Level 1 rich learning activities

In the delivery of the Mathematics and Statistics learning area of the NZC, students should be exposed to a variety of teaching and learning strategies. While we can't expect them to synthesise new techniques, they should be given opportunities to apply their knowledge and skills of mathematics in unfamiliar settings. This can occur with a careful balance of:

• introducing 'building blocks' of knowledge, skills and techniques
• building skills, confidence and competence with 'reinforcement' exercises, where the students are employing a method they have been shown to solve a series of similar problems
• building an understanding of mathematical thinking, such as using a chain of reasoning and independently attempting open-ended problem solving. Examples of this latter group of activities are given in the rich learning activities.

The prior knowledge and skills needed for each rich learning activity are listed under the heading 'Description of Mathematics' in each activity.

These rich learning activities have been constructed for use with students working at NZC Level 1. The types of thinking employed by students in the process of attempting these activities are a consequence of their knowledge and skills within mathematics. At any particular point in time, a typical class might have students falling into several different groupings of approach to problem solving. Whilst the students can all be given the same problem to solve, it is unlikely that they would all attempt to solve the problem using the same process. The teacher may take different approaches in guiding each student towards success in solving the problem, but without removing their opportunity for ownership of the solution. Prompts the teacher might give, have been broken down into categories appropriate to the nature of the achievement objective and the activity at hand.

• The arithmetic approach: recognises that students may be very capable numerically and feel most comfortable using techniques such as counting on and skip counting to solve a problem. Often they appreciate seeing a numerical example or model to accept a process. Classroom observation can help identify students who are taking such an approach to their mathematical exploration. These are the students who might:
  • initially try to solve a problem with 'guess and check' techniques. 
  • appreciate being given numerically equivalent statements to illustrate a concept.
• The visual approach: is aimed at students who have a need to ‘see it to believe it’. They may use materials, images or other such visual representations of a problem in order to attempt a solution. Often, they will combine these visual stimuli with arithmetic in reaching their solution. These are the students who might:
  • focus on ‘picturing’ the problem at hand, perhaps needing clarification where they see apparent ambiguities.
  • make the link between problem and solution within the structure of their model
• The procedural approach: is aimed at students who are competent numerically are able to use appropriate operations and techniques to solve defined problems. These are the students who might:
  • solve problems by following familiar processes. 
  • appreciate having new concepts explained by appealing to familiar rules.

• The conceptual approach: attempts to develop and extend the skills of students who can apply their knowledge and skills in unfamiliar settings and may be able to generalise. These are the students who:

  • make the link between calculation and context.
  • can independently construct equations to represent mathematical information.
  • can link mathematical ideas to form a reasoned argument.
  • understand when all the possible solutions to a problem have been found.

  Students who are able to solve problems in this manner are often, but not always, those identified as needing extension work because they are quick to 'get' a new concept. Rather than being given more work, or being introduced to further concepts (including those from higher levels of the NZC), they should be given the opportunity to develop the depth of their thinking and reasoning and the chance to generalise. The skills they can be encouraged to develop, by following the 'conceptual' approach, form the foundations of the abstract thinking required for application of mathematics.

In all the approaches outlined above, the students should be working at Level 1 of the NZC, covering the concepts and processes outlined in the AOs. As a student develops their ability to generalise, to devise a solution or method of solution and/or to choose the most effective tools for the problem at hand, they may very well follow a different approach. It is therefore important to avoid labelling a student, for example, as 'arithmetic'.