This study investigates how students’ reasoning contributes to their utilization of computer-generated feedback. Sixteen 16-year-old students solved a linear function task designed to present a challenge to them using dynamic software, GeoGebra, for assistance. The data were analysed with respect both to character of reasoning and to the use of feedback generated through activities in GeoGebra. The results showed that students who successfully solved the task were engaged in creative reasoning and used feedback extensively.
After 40 years of ethnomathematics research in Papua New Guinea and policies encouraging ethnomathematics in schools, it was time to look at professional learning for teachers so they could effectively implement the policies. Within a design-based research methodology, we designed a set of interlinked principles, tried them in several workshops for teachers, and revised the principles to take account of needs based on reflexivity and evaluations. We developed a manual to use in the workshops. We are continuing this research through several different phases, moving from direct delivery of the professional learning to teachers in various provinces and ecologies to delivery to trainers who then teach the teachers in three provinces, and finally by technology delivery. Early evaluation data suggest that the key principles showing the importance of culture, language and mathematical thinking in the teaching of early mathematics are sound. Workshops have been well received as teachers inquire into the mathematics of their own cultures. The need for a stronger understanding of early mathematics learning in general has been identified. The use of video of cultural practice and of young children learning to count and investigate has had a significant impact.
In this paper the issue of the authenticity of applied mathematical school tasks is discussed. The paper includes a description of a framework for reflection, analysis and development of authentic tasks. Its possible uses are exemplified by two studies. One of the studies is an analysis of in what way and to what extent the applied tasks included in the national assessments in Finland and Sweden are, or are not, authentic. In the second example the framework is used in a study of the impact of authenticity on the students' sense making in word problem solving.
Mathematical communication, oral and written, is generally regarded as an important aspect of mathematics and mathematics education. This implies that oral mathematical communication also should play a part in various kinds of assessments. But oral assessments of subject matter knowledge or communication abilities, in education and elsewhere, often display reliability problems, which render difficulties with their use. In mathematics education, research about the reliability of oral assessments is comparably uncommon and this lack of research is particularly striking when it comes to the assessment of mathematical communication abilities. This study analyses the interrater reliability of the assessment of oral mathematical communication in a Swedish national test for upper secondary level. The results show that the assessment does suffer from interrater reliability problems. In addition, the difficulties to assess this construct reliably do not seem to mainly come from the communication aspect in itself, but from insufficiencies in the model employed to assess the construct.
This paper outlines the research program for the formative assessment group at Umeå Mathematics Education Research Centre. The program was presented in a symposium at the conference, and focuses on the study of the development and impact of formative assessment. The main purpose of the research carried out by the research group is to provide research results that will be used outside the research community for educational decisions on systemic level, or as support for improved teaching and learning at classroom level. The paper outlines the fundamental ideas of the program, current studies, and examples of completed studies.
Research reviews show that formative assessment has great potential for raising student achievement in general, but there is a need for reviews of formative assessment in individual subjects. This review examines its impact on student achievement in mathematics through an assessment of scientific journal articles published between 2005 and 2014 and indexed in Web of science. Through the use of search terms such as ”formative assessment”, ”assessment for learning” and ”self-regulated learning”, different approaches to formative assessment were included in the review. While varying in approach, they all share the defining characteristic of formative assessment: agents in the classroom collect evidence of student learning and, based on this information, adjust their teaching and/or learning. The results show positive relations between student achievement in mathematics and the ways of doing formative assessment included in the review.
We investigate the mathematical reasoning required to solve the tasks in the Swedish national tests and a random selection of Swedish teacher-made tests. The results show that only a small proportion of the tasks in the teacher-made tests require the students to produce new reasoning and to consider the intrinsic mathematical properties involved in the tasks. In contrast, the national tests include a large proportion of tasks for which memorization of facts and procedures are not sufficient. The conditions and constraints under which the test development takes place indicate some of the reasons for this discrepancy and difference in alignment with the reform documents.
This study focuses on gender differences in the extent to which students take real-world considerations into account when working with word problems in mathematics. Previous studies have found that students have a tendency to neglect an appropriate use of real-world knowledge in their word-problem solving, which leads to solutions that are inconsistent with the ‘real' situations described in the tasks. Research has also shown that the authenticity of the word problems can influence students' activation of their knowledge of the ‘real' situations described in the tasks, as well as their use of this real-world knowledge in order to provide ‘realistic' solutions. The study reported here investigates whether there are gender differences in the students' activation and use of real-world knowledge when working with word problems in mathematics. In addition, it investigates whether task authenticity influences boys and girls differently with respect to these real-world considerations. The results show that even though some of the tasks used in the study affected boys and girls differently, across all tasks and students no evidence of gender differences with respect to real-world considerations were found.
Error analysis is a basic and important task for mathematics teachers. Unfortunately, in the present literature there is a lack of detailed understanding about teacher knowledge as used in it. Based on a synthesis of the literature in error analysis, a framework for prescribing and assessing mathematics teacher knowledge in error analysis was formulated. The major constructs incorporated in this framework were the nature of mathematical error and the phrase of error analysis. The framework was validated through analysis of teachers' documents by two empirical examples.
This study aims to examine values in effective mathematics lessons in Sweden from the perspectives of students in different groups and their teachers. By using methods with lesson observations, student focus group interviews and teacher interviews, it shows that instructional explanation and classroom atmosphere with quietness are shared-values of students and their teachers. The findings propose some crucial issues which related to how mathematics teaching could be adjusted to different students’ learning conditions and whether it needs more instructional explanation in mathematics teaching in Sweden.
This paper looks at proof production in the midst of classroom interaction. The setting is a collegelevel geometry course in which students are working on the following task: Prove that two paralleltransported lines in the plane are parallel in the sense that they do not intersect. A proof of this statement istraced from a student's idea, through a small group discussion, to a large class discussion moderated by ateacher. As the proof emerges through a series of increasingly public settings we see ways in which the keyidea of the proof serves to both open and close class discussion. We look at several examples of openingand closing, showing how not only the key idea, but the warrants and justifications connected to it, play animportant role in the proof development.
Beauty, which plays a central role in the practice of mathematics, is almost absent in discussions of school mathematics. This is problematic, because students will decide whether or not to continue their studies inmathematics without having an accurate picture of what the subject is about. In order to have a discussion about how to introduce beauty into the school mathematicscurriculum, we need to have a clear idea about what beauty means. That is the aim ofthis study, with a focus on characterising beauty in mathematical proof.
We present two different proofs of Pick’s theorem and analyse in what ways might be perceived as beautiful by working mathematicians. In particular, we discuss two concepts, generality and specificity, that appear to contribute to beauty in different ways. We also discuss possible implications on insight into the nature of beauty in mathematics, and how the teaching of mathematics could be impacted, especially in countries in which discussions of beauty and aesthetics are notably absent from curricular documents.
Compactness is a central notion in advanced mathematics, but we often teach the concept without much historical motivation. This paper fills in many of the gaps left by the standard textbook treatment, including what motivated the definition, how did the definition evolve, and how can compactness be expressed in terms of nets and filters.
This paper poses two questions:
These questions are difficult, and subjective, and will not be settled in this paper, but the purpose in raising them is to discuss if we agree that the answer to (1) is yes, then how can we make progress on question (2).
This paper presents a cross-cultural analysis of how authors of elementary mathematics curriculum materials communicate with teachers and what they communicate about, focusing on six teacher’s guides from three distinct school systems, Flanders, U.S. and Sweden. Findings revealed distinct differences between approaches common to each cultural context that relate to different educational traditions. These findings point to differing assumptions about the knowledge needed by teachers to enact instruction. Further research is needed to explore these patterns qualitatively and consider teachers’ use of these materials when planning and enacting instruction.
We investigated the factorial structure of the Perceived Stress Questionnaire (PSQ-recent; Levenstein, Prantera, Varvo et al., 1993) in a large (N = 1516; 35-95 years) population-based Swedish sample (Nilsson, Adolfsson, Backman et al., 2004; Nilsson, Backman, Erngrund et al., 1997). Exploratory principal components analysis (PCA) was conducted on a first, randomly drawn subsample (n = 506). Next, the model based on the PCA was tested in a second sample (n = 505). Finally, a third sample (n = 505) was used to cross-validate the model. Five components were extracted in the PCA (eigenvalue > 1) and labeled "Demands," "Worries/Tension," " Lack of joy," " Conflict," and " Fatigue," respectively. Twenty-one out of the 30 original PSQ items were retained in a confirmatory factor analysis (CFA) model that included the five (first-order) factors and, additionally, a general (second-order) stress factor, not considered in prior models. The model showed reasonable goodness of fit [chi(2)(184) = 511.2, p < 0.001; CFI = 0.904; RMSEA = 0.059; and SRMR = 0.063]. Multigroup confirmatory factor analyses supported the validity of the established model. The results are discussed in relation to prior investigations of the factorial structure of the PSQ.
This paper discusses the Vygotskian construct of social situation of development in relation to two students we call Nelly and Brian, as they learn mathematics in respective grade four classrooms in Sweden and USA. Nested within larger action research projects, both students are found to exhibit developmental crisis, as they transit from familiar environments in primary grades and meet newer demands at middle school. The construct of social situation of development, treated herein as an exploratory framework, is understood in terms of instructional environments of students, their emotional experience of these, leading activities they participate in, social positions expected of them and internal positions with respect to the same. Examining such a framework enables us to shed light on the very nature of occurrence and ways with which to ameliorate periods of crisis, which students seem to routinely face while learning mathematics at school.
Students only learn what they get the opportunity to learn. This means, for example, that students do not develop their reasoning- and problem solving competence unless teaching especially focuses on developing these competencies. Despite the fact that it has for the last 20 years been pointed out the need for a reform-oriented mathematics education, research still shows that in Sweden, as well as internationally, an over-emphasis are placed on rote learning and procedures, at the cost of promoting conceptual understanding. Mathematical understanding can be separated into procedural and conceptual understanding, where conceptual understanding can be connected to a reform oriented mathematics education. By developing a reasoning competence conceptual understanding can also be developed. This thesis, which deals with students’ opportunities to learn to reason mathematically, includes three studies (with data from Swedish upper secondary school, year ten and mathematics textbooks from twelve countries). These opportunities have been studied based on a textbook analysis and by studying students' work with textbook tasks during normal classroom work. Students’ opportunities to learn to reason mathematically have also been studied by examining the relationship between students' reasoning and their beliefs. An analytical framework (Lithner, 2008) has been used to categorise and analyse reasoning used in solving tasks and required to solve tasks.Results support previous research in that teaching and mathematics textbooks are not necessarily in harmony with reform-oriented mathematics teaching. And that students indicated beliefs of insecurity, personal- and subject expectations as well as intrinsic- and extrinsic motivation connects to not using mathematical reasoning when solving non-routine tasks. Most commonly students used other strategies than mathematical reasoning when solving textbook tasks. One common way to solve tasks was to be guided, in particular by another student. The results also showed that the students primarily worked with the simpler tasks in the textbook. These simpler tasks required mathematical reasoning more rarely than the more difficult tasks. The results also showed a negative relationship between a belief of insecurity and the use of mathematical reasoning. Furthermore, the results show that the distributions of tasks that require mathematical reasoning are relatively similar in the examined textbooks across five continents.Based on the results it is argued for a teaching based on sociomathematical norms that leads to an inquiry based teaching and textbooks that are more in harmony with a reform-oriented mathematics education. Elever kan bara lära sig de det de får möjlighet att lära sig. Detta innebär till exempel att elever inte utvecklar sin resonemangs- och problemlösningsförmåga i någon större utsträckning om inte deras undervisning fokuserar på just dessa förmågor. Forskning, nationellt och internationellt visar att det finns en överbetoning på utantillinlärning och på procedurer. Detta verkar ske på bekostnad av en konceptuell förståelse, trots att det under 20 års tid pekats på behovet av en reforminriktad matematikundervisning. Matematisk förståelse kan delas in i procedurell- och konceptuell förståelse där en konceptuell förståelse kan kopplas till en reforminriktad matematikundervisning. Genom att utveckla förmågan att resonera matematiskt utvecklas också den konceptuella förståelsen. Denna avhandling, som inbegriper tre studier (med empiri från gymnasiet år ett och matematikläroböcker från tolv länder) behandlar elevers möjlighet att lära sig att resonera matematiskt. Dessa möjligheter har studerats utifrån att undersöka vilka möjligheter läroboken ger att lära sig matematiska resonemang, dels via en läroboksanalys och dels genom att studera elevers arbete med läroboksuppgifter i klassrumsmiljö. Elevers möjligheter att lära sig att resonera matematiskt har också studerats genom att undersöka relationen mellan elevers matematiska resonemang och deras uppfattningar om matematik. Ett analytiskt ramverk (Lithner, 2008) har används för att kategorisera och analysera resonemang som använts för att lösa uppgifter och som behövs för att lösa en uppgift.Resultaten från studierna har givit stöd åt tidigare forskning vad gäller att undervisning och läroböckerna inte nödvändigtvis harmonierar med en reforminriktad matematikundervisning. Och att elever har uppfattningar om matematik som bygger på osäkerhet, förväntan på ämnet och sin egen förmåga samt motivation och att dessa uppfattningar delvis kan kopplas till att eleverna inte använder matematiska resonemang för att försöka lösa icke-rutinuppgifter. Det vanligaste sättet att lösa läroboksuppgifter var att välja andra strategier än att använda sig av matematiska resonemang. Ett vanligt sätt att lösa uppgifter var att låta sig guidas, av främst en annan elev. Eleverna arbetade framförallt med de enklare uppgifterna i läroböckerna. Bland dessa enklare uppgifter var det mer sällsynt med uppgifter som krävde matematiska resonemang för att lösas relativt de svårare uppgifterna. Resultaten visade även att det fanns en negativ relation mellan en uppfattning av osäkerhet hos elever och ett användande av matematiska resonemang. Resultaten visade vidare att fördelningen av uppgifter som krävde matematiska resonemang var relativt lika i alla undersökta läroböcker från fem världsdelar.Utifrån resultaten argumenteras för en förändrad undervisning mot en undersökande undervisning och läroböcker som är mer i harmoni med en reforminriktad matematikundervisning.
To characterize teaching designs intended to enhance students’ problem solving and reasoning skills or to develop other mathematical competencies via problem solving and reasoning, a literature review was conducted of 26 articles published in seven top-ranked journals on mathematics education from 2000 to 2016. Teaching designs were characterized by a) the educational goals of the designs, b) the claims about how to reach these goals, and c) the empirical and theoretical arguments underlying these claims. Thematic analysis was used to analyze the retrieved articles. All but two studies had goals concerned with developing students’ mathematical competencies. The overarching ideas of the identified emergent claims regarding the achievement of stipulated goals, concerned scaffolding students’ learning and letting students construct their own mathematics. Four recurring theoretical arguments were found to support emergent claims: hypothetical learning trajectories, realistic mathematics education, theory of didactical situations and zone of proximal development.
In mathematics education, there is generally too much emphasis on rote learning and superficial reasoning. If learning is mostly done by rote and imitation, important mathematical competencies such as problem-solving, reasoning, and conceptual understanding are not developed. Previous research has shown that students who work with problems (i.e. constructs a new solution method to a task), to a greater extent increase their mathematical understanding than students who only solve routine tasks.
The aim of the thesis was to further understand why teaching is dominated by rote learning and imitation of procedures and investigate how opportunities for students to solve tasks through problem-solving could be improved. This was done through the following studies. (1) Investigating the relation between types of solution strategy required, used, and the rate of correct task solutions in students’ textbook task-solving. (2) Studying the relationship between students’ beliefs and choice of solution strategy when working on problems. (3) Conducting a textbook analysis of mathematics textbooks from 12 countries, to determine the proportions of tasks that could be solved by mimicking available templates and of tasks where a solution had to be constructed without guidance from the textbook. (4) Conducting a literature review in order to characterize teaching designs intended to enhance students to develop mathematical understanding through problem solving and reasoning. (5) Conducting an intervention study were a teacher guide, structured in line with central tenets of formative assessment, was developed, tested, and evaluated in real classroom settings. The teacher guide was designed to support teachers in their support of students’ in their problem-solving process. Studies I, II and V were conducted in Swedish upper secondary school settings.
The students’ opportunities to solve tasks through problem-solving were limited: by the low proportion of problems among the easier tasks in the textbooks; by the students' choice of using imitative solution strategies; and by the guidance of solution methods that students received from other students and their teachers. The students’ opportunities were also limited by the students' beliefs of mathematics and the fact that a solution method of problem tasks was not always within reach for the students, based on the students' knowledge. In order to improve students’ opportunities, teachers should allow students to work with more problems in a learning environment that lets students engage in problem-solving and support students' work on problems by adapting their support to students' difficulties. The results also give implications for the construction and use of textbooks and how the use of the teacher guide could be part of teachers’ professional development and a tool that teacher students may meet within their education.
To a large extent school mathematics consists of procedures to be memorized without understanding. Using procedures without understanding is one of the main causes behind the difficulties when learning mathematics. By developing mathematically founded reasoning, other mathematical competences are also developed; problem solving and conceptual understanding. The textbook is important in mathematical education. This study reports an analysis of students’ textbook task solving in Swedish upper secondary school were the relation between types of mathematical reasoning used and the success and failure to complete tasks was studied. Results show that using rote learning and superficial reasoning is common, although only about 50 % of the tasks where successfully solved using such strategies. In the case that mathematically founded reasoning was used, the tasks were successfully solved. Conclusions from the study are that students’ work with textbook tasks mainly let the students develop rote learning and superficial reasoning and students are not enhanced to develop mathematically founded reasoning in their work with textbook tasks. Students have to work with textbook tasks in other ways or be given other additional opportunities in order to develop mathematically founded reasoning.
The purpose of this intervention study was to develop, test, and evaluate a teacher guide structured in line with central tenets of formative assessment in a real classroom setting. The teacher-guide was designed to support teachers’ diagnosis of student difficulties and their choice of feedback to help students to continue the construction of solution methods during problem- solving if they become stuck. By using an approach inspired by design research, five teachers used the teacher guide for two plus two weeks in 12 mathematics courses in upper secondary school with revisions of the teacher guide in between the iterations. Ninety-six teacher-student interactions were observed, and teacher interviews were conducted. The results showed that the teacher guide supported the teachers in providing less algorithmic information and instead focusing on the problem-solving process, and by that helping the students to themselves construct solutions during their problem-solving activity. The use of the teacher guide was sometimes constrained by the type of tasks the students were working on, by difficulties in making reasonable diagnoses of students’ difficulties, and by students’ insufficient ability and/or willingness to communicate.
This study reports on an analysis of students' textbook task-solving in Swedish upper secondary school. The relation between types of mathematical reasoning required, used, and the rate of correct task solutions were studied. Rote learning and superficial reasoning were common, and 80% of all attempted tasks were correctly solved using such imitative strategies. In the few cases where mathematically founded reasoning was used, all tasks were correctly solved. The study suggests that student collaboration and dialogue does not automatically lead to mathematically founded reasoning and deeper learning. In particular, in the often common case where the student simply copies a solution from another student without receiving or asking for mathematical justification, it may even be a disadvantage for learning to collaborate. The results also show that textbooks' worked examples and theory sections are not used as an aid by the student in task-solving.
This paper reports on a research project relating to the newly implemented mandatory Swedish national mathematics tests for third-grade students (aged nine and ten). The project’s main research concerns the students’ ideas about and reactions towards these tests and how the specific test situation affects their perception of their own mathematical proficiency. Drawing on theories which suggest that identities are more fluid than static, we want to understand how students with special needs are ‘created’. The specific aim of this paper is to discuss how our research methods have been refined during the various phases of data collection and the resulting implications. It discusses issues surrounding child research and how methods involving video recording and video stimulated recall dialogue (VSRD) can contribute to research on children’s experiences. Particular attention is given to methodological and ethical issues; how to disrupt power relations, for example. In this paper we argue that the context of the test situation not only impacted upon the students but also affected how we changed, developed and adapted our approaches as the project evolved.
Practicing mathematics is not possible without the use of language. To communicate mathematical content, not only words in natural language are used but also non-verbal forms of communication such as mathematical symbols, graphs, and diagrams. All these forms of communication can be seen as part of the language used when doing mathematics. When mathematics tasks are used to assess mathematical competence, it is important to know how language can affect students’ possibility to solve the task. In this thesis, two different but related aspects of the relation between language and mathematics tasks are investigated. The first aspect concerns linguistic features of written mathematics task that can make the task more difficult to read and/or to solve. These features may result in unnecessary and unwanted reading demands, that is, the task then partially assesses students’ reading ability instead of their mathematical ability. The second aspect concerns differences between different language versions of mathematics tasks used in multilanguage assessments. These differences may cause inequivalence between the language versions, that is, the task may be more difficult to solve for students of one language group than students of another. Therefore, the purpose of this thesis is to investigate some of the effects that language can have on written mathematics tasks, in particular, on the validity of mathematics assessments. The thesis focuses on unnecessary reading demands and inequivalence in multilanguage assessments. The data in this thesis are obtained from tasks of the Programme for International Student Assessment (PISA) 2012. The task texts and the student results on these tasks are analyzed quantitatively to identify the occurrence and possible sources of unnecessary reading demands and inequivalence. Think-aloud-protocols and task-based interviews of students who had worked with some of the tasks, serve to qualitatively identify possible sources of reading demands and inequivalence, respectively.
The results showed both unnecessary reading demands and inequivalence in some of the tasks. Some linguistic features were identified as possible sources of these reading demands, while others were not related to them. For example, sentence length was not related to reading demands of tasks in Swedish, whereas sentence structure was identified as a possible source of unnecessary reading demands. Some linguistic differences between different language versions of mathematics tasks were also identified as possible sources of inequivalence, and in addition there were curricular differences that were such potential sources. The findings of this thesis have implications for designing mathematics tasks both in one language and in multilingual settings. They may help to ensure validity of mathematics assessments, but also to make mathematics texts easier to understand for students in general.
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.