In this study, we present a mathematical analysis distinguishing two conceptions of equivalence: proportional equivalence and unit equivalence. These two conceptions have distinct meanings in relation to equivalent fractions: one is grounded in proportionality, while the other is grounded in equal wholes. We argue that (a) the distinction of equivalence gives a unified framework of equal fractions that has not previously been described in the literature; (b) a conceptual understanding of both fraction equivalences is integral to understanding rational numbers; and (c) knowledge of both conceptions of equivalence is important for developing a conceptual understanding of fraction arithmetic. Past research has largely overlooked the distinction between the two types of equivalence. However, this may provide an important foundation for central topics that build on equivalence, and a better understanding of these two types of equivalence may support a more flexible understanding of fractions. Last, we propose future directions for teaching equivalence in mathematics.

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General rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.-Users may download and print one copy of any publication from the public portal for the purpose of private study or research. -You may not further distribute the material or use it for any profit-making activity or commercial gain -You may freely distribute the URL identifying the publication in the public portal -V Content Chapter 1: Introduction 1.1 Students with mathematical learning difficulties 1.2 Fractions 1.3 Presentation of the PhD project 1.4 Aim and research questions 1.5 Overview of the dissertation Chapter 2: Methodology 2.1 The nature of pragmatism 2.2 Experience as the bridge 2.3 Enquiry: the basis for the project 2.4 The phases of enquiry Chapter 3: What are fractions? And how to understand them 3.1 Terminology of fractions 3.2 The historical development of the fractions notation 3.3 The multi-faceted construct of the fraction concept 3.3.1 The part-whole subconstruct 3.3.2 The quotient subconstruct 3.3.3 The measure subconstruct 3.3.4 The ratio subconstruct 3.3.5 The operator subconstruct 3.4 Summary Chapter 4: What is fraction learning and understanding? 4.1 Method used for making the reviews 4.2 Review (1): Mathematical knowledge and fraction proficiency 4.2.1 Conceptual knowledge 4.2.2 Procedural knowledge 4.2.3 The interaction between the two types of knowledge 4.2.4 Fraction proficiency 4.2.5 Summary of fraction proficiency 4.3 Review (2): Natural number bias CONTENT 2 4.3.1 Natural number bias 4.3.2 Who is affected by the natural number bias? 4.3.3 Different aspects of the natural number bias 4.3.4 Summary of natural number bias 4.4 Review (3): Number knowledge development 4.4.1 Conceptual change theory 4.4.2 Integrative theory 4.4.3 The dissertation's theoretical perspective on number knowledge development 4.5 Review (4): Fraction interventions Chapter 5: Fractions in the Danish elementary school 5.1 Official curriculum 5.1.1 After third grade 5.1.2 After sixth grade 5.1.3 After ninth grade 5.2 Analysing the content of the instructional materials 5.2.1 Matematrix 4 (2006) 5.2.2 Multi 4 (2011) 5.2.3 Kontext+ 4 (2014)

This study aimed to examine whether a computational thinking (CT) intervention related to a) number knowledge and arithmetic b) algebra, and c) geometry impacts students' learning performance in primary schools. To this end, a quasi-experimental, nonequivalent group design was employed, with 61 students assigned to the experimental group and 47 students to the control group (n = 108). The experimental group comprised students in primary school who were to be followed across the second and third grades. The experimental group underwent work with digital CT activities, while the control group did not receive such interventions. A one-way analysis of variance (ANOVA) was performed on the data gathered to assess the ability differences between students from the experimental and control groups. The pre-and post-test results revealed that the experimental group's performance was significantly better than the control group's performance for the content areas involving CT activities.