LUMAT: International Journal on Math, Science and Technology Education

The three worlds and two sides of mathematics and a visual construction for a continuous nowhere differentiable function

Juha Oikkonen, Jani Hannula — Thu, 30 Jun 2022 00:00:00 +0300

A rigorous and axiomatic-deductive approach is emphasized in teaching mathematics at university-level. Therefore, the secondary-tertiary transition includes a major change in mathematical thinking. One viewpoint to examine such elements of mathematical thinking is David Tall’s framework of the three worlds of mathematics. Tall’s framework describes the aspects and the development of mathematical thinking from early childhood to university-level mathematics. In this theoretical article, we further elaborate Tall’s framework. First, we present a division between the subjective-social and objective sides of mathematics. Then, we combine Tall’s distinction to ours and present a framework of six dimensions of mathematics. We demonstrate this framework by discussion on the definition of continuity and by presenting a visual construction of a nowhere differentiable function and analyzing the way in which this construction is communicated visually. In this connection, we discuss the importance to distinguish the subjective-social from the objective side of mathematics. We argue that the framework presented in this paper can be useful in developing mathematics teaching at all levels and can be applied in educational research to analyze mathematical communication in authentic situations.

Enablers and obstacles in teaching and learning of mathematics

Fatma Kayan Fadlelmula — Thu, 30 Jun 2022 00:00:00 +0300

This paper presents results of a systematic review of papers published at the LUMAT journal on the current issues positively and negatively affecting teaching and learning in mathematics, in concurrence with the Preferred Reporting Items for Systematic Reviews and Meta-Analyses (PRISMA) guidelines. The analysis also offers insight into the most studied topics in mathematics education research, including key demographic and methodological characteristics such as year of publication, participants, education level, research methodologies, and research focus. Data was gathered from the studies published in the LUMAT: International Journal on Math, Science and Technology Education, starting from its first volume in 2013. So far, 225 articles were published in this journal, with 133 studies written in English and 51 studies related to mathematics. Although earlier studies support the notion that mathematics education is mostly traditional, this review suggests current research has thorough and positive outcomes, such that mathematics educators are likely to implement non-traditional approaches, encouraging student engagement, peer collaboration, and mathematical discourse. Certainly, in such learning environments, students tend to feel more motivated and less anxious about learning mathematics. They may also be more active and responsible in their learning, collaborate with peers, and get into mathematical discussions. Yet, there are also a number of difficulties and obstacles highlighted both in teaching and learning of mathematics. The findings might inspire several instructional implications for mathematics educators, curriculum developers, and researchers. Recommendations are given to add into what the existing literature claims and offer greater empirical evidence to support the verdicts.

Rudimentary stages of the mathematical thinking and proficiency

Jari Metsämuuronen, Annette Ukkola — Thu, 30 Jun 2022 00:00:00 +0300

A national-level dataset (n = 7770) at grade 1 of primary school is re-analyzed to study preconditions in proficiency in mathematical concepts, operations and mathematical abstractions and thinking. The focus is on those pupils whose preconditions are so low that they are below the first measurable level of proficiency in the common framework with reference to mathematics (CFM). At the beginning of school, these pupils may not be familiar with, e.g., the concepts of numbers 1–10, they may not be aware of the consecutive nature of numbers, and they have no or very limited understanding of the basic concepts of length, mass, volume, and time. A somewhat surprising finding is that the key factor explaining the absolute low proficiency in mathematics appeared to be a low proficiency in listening comprehension. This variable alone explains 41% of the probability of belonging to the group of pupils who are not able to show proficiency enough to reach the lowest level in any of the criteria. It is understandable that, if language skills are underdeveloped in general, a child is not expected to master the specific mathematical vocabulary either and, hence, the low score in a test of preconceptions in mathematics too. Other variables predicting the absolute low level or preconditions of mathematics are the decision on intensified or special support, status of Finnish or Swedish as second language, and negative attitudes toward mathematics.

Identifying and promoting young students’ early algebraic thinking

Sanna Wettergren — Thu, 30 Jun 2022 00:00:00 +0300

Algebraic thinking is an important part of mathematical thinking, and researchers agree that it is beneficial to develop algebraic thinking from an early age. However, there are few examples of what can be taken as indicators of young students’ algebraic thinking. The results contribute to filling that gap by analyzing and exemplifying young students’ early algebraic thinking when reasoning about structural aspects of algebraic expressions during a collective and tool-mediated teaching situation. The article is based on data from a research project exploring how teaching aiming to promote young students’ algebraic thinking can be designed. Along with teachers in grades 2, 3, and 4, the researchers planned and conducted research lessons in mathematics with a focus on argumentation and reasoning about algebraic expressions. The design of teaching situations and problems was inspired by Davydov’s learning activity, and Toulmin’s argumentation model was used when analyzing the students’ algebraic thinking. Three indicators of early algebraic thinking were identified, all non-numerical. What can be taken as indicators of early algebraic thinking appear in very short, communicative micro-moments during the lessons. The results further show that the use of learning models as mediating tools and collective reflections on a collective workspace support young students’ early algebraic thinking when reasoning about algebraic expressions.

“Learning models”

Inger Eriksson, Natalia Tabachnikova — Thu, 30 Jun 2022 00:00:00 +0300

The overarching aim of this article is to exemplify and analyse how some algebraic aspects of equations can be theoretically explored and reflected upon by young students in collaboration with their teacher. The article is based upon an empirical example from a case study in a grade 1 in a primary school. The chosen lesson is framed by the El’konin-Davydov curriculum (ED Curriculum) and learning activity theory in which the concept of a learning model is crucial. Of the 23 participating students, 12 were girls and 11 boys, approximately seven to eight years old. The analysis of data focuses on the use of learning models and reflective elaboration and discussions exploring algebraic structures of whole and parts. The findings indicate that it is possible to promote the youngest students’ algebraic understanding of equations through the collective and reflective use of learning models, and we conclude that the students had opportunity to develop algebraic thinking about equations as a result of their participation in the learning activity.

Understanding “proportion” and mathematical identity

Kazuyuki Kambara — Thu, 30 Jun 2022 00:00:00 +0300

Studies have found that problems exist with respect to elementary school teachers’ understanding of proportions and their knowledge of the appropriate methods for teaching the concept. This study aims to help aspiring elementary school teachers form a healthy mathematical identity and deepen their understanding of mathematics. This quantitative study employed the descriptive-research survey method, surveying 86 students in 2019 and 110 students in 2021. Data were gathered using a survey questionnaire designed by Kumakura et al. (2019), with minor modifications made by the author. A major finding was that many students want to become elementary school teachers but are uncomfortable with the concept of proportions. Another important finding is that it is a challenge for students who wish to become elementary school teachers at a traditional school (University A) to hone their ability to use mathematical expressions and develop their sense of quantity. The findings suggest that it is important to help such students understand the content and refine their expressions.

Student teachers’ common content knowledge for solving routine fraction tasks

Anne Tossavainen — Thu, 30 Jun 2022 00:00:00 +0300

This study focuses on the knowledge base that Swedish elementary student teachers demonstrate in their solutions for six routine fraction tasks. The paper investigates the student teachers’ common content knowledge of fractions and discusses the implications of the findings. Fraction knowledge that student teachers bring to teacher education has been rarely investigated in the Swedish context. Thus, this study broadens the international view in the field and gives an opportunity to see some worldwide similarities as well as national challenges in student teachers’ fraction knowledge. The findings in this study reveal uncertainty and wide differences between the student teachers when solving fraction tasks that they were already familiar with; two of the 59 participants solved correctly all tasks, whereas some of them gave only one or not any correct answer. Moreover, the data indicate general limitations in the participants’ basic knowledge in mathematics. For example, many of them make errors in using mathematical symbol writing and different representation forms, and they do not recognize unreasonable answers and incorrect statements. Some participants also seemed to guess at an algorithm to use when they did not remember or understand the correct solution method.

Preschoolers’ ways of experiencing numbers

Camilla Björklund, Anna-Lena Ekdahl, Angelika Kullberg, Maria Reis — Thu, 30 Jun 2022 00:00:00 +0300

In this paper we direct attention to 5–6-year-olds’ learning of arithmetic skills through a thorough analysis of changes in the children’s ways of encountering and experiencing numbers. The foundation for our approach is phenomenographic, in that our object of analysis is differences in children’s ways of completing an arithmetic task, which are considered to be expressions of their ways of experiencing numbers and what is possible to do with numbers. A qualitative analysis of 103 children’s ways of encountering the task gives an outcome space of varying ways of experiencing numbers. This is further analyzed through the lens of variation theory of learning, explaining why differences occur and how observed changes over a prolonged period of time can shed light on how children learn the meaning of numbers, allowing them to solve arithmetic problems. The results show how observed changes are liberating new and powerful problem-solving strategies. Emanating from empirical research, the results of our study contribute to the theoretical understanding of young children’s learning of arithmetic skills, taking the starting point in the child’s lived experiences rather than cognitive processes. This approach to interpreting learning, we suggest, has pedagogical implications concerning what is fundamental to teach children for their further development in mathematics.

Developing mathematical problem-solving skills in primary school by using visual representations on heuristics

Susanna Kaitera, Sari Harmoinen — Thu, 30 Jun 2022 00:00:00 +0300

Developing students’ skills in solving mathematical problems and supporting creative mathematical thinking have been important topics of Finnish National Core Curricula 2004 and 2014. To foster these skills, students should be provided with rich, meaningful problem-solving tasks already in primary school. Teachers have a crucial role in equipping students with a variety of tools for solving diverse mathematical problems. This can be challenging if the instruction is based solely on tasks presented in mathematics textbooks. The aim of this study was to map whether a teaching approach, which focuses on teaching general heuristics for mathematical problem-solving by providing visual tools called Problem-solving Keys, would improve students’ performance in tasks and skills in justifying their reasoning. To map students' problem-solving skills and strategies, data from 25 fifth graders’ pre-tests and post-tests with non-routine mathematical tasks were analysed. The results indicate that the teaching approach, which emphasized finding different approaches to solve mathematical problems had the potential for improving students’ performance in a problem-solving test and skills, but also in explaining their thinking in tasks. The findings of this research suggest that teachers could support the development of problem-solving strategies by fostering classroom discussions and using for example a visual heuristics tool called Problem-solving Keys.

Supporting argumentation in mathematics classrooms

John Francisco — Thu, 30 Jun 2022 00:00:00 +0300

Reform movements in mathematics education advocate that mathematical argumentation play a central role in all classrooms. However, research shows that mathematics teachers at all grade level find it challenging to support argumentation in mathematics classrooms. This study examines the role of teachers’ mathematical knowledge in teachers’ support of argumentation in mathematics classroom. The study addresses a documented need for a better understanding of the relationship between mathematical knowledge for teaching and instruction by focusing on how the knowledge influences teachers’ support of argumentation. The results provide insights into particular aspects of teachers’ mathematical knowledge that influence teachers’ support of students’ development of valid mathematical arguments in mathematics classrooms and suggest implications for research and practice.

Languaging and conceptual understanding in engineering mathematics

Kirsi-Maria Rinneheimo, Sami Suhonen — Thu, 30 Jun 2022 00:00:00 +0300

The ability to apply mathematical concepts and procedures in relevant contexts in engineering subjects sets the fundamental basis for the mathematics competencies in engineering education. Among the plethora of digital techniques and tools arises a question: Do the students gain a deep and conceptual enough understanding of mathematics that they are able to apply mathematical concepts in engineering studies? This paper introduces the use of languaging exercises in the engineering mathematics course ‘Differential Calculus’ during the spring semester 2020, at Tampere University of Applied Sciences, TAMK. In this study, the students’ conceptual understanding and learning of differential calculus is researched. In the learning process, the languaging method is used to deepen the conceptual understanding of the concepts of differential calculus. Pre-test/post-test setup was used to see the possible gain in conceptual understanding. During the course, students did online assignments, which included languaging exercises. Students described the concepts of differential calculus using natural language, pictures, or a combination of them. The students were also asked to fill in a self-evaluation form to collect their perception of their own knowledge of mathematical skills. Mid-term and final exams summarized the acquired knowledge. The study aimed to enhance the learning outcomes and to gain a deeper understanding of mathematical concepts by exploiting the languaging method.

Mathematical thinking and understanding in learning of mathematics

Jorma Joutsenlahti, Päivi Perkkilä — Thu, 30 Jun 2022 00:00:00 +0300

The concept “mathematical thinking” can be found in several studies of mathematics education, in national curricula or in media during the decades all over the world. We searched words “mathematical thinking” from the database of international scientific articles, and we found 456 707 mentions at first time. These are the main reasons why we have chosen “mathematical thinking” as the central concept of the Special Issue. The other interesting question from our point of view is how a student can express his/her mathematical thinking? By answering this question, we have made simple model for the teacher education purposes, and we call it “languaging” (of mathematical thinking). The articles of the Special Issue gives some answers to the questions: “What is mathematical thinking and how we can express it?” and “What are the relationships between conceptual understanding and mathematical thinking?”.