EXPLORING AND UNDERSTANDING THE ABSTRACT

BY DIRECT MANIPULATION OF THE CONCRETE

Oksana Arnold

, Jun Fujima

, Klaus P. Jantke

and Yuzuru Tanaka

Erfurt University of Applied Sciences, Department of Applied Computer Science, Altonaer Str. 25, 99085 Erfurt, Germany

Fraunhofer IDMT, Children’s Media Department, KinderMedienZentrum, Erich-K¨astner-Str. 1a, 99094 Erfurt, Germany

Hokkaido University Sapporo, Meme Media Laboratory, Kita-13, Nishi-8, Kita-ku, Sapporo, 060-8628 Japan

Keywords:

Computability Theory, Webble Technology, Direct Manipulation, Exploratory Learning, Playful Learning.

Abstract:

In schools and in universities, there will always remain a certain very abstract, but highly relevant content.

Remarkable percentages of learners, therefore, are facing severe difﬁculties and need some substantial support.

The paper demonstrates an involved case of learning some of the most abstract contents in higher education by

means of exploratory direct manipulation. Direct manipulation technologies allow for touching media objects,

for moving, modifying and combining them, and for–even playfully–exploring the behavior of mechanisms.

Webble technology has been used to represent content and to implement direct manipulation functionalities.

The implementation reported is in practical use within some moodle environment.

1 INTRODUCTION

Recursion theory also known as the theory of effective

computability (Rogers jr., 1967) is among the most

abstract areas of computer science. The basic results

of recursion theory are explaining the principle reach

and limitations of computers. They are underlying the

fundamental insights of logic–from the classics to the

present–into decidability and undecidability. Last but

not least, recursion theory is determining the reach

of the automation of human reasoning (Siekmann and

Wrightson, 1983a; Siekmann and Wrightson, 1983b).

A correct and comprehensive treatment requires

enormous efforts of understanding abstract concepts

and brings with it notations of a high complexity like

(m+n)

, ... , y

, z

, ... , z

) = ϕ

(n)

(x,y

,...,y

)

, ... , z

)

This equality holds for any G

ODEL numbering ϕ,

where for any two natural numbers m and n there does

exist a certain general recursive encoding function s

accordingly ((Rogers jr., 1967), §1.8, p. 23).

Despite awareness of the discipline’s relevance,

many students are reluctant to memorizing, under-

standing, and actively using formalisms like this.

What are the potentials of media technologies and

media didactics to alleviate the bottleneck of calculi

of effective computability? In particular, what are

the potentials of direct manipulation and of intuitive

interfaces? If technologies may be of any help, how?

How to learn the abstract by concrete manipulation?

2 TOWARD MEDIA DIDACTICS

FOR TEACHING ABSTRACT

CONCEPTS BY MEANS OF

CONCRETE MANIPULATION

There is some scientiﬁc evidence that teaching resp.

learning abstract concepts may be substantially sup-

ported by concrete activities (Barsalou and Wiemer-

Hastings, 2005). However, concepts considered in the

present approach are by far more formal than those

studied by Barsalou and Wiemer-Hastings (2005).

One of the problems with the abstract concepts

and relations of computability theory is that starring at

a formalisms like the s-m-n-theorem on the left does

not help at all. Those formalisms have some opera-

tional meaning, but they are not “running” in any way.

In particular, it helps to learn and, ﬁnally, master

the formalisms by studying the formulas’ meaning for

a larger number of speciﬁc input data. Unfortunately,

given any particular data, such an exploration of some

formula’s meaning is tedious and time-consuming.

Learners have to perform by hand lengthy formal and

mostly trivial, but error-prone transformations.

100

Arnold O., Fujima J., P. Jantke K. and Tanaka Y..

EXPLORING AND UNDERSTANDING THE ABSTRACT BY DIRECT MANIPULATION OF THE CONCRETE.

DOI: 10.5220/0003916001000107

In Proceedings of the 4th International Conference on Computer Supported Education (CSEDU-2012), pages 100-107

ISBN: 978-989-8565-07-5

 2012 SCITEPRESS (Science and Technology Publications, Lda.)

Roughly speaking, there is the question for media

environments in which the abstract has some concrete

representations that allow for a playful manipulation

by hand. Operational meanings “just take place” and

are observed to provoke thought and interpretation.

3 COMPUTABILITY THEORY

This section is intended to present the domain in some

detail to provide a ﬁrm foundation of the subsequent

investigation.

About hundred years ago, a large community of

scientists did strongly believe in the solvability of any

problem supposed the problem has been expressed

sufﬁciently clear and formal. This assumption was

underlying David Hilbert’s famous and remarkably

inﬂuential speech to the 2

International Congress of

Mathematicians, Paris, August 1900 (Gray, 2000).

However, a large variety of problems such as the

difﬁculty to ﬁnd a sound foundation of set theory lead

to severe doubts. Could it, perhaps, be the case that

there were some generally unsolvable problems?

For strictly proving the unsolvability of any par-

ticular problem, i.e. to demonstrate that no algorithm

ever will be able to provide a solution, one needs to

have a sufﬁciently clear understanding of deﬁnitely

all algorithms. This insight lead to the emergence of

a general theory of algorithms or, in other terms, of

effective computability (Rogers jr., 1967).

There are largely varying approaches primarily

by authors such as Alonzo Church (Church, 1936),

Stephen C. Kleene (Kleene, 1936), Emil Post (Post,

1936), and Alan M. Turing (Turing, 1936). All

these attempts to speciﬁcy computability, in general,

turn out to be provably equivalent (see (Machtey and

Young, 1978), e.g.)–an evidence for the so-called

CHURCH’s Thesis that each formalization correctly

reﬂects the true nature of computability (see (Church,

1936), footnote 2, page 346).

Relying on CHURCH’s Thesis, it is sufﬁcient to

study one of these approaches in detail. For the pur-

pose of the research and the applications in higher

education reported in this paper, the authors have

selected the concept of partial recursive functions

originating from (G¨odel, 1931) and (Kleene, 1936)

(see also (P´eter, 1981)).

The underlying key idea of this approach is to be-

gin with a few obviously computable functions and to

deﬁne the class of all computable functions to be the

closure under a few obviously computable operators.

• Every constant function is computable.

• The successor function is computable.

• The projection on an argument is computable

• Substitution inherits computability.

• Limited recursion inherits computability.

• The minimum operator inherits computability.

The ﬁrst three items above specify the basic functions,

whereas the following three items name the operators.

Illustrative applications will follow in section 6.

4 DIRECT MANIPULATION

Conventionally, studies of recursive function theory

involve much writing of terms and term equations

describing certain functions. Suppose s denotes the

successor function and p

(for any m and n with

n ≤ m) denotes the m-ary projection function which

selects the nth element of any m-tupel.

add(x, 0) = p

(x) = 1 (1)

add(x, s(y)) = s(p

(x, y, a(x, y))) (2)

is a deﬁnition of addition add by means of limited re-

cursion which is the abstract form of inductive deﬁni-

tion (the terminus technicus is “primitive recursion”).

Imagine that, instead of writing formulas as shown

above, functions appear as media objects. Imagine

the same applies to operators for the construction of

new functions. Instead of writing equations, you may

grasp an operator media object (very much in the

sense of grasping in (MacKenzie and Iberall, 1994))

and, in addition, you may grasp some function objects

to plug them into the input slots of the operator object.

In the end, you get a new media object describing a

certain new function.

The enormous advantage of directly manipulat-

ing objects is that the newly constructed one at the

learner’s ﬁngertips is operational.

Figure 1: Two function objects and an operator object in

use; the learner plugging the Projection object into the

Primitive Recursion object by dragging it with her ﬁngers

into the slot of the operator object (at the ﬁgure’s bottom).

EXPLORINGANDUNDERSTANDINGTHEABSTRACTBYDIRECTMANIPULATIONOFTHECONCRETE

101

Pondering the correctness of your construction,

there is no need to stare at a formula. Instead, you

simply play with your construct putting some values

in and looking for the outcome.

In particular, error-prone symbol manipulations

are avoided. Instead, learners can concentrate on the

key phenomena of setting up some new deﬁnition. In

particular, variants can be explored easily. Last but

not least, it may be more or less fun to manipulate ob-

jects physically on a touch screen or on an interactive

whiteboard. In contrast, it has been rarely reported

that writing lengthy formulas is considered any fun.

5 MEME MEDIA TECHNOLOGY

Independent of computer science and engineering and

of media technologies, Richard Dawkins (1976) has

published an inﬂuential book on the evolution of

thought. Excited by these ideas, Susan Blackmore

(Blackmore, 1999) is discussing the philosophical

reach of Dawkins’ perspective.

Dawkins’ key concept is named Meme used to de-

note the building blocks of human thought somehow

similar to the genes building the characteristics of the

whole human being. Yuzuru Tanaka has accepted the

challenge, so to speak, from a computer science and

media technology perspective (Tanaka, 2003). His

pioneering work aims at concepts and implementa-

tions that result in environmental conditions for the

computer-supported evolution of human knowledge–

memes `a la Dawkins being digitalized and prepared

for digital evolution. Accordingly, Tanaka speaks

about Meme Media.

From a somehow technocratic point of view, par-

ticular meme media technologies as described in

(Tanaka, 2003) establish some middleware which is

particularly suitable for the development of systems

in which knowledge evolution plays a prominent role.

Pads are the meme media object encapsulating knowl-

edge. Knowledge units can be subject to mutation or

cross-over. Though these manipulations are executed

by humans, the overall outcome of such a knowledge

evolution process is usually unforeseeable.

Early versions of meme media middleware have

been named IntelligentPad stressing that meme me-

dia objects appear like pads on a screen which can be

directly manipulated by dragging and dropping and,

thus, combining them to form new composite pads.

The most recent variant of meme media middle-

ware is named Webble to abbreviate “Web-based life-

like entities”. Webbles, so to speak, live in browser

windows as illustrated by means of ﬁgure 2. The roots

of the technology are due to Tanaka and Kuwahara

Figure 2: A virtual laboratory based on Webble technology.

(Kuwahara and Tanaka, 2009).

6 COMPUTABILITY WEBBLES

The authors have set up some environment to host

a variety of media objects for computable functions

and for operators which allow for the composition of

any further computable functions. As will be shown

below, the environment contains all basic functions

and all operators necessary to deﬁne any computable

function (Rogers jr., 1967), (Machtey and Young,

1978), (P´eter, 1981).

Hence, this little environment is capable of re-

presenting anything computable. It implements the

fundamentals of a full recursion theory accessible by

means of direct media manipulation–computability

theory at your ﬁngertips.

6.1 Webbles for Basic Functions

The functions needed initially are

• the unary successor function s,

• all constant functions c

of any arity n,

• all projection functions p

selecting the mth ele-

ment out of n given arguments (for any m ≤ n).

The repository originally contains a few basic

functions as on display in ﬁgure 3.

In case other initial functions are needed, they are

easily generated by duplicating one of the original

functions and, successively, setting parameters. For

illustration, the lowest Webble shown represents p

It may be easily edited to select the second instead of

the ﬁrst input, that way becoming p

, or to work for

one more argument, that way becoming p

CSEDU2012-4thInternationalConferenceonComputerSupportedEducation

102

Figure 3: The initial functions of Primitive Recursion.

6.2 Webbles for Operations

The set of all computable functions, usually named

the partial recursive functions (Rogers jr., 1967), may

be generated, if the three operators

• substitution of functions,

• limited, i.e. so-called primitive, recursion,

• minimum operator

are available.

Note that in ﬁgure 4 the values “-1” in the output

slots indicate some failure, because the operators are

not in use.

Substitution as shown above is prepared for the

substitution of one unary function into another one.

One may easily edit the scheme, for instance, to al-

low for the substitution of one binary function into

one unary function or, alternatively, to allow for the

substitution of two unary functions into some binary

one. These two examples are on display in ﬁgure 5.

The other operators may be easily edited similarly.

6.3 Direct Manipulation of Deﬁnitions

Direct manipulation, loosely saying, means that you

are able to take a media object directly, move it to

a place where you would like to have it, and per-

form what you want to do–turning around, rescaling,

setting parameters, and the like–immediately. Using

modern interfaces, you can do it with your own hands.

Figure 4: Operators necessary and sufﬁcient for generating

the complete collection of all Partial Recursive Functions.

Figure 5: Two more variants of the substitution operator.

When students shall learn that for the purpose of

deﬁning a particular target function, one has to substi-

tute some given original into another function given

as well, they should be allowed to do it by hand, so

to speak. The physical activity is assumed to support

cognition (Barsalou and Wiemer-Hastings, 2005).

When using Webbles for direct manipulation, a

standard activity is to drag and drop the one media

object over the other to plug it in. See the ﬁgure 6

below for a simple illustration.

The ﬁgure displays a real screenshot of the authors’

learning environment within moodle in which three

subsequent phases are overlayed in a single window,

for the sake of comparison.

In phase one, the learner takes some media object

(the one with the big dot at its left upper corner) to

drag and drop it over another composite object into

its apparently empty slot. Then, the object is latch-

ing automatically. The result is shown as a composite

media object on the right side of ﬁgure 6–phase two

EXPLORINGANDUNDERSTANDINGTHEABSTRACTBYDIRECTMANIPULATIONOFTHECONCRETE

103

Figure 6: Phases of a function deﬁnition by drag & drop.

of the construction. With some click to the object, the

learner may open a menu offering further opportuni-

ties. There is the action of grouping or, so to speak,

hiding details. After performing this activity as phase

three of the overall assembly process, the composite

object appears as displayed at the bottom.

Learners may deﬁne any computable function in

this way simply dragging and dropping elementary

objects or those constructed before as intermediate

stages of a more complex construction process.

To automatically lock objects in place is a very

convenient support to the human learner, but several

alternatives exist (Fujima, 2010).

According to the authors’ experience (varying up

to 25 years) in teaching computer science courses, a

remarkable percentage of students have difﬁculties in

understanding partiality of recursive functions theory.

It is always a sticking point to deﬁne one’s ﬁrst partial

recursive function that is not total. One has to think of

processes that terminate sometimes, but not always.

It is crucial to employ the minimum operator–

usually denoted by µ–appropriately. In the case of

deﬁning some unary function f , the terminology

f(x) = µy[g(x, y) = 0] (3)

means the value of f on input x is deﬁned to be the

smallest y such that g(x, y) equals 0. Obviously, in

case g(x, y) is always greater than zero, the value f(x)

remains undeﬁned. The minimum operator represents

the most abstract concept of unbounded search.

In several exercises, the function exact is deﬁned

as follows

exact(x, y) = (sq(y) −

x) + (x−

sq(y)) (4)

where sq(y) is the square of y and the subtraction in

use yields zero if the second argument is not smaller

than the ﬁrst one. Students may be encouraged to play

Figure 7: Exploring how the minimum operator works.

with some functions like this and with the minimum

operator (see ﬁgure 7 above).

The advantage of direct manipulation becomes

obvious. Learners can playfully explore the theory.

7 SCENARIOS OF PLAYFUL

EXPLORATORY LEARNING

From some methodologicalpoint of view, the authors’

approach towards learning of abstract concepts and

of their properties and mutual relations may be seen

as an experiment to support Barsalou’s and Wiemer-

Hasting’s fourth hypothesis that “the content of ab-

stract concepts could, in principle, be simulated” (see

(Barsalou and Wiemer-Hastings, 2005), pp. 136/137).

From a technological point of view, the approach may

be understood as just another case of using Webble

technology for playful learning (Fujima et al., 2010).

Direct manipulation may be fun (Xie et al., 2008)

beneﬁcal to learning–evenextremely abstract content.

7.1 Exploratory Learning

Let us take the illustration from ﬁgure 7 in the column

on the left as an easy case study. The function object

(top right) which is plugged into the minimum oper-

ator object has been constructed beforehand. Taking

CSEDU2012-4thInternationalConferenceonComputerSupportedEducation

104

the requirements of partial recursive function theory

deﬁnitions (P´eter, 1981) seriously, a formally com-

plete speciﬁcation of the mentioned function object

requires an equational speciﬁcation as shown in the

Appendix.

Conventionally, the application of the minimum

operator to the function exact would be written as

f(x) = µy[exact(x, y) = 0] (5)

Would you as a reader, personally, like to compute

the value f(225), e.g., according to this deﬁnition tak-

ing the ten equations from the appendix as a basis?

How easy is it to derive characteristics of the function

f from its equational speciﬁcation?

The authors are clearly in favor of exploratory

learning which is supported by the operationality of

composite media objects such as the one next to the

bottom of ﬁgure 7.

Experimenting with media objects at your ﬁnger-

tips, you can easily ﬁnd out that on input 225 you get

the output 15 as shown in ﬁgure 7. Some additional

explorations yield further insights such as, e.g., that

there are outputs in response to the inputs 1 and 4,

but no response to 2 and 3.

Interested learners might now go further and

explore why there are no outputs in these cases.

Some might inspect the deﬁning equations in detail.

Others might, instead, decompose the media object

and inspect the data ﬂow and transformation in its

components–direct manipulation and exploration.

7.2 Game-based Learning

Webbles implementing partial recursive functions of-

fer a large number of opportunities for competitiveex-

ercises and playful exploration, although the domain

is extremely abstract and formulas are frequently

cumbersome.

It is somehow exciting that most–if not all–of the

deeper insights into computable functions theory lead

to numerous speciﬁc exercises.

The fundamental KLEENE normal form theorem

(see (Rogers jr., 1967), §1.10, p. 30) may serve as a

convenient introductory example. Roughly, although

applications of the minimum operator may be iterated

several times, this basic theorem says that for any

computable function there is no need for more than

just one call of the minimum operator. The theorem

may be summarized by the following simple equality

for any computable function f of n arguments.

f(x

, . . . , x

) = α( µy[ β(i, x

, . . . , x

, y) = 0 ] ) (6)

Here, α and β are two universal functions deﬁnable

without any application of the minimum operator and

i is some index speciﬁc for f. By the way, α and β are

just primitive recursive decodings and encodings.

It follows a certain generic exercise based on the

KLEENE normal form theorem:

Provide any function deﬁnition using two or more

calls of the minimum operator. Ask students for the

simplest alternative Webble media object describing

the same particular function, but using the minimum

operator Webble (as shown in ﬁgure 7) at most once.

Competitions of this type are complicated by the

problem to demonstrate that a certain proposed alter-

native is correct. In general, the equivalence problem

for deﬁnitions of partial recursive functions is known

to be undecidable (Rogers jr., 1967).

To sum up, competitive exercises as sketched here

may lead to a variety of deeper problems such as, e.g.,

program equivalence which may be experienced by

the learners in a quite playful manner. In this way, we

return from competition to exploration.

In another game, one might proceed as follows:

• Peel off an inner media object.

• An opponent has to ﬁnd another object for plug-in

leading to an inequivalent deﬁnition.

• If the opponent fails, he looses. Otherwise, the

players’ roles are exchanged and it becomes the

opponent’s turn to peel off some object.

• The game ends when constructions are repeated.

To complete the present section, readers should

recall that the current practice of teaching recursive

function theory is far from being fun on both sides–

the teachers and the students.

8 DIRECT MANIPULATION FOR

LEARNING THE ABSTRACT IN

RECENT HIGHER EDUCATION

Direct manipulation, in its early years, has been

seen as a paradigm for alleviating severe bottlenecks

of human-computer interaction (Shneiderman, 1982),

(Shneiderman, 1983), (Hutchins et al., 1985). More

recently, it has developed into a paradigm supportive

to human understanding in highly involved studies.

(Thalheim, 2008) provides some valuable approach

to teaching SQL. Thalheim’s approach, however, is

much less operational than the present one, because

his Visual SQL queries are not functional. They still

need to be translated. In contrast, the authors’ partial

recursive functions Webbles are ready to run and they

respond to any data input by processing the data.

The implementation of computability theory by

means of Webble technology described in the paper’s

EXPLORINGANDUNDERSTANDINGTHEABSTRACTBYDIRECTMANIPULATIONOFTHECONCRETE

105

preceding sections has been embedded into some

Moodle learning environment for seamless curricular

integration. It is currently in use in university level

courses on Theoretical Computer Science.

In this environment, the students ﬁnd a certain

repository of Webbles sufﬁcient to construct arbitrary

computable functions. This provides, so to speak,

computability theory at the students’ ﬁngertips.

There are exercises that enforce students’ collabo-

ration. In particular, students are encouraged to work

together on a project exchanging composite Webbles.

In some sense, this may be seen as a Web 2.0

approach to the study of computability theory, be-

cause learners contribute their own constructs to some

repository which may be used within the community.

In particular, there are tasks that establish con-

tests and encourage students to compete for optimal

solutions. That way some of the exercises are get-

ting close to games and studying computability theory

may become a bit playful.

Foremost, the implementation provides a practical

case study in which learning of rather involved and far

reaching abstract content is substantially supported by

active learning based on direct media manipulation,

exploration, and play.

Within the oral presentation of the present paper

as well as in the related discussion, the authors will be

able to report their recent experiences from teaching

during the Summer term 2012.

However, it is the present authors’ strong believe

that experimentation and systematic evaluation, in

particular, is a ﬁeld of scientiﬁc work in its own right.

This paper is a contribution to e-learning technologies

introducing Webbles as a middleware for exploratory

and playfully learning studying an abstract domain.

9 CONCLUSIONS & OUTLOOK

The present contribution is intended to set the stage

for direct manipulation approaches to studies of

highly abstract domains such as computability theory.

The basics have been prepared and some complete

implementation–including its integration into some

Moodle environment–has been provided. Further-

more, a couple of scenarios of exploration and playful

competition have been developed.

This way, the authors are facing now systematic

experimentations and evaluations. The results should

be published in some subsequent paper, but go beyond

the limits of the present short paper.

ACKNOWLEDGEMENTS

The authors gratefully acknowledge the cooperation

with a larger number of scientists and engineers who

contributed to the development of IntelligenPad and

Webble technologies providing cases of application.

Micke Kuwahara deserves particular thanks.

Furthermore, they gratefully acknowledge the

support by colleagues to introduce the development

reported in the present paper into higher education

practice including some students providing feedback.

Andr´e Schulz is the one who has always kept our

computability Webbles running in Moodle.

REFERENCES

Barsalou, L. W. and Wiemer-Hastings, K. (2005). Situating

abstract concepts. In Pecher, D. and Zwaan, R. A.,

editors, Grounding Cognition: The Role of Perception

and Action in Memory, Language, and Thought, pages

129–163. New York: Cambridge University Press.

Blackmore, S. (1999). The Meme Machine. Oxford Univer-

sity Press.

Church, A. (1936). An unsolvable problem of elementary

number theory. The American Journal of Mathemat-

ics, 58:345–363.

Dawkins, R. (1976). The Selﬁsh Gene. Oxford University

Press.

Fujima, J. (2010). Auto-connection mechanisms for ed-

ucational virtual laboratory. In The IET Interna-

tional Conference on Frontier Computing, Proceed-

ings (DVD), Taichung, Taiwan, August 4-6, 2010,

pages 396–401.

Fujima, J., Hawlitschek, A., and Hoppe, I. (2010). Living

machinery – Advantages of Webble technologies for

teaching and learning. In Proc. 2nd International Con-

ference on Computer Supported Education, CSEDU

2010, Valencia, Spain, April 7-10, 2010.

G¨odel, K. (1931).

Uber formal unentscheidbare S¨atze

der Principia Mathematica und verwandter Systeme I.

Monatshefte f¨ur Mathematik und Physik, 38:173–198.

Gray, J. (2000). The Hilbert problems 1900-2000. Newslet-

ter of the European Mathematical Society, 36:10–13.

Hutchins, E. L., Hollan, J. D., and Norman, D. A. (1985).

Direct manipulation interfaces. Human-Computer In-

teraction, 1:311–338.

Kleene, S. C. (1936). General recursive functions of natural

numbers. Mathematische Annalen, 112(5):727–742.

Kuwahara, M. and Tanaka, Y. (2009). Webble portal.

http://www.meme.hokudai.ac.jp/WebbleWorldPortal/.

Machtey, M. and Young, P. (1978). An Introduction to the

General Theory of Algorithms. New York, Oxford,

Shannon: Elesevier North-Holland.

MacKenzie, C. L. and Iberall, T. (1994). The Grasping

Hand. Amsterdam: North-Holland.

CSEDU2012-4thInternationalConferenceonComputerSupportedEducation

106

P´eter, R. (1981). Recursive Functions in Computer Science.

Akad´emia Kiad´o, Budapest.

Post, E. L. (1936). Finite combinatory processes. formula-

tion I. The Journal of Symbolic Logic, 1:103–105.

Rogers jr., H. (1967). Theory of Recursive Functions and

Effective Computability. McGraw-Hill.

Shneiderman, B. (1982). The future of interactive systems

and the emergence of direct manipulation. Behavior

and Information Technology, 1:237–256.

Shneiderman, B. (1983). Direct manipulation: A step

beyond programming languages. IEEE Computer,

16:57–69.

Siekmann, J. and Wrightson, G., editors (1983a). Automa-

tion of Reasoning 1. Classical Papers on Computa-

tional Logic 1957–1966. Berlin, Heidelberg, New

York: Springer-Verlag.

Siekmann, J. and Wrightson, G., editors (1983b). Automa-

tion of Reasoning 2. Classical Papers on Computa-

tional Logic 1967–1970. Berlin, Heidelberg, New

York: Springer-Verlag.

Tanaka, Y. (2003). Meme Media and Meme Market Archi-

tectures: Knowledge Media for Editing, Distributing,

and Managing Intellectual Resources. IEEE Press &

Wiley-Interscience.

Thalheim, B. (2008). Visual SQL: Towards ER-based

object-relational database querying. In Li, Q., Spac-

capietra, S., Yu, E., and Oliv´e, A., editors, Proceed-

ings ER 2008, 27th International Conference on Con-

ceptual Modeling, Barcelona, Spain, October 20-24,

2008, number 5231 in LNCS. Springer Verlag.

Turing, A. M. (1936). On computable numbers with an ap-

plication to the Entscheidungsproblem. Proceedings

of the London Mathematical Society, 42(7):230–265.

Xie, L., Antle, A. N., and Motamedi, N. (2008). Are tan-

gibles more fun? Comparing children’s enjoyment

and engagement using physical, graphical and tan-

gible user interfaces. In Proceedings of the Second

International Conference on Tangible and Embedded

Interaction (TEI’08), Feb 18-20, 2008, Bonn, Ger-

many, pages 191–198. New York, NY, USA: ACM.

APPENDIX

add(x, 0) = p

(x) (7)

add(x, s(y)) = s(p

(x, y, add(x, y))) (8)

mult(x, 0) = c

(x) (9)

mult(x, s(y)) = add(p

(x, y, mult(x, y)), p

(x, y, mult(x, y))) (10)

sq(x) = mult(x, x) (11)

pred(0) = 0 (12)

pred(s(y)) = p

(y) (13)

subtr(x, 0) = p

(x) (14)

subtr(x, s(y)) = pred(p

(x, y, subtr(x, y)) (15)

exact(x, y) = add(subtr(p

(x, y), sq(p

(x, y))), subtr(sq(p

(x, y)), p

(x, y))) (16)

EXPLORINGANDUNDERSTANDINGTHEABSTRACTBYDIRECTMANIPULATIONOFTHECONCRETE

107