Ronald Keijzer

Learning for mathematical insight: a longitudinal comparative study on modelling

Abstract This paper reports on a longitudinal study of teaching and learning the subject of fractions in two matched groups of ten 9–10-year-old students. In the experimental group fractions are introduced using the bar and the number line as (mental) models, in the control group the subject is introduced by fair sharing and the circle-model. In the experimental group students are invited to discuss, in the control group students work individually. The groups are compared on several occasions during one year. After one year, the experimental students show more proficiency in fractions than those in the control group.

AUDREY'S ACQUISITION OF FRACTIONS: A CASE STUDY INTO THE LEARNING OF FORMAL MATHEMATICS

National standards for teaching mathematics in primary schools in the Netherlands leave little room for formal fractions. However,a newly developed programme in fractions aims at learning formal fractions. The starting point in the development of this curriculum is the students’ acquisition of `numeracy infractions’. In this case study we describe the growth in reasoning ability with fractions of one student in this newly developed programme of 30 lessons during one whole school year. In the study we found indications that the programme and its teaching stimulated the progress of an average performer in mathematics. Moreover we found arguments as to what extent formal operations with fractions suits as an educational goal.

Designing Non-routine Mathematical Problems as a Challenge for High Performing Prospective Teachers

Designing non-routine mathematical problems is a challenging task, even for high performing prospective teachers in elementary teacher education, especially when these non-routine problems concern knowledge at the mathematical horizon (HCK). In an experimental setting, these prospective teachers were challenged to design non-routine HCK problems. Interaction with peers, feedback from experts, analyzing HCK problems to find criteria, building a repertoire of prototypes, a cyclic design process, experts who are themselves struggling in designing problems were the most important and effective aspects of the learning environment to rise from this explorative study.

Measurement and Geometry in Upper Primary School

The focus of this book is on measurement and geometry in the upper grades of primary education. Measurement and geometry are important topics which perhaps do not get the emphasis they deserve. They build, in a manner of speaking, a bridge between everyday reality and mathematics. Measurement concerns the quantification of phenomena; consequently, it makes these phenomena accessible for mathematics. Geometry establishes the basis for understanding the spatial aspects of reality.

Geometry attainment targets

Pupils can interpret sketches, floor plans and maps that vary regarding the aspects of scale preservation, detail and orientation. Pupils can indicate the position of people or objects in space. During this process, they make use of sketches, floor plans and maps, and of informal and formal coordinate systems.

A description of the domain of measurement

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 reality. 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.

Concise learning-teaching trajectories and intermediate attainment targets for measurement

In this chapter we will provide a concise summary of the learning-teaching trajectories and targets for measurement. In very broad terms, during the upper grades of primary school, measurement education focuses on flexible manipulation of units and reasoning with units. Activities therefore do not focus on imprinting relationships and calculation rules, but on searching for such relationships based on known unit meanings. In this chapter we will work out this approach for ‘benchmarks and measurement instruments’, ‘prefixes and units of length, ‘units of area’, ‘units of volume’, ‘weight’, ‘time’, ‘temperature’ and ‘speed and other composite quantities’. For the purpose of clarity, we will isolate these learning-teaching trajectories. Of course, this does not mean that these are isolated learning processes. Quite the contrary. Virtually all measurement situations can be placed in two or more of the above-mentioned clusters.

Learning-teaching trajectories in geometry

In this chapter we describe the learning-teaching trajectories in geometry for each of the three terrains ‘Spatial sense’, ‘Plane and solid figures’, and ‘Visualization and representation’. Every learning-teaching trajectory description is connected to an example activity that is described, analyzed and discussed based on the core insights.

The domain of geometry

Babies come into contact with geometry while they are still in the cradle. This happens sooner than with any other aspect of mathematics. Freudenthal, among other authors, has written beautifully about this topic in his book ‘ Mathematics as an Educational Task’. He is one of the founders of Realistic Mathematics Education and writes ‘about the truncated cone that a drinking cup actually is’, ‘about the ball that the still incapable fingers try to hold’ and ‘growing up with geometry’. During this phase, the space for the child is still limited to the cradle, the playpen, the changing table, the baby bath and the baby’s room. But as the child grows older, this space continues to expand. The room turns out to be part of a house. The house is located on a street. There are other houses on the street. Grandma lives far away, and there are various ways to go there.

Introduction and outline

Arithmetic, measurement and geometry are closely related. You could even say that measurement and geometry build a bridge between everyday reality 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 aspects 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.