Gravemeijer (1999) notes that the final outcome of a design study is to formulate an instructional theory based on “theoretical deliberations and empirical assessments” (p. 157). The instructional theory explains how the learning of a specific topic can be supported. The key features of a first level number pattern generalisation theory provides guidance for this support. The goal of the RME theory is to establish means by which learners can reinvent mathematics (Gravemeijer, 1999). This reinvention principle occurs when learners construct models through the process of mathematising. Table 5.4 outlines the key features of the instructional design for a number pattern theory by referring to the RME principles that will ensure emergent modelling.
Key features of the instructional design RME principle
Learners are active participants in their learning Activity principle The curriculum is directed towards pattern generalisation: Learners
give meaning to patterns in various kind of situations that are real to them
Reality principle
The curriculum contains a teaching strategy in progressive stages in which pattern generalisation is developed through model building: searching for and extending patterns (linear, quadratic and exponential), generalising linear patterns, generalising quadratic number patterns, generalising exponential number patterns
Level principle
Learners are given the opportunity to use pre-established knowledge and tools to solve problems
Intertwinement principle
Group and class discussions are used to share ideas and evoke reflection to construct meaning
Interaction principle
Learners are offered the opportunity to reinvent mathematics through guidance in the form of well-structured and selected learning activities
Table 5.4: Key features of a number pattern theory
These key features can be noted throughout the learning activities’ analyses and rationales. The planning phase of the teaching experiment, the development of the teaching experiment and the retrospective analysis have provided the reader with the basis of an instructional theory and has incorporated the RME design heuristics throughout.
During the phenomenological analysis in Section 3.2 the goals for the content of the study were formulated. The second goal of the study (see Section 3.4) was the notion of generalisation. The historical and didactical phenomenological analyses indicated that the process of generalisation is a problem area for learners. The South African mathematics curriculum provided teaching guidelines that focused on investigating and generalising linear number patterns in Grade 10 and quadratic number patterns in Grade 11. The LT curriculum therefore focused on a generalisation goal. The results from the study show (see Tables 5.2 & 5.3) that a mixed ability group of Grade 10 learners are able to generalise linear, quadratic and exponential number patterns when their learning is predicted and adequately supported throughout the teaching experiment.
The mathematical modelling approach to the teaching and learning of number patterns can successfully be integrated into a mathematics curriculum. It will ensure the development of sophisticated models, the opportunity to reinvent formal mathematics and create meaningful mathematical experiences. Table 5.5 gives an overview of the role of the suggested learning activities in an instructional sequence by referring to the progression stages (see Table 5.4) and topics of mathematical discourse noted from the teaching experiment.
Gravemeijer (1999, 2004) summarises that the local instructional theory is the whole instructional sequence, the general theory, and the framework against which teachers can develop a HLT to fit their classrooms. The suggested learning activities produced the first attempt to such an instructional sequence. The general theory was explained in the literature study, applied in the preparation phase and implemented in the teaching experiment.
The developmental nature of the study provides the teacher with a framework to develop a LT and guidance to successfully support learners’ learning by following a mathematical modelling perspective. The requirements of LIT are evident in the theory that has been developed for
number patterns.
Suggested learning activities
Progression stages Potential activity and mathematical discourse topics Learning activity 1
Broken eggs
Searching for patterns Grouping
Using multiples of seven Learning activity 2
More broken eggs
Extending linear number pattern Generalising linear number pattern
Many solutions follow the same rules
Finding a shorter way to get more solutions
Learning activity 3 Marcella’s
doughnuts
Searching for patterns Trying different mathematical strategies to solve the problem Backtracking to find the solution
Learning activity 4 Extended doughnuts
Extending linear number pattern Generalising linear number pattern
Dependence of one value on another
The same process was repeated three times
Learning activity 5 Thinking diagonally
Extending quadratic number pattern Generalising quadratic number pattern
What is a diagonal?
Absence of a constant difference Relating the diagonals to the shape’s sides
Learning activity 6 Squares
Extending quadratic number pattern Generalising quadratic number pattern
Absence of a constant difference Relating the squares to the stacks
Learning activity 7 Consecutive sums
Extending quadratic number pattern Generalising quadratic number pattern
What is consecutive? Noticing relationships
Different consecutives to be tested
Learning activity 8 Extending exponential number patterns
Generalising exponential number patterns
Folding paper
Using differences to search for a pattern
Table 5.5: Role of the suggested learning activities in the instructional sequence
5.13 SUMMARY
DBR study. A task oriented comparison was essential to compare the conjectured learning with the actual learning that was observed by the researcher based on the data. A data matrix analysis presented these comparisons and matched the accuracy of the predictions in Table 5.1. Table 5.2 showed a results summary. A longitudinal analysis was implemented in Sections 5.3 to 5.11. The analysis provided evidence for the learners’ mathematising competencies as they worked through the learning activities. This analysis also contributed to the rationale for the modelling sessions. This analysis was in line with Treffers’ (1987) aim of a holistic three-dimensional goal
description: it provides informed guidelines to support teachers by giving a constructive analysis of the learning materials and didactics. Each selected activity in the LT has played a significant role in the learners’ learning of the modelling process and the development of number pattern competencies.
The confirmation of the conjectures in Table 5.2, the development of the mathematising competencies for number patterns and the explanation of the activities in the analysis have formed the basis of the LIT. Section 5.12.1 explained how the RME theory was used throughout the study to form a theory for number patterns. The LIT for number patterns was discussed in Section 5.12.2. Chapter 6 will provide the conclusions for this study and will include the limitations of the study and recommendations for further areas of study.