The potential of using formative assessment is well demonstrated, but studies about the use of formative assessment from a special education perspective are lacking. This study adds to this gap by investigating the view of formative assessment in a group of 39 special education teachers in mathematics (SETMs) who had learned about formative assessment within the SETM-program 2–6 years earlier. Five respondent interviews were used to design a questionnaire answered by the rest of the group. The SETMs had perceived formative assessment beneficial and useful in all their common sub-responsibilities and reported experiences of benefits as well as challenges. The article discusses the importance of reaching an inclusive formative assessment practice in mathematics education.
Research shows that substantial learning gains are possible through the use of formative assessment. However, little is known about Swedish mathematics teachers’ use of formative assessment, and thus about the possible value of professional development programmes. This study uses teacher interviews and classroom observations to examine the classroom practice of 38 randomly selected primary and secondary school teachers in a mid-sized Swedish municipality. A framework of formative assessment comprising one big idea and five Key strategies structured the analysis. The study identifies characteristics of current formative assessment practices. The results show that the teachers do use a variety of formative assessment activities, but also that there is much room for development towards a more effective formative classroom practice.
Artikeln beskriver ett försök där gymnasieelever får undersöka faktorisering av andragradspolynom med hjälp av grafiska representationer av funktioner. Eleverna leds in i ett för dem nytt arbetssätt, där de tillsammans med en lärare får arbeta med ett antal uppgifter med hjälp av en grafräknare. Resultaten visar att eleverna kommer med egna hypoteser och använder grafräknaren på eget initiativ i vissa situationer. Resultaten visar också att eleverna i försöket i viss mån kunde använda grafräknaren i ett undersökande arbetssätt.
This study is based on a framework of algorithmic and creative mathematical rea- soning and focuses on students’ strategy choices in both practice and test. Previous research indicates that students that practice mathematics with tasks with given solution methods are outperformed in later test by students that have to construct solution methods during practice. Video recordings, students’ written solutions, and student interviews from ten university students provides data on strategy choices. The analysis was carried out to capture students’ strategy choices and reasons for these choices. The results showed that there was no real difference in how the stu- dents solved the tasks in the test. Regardless of practice condition, more or less the same solution strategies were used in the test situation.
This study concerns mediating activities in student discussions during collaborative work with self-explanation prompts (SEPs). While the aim of most other tasks, from the students’ perspective, can be perceived as finding the correct answer, discussions supported by SEPs require a different approach, because students must engage in mathematical discussions, and explain their insights into the mathematics at hand. In this study, we explore activities that are fostered by SEPs. The analysis of the activities taking place, reveal five mediating activities to promote in teaching, but also potential hinders for the intended discussion to occur.
One part of being proficient in mathematics is to be able to read and solve mathematics tasks where mathematics is represented using different semiotic resources (i.e. natural language, mathematical notation, and different types of images). In the current study, statistical methods are used to investigate the potential meaning that the presence and co-occurrences of semiotic resources have for how demanding a mathematical task is to read and solve. The results reveal that the number of different semiotic resources in a mathematical task is not related to difficulty, but that difficulty is related to the particular combinations of semiotic resources where pictorial images are one of the resources. The results also indicate that the difficulty related to these semiotic characteristics is not related to an unnecessary reading demand.
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.
Beyond understanding the Vygotskian construct of zone of proximal development or ZPD with reference to an individual student, this paper explores the formation of ZPD within the pedagogical constructs of cooperation, wherein students cooperate with each other within their groups; as well as collaboration, wherein students from different groups that constitute the classroom collaborate with each other. Identified on the basis of functions that are in the process of maturing, the formation of either ZPD is exemplified from a socio-cultural-historical study at an upper secondary mathematics classroom in Norway. An emphasis on what distinguishes events in instruction that are educational from those that are not is also explored, before illustrating what nature of ZPD is established. The role of guidance received, imitation and cultural resources in the development of higher mental functions is understood as these ZPD are formed, enabling students to act independently within the classroom teaching-learning of mathematics.
Mathematics textbooks are teaching tools used by most students studying mathematics worldwide. In this descriptive textbook analysis, all Swedish mathematics textbooks for Year 1, both digital and printed, were mapped out according to the resources used for communication. For delimitation, a focus on subtraction as an arithmetic operation was chosen. The result shows large differences between the 17 textbook series, concerning both the type of subtraction exercises offered and the use of different resources for communication and learning, such as writing, images, and mathematical symbols. Digital textbooks were largely similar to the printed textbooks, except for one tablet-based textbook. Altogether, the study shows that the choice of mathematics textbooks affects how subtraction is presented to students and, by extension, the learning situations students encounter when working with mathematics textbooks.
The present study investigates how to support students’ creative reasoning when they need assistance in solving non-routine tasks. Two groups of 11–12-year-old students solved the same tasks, one group receiving feedback directed at the task solution and the other feedback directed at their thinking processes. The results showed that students who received feedback directed at their thinking processes expressed reasoning based on their attempts to solve tasks while the other group often repeated the researcher’s suggestions for solutions. However, there were some instances in which feedback on task level entailed students engaging in creative reasoning.
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.
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.
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.
This paper focuses on students in need of special education in mathematics (SEM students) and highlights teachers’ and principals’ reflections upon these students’ construction of knowledge in relation to two educational settings: the regular teaching setting and the test setting. The findings indicate that SEM students’ knowledge is legitimized only when displayed. However, there appear to be differences according to the specific setting. Different settings imply different knowledge representations, norms, and practices that need to be taken into account when reflecting, planning, and carrying out teaching in mathematics in relation to SEM.
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.
This article reports on a Swedish research project on the reintroduction of national tests in mathematics for nine- to ten-year-old pupils. Data were collected over a period of three years (2010-2012) by video recording test situations in different classrooms and by conducting video-stimulated recall interviews with children. The aim is to explore and analyse the testing situation and how it creates different positions for children. We conclude that discourses of testing, caring and competition, sometimes strengthening and sometimes shadowing each other and thereby, produce knowledge in children about success and failure in mathematics, positioning children as ‘winners’ or ‘losers’. The tests are interpreted as a technology – a form of disciplinary power that functions at the level of the body (Foucault, 1980).
The mathematical ideas that emerge in children’s free and guided play can be both complex and sophisticated, and if they are linked to formal mathematics, they can be a powerful basis for mathematical development. To form such links, one needs knowledge of how children use and express these ideas. This is especially true in the intersection of arithmetic and geometry, where the intermingling of numerical and spatial concepts and skills is not yet fully understood. This study aims to gain understanding of children’s mathematical practices by describing the interplay of key mathematical ideas, and more specifically how young children exercise mathematical competencies in the intersection of early arithmetic and geometry. The results show that children can use spatial representations when reasoning about numbers, and that they are able to connect spatial and numerical structures. Furthermore, it is shown that children not only use and invent effective procedures, but also are able to explain, justify and evaluate such procedures.
Implementing teaching through mathematical problem-solving entails substantial challenges and calls for sustained teacher-researcher collaboration. The joint research and development project ”Teaching that supports students’ creative mathematical problem-solving” has a fundamental ambition to be symmetric in that both teachers’ and researchers’ needs and conditions are attended to and complementary in that their different areas of expertise are utilised and valued. In this paper we show how the interplay and development of symmetry and complementarity can function as a means for studying teacher-researcher collaborations.
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.
The main question discussed in this paper is whether students need to learn how to read mathematical texts. I describe and analyze the results from different types of studies about mathematical texts; studies about properties of mathematical texts, about the reading of mathematical tasks, and about the reading of mathematical expository texts. These studies show that students seem to develop special reading strategies for mathematical texts that are not desirable. It has not been possible to find clear evidence for the need of a specific ”mathematical reading ability”. However, there is still a need to focus more on reading in mathematics teaching since students seem to develop the non-desirable reading strategies.
A common problem in belief research seems to be a missing link between aspects of theory and empirical analyses and results. This issue highlights a question of how dependent empirical studies about beliefs actually are on the theoretical perspective described in the study. In this paper, I examine relationships between two different perspectives. One perspective focuses on belief change, and seems to rely on a type of cognitive perspective, where beliefs can be characterized as mental objects. The other perspective argues for moving away from such cognitive perspective and instead to adopt a participatory perspective in the analysis of mathematics teaching. The results show that the study about belief change is not dependent on seeing beliefs as mental objects, but that this study could as well have been located within a participatory perspective.
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.