Working mathematically
Working mathematically

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Working mathematically

Appendix: processes uncovered

The following listing provides possible components for the six processes (Source: Begg, 1994, pp. 183–92).

Problem solving (including investigating)

(i) plan

  • identify, describe and solve problems;
  • inquire, explore, generate, design, measure and make;
  • formulate a plan and identify sub-tasks;
  • decide whether sufficient information is known;
  • locate, gather and retrieve information;
  • distinguish between important and irrelevant information.

(ii) strategies

  • guess and check;
  • make a list; draw a picture, table or graph;
  • find a pattern, a relationship, and/or a rule;
  • make a model;
  • solve a simpler problem first;
  • work backwards;
  • eliminate possibilities;
  • try extreme cases;
  • write a number sentence;
  • act out a problem;
  • restate the problem;
  • check for hidden assumptions;
  • change the point of view;
  • recognise appropriate procedures and justify them.

Mathematical modelling

(i) general
  • use concrete materials;
  • use Cartesian and other graphs to model change;
  • use lines, networks, and tree diagrams to represent relationships and sequences;
  • use flow diagrams to represent procedures;
  • use formulae to model relationships;
  • use diagrams and three dimensional models to model geometric situations; and
  • apply the process of mathematical modelling to real world problems.
(ii) translating
  • restate the real problem as a mathematical problem;
  • use estimation to check the solution;
  • verify and interpret results with respect to the original question; and
  • recognise that the best maths solution may not be the best real solution.

Reasoning

(i) classification and description
  • sort and classify objects;
  • describe objects and procedures unambiguously and give definitions; and
  • organise information to support logic and reasoning.
(ii) inferring
  • make and evaluate mathematical conjectures;
  • infer, interpolate, and extrapolate;
  • make and test hypotheses;
  • make generalisations; and
  • make appropriate and responsible decisions).
(iii) reasoning
  • draw logical conclusions;
  • use models, facts, properties and relationships to provide reasons;
  • justify answers and procedures;
  • use patterns and relationships to analyse situations;
  • develop confidence with spatial reasoning;
  • develop confidence with graphical reasoning (interpretation of graphs);
  • recognise the meanings of true, false and not proven; follow logical arguments;
  • judge the validity of arguments;
  • construct simple valid arguments;
  • recognise and apply deductive reasoning;
  • recognise and apply inductive reasoning;
  • formulate counter examples;
  • appreciate the pervasive use and power of reasoning as a part of mathematics.
(iv) proving
  • appreciate the axiomatic nature of mathematics; and
  • construct proofs for mathematical assertions including indirect and inductive proofs.

Communicating

(i) personal
  • relate to others; and
  • work cooperatively.
(ii) general
  • understand what needs to be done in broad terms;
  • reflect on and clarify thinking;
  • relate everyday language to mathematical language, understand mathematical vocabulary;
  • formulate definitions;
  • express generalisations.
(iii) listening and speaking
  • follow instructions from the teacher;
  • discuss difficulties;
  • ask questions;
  • present and explain results to others;
  • discuss the implications and accuracy of conclusions;
  • discuss other possible interpretations of conclusions.
(iv) reading and writing
  • follow instructions from a text or a computer;
  • debate possible courses of action with others;
  • use reference material; and
  • make a report.
(v) representing
  • use graphs and diagrams to depict mathematical ideas visually;
  • use symbols to represent ideas precisely;
  • explore problems and describe results using graphical, numerical, physical, algebraic, and verbal mathematical models or representations;
  • appreciate the economy, power, and elegance of mathematical notation.

Making connections

(i) within mathematics
  • link concepts, procedures, and topics in mathematics;
  • relate various representations of a concept or procedure to one another;
  • recognise equivalent representations;
  • see mathematics as an integrated whole.
(ii) other curriculum areas
  • use mathematics in other school subjects
(iii) everyday life
  • use mathematics in everyday life, in work and leisure activities;
  • relate results to one’s everyday experienced world.
(iv) general
  • apply mathematics to familiar and unfamiliar situations;
  • value the relationship between mathematics and history, culture and society.

Using tools

(i) instruments
  • use measuring and drawing instruments.
(ii) calculators
  • use simple, scientific, graphical, symbol manipulating calculators.
(iii) computers
  • use computers, and general applications such as word processing, databases, spreadsheets;
  • use the Web as a resource for information;
  • use specialist packages for handling and graphing data, for symbol manipulation and graphing, and for dynamic geometry.
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