Assessing the Effectiveness of the APOS/ACE Method for Teaching

Mathematics to Engineering Students

MICHAEL GR. VOSKOGLOU

Mathematical Sciences, School of Engineering,

University of Peloponnese (ex T.E.I. of Western Greece),

26334 Patras,

GREECE

Abstract: - The present work focuses on a classroom application for evaluating the effectiveness of the

APOS/ACE instructional treatment for teaching mathematics to engineering students. Using linguistic

(qualitative) grades, the assessment of the mean student performance is realized with the help of grey numbers

and the assessment of their quality performance by calculating the Grade Point Average (GPA) index. A

neutrosophic assessment method is also applied for evaluating the overall student performance, because the

instructor had doubts about the accuracy of the qualitative grades assigned to some students.

Key-Words: - APOS/ACE instructional treatment, fuzzy assessment methods, grey numbers (GNs), GPA index,

neutrosophic sets (NSs), neutrosophic triplets.

Received: June 12, 2022. Revised: February 14, 2023. Accepted: March 11, 2023. Published: April 12, 2023.

1 Introduction

Computers have become nowadays a valuable tool

for Education The animation of figures and

representations, achieved by using the proper

software, develop the students’ imagination and

enhance their problem-solving skills. In the

forthcoming era of the Fourth Industrial Revolution

computers will provide, through the advanced

Internet of Things (IoT), a wealth of information for

students and teachers.

Several didactic methods have been already

developed in which computers play a dominant role.

One of them is the APOS/ACE instructional

treatment for teaching Mathematics, developed in

the USA by Ed Dubinsky and his collaborators

during the 1990’s [1-3]. In earlier works we have

applied this approach for teaching the rational

numbers [4], the polar coordinates [5, 6] and the

derivative [7, 8] at university level. The present

paper focuses on a classroom application for the

assessment of the effectiveness of the APOS/ACE

approach for teaching Mathematics to engineering

students using qualitative (linguistic) grades.

The rest of the paper is formulated as follows:

Section 2 is devoted to a brief presentation of the

basic principles of the APOS/ACE theory. The

necessary mathematical background about fuzzy

sets neutrosophic sets and grey numbers, needed for

the purposes of this work, is presented in Section 3.

The fuzzy methods used for the assessment of the

effectiveness of the APOS/ACE instruction with

qualitative grades are developed in Section 4 and the

classroom application is presented in Section 5. The

paper closes with the final conclusions and some

hints for future research, which are included in the

last Section 6.

2 The APOS/ACE Method

Dubinsky had already spent twenty- five years

doing research in Functional Analysis and teaching

undergraduate mathematics before starting on

figuring out pedagogical strategies that help students

to be more successful in learning mathematics.

APOS is a theory based on Piaget’s principle that an

individual learns by applying certain mental

mechanisms to build specific mental structures and

uses these structures to deal with problems

connected to the corresponding situations [9].

According to the APOS, these mechanisms involve

interiorization and encapsulation, while the

cognitive structures involve Actions, Processes,

Objects and Schemas. The first letters of the last

four words form the acronym APOS.

A mathematical concept begins to be formed as

one applies transformations on certain entities to

obtain other entities. A transformation is first

conceived as an action. For example, if an

individual can think of a function only through an

explicit expression and can do little more than

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substitute for the variable in the expression and

manipulate it, he (she) is considered to have an

action understanding of functions.

As an individual repeats and reflects an action it

may be interiorized to a mental process. A process

performs the same operation as the action, but

wholly in the mind of the individual, enabling

her/him to imagine performing the transformation

without having to execute each step explicitly. For

example, an individual with a process understanding

of a function thinks about it in terms of inputs,

possibly unspecified, and transformations of those

inputs to produce outputs.

If one becomes aware of a mental process as a

totality and can construct transformations acting on

this totality, then he/she has encapsulated the

process into a cognitive object. In the case of

functions, for example, encapsulation allows one to

form sets of functions, to define operations on such

sets, to equip them with a topology, etc. Although a

process is transformed into an object by

encapsulation, this is often neither easy nor

immediate. This happens because encapsulation

entails a radical shift in the nature of one’s

conceptualization, since it signifies the ability to

think of the same concept as a mathematical entity

to which new, higher-level transformations, can be

applied. Many mathematical situations, however,

require one to de-encapsulate an object back to the

process that led to it. This cycle may be repeated

one or more times, e.g. going back from a composite

function to its component functions for the better

understanding of the rule of derivation of a

composite function, going back from the derivative

to the initial function in order to understand the

process of the integration of a function, etc.

A mathematical topic often involves many

actions, processes and objects that need to be

organized into a coherent framework that enables

the individual to decide which mental processes to

use in dealing with a mathematical situation. Such a

framework is called a schema. In the case of

functions, for example, the schema structure is

used to recognize the need of using a specific

function in a given mathematical or real-world

situation.

The implementation of the APOS theory as a

framework for learning and teaching mathematics

involves three stages. First, a theoretical analysis,

called genetic decomposition (GD) of the concepts

under study, is performed. The GD comprises a

description that includes actions, processes and

objects and the order in which it may be best for

learners to experience them. Then instructional

sequences based on the GD are developed and

implemented and finally data are collected and

analysed in order to test and refine the GD and the

pedagogical strategies that have been employed.

The main contribution obtained from an APOS

analysis is the increased understanding of an aspect

of human thought. However, explanations offered

by such analyses are limited to descriptions of the

thinking that an individual may be capable of and

not of what really happens in an individual’s mind,

since this is probably unknowable. Moreover, the

fact that one possesses a certain mental structure

does not mean that he/she will necessarily apply it

in a given situation. This depends on other factors

regarding managerial strategies, prompts, emotional

state, etc.

The APOS theory has important consequences

for education. Simply put, it says that the teaching

of mathematics should consist in helping students

use the mental structures they already have to

develop an understanding of as much mathematics

as those available structures can handle. For

students to move further, teaching should help them

to build new, more powerful, structures for handling

more and more advanced mathematics.

Dubinsky and his collaborators realized that for

each mental construction that comes out from an

APOS analysis, one can find a computer task such

that, if a student engages in that task, he (she) is

fairly likely to build the mental construction that

leads to the learning of the corresponding

mathematical topic. As a consequence, the

pedagogical approach based on the APOS analysis,

known as the ACE teaching cycle, is a repeated

cycle of three components: Activities on the

computer (A), Classroom discussion (C) and

exercises (E) done outside the class.

In applying the ACE cycle the mathematical

topic under consideration is divided into smaller

subtopics and each iteration of the cycle

corresponds to one of the above subtopics. The

computer activities, which form the first step of the

ACE approach, are designed to foster the students’

development of the appropriate mental structures.

The students do all of their work in computer

laboratories divided in cooperative groups.

In the classroom the teacher guides the students

to reflect on the computer activities and their

relation to the mathematical concepts being studied.

They do this by performing mathematical skills

without using the computers. They discuss their

results and listen to explanations, by fellow students

or the teacher, of the mathematical meanings of

what they are working on.

The homework exercises are fairly standard

problems related to the topic being studied. Students

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reinforce the knowledge obtained in the computer

activities and classroom discussions by applying it

in solving these problems.

The implementation of the ACE cycle and its

effectiveness in helping students make mental

constructions and learn mathematics has been

reported in several research studies of Dubinsky’s

team, e.g. [3, 10, 11],etc.

3 Mathematical Background

3.1 Fuzzy Sets and Logic

The development of human science and civilization

owes a lot to Aristotle’s (384-322 BC) bivalent logic

(BL), which was in the center of human reasoning

for centuries. BL is based on the “Principle of the

Excluded Middle”, according to which each

proposition is either true or false. Opposite views,

however, appeared also early in the human history

supporting the existence of a third area between true

and false, where these two notions can exist

together; e.g. by Buddha Siddhartha Gautama

(India, around 500 BC), by Plato (427-377 BC),

more recently by the Marxist philosophers, etc.

Integrated propositions of multi-valued logics

reported, however, only during the early 1900’s,

mainly by Lukasiewicz and Tarski [12, Section 2].

According to the Lukasiewicz’s “Principle of

Valence” propositions are not only either true or

false, but they may have intermediate truth-values

too.

Zadeh, replacing the characteristic function of a

crisp subset of the universe U with the membership

function m: U→ [0, 1], introduced in 1965 the

concept of fuzzy set (FS) [13], in which each

element x of U has a membership degree m(x) in the

unit interval. The closer m(x) to 1, the better x

satisfies the characteristic property of the

corresponding FS. For example, if A is the FS of the

tall men of a country and m(x) = 0.8, then x is a

rather tall man. On the contrary, if m(x) = 0.4, then

x is a rather short man. Formally, a FS A in U can

be written as a set of ordered pairs in the form

F = {(x, m(x)): x



U} (1)

Zadeh also introduced, with the help of FS, the

infinite-valued in the unit interval fuzzy logic (FL)

[14], on the purpose of dealing with the existing in

the everyday life partial truths. FL, in which truth

values are modelled by numbers in the unit interval,

embodies the Lukasiewicz’s “Principle of Valence”.

Uncertainty can be defined as the shortage of

precise knowledge or complete information on the

data that describe the state of a situation. It was only

in a second moment that FS theory and FL were

used to embrace uncertainty modelling. This

happened when membership functions were

reinterpreted as possibility distributions [15, 16].

Zadeh [15] articulated the relationship between

possibility and probability, noticing that what is

probable must preliminarily be possible.

Probability theory used to be for a long period

the unique tool in hands of the specialists for

dealing with problems connected to uncertainty.

Probability, however, was proved to be suitable only

for tackling the cases of uncertainty which are due

to randomness [17]. Randomness characterizes

events with known outcomes which, however,

cannot be predicted in advance, e.g. the games of

chance. FSs, apart from randomness, tackle also

successfully the uncertainty due to vagueness,

which is created when one is unable to distinguish

between two properties, such as “a good player” and

“a mediocre player”. For general facts on FSs and

the connected to them uncertainty we refer to the

book [18].

3.2 Neutrosophic Sets

Several generalizations and extensions of the theory

of FSs have been developed during the last years for

the purpose of tackling more effectively all the

forms of the existing in real world uncertainty. The

most important among them are briefly reviewed in

[19].

Atanassov in 1986, considered, in addition to

Zadeh’s membership degree, the degree of non-

membership and extended FS to the notion of

intuitionistic FS (IFS) [20]. Smarandache in 1995,

inspired by the frequently appearing in real life

neutralities - like <friend, neutral, enemy>, <win,

draw, defeat>, <high, medium, short>, etc. -

generalized IFS to the concept of neutrosophic set

(NS) by adding the degree of indeterminacy or

neutrality [21]. The word “neutrosophy” is a

synthesis of the word “neutral´ and the Greek word

“sophia” (wisdom) and means “the knowledge of

the neutral thought”. The simplest form of a NS is

defined as follows:

Definition 1: A single valued NS (SVNS) A in

the universe U is of the form

A = {(x,T(x),I(x),F(x)): x



U, T(x),I(x),F(x)



[0,1],



T(x)+I(x)+F(x)



3} (2)

In equation (2) T(x), I(x), F(x) are the degrees of

truth (or membership), indeterminacy (or neutrality)

and falsity (or non-membership) of x in A

respectively, called the neutrosophic components of

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x. For simplicity, we write A<T, I, F>.

Indeterminacy is defined to be in general everything

that exists between the opposites of truth and falsity

[22].

Example 1: Let U be the set of the players of a

soccer club and let A be the SVNS of the good

players of the club. Then each player x is

characterized by a neutrosophic triplet (t, i, f) with

respect to A, with t, i, f in [0, 1]. For example,

x(0.7, 0.1, 0.4) ∈ A means that there exists a 70%

belief that x is a good player, but at the same time

there exist a 10% doubt about it and a 40% belief

that x is not a good player. In particular, x (0, 1, 0) ∈

A means that we do not know absolutely nothing

about the quality of player x (new player).

If the sum T(x) + I(x) + F(x) < 1, then it leaves

room for incomplete information about x, if it is

equal to 1 for complete information and if it is >1

for inconsistent (i.e. contradiction tolerant)

information about x. A SVNS may contain

simultaneously elements leaving room to all the

previous types of information. All notions and

operations defined on FSs are naturally extended to

SVNSs [23].

Summation of neutrosophic triplets is equivalent

to the union of NSs. That is why the neutrosophic

summation and implicitly its extension to

neutrosophic scalar multiplication can be defined in

many ways, equivalently to the known in the

literature neutrosophic union operators [24]. For the

needs of the present work, writing the elements of

a SVNS A in the form of neutrosophic triplets and

considering them simply as ordered triplets we

define addition and scalar product as follows:

Definition 2: Let (t1, i1, f1), (t2, i2, f2) be in A and

let k be a positive number. Then:

 The sum (t1, i1, f1) + (t2, i2, f2) = (t1+ t2, i1+

i2, f1+ f2) (2)

 The scalar product k(t1, i1, f1) = (kt1, k i1,

kf1) (3)

Remark 1: Summation and scalar product of

the elements of a SVNS A with respect to Definition

2 need not be closed operations in A, since it may

happen that (t1+ t2)+(i1+ i2)+(f1+ f2)>3 or kt1+k i1 +

kf1>3. With the help of Definition 2, however, one

can define in A the mean value of a finite number of

elements of A as follows:

Definition 3: Let A be a SVNS and let (t1, i1, f1),

(t2, i2, f2), …., (tk, ik, fk) be a finite number of

elements of A. Assume that (ti, ii, fi) appears ni times

in an application, i = 1,2,…., k. Set n =

n1+n2+….+nk. Then the mean value of all these

elements of A is defined to be the element (tm,im,fm)

of A calculated by

[n1(t1, i1,f1)+n2(t2,i2,f2)+….+nk(tk,ik, fk)] (4)

3.3 Grey Numbers

The theory of grey systems [25] introduces an

alternative way for managing the uncertainty in case

of approximate data. A grey system is understood to

be any system which lacks information, such as

structure message, operation mechanism or/and

behavior document.

Closed real intervals are used for performing the

necessary calculations in grey systems. In fact, a

closed real interval [x, y] could be considered as

representing a real number T, termed as a GN,

whose exact value in [x, y] is unknown. We write

then T ∈ [x, y]. A GN T, however, is frequently

accompanied by a whitenization function f: [x, y] →

[0, 1], such that, if f(a) approaches 1, then a in [x, y]

approaches the unknown value of T. If no

whitenization function is defined, it is logical to

consider as a representative crisp approximation of

the GN T the real number

V(T) =

x+y

(3)

The arithmetic operations on GNs are introduced

with the help of the known arithmetic of the real

intervals [26]. In this work we are going to make

use only of the addition of GNs and of the scalar

multiplication of a GN with a positive number,

which are defined as follows:

Definition 2: Let A ∈ [x1, y1], B ∈ [x2, y2] be two

GNs and let k be a positive number. Then:

 The sum: A+B is the GN A+B ∈ [x1+y1,

x2+y2] (4)

 The scalar product kA is the GN kA ∈ [kx1,

ky1] (5)

4 Fuzzy Assessment Methods with

Qualitative Grades

In many cases it is a common practice to assess the

student performance by using qualitative (linguistic)

instead of numerical grades. A widely accepted

scale of such grades is the following: A=excellent,

B=very good, C=good, D=mediocre and

F=unsatisfactory. Here we present a series of fuzzy

assessment methods {see also [27]) that we are

going to use in this work for assessing the overall

performance of a student group.

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4.1 Mean Performance

In case of using qualitative grades the mean

performance of a student group cannot be evaluated

with the classical method of calculating the mean

value of the student individual grades. To overcome

this difficulty, we assign to each grade a GN,

denoted for simplicity with the same letter, as

follows: A = [85, 100], B = [75, 84], C = [60, 74], D

= [50, 59],

F = [0, 49]. The choice of the above GNs, although

it corresponds to generally accepted standards, is not

unique. For example, for a more strict assessment,

one may choose A= [90, 100], B = [80, 89], C= [70,

79], D = [60, 69], F = [0, 59], etc. Such changes,

however, does not affect the generality of our

method

Assume now that, from the n in total students of

the group, nX obtained the grade X=A, B, C, D, F.

It is logical then to accept that the crisp

approximation V(M) of the GN

M =

A B C D F

1(n n B nA C n D )+ + nF

n

(5)

can be used for estimating the mean performance of

the student group.

4.2 Quality Performance

A very popular in USA and other countries method

for evaluating the quality performance of a group is

the use of the Grade Point Average (GPA) index

[28, p.125], which is calculated by the formula

GPA =

F D C B A

0n +n +2n +3n +4n

(6)

In other words, the GPA index is a weighted

average in which greater coefficients (weights) are

assigned to the higher grades. Note that, since in the

worst case (n=nF) is GPA=0 and in the ideal case

(n=nA) is GPA=4, we have in general that

0≤GPA≤4 (7)

When two groups have the same GPA index,

however, this method is not sufficient to show

which of them performs better. In such cases the

Rectangular Fuzzy Assessment Model (RFAM),

which is based on the Center of Gravity (COG)

defuzzification technique can be used [27, pp. 126-

130].

4.3 Neutrosophic Assessment

Frequently in practice the teacher has doubts about

the grades assigned to some students, either because

he/she had not the opportunity to evaluate their

skills explicitly during a course, or because they

didn’t clarify their answers properly in a written

test. In such cases, the most suitable method for

assessing the overall performance of a student group

is to use NSs as tools. Considering, for example, the

NS of the good students of the group, one introduces

neutrosophic triplets characterizing the individual

performance of each student and then calculates the

mean value of all these triplets with the help of

equation (4) in order to obtain the proper

conclusions about the group’s overall performance.

In order to have complete information for each

student’s performance, the sum of the component of

each triplet must be equal to 1.

5 The Classroom Application

The purpose of the following classroom application

was to evaluate the effectiveness of the APOS/ACE

approach for teaching mathematics to engineering

students. The subjects were the first term students of

two departments of the School of Engineering of my

university during the teaching of the course “Higher

Mathematics I”, which includes Complex Numbers,

Differential and Integral Calculus in one variable

and elements from Linear Algebra. According to the

grades obtained in the PanHellenic examination for

entrance in Higher Education, the potential of the

two departments in mathematics was about the

same. The course’s instructor was also the same

person, but the teaching methods followed were

different. Namely, the APOS/ACE approach was

applied for teaching the course to the 60 students of

the first department (experimental group), whereas

the classical method with lectures on the board was

applied for the 60 students of the second department

(control group).

The results of the final examination, after the end

of the course, were the following:

 Department I: A: 9 students, B: 15, C: 18,

D: 12, F: 6

 Department II: A: 12, B: 15, C: 9, D: 12, F:

Therefore, applying the assessment methods of

Section 4, one evaluates the performance of the two

departments as follows:

Mean performance

By equation (5) one finds that

MI =

+15[75,84]+18[60,74]+1(9[85 2[50,,100] 59]+

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+6[0,49])

[3570,4994]≈[59.5,83.23].

Therefore, equation (3) gives that V(MI)≈71.36,

which shows that the experimental group

demonstrated a good (C) mean performance.

In the same way one finds that V(MII)≈62.56,

which shows that the control group also

demonstrated a good (C) mean performance, which,

however, was 8.8% worse than that of the

experimental group.

Quality performance

Equation (6) gives that

GPAI=

12+2*18+3*15+4*9

=2.12and similarly

GPAII=2.05, which shows that the experimental

group demonstrated a slightly better quality

performance. In fact, with the help of equation (7) it

is easy to check that the superiority of the

experimental group in this case is only

0.07*25=1.75%.

Note that, some of the student answers in the

final examination were not clearly presented or well

justified. As a result, the instructor was not quite

sure for the accuracy of the grades assigned to them.

For this reason, we decided to apply the

neutrosophic method of section 4.3 too for the

assessment of the two departments’ overall

performance. For this, starting from the students

with the higher grades, let us denote by Si,

i=1,2,….,60, the students of each department.

Considering the NS of the good students, we

assigned neutrosophic triplets to all

students of the two departments as follows:

 Department I: S1-S32: (1,0,0), S33-S38:

(0.8,0.1,0.1), S39-S42: (0.7,0.2,0.1), S43-S46:

(0.4,0.2,0.4), S47-S50: (0.3,0.2,0.5), S51-S53:

(0.2,0.2,0.6), S54-S55: (0.1,0.2,0.7), S56-S57:

(0,0.2,0.0.8), , S578-S60: (0,0,1).

 Department II: S1-S31: (1,0,0), S32-S35:

(0.8,0.1,0.1), S36: (0.7,0.1,0.2), S35-S43:

(0.4,0.1,0.5), S44-S46: (0.3,0.2,0.5), S47-S50:

(0.2,0.2,0.6), S51-S52: (0.1,0.2,0.7), S53-S58:

(0,0.3,.0.7), , S59-S60: (0,0,1).

Then, by equation (4), the mean value of the

neutrosophic triplets of Department I is equal to

[32(1,0,0)+6 (0.8,0.1,0.1)+ 4(0.7,0.2,0.1)+4 (0.4,

0.2,0.4)+ 4(0.3,0.2,0.5)+3(0.2,0.2,0.6)+ 2(0.1,0.2,

0.7)+2(0,0.2,0.0.8)+3 (0,0,1)≈(0.72, 0.07, 0.21). In

the same way one finds that the mean value of the

neutrosophic triplets of Department II is equal to

(0.65,0.08,0.27).

Thus, the probability for a random student of

Department I to be a good student is 72%, but at the

same time there exists a 7% doubt about it and a

21% probability to be not a good student. Also, the

probability for a random student of Department II to

be a good student is 65%, with a 8% doubt about it

and a 27% probability to be not a good student.

Consequently, the experimental group, despites the

doubts of the instructor for the grades assigned to

the students, demonstrated a better overall

performance.

6 Conclusion

The classroom application presented in this work

demonstrated a superiority of the experimental

(APOS/ACE) group with respect to the control

group. This superiority was significant concerning

the two groups’ mean and overall (in terms of the

neutrosophic method) performance, but rather

negligible concerning their quality performance.

This gives a strong indication that the application of

the APOS/ACE method benefits more the mediocre

and the weak in mathematics students, but less the

good students. Much more experimental research is

needed, however, for obtaining safer conclusions.

References:

[1] Asiala, M., Brown, A., DeVries, D.J., Dubinsky,

E., Mathews, D. & Thomas, K., A framework for

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Issues in Mathematics Education, 6,1996, pp. 1-

32.

[2] Dubinsky, E. & McDonald, M. A., APOS: A

constructivist theory of learning in undergraduate

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al. (Eds.), The Teaching and Learning of

Mathematics at University Level: An ICMI Study,

pp. 273-280, Kluwer Academic Publishers,

Dordrecht, Netherlands, 2001.

[3] Arnon, I. Cottrill, J. Dubinsky, E., Oktac, A.,

Roa, S., Trigueros , M. & Weller, K., APOS

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[4] Voskoglou, M. Gr., An application of the

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numbers, Journal of Mathematical Sciences and

Mathematics Education, 8(1), 30-47, 2013.

[5] Borji, V. & Voskoglou, M.Gr., Applying the

APOS Theory to Study the Student Understanding

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WSEAS TRANSACTIONS on ADVANCES in ENGINEERING EDUCATION

DOI: 10.37394/232010.2023.20.6

Michael Gr. Voskoglou

E-ISSN: 2224-3410

Volume 20, 2023