The Image is a fourteenth century painting by deVoltolina, The Lecture. The essential structure of a teacher-centered classroom has not changed for many centuries. Neither has the behavior of students, some of whom in the image are shown to be sleeping, chatting, or generally distracted. Everything is changing now, of course, with the advent of digital convergence and the ubiquitous information provided by the internet. Iconic mathematics explores what Lee Shulman calls the pedagogical content knowledge of mathematics, that particular form of mathematics that is most germane to its learnability. What makes mathematics comprehensible to a novice? Which mathematical forms afford the fewest errors in understanding both structure and meaning? Which transformation rules and axioms are more error-prone? When should symbolic models of mathematical concepts be augmented with manipulatives? And in particular, when will we reconnect math to physical reality in order to enhance concept learning in mathematics classrooms?
Hilbert’s Program
A century ago, well prior to the advent of digital display technologies, David Hilbert used his considerable influence to exclude diagrams and spatial intuition from formal mathematics and consequently from the mathematics curriculum [1]. Hilbert’s agenda has been rigorously pursued, so that today “Mathematics is a game played according to simple rules with meaningless marks on paper.” (David Hilbert, c. 1900)
“…proof is a syntactic object consisting only of sentences arranged in a finite and inspectable array.” [3]
“…despite the obvious importance of visual images in human cognitive activities, visual representation remains a second-class citizen in both the theory and practice of mathematics.” [4, p3]
The fundamental structures of formal mathematics do not necessarily align with the goals of mathematics education. Higher mathematics is expressed almost exclusively in symbolic string languages designed by experts for experts, not for novices who may have never considered that a letter could stand in place of a set of numbers. The symbolic approach is excellent for digital computers but quite inappropriate for students who must master not only mathematical concepts, but also their notations which are, by design, independent of the concepts.
Much of elementary mathematics dwells on specific algorithms for a singular specific system of representation (the decimal place-value system). Kaput [5] is directly critical of the emphasis of form over content in elementary mathematics, agreeing that the predominance of math education addresses only a particular set of representations and algorithms. Place-value notation, for example, is nearly universal. Place-value algorithms mix arithmetic operations with the maintenance of the place-value representation, and thus confound semantics (mathematical concepts) with syntax (the representation of mathematical ideas). Symbolic addition is purely syntactic. Addition is defined by the three mechanisms that maintain the place-value representation: digit combination facts, alignment of place-values, and carrying. Adding It Up: Helping Children Learn Mathematics points to an inherent tension between the abstract and concrete aspects of mathematics. “This tension is a fundamental and unavoidable challenge for school mathematics.” [6, p74]. The pedagogical shift from early childhood manipulatives to symbolic computation is a shift from semantics to syntax, from construction of meaning to management of a particular representation.
What is missing to date are rigorous spatial and iconic systems for numerics, systems that are inherently interactive and manipulable while at the same time meeting formal criteria of rigor. The iconic systems on this site meet these criteria. They are based on visual/interactive axioms, and provide what Philip Davis calls visual theorems:
Kempe’s 1886 goal: “…to separate the necessary matter of exact or mathematical thought from the accidental clothing — geometrical, algebraical, logical, etc.” [8] was admirable at the time. However, it is not possible to separate mathematics education from the pedagogical qualities that make a mathematical formalism comprehensible and learnable.
Learning Formal Math
In mathematics education, older students learn symbolic rules to manipulate numerals and algebraic expressions, while younger students use manipulatives to understand how numbers and abstractions work. Concrete and virtual manipulatives rely upon spatial models and transformations. Symbolic calculation and modern algebra rely upon textual, string-based models and transformations. This difference leads to a significant discontinuity in the teaching of mathematics, the change in perspective that students face when moving from an early elementary curriculum grounded in concrete experiences to a late elementary curriculum grounded in abstract concepts. This discontinuity is rarely brought into question even though it arises from how we write mathematics down as communication, not from the nature of mathematics itself. The absence of a mapping between concrete manipulatives and symbolic abstractions may manifest in a variety of student errors, including failure to connect the two systems, slips in symbol manipulation, misapplication of tools and algorithms, partial and fragmented integration of concepts, and confused or muddled cognitive models. More visual, meaning-laden notations may aid mathematics understanding for concrete, visual, tactile and experiential learners, and for those who have disabilities, disadvantages, or cognitive and behavioral resistance to the conventional symbolic structure of mathematical expressions.
The feasibility of incorporating manipulative formal systems into the math curriculum depends upon computer-based display and animation. Only recently, with the advent of web-based virtual manipulatives, have display and interaction technologies become available to promote diagrammatic and iconic mathematical systems from a second-class role as informal aids to a first-class role as rigorous formal tools. Virtual interaction and manipulation provides flexibility, extensibility, goal-directed guided interactivity, and tractable cost. A possible avenue of research would be to compare the benefits and costs of symbolic and iconic arithmetic by presenting both to students and measuring their comprehension. Comparative axiomatics does not yet exist, but it is time to recognize that some approaches to mathematics may bear fruit for mathematicians while others may be far better for novices.
Educational Motivation
It is well known that America’s students are underperforming in mathematics education. On the Washington [State] Assessment of Student Learning (WASL) in 2006, for example, half of the students in Grades 6 through 10 failed to meet grade-level standards of performance for mathematics [9]. The 2006 overall current failure rate of 50% incorporates a 75% failure rate for Washington State minorities and a 70% failure rate for students living at the poverty level, a group composed primarily of white and Asian students [10]. The WASL is based on the Principles and Standards for School Mathematics developed by the National Council of Teachers of Mathematics (NCTM) [11]. After a decade of extensive effort, the 2006 results represent a significant improvement, having increased from a 33% pass rate in 1999.
The Additive Principle
Early primary school, as well as early recorded history, associates addition with the Additive Principle [12]. This principle is exemplified by unit-ensemble arithmetic, the stroke or tally arithmetics within which units are represented by individual marks or objects, and numbers are simply a collection of identical replicated units [13]. Addition in unit-ensemble arithmetic is achieved by the activity of placing representations of units together in the same space. The Additive Principle might be considered to be the definition of the addition operation. Even the modern Peano axiomatization of arithmetic defines addition recursively as successively adding a single unit.
Addition is introduced in early elementary school as the physical activity of joining together collections. Symbolic addition, also introduced in early elementary school, follows completely different principles. The symbols themselves are arbitrary and make no inherent sense. Instead, students memorize number facts and algorithms to determine a sum. Of course, symbolic numerals can be decomposed into units, reverting their additive behavior back to that of unit-ensemble arithmetic, but this fact is rarely acknowledged or taught.
Teacher training texts recognize the importance of additive systems throughout lower elementary mathematics. However, these texts explain the meaning of addition in terms of symbol manipulation, not in terms of the spatial intuitions of the Additive Principle. The actual structure of iconic additive systems is redefined to fit the group theoretic structure of symbol-based systems. For example, an otherwise excellent teacher training book completely confounds the spatial freedom of unit-ensembles by imposing group theoretic commutativity on collections:
Unit-ensemble addition does not incorporate or require a temporal ordering of actions. When two or more collections of units share the same table, children can add them by pushing the piles together all at the same time. One collection simply does not have temporal or positional precedence over another. The same idea applies to parallel addition of three or more ensembles; the number of collections are being added together is simply not a relevant concept. Further the statement above embodies a conceptual confusion. Sets are defined as being composed of unique members, while the collections being added in elementary schools are usually composed of identical units. Sets do not support identical members, the members of a set must be different. Yet counting objects such as beans or blocks or pennies are collections of virtually identical units. This makes understanding what is going on with putting things in a dish quite confusing for learning math concepts.
Manipulatives
Concrete physical manipulatives have a long history in the classroom. Elementary school teachers use manipulative and diagrammatic techniques widely [15]. Due to the pragmatic limitations of physical objects, physical manipulatives have been constrained to “concrete examples”, for example a triangle drawn in a geometry class is considered to be only a particular triangle. However, with new software tools such as Geometer’s Sketchpad [16], a student can “draw” a generic triangle, and use the diagram itself as an object of computation. Digital technology has expanded the domain of representation of mathematical concepts from typographical strings to spatial forms and to virtual manipulatives.
Conventional manipulatives are analogical, relying on an underlying geometry or metric to construct a mapping between spatial form and mathematical concept. One problem with assessing the effectiveness of concrete manipulatives is that each different manipulative relies upon a different analogical connection to the abstract concepts it exemplifies. This assures that students will develop different understandings and different errors. By presuming that symbolic math is the only legitimate standard for conveying mathematical ideas, spatial and iconic approaches are assured to fail. A student who counts on his fingers, for example, is defined to be using an inferior process compared to memorizing relationships between numerals.
Formal iconic systems provide a rigorous basis for iconic and diagrammatic representation of mathematical concepts. It is possible to isolate manipulative errors associated with the violation of specific structural axioms in both symbolic systems (the axioms of group theory) and iconic systems (the axioms of additive mathematics). Not only should we be able to identify which rules and axioms are associated with learning difficulties, but we should also be able to localize the specific transformation rules for which one approach (textual/symbolic or spatial/manipulative) might improve upon the other.
There is abundant evidence that interactivity can assist students who are having difficulty learning abstract material [17][18]. Students find it easier to learn math if it is made concrete through the use of manipulatives [19]. Use of manipulatives produces greater gains in achievement [20]. Iconic representations enhance understanding, since expressing ideas spatially, visually and tactilely allows information to be analyzed more effectively by parallel perceptual processes than by linear cognitive processes [21][22]. Many different ways of making mathematical concepts more concrete have been shown to be effective in learning algebra [23][24]. However, manipulatives must be clearly and explicitly connected to the abstract concepts they represent [25]. And the structure of the manipulative tool may itself distract from or interfere with the mathematical concepts it is associated with.
Virtual Manipulatives
A virtual manipulative is “an interactive, Web-based visual representation of a dynamic object that presents opportunities for constructing mathematical knowledge” [26, p373]. Reimer [27] and Clements [24] discuss potential benefits of virtual manipulatives for learning. Today, web-based applets that provide virtual models of mathematical concepts and computations are burgeoning. Extensive collections of free virtual manipulatives software for mathematics are maintained by the math archives at Drexel University [28], by the National Library of Virtual Manipulatives at Utah State [29], by the National Council of Teachers of Mathematics Electronic Examples [30], and by public [31], commercial [32] and home-schooling [33] interest groups.
Virtual manipulatives provide iconic models that simulate concrete manipulation [34]. They currently bare a strong resemblance to concrete manipulatives such as Cuisenaire rods, base-ten blocks, pattern blocks, rulers, number lines, logic blocks, fraction pieces, and geoboards. Virtual manipulatives are decidedly constructivist. Students construct meaning by using computer input devices, such as the keyboard, mouse, touch-screen, joystick, etc., to control apparently physical actions of virtual objects through translation, rotation, flipping and other spatial transformations. For example, graphing linear equations can be made interactive both by generating a graph given student input of a linear equation, and by generating a linear equation that corresponds to student dragging and rotating the line graph itself [35].
Today the libraries of virtual manipulatives are organized around specific content and grade levels. Very few manipulatives are targeted toward specific needs and miscomprehensions. If a student is having difficulties with the distributive rule, we do not know which manipulative techniques will address that difficulty, how a particular technique might correct the difficulty, or when a teacher should elect to change from a symbolic to a physical presentation.
Spatial Analogs
Spatial analogs are lines, scales, dials, etc. that support the concept of number within their geometry. Visual and spatial analogs and models are elevated to a curriculum design principle in How Students Learn: Mathematics in the Classroom:
Often though, representations based on spatial analogs fail to adequately model the concept of multiplication. The additive spaces of dials and scales do not support multiplicative concepts. Cuisenaire blocks achieve multiplication via an increase in dimension, reaching a pragmatic limit for base-10 polynomials of degree three. Spatial analogs provide a visual model but not an algorithmic understanding.
A primary difficulty with spatial analogs is that we do not know how the structure of the analog will interact with the abstract concept being taught. Uttal [25] for example, has observed that students identify with familiar but incidental attributes of a manipulative form, and thus make the wrong generalizations, connecting incidental characteristics to the taught concepts. Some manipulatives cannot model the appropriate levels of abstraction. More generally, we do not know how manipulative techniques interact with specific mathematical concepts.
Multiple Representations
Number systems represent abstract numerical concepts; in turn, abstraction supports multiple concrete interpretations. Multiple perspectives and representations are known to improve concept learning [38]. “The usefulness of numerical ideas is enhanced when students encounter and use multiple representations for the same concept.”[6, p2]
Bruner [39] distinguishes three types of representation for mathematical operations. Each plays an essential role in mathematics understanding. Enactive addition, for example, is concrete; collections of objects are placed together physically. Symbolic addition abstracts the cardinality of a set of objects into a symbolic name such as “3″ or “7″. Rules for combining symbolic names guide the determination of their sum. Iconic addition is most commonly presented as pictures of groups of objects with specific cardinality. The value of multiple representations suggests the possibility of concurrent use of concrete, symbolic, and virtual forms. This in turn calls for a deeper understanding of the relative pedagogical merits of each form of representation.
Since Bruner’s writing, iconic systems have acquired a richer meaning, they can be dynamic and animated. Virtual manipulatives permit iconic forms to illustrate and to enact computation. However, web-based diagrammatic and manipulative tools further confound the distinction between concrete application and symbolic abstraction. Manipulatives must be related explicitly to relevant concepts [40], otherwise students learn different but unconnected techniques. But the confusion between concrete and abstract representation goes deeper, implicating our understanding of what the fundamental concepts of mathematics are. Web-based manipulatives provide access to multiple representations to a much greater degree than the physical and even the conceptual tools of a decade ago. The ability to quickly change representations can also be seen as a unifying force, integrating haptic, visual, conceptual and behavioral interpretations [41]. This though requires a broader perspective on what math is, the symbolic standard can no longer maintain its dominance. Similar to the way that the web has forced a redefinition of film, music, and entertainment, mathematics itself must undergo reconceptualization.
Algorithms are known to depend upon representation. In silicon computation, for example, decimal numerals are abandoned entirely in favor of binary numerals. In math education, the algorithms for manipulation of fractions differ from those of decimals, although fractions and decimals can express the same abstract concept of ratio of magnitudes. Operations on diagrams representing fractional quantities are again of a fundamentally different, concrete nature than the symbolic algorithms for both fractions and decimals.
Embedding Mathematics in a Virtual World
Traditionally the domain knowledge of science and mathematics is represented symbolically and taught through manipulation of algebraic equations. The symbol systems taught in school exhibit many difficulties: they are difficult for novices to understand; they are abstract rather than contextually anchored; the operational meaning of many symbols is often unclear, ambiguous and context dependent; and mastery of a symbol system is usually confused with mastery of the system’s referential content.
Virtual environments permit symbolic processes to be hidden from a participant, while the behavior of objects in the virtual environment provides direct evidence of the consequences of those symbolic processes. A virtual ball, for example, programmed to obey the laws of motion, will travel along the expected trajectory, mimicking physical behavior. The computation of the trajectory is hidden but evident, it is embedded within the virtual object. In the virtual world, however, the ball’s simplest behavior is not in accord with the laws of motion. The non-programmed ball, having no substance, sticks to the exact spot of its release. Inertia, gravity, and force must be programmed in as symbolic computations. This apparent weakness, that virtual environments are heavily dependent on symbolic computation, is also a strength, for it permits complete computational control over observed behavior. In particular, we can program virtual worlds so that the behavior of objects in the environment embeds both symbolic and pedagogical processes. Students could learn literally though direct experience of behavioral consequences (in simulated environments with embedded symbolic processes).
Embedded computation means more than interaction with a process. Embedding implies natural interaction with realistic (or at least naturalistic) objects in an environment, objects which react and respond to environmental perturbations (such as a participant’s query), objects which met the expectations of interacting agents. Embedding is semantic rather than syntactic; interaction is well-intended rather than well-formed.
A handheld calculator embeds numerical computation, it’s addition button performs as expected. The keypad enforces well-formed input by limiting transactions to keyed operations on digits. But people have difficulty sequencing complex numerical calculations on a calculator because it does not embed computational strategies, teaching techniques, or naturalistic language domains. Thus, the simple calculator provides a very weak embedded functionality. Stretching an environmental concept, we might say that embedded capabilities are the affordances of an object, they are provided by the object as part of its interactional character.
A computational interface has natural semantics when the display has a meaning that does not require explanation. Consider a house. A textual description requires reading skills, a procedural database (lists of coordinates) requires decoding, a picture can be recognized immediately but is not interactive. A house in a virtual environment is most like a physical house, you can walk in the front door and explore each room while looking around. Transformations can be expressed as natural behavior. Opening a window, for example, can be achieved by grabbing it and pulling it open, rather than by editing the symbolic attributes of the window to change the values that represent the location of the window. A virtual house has natural semantics, no one needs to explain it. Natural semantics is what a child learns before symbolic schooling. Most sciences have natural semantics, most symbolic studies (the three Rs) do not.
Embedded mathematics presents symbolic computational processes as behavior with natural semantics. Virtual objects are constructed so that they perform both the physical and the computational behaviors expected of them. Virtual objects which embedded mathematics can include word processors, checkbooks (the spreadsheet application), lumber (which could generate its own bill of lading when assembled into a house), most scientific visualizations (such as pollution clouds, deep-sea rifts, four-dimensional cubes, and galactic collisions), and intelligent software agents.
The Problems with Symbolic Algebra
The difficulties students have when they begin to learn algebra are well documented [42][43][44][45] [46]. Thwaites [47] found that students are often baffled by algebra’s non-visual nature, its apparent arbitrariness, its complexity, and how problems are expressed through its symbols. Students often do not understand what variables are, how letters are used to represent them, or how equality can be used [5][48]. If students fail to understand algebraic representation, then the only way they can solve algebra problems is by the rote application of procedures they have memorized. This memorization is brittle, often both over- and under-generalized, and elaborated by motivations independent of the content of algebra [42]. Unfortunately, the rule-based approach to algebra is the one that is often taught, even though this does not promote the development of good conceptual models of algebra [47][48][49][50].
Many attempts to address symbolic abstraction make it concrete through the use of manipulables, a positive approach supported by many experimental studies [51]. The symbols of algebra are typically reified either in physical objects or in computer representations or simulations of problems and problem-solving. Thus, showing numbers as sets of objects that students can actually count makes it easier for students to solve arithmetic word problems. In the area of equation solving, we find studies of the effectiveness of using pan balances [52], spreadsheets [53], computer-based graphing [54], and a variety of other strategies [55].
Technology has matured so that it is possible to develop virtual worlds with embedded objects which both obey the rules of mathematics and can be used as tools for problem solving. The flexibility of virtual objects allows us to transfer our semantic intentions more easily, to make them look and act like we expect. Virtual objects are situated in virtual environments, permitting us to design contextual cues, object interactions, and circumstantial affordances to assist meaning making. As an example, imagine a student trying to solve the standard two unknown problem of the ages of two people, when the sum and product are known (say,
The Reasons for Learning in Virtual Environments
Expectations for the success of VR to help students learn abstract material, and the guidelines that direct our design of virtual environments, are based on the theories of constructivism, multiple intelligences, and representation. Constructivism leads us to expect that first-person experience will create deeper understanding of abstract concepts. Multiple intelligences suggest that learning has many pathways and many modalities. The theory of representation provides us with display and interaction techniques which remove the disassociation of abstract symbols while maintaining formal semantics for computation.
Non-symbolic Representation
Virtual environments finesse the semantics/syntax barrier by providing the representational flexibility to construct symbols which reflect meaning as well as abstract patterns [56]. Interaction is direct and not mediated by an interface [57]. Objects in algebraic relationships can quite literally be picked up and moved around algebraically, while the student observes the consequences [58].
Multi-modal Learning
The comprehension of abstract rules and concepts can be significantly improved when represented by several sensory modalities. Animated computer graphics and graphs make it easier to learn concepts and rules. Adding sound effects to graphic representations improves comprehension of complex data sets. Inclusive environments have been shown to be emotionally involving and extremely easy to use [57][59].
Knowledge Construction
Students construct their own meaning by interacting with material rather than being taught something explicitly [60]. To specify a particular content organization or instructional strategy is counterproductive. These techniques are exemplified by the hypermedia system developed by Spiro and his colleagues [61] that lets students learn problem-solving through the exploration of ill-structured domains such as literary criticism, military strategy, and cardio-vascular medicine; and by the interactive videodisk materials developed by Bransford et al. [62] and the Cognition and Technology Group at Vanderbilt University [63] that facilitate the solving of complex mathematics problems by allowing children to interact with dramatically presented adventures. Some key theoretical elements behind constructivism are contained in Cognitive Complexity Theory [61], which proposes a number of strategies to promote the acquisition of flexible knowledge.
Pros and Cons of Iconic Math for Education
The potential benefits of studying iconic mathematical systems in the classroom are compelling.
• An alternative iconic notation may assist the teaching and learning of concepts for some students, since diagrammatic and spatial formal systems permit visual, manipulative and physical interaction with mathematical abstractions.
• New representations and axiom bases can broaden mathematics understanding. A comparative axiomatics may enrich our teaching of mathematical concepts.
• Comparison of the errors made in symbolic vs iconic systems can guide research insight into the semantic and syntactic sources of mathematics miscomprehension.
• Spatial representation may help to bridge the gap between concrete understanding and abstract symbol manipulation, by providing tools that more closely align the abstract formality of mathematics with our concrete experience of real objects.
Here are some of the potential disadvantages of innovating math education with iconic formal systems.
• Non-standard axioms, even if they incorporate significant pedagogical advantages, do not conform with the prevailing group theoretic basis for modern algebra. This may create problems for communication of knowledge.
• Students may become confused when presented alternative perspectives on the foundations of mathematics.
• Teachers will not be familiar with the availability of alternative foundations.
• Computer-based manipulatives can blur the distinction between concrete and abstract.
• The community of scholars knowledgeable about additive axiomatics is small.
• The software systems used to prototype the curriculum materials may not be available in all classrooms.
Each of these concerns can be adequately addressed. Briefly:
Communication of knowledge: Additive systems are intuitive, are taught in early grade school mathematics, and can be presented as supplemental to understanding the symbolic mathematics of middle school and high school. The non-standard basis is widely taught, it is just not associated with axiomatic rigor. Iconic math adds formal models for already existing practices.
Alternative perspectives: Students are already exposed to both manipulative and symbolic systems, and already exhibit confusion. The explicit formalization of iconic and interactional math simply puts existing practices on the same rigorous footing.
Unfamiliarity: Software tools can provide interactive access to unfamiliar mathematics and permit evaluation of the effectiveness of alternative foundational approaches. Teachers may welcome an option to the known difficulties of teaching symbolic math.
Abstract manipulatives: Rather than withdraw from the impact of computer-assisted mathematics, an alternative is to accept it is inevitable. Virtual manipulatives have been widely funded by the National Science Foundation; iconic math contributes to their understanding.
Community of scholars: Mathematical foundations in general attract little research or pedagogical interest.
Software availability: Modern software systems include free downloadable players available over the Internet. An increasing tread, especially in education, is to provide free software and charge only for development and services.
More generally, there are several advantages to teaching iconic math that are consistent with a modern understanding of good teaching:
• Spatial models are rigorous yet simple and familiar, permitting less opportunity for student error [42].
• Learning through concrete interaction with iconic, visual and interactive forms can enhance mathematics understanding, particularly when the structure of the manipulative representation closely aligns with the abstract concepts it is intended to represent [25].
• Representational variety strengthens both mathematical content and reasoning [64].
• Identification of the relative strengths and weaknesses symbolic and iconic approaches may guide better structuring of classroom curricula and materials.
• Students failing to comprehend symbolic form and the associated symbolic transformation rules may welcome a spatial, interactive alternative for success in math.
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