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Textual configurations as an approach to evaluate textual difficulties of mathematical tasks
David Bednorz, Michael Kleine
To cite this version:
David Bednorz, Michael Kleine. Textual configurations as an approach to evaluate textual difficulties of mathematical tasks. Eleventh Congress of the European Society for Research in Mathematics Education, Utrecht University, Feb 2019, Utrecht, Netherlands. �hal-02435236�
Textual configurations as an approach to evaluate textual difficulties of mathematical tasks
David Bednorz and Michael Kleine
Bielefeld University, Germany; david.bednorz@uni-bielefeld.de
Language is a major factor in learning mathematics. Language learners or those with a low socioeconomic status have special difficulty solving mathematical tasks with high language demands. The project evaluates linguistic task difficulties in mathematics. This paper discusses the first step of the project: empirical evaluation of textual configurations and analysis of the lexical and grammatical structures of a corpus of 348 tasks. An explorative factor analysis is used to analyze the configuration of 17 linguistic variables. The analysis revealed a reduction of the variables to five factors, which are interpreted as different textual configurations of mathematical tasks based on their obligatory allocation of typical situational use. The structure offers potential for an empirical model to evaluate the difficulties.
Keywords: Language, language variation, mathematics education.
Introduction
Language is omnipresent in human life; it helps us to create a cultural base for human interaction. Language enables a direct interaction to make meaning between people. To define what language is and what potential relevance language has to learn math, this paper introduces the systemic functional linguistic (SFL) theory. The important theoretical terms to describe language for the analysis are the instantiation and realization of language.
Furthermore, the notions of register and the contextual configuration are presented in the theoretical part of this paper. Registers and contextual configurations are only considered in terms of the objectives of the empirical results in this paper. Afterwards, the research questions of the project “Language Difficulties in Mathematical Tasks” (LIMiT) are presented, followed by the empirical method and the results of the factor analysis. Finally, a prospectus is made of the next goals of the project LIMiT.
Theoretical framework
The framework conditions for successful pedagogical processes are constantly changing, and the importance of language is becoming more relevant for mathematics classrooms. In addition to the content specific requirements of mathematical tasks, language requirements are a factor that determines the complexity of mathematical tasks (Abedi & Lord, 2001). To solve tasks with content related subject, on the one hand students need to have contextual knowledge, which are essential to solve the problem which are presented in the tasks. On the other hand, the grammatical and lexical textual basis of mathematical tasks are important factors of difficulties for understanding (Abedi & Lord, 2001). In particular, language learners and learners with a low socioeconomic status show special difficulty in mastering mathematical tasks (Abedi, Hofstetter, & Baker, 2001; Haag & Heppt, 2015; Martiniello, 2008). The academic language and mathematical language are theoretical constructs to explain the language requirements in mathematics tasks (Maier & Schweiger, 1999; Morek &
Heller, 2012; Prediger, Wilhelm, Büchter, Gürsoy, & Benholz, 2015). On the perspective of academic and mathematical language as register, the difficulties can be explained by the variation of language. Register tend to vary, especially in lexical and grammatical features (Morek & Heller, 2012).
Language in mathematics classrooms
With the focus on language variation, a possible theoretical approach to explain the relevance of language in mathematics classrooms is the SFL theory (Halliday, 2014; Halliday & Hasan, 1989). The SFL theory describes, theorizes and analyzes language through the meaning of the environment (Halliday, 2014). Language is considered in a systemic functional approach as a resource for making meaning. To develop meaning, humans use different kinds of semiotic systems which are useful or functional in a specific context (Eggins, 2004).
Instantiation of language
According to Halliday (2014, p. 28) language is described through a “cline of instantiation”, with two different poles, one potential pole and one instance pole. On the instance pole, a specific textual instance is embedded in the context of a situation. The context of a situation is characterized after Halliday (2014) with three variables called field, mode and tenor.
According to Eggins (2004, p. 90), field refers “what the language is being used to talk about”, mode describes the way language functions in interaction and tenor focuses on the relationship of interactants involved in the discourse which are a short description of the three variables. The potential of a particular language is described by the contextual conditions within a culture (Halliday, 2014). Halliday (2014, p. 33) defines this contextual potential of a community as the “context of culture”. The context of culture describes what members of a community may mean by cultural terms. Culture is described as a system of higher importance, as an environment of meanings in which different semiotic systems function.
This describes a “multi-dimensional semiotic space” (Halliday, 2014, p. 34) that we also find in mathematics classrooms. The perspective for exploring culture is the reduction to specific cultural domains, such as analyzing different contexts of situations within an institution. The isolation of culture relative to the institution allows the linguistic analysis of language used within the institution.
The concept of register
It is clear that situations vary depending on the location, time and participants. It is not quite so clear that at the same time, language varies corresponding to various situations. The notion of a register describes the variation in language in relation to the context of situation (Eggins, 2004). One viewpoint to define registers, especially for a quantitative purpose is that registers are systemic probabilities of the use of language in specific situations or a set of possibilities for the entry in a situation (Halliday, 2014; Halliday & Hasan, 1989). For example, symbolic notation will occur more frequently in mathematical texts than in stories, and the graphic proof of the Pythagorean theorem will occur more frequently or even exclusively in mathematical texts compared to geography textbooks. According to Morek and Heller (2012) as mentioned above, registers vary according to usage and tend to vary within semantics, especially in grammar and vocabulary. This leads us to Table 1, which shows a possible
categorization for the variation of language in typical situations (Eggins, 2004). Furthermore, the three variables field, tenor and mode constitute the language used in a situation. Because of the changing situation, these three variables are also called register variables (Eggins, 2004). The value of each register variable defines the language used in a text. Eggins (2004) combined the three variables with typical situations of use which enables a lexical and grammatical fixation of the systemic probability of specific lexical features.
Register variables
Field Mode Tenor
Typical situations of language use
Everyday situation
Technical situation
Spoken language
Written language
Informal Formal
Selection of characteristic
features
everyday terms (words we all understand), standard syntax
technical terms (words only
‘insiders’
understand), acronyms, abbreviated syntax
everyday lexis, content- dependent, low lexical density, open- ended, non- standard grammar
prestige lexis, context independent, high lexical density, closed, standard grammar
attitudinal lexis, colloquial lexis, typical mood choices
neutral lexis, formal lexis, incongruent mood choices
Table 1: Summary of register variables and characteristic features (Eggins, 2004) Contextual configuration
Within cultural practice, similar linguistic structures or obligatory linguistic features are used among other things for pragmatic reasons. Halliday and Hasan (1989) describes these structures as contextual configurations which are recurring in similar situations. Similar situations, framed by a context of culture or an institutional basis, lead to the realization of specific values for field, tenor and mode, which can be described as a contextual configuration. The contextual configurations do not imply a concrete situation, but a type of related situations. As such, the contextual configuration instantiates through a multiplicity of instances of related types which can be classified by specific values for a particular purpose.
Applied to mathematical learning processes, this means that the contextual configuration of specific, similar situations is characterized by the inclusion of recurring elements or structures. Thus, the introduction to a mathematical problem differs from the solving of the task and each sequence has a set of possibilities for the realization of the register variables.
Theoretical model and research question of LIMiT
For mathematical tasks, as a standard mathematical classroom situation, the same conclusions apply to contextual configurations. A mathematical task is contextualized with a textual structure which forms the textual basis of a mathematical task. In addition to the textual structures, there are additional activities that determine the contextual configurations in the
T E X T U A L B A S I S
P R O D U C T I V E P R O C E S S E S R E C E P T I V E
P R O C E S S E S
C O N T E X U A L C O N F I G U R AT I O N
T E X T U A L C O N F I G U R AT I O N
( s e l e c t i o n o f l e x i c a l a n d g r a m m a t i c a l f e a t u r e s )
M AT H E M AT I C A L T A SK S
Figure 1: Theoretical model of LIMiT
process of solving a task, including receptive processes such as extracting information and productive processes such as creating a solution. However, the textual basis of mathematical tasks forms the initial structure of mathematical problems which can be interpreted as particularly crucial for the further processes of problem solving. The starting point of the study is the analysis of textual configurations of mathematical tasks and the interpretation of the specific values for field, mode and tenor, based on a selection of grammatical and lexical features as seen in Figure 1. The model lies in terms of the cline of instantiation between the system pole and the instance pole. Considering the fact that registers vary in lexical and grammatical structures in particular, and this variation may present difficulties for learners, textual structures should be analyzed in the project. This model is preliminary in the way that it serves to illustrate the empirical analysis. For the project, a further extension of this model
based on the metafunctions ideational,
interpersonal and textual function. The textual
configurations will be analyzed, and the
following questions emerge: Which textual
configurations can be extracted from
mathematical tasks? How can the extracted
structures be interpreted?
Method
The empirical analysis was based on 348 tasks from nine different secondary-school textbooks for mathematics classrooms. The textbooks are designed for students aged between 10 and 16 years. For each task, 17 different lexical and grammatical features were determined. A part of the lexical and grammatical features was quantified by expert ratings, such as mathematical terms, other parts of the data was quantified using corpus analysis, such as determining the frequency of words in a comparable corpus to determine the rarely used words (Michalke, Brown, Mirisola, Brulet, & Hauser, 2017). The selection of the lexical and grammatical features was made because of their relevance in the literature. The findings on difficulty-generating traits indicate that there are particular difficulties, especially in
vocabulary, the presence of compressed structures but also in relational connections (Caplan
& Waters, 1999; Kintsch, 2007; Leisen, 2013; Maier & Schweiger, 1999; Ozuru, Rowe, O'Reilly, & McNamara, 2008). Table 2 shows the chosen association between lexical and grammatical features and the typical situations of language use. Compared to Table 1 some changes were made. Thus, the category relator was added in mode, which can be regarded as essential for math problems, and furthermore, the everyday situations were excluded from consideration for the empirical analysis, since there is no clear evidence regarding difficulty- generating traits.
Register variables
Field Mode Tenor
Typical situations of
language use
mathematical situations
spoken language
written language
relator informal formal
Lexical &
grammatica l features
mathematical terms, mathematical symbols, discontinuous text, numbers
compound s, lexical density, synonyms, passive voice
filler words, present perfect
prepositions, conjunctions
modal verbs
nominalization, propositional density, rarely used words, impersonal language
Table 2: Selection and arrangement of lexical features Results
The suitability of the correlation matrix for the use in a factor analysis was checked with the Bartlett test of sphericity, the test criteria have been met (χ2 = 454.31, p <0.001). The KMO test for checking the adequacy of the data indicates only a moderate fit (overall MSA = 0.67).
The moderate result can be explained by the fact that in the case of lexical and grammatical variables, a low share of common variance can be assumed. Accordingly, the present MSA may be considered sufficient for the adequacy of the data. Overall, a potential five-factor solution, can achieve a variance explanation of 45%, with a reduction of totally 70.6 % of the variables. The scree plot in Figure 2 shows the possible number of factors. To select the number of factors, the parallel analysis, shown in Figure 2, was used. According to the
parallel analysis, a choice of 5
factors is sufficient.
Figure 2: Scree plot to determine the factor solutions
Figure 3 and 4 presents the five-factor solution in different versions. Figure 3 shows the loadings of the variables for each factor. This figure illustrates the specific characteristics variation of the variable for each factor. To illustrate which are the central variables with high loadings, the simplified factor diagram in Figure 4 is used. The figures illustrate that factor 5 and factor 3 have a simple structure, which means that on these factors only single variables have high loadings. For factor 5, the variable mathematical symbols has the highest loading.
In addition to mathematical symbols, the variable mathematical terms have also a characteristic loading on factor 5. Small loadings on factor 5 also show prepositions, numbers, and lexical density. The second factor with a simple structure is factor 4.
Particularly relevant variables for this factor are the lexical density, rarely used words with a high negative loading numbers.
Figure 3: Factor loading diagram Figure 4: Simplified factor diagram
The remaining three factors have more complex structures. At factor 3, the three variable synonyms, modal verbs and perfect are the variables with the highest loadings. In addition to the three factors, mentioned in the simplified factor diagram, lexical density, and numbers are variables with characteristic loadings. Furthermore, as can be seen in Figure 3, mathematical terms, and discontinuous text have relevant high negative loadings. For Factor 2, the variables compounds, passive constructions, lexical density and discontinuous text loading high on this
−1
−0.8
−0.6
−0.4
−0.2 0 0.2 0.4 0.6 0.8 1
FA1 FA2 FA3 FA4 FA5
conjunctions mathematical terms prepositions
numbers propositional density rarely used words lexial density compounds
symbols discontinuous text passive constructions synonyms nominalizations
perfect filler words modal verbs impersonal language
conjunctions prepositions impersonal language
nominalization numbers mathematical terms
filler words compounds passive constructions
lexical density discontinuous text
synonyms modal verbs perfect rarely used words propositional density
symbols
FA1 0.67
0.59 0.55 0.54 0.48
0.45
0.44 FA2
0.75 0.47 0.45
0.44
FA4
0.58 0.43 0.36
FA3
0.63
0.54 FA5
0.86
factor. Other loadings with characteristic loadings are propositional density, nominalization, and modal verbs. The results of Factor 1 show high loadings of seven variables. On this factor, conjunctions, prepositions, impersonal language, nominalization, numbers, mathematical terms and filler words are loading high.
Interpretation
To interpret the results of an exploratory factor analysis, simple structures are the easiest way to choose appropriate names. However, in the present results of the explorative factor analysis, three out of two factors do not show a simple structure, and the designation of the factors may be problematic. To simplify the problem, Table 3 is used to assign the different variables to the typical situations of language use. To name the factors, the variables with the highest loadings are considered and to which situation of the language use they can be assigned to.
The factor 5 is referred to mathematical textual configurations due to the high loadings of variables from typical mathematical situations of language usage. The variables from typical formal language situations load high on factor 4. This results in the naming formal textual configuration for factor 4. The term informal written textual configuration for factor 3 derives from the association of high loading variables from informal and written situations of language use. For factor 2, situations of written and mathematical use are central. However, it should be noted that the assignment applies only to discontinuous text. Accordingly, the term written discontinuous textual configuration for factor 2 is chosen. As expected, Factor 1 has the greatest variety of characteristic situation of language use. Here are relational, mathematical and formal situations of language use characteristic elements. Hence the term formal relational mathematical textual configuration was chosen.
In summary, there are five textual configurations due to the loadings and the variations of the variables on the different factors: mathematical TC, formal TC, informal written TC, written discontinuous TC, and formal relational mathematical TC.
Outlook
The naming of the factors shown above are qualitatively further specified in the present project in more details by individual instances. This means that regression scores are calculated for the individual variables and extreme cases are used to allow a further qualitative specification of the factor names. However, this specification is not possible at this point due to the focus of the paper. Rather, this paper should show that lexical and grammatical variables can be reduced to factors on which each variable varies.
In summary, the SFL-theory shows potential to analyze linguistic difficulties of learners in mathematical classroom. With SFL theory, the patterns of variation in the empirical study can be explained theoretically and thus, it is possible to present relevant implications for teachers.
Moreover, the analysis of language on an SFL-perspective is content specific and therefore, more concrete for mathematics classes and provides pedagogical accessibility. The factor analysis, as an empirical method, is useful to identify textual configurations, but also contextual configuration in general of mathematical tasks. As shown above five factors can be identified, which provides an approach, on the one hand, to evaluate the difficulties. On the
other hand, on this basis, a model for language proficiency in mathematical classrooms can be developed and implications for the practice of teachers can be made. The analysis provides an approach to determine the difficulty for each factor. The next steps of the project LIMiT are the theory-based interpretation and designation of the five textual configurations of the mathematical task. This is to be followed by the use of a further empirical method to evaluate the difficulties. Based on the results, a conception of an empirical model of task difficulties in mathematics and the construction of a pedagogical model for language proficiency in mathematics classrooms will be carried out.
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