The educational objectives represent the precise statements of what we expect or intend students to learn as a result of education. We have conducted an analysis of the educational tasks and objectives system within a formal context with respect to the collected real data on an array data structure of ve teachers in the eld of computer science. We submitted a report and the corresponding concept lattice to each individual teacher and explored their additional feedback. In addition, we formulate the general observations and present the feasible set of tasks and objectives of an array data structure. The results are expected to annotate in the future formation of the curricular documents as supplement to the National Education Program in Slovak republic which is formulated concisely.
The scope of the computer science education in Slovakia is o cially declared in The National Education Program of Slovak republic as the supreme curricular document. The Slovak National Education Program de nes the main principles and general objectives on which education and training in computer science is based. The education of computer science at secondary schools in Slovakia includes ve areas: c) Procedures, problem solving and algorithmic thinking (algorithms, program, programming languages, etc.), d) Principles of ICTs operations (software, hardware, architectures, etc.), e) Information society (e-learning, licenses, risks, etc.).
Formal concept analysis [
In this paper, our aim is to provide the system of objectives and tasks that is
expected to ll in the gap of the National Education Program in Slovak republic.
In general, the National Education Program is formulated concisely and we
put emphasis in a long term to particularize other supplementary curricular
documents and express the educational objectives more explicit in various areas.
Therefore, we have focused on an algorithmic thinking area and chosen an array
data structure as an educational content in which we have fruitfully applied
formal concept analysis. Simultaneously in this area, we focus on algorithms
including searching, sorting or text processing. In other countries, the national
curricular documents and other standards de ne the educational objectives in
the various levels of speci cation, see [
An array data structure, as a collection of indexed elements, plays an important role in the education of programming. An array or its equivalent as a kind of data type is implemented in the most of programming languages. The term is also used in a theoretical computer science as abstract data type.
We aim at specifying the particular and relatively precise objectives of an array data structure education in the algorithmic thinking area. Regarding our long-term cooperation with the teachers in the eld, we declare some input set of objectives of an array data structure: 1) to specify an array as the structured homogeneous data type with elements denoted by single identi er, 2) to appoint the real examples of one-dimensional array data structure (e.g.
rooms in a hotel, seats in a plane, etc.), 3) to interpret the notions of an array index (an array key) and an array element and to explain the di erence between them, 4) to di er an array index type and an array element type, 5) to reason that an array index type is an ordinal type (numbers, characters, other enumerations), 6) to declare a variable of array, 7) to read and to write out the array elements, 8) to manipulate with the array elements, to assign the array element to the other variables, to increment the array elements, 9) to appoint the common errors related to an array data structure (incorrect index type, over ow, incompatibility of the types), 10) to apply an array data structure in the simple issues (e.g. to store an array, to nd the maximal value, to modify the elements of array, etc.), 11a) to apply an array data structure in a searching, 11b) to apply an array data structure in a sorting, 11c) to multiple access to the array elements, 11d) to apply an array data structure in a text processing, 11e) to apply an array data structure in a simple game programming, 12) to recognize the issues in which can be applied an array data structure e ectively, to appoint the advantages and disadvantages of an array in comparison with other simple data structures (an access to elements, a space complexity).
The speci ed aims are enumerated by the revised taxonomy of Bloom [
We submitted the previous list of aims to the teachers in the secondary schools in Slovakia. The teachers were instructed to appoint the tasks which they usually apply in an educational process of an array data structure in programming. Teachers were not limited by the number of tasks and moreover, it was possible to add some additional aims (13, 14, : : :) if they required. Having such instructions, every teacher was asked to ll in the following table: N. task / aim 0 2nd max. element 1 2 ...
1 2 3
We have obtained the data of ve teachers who proposed overall 92 tasks, some of them equal with respect to the aims. Four teachers launched the additional educational aims including the applying of an array data structure as the parameter of the procedure, a dynamic array, two-dimensional arrays, the issue of indexing the rst element, reasoning initializing errors (seven aims in overall). For each individual teacher we have constructed the concept lattice from the collected data using ConExp1 software. We use a concept lattice with reduced labeling (labeled line diagram) regarding own objects and own attributes which is accessible to human reasoning. An example of a concept lattice shown to one of the teachers follows in Fig. 2. In e ort to explore the task di culty, we have assigned the degree to each individual aim from the set f0:25; 0:5; 0:75; 1g depending on its dimension in Bloom revised taxonomy. A value of the task di culty (computed as the sum of the degrees of involved aims) is shown in a concept lattice as a supplement of the particular task label (i.e. own object).
In our feedback given to teachers, we have added the comments and some additional questions. The comments include the aims which were obtained in all tasks (Aim6 and Aim7 in Fig. 2), the tasks which contain the unique aims (for instance Aim11b is included only in Task6) and the following instructions how to read a concept lattice to help teachers analyze the results: { tasks in the rst row are the representatives of the teacher's system of tasks (there is no task with the superset of aims), { the shaded labels linked to a node in the rst row represent the attributes introduced uniquely in the task, { the aims introduced by a particular task one can obtain by collecting the shaded labels on all paths leading up from the selected task node, 1 Concept Explorer, version 1.3, website: http://conexp.sourceforge.net. Nevertheless, we have successfully tested some other formal concept analysis software tools, for instance FcaStone, Lattice Miner, ToscanaJ, FCART, as well. { the tasks that involve a particular aim one can acquire by tracing task labels leading down from the selected aim node, { the top element of a concept lattice introduces the aims obtained in all tasks, { a task in a higher row of a concept lattice is appropriate for the rst acquisition phase of an educational process (not compulsory), { a task in a lower row of a concept lattice is appropriate for the xation or systematization phase of an educational process (not compulsory).
We state that not all of the submitted aims were used by the engaged teachers (for instance Aim9 is not introduced in Fig. 2). In contrary, some of the teachers have added their additional aims. Namely, we have rst analyzed the systems of tasks from the viewpoint of each individual teacher. Otherwise, not introduced aims would be gured at the bottom element of a concept lattice.
For each concept lattice, we have calculated the degree of tasks and aims system gradation level as the proportion of the number of task nodes (as own objects) in the longest path and the total number of the tasks in a concept lattice. This indicator shows how gradated are tasks of teacher's system. The smaller number indicates the more diversi ed system, the higher number expresses the more gradational system. The obtained results and concluding remarks follows: { the minimal number of tasks was 9, the maximal 27 in teacher's set of tasks, { the most frequently introduced aims are Aim6, Aim7 in order to declare a variable of array and to read/write out elements of an array, { two systems contain the aim(s) introduced in every task, { two systems include the set of equal tasks with respect to the aims (i.e., at least two tasks equal), { the gradational level in the systems takes the values from 0.13 to 0.22, { the average task di culty in teacher's set of tasks takes the values from 1.97 to 4.09; lower value indicates that the set of tasks is appropriate more for beginners, the higher value expresses focusing on advanced students.
We were interested in a feedback of the teachers in relation with the obtained results. One of the teachers con rmed that his/her set of tasks was used for advanced students (the average task di culty is 3.22). The teachers have explained the reasons to add some new aims, reported the tasks which they used to apply in a diagnostic phase, declared the most problematic aims for students, etc. These issues and some other recommendations will be still discussed with the teachers and other respondents in a formal and an informal way. 4
Beside the concept lattice diagram one can examine the implications between attributes valid in a teacher's tasks and aims tables. For instance, the implication fAim7, Aim8g ! fAim6g shows that the following rule holds in a table of tasks and aims: Aim6 is introduced in every task that includes Aim7 and Aim8 together. This means that the task focused on reading, writing out and manipulation with the elements of an array will also satisfy the aim of the declaration of an array data type. We have veri ed that this natural implication holds for the all ve collected set of tasks (con dence of this association rule is 100%). On the other hand, the implication fAim8g ! fAim7g does not hold in general, because there are some tasks (the counterexamples) focused on manipulation with the array elements, but do not read and write out the array elements (con dence of this association rule is 66%). Another natural implication fAim4, Aim5g ! fAim3g means that every task focused on recognizing a di erence between an array index type and an array element type will satisfy the aim of interpretation of an array index and array element. This association rule holds in the tasks of teachers with con dence 86%.
The implications one can read o from the concept lattice with reduced
labeling, it is su cient to check whether the each attribute's node from an
implication's conclusion is above (or equal to) the in mum of all attributes nodes
from a premise. In addition, one can compute the Duquenne-Guigues basis of
implications, which is optimal in terms of its size and includes a minimum cover
of all valid attribute implications, for more information see [
The knowledge acquisition method called attribute exploration is described
in general by [
First question in Fig. 3 indicates that all tasks cover the aims Aim6, Aim7. We have answered this question yes that means that we agree to preserve this implication in a teacher's set of tasks. In contrary, fth question expresses that if a task has aims Aim3, Aim6, Aim7, then it also has aims Aim1, Aim4. We do not agree to preserve this implication and advise to add a new task having aims Aim3, Aim6, Aim7, Aim10. It is advised to distinguish Aim3 and Aim4. Actually, Aim3 is introduced in a task having, for instance, integers as an array index type and also as an array element type. Nevertheless, Aim4 requires di erentiation of an array index type and an array element type to achieve this goal absolutely. The separation of these two aims in at least one educational task is helpful to encourage students to understand an array data type. Hence, we advise in addition a counterexample of a task with aims Aim3, Aim6, Aim7, Aim10 by the educational aim exploration shown in Fig. 3. This task removes the undesirable implication from the basis of all implications. Remind that a counterexample must not contradict the implications we have con rmed so far (the rows 1,2,3,4 in Fig. 3). However, if the counterexamples are added into the table, the concept lattice is modi ed. 5
Regarding ve teachers data and one additional set of 10 tasks proposed by two of the authors, we have analyzed 102 tasks and 23 educational aims obtained in this research. We have generated the summary concept lattice and found the following observations: { 45 tasks (the rst row in the summary concept lattice) are the representatives; i.e. every task includes the unique set of aims and there is no task that introduces the superset of these aims, { 5 tasks (from 45 representatives) are such that every task includes the unique set of aims and there is no task that introduces neither superset nor subset of these aims, { 3 aims (the rst row in the summary concept lattice) are unique, i.e. the aim is introduced only by one task, { the most frequent aims in general are Aim6 (80% of tasks), Aim7 (63%), Aim10 (51%), Aim8 (46%), Aim4 (35%) which represent the basic declaration, read, write out, manipulation and applications of an array data type in the simple tasks, { the most frequent aims including applications of an array data type in the more di cult situations are Aim11c (22%), Aim11d (22%), Aim12 (20%) and Aim11a (19%); the applications in a sorting and a simple game programming are the least represented from these group of aims.
In e ort to prepare the graduated sets of tasks, we have explored the longest paths extracted from the summary concept lattice with reduced labeling of all 102 tasks. Some of the longest paths are shown in Fig. 4. Every path contains the graduated system of tasks depending on the nal task we want to achieve in conclusion. The object label, for instance 3.5, corresponds to the fth task of third teacher. The set of tasks labeled 6.1 { 6.10 comes from the authors.
The paths have di erent lengths, because there are nodes in the summary concept lattice with reduced labeling, which do not contain neither task or aim label (own object or own attribute). Therefore, these nodes are omitted in the extracted longest paths shown in Fig. 4. Nevertheless, some of the paths can have the same length and can di er only in a small number of tasks. To capture two or more paths (which are similar in this sense) by one gure, we display also not linear cases (b), (c) in Fig. 4.
The longest paths are recommended to apply in a diagnostic phase of an educational process. For instance, the longest path (a) illustrates that if a student has a problem with Task6.5, we ask him/her to solve Task3.12. Moreover, if we have found that a student has a problem with Aim6 in Task6.5, we give him/her to xation one task (or more) from the node which contains the set of equal tasks Task1.3, Task1.4, Task1.5, Task1.8. In contrary, if a student has no problem with Task6.5, we suppose that he/she will pass also the Task3.12. There is only one path including seven task nodes, however paths with six nodes appear in the summary concept lattice several times. The cases (b) and (c) were chosen to cover the most frequent aims by the combination of (a),(b),(c) cases. Moreover, the case (b) shows that if a student has a problem with Task6.4, we can choose either Task1.3 or Task 4.1 in order to cover the aim that was not ful lled by a student. We can also extract some other (not compulsory the longest) paths starting with other initial aims and di erent initial tasks extracted from the summary concept lattice. The path (d) presents an example of the ve nodes path starting with Task 4.3 in contrary with an initial task of the previously described paths.
As a conclusion, we propose to supplement four of the input objectives and to add two additional objectives (mainly for advanced students) into the input system of objectives as follows: 5) original form supplemented by: the rst array index is not necessary 0 or 1, 8) original form supplemented by: nd a presence of some value in a an array, 9) original form supplemented by: errors related to a clear of an array, 10) original form supplemented by: an array as a parameter of procedure, 13) to apply a dynamic array in the simple issues, 14) to apply a two-dimensional array in the simple issues.
Moreover, we present some interesting educational tasks which appear in the summary concept lattice mostly in the rst row and one can advise them to apply in the educational process related to an array data structure. The formulations are shortened in comparison with the original author's texts.
Propose the way how to denote the parking places in front of a hotel. How are the train carriages enumerated? How would you denote the overall and nal results of six teams in the television knowledge contest? We have observed GPS data containing ten altitudes on our tourist route. Write a program to print out the altitudes on a reverse route.
Imagine that you have received SMS from your friend. Write a program to count the number of words in your text message.
A musical instrument, like a piano, can be simulated by a computer program. Some of the keys will have assigned a particular tone frequency. Write a program to play a tone when the particular key is pressed.
Consider the starting sequence of children names and the nal shift of Ferris wheel as the input. Write a program to make a list of the children names in the sequence in which they will get out Ferris wheel.
Write a program to generate twelve random values expressing the number of your website visits in a particular month. Draw a histogram, highlight the maximum and minimum and show an average value as a horizontal line. 6
We have fruitfully applied formal concept analysis as a powerful tool in a simultaneous analysis that involves the teaching of programming skills in an opensource web application App Inventor 2. The tutorial website2 provides materials in the form of learning cards for building the basic applications, but one of the authors of this paper has prepared the set of ten complex educational tasks which in summary cover 129 elements (components and their elements, event handlers, call, set instructions, get instructions, data structures, etc.) available at the present time. The added value includes the proposal of the introductory set of complex tasks and its further modi cation in e ort to teach and learn the di erent target groups. The talented lower secondary school's pupils participate in our optional university courses and the teachers of secondary schools attend the didactic workshops at our university. Our results are concerned with the inclusion of the programming language elements (available at the App Inventor website at present) in the complex educational tasks and the e ort to extract the appropriate tasks for the di erent types of an educational process. The formal context contains 10 tasks as the set objects and 129 App Inventor programming elements as the set of attributes. 2 http://www.appinventor.org/
Exploring own attributes, the resulting concept lattice and its attribute labels shown in Fig. 5 give an information about the elements introduced uniquely by a particular task. As conclusion, we recommend the following methodology: { a task with a high ratio of the own elements and the low total number of elements is advised to use in a rst acquisition phase of an education, { a task with a low ratio of the own elements and the low total number of elements is recommended to use in a xation phase of an educational process, { a task with a low ratio of the own elements and the high total number of elements is suggested in a systematization or diagnostic educational phase, { a task with a high ratio of the own elements and the high total number of elements is the least appropriate for an educational process, because it brings many new elements without their introduction in a more simple task.
We have investigated the educational tasks and objectives of ve real teachers
giving lessons in computer science. Extracted paths from the summary
concept lattice seem to be based on the similar idea as in the learning paths from
knowledge space theory introduced by Doignon and Falmagne [
The learning process of students in computer science is also concerned in
the work of Uta Priss [