Number pattern competencies for mathematising

In Section 3.5.1 number pattern competencies have been mapped (Table 3.1) so that mathematising competencies for number patterns can be developed. Table 3.2 separates horizontal and vertical mathematising competencies and focuses each competency on one or more sub-competency. Ellis’ taxonomy (2007a, 2007b) uses various activities to characterise a learner’s generalisation activities into specific levels. Using these levels and explanations, competencies can now be identified by looking at a learner’s external representations, i.e. what the learner does, says, makes and writes. These activities serve as indicators to acknowledge whether horizontal and vertical competencies are utilised when working through a number pattern modelling problem.

Internalising is the horizontal mathematising competency when a learner explores the real

problems so that he understands the problem and is able to simplify the problem. The

competency internalising can be identified when a learner rephrases the problem into his own language, when he explains or notes important information. According to the generalisation taxonomy (Ellis, 2007a, 2007b) a learner relates back to previous problems when attempting to make sense of a current problem. During the competency interpreting, the learner recognises quantities that influence the situation and make assumptions to note conditions that will work or not work for a problem. When a learner is interpreting he recognises quantities that influence the situation. In Ellis’ taxonomy relating objects can be compared with interpreting while searching can be compared with the competency structuring. Structuring involves setting up a real model based on relationships and patterns. The external representations of a learner when he is

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Table 3.2: Number pattern competencies for mathematising

Competencies Sub-competencies What the learner does, says, makes or writes

Horizontal Internalising  Understanding the problem

 Distinguishing between relevant and irrelevant information

 Simplifying the situation

 Learner states the problem in language he understands

 Learner notes/explains important information

 Learner notes/explains/relates a previous problem that is similar to the current one

Interpreting  Making assumptions

 Identifying conditions

 Identifying constraints

 Recognising quantities that influence situation

 Learner makes assumptions

 Learner notes conditions that will work/not work for a problem

 Learner recognises quantities that influence the situation

Structuring  Setting up a real model

 Naming quantities

 Identifying key variables

 Recognise patterns

 Recognise relationships

 Learner looks for a pattern/relationship

 Learner notes a recurring value or situation in the problem

 Learner recognises a pattern/relationship

 Learner states the relationship or pattern

Symbolising  Choosing appropriate mathematical symbols

 Using symbols

 Setting up a mathematical model

 Switching between symbolisations

 Learner draws pictures to represent the problem

 Learner draws pictures to show the relationship/pattern

 Learner uses objects to build the pattern

Vertical  Learner uses symbols to represent his pictures/patterns

 Learner forms a pattern using symbols

 Learner extends his pattern

 Learner formulates a rule using symbols

 Learner creates a model of Adjusting  Rephrasing the problem

 Refining

 Using and switching between operations

 Learner adapts his pattern so that it makes sense for the situation

 Learner tests his pattern

 Learner refines his pattern after testing it

 Learners reflects back to the pattern/symbols

 Learner reflects back to the real problem

 Learner creates a model for Organising  Viewing problem in a different form

 Use mathematical knowledge to solve problem

 Using heuristics

 Combining

 Integrating

 Learner constructs a rule that works for all elements

 Learner reflects back to the real problem

 Learner uses the rule to solve a problem

 Learner validates his solution

 Learner creates a model for Generalising  Establishing similar relationships in different

problems

 Independent reasoning and acting

 Learner uses deductive reasoning to prove his rule

 Learner uses/adapts the rule for another situation

The competency symbolising can be a horizontal or a vertical competency. Activities that identify horizontal competencies can be: the learner draws pictures to represent the problem, he shows the relationship or pattern in the problem, he uses objects to build the pattern. Vertical mathematising can be identified when learners use symbols to represent the problem. When a learner is adjusting, the emphasis is on refining the symbolisations. Adjusting can be noted when a learner refines his symbolisations. Ellis’ taxonomy (2007a, 2007b) summarises the

identification or statement to be focused around a continuing phenomenon, a statement of commonality or similarity or a general principle. This means the act of finding a pattern or rule that symbolises a common element. The competency organising is identifiable when a learner constructs a rule that works for all elements. Generalising involves independent reasoning and acting and a learner can now use or adapt the rule for another situation.

The third goal in the phenomenological analysis was to develop mathematising competencies specifically for number patterns. The number pattern competency continuum will be used in Chapter 5 as a tool to identify competencies during the teaching experiment.

3.6 GOAL 4: ESTABLISHING THE LEARNERS’ PRE-KNOWLEDGE

The starting point of the HLT is dependent on the learners’ prior knowledge. A baseline assessment is an effective tool to establish a learner’s mathematical skills and knowledge. The baseline assessment provides a teacher the opportunity to effectively plan the learning process. Kyriakides (2002) notes that the baseline assessment is used to identify what the learner can and cannot do so that differentiated learning needs can be targeted. The baseline assessment also serves as a basis for measuring future progress. The baseline assessment will provide important information about groups and individuals in the mathematics classroom and how the groups in a class can be structured. It will focus the HLT.