Studies in Philosophy and Education

, Volume 26, Issue 2, pp 127–146

From Searle’s Chinese room to the mathematics classroom: technical and cognitive mathematics

Authors

    • MathematicsPedagogical Institute-Greece
Original Paper

DOI: 10.1007/s11217-006-9018-y

Cite this article as:
Gavalas, D. Stud Philos Educ (2007) 26: 127. doi:10.1007/s11217-006-9018-y
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Abstract

Employing Searle’s views, I begin by arguing that students of Mathematics behave similarly to machines that manage symbols using a set of rules. I then consider two types of Mathematics, which I call Cognitive Mathematics and Technical Mathematics respectively. The former type relates to concepts and meanings, logic and sense, whilst the latter relates to algorithms, heuristics, rules and application of various techniques. I claim that an upgrade in the school teaching of Cognitive Mathematics is necessary. The aim is to change the current mentality of the stakeholders so as to compensate for the undue value presently attached to Technical Mathematics, due to advances in technology and its applications, and thus render the two sides of Mathematics equal. Furthermore, I suggest a reorganization/systematization of School Mathematics into a cognitive network to facilitate students’ understanding of the subject. The final goal is the transition from mechanical execution of rules to better understanding and in-depth knowledge of Mathematics.

Keywords

Chinese roomMathematics classroomEducationCognitive mathematicsTechnical mathematics

Searle’s ‘Chinese Room’

Searle (1980) proposes the ‘thought experiment’, as he calls it, of the Chinese Room, to show that cognition and cognitive functions do not simply amount to a computing-syntactic procedure that executes a set of rules-program, whether it takes place within a machine or a human brain. The thought experiment was intended as a response to supporters of the view that a syntactic machine, equipped with the appropriate rules and algorithms, also understands and possesses sense.

The thought experiment goes as follows: Let us suppose that an individual, who does not speak Chinese but understands her native tongue, for instance, Greek, is in a room. The room has two letterboxes – let us call them I (Input) and O (Output) respectively. Input I is used for sending through papers with Chinese symbols. Inside the room there is a book written in Greek, wherein one can find all necessary rules such as: “When you receive a piece of paper from letterbox I with the Chinese symbol X, follow the corresponding rules, write on another piece of paper the Chinese symbol Y, and pass it through letterbox O”. These rules allow the correlation of one set of typical symbols with another set of typical symbols, where ‘typical’ means that one can recognize symbols only by virtue of their form. The book also contains instructions on how the person inside the room can design certain Chinese symbols to respond to certain types of characters she is given. The people who provide the symbols use a variety of technical terms on the papers, such as ‘writing’, ‘vocabulary’, ‘syntax’, ‘questions’, without the person inside the room knowing it. Additionally, they use the term ‘responses’ for the symbols they receive through letterbox O and ‘program’ for the set of rules and instructions in Greek they have provided the person inside the room with. In this manner each time a sequence of Chinese symbols enters the room, the person processes them according to the rules she has at her disposal and exports a suitable sequence of Chinese symbols.

After a certain period of time, the person inside the room becomes so capable of executing the operating rules of the Chinese symbols, and the people outside the room become so capable of providing her with the right rules, that to an external observer the person’s responses seem not only right but also similar to those of a native Chinese speaker. However, in the case of Chinese, and contrary to what is the case in the person’s native tongue, in our instance Greek, the responses are given by means of managing typical symbols that are un-interpreted to her. Thus, as far as Chinese is concerned, the person’s behavior is similar to a computer program, in that she executes computing operations on data configured in a typical fashion and she is, therefore, only an individualization of a computer program.

Consequently, the external observer’s assumption, that the person inside the room speaks Chinese, is wrong. This person simply behaves as if she speaks Chinese without in reality understanding it. What she does is managing symbols with the help of rules without knowing what the symbols mean. Thus, the Chinese Room corresponds to the operation of a computer and using this example Searle proves that the execution of rules can never be considered to be understanding or thought, since both the person inside the room and the computer merely operate symbols according to their form and using the appropriate rules.

Searle (1980, 1984) postulates the following to corroborate his argument:
  1. (1)

    The brain induces understanding. Cognitive operations are generated from operations taking place inside the brain.

     
  2. (2)

    Form-syntax is not sufficient for sense-meaning.

     
  3. (3)

    Programs have only a syntactical structure. In other words, they are only sensitive to the form of the symbols they employ.

     
  4. (4)

    True understanding (and thus thought) is sensitive to the sense-meaning of symbols.

     

With this last postulate Searle explains that thoughts, beliefs and desires refer to something or deal with state of affairs of the world (intentionality). This happens because their content directs them to the state of affairs of the world. From postulates (2), (3), (4) above, it follows that:

Inference 1

There is no set of rules that is sufficient to assign cognitive states to a system. In summary, the set of rules-program is not a mind nor is it sufficient to be considered a mind.

This is a very valuable inference in that it shows that any attempt to induce cognitive states based only on the design of rules-programs, is destined to failure. It also concerns any level of complexity for rules-programs. This is a purely logical inference that follows from claims-postulates (1–4). The latter are widely accepted, even by those who have reservations for Searle’s views. Even the most ardent supporters of the view that rules are sufficient for understanding and thought agree that, biologically, cerebral activities induce cognitive states and also acknowledge that programs are defined only typically syntactically. However, the above inference proves that, the belief that understanding is achieved through the implementation of a set of rules is not valid. From postulate (1) and inference 1 above, Searle concludes that:

Inference 2

The way by which the cerebral operations induce cognitive states, cannot be owed to the execution of a set of rules or a computer program.

Since the brain induces cognitive states and computer programs are not sufficient to perform this function, it is obvious that cognitive states cannot be induced simply by executing a computer program. This is again a very important inference in that, in effect, it claims that the brain is not a computer and that the computational properties of the brain are not sufficient to explain how the latter induces cognitive states. What is vital is that the brain is a biological machine rather than what the supporters of the view that rules are sufficient for understanding and thought claim, namely that the mind just happens to be materialized inside a human brain. From postulate (1) Searle also concludes the following:

Inference 3

Whatever induces cognitive states must have causal powers, at least equivalent to those of the brain.

It is possible that another system is capable of inducing cognitive states using different biochemical or chemical characteristics from those used by human brains. Let us assume, Searle argues, that Martians arrive on Earth and we determine that they have cognitive states. However, when we examine their brain we find that it consists of a green slimy substance. For this substance to induce cognition, and indeed any characteristic of a cognitive life, it must also possess causal powers equivalent to those of the human brain. Finally, from inference 1 and 3 above, Searle concludes that:

Inference 4

Whatever creation we build, which has cognitive states equivalent to those of a human, its materialization only by means of a set of rules is not sufficient. On the contrary, our creation must possess powers equivalent to those of the human brain.

The goal of the above discussion has been to remind the reader that cognitive states are biological phenomena. Consciousness, intentionality, subjectivity, and mental causality are as much part of our biological history as are development, reproduction, and digestion. The essence of Searle’s account consists in postulate 2 above and, more specifically, in his differentiation between syntax and semantics. In concluding his thought experiment, Searle claims that cognition is more than just the syntactical operation of symbols according to certain rules.

At this point I must stress that, not all philosophers might accept Searle’s argument. There are many replies to this that can be classified as follows: The systems reply, the robot reply, the brain simulator reply, the combination reply, the other minds reply, the many mansions reply, the connectionist reply. For a detailed discussion of the latter the reader can refer to Searle (1980, 1984, 1989, 1990), Churchland and Churchland (1990), Harrison (1999), Hauser (1996), Kim (1996), Rey (1997), Sterenly (1990), etc. It is outside the focus of the present paper to discuss Searle’s argument per se. My intention is to use it as an analogy with regard to the discussion of the Mathematics classroom that follows and to draw some useful conclusions from it. Taking this opportunity I present some thoughts about teaching and understanding of Mathematics and I discuss some parallel views.

Before we go on to the study of this analogy, I want to underline that in an analogy the objects in the two domains can have dissimilar features but similar relations and that some relations are essential for the operation of the system whereas others are not (Mayer, 1992). In Gentner and Markman (1997) we also find that there is, in general, an indefinite number of possible relations that an analogy could pick out and most of these are ignored.

The mathematics classroom

According to Systems Theory (Gavalas, 2000; Heylighen & Joslyn, 1992), since the elements that constitute the Chinese Room interact among them, it is legitimate to consider the overall device as a system (CR System). For the same reason, the Mathematics Classroom is also a system (MC System). I compare, considering their structure and function, these two systems in the following table:

The CR system

The MC system

(1) Structure

Input (I): Input Operators: People outside the room Input Data: String of Chinese symbols.

Input (I): Input Operators: Teacher Input Data: String of mathematical symbols.

System under study (Plant): Person inside the room + Book of rules.

System under study (Plant): Student inside the classroom + Book/verbal instructional rules.

Output (O): Appropriate answers to input data (Chinese symbols).

Output (O): Appropriate answers to input data (mathematical symbols).

(2) Function

People outside the room introduce the string of Chinese symbols in the system under study.

Teacher by teaching introduces the string of mathematical symbols in the system under study.

Process/Transfer Function: The person inside the room with the aid of the book processes the input information and responds accordingly.

Process/Transfer Function: The student inside the classroom with the aid of the book/verbal instructions processes the input information and gives the answer.

The response of the system is the appropriate Chinese symbols that exit the room.

The response of the system is the appropriate mathematical symbols that the student gives by various examinations.

The person’s process inside the room with

the aid of rules (T)

of the Chinese symbols (I)

gives the response (O):

T(I) = O

The student’s process inside the classroom with

the aid of rules (T)

of the mathematical symbols (I)

gives the response (O):

T(I) = O

This first close correspondence between the CR and MC systems allows us to put a second more specific correspondence:

Chinese Room

 

Mathematics Classroom

People outside the room

Mathematics Teacher

String of Chinese symbols

String of mathematical symbols

Person inside the room

Mathematics student

Book of rules – instructions

Mathematical rules – instructions

External observer

Examiner/Examinations

Considering the previous systems some notes are necessary: (i) Each one of the systems is controllable and observable, since there is an external observer. (ii) The systems have a negative feedback, because after a certain period of time, the person inside the room/student becomes so capable of executing the operating rules of the Chinese/mathematical symbols, and the people outside the room/Mathematics teachers become so capable of providing her with the right rules, that to an external observer/examiner the person’s/student’s responses seem entirely right. (iii) This negative feedback maintains the stability of each system and renders it self-organizing and self-regulative i.e. system and not static machine. (iv) This is an analog simulation of discrete time systems.

Many researchers (Anderson, 1993; Holland et al., 1986; Rosenbloom et al., 1993) argue that higher order cognitive processes, such as Mathematics, are rule-based and discuss various types of learning in terms of rule-based systems. I will claim that only the formal and rule governed features of Mathematics correspond to the formal and rule governed features of Searle’s Chinese Room.With this in mind let us move to some conclusions.

Inside the framework of the analogy and limited to those that come from it, I can say the following: As far as the Mathematics Classroom is concerned, what we have above is a set of ‘one to one’ correspondences that allows us to reach conclusions similar to those reached by Searle regarding the Chinese Room. More specifically, a student’s responses in Mathematics seem right and similar to those given by a person who has in-depth knowledge and understanding of the subject (we naturally only consider those who provide right answers). However, in the case of Mathematics, the responses are provided by managing typical symbols, usually un-interpreted to the student. With regard to Mathematics, the student is comparable to a computer, in that she executes computational operations on data defined in a typical fashion and she is, therefore, an individualization of a computer program. Consequently, whilst the exams might show that the student does understand Mathematics, in reality, this fact is not valid. The student simply behaves as if she understands Mathematics without in reality understanding it at all. In effect, the student merely operates symbols using a set of rules, without any understanding of the meaning of these symbols. As a result, the Mathematics Classroom, like the Chinese Room, can be compared to a computer. We can, therefore, claim that the execution of rules cannot be considered to be equivalent to understanding or thought, since the student operates symbols based on their form and using the appropriate rules. Thus, since cognition is more than the syntactical operation of symbols according to a set of rules, and the student is confined to perform solely this function, then it follows that we cannot indisputably argue that the student knows and understands Mathematics. This theoretical conclusion can also be practically substantiated by anyone who has extended experience of the everyday reality of a Mathematics Classroom.

Taking this opportunity, let us remember that thirty years ago Skemp (1976) from another path reaches analogous findings. He speaks about two meanings of the word ‘understanding’ i.e. ‘instrumental understanding’ and ‘relational understanding’. The first term means what is now known as ‘rules without reasons’; in this case one has fixed plans, by which he/she can go from the problem’s data to the desired inferences. The second term means knowing both what to do and why; in this case one masters a conceptual structure or schema from which he/she can produce a number of plans for getting from any starting point within his/her schema to any finishing point. We need to stress here that without a theory of learning one can only have instrumental understanding. Professional educators need relational understanding of the complex task in which they are engaged. For further study on Skemp’s terms ‘instrumental’ and ‘relational’ see discussion in Section "Discussion".

At this point it is imperative to say that the term ‘mathematical symbol’ is not limited only on x or y which represent the unknown variables in an equation, but also includes symbols from Calculus (:calculate, calculations), Algebra and Set Theory such as ∀, ∃, ∈, \(\subseteq \), ∪, ∩, \(\smallint\), ′, ′′ etc. Students just learn the rules and apply them mechanistically even to complex issues like the calculation of derivatives with the aid of de l’ hôpital Rule, systems solution with Cramer’s Rule, calculation of integrals etc. It is common experience for Mathematics teachers that their students can solve exercises with absolute value, derivative, integral, but most of the times cannot say anything about the corresponding concepts of distance, rate of change, area etc. All calculations are algorithmic, that is why they are solved easily by a computer program. Symbols themselves are empty of meaning. The meaning is conveyed by the subject, when he/she connects the particular symbol with the corresponding concept. This concept is created and is being accepted by the mathematical community. If the student does not have conceptual understanding, then the symbol continues to be empty for him/her. The only thing that is left to do is to operate it according to the rules. Here, Skemp is absolutely right in saying: “Instrumental understanding I would until recently not have regarded as understanding at all”.

The question I want to ask at this stage is: What must we do to move away from the unilateral managing of symbols using a set of rules, toward better understanding and in-depth knowledge of Mathematics? My view is that today we need a harmonic co-existence and co-functioning of both tensions.

Technical and cognitive mathematics

Let us begin with the history of Thought. In Rucker (1988) we find that the Greeks frequently organized their thoughts in terms of dyads, a pair of opposing concepts. The Pythagoreans often drew up ‘Tables of Opposites’ listing related dyads. Which of the two takes intellectual priority in a dyad? Neither. There is no real priority; the two modes of existence are complementary aspects of reality. The term ‘complementarity’ was first introduced into Science and Philosophy by the quantum physicist Niels Bohr who used this expression to sum up his belief that basic physical reality is both spotty and smooth. At the deepest level of physical reality, things are not definitely spotty or definitely smooth. It seems that reality is one; mind and language introduce distinctions that are needed for our understanding.

Leibniz’s relationism thought of the space-time dyad as being an essential part of reality. Bohr thought of the number-space dyad in the same way. Given a dyad, there is always the temptation to believe that if we could only dig a little deeper, we could find a way of explaining one half of the dyad in terms of the other, but the philosophy of complementarity says that there doesn’t have to be any single fundamental concept. Some aspects of the world are spread out and spotty, like the counting numbers; some other aspects are smooth and connected, like space. Complementarity tells us not to try to make the world simpler than it actually is.

It is interesting to realize that two complementary world views seem to be built into our brains. I am referring here to the well-known human brain’s allocation of different functions to its two halves. Many of the intellectual tasks a brain performs can be thought of as primarily digital or primarily analog. Each of the first kind is a ‘step-at-a-time’ process, while in the second kind brain seems to see e.g. a picture ‘all at once’ rather than to divide it up into lumps of information. The left brain is in charge of digital manipulations and the right brain is in charge of analog activities. In other words, the left half of the brain thinks in terms of number (the analytical left brain), and the right half of the brain thinks in terms of space (the synthesizing right brain) (Dehaene, 1997; Eysenck & Keane, 2000).

A dyad is a basically static grouping of concepts. One of Hegel’s (1969) contributions to Philosophy is the idea of grouping concepts into triads, which consist of three concepts arranged in the well-known thesis-antithesis-synthesis or dialectic pattern. The triad is an essentially dynamic grouping, for each synthesis can become the thesis for a new antithesis. Just as Hegel goes a step beyond the Greeks, the psychologist Jung (1953–1979) goes a step beyond Hegel. For Jung, the fundamental pattern of thought is not the triad, but the tetrad, a balanced arrangement of four concepts, also known as the quaternity. Until now we have been drawing tetrads as the corners of a square; one might also think of a tetrad as comprising the corners of a three-dimensional tetrahedron. Dyads, triads, and tetrads are archetypes, where ‘archetype’ means ‘a recurrent form of human thought’. Plato, for instance, frequently reasoned dialectically, fitting theses, antitheses, and syntheses together into chains of triads.

Let us pass now to our dyads, triads and tetrads. According to Davies and Hersh (1984), in some areas of Mathematics it is necessary to apply an algorithm, in others, new techniques must be devised to solve a particular problem, in still more the main concern is concepts and meanings, and finally in others, within a limited framework of principles, the existence ofentities and solutions can be proven without an interest in the chosen approach. The first area is called algorithmic Mathematics, the second heuristics, the third conceptual Mathematics, and the fourth existential. Algorithmic and heuristicsMathematics, on the one hand, are concerned with the finding and application of techniques, whilst conceptual and existentialMathematics, on the other, are concerned with concepts, meanings, knowledge, understanding, logic, and sense. I will term the former category TechnicalMathematics and the latter Cognitive Mathematics.

Taking into account the above and the mathematical experience and results, we can choose two basic dialectical schemes, which seem to characterize the situation. From my point of view, I tend to accept the claims of Davis and Hersh, as having high consistency and harmony with the mathematical experience. Consequently, our basic dialectic schemes will be the following:
https://static-content.springer.com/image/art%3A10.1007%2Fs11217-006-9018-y/MediaObjects/11217_2006_9018_Sch1_HTML.gif
Fig. 1

Total mathematical activity/synthetic view

From our point of view, Mathematics seems to be the unity of opposites of the basic dialectic schemes, i.e. the Greek’s pemptusia/quintessence. The total mathematical activity can be assumed as being divided into two sub-divisions: To Cognitive Mathematics and to Technical Mathematics. These two sub-divisions are mutually dependent and interact with each other, so that the one leads to the other.

In the next table we clarify the two terms as complementary views of the total mathematical activity:

Technical Mathematics

Cognitive Mathematics

Heuristics, algorithms, methods

Concepts, meaning, logic, sense

Name, symbol, signifier

Reference, signified

Syntactic

Semantic

Know how

Know why

All these are discriminations of the mind: The two views co-exist, co-function and determine one another; otherwise neither of them can exist: one exists in reference to the other. Indeed, Cognitive Mathematics is a term which is used in contrast to Technical Mathematics and vise versa. If somebody cannot accept the one concept as valid, then the other cannot stand on its own. The two concepts owe their existence to their interdependence. Only by abandoning this dualistic view can we reach the holistic/synthetic reality.

The above schema shows that the two opposite views can be unified, since their existence is relative, that is the one exists in reference to the other and both are different expressions of the total mathematical activity. Cognitive Mathematics and Technical Mathematics as opposite concepts are at the edges of a continuous spectrum and between them lay practically infinite situations. The phenomenon is analogous to the passing from the two-valued logic of the set {0, 1} to the many-valued logic of the interval [0, 1], where the concept of degree is introduced. An analogue scheme holds for mathematical understanding with the dyad being instrumental and relational understanding.

For the purposes of the present article I use the above scheme. Besides this scheme, which comes from the manipulation of the views expressed by Davis and Hersh, there are other parallel ones. Such a scheme comes from the synthesis, on the one hand, of the dyad proposed by Bradshaw–Nettleton (1981): (a) Analytic-elementwise vs. holistic-structural, and, on the other hand, from the one that derives from Lawvere’s (1976) work: (b) Logic vs. Geometry. We can have an analogue quaternity as above. On the first dyad we notice that the analytic thought analyses everything in elements, so it results to the analytic-elementwise, while the holistic thought captures the structure ‘all at once’, so it results to the holistic-structural. On the second dyad we notice that under the term Geometry we can condense concepts like: visual-spatial, extensional, sensuous and empirical, semantic and signified. On the other hand, Logic summarizes concepts like: rational, intentional, symbolic and signifier. For details in the use of such a scheme one can see Drossos (1987) and Gavalas (1999).

The prevalence of the technical aspect of Mathematics, primarily due to advances in technology and machines as well as to a need for application development, results in an underestimation of other aspects of Mathematics and a consequent smothering of its semantic and cognitive sides. Thus, this dichotomy between the ‘body’ and ‘mind’ of Mathematics, once again, favors the former and hampers the latter. There exists a distinction between the technical and conceptual aspects of Mathematics, between formalism and reflection, syntactic–linguistic, and semantic–cognitive aspects. The attachment to the technical aspect of Mathematics, which is an attachment to the signifier and disregard for the signified, naturally leads to loss of meaning. However, opposites are complementary with regard to the whole. I, therefore, argue for a need to render the two aspects of Mathematics equal and consequently for a practical development of its cognitive aspect. The conceptual framework within which the basic mathematical concepts and their interrelationships are placed must precede the finding and application of techniques, because in this manner we assign meaning to the latter and render them natural and justified, rather than arbitrary and lacking of content.

In Davis and Hersh we also find that Mathematics of ancient Orient was technical. Conceptual Mathematics began in ancient Greece, when the two aspects co-existed. Only in recent times, and especially after Cantor, do we find Mathematics with a somewhat restricted technical content, that we can aptly call conceptual. Whilst for most of the 20th century Mathematics has focused more on its conceptual rather than its technical side, a recent regression to its technical side is evident, due to advances in technology and the need for application development. The possible future development of cognitive machines will, at least initially, render the two aspects equal, but will later lessen the significance of the technical aspect, since, not only does this seem to be the state of affairs, but also, in the incoming era, there seems to be a need for the pursuit of meaning in a world characterized by mechanization.

The technical side of Mathematics constitutes a means for the solving of problems. Within this side the rules might differ depending on the problem and the available computing equipment. It is an area where action is necessary and results are guaranteed. The conceptual side, on the other hand, is logical, statements are liable to the criterion of truth, and objects have specific properties and are liable to the criterion of existence. The principles are definite and are universally accepted. This is an area where thought and reflection are necessary to achieve understanding and in-depth knowledge and where everything seems to have meaning.

Teaching of mathematics in today’s school

According to Bloom (1956), Krathwohl, Bloom, and Masia (1956), Krathwohl, Bloom, and Bertram (1973), there are three domains of educational activities: (i) Cognitive: mental skills (knowledge), (ii) Affective: growth in feelings or emotional areas (attitude), (iii) Psychomotor: manual or physical skills (skills). So, we have:
  1. (i)

    Cognitive: The cognitive domain involves knowledge and the development of intellectual skills. This includes the recall or recognition of specific facts, procedural patterns, and concepts that serve in the development of intellectual abilities and skills. There are six (6) major categories, which are listed in order below, starting from the simplest behavior to the most complex. The categories can be thought of as degrees of difficulty. That is, the first one must be mastered before the next one can take place: Knowledge, Comprehension, Application, Analysis, Synthesis, and Evaluation.

     
  2. (ii)

    Affective: This domain includes the manner in which we deal with things emotionally, such as feelings, values, appreciation, enthusiasm, motivations, and attitudes. The five (5) major categories listed the simplest behavior to the most complex: Receiving Phenomena, Responding to Phenomena, Valuing, Organization, and Characterization by a Value or Value Complex/Internalizing values.

     
  3. (iii)

    Psychomotor: The psychomotor domain includes physical movement, coordination, and use of the motor-skill areas. Development of these skills requires practice and is measured in terms of speed, precision, distance, procedures, or techniques in execution. The seven (7) major categories listed the simplest behavior to the most complex: Perception, Set, Guided Response, Mechanism, Complex Overt Response, Adaptation, and Origination. The psychomotor domain discussed above is by Simpson (1972). There is another popular version by Dave (1975): Imitation, Manipulation, Precision, Articulation, and Naturalization.

     

After these general and classical views about Learning Domains, let us see what happens in the contemporary Mathematics education domain. According to the relevant research which took place the last twenty five years in this domain, mathematical understanding appears to be a function of several interdependent factors: Knowledge, control, beliefs, sociocultural context, that interact in a variety of ways. There is moreover the influence of various affective dimensions such as attitudes, emotions, etc. Control and metacognition are important too and interact with other attributes like resources, heuristics, affect. But what do all these mean?

Resources are the conceptual understanding, knowledge, facts and procedures used during mathematical activity. Resources consist in formal and informal knowledge about the content domain including facts, definitions, algorithmic procedures, routing procedures and relevant competence about rules of discourse. The utility of resources depends on the factor of control: Even when individuals appear to possess the resources to solve a particular problem, they often do not access those resources in the context of producing a problem solution. According to Shoenfeld (1992), it’s not just what you know, it’s how, when and whether you use it.

Metacognition has also been a focus of studies (Lester, 1994) and has been defined to include knowledge about and monitoring of one’s thought processes and control during problem solving. According to Shoenfeld (1992), the term monitoring means the mental actions involved in reflecting on the effectiveness of the problem solving process and products. Research has also substantiated that affective variables such as beliefs, attitudes and motions have a powerful influence on the behavior of the problem solver. Metacognitive behaviors include reflection on the efficiency and effectiveness of the cognitive activities and subsequent self-regulatory behaviors. Control refers to the metacognitive behaviors and global decisions that influence the solution path as well as the decision of strategies and the exploitation of techniques. Methods describe the general strategies used when working a problem, while heuristics describe more specific procedures and approaches. Affect includes attitudes, beliefs, emotions and values/ethics, but includes mathematical intimacy and integrity too (Carlson & Bloom, 2005).

Building on the existing body of literature I focus my study on some other factors that I consider them as very important. As was discussed in the above sections, it is vital that a bold change of direction against the established mentality in the teaching of Mathematics takes place. Students’ indifference and phobia of Mathematics, especially in view of the way it is currently taught, relates to the terror of mechanization and of lack of meaning, as it usually happens with Technical Mathematics. The depressing and unconstructive conditions both in social and school environments – for instance human remoteness, the lack of emotion, the demand solely for achievements and results, the psychological and material exploitation of all difficulties – render one of humankind’s highest achievements, namely Mathematics, inaccessible and objectionable amongst young people. The majority of students, on the one hand, do not have a natural inclination or an interest in Mathematics and, on the other, retain wrong impressions about its true nature, have difficulties in understanding its essence, and appreciate its beauty. What is more, when they become teachers and parents themselves, they reproduce this erroneous stance and ignorance. The above mentality has lost the objective, because Mathematics is not an end in itself but primarily an instructional means, a means for intellectual and emotional development. We must therefore, according to Kofman and Senge (1995), shift our minds. Indeed, the emphasis on making choices, and its ensuing competition amongst students owing to examinations, renders ‘seeming’ good more important than ‘being’ good. The fear of failure and not seeming good are key enemies of learning. In order to learn we need to accept that there is something we do not yet know, and to act specifically on the areas we are not competent. The main malfunction of today’s school is, in reality, a consequence of its successes in the past. Rather than being a problem that needs to be addressed, this malfunction resembles crystallized modes of thinking that need to be dissolved. The dissolver is a new mode of thinking, indeed a shift of mind and mentality. In this new mode of thinking we must move away from the part toward the whole, away from the mechanized toward the cognitive–semantic.

According to Skemp (1976), the main cause for the negative attitude and the fear of so many individuals about Mathematics is the wide failure to teach relational Mathematics. On the occasion of Skemp’s article, one observes that since the mathematical concept is formed in student’s mind – that is he/she knows the characteristics that comprise its depth as well as the objects that comprise its extension – he/she can distinguish it in a specific framework or to take it to another similar framework and find appropriate examples of instrumental type for it. Conversely, beginning from examples of instrumental type, if the student can make some regular and permanent inferences, then he/she has the ability by abstracting and generalizing to attain the corresponding concept. Therefore, the two views are interconnected/interdepended and together form the two poles of a procedure; this procedure -being a holistic dynamical phenomenon- consists the total activity of the mathematical understanding. So, the teacher must lead students to attain both instrumental and relational understanding. This is the only way to achieve the total mathematical understanding; otherwise their learning is one-sided, since the other side is missing.

A serious problem that in general faces the teaching of Mathematics is the lack of meaning. The major part of students learns techniques and mechanisms to solve problems and exercises without understanding their meaning. It is a common phenomenon for a student to study a subject without realizing completely what exactly he/she is doing and in what context all his/her actions are included. The consequence of this tactic is that mathematics for the majority of students is a bunch of techniques and mechanisms which solve problems without meaning.

For instance, instead of prompting the student to think what he/she has to do in a specific problem or what strategies he/she already knows, we ask him/her to answer in a series of typical problems using a readymade methodology in order to have an immaculate technique. It’s easy therefore for the student to lose his/her orientation in a world of signs and symbols, which have their own autonomy, rules and algorithms which are the only objective. For every fundamental mathematical concept, it is almost obligatory to teach its role and value before the student learns the techniques, mechanisms and algorithms around it. The quick passing to the algorithms destroys the meaning. We know very well from the didactic experience that the lack of meaning is disincentive which leads to indifference.

According to Yalom (2002): “We humans appear to be meaning-seeking creatures that have the misfortune to be thrown into a world devoid of intrinsic meaning. (...) The problem of life meaning plagues all self-reflective beings. (...) A man in Rodin’s ‘thinker’ posture and chanting to himself “What’s it all about? What’s it all about?”. (...) It seems evident to me that life-purpose projects take on a deeper, more powerful significance if they are self-transcendent – that is, directed to something or someone outside themselves – the love of a cause, the creative process, the love of others”.

Hence, our objective is not to teach students how to solve numerous different exercises in a mechanical fashion after years of studying, facing difficulties to understand exactly what they are doing and what the meaning of their actions is, but rather to help them to create a conceptual framework using those basic mathematical concepts that are the bearers of meaning, and to form a mode of thinking that will be useful to them and to science itself. Our aim is to equip and arm their cognitive arsenal with the appropriate concepts that will help them to confront the real problems in life. We can turn Mathematics into a precious intellectual journey, full of meaning and artistic value, rich in instructive findings. There comes no benefit from making it a nightmare for students.

The problem which concerns the lack of understanding in Mathematics shows up sometimes in students’ behavior, when they are unable to explain their answers. But there are times when we must ask whether someone, whose behavior shows that he/she understands Mathematics, really understands it. In an exam for example, no one can establish whether the student, has provided the right answer in a mechanistic fashion or rather because he/she has constructed in his/her mind a conceptual understanding of the subject in question.

At this point please allow me to say a few things about education in my country, which have a wider interest. In Greece the whole educational system is oriented to lead students entering university. For the teaching of Mathematics the traditional teaching model is followed. In the official directives book of the Greek Pedagogical Institute, concerning the teaching of Mathematics in school, we read: “According to the traditional teaching model, the mathematics teacher begins the teaching by presenting a technique followed by exercises for practicing and application. The main purpose of this teaching focuses on how students will acquire those skills that are presented to them as well as to give exact and fast answers. This model works under the following assumption: The set of techniques that the students possess for solving exercises constitutes the body of knowledge they must have. Therefore, the students’ facility in these techniques shows if they have learnt Mathematics or not. In this model, knowledge is personal affair of each student who works alone. The knowledge and the student are considered as two separate entities, as independent factors. Under this claim, the student cannot affect knowledge and his/her only move is to learn it. On the other hand, problem and problem-solving in this model have specific character which is limited to a learning criterion. This means that the teacher teaches an algorithm and afterwards, in order to detect whether the student has learnt it, he/she must be able to solve the exercises and the problems that are listed at the end of the unit or chapter”. One easily can see the similarities between this traditional model and the Chinese Room. In Greece, there is not problem-solving, only exercises, and co-operative learning, projects etc are not followed. Each student works alone and is antagonistic to the other ones.

Senge et al. (2003), on the other hand, discuss the ‘Industrial-age Assumptions about School’. In a paragraph titled ‘Learning is Primarily Individualistic and Competition Accelerates Learning’ they say: “Because we see knowledge as something that teachers have and students are supposed to get, we see it as possessed by individuals, and we tend to see the learning process as being similarly individualistic. Yet the traditional classroom focuses almost exclusively on the individual perspective. Individual learners are supposed to master subject matter. Individuals are tested for their comprehension and individuals compete with one another to determine how well they do”.

In Rucker (1988) we find that in human relations we can emphasize either the roles of various individuals or the importance of the overall society. In Psychology we can talk about either various elementary perceptions or the emotions that link them. The world can be viewed as a collection of distinct things or as a single organic whole. One might also ask whether a person is best thought of as a distinct individual or as a nexus in the web of social interaction. No person exists wholly distinct from human society, so it might seem best to say that the space of society is fundamental. On the other hand, each person can feel like an isolated individual, so maybe the number-like individuals are fundamental. Complementarity says that a person is both individual and social component, and that there is no need to try to separate the two.

So, we can see the procedure of education in general as an individualistic one and/or as a holistic one. A representative case of the first view is Piaget (1978) who studies individuals while of the second is Vygotsky (1978) who cares about the social dimension of this procedure. In the first case we apply the method of analytical thought which fits better to the traditional education, while in the second case we apply the method of systemic thought which fits better to contemporary systemic education (Senge et al., 2003).

It is obvious that the initial discussion about CR and MC systems concerns the traditional education, because in a high degree we still have, and especially in my country, the industrial-age traditional school. This is exactly what must change, because it is in non-correspondence with the information and globalization era. In the spirit of this era, conceptual understanding, as opposed to mechanistic learning, could be evaluated if we transferred a concept from one field to another – for example, if we transferred the concept of vector from Mathematics to Physics – in other words, if we used transdisciplinary or systemic methodology. However, this is not the case in reality.

Systematization of the subject matter/conceptual network

One way to help students better understand, retain and reproduce the acquired information and knowledge more effectively is by the organization and systematization of School Mathematics. This suggestion is not of course a panacea. It has been applied first to Sciences and my personal experience in the classroom shows that it is one of the factors that can help the situation. I, therefore, propose that at the beginning of each grade, some time must be allocated toward the construction of a reference framework for School Mathematics. This framework will primarily consist of those basic elements the student has already been taught, organized and inter-connected within this new structure. In this way these basic elements are clarified and put forward as an organized set wherein we can introduce basic new elements, such as the fundamentals of logic, updates on probative methods, insights into what mathematical objects are and a general overview of how Mathematics works. The organization of such elements into a unified set produces a general reference framework or a map of School Mathematics that is continually updated and enriched with new features as the teaching of the subject progresses. This method of constructing the reference framework is cooperative. The teacher carefully guides the students to place on the map the features they consider important, organize them in a network, and find the relations that bind these features together. The students constantly add new features to the initial network, which are in some way related to the already existing ones. I also propose that knowledge obtained after the completion of every chapter is systematized. This amounts to arranging the material in a conceptual network, placing the basic concepts of every chapter and their interrelations into a unified set, together with a monitoring mechanism for the comprehension of techniques-algorithms. Systematization–organization of the material taught in the course of the full school year is also possible.

This process of organizing the subject matter and generating a reference framework, helps the students, who are its main designers, to easily discern the essential from the redundant, the primary from the secondary. Despite the advantages of cognitive classification and a feeling of security, the student is also facilitated in his/her ability to instantly assess whether he/she understands or not what he/she needs. This organization–systematisation can be done in a couple of pages per chapter or with the help of an illustration–diagram, so that a few pages and diagrams offer the student an overview of the basic concepts of the book, in other words, a conceptual network. The knots of the network represent the basic concepts the student has learnt whilst its interconnecting branches (when they exist) correspond to the relations amongst the concepts. This allows the student to recognize that the different chapters of School Mathematics do not contain scattered knowledge but, on the contrary, constitute a unified system that has continuity and an organic connection amongst its constituent parts. The student discovers that when he/she does not know something, what he/she does not know is not only a specific feature but also all other features that are closely related to it, since, in the realm of Mathematics, almost everything is interconnected and what the student has at his/her disposal is a map whereby he/she can orientate himself/herself and investigate the area (Entrekin, 1992; Hasemann & Mansfiled, 1995). This way of perceiving the map implies that knowledge representations, which become standardized with the use of theoretical structures, can be of geometrical nature. We can introduce the basic pattern of concepts, in other words, the category, either in its conceptual or mathematical sense (Lawvere & Schanuel, 2001) and assess its function in knowledge representation. The dialectical scheme, constant–variable, of such patterns can be discussed in geometrical terms, as argued above.

This straightforward proposal for the generation of mathematical networks (Mac Lane, 1986) in schools, also leads us to the problem of knowledge representation. The study of Science in general, and that of Mathematics in particular, offer a plethora of information but only the information organized into structured systems can it become knowledge. When one has to manage this information he/she needs the appropriate filters; additionally he/she turns to the organization and the processing of the information using computers. To develop suitable computer programs we need to set standards using exact terms, since incomplete or ambiguous programming commands are not intelligible to the computer. It is, thus, necessary to classify knowledge into well-defined conceptual or even logical categories that can be correctly interpreted to a programming language. This implies that we need a posteriori typical knowledge representation rather than a priori mechanistic information processing, which is what is currently taking place in a Mathematics Classroom, where the order of things has been reversed.

Knowledge representation

Knowledge representation is a human endeavor that commenced at the same time as scientific thought. This effort has, sometimes, resulted in a mathematical representation, as in the case of Newton, or has, in others, been expressed differently, that is without using Mathematics. Even in the latter case, a vast amount of information can be classified in a type of structure, be it a conceptual structure, a network, or a system. An example of this attempt is that of Aristotle, or Kant, or that of Linnaeus, on a more practical level, in the area of botanical classification. The need for typical structures appears when, like today, we have to manage complex systems that exhibit great diversity, high variation, and varied interactions amongst their members. This need has led to the development of Systems Theory and its equivalent Systemic Methodology as well as to that of Model Theory. From a technological perspective we also have the much broader area of Artificial Intelligence.

According to Grenander (1997), theoretical knowledge representations using models and schemes/shapes are not always possible. The former use structures whilst the latter use geometrical forms. There are also cases where representation is not at all feasible. However, the objective of such representations as those discussed here is fundamentally different from that of Artificial Intelligence. The aim of the latter is to imitate human intelligence in particular, and knowledge in general, using computational algorithms. The increasing effort in the area of Artificial Intelligence has innumerable consequences for the development and restructuring of human knowledge. We have witnessed the most extensive study of human biological and cognitive make up. Yet, the most important question that still remains is: What does human intelligence consist in?

In the course of the industrial era, and even before that, precision and clarity were considered to be virtuous properties, as the famous saying `clear is wise’ implies. However, the move to our Informational Society in conjunction with research in the areas of Artificial Intelligence and Machine Intelligence has had the following consequences:
  1. (i)

    Although computers are equipped with 0–1 logic (absolute clarity) and have reached astonishing speeds, they still exhibit remarkable stupidity. This leads us to the opposite direction of that implied, by `wise is what is suitably ambiguous’. Besides, all sayings that do possess wisdom are expressed in a more or less poetic and vague manner, in other words, they are polysemous.

     
  2. (ii)

    The boundless expressiveness of natural languages is due to appropriate handling and use of ambiguity.

     

Observations such as those above show that the notion that intelligent cognition exists only when there is, at least, appropriate use of ambiguity, is becoming more widely accepted. Consequently, for an entity to be called intelligent, at least, the following must be present: (a) Plasticity and flexibility, in other words, self-regulation, (b) Ability to learn and remember, even under circumstances when limited information is provided, (c) Multi-leveled organization and an overall ability to create a consistent and effective system for managing ambiguity and partiality.

It is obvious that the mechanistic usage of ‘sense-less’ mathematical symbols by the students does not lead either to self-regulation or adequate handling of ambiguity which, from another perspective, also implies lack of in-depth knowledge and understanding of Mathematics.

Discussion

Aristotle in Metaphysics makes a distinction between craftsmen and master craftsmen regarding knowledge. He says: “For men of experience know that the thing is so, but do not know why, while the others know the ‘why’ and the cause” (Book A, chapter 1, paragraph 981a, lines 29–30). In this Aristotelian distinction one can find the source of the ‘technical’ and ‘cognitive’ understanding dichotomy. This discussion is so fruitful that many others used this view of knowledge in favor of their own arguments. In contemporary philosophy e.g. Wittgenstein uses an analogue to Aristotle’s craftsman, specifically a builder, to support his argument about language (Charles, 2001).

Skemp (1972, 1976), in the Mathematics education domain, makes a distinction between ‘instrumental’ and ‘relational’ understanding, which is essentially the same as Aristotle’s in more modern disguise although Skemp does not cite Aristotle. Skemp (1972) defines instrumental understanding as the ability to apply an appropriate remembered rule to the solution of a problem without knowing why the rule works. Relational understanding is the ability to deduce specific rules or procedures from more general mathematical relationships. Skemp (1987) argues that when these differing concepts of understanding are applied to the teaching of mathematics they result in such differing kinds of knowledge that there is a strong case for considering them as different kinds of mathematics: “I now believe that there are two effectively different subjects beingtaught under the same name, ‘mathematics’. If this is true, then this difference matters beyond any of the differences in syllabi which are so widely debated”.

Using Skemp’s distinction several Mathematics educators have expressed their own views. Tall (1978) says that: “Understanding which comes about through search for coherence I would term ‘relational understanding’. On the other hand, ‘Instrumental understanding’ can simply be an exercise of the memory, or worse, be characterized on occasions by compartmentalization of ideas, not wishing to make an overall pattern and preferring the comfort of a limited closed system. This closure manifests itself as ‘rules without reason’ in Skemp’s description. I feel personally that the distinction between these two types of understanding is often one of attitude – the desire to make a coherent pattern out of different pieces of information distinguishing the one from the other. Note that relational understanding, according to this viewpoint, can occur in very rudimentary situations”.

Byers and Herscovics (1978) suggest the ‘tetrahedron of understanding’ that is helpful to consider their four kinds of understanding (instrumental, relational, intuitive and formal) as vertices of a tetrahedron. A research, also, was conducted by Zeleke and Lee (2002) in order to find out how students relate the Strong Law of Large Numbers (SLLN) and the Central Limit Theorem (CLT) to ‘variation’. Analysis of the results shows that most students were able to list adequate interpretations or applications about ‘variation’. This suggests that students, in general have instrumental understanding of the concept. However, the second question seems to bring lots of confusion and difficulties to students. Less than half of students were able to describe correctly in their own words the CLT. Only a few were able to explain the ‘variation’ using CLT and SLLN. Some even suggested that the concepts are not related at all. This indicates that most students’ understanding of ‘variation’ is not at the level of relational understanding 1.

In order to avoid misunderstandings, I have to clarify two concepts: ‘Cognitive Mathematics’ is a term which deals with concepts, logic, semantics, profound meaning etc (as was discussed in section "Technical and cognitive mathematics"). One can also come across the term ‘Teaching Mathematics for Understanding’ which refers to the teaching-learning process that allows the student to make connections between informal and formal learning contexts. In other words, ‘Teaching Mathematics for Understanding’ is more appropriate for didactics and pedagogy, whilst ‘Cognitive Mathematics’ is more appropriate for Mathematics. ‘Teaching Mathematics for Understanding’ becomes a relevant subject only if we want to teach students Cognitive Mathematics. Same observations hold for Skemp’s distinction. Understanding is an act of human mind and not Mathematics itself, anyway. For a more detailed discussion of ‘Teaching Mathematics for Understanding’ one can refer to Carpenter, Fennema, and Romberg (1992), Grouws (1997), Fennema and Romberg (1999), NCTM (2000), Wilson (1993).

Conclusion

Using Searle’s argument as an analogy to the Mathematics Classroom, and specifically the ways it functions in today’s school environment, I conclude that the student is confined to manipulating symbols following a set of rules. In my view, this does not constitute understanding or a conceptual change of Mathematics. To remedy the situation I propose the coexistence of the two sides of Mathematics, namely Technical and Cognitive. The objective is not the mechanistic solving of exercises but rather in-depth understanding of a conceptual framework, which leads to problem solving in many branches of knowledge. For this purpose, I propose the organization and systematization of mathematical knowledge in a network for school use.

From a practical point of view, the Mathematics teacher, using School Mathematics and its methods, can help the student to understand his/her conceptual tools (Cognitive Mathematics) and their corresponding techniques (Technical Mathematics). The student, in his/her turn, can employ the acquired knowledge in solving life’s and science’s real problems. More specifically, the implementation of the views expressed here can contribute, amongst others, to the empirical-experiential understanding of mathematical concepts, to conceptual and technical knowledge, and to the development of an accurate mathematical sense. Furthermore, it can restore the true meaning of Mathematics, by helping to avoid one-sidedness and offering a more comprehensive and holistic view of it. Finally, it can assist the transition from the handling of symbols to in-depth knowledge and understanding of Mathematics.

Footnotes
1

For a contemporary debate about Skemp’s views one can see Tall and Thomas (2002), which is a tribute to Richard Skemp.

 

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© Springer Science + Business Media B.V. 2006