Contents Preface VII Chapter 1. Introduction and outline 1 Measurement 2 Geometry 3 Graphs 5 Applets and poster 5 Chapter 2. A description of the domain of measurement 7 Gaining control of reality 7 Overview 10 Basic principles, justification and characterization 11 Prefixes 21 Length as the basis 24 Area 26 Circumference and area 33 Volume 38 Enlarging and reducing 44 Meaning and relationships 45 Weight, time speed and other quantities 47 In conclusion 54 Chapter 3. Concise learning-teaching trajectories and intermediate attainment targets for measurement 55 Benchmarks and measurement instruments 55 Prefixes and length 57 Constructing units of area 58 Constructing units of volume 60 Weight 62 Time 63 Temperature 64 Speed and other composite qualities 65 V.
Preface This book is the fourth – and final – publication in the TAL project series (Tussendoelen Annex Leerlijnen - Interim goals and learning-teaching tra- jectories). This TAL project was initiated by the Dutch Ministry of Educa- tion, Culture and Science. Its aim is to improve the quality of mathematics education by providing a perspective on didactic goals and learning-teach- ing trajectories, and the relationship between them.
1 Introduction and outline Arithmetic, measurement and geometry are closely related. You could even say that measurement and geometry build a bridge between everyday re- ality on one side and mathematics on the other. Measurement is what we do when we quantify reality, i.e. when we allocate numbers in order to acquire a grip on reality. With these numbers, we can calculate and make comparisons and predictions. For example, we can determine how much of something we need, how long something will last, or how much something will cost. Geometry establishes the basis for understanding the spatial as- pects of reality. We use geometric knowledge even without being aware of it, for instance when we plan a route, furnish a room or interpret a plan. In geometry education, we try to expand this informal knowledge.
Graphs have a clear geometric character, but at the same time they are im- portant tools for displaying measurement results. We see graphs primarily as a supplement to the components of measurement and geometry, which are subjects of this book. We therefore do not describe any explicit learn- ing-teaching trajectories for this component.
system, so that they can ultimately reconstruct the relationships within the system themselves. Instead of memorizing rules, we want the pupils to un- derstand how they can reason out the relationships between various meas- urements. To this end, they must understand the meaning of prefixes such as ‘centi’ and ‘kilo’, and be able to depict the various measurements them- selves. They should also be able to refer to the measurement knowledge that is based on reference points in reality. A well-known example in this context is the realization that a door is generally 2 m tall. These basic principles are worked out in greater detail in Chapter 3, which also provides a sketch of measurement education. The idea of seeing meas- urement and the development of the metric system as mathematical organi- zation and the development of mathematical tools are worked out sequen- tially for the concepts of length, area and volume. Before this, attention is focused on the prefixes with which any metric unit can be expanded to become a series of measurements with a virtually unlimited range. The discussions on length, area and volume end with an explanation of their meaning and relationships Attention is also paid to the intended coherence and nature of the knowledge that the pupils should develop. After this, we focus on weight, time, speed and other quantities. This part of the book ac- tually sketches out the rationale and structure behind the learning-teaching trajectories for measurement. In Chapter 3, we provide a brief summary of these learning-teaching trajectories and add a description of the intermedi- ate attainment targets.
in a learning-teaching trajectory and what this learning-teaching trajectory looks like. Or more accurately, what it can look like. After all, differences in accents and sequences are possible, specifically because geometry is such a rich area. The most important aspects are the substantive analysis and the coherence between the various concepts, insights and techniques. In geometry education, we believe it is essential to carefully consider the bigger picture that surrounds a specific activity: What are the implications of this activity? What precedes this activity? What can follow this activ- ity? The answers to these questions provide support for implementing the activity, provide insight into the support that you can offer the pupils and indicate the topics on which the pupils can reflect.
in a learning-teaching trajectory and what this learning-teaching trajectory looks like. Or more accurately, what it can look like. After all, differences in accents and sequences are possible, specifically because geometry is such a rich area. The most important aspects are the substantive analysis and the coherence between the various concepts, insights and techniques. In geometry education, we believe it is essential to carefully consider the bigger picture that surrounds a specific activity: What are the implications of this activity? What precedes this activity? What can follow this activ- ity? The answers to these questions provide support for implementing the activity, provide insight into the support that you can offer the pupils and indicate the topics on which the pupils can reflect.
2 A description of the domain of measurement Gaining control of reality Measurement is a way to gain control of reality. For example, we talk about how big something is or how heavy it is. Or we wonder how far away something is, how much it costs, how sweet it is, how hot it is, or how long something lasts. Measurement is a specific mathematical approach to real- ity. If we want pupils to learn to look at reality in a similar fashion, we must encourage them to structure and quantify situations in reality. For many children, this appears to be difficult. A periodic assessment conducted in the Netherlands (Cito 2005, Balans (32) van het reken-wiskundeonderwijs aan het einde van de basisschool 4) showed that children at the end of primary school often have difficulty, especially when working with area measurement. Pupils who have weak mathematics skills at the end of pri- mary school do not understand area. Volume also causes problems. For pupils with weak mathematics skills, relating various liter measurements to each other is also difficult. Pupils with good mathematics skills are able to do this, but they also tend to become stymied when they start to work with cubic measurements.
2 A description of the domain of measurement Gaining control of reality Measurement is a way to gain control of reality. For example, we talk about how big something is or how heavy it is. Or we wonder how far away something is, how much it costs, how sweet it is, how hot it is, or how long something lasts. Measurement is a specific mathematical approach to real- ity. If we want pupils to learn to look at reality in a similar fashion, we must encourage them to structure and quantify situations in reality. For many children, this appears to be difficult. A periodic assessment conducted in the Netherlands (Cito 2005, Balans (32) van het reken-wiskundeonderwijs aan het einde van de basisschool 4) showed that children at the end of primary school often have difficulty, especially when working with area measurement. Pupils who have weak mathematics skills at the end of pri- mary school do not understand area. Volume also causes problems. For pupils with weak mathematics skills, relating various liter measurements to each other is also difficult. Pupils with good mathematics skills are able to do this, but they also tend to become stymied when they start to work with cubic measurements.
a natural unit of measurement. After this, a standard is introduced – the centimeter – and the pupils learn to work with a ruler and a tape measure. In a comparable fashion, volume and weight are introduced as standard units. For an extensive description of learning measurement in the lower grades of primary school, see the publication ‘Young children learn meas- urement and geometry’ (2004).