Causal interconnection between informatical and mathematical competences in elementary education

U suvremeno bi doba nastavni programi trebali biti usklađeni prema uzročno posljedičnim povezanostima srodnih predmeta, osobito predmeta iz STEM (Science, Technology, Engineering i Mathematics) područja. Kauzalna povezanost informatičkih i matematičkih kompetencija nije dosad posebno ispitivana, a posebno ne kod učenika osnovne škole u Republici Hrvatskoj. Cilj je doktorskog istraživanja izrada kauzalnog modela povezanosti informatičkih i matematičkih sadržaja u osnovnoj školi što će pridonijeti usustavljivanju potrebitih kompetencija kod učenika. Svrha je tako izrađenog modela stvoriti pretpostavke za izradu programa informatičkih i matematičkih sadržaja u osnovnoj školi, slijedeći jedni druge logički i razvojno. Model će također omogućiti uspoređivanje kvalitete usvojenosti matematičkih i informatičkih kompetencija učenika, kao i njihove povezanosti s ostalim kompetencijama učenika osnovne škole. Očekivani znanstveni doprinos unutar informacijskih i komunikacijskih znanosti jest izrada kauzalnog modela povezanosti informatičkih i matematičkih kompetencija u osnovnoj školi koji može imati veliki značaj pri oblikovanju budućih nastavnih programa i može unaprijediti obrazovni proces u nastavi Informatike i Matematike. Ovaj će model omogućiti kauzalnu povezanost s IKT standardima u osnovnoj školi što je nastojanje onih obrazovnih sustava u svijetu koji nemaju nastavni predmet Informatika u osnovnoj školi.Contemporary educational programs should be aligned with causal relations between school subjects, especially STEM subjects (Science, Technology, Engineering and Mathematics). So far, there has not been a separate research on causality between competence in computer science and competence in mathematics. This research attempts to help systematize the aforementioned competencies and provide the causality model illustrating the connection between computer science and mathematics. The causality model will enable the comparison of competence levels and contribute to developing educational programs for computer science and mathematics in primary school. The research, along with the causality model illustrating the connection between computer science and mathematics, attempts to help systematize the aforementioned competencies. The scientific contribution of this research to information and communication science can be seen in development of the model of causality between informatical competence and mathematical competence in primary school. This model can also have a significant role in developing future educational programs and improving the educational process. 'Informatics' is a concept which has been first suggested by Karl Steinbuch in 1957. At the time the concept has been often translated into English under the term 'computer/computing' science. Nowadays the concept itself as well as its variations have been in use, broadening as well as focusing on specific aspects of the scientific field (e.g. informatics, information technology, information-communication technology, computing science). Europe explains the Informatics concept in the following way: „the concept of informatics depends on the country which is using it“. However, the variety and richness of both definitions and contexts has given rise to difficulties in studies comparison. Informatics in the context of primary education in the Republic of Croatia can also be studied through the expected and achieved informatics competences, which are defined by the National Curriculum Framework. With regard to mathematics within the context of primary education worldwide in comparison to that in the Republic of Croatia, the situation is clearer. Stern and Döbrich (1999) emphasise three types of mathematical competences: 1) strategies used for arithmetic problem solving, 2) strategies used for arithmetic word problem solving and 3) proportional thinking (Stern & Döbrich, 1999). Preschool mathematical competencies not only have an impact on the competences which the children will exhibit in elementary school but can also serve as predictors of either well-developed competences or difficulties (Aunola et al, 2004; Bailey et al, 2014; Claessens i Engel, 2013; Claessens et al., 2009; Duncan et al., 2007; Jordan et al., 2009; Watts et al., 2014). Furthermore, taking into consideration that the development of mathematical competences and skills is hierarchical, Andersson (2008) emphasises eight mathematical fields with appropriate competences: 1) arithmetic fact retrieval, 2) written multidigit arithmetic calculation, 3) approximate arithmetic, 4) place value, 5) calculation principles, 6) one-step mathematic word problems, 7) complex multistep mathematic word problems and 8) time telling which competences have been outlined because the Author argues one competence implies another and that a much broader spectrum of competences need to be examined in order to have a clearer picture (Andersson, 2008). On the other hand, the picture with informatics competences is not so clear. One of the concepts often used in this context is digital literacy. Erstad (2010) argues that the concept of digital literacy is too limited for modern age and should be expanded to media literacy, with the following media literacy aspects parts of school curriculum: 1) basic skills, 2) media as a focus of analysis, 3) knowledge gathered within specific domains i.e. school subjects, 4) cultural competence (Erstad, 2010). Literacy skills of the 21st century is another concept which is in use, which Jenkins et al (2006) explain through its aspects: play, performance, simulation, appropriation, multitasking, disturbed cognition, collective intelligence, judgement, transmedia navigation, networking and negotiation (Jenkins et al, 2006). Furthermore, digital competencies are mentioned as one of the eight key competencies in the document ‘Key competences for Lifelong learning’ issued by the European Union in 2010, which is defined in the following way: “digital competence includes confident and critical use of the ICT for work, fun and communication. It is supported by the basic ICT skills: the use of computers in order to gather information, evaluate it, store, present and exchange information, communicate and participate in networks (European Union, 2010). One of the questions which arises from the study of informatics and mathematical competences within the context of primary education is how the two of them are connected, which mathematical competences imply informatics competences and lastly, what all of this means for primary education? It can be said that logical thinking, formal writing and mathematical logic are very important for informatics, just as the use of mathematical software in classrooms can improve deeper understanding of mathematical concepts as well as offer the pupils a tool for learning through experimentation. Problem-solving as a mathematical competence is closely related to computational thinking (Wing, 2011), whereas some authors consider that even though computational thinking is closely related to elements of engineering, mathematics and other natural sciences, it is different due to its focus on information processing (Denning & Freeman, 2009; pg. 30). Joint Research Centre, which is a part of the European Committee for science and knowledge emphasises the need to integrate computational thinking into primary education. The authors outline several basic skills of which the parts of computational thinking are: subtraction, algorithmic thinking, automation, decomposition and generalisation (Bocconi et al., 2016). Given all the research done on informatics and mathematical competences the implication would be to create models which would graphically represent their interconnectedness as well as which competences imply which. In 2010 Cartelli, Dagiene and Futschek created a model in order to assess digital literacy as a part of the Bebras contest (an IT/ICT contest for children). The authors developed a questionnaire which they called ‘digital literacy assessment’, focusing on the following areas: 1) recognising possible solutions to technological problems and choosing the appropriate one, as well as differentiating between real and virtual phenomena, 2) working with text, data organisation, choosing and interpreting text as well as assess relevance and reliability of information, 3) respect toward others within the virtual community and understanding of social and technological differences (Cartelli et al, 2010). Haspekian and Bruillard (2010) on the other hand, focused, within the scope of DidaTab (spreadsheet didactics), on examining the connection between the use of spreadsheets and mathematics and the pupils’ competences. Comparing the elements of algebra and the elements found in work with spreadsheets (such as e.g. MS Excel) the authors outline several equivalents: 1) objects: unknowns and equations/variables and formulae, 2) pragmatic potential: problem solving tools/generalisation tools, 3) solving process: the application of algebraic rules/arithmetic process of trial and error and 4) the nature of solution: correct solutions/correct or approximate solutions. The sample consisted of 13 pupils aged 17 attending vocational school and the authors administered a test to be done on computers with the aim to assess the pupils’ abilities to work with spreadsheets. After analysing the results, the authors came to the conclusion that there was a lack of understanding of algebraic concepts because the use of formulae in spreadsheets requires understanding of the sheer concept of a variable within the same environment (Haspekian & Bruillard, 2010). Similarly, Haspekian did a research within the scope of which he presented introduction to algebra through the use of spreadsheets to a sample of 7th grade elementary school pupils. He outlined the following difficulties: understanding formulae and the increment tool (sequence continuation). The author argued that algebraic i.e. mathematical concepts were kept within the frames of traditional mathematics education and that the concepts themselves are not explained nor analysed in depth which is clearly observable on the pupils’ inability to apply them in a different environment (i.e. a spreadsheet) (Haspekian, 2005). In statistics nowadays the use of the term causality is being used more and more often as a tool for explaining interconnectedness. Causality can be described as a difference between actual results and results which did not occur (Clemens, 2017). By studying causal connection between variables we arrive to a causal structure, which is defined as an oriented graph on which each edge between e.g. variables X and Y, oriented as X->Y represents a causal connection between variable X as cause and Y as effect. Simplified research which studies causality demands having a sample, which needs to be big enough as well as meet the conditions to be representative of population, divided into control and experimental group. A randomised experiment demands of the variables to be directly observable, that it is possible to intervene and that the sample is large enough in order to correctly show the effect of opposite action. When it is impossible to intervene or change variables, but which are observable, statistics resorts to the concept of correlation which determines whether two variables within a data set are behaving in a similar way, but correlation does not imply causality (Clemens, 2017). One of statistical models often in use is Bayes’ model. Bayes’ classifiers predict the probability of belonging to a certain class, with one of the reasons for their popularity having a high accuracy percentage as well as speed when applied to large data sets. Bayes’ networks are graphic models which enable the representation of dependence between attribute subsets, but which, at the same time ,can be used for classification. They enable creating a graphic model of causal connections (Han, Pei & Kamber, 2011). The advantage of Bayes’ concepts and methods are that they can include prior information and technical concepts without appropriate data and enable studying the cause and effect connections, apart from the simple studying of correlation (Wang & Amrhein, 2018). During the past thirty years researchers have used Bayes’ networks in order to uncover causal structures within the mass of statistical data, contrary to the presumption that this is feasible only within the environment of carefully controlled experiments (Pearl & Verma, 1995; Spirites et al, 2000; Pearl, 2003). The sample for our research comprised of 21 elementary schools in the Republic of Croatia, with 9 from the rural and 12 from urban areas. The participants were 1000 pupils attending 5 th, 6th, 7th and 8th grades, under the presumption that there were at least 10-15 pupils within a school per each generation attending informatics as an elective subject. The pupils were 10-11 years old (5th grade), 11-12 years old (6th grade), 12-13 years old (7th grade) and 13-14 years old (8th grade) of both genders. Background questionnaire has been administered among teachers in spring 2018 for the school year 2016/2017, which consisted of socio-demographic data (age, gender, rural/urban school), final grades in all subjects, average final grades, grades achieved on mathematical and informatics tests in 5-6 individual topics according to the National Plan and Programme for the subject mathematics and elective subject informatics. The basic research aim has been to measure the frequency, interdependency, dependence and interconnectedness between informatics and mathematical competences. The hypotheses were: 1) to examine whether a statistically significant correlation between final grades in informatics and mathematics within each grade exists, 2) to examine whether a statistically significant correlation between grades in specific mathematics and informatics topics within each grade exists, 3) to examine whether statistically significant difference between the final grade in mathematics and final grade in informatics exists as well as correlation between the final grade in mathematics and final grades in other subjects and average final grade exist within each grade, 4) to examine whether causal connection between a) informatics and mathematics grade within each grade exists, b) mathematics grade and average final grade within each grade exists, c) informatics and average final grade within each grade exists and d) specific topics within the mathematics and informatics curriculum exists. The pupils achieve statistically significantly higher average grades in educational compulsory subjects, with average higher grades in informatics following. No correlation between the final grades in informatics and mathematics in each grade has been found. Furthermore, the hypothesis ‘correlation between final mathematics and informatics grades is statistically significantly higher than the correlation between the final mathematics grade and final grades in other subjects, as well as average final grade’ has not been corroborated. In fifth grade, no correlation has been found between average final grade and final grade in informatics, but has been found between average final grade and final grades in other subjects. In sixth, seventh and eighth grade correlation has been found between average final grade and final grade in informatics. With regard to correlations between acheivements in chosen topics in mathematics and informatics, a positive correlation between grades in ‘drawing with the help of computer’ and ‘natural numbers’ in fifth grade as well as a small but negative correlation between the grades in ‘text processing’ and ‘linear equations with one unknown’ in sixth grade. Therefore the hypothesis that a statistically significant correlation between grades in specific topics in informatics and mathematics exists has not been corroborated. Statistically significant difference in average grade in informatcs has been found in fifth grade which is higher in rural schools. Comparing the created causal model for chosen topics in informatics and the school curriculum, several discrepancies have been found. In fifth grade, discrepancies have been found with regard to the following causal connection: basic 2D geometry -> drawing with the help of computer, decimal numbers (mathematics) -> programming (LOGO) (informatics), basic 2D geometry -> first steps in working with computers, basic 2D geometry -> storage and computer equipment. In sixth grade, only one discrepancy has been found: quadrangle -> multimedia, whereas in seventh grade the following discrepancies have been found: linear equations systems -> programming (LOGO), similarities and polygons -> programming (LOGO), proportionality and inverse proportionality -> programming (LOGO), similarity and polygons -> coordinate system, linear functions -> spreadsheets and linear functions -> linear equation systems. And finally, in eighth grade, discrepancies with regard to the following causal connections have been found: presentations -> basic 3D geometry, real numbers -> programming (LOGO), isometry -> programming (LOGO), internet -> databases, Pythagorean theorem -> fundamentals of informatics, geometrical bodies -> programming and isometry -> programming. Scientific contribution of this doctoral thesis is the creation of the causal model for the chosen topics from the curriculum for mathematics and informatics, which contributes to theoretical understanding of the mutual connection between school topics, as well as enabling a better logical and organisational structuring of school topics within the subjects of mathematics and informatics during macro (school curriculum) or micro (specific topics within the school subjects) planning. Furthermore, the created causal model offers specific examples of discrepancies between certain school topics between the subjects of mathematics and informatics, but also within the subjects of mathematics and informatics, which has been shown by the causal models. Similar causal models can be created for other school subjects, which would guarantee exact criteria for checking quality of the created curriculum which would be more adapted to the pupils’ age as well as their cognitive abilities, but also expose the need to remove some topics from the curriculum because they bring connection with other topics into question and to structure the topics via micro and macro planning. This is especially important for introducing new information into the educational system in order to check the connection with other topics by creating causal models, as well as testing the mutual connection of topics and information. The information science therefore gain a new instrument – causal model, in order to ensure the quality of planning and information structuring as well as context within the educational process