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Original Article

Evaluation of Cognitive Complexity of

Tasks for the Topic Hydrogen Exponent

in the Solutions of Acids and Bases

Saša Horvat*, Dušica D. Rodić, Mirjana D. Segedinac, 
Tamara N. Rončević

Department of chemistry, biochemistry and environmental protection,

University of Novi Sad, Faculty of sciences, Serbia

*Email: sasa.horvat@dh.uns.ac.rs

Abstract

The aim of this study was evaluation of cognitive complexity of tasks for the topic hydrogen exponent in the so-
lutions of acids and bases and its validation. The created procedure included an assessment of the difficulty of
concepts and an assessment of their interactivity. There were 48 freshmen students enrolled in the study pro-
gram Basic academic studies in chemistry. As a research instrument for assessing performance, test of knowl-
edge was specifically constructed for this research. Each task in the test was followed by a seven-point Likert
scale for the evaluation of invested mental effort. The evaluation of cognitive complexity was confirmed by a
series of linear regression analysis where high values of correlation coefficients are obtained among the ex-
amined variables: student’s performance and invested mental effort (dependent variables) and cognitive com-
plexity (independent variable).

Keywords: mental effort; performance; pH

Introduction

Chemistry as a teaching subject is difficult to understand and master at all levels of stu-
dents´ education. In the Republic of Serbia, students met chemistry for the first time in
the seventh grade of elementary school, when they are already expected to adopt and
understand a large number of unknown chemical concepts. Students need to understand
the ways in which chemical processes occurre, which are not available to direct sensory
observation, as well as many chemical laws. A difficult understanding of chemistry could
be observed in its abstraction and in the specific chemical language, which often contains
terms from everyday life (Markić & Childs, 2016). Besides, chemistry is closely related to
many natural sciences and mathematics. In addition to these difficulties, students also
encounter numerical problem tasks of high complexity. Numerical tasks in chemistry often
include some advanced mathematical functions, such as logarithm, exponents, rooting
etc. These functions are included in the problems with the hydrogen ion exponent, the

Journal of Subject Didactics, 2017

Vol. 2, No. 1, 33-45, DOI: 10.5281/zenodo.1238972



reaction between acids and bases and acid-base equations, which are part of the Acid
base chemistry. This topic is closely related to other areas of chemistry and it contains a
large number of concepts such as: electrolytic dissociation, chemical equilibrium, chem-
ical reactions, stoichiometry, limiting reactant and solutions. While most simple problems
are easy to solve, problems that include concepts from multiple domains often involve
long-lasting and strenuous numerical calculations (Chetan et al., 2005). The teaching
topic of acids and bases is important in the teaching of chemistry and it is studied from
elementary school to university level.

According to teachers, students rather use algorithmic approach in solving acid base
numerical calculations which is not based on conceptual understanding (Curtright et al.,
2004). Namely, students often only include numerical data in formulas instead of trying
to conceptually understand the problem (Bransford et al., 1999 cited in Watters & Watters,
2006). 

Some of the difficulties encountered in this area are the following: students think that
acidity and pH are the same terms, as well as strength and concentration, then they be-
lieve that the pH scale is unique and does not depend on the temperature and nature of
the solvent (Alvarado et al., 2015). It is interesting to mention that in Tümay's study, the
most of students (66%), when considering the concept of the strength of acids and bases,
claim that a strong acid or base is 100% dissociated in water (Tümay, 2016). Inappropri-
ate conceptual understanding is also noticed in the definition of acids and bases (83%).
Namely, students believe that all acids and bases dissolve in water producing hydrogen
ions and hydroxide ions, without taking into consideration modern acid base theories.
Numerous misconceptions occur in this area and they refer to the understanding of acids
and bases, the strength of acids and bases, the recognition of the acid or base character
of the substance, as well as the reactions between them (Cooper et al., 2016). 

The concept of hydrogen exponent is very important concept especially in analytical
chemistry, in the teaching theme volumetry. Understanding this concept requires the un-
derstanding of acid-base reactions at a particle level. The concept of pH is difficult for
students, as they often believe that strong acids have high pH values, while weak acids
are characterized by low pH values (Ouertatani et al., 2006). 

It seems that students encounter with problems on two levels of conceptual under-
standing. The first problem relates to the conceptualization of acids and bases on a mean-
ingful and integrated level of understanding, and the second problem relates to the use
of mathematics in order to successfully apply their knowledge (Watters & Watters, 2006).
It has been observed that most students could define the pH value and calculate it by
using the calculator, but they do not understand it conceptually. Solving numerical calcu-
lations, related to pH, are based on understanding the exponential numbers and the use
of algorithms that are fundamental concepts of calculations. The pH concept, that it in-
troduced by Sørensen (1909), is described as: -log[H+] (Sørensen, 1909 cited in Watters
& Watters, 2006). In order to understand this relationship, students have to know the
meaning of “minus” and the notion of concentration, as well as to know how to calculate
the concentration of hydrogen ions expressed in the exponential value from the given
pH value, and vice versa. Understanding the concept of acidity and pH is also incompre-
hensible to students because the numerical values of hydrogen ion concentration and
pH is in the indirect proportion, that is to say, the higher numerical pH value corresponds
to a lower numerical value of the concentration of hydrogen ions. It has been found that
students do not know what the logarithmic function means, but they know where the “log”
button is on their calculator (Watters & Watters, 2006). 

34



Many factors may be the cause of low performance in quantitative chemical problems.
The complexity of tasks is often used as a variable that affects students' performance
(Wood,1986). Complex tasks impose high load on the students' working memory, due to
which students have to invest a great mental effort in order to solve them. Numerous
studies have shown that the capacity of the working memory is a factor that must be
taken into account in solving quantitative chemical problems (Johnstone & El-Banna,
1983; Niaz, 1996). The basic assumption is that short-term memory requires individual
resources that are necessary for achieving the goals of specific cognitive activities in cer-
tain situations, which makes the basis for defining the theory of cognitive load (Sweller,
1988). Cognitive load is a multidimensional concept which is comprised of three compo-
nents: mental effort, mental load and student performance (Pass, 1992, Pass & van Mar-
rienboer 1994). The mental load refers to the students' cognitive capacity needed to solve
the problem tasks. Mental effort is related to working memory resources that must be
used to achieve the requirements of problem solving, and students' performance include
students' achievement on tasks. In order to overcome overload of the working memory,
which is detrimental to learning due to high cognitive load, the mental effort should be in-
vested in processes that are essential for learning (Kalyuga, 2009).

With the aim of improving teaching, many researchers have worked to effectively
measure the cognitive load. Recent research is based on a combined measurement of
the student's mental effort that is invested and student's performance that is achieved by
solving the numerical calculation (Pass & Merrienboer, 1993). 

One of indicators of cognitive load, which has been recently used as an objective
measure is the cognitive complexity (Raker et al., 2013; Harris et al., 2013). The concept
of cognitive complexity was the first mentioned by George Kelly in his Personal Construc-
tive Theory (Kelly, 1955). According to this theory, individuals are able to understand,
predict, and control events in the same way as scientists work, building their own systems
of (personal) constructs, using them as "cognitive patterns" to understand the world.
Based on Kelly's personal constructive theory, Bieri (Bieri, 1955) proposed the concept
of cognitive complexity that represents the degree of differentiation in the constructive
system of an individual, or the relative number of different dimensions based on the de-
cisions that can be made. The more complex the cognitive ability of an individual is, more
the construct that this person uses to understand the world, will be differentiated (Bieri,
1955; Xin & Chi, 2007). Therefore, cognitive complexity may relate to an individual, but
it can also refer to the task. 

Cognitively complex tasks should be designed in a way that they encourage students
to think about the problem, develop strategies, methods, and procedures for solving
tasks. Students should not only reproduce the answer, but should also know and explain
it in an adequate way. Tasks should also be constructed to enable a large number of so-
lution methods to (Magone et al., 1994).

What is crucial in assessing cognitive complexity is to consider the difficulty of the el-
ements - the concepts represented in the task and their interactivity. In order to assess
the cognitive complexity, an estimation of the difficulty of the concepts represented in the
assignment should be made. It is very important to establish which are the key factors
that determine how complexity is elaborated, and the most important factor in the as-
sessment is to include the degree of interactivity of the elements (Sweller, 1988, Knaus
et al., 2011). Halford et al. (Halford et al., 1998) considered that the limiting factor in as-
sessing the complexity of tasks is not the number of items or the amount of information,
but the relationship between the entities. The problem becomes more complex with an

35



increase in the number of interacting factors. Complexity can be measured by dimen-
sioning relationships or the number of associated variables. If the task contains two items
that are interacting in a binary relation, it is simpler in comparison to the task containing
the ternary relation, wherein three things are in interaction, and it is again simpler than
the one that includes four interconnected items, that is, one that has quaternary relation
and so on. The idea of relational complexity is analogous to the number of factors in the
experimental design that is considered to be the set of dependent and independent vari-
ables. According to concepts of load and resource, this relational complexity is usually
used to calculate performance in cognitive psychology.

The researchers designed special instruments for the assessment of cognitive com-
plexity, which are based on the creation of the Rubrics which helped in determining nu-
merical value of cognitive complexity (Knaus et. al., 2011; Raker et al., 2013; Horvat et
al., 2016). This type of instrument is based on assigning numerical values of the cognitive
complexity of a given task. The estimation of the number of elements, the expert's estima-
tion of the difficulty of the concepts represented in the task, and the assessment of their in-
teractivity are the three steps during obtaining a numerical rating of cognitive complexity.

It is important to note that every rating system of complexity contains a certain dose
of subjective components, since it relies on expert assessments. For this reason, the cre-
ation of Rubrics must be carried out in such a way that the results are valid and reliable,
so that the expert method for the complexity rating can serve as a substitute for the ob-
jective complexity (Knaus et al., 2011). 

The instrument for assessing the cognitive complexity of chemical tasks that proved
to be valid and reliable was first constructed by Knaus et al. (Knaus et al., 2011) who
called it the "Rubric for the Cognitive Complexity of Chemical Problems". The validity of
the instrument has been confirmed in two ways: (1) a statistically significant correlation
between the cognitive complexity by experts and student performance, and (2) a statis-
tically significant correlation between the expert assessment of cognitive complexity and
students’ assessment of mental effort. Calculation of the numerical value of the cognitive
complexity rating is based on the principle of additivity of the difficulty rating of the skills
and concepts represented in the task and the factor of interactivity between these con-
cepts (Knaus et al., 2011; Raker et al., 2013; Horvat et al., 2016). The great advantage
of the Rubrics for cognitive complexity rating is that they rely on the assessment of inter-
activity among the concepts that are present in the task - the aspects of cognitive load
theory, more precisely on intrinsic cognitive load (Sweller et al., 1998).

The creation of a Rubric has repeatedly proven to be a good method for calculating
the cognitive complexity rating. The reason lies in the fact that the subjectivity of an expert
is reduced to a minimum. 

Methods

Aim of the research

The aim of this paper is to design and validate the Table for assessing difficulty of con-
cepts and their interactivity for topic hydrogen exponent in the solutions of acids and
bases.

Research sample

The sample of this research consisted of 48 freshmen students enrolled in the study pro-
gram Basic academic studies in chemistry at the Faculty of Sciences in Novi Sad. All re-

36



spondents previously completed different profiles of secondary education and were aged
19 to 20 of mixed socioeconomic status. All of them agreed to voluntary participate in
this research. The research was conducted in October 2017/18 academic year.

Research instruments

As a research instrument, we used a Test, which was specifically constructed for this re-
search. The time available for test solving is 45 minutes. Respondents previously studied
all the concepts presented in the tasks during regular chemistry classes within secondary
school. A test contained six tasks. Each correct answer is scored by one point, so that
the maximum total score on the test was six points.  Incomplete tasks were not taken
into consideration and scoring. 

In addition to performance, students' mental effort was also evaluated. Assessment
of invested mental effort was measured by a subjective technique with the use of seven-
point Likert scale. After each task accomplished, students were asked to express an as-
sessment of the mental effort they had invested in solving the task by choosing the
appropriate descriptive grade on the scale. During statistical data processing, numerical
values of estimates have been assigned to descriptive estimates.  'Extremely easy' - as-
signed a numerical value of 1 while 'extremely difficult' - the numerical value 7. 

The quality of the test, was evaluated by pre-test and post-test assurance parameters.
The obtained results were processed by statistical software programs Stat Graphics Cen-
turion XVI and IBM SPSS Statistics 22.

Instrument psychometric

Pre-test assurance parameters were determined by experts in the field Didactics of
Chemistry. Due to the compliance of tasks with a valid curriculum and proposed text-
books, the authors concluded that the test is valid for this testing. The tasks on the test
are defined by experts as diverse, with clearly defined requirements.

Post-test assurance parameters of quality are defined as basic statistical parameters:
indices of item difficulty, index of test difficulty, the coefficient of reliability, discrimination
indices, as well as discrimination index of the test. Created test showed good metric char-
acteristics. Reliability is calculated as a measure of internal consistency and expressed
as a Cronbach α coefficient of 0.61 for performances, and 0.84 for self-assessed mental
effort thus indicating good reliability. Cronbach α values above 0.6 are acceptable when
a small number of tasks are represented on the test (Moss et all., 1998; Loewenthal,
2004). Indices of taks’ difficulty are in the range of 5.26% to 78.95% (average value of
test difficulty is 39.04%, which makes it a test of moderate difficulty). One task has a dif-
ficulty index less than 25% which makes it difficult, while one task has a difficulty index
greater than 75% which makes it easy task (Towns, 2014). Discrimination indices are in
the range of from 0.10 to 0.80 (average of 0.55 which is an excellent discrimination index).
Five tasks have an excellent discrimination index, greater than 0.4, while only one task
has a poor index of discrimination (0.10), so it should be revised for future usage.

The basic statistical test parameters are shown in Table 1.
Validation of instrument for assessment the mental effort was also confirmed by linear

regression related to observing the dependence of student performance and self-invested
mental effort. Graphic dependence and statistical parameters of regression analysis are
shown in Figure 1 and Table 2.

37



This dependence describes a very strong correlation (R=-0.80; P=0.03). P - value is
less than 0.05, indicating a statistically significant correlation between mental effort as
dependent variables and performances as independent variables at the confidence level
of 95%. 

38

Figure 1. Correlation of students' performances and students' evaluation of invested mental effort.

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Table 1. Descriptive statistics for the students' performance and mental effort.



Results and Discussion

In order to successfully evaluate the cognitive complexity it is necessary to validate the
procedure for determining cognitive complexity of the problems with the hydrogen expo-
nent in the solutions of acids and bases. In order to ensure objectivity in assessing cog-
nitive complexity, a Table for assessing difficulty of concepts and their interactivity has
been developed. The basic concepts that were considered were: Dissociation degree of
acids and bases, Dissociation constant of acids and bases, and Ostwald’s dilution law.
In accordance with the Rubric developed by Knaus et al. (Knaus el al., 2011), assessment
of difficulty of the concepts was made. Concepts were estimated as easy, medium and
difficult:

1. The concept Dissociation degree is structured in three levels: the calculation of
pH (pOH) from dissociation degree in the solution of monobasic acid and
monoacidic base, which is an easy concept when calculating the pH in the acid
solution, or pOH in the solution of the base. The concept is of medium difficulty
if it is necessary to calculate the pOH value in the acid solution, or pH value in
the solution of the base. This concept is also of medium difficulty if it is necessary
to calculate the pH value in the polybasic acid solution, or pOH in the solution
of the polyacidic base. If it is necessary to calculate the pOH value of the poly-
basic acid, or the pH value of the polyacidic base, the concept is difficult. 

2. The concept of Dissociation constant of acids and bases is structured in two
levels. It is medium when calculating the pH from Ka in a solution of weak
monobasic acid, or pOH from Kb in a solution of a weak monoacidic base. If the
pH is calculated in a base solution based on Kb, or pOH in the acid solution
based on the Ka concept is difficult.

3. Ostwald's dilution law is a difficult concept because it links dissociation degree,
dissociation constant and the concentration of the acid or base solution.

In addition, in the Table for assessing the difficulty of concepts, the concept of a solu-
tion is presented as an additional concept, which is numerically expressed in the rating
of cognitive complexity and it contributes to the increase of interactivity. 

If it is necessary to calculate the molar concentration from the mass concentration or
from the mass and the volume of the solution it has additivity value 1. If it is necessary
to calculate the molar concentration from the mass fraction and the density it has additivity
value 2.

Interactivity is evaluated based on the number of concepts in the task. If one concept
is represented in the task, interactivity is evaluated with a value 0. If the task contains
two concepts, interactivity has value 1, and if the task contains three or more concepts
interactivity is evaluated by a numerical value 2. 

In the Table 3, Table for assessing difficulty of concepts and their interactivity in the

39

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Table 2. Statistical parameters of the regression analysis of students’ performance and students'
evaluation of invested mental effort.



tasks with the hydrogen exponent in solutions of acids and bases is presented.
The Table is simple and objective to use. The tasks used in the test have different lev-

els of cognitive complexity. The method for calculation of cognitive complexity rating will
be shown in the following examples:

Task 1. Calculate pOH of a solution of sodium hydroxide, concentration of which is

0.001 mol/dm3. Consider that the dissociation of sodium hydroxide is complete.

To solve this task, the student needs to know that the concentration of hydroxide ions
can be determined from the total concentration of sodium hydroxide, because there is a
complete dissociation in the solution of sodium hydroxide, i.e. α = 1.

From Table 3, it can be seen that the task contains an easy concept and refers to the
calculation of the hydrogen exponent from the degree of dissociation. According to the
Rubric from Knaus et al. (2011), the rating of cognitive complexity that contains only one
easy concept is 1. According to this procedure, to assess the difficulty of concepts, it is
also necessary to determine the degree of interactivity of the elements, and since this
task contains only one concept, the interactivity is 0 and thus the overall complexity 1.

In addition, there are tasks that contain additional concepts that increase the overall
cognitive complexity of tasks. In the following example we present a task which contain
an additional concept.

Task 4. Calculate pOH value of ammonia solution which was prepared by introducing

1.7 grams of gaseous ammonia in 2 dm3 of distilled water. Consider that all the mass of

ammonia is dissolved in distilled water, and the change in volume due to dissolution is

negligible.  The ammonia base constant is Kb = 1.8 x 10 
-5.

In this task, two concepts, one basic and one additional concept, are represented.
The basic concept is the calculation of pOH of weak monoacidic base, which is of medium
difficulty, and the additional concept is related to calculation of molar concentration of

40

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Table 3. Table for assessing difficulty of concepts and their interactivity in the tasks with the
hydrogen exponent in solutions of acids and bases.



ammonia from the mass and volume. According to the Rubric (Knaus et al., 2011), the
cognitive complexity of this task is 3, because the task contains one basic concept of
medium difficulty (2) and one additional concept and because of the presence of two con-
cepts the interactivity is 1.

The results for all tasks from the test are summarized in Table 4.

Procedures for assessing the cognitive complexity of tasks are validated by comparing
with student performance measures and measures of mental effort (Knaus et al., 2011).

In his paper, the correlation of the estimated cognitive complexity of tasks with student
performance and self-assessment of mental effort is described by graphic dependences
and basic statistical parameters.

In the first phase, the regression analysis of the dependence of students' performance
(dependent variable) from the estimated cognitive complexity (independent variable) was
done. Linear regression is applied in accordance with previous studies (Knaus et al.,
2011; Raker et al., 2013; Horvat et al., 2016). The results of the regression analysis, are
presented graphically in Figure 2 and tabular in Table 5.

It is worth to mention that in the correlation analysis the student’ performance was
calculated as the average performance value of all students in the test tasks. Observing
the P-value which is greater than 0.05, no statistically significant linear correlation be-
tween the dependent and the independent variable was found, even though the correla-
tion coefficient (R = -0.71) indicated a strong, negative correlation between the variables

41

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Table 4. Cognitive complexity ratings for test tasks.

Figure 2. Correlation of students' performance with cognitive complexity.



(Evans, 1996). The reason for such results might be in the students’ low performance in
the several tasks that were estimated with relatively low cognitive complexity. Further
analysis of our students’ written responses revealed a number of conceptual misunder-
standings which might cause low performance in these tasks. For example, students pos-
sessed the misunderstandings about the basic concept of the dissociation degree of acids
and bases, i.e. about the strength of acids and bases. They observed poor acids - vinegar
and butyric acid, as strong acids, which reflected on their calculations. Namely, they be-
lieved that such acids are 100% dissolved on ions in the aqueous solutions, not taking
into account the value of the acid constants given in the text of the tasks. They made the
similar mistake in the calculations in which the aqueous ammonia solution was observed
as a strong base. Also, in the tasks for determining the pH value of weak acids and bases,
students forgot to apply the mathematical operation of the roots, which exists in the ex-
pression for the concentration of hydrogen ions, Therefore, they got the
incorrect value for concentration, and the obtained pH value was greater than 7 in the
solution of weak acid. According to this, it could be assumed that students memorized
and used algorithmic approach to solve such problems. Additionally, students calculated
the pOH value of a solution of a weak ammonia base using the formula
believing that the concentration of hydroxide ions can be obtained by
rooting the product of the acidity constant and the concentration of the base. Students
also possessed misunderstandings about the calculation of pOH value in the solution of
strong bases. For example, they calculated pH value 3, or pOH value 11, in the sodium
hydroxide solution concentration of 0.001 mol/dm3. In addition, students observed poly-
basic acids as monobasic, regardless the notation in the task text about the complete

42

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Table 5. Statistical parameters of the regression analysis of students’ performance 
and cognitive complexity.

Figure 3. Correlation of students' mental effort with cognitive complexity.

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dissociation of sulfuric acid in both stages. The similar results were previously obtained
by Watters & Watters. (Watters & Watters, 2006).  

In the second phase, the regression analysis of the dependence of invested mental
effort (dependent variable) from the estimated cognitive complexity (independent variable)
was done. Statistical parameters and graphical dependency are shown in Figure 3 and
in Table 6.

The correlation coefficient (R = 0.84) and the P-value (P = 0.03) indicated a very
strong correlation between the mental effort as dependent variable and the numerical
value of the cognitive complexity rating as independent variable. The positive value of
the correlation coefficient indicated that with increased cognitive complexity, students
have to invest more mental effort to solve the task. 

Conclusions

In this study, the Table for assessing difficulty of concepts and their interactivity for topic
hydrogen exponent in the solutions of acids and bases was developed. The procedure
for estimating the cognitive complexity of the tasks was evaluated by a series of regres-
sion analyzes of the dependence of students' performance, as well as invested mental
effort, from cognitive complexity. During the evaluation, several students’ conceptual mis-
understandings were observed and described, in accordance with some previous studies. 

The main disadvantage of this research is reflected in relatively small number of tasks
on the test. Although studies with a similar number of tasks could be found in the litera-
ture, it is recommended that the number of tasks in the test should be greater. A small
number of tasks might have affected the obtained statistical data to a certain extent, and
therefore, for the next usage, the test should be extended with new tasks.

Henceforth, the main contribution of the constructed Table is to help teachers to create
the tasks with different cognitive complexity levels, in order to affect the cognitive devel-
opment of each student. 

Regarding implications for future research, we suggest the application of new methods
for assessing cognitive complexities, such as, for instance, the Knowledge space theory,
which can additionally confirm the validity of the design Table.

Acknowledgements

Presented results are part of the research conducted within the Project “Infrastructure
for electronic supported learning in Serbia”. Grant No. 47003 of the Ministry of Education,
Science and Technological Development of the Republic of Serbia.

43

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Table 6. Statistical parameters of the regression analysis of students’ mental effort 
and cognitive complexity.



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Received: November 16, 2017
Accepted: December 13, 2017

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