys (as in Figure 13) to make the whole. If help is requested, the part is partitioned as shown in Figure 18. A helpful answer feedback is shown in Figure 19.

 

Figure 17

Figure 18

Figure 19

 

Figure 20 is similar to Figure 17 except that discrete quantities are emphasized rather than continuous quantities. A help is shown in Figure 21.

 

Figure 20

Figure 21

 

Figure 22 goes beyond comparing pairs of fractions; the student is asked to generate fractions equivalent to a given function. Figure 23 shows the additional information given if help is requested.

 

Figure 22

Figure 23

 

Figure 24 focuses on whole sets of equivalent fractions, and Figure 25 shows one of a sequence of hints that can be given. Figures 26 and 27 show how the types of diagrams in Figures 13 through 25 can be used to go beyond relationships among fractions to deal with operations with pairs of fractions, or with more complex relationships among fractions. Similar diagrams and procedures can also be used to introduce the concepts of decimals, ratios, rates, percent, or proportions.

Figure 24

Figure 25

Figure 26

Figure 27

 

The examples in Figures 13 through 27 show how youngsters can be led to conceptually "take apart" and "re-assemble" rational number diagrams by systematically focusing attention on different relationships and attributes. The goal in such activities is to help students gradually use the relevant relational systems in an organized way.

Helping students construct a system of mathematical relationships is similar to helping students coordinate systems of overt activities like those involved in playing tennis or riding bicycles; that is, the student begins in situations in which the complexity of the system and the degree of coordination are minimal (e.g., all of the balls come waist high on the forehand side just within arm’s reach) and gradually progresses to situations that require more complex and well-coordinated systems (e.g., where "getting in position" is important). In general, building more complex systems involves:

  1. integration - i.e., simple systems are linked together to build more complex systems, as when a tennis serve is built up by gradually linking together the toss, the hit, the follow-through, etc.
  2. differentiation - i.e., a single system is differentiated to produce two or more distinct variations, as when a forehand volley is varied slightly to produce topspin or backspin.

Poorly integrated -mathematical systems are also similar to poorly coordinated behavioral systems because:

  1. The student will not "read out" all of the available information - e.g., when first learning to ride a bicycle or hit tennis balls, a great deal of relevant information is not noticed.
  2. The student "reads in" interpretations that are not objectively given e.g., when first learning to ride a bicycle or hit tennis balls the student’s description of an activity is often distorted and biased.

Both of these factors also appear when, for example, an "eyewitness" to an accident interprets given information in a way that is biased (because only selected pieces of information are noticed) and distorted (because what "made sense" and what was "expected" influenced the interpretation of what actually happened). Similar biased and distorted interpretations also influence students’ mathematical judgments in graphics-related problems like the examples in this section.

Next, a few examples will be given to show how the basic approach of "taking apart" and "re-assembling" mathematical ideas can be extended to basic algebraic concepts. We will focus on "unpacking" the systems of operations, relations, and transformations that underlie the basic concepts of linear equations and simple polynomials.

The activities that follow are based on a symbol-manipulator/function-plotter called SAM that WICAT developed to enable students to write, graph, transform and solve algebraic expressions and equations. In lessons, SAM helps students learn some of the most important basic ideas in algebra or calculus, and the algebra ideas can make SAM more useful for problem-solving situations that students want to address. However, SAM is more than a calculator; it has the following characteristics:

  1. SAM can serve as an expression checker. We don’t have to wait until students give final answers to know whether they are proceeding along correct solution paths. We can, for example, assess whether they "set up" the equations correctly.
  2. SAM is LISP-based, so it not only generates answers, it can also produce solution path "traces" that create many instructional capabilities. For example, it allows us to: (a) generate hints by gradually revealing solution steps one at a time, (b) monitor individual steps in students’ solution paths, (c) let students examine processes as well as products of solution attempts, and (d) give students the capability to build/edit/store equation solving routines (like the quadratic formula) in a LOGO-like fashion.
  3. SAM’s symbol manipulation capabilities interact with its function plotter to produce graphic interpretations of transformations leading to solutions. This gives students ways to visualize symbol transformations, and (in yet another way) to focus on processes as well as "answers" during solution attempts.
  4. SAM can reduce answer-giving phases of problem solving so that attention can be focused on "nonanswer-giving" phases (e.g., problem formulation, trial solution evaluation, the quantification of qualitative information, the examination of alternative possibilities, etc.) where "second order" (i.e., thinking about thinking) monitoring and assessing functions often are especially important. So, SAM is not simply an answer-giver; it can help students to go beyond thinking to think about thinking.

For polynomials, it is easy for students to use SAM to carry out the following kinds of investigations:

1. Pick a value for n, between -10 and 10, and investigate the changes that this value produces in the graph of the, linear expression nx.
2. Plot the graph of the squared term x2; then plot the graph of the linear term nx (as in step #1 above); and finally, plot the graph of the polynomial x2 + nx. Notice that the polynomial crosses the x-axis at the points zero and -n.

 

For example, Figure 28 shows the graphs of x2 and 4x. Figure 29 also shows the graph of the polynomial x2 + 4x.

 

Figure 28

 

Figure 29

 

After repeating step 2 f or a series of different values for n, it is easy f or students to notice that the effect of adding x2 and nx is to "slide the graph of x2 downhill along the line nx." Furthermore, it is easy for students to notice that the amount of the slide is just enough to make the polynomials graph pass through the points zero and -n.

 

3.

Polynomials from step 2 can be factored into the form x(x + b), and each of the linear factors can be graphed as shown in Figure 30. Then notice that the two lines pass through the points zero and -n.

 

Figure 30

 

Step 3 shows why polynomials can be solved by: factoring, setting each of the linear factors equal to zero, and then solving these linear equations. The linear terms are equal to zero at exactly the same places as the original polynomial.

Using a symbol-manipulator/function-plotter like SAM, sequences of activities like those in steps 1-3 above can quickly and easily be extended to include more terms and more complex polynomials. For example:

  1. First, graph three monomials, like (-3/8)x2 and (1/4)x and 15 (see Figure 31).
  2. Next, graph the polynomial formed by taking the sum of the monomials in step 1 (see Figure 32).
  3. Then, change the coefficients of the monomials in step 1, and investigate what changes these induce in the polynomial in step 2.
  4. Finally, factor the polynomials in steps 2 and 3 and graph each of the linear factors (see Figure 33).

Or reverse the preceding four steps:

  1. Graph a pair of linear equations like those shown in Figure 33.
  2. Graph the polynomial formed by the product of the linear expressions in step I (see Figure 32).
  3. Change the linear expressions in step 1, and investigate induced changes in the polynomial at step 2.
  4. Graph each of the monomial terms for the polynomials in step 3 (see Figure 31).

Figure 31

 

Figure 32

 

Figure 33

 

In the preceding examples, the two models involved were: (a) written symbols which (although they are on a computer screen) are like those that mathematics teachers write on blackboards, and (b) computer graphics, consisting of graphs of equations in a rectangular coordinate system. Nonetheless, the computer-based activities that use these representations can be based on direct applications of Dienes’ instructional principles. For example:

  • The constructive principle is involved when we "take apart and then reassemble" complex mathematical systems related to polynomials.
  • The multiple embodiment principle is involved when we focus on mappings between two given models (i.e., written symbols and graphs of equations).
  • The dynamic principle can be used to show how transformations performed on algebraic equations are reflected in changes in the graphs of the equations at each step. For example, in the next section, we will show how a slight variation on the preceding sequence of activities can be used to show why the "completing the square" process works in the derivation of the quadratic equation.

The preceding kinds of activities can be used in much the same way as rational number pictures and diagrams to help students build "concrete feelings and imagery" for abstract mathematical ideas. So even though the "materials" used in these examples are computer-based graphics rather than "concrete materials" in the usual sense of this word, the activities can indeed involve overt actions that students can apply to "objects" that they can see and manipulate; and for the first time Dienes’ instructional principles can be applied to content areas like "polynomials" which did not seem to lend themselves to a "mathematics laboratory" form of instruction.

Dienes’ Dynamic Principle

Earlier in this paper we noted that in classroom activities involving concrete materials students seldom have opportunities to investigate transformations within even a single model. Almost never do they investigate transformations within a second model, or correspondences between transformations in the two models.

The primary point of this section is that computers possess the powerful instructional capabilities to allow students to investigate manipulations within one mathematical model and immediately see the corresponding manipulation in a second or third model.

To begin, it is useful to give an example in which algebraic equations model a typical textbook word problem and then coordinated graphs model the equations. The problem deals with simultaneous linear equations.

Traveling upstream on a river, a boat takes two hours to reach its destination eight miles away. The return trip downstream takes one hour and twenty minutes. What is the speed of the river current?

To translate this problem into an algebraic description, a student might let x represent the speed of the boat and y represent the speed of the current. Then the given relationships can be modeled (or described) using the following two algebraic sentences:

 

2(x - y) = 8
and
(1 1/3)(x + y) = 8

 

Such equations are models because they are useful simplifications of reality. Once the problem is translated into algebraic sentences, algebraic transformations can be used to solve for x and y, or the algebraic model can be converted to a geometric model by graphing each of the two equations. Then the (x, y) values that satisfy both equations will be given by the coordinates of the point located on both graphs. The solution (5, 1) can be read directly from the coordinates on the graph as shown in Figure 34.

 

Figure 34

 

In this case, the graph shows that the algebraic solutions are x = 5 and y = 1, which means that the speed of the current was one mile per hour, and the speed of the boat was five miles per hour. The preceding solution involves three significant translations: (1) describing the problem situation with algebraic sentences, (2) translating the equations into graphic form and reading the solution from the graph, and (3) interpreting the graphic solution in the context of the original problem situation (see Figure 34 above).

Models like coordinate graphs, or systems of linear equations, can be considered to be "conceptual amplifiers" because they help students use their ideas more effectively. They are not simple inert systems that have no meaning; once students learn to meaningfully embed mathematical systems (ideas and principles) or problem situations within them, students are able to "read out" additional meaning. For example, in the preceding graphing-based solution, once students had learned to use equations to describe the problem situation, and to graph these equations, they could solve many problems immediately by simply reading the graphs.

Through most of this paper we have emphasized how ideas can evolve within particular concrete situations when youngsters are helped to construct relevant structural models. However, basic ideas do not always come into being through gradual development within an environment that corresponds to a particular concrete model. Instead, students often learn new ideas in new contexts by mapping to an old model and by relying on meanings that have already developed in the context of the old model to give meaning to the new idea in the new context.

The next example illustrates how the preceding process might work, using a computer-based symbol-manipulator/function-plotter utility to teach the new idea of solving pairs of linear equations with two unknowns. Two old ideas are involved:

  1. the ability to solve simultaneous linear equations and single linear equations, and
  2. the ability to find the coordinate graphs of linear expressions.

Suppose that students have learned to graph linear equations of the form y = mx + b, and that a computer will henceforth graph such equations for them automatically, whenever they want. Furthermore, suppose that the students are learning to perform algebraic transformations by giving commands to a computer such as "add 3 to both sides of the equation," or "substitute x - 4 for y in the current expression." Then the algebraic solution to the "boat" problem might look like this:

(1) Divide by 2 to simplify
2(x - y) = 8
to
x -y = 4.
  Multiply by 3/4 to simplify
4/3 (x + y) = 8
to
x + y = 6.
(2) Add y and subtract 4 to convert
x - y = 4
to
x - 4 = y.
  Subtract x to convert
x + y = 6
to
y = -x + 6.
(3) Graph the pair of equations
x - = y
and
y = -x + 6.
(4)

Substitute x - 4 for y from the first equation into the second equation to get

x - 4 = -x + 6.

Then, notice that the two line graphs from step 3 can be thought of as graphs of the two sides of this equation (see Figure 35).

 

(5)

Add 4 to both sides of the previous equation to get

x = -x + 10

and plot the graphs of the two sides of the equation (see Figure 36)

(6)

Then add x to both sides of the previous equation to get

2x = 10

and plot the graphs of the two sides of the equation (see Figure 37).

(7)

Divide by 2 on both sides to get

x = 5

and again plot the graphs of the two sides of the equation (see Figure 38).

 
After working on activities of the type described above, many students have noticed that for any given problem, intersection points for the pairs of lines at each solution step always He on a single vertical line: the solution for x. Some students have even gone on to think about and describe why this invariant feature occurs. So this dynamic representation system, once constructed, actually helps students generate significant new questions and sophisticated solutions related to two of the most fundamental ideas in algebra; that is, our students have used informal language to describe rather deep principles related to: (1) invariance under mapping among isomorphic systems, and (2) invariance under transformations within a given system.

The examples in this section illustrate how computer environments are well suited to Dienes’ dynamic principle. Whether we are dealing with linear equations and graphs, fraction diagrams and simple proportional reasoning questions, or polynomials, computers make it easy for the student to manipulate one model and immediately see corresponding transformations in one or more other models.

Let’s end this section with one more series of examples which win be left as a sort of exercise for the interested reader to interpret. The example has to do with the process of "completing the square," which can be used (prior to using the quadratic formula) to find the roots or factors of quadratic equations like x2 + 2x - 3 = 0. Figure 39 shows the graph of x2 + 2x - 3 = y and y = 0. Figure 40 shows the graph of x2 + 2x = y and y = 3. Then, Figure 41 shows the graph of x2 + 2x + 1 = y and y = 4. Notice that the tip of the parabola just touches the x-axis. (Is this significant? Would it happen for other quadratic equations? Which kinds?) Figure 42 shows the graphs of x + 1 = y, y = +2, and y = -2. Notice that the diagonal line goes through the x-axis at the same point where the parabola had touched. (Is this significant?) Figure 43 moves the graphs in Figure 35 so that the diagonal line goes through the origin of the graph. (Is this significant?)

 

Figure 39

Figure 40

Figure 41

Figure 42

Figure 43

 

CONCLUSIONS

In general, we are in sympathy with those LEGO BEFORE LOGO proponents who believe that children’s mathematical abstractions should be built on a firm foundation of experiences with real manipulable models and realistic problem-solving situations. However, we also know that even real concrete objects often are used only in very abstract ways and that very few teachers successfully use concrete activities as a significant instructional tool. On the other hand, we have seen that when students use the kind of computer-based activities described in this paper (many of which are electronic versions of the kinds of concrete models that we really hope students will have the opportunity to explore), their teachers actually become more likely to use "mathematical laboratory" activities with real concrete materials. This increased use of real concrete activities seems to occur because computer-based simulations of mathematics laboratories tend to minimize the obstacles to teachers’ use of concrete mathematics laboratory principles.

Throughout this article we have illustrated how computer-based activities help teachers implement mathematical laboratory principles such as those described by Dienes. In fact, principles like Dienes’ dynamic principle are perfectly suited to computer environments and have never seldom successfully implemented in any other context.

Some factors that favor the use of computer-based mathematics laboratory activities include the following:

Not only can computer utilities serve as powerful tools to help students acquire mathematical ideas and increase their meaningfulness, they can also amplify the power of the acquired concepts. Problem solving in the presence of computer-driven "conceptual amplifiers" (such as the symbol-manipulator/ function-plotter referred to in this paper, or even familiar tools like VisiCalc) are becoming indispensable in science, mathematics, and engineering. One can no longer assume that the problem solver is working alone with only a pencil and paper for tools. New "realistic" problem types, such as those involving non-deterministic answers and "stochastic refinement" solution processes, will also be increasingly important (Lesh 1985).

Due to the availability of powerful new computer utilities (like WICAT’s symbol-manipulator/function-plotter), realistic applications can now be used to introduce a wide variety of mathematical topics. Rather than first attempting to "teach" a given idea and then introducing applied problems involving the idea and associated procedure, computer utilities allow us to "give" the procedural capabilities to the student at the beginning of instruction. Realistic applications can then be used to gradually guide students to build their own conceptualizations of the underlying idea (Fey 1985).

By minimizing tedious answer-giving procedures, computer utilities can focus students’ attention on nonanswer-giving phases of problem solving, where higher-order thinking activities related to data gathering, information filtering, problem formulation, and trial solution evaluation are involved. By focusing attention on the underlying conceptualizations of problem situations and on the sensibility of products of thought, the subtle meanings of relevant ideas become apparent. Also, by reducing the conceptual energies devoted to "first-order thinking," higher-order "thinking about thinking" becomes possible. Otherwise, students frequently become so embroiled in "doing" a problem that they are unable to think about what they are doing and why.

Results of a recent Applied Mathematics Problem Solving project (Lesh and Akerstrom 1983; Lesh, Landau, and Hamilton 1983) support the view that the underlying meanings of mathematical ideas tend to be emphasized during nonanswer-giving phases of problem solving; and using computer utilities, nonanswer phases can be addressed even in conceptual areas characterized by complex number-crunching routines or sophisticated conceptualizations. Consequently, computer utilities enable us to use realistic applications to introduce whole new categories of sophisticated mathematical concepts.

Notes

  1. The research of the RN, PR, and AMPS projects was supported in part by the National Science Foundation under grants SED 79-20591, SED 80-17771, and SED 82-20591. Any opinions, findings, and conclusions expressed in this report are those of the authors and do not necessarily reflect the views of the National Science Foundation.
  2. Perhaps, we should state one of our prejudices about the nature of mathematics for readers who hold different biases. In our opinion, although most mathematical ideas, from early number concepts to algebraic topology, often have computation-type procedures associated with them, "doing the procedure" frequently has little to do with "doing mathematics," nor is it necessarily a good indicator of a student’s understanding about the underlying ideas.

    Consider the mega-ideas underlying differentiation or integration in calculus. The procedures needed to compute a given derivative or integral bear virtually no resemblance to the network of relations that define the underlying ideas and constitute the meanings. Unfortunately, the "network of relationships" that characterize (psychologically define) most mathematical ideas are quite difficult to specify. For certain elementary mathematical ideas, considerable progress has been made in this regard in recent years. However, with the exception of certain early number concepts, rational number concepts, and simple spatial/geometric concepts, psychologists and mathematics educators have only begun to map out the specific "conceptual models" that underlie most fundamental mathematical ideas. Certainly, very few details can be given about what it means to "understand" a topic like simultaneous linear equations. Consequently, by default, understanding almost exclusively tends to be assessed in terms of students’ abilities to carry out procedures, even though "taking a derivative," say, or solving "one rule under your nose" style word problems, is quite easy to "master" with only the foggiest notion of the underlying ideas.

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