Ratio Mathematica                                        Volume 39, 2020, pp. 111-136   

 

              

 

111 

 

Teaching as a decision-making model: 

strategies in mathematics from a practical 

requirement 
 

 

Viviana Ventre* 

Eva Ferrara Dentice 

 Roberta Martino† 
 

 

Abstract 

The need in the current social context to adopt teaching methods that 

can stimulate students and lead them towards autonomy, awareness 

and independence in studying could conflict with the needs of 

students with specific learning disorders, especially in higher 

education, where self-learning and self-orientation are required. In 

this sense, the choice of effective teaching strategies becomes a 

decision-making problem and must, therefore, be addressed as such. 

This article discusses some mathematical models for choosing 

effective methods in mathematics education for students with 

specific learning disorders. It moves from the case study of a student 

with specific reading and writing disorders enrolled in the 

mathematical analysis course 1 of the degree course in architecture 

and describes the personalised teaching strategy created for him. 

Keywords: decision-making; inquiry model; social skills; 

personalised didactic strategy; Analytic Hierarchy Process. 

2010 AMS subject classification: 91B06, 97D60.‡ 

 

 
*. Corresponding author: viviana.ventre@unicampania.it 

†. All the authors belong to the Department of Mathematics and Physics of the 

University of the Study of Campania “L. Vanvitelli”, viale A. Lincoln n.5, I-81100, 

Caserta (CE), Italy. 

‡ Received on November 20th, 2020. Accepted on December 22th, 2020. Published on 

December 31st, 2020. doi: 10.23755/rm.v39i0.559. ISSN: 1592-7415. eISSN: 2282-

8214. ©Ventre et al. This paper is published under the CC-BY licence agreement. 



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1  Introduction 
In recent years the institutions have increased their interest in a very 

worrying phenomenon that concerns Italy and generally speaking the countries 

in the Western world: the growth of 'disaffection' towards mathematics due to a 

traditional didactic approach to the subject (Piochi, 2008). Young people 

coming out of secondary schools often have the idea that mathematics consists 

of mechanical processes, seeing it as an arid and pre-packaged discipline whose 

understanding and description seem impersonal. The mathematics one learns at 

school is very often a set of basic notions, axioms and definitions given by the 

teacher and practically impossible to discuss, causing the view of a subject that 

is “already done” and immutable (Castelnuovo, 1963). The experience of 

mathematicians, on the other hand, is very different: mathematics is something 

extremely changeable whose results are the result of hard work, debate and 

controversy. So, axioms and definitions first presented in textbooks come into 

reality only at the end, when the whole structure of the problem is understood. 

Then, the following question arises: what is mathematics? Definitions such as 

“mathematics is the science of numbers and forms” accepted 200 years ago is 

now reductive and ineffective because mathematics has developed so rapidly 

and intensely that no definition can take into account all the multiple aspects 

(Baccaglini-Frank, Di Martino, Natalini, Rosolini, 2018 (A)). The list of 

applications of this discipline in daily life could be endless, and so could the list 

of motivations that could be given to pupils to convince them to study.  

About this matter it is really important the following statement: "No doubt, 

mathematical knowledge is crucial to produce and maintain the most important 

aspects of our present life. This does not imply that the majority of people should 

know mathematics." (Vinner, 2000). Mathematics can also cause terror in 

students (the phenomenon of “fear for mathematics” (Bartilomo and Favilli, 

2005)) or a state of dissatisfaction with the common conception that “you have 

to be made for it” so much that even great professionals boast that they have 

never understood anything about mathematics. So, is it necessary to teach 

mathematics to everyone? The answer is simple: apart from the fact that having 

a basis in mathematics is a cultural question regardless of the future job, 

mathematics teaches to evaluate multiple aspects of a question, and provides 

knowledge and skills in order to consciously face a discussion defending one's 

own positions with responsibility and respect for the arguments of others 

(National Indications, 2007). 

The key role of mathematics education in the development of rational 

thinking and with it the responsibilities of mathematics teachers at all levels is 

therefore underlined. Already in 1958, the theme of the congress of the Belgian 



Teaching as a decision-making model: strategies in mathematics from a 

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Mathematical Society was entitled “The human responsibility of the 

mathematics teacher” (Castelnuovo, 1963). 

The most effective way to bring students closer to mathematics is, therefore, 

the image of a “method for dealing with problems, a language, a box of tools 

that allows us to strengthen our intuition” (Baccaglini-Frank, Di Martino, 

Natalini, Rosolini, 2018 (A)).  

 

2 Mathematical education: theories and models 
2.1 The concept of error and the inquiry model: mathematics 

as a humanistic discipline  
Mathematics is one of the disciplines in which many 'students' manifest 

difficulties that compromise the relationship with the subject. A student who 

comes out of secondary school has a long series of 'failures' accompanied by the 

conviction that she can never do mathematics because she is not good at it. The 

problem lies in identifying errors and difficulties in mathematical learning with 

the conviction that the absence of errors certifies the absence of difficulties and 

on the other hand the absence of difficulties guarantees the absence of errors 

(Zan, 2007 (A)).  

This identification leads to the didactic objective of obtaining the greatest 

number of correct answers by nourishing the "compromise of correct answers" 

(Gardner, 2002): on the one hand the teacher chooses activities that are not "too" 

difficult and on the other hand the students elaborate the answers expected by 

the teacher in a reproductive way. Of course, this method does not guarantee 

any learning, revealing itself dangerous and counterproductive (Di Martino, 

2017). Moreover, with it the fear of making mistakes arise and also the 

conviction that mathematics is not for everyone (for instance: you can't study 

mathematics if you do not have a good memory!) (Zan, Di Martino, 2004).  In 

order to face the identification of difficulty-error, there is, therefore, a need to 

revolutionise the conception of error and to convey to students that ''making a 

mistake at school may not be perceived as something negative to avoid at all, 

because it could be an opportunity for new learning (and teaching) 

opportunities to be exploited'' (Borasi,1996).  

The Inquiry model is a teaching-learning model that proposes a positive and 

fundamental role of errors in mathematics teaching. This model sees knowledge 

as a dynamic process of investigation where cognitive conflict and doubt 

represent the motivations to continuously search for a more and more refined 

understanding. Therefore, instead of eliminating ambiguities and contradictions 

to avoid confusion or errors, these elements must be highlighted to stimulate 

and give shape to ideas and discussions. Questions such as "what would happen 

if this result were true?" or "under what circumstances could this error be 

corrected?" lead to a reformulation of the problem where the error is only the 



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starting point for a deeper understanding. Communication in the classroom 

plays a fundamental role, and so does the conception of mathematics as a 

humanistic discipline: the teacher provides the necessary support for the 

student's autonomous search for understanding, who in turn is an active member 

of a research community (Tematico, Pasucci, 2014).  Learning turns out to be a 

process of constructing meanings, and in this way, the students also understand 

that what is written in textbooks is the result of debates and arguments and not 

simply something for its own sake ("falling from the sky"). 

 

2.2 Cooperative learning: development of disciplinary and social 

skills 
Some studies highlight the need to build learning teaching models that take 

into account students' emotions, perceptions and culture based on the idea that 

human learning has a specific social character (Radford, 2006). The 

collaborative group and peer tutoring are two models that take on both the 

disciplinary dimension and the affective and social dimension and facilitate 

discussion in the classroom. In fact, in most cases, the teacher cannot give 

everyone the opportunity to express themselves, nor is he able to solicit the 

interventions of those who are not used to intervene.  

Collaborative learning, instead, sees the involvement of all the students in 

two successive moments: first within the individual group and then in the final 

discussion in class. The necessary conditions for such learning are positive 

interdependence and the assignment of roles: the first is reached when the 

members of the group understand that there can be no individual success without 

collective success; the second condition allows the distribution of social and 

disciplinary competences among the various members of the group favouring 

collaboration and interdependence. The recognition of roles also helps to 

overcome problems such as low self-esteem or a sense of ineffectiveness, 

allowing social skills to grow: knowing how to make decisions, how to express 

one's own opinions and listen to those of others, how to mediate and share, how 

to encourage, help and resolve conflicts are skills that the school must teach with 

the same care with which disciplinary skills are taught.  

Dialogue among peers guarantees greater freedom and spontaneity: the 

majority of students identify that among peers there is no fear of expressing 

doubts and perplexities, the main motivation that justifies the effectiveness of 

such models (Baldrighi, Pesci, Torresani, 2003; Pesci, 2011). 

 

2.3 Recovery and enhancement interventions: breaking the 

educational contract 
The variety of possible processes, the fact that behind correct answers there 

can be difficulties and that some mistakes can come out of significant thought 

processes, brings important elements to support the criticism of the 



Teaching as a decision-making model: strategies in mathematics from a 

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identification between mistakes and difficulties.  For example, the incorrect 

resolution of a problem is not necessarily due to the inability to manage the 

mathematical structure of the proposed situation but is probably due to a lack of 

understanding of the problem itself. The understanding of a text is not always 

immediate because it involves the student's personal knowledge of common 

words and scripts. Understanding is, therefore reduced to a selective reading that 

aims to identify the numerical data and the right operations suggested by 

keywords (Zan, 2012).  

Recovery interventions must therefore be based on the analysis of the 

processes that led the student to make mistakes, shifting the attention from the 

observation of errors to the observation of failed behaviour with the sole 

objective of change. The student, in turn, must take responsibility for her own 

recovery and therefore there is a need for teaching that makes her feel that she 

is the protagonist of new situations and not simply the executor of procedures to 

be applied to repetitive exercises (Zan, 2007 (B)).  

The teacher must propose exercises and problems that do not favour a 

mechanical approach but question the rules that pupils are used to use and that 

form part of the so-called teaching contract (D'Amore, 2007; D'Amore, 

Gagatsis,1997). The idea of a didactic contract was born to explain the causes 

of elective failure in mathematics, that is, the kind of failure reserved only for 

mathematics by students who instead do well in other subjects. The didactic 

contract holds the interactions between student and teacher and is made up of 

"the set of teacher's behaviours expected by the student and the set of student's 

behaviours expected by the teacher" (Brousseau, 1986).  

This explains the students' belief that a problem or exercise always has a 

solution because it is the teacher's job to make sure that there is only one answer 

to the proposed question and that all the data is necessary (Baruk, 1985). 

In Bagni (1997) the following goniometry test is proposed to fourth-year 

students in three classes of scientific high school (students aged 17 to 18). 

Determine the values of x belonging to ℝ for which it results: 
 

a) sinx = 1 2Τ  b) cosx = 1 2Τ  

c) sinx = 1 3Τ  d) tgx = 2 

e) sinx = 𝜋 3Τ  f) cosx = 𝜋 2Τ  

g) sinx = ξ3 h) cosx = ξ3 3Τ  

Table 2.1 Experiment in Bagni, 1997. 

Remember that the goniometric functions are often introduced by making 

initial reference to the values they assume in correspondence to relatively 

common angles of use, so we have the well-known table shown in the next page 

(Table 2.2.) 



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116 

 

x 0 𝜋 6Τ  𝜋 4Τ  𝜋 3Τ  𝜋 2Τ  2 𝜋 3Τ  3 𝜋 4Τ  5 𝜋 6Τ  𝜋 … 
sinx 0 1 2Τ  ξ2 2Τ  ξ3 2Τ  1 ξ3 2Τ  ξ2 2Τ  1 2Τ  0 … 

cosx 1 ξ3 2Τ  ξ2 2Τ  1 2Τ  0 − 1 2Τ  − ξ2 2Τ  − ξ3 2Τ  -1 … 

tgx 0 ξ3 3Τ  1 ξ3 n.d. −ξ3 -1 − ξ3 3Τ  0 … 

… … … … … … … … … … … 

Table 2.2 Values assumed by common angles where “n.d.” is “not define”. 

Let us now examine the test: agreed time 30 minutes and pupils were not 

allowed to use protractor tables nor scientific calculator. It has been conceived 

with: 

• two "traditional" questions (a), (b); 

• two possible questions, but with the results not included between the values 
of x "of common use" (c), (d);  

• two impossible questions (e), (f) but with values (of sinx and cosx) that 
recall the measurements in radians of "common use" angles (𝜋/3, 𝜋 /2); 

• two questions (g), (h) where the first impossible and the second possible. 
They propose instead values (of sinx, cosx) that are included in the table 

referred to the angles "of common use" but in relation to other goniometric 

functions (tgx, cotgx). 

Well, as far as the answers to the questions (e), (f) are concerned, the 

didactic contract has led some pupils to look for solutions anyway; and the 

"solutions" that most spontaneously presented themselves to their mind are the 

ones that they see associated, in the case of the sinus function, the two values 

𝜋/3 and ξ3 2 Τ and, in the case of the cosine function, the two values 𝜋 /2 and 0. 
So we have, for instance, the following errors: 

   sinx = 𝜋 /3      so      x = ξ3 2 Τ  
          cosx = 𝜋 /2     so      x = 0 
As far as the answers to questions (g), (h) are concerned, the reference to 

the tangent function was clearly expressed in the answers of some students: also 

in this case, some students, not finding the proposed values among those 

corresponding to the most frequently used x values (for the sine and cosine 

functions, in the table above), were induced to look for another correspondence 

in which the proposed values are involved. We then find errors such as: 

          if sinx = ξ3 , then x = 𝜋 /3 

          if sinx = ξ3 , then x = 𝜋 /3+k 𝜋 
What has now been pointed out obliges us to conclude that the need that 

leads the student to always and in any case look for a result for each proposed 

exercise is unstoppable: breaking the teaching contract can be used as a 



Teaching as a decision-making model: strategies in mathematics from a 

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117 

 

teaching strategy to overcome the mechanical approach used by the students 

and enhance knowledge (Bagni, 1997).  

 

3  Concept image ad concept definition 
These notions were developed to analyse the learning processes of 

mathematical definitions (Tall and Vinner, 1981). Concept image is the whole 

cognitive structure related to the concept and includes all mental images, the 

properties and processes of recall and manipulation associated with a concept, 

bringing into play its meaning and use. It is built through years of experience of 

all kinds, changing with the encounter of new stimuli and the growth of the 

individual. The concept definition is the set of words used to specify a concept 

and turns out to be personal and can often differ from the formal definition 

because it represents the reconstruction made by the student and the form of the 

words he uses to explain his concept image. It can change from time to time and 

for each individual the concept definition can generate its own concept image 

which can be called in this case concept definition image. The acquisition of a 

concept occurs when a good relationship is developed between the concept 

name, the concept image and the concept definition. Students tend to learn 

definitions in a mechanical way and this can lead to conflict factors when 

concept image or concept definition are invoked at the same time which conflict 

with another part of the concept image or concept definition acquired on the 

same concept. To explore this topic a questionnaire was administered to 41 

students with an A or B grade in mathematics. They were asked: "Which of the 

following functions are continuous? If possible, give your reason for your 

answer."  

 

 
Figure 3.1 Images from Tall and Vinner, 1981. 

We see that the concept image of this topic comes from a variety of 

resources such as the colloquial use of the term “continuous” in phrases such as 

“It rained all day long”. So, often the use of the term “continuous function” 

implies the idea that the graph of the function can be drawn continuously. The 

answers are summarised in the table shown in Figure 3.2.  



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118 

 

 

 
Figure 3.2 Tables from Tall and Vinner, 1981. It Summarises the results of the 

experiment.  

The reasons given to justify the discontinuity of f2(x) and f4(x) are of the 

type: “The graph is not in a single piece”, “There is no single formula”. In these 

answers, we see that many students invoked a concept image including a graph 

without any interruption or a function defined by a “single formula”. Instead, 

there are many continuous functions that conflict with the concept images just 

mentioned as the following:  

𝑓ሺ𝑥ሻ = ൜
0 ሺ𝑥 < 0 𝑜𝑟 𝑥2 < 2ሻ

1ሺ𝑥 > 0 𝑜𝑟 𝑥2 > 2ሻ
 

whose graph is: 

 

 
Figure 3.3 Image from Tall and Vinner, 1981. It represents the function defined 

above. 

The idea that emerges from similar issues is that mathematical concepts 

should be learned in the everyday, not technical, way of thinking, starting with 

many examples and non-examples through which the concept image is formed 

and then arriving at a formal definition. Students should use the formal 

definition, but in order to internalise the concept it is necessary to aim at 

cognitive conflicts between concept image and concept definition. To do this it 

is necessary to give tasks that do not refer only to the concept image for a correct 

resolution, inducing the students to use the definition (Baccaglini-Frank, Di 

Martino, Natalini and Rosolini , 2018 (B)).  

 

4  Teaching as a decision problem  
Today more than ever, the world of education has to work on the 

construction of personalities that can favour to all the students with freedom of 

choice and reactivity. The social context in which we live is complex because it 

comprehends factors of unpredictability and uncertainty: the educational 

systems have the job to provide a path that aims to thought and action autonomy. 



Teaching as a decision-making model: strategies in mathematics from a 

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The school, therefore, has an orientation character, where the term 

“orientation” indicates a continuous and personal process that involves 

awareness, learning and education in choice (Biagioli, 2003). In particular, 

placing orientation as the main purpose of teaching "means developing 

strategies, methodologies and contents aimed from the acquisition of awareness 

to understanding the complex society and the mechanisms that govern the world 

of studies and work" (Guerrini, 2017). To give the proper and necessary 

instruments to the student, in order to activate the auto-orientation processes, the 

teacher has to chose what the best didactic strategy is. Therefore, on an 

operational point of view, the teaching is a decisional problem and has to be 

faced as it is. Indeed, we can speak of decision when in a situation there are: 

alternatives (being able to act in several different ways), probability (the 

possibility that the results relating to each alternative will be achieved) and the 

consequences associated with the results. Such factors are characteristic of the 

school world. So, to realise the best didactic strategy it is necessary to start with 

a representation of the problem: only through the calculation of the expectations 

and the evaluation of the results, it is possible to choose the right option. 

Decisions can be studied in terms of absolute rationality or limited 

rationality. The first model ideally combines rationality and information by 

preferring the best alternative; the second recognises the objective narrowness 

of the human mind by proposing the selection of the most satisfactory alternative 

(Lanciano, 2019-20).  

It is important to emphasise that the consequences of a decision are 

determined also by the context in which the decision-making process is 

developed. On the basis of the decision maker's knowledge of the state of nature. 

we distinguish various types of decisions: 

• decisions in a situation of certainty: when the decision-maker knows the 
state of nature; 

• decisions in risk situations: when the decision-maker does not directly know 
each state of nature, but has a probability measure for them; 

• decisions in situations of uncertainty: when the decision maker has neither 
information on the state of nature nor the probability associated with it. 

The decision maker can adopt two kinds of approaches: 

• Normative approach. which bases the choice with reference to rational 
decision-making ideals; 

• Descriptive approach which analyses how to make a decision based on the 
context. 

So, the teacher has to consider on the basis of the objectives and the 

context the various alternatives, and for each one of them, the possible 

consequences. For each pair (alternative, circumstance) the teacher obtains a 

result according to a utility function. 



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However, the decision is subjective: it is based on the criterion of obtaining 

a maximum value for the utility function. Moreover, even if the choice is 

rational, it is made in terms of limited rationality because, in general, there are 

few alternatives, but it increases as the teacher expands his/her culture and 

experience (Delli Rocili, Maturo, 2013; Maturo, Zappacosta, 2017). 

 

4.1 A model for evaluating educational alternatives 
Multi-criteria decision analysis (MCDA) provides support to the decision 

maker, or a group of decision makers, when many conflicting assessments have 

to be considered, especially in data synthesis phase while working with complex 

and heterogeneous pieces of information. 

Let A = {A1, A2, ..., Am} be the set of the alternatives, i. e.  the possible 

educational strategies. Let O = {O1, O2, ..., On} be the set of the objectives that 

we want to achieve. Let D = {D1, D2, ..., Dk} be the set of the decision making 

processes. The first phase consists of the establishment of a procedure that is 

able to assign to each couple (alternative Ai, objective Oj) a pij score. In this way, 

the responsible for the decision measure the grade in which the alternative Ai 

satisfies the objective Oj. Assume that pij is in [0, 1], where: 

• pij = 0 if the objective Oj is not at all satisfied by Ai; 

• pij = 1 if the objective Oj is completely satisfied by Ai. 
At the end of the procedure we obtain a matrix P = [pij] of the scores which 

is the starting point of the elaborations that lead to the choice of the alternative, 

or at least to their ordering, possibly even partial (Maturo, Ventre, 2009a, 

2009b). There may be constraints: it could be necessary to establish for each 

objective Oj a threshold j > 0, with the constraint pij ≥ j, for each i. 

Furthermore, through a convex linear combinations of alternatives Ai it is 

possible to take into consideration mixed strategies that will have the following 

form: 

A(h1, h2, ..., hm) = h1 A1 + h2 A2 + ... + hm Am 

with: 

• h1, h2, ..., hm non-negative real numbers; 

• the hi’s are such that h1 + h2 + ... + hm = 1; 
The number hi can represent the fraction of time in which the teaching 

strategy Ai is adopted. If we consider also the mixed strategies, then the single 

alternatives Ai are called pure strategies. The mixed strategies are particularly 

considered in presence of “at risk” alternatives: these situations have high scores 

for certain objectives and low for others (possibly below the threshold).  

It is appropriate to construct a ranking of the alternative educational plans, 

i. e., a linear ordering of the alternatives that takes into account the objectives 

which contribute to the most suitable formation of the student. Such a ranking 

can be usefully obtained by means of the application of the Analytic hierarchy 

process, a procedure due to T. L. Saaty (1980, 2008). 



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4.2  The Analytic Hierarchy Process: attributions of weights and 

scores 
The Analytic Hierarchy Process (AHP) is both a method and a technique 

that allows to compare alternatives of different qualitative and quantitative 

nature, not easily comparable in a direct way, through the assignment of 

numerical values that specify their priority. The first thing to do is represent the 

elements of the decision problem through the construction a hierarchical 

structure. Indeed, the Analytic Hierarchy Process is based on the representation 

of the problem in terms of a directed graph G = (V, A). Let us recall that (Knuth, 

1973): 

• a directed graph, or digraph, is a pair G = (V, A), where V is a non-empty 
set whose elements are called vertices and A is a set of ordered pairs of 

vertices, called arcs; 

• a vertex is indicated with a Latin letter; for every arc (u, v) u is called the 
initial vertex and v the final vertex or end vertex;  

• an ordered n-tuple of vertices (v1, v2, ..., vn), n > 1, is called a path with 
length n -1, if, and only if, every pair (vi, vi + 1), i = 1, 2, …, n-1, is an arc of 

G. 

Furthermore, in our context, we assume the following conditions be 

satisfied from a directed graph: 

• the vertices are distributed in a fixed integer number n ≥ 2 of levels; each 
level is indexed from 1 to n; 

• there is only one vertex of level 1, called the root of the directed graph; 

• for every vertex v different from the root there is at least one path having 
the root as the initial vertex and v as the final vertex; 

• every vertex u of level i < n is the initial vertex of at least one arc and there 
are no arcs with the initial vertex of level n; 

• if an arc has the initial vertex of level i<n, then it has the end vertex in the 
level i+1. 

Let us describe, considering for example n=3, functional aspects of each 

level:  

• the level 1 vertex is called the general objective and denoted GO. It indicates 
the objective of the entire decision making process; 

• the vertices of level 2 are called criteria. With this level we indicate the 
parameters used to evaluate the alternatives; 

• vertices of level 3 are called alternatives that represent the various ways of 
reaching the GO. 

So we have the structure shown in the next page (Figure 4.1). 



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Figure 4.1 The Digraph of the Functional Aspects.  

For instance, to build up a decisional model for the mathematical didactic 

the elements of the hierarchy might be the following: 

• General objective: to guarantee a route that, through education, ensures to 
all students  the formation of knowledge and the development of action and 

communication skills; 

• Criterion 1: acquisition of the objectives set out in the educational plan; 

• Criterion 2: ability to measure oneself against peers in a safe and rational 
way while respecting the ideas of others; 

• Criterion 3: internalisation of the objectives set by the didactic plan. With 
the term “internalisation” we indicate the ability to re-elaborate knowledge 

from a critical and personal point of view; 

• Criterion 4: ability to cope with trials, planned or not, without negative 
moods, anxiety and terror of judgement; 

• Alternative 1: new didactics, inspired by the inquiry model that puts the 
pupil and her emotions at the center of the context with cooperative learning 

experiences;  

• Alternative 2: traditional didactic, that is a model of theaching-learning that 
prefers frontal lessons. In terms of learning this alternative hypothesises that 

the acquisition and internalisation take place at the same time and that the 

error is the manifestation of the failure to complete one of the two processes. 

• Alternative 3: distance learning, inspired by the inquiry model mediated 
through the use of technological tools characterised by a total absence of 

sharing the same physical space between student and teacher. 

A decision-maker assigns a score to each arc following the AHP procedure 

(Saaty, 1980, 2008; see also: Maturo, Ventre, 2009a, 2009b). So, the second step 

consists in determining the ratios of preference of the elements of a level over 



Teaching as a decision-making model: strategies in mathematics from a 

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123 

 

any of the higher, i.e. previous, level: we, therefore, compare alternatives A1, 

A2 and A3 with respect to each criterion and the individual criteria with respect 

to the general objective. In order to determine the value of each comparison, the 

following scale of evaluations is used: 

 

Value aij Interpretation 

1 i and j are equally important  

3 i is a little more important than j 

5 i is quite more important than j 

7 i is definitely more important than j 

9 i is absolutely more important than j 

1/3 i is a little less important than j 

1/5 i is quite less important than j 

1/7 i is definitely less important than j 

1/9 i is absolutely less important than j 

Table 4.1 Scale of evaluations. 

Matrices are called pairwise comparison matrices and they represent 

quantitative preferences between criteria or between alternatives and satisfy the 

followings: 

• if alternative i assumes the value x in comparison with alternative j with 
respect to a criterion, then the comparison of alternative j with alternative i 

with respect to the same criterion assumes the value 1/x. Analogous is the 

procedure to assign values when comparing couples of criteria; 

• since equally important alternatives correspond to value 1, the diagonal of 
the matrices are composed entirely of unit values.  

In our case, we obtain the matrices shown in the next page (from Table 4.2 

to Table 4.6) whose values have been assigned due to the following 

considerations: 

• acquisition and internalisation are two different processes and only through 
a mutual combination of them the full formation of the student can be 

guaranteed; 

• traditional didactic is far from the social character of human learning; 

• the relationship with others is a fundamental space for personal and social 
development, necessary for the student to learn to respect rules and roles; 

• distance learning offers insufficient physical interaction between student-
teacher and student-student: expressions and gestures make the difference 

in the learning process; 

• exercising young people to face tests in a lucid way is a fundamental aspect 
for the construction of a personality that faces in a competitive way the 

working challenges of a competitive society.  

 

 



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O C1 C2 C3 C4 

C1 1 5 1 1 

C2 1/5 1 1/5 1/7 

C3 1 5 1 1 

C4 1 7 1 1 

Table 4.2 Matrix M1: comparison of criteria by the ratio of preference with 

respect to the general objective   

 
C1 A1 A2 A3 

A1 1 5 1 

A2 1/5 1 1/3 

A3 1 3 1 

Table 4.3 Matrix M2: comparison of alternatives by the ratio of preference with 

respect to the criterion 1 

 

C2 A1 A2 A3 

A1 1 5 7 

A2 1/5 1 3 

A3 1/7 1/3 1 

Table 4.4 Matrix M3: comparison of alternatives by the ratio of preference with 

respect to the criterion 2 

 

C3 A1 A2 A3 

A1 1 7 5 

A2 1/5 1 3 

A3 1/7 1/3 1 

Table 4.5 Matrix M4: comparison of alternatives by the ratio of preference with 

respect to the criterion 3 
 

 



Teaching as a decision-making model: strategies in mathematics from a 

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125 

 

C4 A1 A2 A3 

A1 1 3 7 

A2 1/3 1 3 

A3 1/7 1/3 1 

Table 4.6 Matrix M5: comparison of alternatives by the ratio of preference with 

respect to the criterion 4 

 

After constructing the pairwise comparison matrices, we have to fix the 

weights of the elements in each level. This step is fundamental to determine 

whether the matrices are relevant through a scale of values ranging from 0 to 1. 

These weights must meet the normality condition: 

w1+w2+…+wn =1 

This procedure supposes that, if the decision maker knew all the actual 

weights of the elements of the pairwise comparison matrix, then it would be: 

𝐴 = ቀ
𝑤𝑖

𝑤𝑗ൗ ቁ = ቌ

𝑤1
𝑤1ൗ ⋯

𝑤1
𝑤𝑛ൗ

⋮ ⋱ ⋮
𝑤𝑛

𝑤1ൗ ⋯
𝑤𝑛

𝑤𝑛ൗ

ቍ 

In this case the weights would be obtained from any of the rows which are 

all multiple of the same row ൫1 𝑤1ൗ ,
1

𝑤2ൗ , … ,
1

𝑤𝑛ൗ ൯. It follows that the matrix 

A has rank 1. Being w= (w1, w2, …, wn)
T we get: 

Aw = nw. 

Thus, from the equation above, n is an eigenvalue of A and w is one of the 

eigenvectors associated with n. Since the elements on the diagonal are all 1, 

denoted with λ1, λ2, …, λn = n the eigenvalues of A, the value of the trace of A 

is: 

tr(A)= λ1 + λ2 + … + λn  = n 

As n is an eigenvalue of A the other n-1 eigenvalues of A must be zero. The 

matrix A satisfies the condition:  

aijajk = aik  

for every i, j, k, called consistency condition, that implies transitivity of the 

preferences, and A is said to be consistent is said to be A.  

In practice the decision maker does not know the vector w: the aij values that 

he assigns according to his judgement may deviate from the unknown wi/wj. So 

the decision maker may produce inconsistent pairwise comparison matrices.  

However the closer the aij values are to wi/wj, the closer the maximum 

eigenvalue is to n and the closer the other eigenvalues are to zero. 

Therefore the vector of the weights w'T = (w'1, w'2, …, w'n) associated to the 

maximum eigenvalue (among the infinite w'T we choose the one for which w'1 

+ w'2 + … + w'n =1) will be an estimate of the vector w
T = (w1, w2, …, wn) the 



Viviana Ventre, Eva Ferrara Dentice, Roberta Martino 

 

126 

 

more precise the more the maximum eigenvalue 𝜆max of A is close to n, what is 
due to the continuity of the involved operations (Ventre, 2019). Where:  

𝐶𝐼 =
𝜆𝑚𝑎𝑥 − 𝑛

𝑛 − 1
 

is defined as the consistency index of A, and n is the order of the matrix itself. 

The consistency index reveals how far from consistency the matrix is. In our 

case with the help of MATLAB we obtain: 

 M1 M2 M3 M4 M5 

𝜆max 4,0142 3,0291 3,0649 3,0649 3,0070 

CI 0,0047 0,0145 0,0324 0,0342 0,0035 

Table 4.7 CI Values for each matrix.  

We can therefore write the local priorities obtained by proceeding with the 

last step of the AHP method which, through the aggregation of the relative 

weights of each level, provides a weighted ranking of the alternatives. 

The third step implements the estimation of local assessments, i.e. the 

weightings that express the relative importance of the elements of a hierarchical 

level over any element of the next higher level. 

 

 
Figure 4.2 The Digraph of the Functional Aspects, with weights. 

We observe that scores are nonnegative real numbers and such that the sum 

of the scores of the arcs coming out of the same vertex u is equal to 1. The score 

assigned to an arc (u, v) indicates the extent to which the final vertex v meets 

the initial vertex u: the score of a path is the product of the scores of the arcs 

that form the path. 

• For every vertex v different from GO the score p(v) of v is the sum of the 
scores of all the paths that start from GO and arrive in v.  



Teaching as a decision-making model: strategies in mathematics from a 

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• For every level, the sum of the points of the vertices of level i is equal to 1.  
The global properties determined will then be: 

• A1=0,42*0,29+0,43*0,06+0,44*0,29+0,65*0,36=0,51; 

• A2=0,28*0,29+0,42*0,06+0,15*0,29+0,26*0,36=0,24; 

• A3=0,3*0,29+0,15*0,06+0,41*0,29+0,09*0,36=0,25. 
We can conclude that, in order to achieve the objective, a new teaching 

strategy is preferred to a traditional one and direct communication and human 

contact are factors not to be overlooked. Therefore, teaching cannot be reduced 

to an online practice because the presence of the student is not attributable to a 

"virtual presence". 

 

5 Mathematics and specific learning disorders  
5.1 Background 

Students with specific learning disorders (SLD) need a personalised 

learning plan that appropriately accommodates their difficulties. At the 

university a student with SLD attending the course of mathematical analysis has 

succeeded, thanks to the semester tutoring activity dedicated to the subject, to 

face his difficulties by passing the exam at the first useful date. The course has 

evolved through a personalised didactic strategy mostly based on the use of 

mind maps and peer comparison activities. Specific learning disorders SLD 

usually involve reading, writing and calculation skills. In Italy, two students 

with dyslexia out of three do not receive an adequate diagnosis of the disorder 

and therefore SLD are one of the main causes of school difficulties with 

important negative repercussions also in the personal sphere of the individual.  

It is therefore clear the importance of a conscious environment able to 

respect the “different” way of learning of a student with SLD whose cognitive 

abilities and physical characteristics are in the norm.  In fact, although they have 

an intelligence appropriate to their age, unlike their peers, these subjects learn 

at a slower pace because during the study they dissipate most of their energy to 

compensate for their disorders. Initiatives promoted by the MIUR, accompanied 

by individual schooling, support the right to study of students with SLD, which 

since the beginning of the year two thousand is also protected at the legislative 

level (MIUR law no.170/2010).  To this end, the Regional School Offices are 

committed to promote the issue of detailed certifications that allow as much as 

possible, together with parents and the figures who follow the student in school 

activities, a Personalised Educational Plan that aims to achieve the same 

objectives of peers through compensatory tools and dispensative measures 

(MIUR, 2011). 

 

 

 



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5.2 SLDs in mathematics 
 

Specific reading disorder 

On the whole, this disorder leads to difficulties in reading, which is slow 

and incorrect, making it difficult to understand the text and to distinguish useful 

parts from those containing additional information (ICD F81.0, 2007). In 

mathematics, these deficits make even the simplest problems complex as the 

student may not be able to distinguish between hypotheses and data. In addition, 

to maximise learning, rather than studying theory from the book, we suggest the 

constant use of mind maps that allow you to correct and rework ideas slowly 

until the subject is mastered. This type of maps is the one that best suits the way 

of learning of students with specific reading disorders because, through images 

and colors, it aims to stimulate visual memory through the insertion of mental 

associations. 

 

Specific writing disorder 

The specific writing disorder is called dysgraphia if it affects writing and 

dysorthography if it affects spelling. The former involves a deficit of a motor 

nature and refers to the graphic aspects of handwriting, the latter is a text 

encoding disorder that therefore involves the linguistic component. Disgraphers 

therefore produce poorly readable texts (even by themselves) with words that 

are often misaligned and characterised by letters of different sizes, while 

disorthographers manifest errors such as inversion of syllables, arbitrary cuts of 

words and omissions of letters in words making the content unclear. In both 

cases the elaboration of a written text is a difficult and long process with serious 

repercussions also in the mathematical field. In fact, mathematics has its own 

language, characterised by symbols, signs and letters of the greek and latin 

alphabets: just think of the use of lowercase greek letters in the geometric field 

to indicate angles and uppercase latin letters to indicate vertices.  In the set of 

symbols, the symbol of belonging “∈” may not be decoded, leading to confusion 
with “E” even though it does not denote, from a didactic point of view, a lack of 

understanding of the set meaning itself.  

Inaccurate writing also causes errors in the resolution of algebraic 

expressions, for example by confusing the letter “s” and the number “5”, or in 

the resolution of a linear system which requires many transcription steps.  

Another difficulty due to alignment is found in the case of powers where base 

and exponent are confused with a multi-digit number (34 instead of 34). Along 

this line, the teacher should take into account the content rather than the form, 

in the process of evaluating the written texts. Errors should not be penalised 

when the concept expressed is clear, whereas oral verifications should acquire 

more weight in the final evaluations.  In order to deal with these problems, the 

teacher, in the process of evaluating the written tests, could take into account the 



Teaching as a decision-making model: strategies in mathematics from a 

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content rather than the form, not penalising errors when the concept expressed 

is clear and giving more weight to oral verifications.  

 

Specific disturbance of arithmetic activities 

The specific disturbance of arithmetic activities disorder implies a deficit 
that concerns the mastery of fundamental calculation skills which generally can 

be divided into two different profiles according to the type of error. The first 
profile is defined as "Weakness in the cognitive structuring of numerical 

cognition components" and summarises both the difficulty in 

comparing/quantifying the elements of one or more sets and the reduced 

counting capacity. This type of disorder is inspired by Butterworth's studies 

(1999; 2005): he hypothesised the existence of a "mathematical brain" 

specialised in classifying the world in terms of numbers. The second profile is 
renamed with "Difficulty in acquiring calculation procedures and algorithms" 

and includes the following three cases (Temple, 1991): dyslexia for digits 

indicating an incorrect reading/writing of the numbers (the student sees the 

number 3 and pronounces 6); procedural dyscalculia indicating the difficulty in 

the choice, application and maintenance of procedures leading to errors in 

borrowing, carry-over or sticking; dyscalculia for arithmetic facts which leads 

to confusion between the rules of rapid access with the consequent compromise 

of the acquisition of numerical facts within the calculation system. For both 
profiles examined the tools to compensate could be tables, diagrams, calculators 

and an extensive collection of procedural examples. 
Objective 

Learning disorders are, therefore varied and create different deficits that 

require different compensatory instruments and dispensation measures. For this 

reason, it is important to experiment the different strategies for teaching in order 

to identify a scheme which, although it can never be universal, can be taken as 

a canvas and then refined according to the personal limits and objectives of the 

student. The aim is, therefore to encourage learning with the aim of making the 

student as autonomous as possible, also increasing the level of self-esteem and 

personal gratification.  Below is the strategy used for a student with specific 

reading and writing disorders during the tutoring activity of the mathematical 

analysis course 1.  

The case study 

The student, after presenting his certificate at the beginning of the course, 

immediately showed interest in possible remedial activities. Although the 

student was initially autonomous, after almost a month, the first difficulties 

began to emerge. This situation prompted the student to make constant use of 

the tutoring hours in which a personalised teaching strategy was constructed. 

The construction of such a strategy was obtained through the procedure 

previously shown: alternatives, objectives and criteria were modified 



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considering the specific difficulties. The method proved to be successful: 

indeed, the student was able to acquire his study methodology, which proved to 

be effective. 

 

Stuff and methods 

From a didactical point of view, it is particularly interesting that problems 

related to short-term memory are a common feature of the different SLDs. 

Baddeley and Hitch (1974) extended this concept, in the field of cognitive 

psychology, to the "working memory" which specifically indicates that set of 

notions necessary for written and oral productions that remain short in the 

student's mind. If this capacity is reduced, then temporary archiving and the first 

management/manipulation of data will be compromised with the following 

consequences: difficulty in taking notes, difficulty in maintaining attention, 

need for longer periods. To cope with this situation, it is necessary to reduce the 

information load by giving priority to fundamental concepts and using support 

tools that favour direct observation and experimentation. 

Some indications for the didactic strategy are: 

• Use of schemes and concept maps; 

• Dispense with reading aloud and mnemonic study; 

• Privileging learning from experience and laboratory teaching; 

• Encourage students to self-assess their learning processes; 

• Encourage peer tutoring and promote collaborative learning; 

• Guarantee longer times for written tests and study; 

• Take an encouraging attitude to improve self-esteem; 

• Evaluate according to progress and difficulties; 

• Use of calculator and digital devices. 
In our case, the implementation of the didactic strategy took place through 

afternoon meetings, lasting two hours, usually held after the lesson held by the 

teacher in the same morning. At first, the meeting was based on the study of the 

last topics explained by the teacher and already at this stage it was evident to 

what extent the characteristic features of dysgraphia hindered learning: the bare 

and confused notes, characterised by large empty spaces, were practically 

impossible to read and to study. In order to try to provide constructivist learning 

and to avoid the use of the book, since reading was slow and incorrect due to 

dyslexia, the theory was flanked by practice or questions to answer. Once the 

new subject was finished, work was done on the previous ones. 

In order to prevent the pupil from distracting himself, while following me 

with interest, I often drew his attention by naming him and repeating the concept 

in different ways. Difficulties began to come up from the first topics when, in 

the numerical set exercises, there was a lot of confusion between the parentheses 

used to describe the intervals. For example: 

• interval from 'a' to 'b' not including the extremes, ]a, b[; 



Teaching as a decision-making model: strategies in mathematics from a 

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• interval from 'a' to 'b' including the extremes, [a, b]; 
To cope with this difficulty, with a bit of imagination, I made him imagine 

drawing two arms 'embracing' the number if the number had to be present in the 

interval or, otherwise, they refused to do so. In this way, together with the use 

of graphic representations, the pupil acquired the competence to distinguish 

between the two types of parentheses and rarely found confusion until the end 

of the course. The mind maps, which we built together, were of great help 

especially in solving equations and inequalities with absolute values.  Initially, 

the student's difficulty consisted in not being able to “visualize” the writing of 

the systems that came out of the procedure. With the use of a map, similar to the 

one below, he was able, after several lessons, to carry out the simplest exercises 

correctly but still presenting difficulties for the more complex ones. The result 

was satisfactory, however, because I believe that these difficulties were linked 

not so much to a lack of understanding of the absolute value function, but rather 

to a lack of ability to concentrate for so long on the same procedure. Example 

of a map for the absolute value function:  

 

 
Figure 5.1 Example of a map built during the activity. 

 

Towards the end of the course, as the exam date was approaching, other 

students also started to attend tutoring lessons on a regular basis. For this motive, 

the last topics of the course, derived and function study, were addressed through 

a collective study during which the student with specific learning disorders 

discussed with his classmates to the point of realising, under my guidance, his 

mistakes. During the lessons we made extensive use of the calculator, reducing 

the material to be memorised as much as possible. For example, to study the 

domain of a function, initially the student made extensive use of the tables 

summarising the domains of elementary functions but, subsequently, using the 

calculator (we get "ERR" if the function is not applied to an elementary of the 



Viviana Ventre, Eva Ferrara Dentice, Roberta Martino 

 

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domain) he learned what this procedure really represented by obtaining, on his 

own, the conditions for simple functions such as roots and logarithms. 

Results 

The boy achieved the minimum objectives, and at the end of the course, he 

passed the exam. The fundamental concepts were acquired and therefore, the 

test was approached independently. The improvements were gradual: as the 

tutoring lessons increased, the student was able to follow more and more. The 

shared study experience with peers was certainly helpful as, through peer 

comparison, he gained such confidence that he could guide his peers in difficulty 

during the exercises. In this regard, it is important to stress that the objectives 

achieved are not only didactic in nature but also personal: the ability to compare 

oneself with peers while respecting the ideas of others has increased the level of 

self-esteem and gratification.  

 

6 Conclusion  
The purpose of this document is, therefore, to underline that by using Saaty's 

hierarchical structure it is possible to choose the didactic strategies that best suit 

the various situations. In fact, all students and in particular students with SLD 

can maximise their learning if followed correctly. In our case, the didactic plan 

allowed for peer learning. This strategy, having been applied to only one student, 

cannot be generalised because each case requires different objectives and tools 

depending on the disturbance and the problems that this entails. To conclude, 

we stress that the affective dimension has played a fundamental role: a serene 

and stimulating environment has been a necessary condition to motivate 

students to improve themselves. However, the suggested indications are a good 

start to deal with different cases. 

 

 

 

 

 

 

 

  



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