This collection of learning activities is designed to provide engaging contexts in which to explore the achievement objectives from Level 5 of the NZC. The activities themselves do not represent a complete coverage of learning at this level. Rather they provide an opportunity to apply the mathematics learned in previous lessons. The activities are intended to reflect the multi-level nature of activities that students will encounter in NCEA internal assessment activities and the range of approaches that make up a differentiated classroom. (show more)
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 to the extent that they are 'reinventing the wheel', they should be given opportunities to derive and to feel an ownership of their mathematics. This can occur with a careful balance of:
- introducing 'building blocks' of knowledge, skills and techniques (teacher-driven including discussion, examples, notes)
- 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 (text books or worksheets)
- building an understanding of mathematical thinking, with structured investigations, proof work and open-ended problem solving. Examples of these activities are given in the rich learning activities.
The sequence of knowledge and skills needed for a rich learning activity are shown in flow chart format, under the heading 'Description of Mathematics' in each activity. An example of the teaching and learning that should occur before students would be given the Parabolic Investigation activity follows. It should be noted that algebraic problem solving at NZC Levels 5-8 are often reducible to solving a quadratic. In the most elegant of these problems, the roots are real (preferably integers) and so may be:
- clearly seen in a graph, and/or
- found by factorising the quadratic to get the form (x-a)(x-b)=0
The 'building blocks' and reinforcement exercises that will prepare the student for this rich learning activity (A parabolic investigation) are shown in a suggested order of treatment below.
These rich learning activities have been constructed for use with year 9/10 students working at NZC Level 5, on track for progressing to NCEA Level 1, and NZC Level 6, Mathematics. They focus on the skills and habits that can prepare students for the investigative and context based problem solving activities of many of the internal assessment activities for NCEA Mathematics.
The types of thinking employed by students in the process of attempting these activities are a consequence of their skill and familiarity with the processes they have mastered in 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. Where the achievement objective aims to advance algebraic technique in problem solving, three main approaches have been highlighted. Such achievement objectives are not exclusively in the Number and Algebra strand.
- The arithmetic approach: recognises that students may be very capable numerically but not necessarily so confident in their independent use of algebra when solving problems. 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. For example, if asked to rearrange a linear equation to solve for x, that student might try substituting in various values for x instead.
- appreciate being given numerically equivalent statements to illustrate a concept. For example, if a teacher is showing the index rule bn.bm = bn+m, then that student appreciates an example such as 22.23=25
- The procedural algebraic approach: is aimed at students who are competent numerically and algebraically and are able to use algebra as a tool to solve defined problems. These are the students who might:
- solve problems by following familiar processes. For example, if asked to solve for x in a linear equation, that student will diligently add the same number to both sides, divide both sides by the same value, or whatever is required to eventually get x as the subject .
- appreciate having new concepts explained by appealing to familiar rules. For example, if a teacher is showing the index rule bn.bm = bn+m, then that student appreciates an expansion such as b2.b3=bbb.bb=b5
- The conceptual algebraic approach: attempts to develop and extend the skills of students who can apply the tools of algebra, with abstract thinking to generalise. These are the students who:
- can independently construct algebraic 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 synthesise. The skills they can be encouraged to develop, by following the 'conceptual algebraic' approach, form the foundations of the abstract thinking required for excellence in NCEA mathematics and for successful application of mathematics beyond.
Where the AO focuses on the application of a mathematical process to a context, two main approaches have been highlighted to advance students' understanding of such achievement objectives, the corresponding rich learning activities have been designed with prompts for the teacher following two distinct learning approaches.
- The procedural approach is for students who can follow the steps of a given mathematical or statistical procedure but might need hints and thoughtful questioning:
- to make the link between calculation and context
- and specifically in the case of the statistical enquiry cycle:
- to comment on the key features of processed data
- to evaluate the validity of the conclusions of an investigation.
- The conceptual approach is for students who can:
- make the link between calculation and context
- and specifically in the case of the statistical enquiry cycle:
- recognise the key features of processed data
- make sensible comments on the validity of the conclusions of an investigation.
In all the approaches outlined above, the students should be working at Level 5 of the NZC, covering the concepts and processes outlined in the AOs. As a student develops their ability to generalise with algebra 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'.
