In mathematics, students’ abilities to make transformations between mathematical representations are fundamental. The recent implementation of digital technologies, such as Dynamic Geometry Environments (DGEs), have changed students’ access to mathematical representations by providing a variety of different representations, available just by pressing a button. Students use of a DGE may change their mathematical communication to become more dynamic and active. However, it is not clear how the use of DGEs and the change in communication style relate to more epistemological aspects of students’ mathematical work. This study, therefore, investigates the interplay between students’ dynamic mathematical communication and their transformations of mathematical representations when using a DGE from a competency perspective. Based on analyses of instances of lower secondary school students’ dynamic mathematical communication, findings indicate that students’ mathematical communication reflect the new ways of engaging with mathematical representations when using a DGE. Hence, the communication becomes dynamic. Theoretically, the complexity of having “continuous” transformations of mathematical representations may reduce the readiness of the mathematical communication competency. The “continuous” transformation may in fact cause students to unintentionally ascribe dynamic properties to mathematical representation.
This study uses a statistical method to identify verbal items among mathematical items from PISA 2003. The verbal items are preliminary analysed and compared to the non-verbal items concerning number of text lines, response types, cognitive level, and competences measured. The results show that the verbal items, to a higher percentage than the non-verbal items, measures the reproduction competency, are straightforward, and of open constructed-response type. These results and proposed further analyses are discussed.
This dissertation consists of two different but connected parts. Part A is based on two articles in mathematics and Part B on two articles in mathematics education.
Part A mainly focus on properties of positive currents in connection to polynomial convexity. Earlier research has shown that a point z0 lies in the polynomial hull of a compact set K if and only if there is a positive current with compact support such that ddcT = μ−δz0. Here μ is a probability measure on K and δz0 denotes the Dirac mass at z0. The main result of Article I is that the current T does not have to be unique. The second paper, Article II, contains two examples of different constructions of this type of currents. The paper is concluded by the proof of a proposition that might be the first step towards generalising the method used in the first example.
Part B consider the types of reasoning that are required by students taking introductory calculus courses at Swedish universities. Two main concepts are used to describe the students’ reasoning: imitative reasoning and creative reasoning. Imitative reasoning consists basically of remembering facts or recalling algorithms. Creative reasoning includes flexible thinking founded on the relevant mathematical properties of ob jects in the task. Earlier research results show that students often choose imitative reasoning to solve mathematical tasks, even when it is not a successful method. In this context the word choose does not necessarily mean that the students make a conscious and well considered selection between methods, but just as well that they have a subconscious preference for certain types of procedures. The research also show examples of how students that work with algorithms seem to focus solely on remembering the steps, and researchers argue that this weakens the students’ understanding of the underlying mathematics. Article III examine to what extent students at Swedish universities can solve exam tasks in introductory calculus courses using only imitative reasoning. The results show that about 70 % of the tasks were solvable by imitative reasoning and that the students were required to use creative reasoning in only one of 16 exams in order to pass. In Article IV, six of the teachers that constructed the analysed exams in Article III were interviewed. The purpose was to examine their views and opinions on the reasoning required in the exams. The analysis showed that the teachers are quite content with the present situation. The teachers expressed the opinion that tasks demanding creative reasoning are usually more difficult than tasks solvable with imitative reasoning. They therefore use the required reasoning as a tool to regulate the tasks’ degree of difficulty, rather than as a task dimension of its own. The exams demand mostly imitative reasoning since the teachers believe that they otherwise would, under the current circumstances, be too difficult and lead to too low passing rates.
Empirical research shows that students often use reasoning founded on copying algorithms or recalling facts (imitative reasoning) when solving mathematical tasks. Research also indicate that a focus on this type of reasoning might weaken the students' understanding of the underlying mathematical concepts. It is therefore important to study the types of reasoning students have to perform in order to solve exam tasks and pass exams. The purpose of this study is to examine what types of reasoning students taking introductory calculus courses are required to perform. Tasks from 16 exams produced at four different Swedish universities were analyzed and sorted into task classes. The analysis resulted in several examples of tasks demanding different types of mathematical reasoning. The results also show that about 70% of the tasks were solvable by imitative reasoning and that 15 of the exams could be passed using only imitative reasoning.
Students often use imitative reasoning, i.e. copy algorithms or recall facts, when solving mathematical tasks. Research show that this type of imitative reasoning might weaken the students' understanding of the underlying mathematical concepts. In a previous study, the author classified tasks from 16 final exams from introductory calculus courses at Swedish universities. The results showed that it was possible to pass 15 of the exams, and solve most of the tasks, using imitative reasoning. This study examines the teachers' views on the reasoning that students are expected to perform during their own and others mathematics exams. The results indicate that the exams demand mostly imitative reasoning since the teachers think that the exams otherwise would be too difficult and lead to too low passing rates.
Over the last decades, there has been an on-going international reform for school mathematics, which has, not surprisingly, been difficult to implement. This study focuses on teachers’ interpretation of formal written curriculum documents, especially whether their interpretations align with how a concept (the concept of problem) is conveyed in the documents (in Sweden). The results show that the formal written documents are vague, but that it to some extent conveys the concept of problem as “a task for which the solution method is not known in advance to the solver.” The interviews show that about 53 % of the teachers interpreted problem as “any task,” and that teachers’ interpretationstherefore are not aligned with how the concept is (albeit vaguely) conveyed in the documents.
More than 15 years after the introduction of a standards-based curriculum reform, the mathematics teachers are positive towards the reform message but have not changed their classroom practice accordingly. To improve the impact of future reforms, it is important to learn from this situation and to better understand the role of the national policy documents. The purpose of this study is therefore to examine how the standards-based reform in mathematics in Sweden was conveyed in the formal written curriculum. The research questions focus on to what extent and how clearly the national policy documents convey the message. The results show that the message is present to a large extent in the policy documents, but that it is vague and formulated with complex wording. The study gives concrete examples and shows in detail in what ways the documents are vague and complex.
In mathematical text and talk, natural language is a constant companion to mathematical symbols. The purpose of this study is to identify different types of relations between natural language and symbolic language in mathematics textbooks. Here we focus on the level of integration. We have identified examples of high integration (e.g., when symbols are part of a sentence), medium integration (e.g., when the shifts between natural and symbolic language occurs when switching to a new line), and low integration (e.g., when symbols and written words are connected by the layout).
In this study, we examine how the use of natural language varies, considering the symbolic language in procedural and conceptual aspects of mathematics.
This paper focuses on relationships between vocabulary in mathematical tasks and aspects of reading and solving these tasks. The paper contains a framework that highlights a number of different aspects of word difficulty as well as many issues to consider when planning and implementing empirical studies concerning vocabulary in tasks, where the aspect of common/uncommon words is one important part. The paper also presents an empirical method where corpora are used to investigate issues of vocabulary in mathematical tasks. The results from the empirical study show that there are connections between different types of vocabulary and task difficulty, but that they seem to be mainly an effect of the total number of words in a task.
This empirical study examines how different types of symbols, familiar and unfamiliar, are processed in working memory; phonologically and/or visuo-spatially.
Våren 2004 infördes på försök två olika versioner av de nationella kursproven i matematik på kurserna B, C och D. På varje kurs skilde sig de båda versionerna åt i några av de ingående uppgifterna, medan övriga uppgifter var identiska. Syftet med denna studie är att undersöka om, hur och varför dessa förändringar i matematikuppgifterna påverkar uppgifternas svårighetsgrad.
Resultaten visar att en förändring av de i uppgifterna ingående talen endast i ett fåtal fall påverkat uppgifternas svårighetsgrad i någon större utsträckning. Dessa fåtal fall studeras vart och ett för sig. När två uppgifters sammanhang och formulering skiljer sig åt, även om det matematiska innehållet är i stort sett identiskt, visar ett exempel på att skillnaden i svårighetsgrad kan vara mycket stor.
The mathematics PISA tasks are primarily supposed to measure mathematical ability and not reading ability, so it is important to avoid unnecessary demands of reading ability in the tasks. Many readability formulas are using both word length and sentence length as indicators of text difficulty. In this study, we examine differences and similarities between English, German, and Swedish mathematics PISA tasks regarding word length and sentence length. We analyze 146 mathematics PISA tasks from 2000–2013, in English, German, and Swedish. For each task we create measures of mean word and sentence length. To analyze if there are any differences between the three language versions of the tasks, we use t-tests to compare the three languages pairwise. We found that in average, the German versions have the longest words, followed by Swedish and then English. Average sentence length was highest for English, followed by German and then Swedish.
The purpose of this study is to deepen the understanding of the relation between the language used in mathematics tasks and the difficulty in reading and solving the tasks. We examine issues of language both through linguistic features of tasks (word length, sentence length, task length, and information density) and through different natural languages used to formulate the tasks (English, German, and Swedish). Analyses of 83 PISA mathematics tasks reveal that tasks in German, when compared with English and Swedish, show stronger connections between the examined linguistic features of tasks and difficulty in reading and solving the tasks. We discuss if and how this result can be explained by general differences between the three languages.
In this paper we suggest a theoretical model of the connection between the process of reading and the process of solving mathematical tasks. The model takes into consideration different types of previous research about the relationship between reading and solving mathematical tasks, including research about traits of mathematical tasks (a linguistic perspective), about the reading process (a psychological perspective), and about behavior and reasoning when solving tasks (a mathematics education perspective). In contrast to other models, our model is not linear but cyclic, and considers behavior such as re-reading the task.
In this study we analyse the communication competence included in two different frameworks of mathematical knowledge. The main purpose is to find out if mathematical communication is primarily described as communication of or about mathematics or if it is (also) described as a special type of communication. The results show that aspects of mathematics are mostly included as the content of communication in the frameworks but the use of different forms of representation is highlighted both in the frameworks and also in prior research as a potential cause for characterising mathematical communication differently than "ordinary" communication.
Det språk vi använder oss av i matematikklassrummet kan fokuseras på många olika sätt. Språket är också nödvändigt att förhålla sig till vid utvecklingen av sitt matematiska tänkande. Författarna diskuterar här relationer mellan språk och lärande.
There is much research on the role of theory in mathematics education research, at least from more overarching or theoretical perspectives. Micro analyses of the role of theory in particular research studies are rarer. We contribute by analysing one empirical study to allow for in-depth analyses and discussions around the role of theory in a specific case, concerning relationships between mathematics and reading. Our results show that studies that do not use an explicit theoretical model can still be strongly influenced by implicit theoretical assumptions. We conclude that it is important to identify existing theoretical assumptions in an empirical research study and try to convey them as clearly as possible, and we discuss specific issues concerning research on relationships between mathematics and reading.
This study is an ongoing project where we only have very preliminary results, and the main aim of this paper is not to present results, but to discuss the methodological issues when using content analysis on the course syllabus for the Swedish compulsory school. Content analysis (Krippendorff, 2004) is in most cases in the literature used on large materials, for example all news articles from a specific year that concern a certain political issue. We will discuss what happens when we try to use the method on a comparatively small document of a very different kind.
The aim of this paper is to discuss what a “strong presence” of a message in a syllabuscould be. The discussion takes a starting point in what we call the reformmessage; that what mathematics is can not only be described in terms of content andprocedures, but must also be defined in terms of competencies, e.g. problem solving,reasoning and communication. The analyzed document is the Swedish syllabus forthe first course at upper secondary school. Different ways, both quantitative andqualitative, of determining what a strong presence of a message could be are presentedand discussed.
This study investigates the impact of a national reform in Sweden introducing mathematical competency goals. Data were gathered through interviews, classroom observations, and online surveys with nearly 200 teachers. Contrasting to most studies of this size, qualitative analyses were conducted. The results show that teachers are positive to the message, but the combination of using national curriculum documents and national tests to convey the reform message has not been sufficient for teachers to identify the meaning of the message. Thus, the teachers have not acquired the functional knowledge of the competence message required to modify their teaching in alignment with the reform. The results indicate that for complex reform messages, such as the competency message, to have intended impact on classroom practice, special attention needs to be put on the clarity of the message. To have high-stakes tests, for example, does not alone seem to be sufficient.
When using a dynamic geometry environment, students’ mathematical communication may becomemore dynamic, shown by adverbs and verbs indicating activity or change. In this paper, three examples of students’ answers when using a DGE template are analysed through Duval’s (2017) semiotic register approach as well as the concept of dynamic mathematical communication. Results exemplify how students’ mathematical communication when using DGE may have a dynamic character (using words such as ‘drag’). Results also indicate that coordinating representations across four different registers is challenging, and students may focus on only performing treatments in one register. Furthermore, the students’ insightful readiness to communicate mathematically may be challenged in DGE settings.
This study reports on the relation between commonness of the vocabulary used in mathematics tasks and aspects of students’ reading and solving of the tasks. The vocabulary in PISA tasks is analyzed according to how common the words are in a mathematical and an everyday context. The study examines correlations between different aspects of task difficulty and the presence of different types of uncommon vocabulary. The results show that the amount of words that are uncommon in both contexts are most important in relation to the reading and solving of the tasks. These words are not connected to the solution frequency of the task but to the demand of reading ability when solving the task.
This study aims to construct a framework of linguistic properties of mathematical tasks that can be used to compare versions of mathematics test tasks in different natural languages. The framework will be useful when trying to explain statistical differences between different language versions of mathematical tasks, for example, differences in item functioning (DIF) that are due to inherent properties of different languages. Earlier research suggests that different languages might have different inherent properties when it comes to expressing mathematics. We have begun with a list of linguistic properties for which there are indications that they might affect the difficulty of a task. We are conducting a structured literature review looking for evidence of connections between linguistic properties and difficulty. The framework should include information about each property including methods used to measure the property, empirical and/or theoretical connections to aspects of difficulty, and relevance for mathematical tasks.
Curricula in many countries include mathematical reasoning as an aim, a competence or proficiency that students should acquire. This inclusion has been supported by wide dissemination of frameworks advocating reform that have arisen from the research community. We present the first part of a project aiming to investigate how ideas about reasoning originating in these frameworks are recontextualised in curricula, textbooks and classrooms. We analyse discourses about reasoning in three such frameworks, identifying how each characterises the nature of mathematical reasoning and the ways students are expected to relate to it. We also examine the extent to which reasoning is construed as a goal of mathematics education or as a means to achieving other goals. In this paper, we explain the methods used for analysing reasoning discourse and identify key findings from the analysis.
Argumentation is a key skill in most school subjects and academic disciplines, including science and mathematics. This study compares explicit argumentation in first-semester university textbooks in biology, chemistry and mathematics in order to increase the understanding of how similarities and differences between disciplines can contribute to, or disrupt, students’ transferrable argumentation skills. Results show that there is significantly more explicit argumentation in the mathematics textbook compared to the biology and chemistry textbooks, and signifycantly more explicit argumentation in the chemistry textbook compared to the biology textbook. Further, the biology textbook contains less argumentation marked by classical argumentative markers such as “since”and “because” and more marked with other, less clear, types of markers such as “which is why” and “when” compared to the other two textbooks. The mathematics textbook contains more complex (recursive) argumentation than the science books. Thereby, the subject-specific languages inthe disciplines have potential to offer students different examples of argumentation. The results will be discussed in relation to students’ development of scientific literacy.
When mathematics tasks are used in multilanguage assessments, it is necessary that the task versions in the different languages are equivalent. The purpose of this study is to deepen the knowledge on different aspects of equivalence for mathematics tasks in multilanguage assessment. We analyze mathematics tasks from PISA 2012 given to students in English, German and Swedish. To measure formal equivalence, we examine three linguistic features of the task texts and compare between language versions. To measure functional equivalence, a Differential item functioning (DIF) analysis is conducted. In addition, we examine statistically if there is a relation between DIF and the differences regarding linguistic features. The results show that there is both DIF and differences regarding the linguistic features between different language versions for several PISA tasks. However, we found no statistical relation between the two phenomena.
When mathematics tasks are translated to different languages, there is a risk that the different language versions are not equivalent and display differential item functioning (DIF). In this study, we aimed to identify possible sources of DIF. We investigated whether differences in some linguistic features are related to DIF between the English (USA), German, and Swedish versions of mathematics tasks of the PISA 2012 assessment. The linguistic features chosen in this study are grammatical person, voice (active/passive), and sentence structure. We analyzed the three different language versions of 83 mathematics PISA tasks in three steps. First, we calculated the amount of differences in the three linguistic features between the language versions. Then, we calculated DIF, using the Mantel-Haenszel procedure pairwise for two language versions at a time. Finally, we searched for correlations between the amount of linguistic differences and DIF between the versions. The analysis showed that differences in linguistic features occurred between the language versions – differences in voice were most common – and that several items displayed intermediate or large level of DIF. Still, there were no statistical significant correlations between differences in linguistic features and DIF between the language versions, that is, there must be other sources of DIF.
When the results of international comparative studies such as PISA or TIMSS get published, they are discussed broadly in media and are used to influence politics and public opinion. To solve mathematics PISA tasks, students have to read and understand the task text. Still, since the mathematics tasks are primarily supposed to measure mathematical ability and not reading ability, it is important to avoid unnecessary demands of reading ability in the tasks. In addition, the different language versions of a task used in PISA might vary in reading difficulty. Such differences can result in differential item functioning (DIF), that is, that students with the same mathematical ability but from different countries have a different probability of answering the item correctly. One reason for DIF between language versions is that linguistic features can differ between language versions. In this study we focus on four different linguistic features that in earlier studies have shown connections to the difficulty of solving mathematics tasks (e.g., Abedi, Lord, & Plummer, 1997).
This study is part of a larger project examining the relation between the language used in mathematics tasks and both the tasks’ difficulty and demand of reading ability. The research questions in this study are: Which differences in the four linguistic features investigated occur between PISA tasks in English, German, and Swedish? Which of these differences are related to DIF between the task versions? The English (USA), German, and Swedish language versions of 83 mathematics tasks of the PISA 2012 assessment are analyzed. The first step of the analysis was to search for differences in the four linguistic features between the different language versions of the tasks. The next steps will be quantitative analyses of the differences, a statistical analysis to detect DIF between the versions, and then statistical analyses to investigate possible relations between the differences and DIF. The first step showed that some differences occur sporadically, for example, the use of third person (he/she/it) in one language version and second person (you) in another language version. Other differences occur much more frequently. For example, differences in word order are quite common, in particular since the finite verb always is at the last position in subordinate clauses in German but not in English and Swedish. The next steps of the analysis are at present (January 2017) ongoing.
A reader of mathematical text must often switch between reading mathematical symbols and reading words. In this study, five different categories of structural connections between symbols and language, which invite such switches, are presented in a framework. The framework was applied in a study of Swedish mathematics textbooks, where 180 randomly selected pages from different educational stages were analyzed. The results showed a significant change in communication patterns as students progress through school. From a predomination of connections based on proximity found in year two, there is a gradual change to a predomination of symbols interwoven in sentences in year eight. Furthermore, more qualitative investigations of the different connections complemented the quantification, both through further explanations of the quantitative results, and through more examples of differences in communication patterns. The implications for readers of mathematics texts are discussed.
In this paper we examine four statistical methods used for characterizing mathematical test items regarding their demands of reading ability. These methods rely on data of students' performance on test items regarding mathematics and reading and include the use of regression analysis, factor analysis and different uses of correlation coefficients. Our investigation of these methods focuses on aspects of validity and reliability, using data from PISA 2003 and 2006. The results show that the method using factor analysis has the best properties when taking into account aspects of both validity and reliability.
This study surveys claims in research articles regarding linguistic properties of mathematical texts, focusing on claims supported by empirical or logical arguments. It also performs a linguistic analysis to determine whether some of these claims are valid for school textbooks in mathematics and history. The result of the survey shows many and varying claims that mainly describe mathematical texts as highly compact, precise, complex, and containing technical vocabulary. However, very few studies present empirical support for their claims, and the few empirical studies that do exist contradict the most common, and unsupported, claims, since no empirical study has shown mathematical texts to be more complex than texts from other subjects, and any significant differences rather indicate the opposite. The linguistic analysis in this study is in line with previous empirical studies and stands in contrast to the more common opinion in the unsupported claims. For example, the mathematics textbooks have significantly shorter sentences than the history textbooks.
In this study we utilize results from Swedish students in PISA 2003 and 2006 to examine what types of task properties predict the demand of reading ability of a task. In particular, readability properties (sentence length, word length, common words, and information density) and task type properties (content, competence, and format) are examined. The results show that it is primarily readability properties of a task that predict the task’s demand of reading ability, in particular word length and to some extent information density (measured through the noun-verb quotient).
The purpose of this study is to contribute to the methodology of research on difficult vocabulary in mathematics tasks. The contribution consists of a framework for the study of difficult vocabulary in mathematics tasks and a literature review of empirical research in the area. The framework includes five main aspects of word difficulty that have been examined in empirical studies and discuss these in the light of theories on reading comprehension. In addition, methodological issues are presented in relation to each main aspect. The literature review examines both methodological aspects of 36 reviewed articles, and synthesizes results on difficult vocabulary. The literature review shows that a commonly used method—to study several word aspects together—is very unfortunate from the perspective of building accumulative knowledge about difficult vocabulary in mathematics tasks. The only well-supported conclusion possible to draw from the synthesis of results from the empirical studies, is that some word aspects are not related to task difficulty.