# 1.2. Problem Solving

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Problem solving is an integral part of all mathematics learning. In everyday life and in the workplace, being able to solve problems can lead to great advantages. However, solving problems is not only a goal of learning mathematics but also a major means of doing so.

Problem solving means engaging in a task for which the solution is not known in advance. Good problem solvers have a “mathematical disposition” – they analyse situations carefully in mathematical terms and naturally come to pose problems based on situations they see. For example, a young child might wonder, “How long would it take to count to a million?”

Systematic reasoning is a defining feature of mathematics. Exploring, justifying, and using mathematical conjectures are common to all content areas. Through the use of reasoning, you will learn that mathematics makes sense. Reasoning and proof must be a consistent part of your mathematical experiences.

Reasoning mathematically is a habit of mind, and like all habits, it must be developed through consistent use in many contexts. You should learn to make effective deductive arguments as well, using the mathematical truths they are establishing.

**Statistical Problem Solving Process**

We would generally collect and work with data to fulfil a specific purpose such as:

- determining trends in societal issues such as crime and health
- identifying relevant characteristics of target groups such as age range, gender, socio-economic group, cultural belief, and performance
- considering the attitudes or opinions of people on issues

In order to establish statistical models we would collect and work with data using various techniques, including:

- the formulation of questionnaires and interviews for a specific purpose,
- the use of interviews and questionnaires to obtain data for surveys and censuses
- selecting a sample from a population with due sensitivity to issues relating to bias
- generating experimental data appropriate to the situation under investigation
- working with dichotomous, discrete, and continuous data
- using a variety of methods to represent statistics including pie charts, bar graphs, histograms, stem and leaf plots, box plots
- calculating measures of centre and spread such as mean, median, mode, range and interquartile range
- using scatter plots to represent the association between two variables

In order to explore probability models, make predictions and study problems, we would:

- organise, sort and classify data for processing and analysis
- distinguish between theoretical and experimental probabilities
- use probability models for comparing experimental results with mathematical expectations
- use tree diagrams in representing and working with events
- distinguish between independent, mutually exclusive and complementary events
- use basic counting techniques to determine the number of ways an event can occur
- make and test predictions about probability in the context of games and real-life situations

Once we have collected, organised, established statistical and probability models we would use statistical and probability concepts in problem-solving and decision making to critically interpret the results and draw conclusions, by:

- Sourcing and interpreting information from a variety of sources including nested or layered tables,
- Critically evaluating arguments based on statistics and probability and make recommendations
- Describing the use and misuse of statistics and probability in society
- Using probabilities to make predictions and judgements

To investigate life and work related problems using data and probabilities, we would have to decide on a procedure to use in order to collect data from which we can draw a calculated conclusion.