One of the main problems with learning difficulties in mathematics is that rote-learning becomes the very foundation of mathematics for many students. Procedural as well as conceptual knowledge is needed to build a broad mathematical competence. Students learn only what they get an opportunity to learn, which means that we must consider what opportunities to learn are given to school students. For the purpose of exploring what opportunities are available to learn to reason mathematically, a well established framework is used to analyze the reasoning required by textbook tasks as well as the reasoning used by students. The framework was used to refine the discussion of what type of knowledge is used by the students. Application of the framework distinguishes between creative mathematical reasoning, where a solution has to be created by the student, and imitative reasoning which is based on rote learning or following an existing template. Opportunities to learn depend on the classroom norms that have been negotiated between students and teacher. These norms are influenced by several factors. This thesis deals with the textbook, both as one of several pictures of the education, and in terms of how it is used in the classroom, as well as students’ beliefs about mathematics. There are three studies included in the thesis. In the first study, tasks in mathematics textbooks used in secondary school around the world are analyzed concerning the reasoning requirements. For the second study an analysis of students reasoning during textbook task solving in the classroom has been conducted. In the third study a thematic analysis has been used to explore students’ beliefs about mathematics and relate these beliefs to the reasoning used.

Results from analyzing textbooks from twelve different countries paint a similar picture when it comes to the proportion of tasks requiring students to use creative mathematical reasoning. On average, only every tenth task required creative mathematical reasoning to a greater extent. Furthermore, students in the Swedish upper secondary school level mainly focus on solving the easier, earlier tasks and also mainly use imitative reasoning. Opportunities for students to use creative mathematical reasoning seem limited. When students guide each other during task solving, the main focus seems to be to reach a conclusion in terms of an answer corresponding to that given in the answer-section of the book. Moreover, guidance from a teacher does not seem to lead to anything other than imitation of a procedure. Students also indicate their beliefs by expressing that most tasks should be possible to solve using imitative reasoning, and that therefore, rote learning is a central part of mathematics education. This places pressure on teachers to carefully reflect on what tasks and textbooks they use in their teaching, and also what types of classroom norms they wish to present. The manner in which students work in the classroom also needs consideration, where a greater focus should be directed toward understanding.

Dalarna University, School of Education, Health and Social Studies, Mathematics Education. Umeå universitet, Umeå forskningscentrum för matematikdidaktik (UFM).

Students' ability to develop their mathematical competency is influenced by the tasks they work with. A routine task is a task that a student can solve by using a familiar method, or by imitating a template. In order to solve a mathematical problem however, the student needs to construct a to her new solution method. To develop their mathematical competency, students need to work with routine tasks as well as mathematical problems. A creative problem-solving skill, as well as a conceptual understanding may be developed through problem solving.

The thesis consists of five studies, of which the purpose of studies 1-3 was to explore the opportunities to work with mathematical problem solving offered to students in secondary school. Tasks in textbooks from 12 countries were analyzed (study 1), and approximately 10 percent of these were mathematical problems. The students worked (study 2) almost exclusively with tasks categorized by the textbook authors as easy. Among these tasks, the proportion of mathematical problems was 4 percent. Nor among tasks categorized as 'problem solving' or 'exploring', mathematical problems were predominant. The proportions were relatively similar in textbooks from the twelve countries. Students' beliefs that routine work is more secure and something that is reasonable to expect in mathematics (study 3) can have an additional impact on their opportunites to mathematical problem solving. Given the positive effects of problem solving, students' opportunities to work with problem solving seem limited. There is potential in an increased proportion of mathematical problems in textbooks, as well as in a more deliberate task selection from these textbooks.

The purpose of studies 4 and 5 was to contribute towards a better understanding of mathematical problems and mathematical problem solving. An analytical framework was developed to identify creative, conceptual and other challenges in students' problem solving. Each challenge was characterized to be able to understand and describe these components of problem solving. Students' work with mathematical problems (study 4) and the, by teachers anticipated challenges students face in problem solving (study 5) were studied. Conceptual and creative challenges proved to be the most central to students' problem solving. Through the characteristics of each of the challenges, the relation between task and challenge, and difficulties in identifying, especially the creative challenge, was discussed.

Dalarna University, School of Education, Health and Social Studies, Mathematics Education.

Task design with a focus on conceptual and creative challenges2019In: Proceedings of the Eleventh Congress of the European Society for Research in Mathematics Education (CERME11, February 6 – 10, 2019). / [ed] Jankvist, U. T., Van den Heuvel-Panhuizen, M., & Veldhuis, M., Utrecht, the Netherlands: Freudenthal Group & Freudenthal Institute, Utrecht University and ERME. , 2019, p. 4234-4241Conference paper (Refereed)

Abstract [en]

Tasks are an important part of the education in mathematics. In an ongoing study, an analytic framework for identifying challenges in students mathematical task solving has been developed, and the conceptual and the creative challenge has been defined. Preliminary results indicate that considerations are needed to include these challenges in mathematical tasks. This paper takes off from there to describe a structure for selection and (re)design of tasks. The aim is to be able to discuss the basis for the structure. A further aim is to develop a support for teachers, test designers, textbook authors and others, in creating tasks with specific learning goals.

4.

Jäder, Jonas

et al.

Dalarna University, School of Education, Health and Social Studies, Mathematics Education. Umeå universitet, Umeå forskningscentrum för matematikdidaktik (UFM).

Lithner, Johan

Umeå universitet, Umeå forskningscentrum för matematikdidaktik (UFM).

Sidenvall, Johan

Umeå universitet, Umeå forskningscentrum för matematikdidaktik (UFM).

A selection of secondary school mathematics textbooks from twelve countries on five continents was analysed to better understand the support they might be in teaching and learning mathematical problem solving. Over 5700 tasks were compared to the information provided earlier in each textbook to determine whether each task could be solved by mimicking available templates or whether a solution had to be constructed without guidance from the textbook. There were similarities between the twelve textbooks in the sense that most tasks could be solved using a template as guidance. A significantly lower proportion of the tasks required a solution to be constructed. This was especially striking in the initial sets of tasks. Textbook descriptions indicating problem solving did not guarantee that a task solution had to be constructed without the support of an available template.

Dalarna University, School of Education, Health and Social Studies, Mathematics Education.

Mathematical reasoning and beliefs in non-routine task solving2015In: Current State of Research on Mathematical Beliefs XX: Proceedings of the MAVI-20 Conference September 29 - October 1, 2014, Falun, Sweden / [ed] Lovisa Sumpter, Falun: Högskolan Dalarna, 2015, p. 115-125Conference paper (Other academic)

Abstract [en]

This paper explores low performing upper secondary school students’ mathematical reasoning when solving non-routine tasks in pairs. Their solutions were analysed using a theoretical framework about mathematical reasoning and a model to study beliefs as arguments for choices. The results confirm previous research and three themes of beliefs are used by the student. These themes are safety, expectations, and motivation. The results also show a connection between beliefs and imitative reasoning as a way to solve non-routine tasks.

7.

Sidenvall, Johan

et al.

Linköpings universitet, Institutionen för samhälls- och välfärdsstudier.

Lithner, Johan

Umeå Mathematics Education Research Centre, Umeå University, Sweden.

Jäder, Jonas

Linköpings universitet, Institutionen för samhälls- och välfärdsstudier.

Students’ reasoning in mathematics textbook task-solving2015In: International journal of mathematical education in science and technology, ISSN 0020-739X, E-ISSN 1464-5211, Vol. 46, no 4, p. 533-552Article in journal (Refereed)

Abstract [en]

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.