Mathematical Inquiries and Proficiencies Study Guide

Mathematical Proficiencies

The Australian Curriculum, Assessment and Reporting Authority (ACARA) defines four primary proficiencies that are essential for mathematical development:

• Understanding: Students develop a ‘robust’ understanding of mathematical concepts. This involve making connections between related concepts and progressively applying learned knowledge and developed skills to new and unfamiliar situations.

• Fluency: Students develop the ability to calculate answers efficiently and accurately. This includes the capacity to recall definitions and facts and to choose appropriate procedures for given tasks.

• Problem-solving: Students develop the ability to formulate, model, investigate, and solve problems.

• Reasoning: Students develop an increasingly sophisticated capacity for logical thought and actions. This includes activities such as analysing, evaluating, explaining, inferring, justifying, and generalising.

Practical Application: Year 6 Proficiency Problem

A practical example of applying these proficiencies involves a geometry and percentage problem for Year 6 students:

• Problem Statement: A school is enclosed by a fence that has sides of equal length. $60\%$ of the whole fence has been painted black.

• Task 1: How many different ways can you draw what the fence might look like?

• Task 2: For each drawing, represent the painted section as a fraction.

• Task 3: Represent the painted section as a decimal.

• Learning Objectives: This task integrates Understanding, Fluency, Problem-solving, and Reasoning by requiring students to visualize spatial relationships and convert between percentages, fractions, and decimals.

A Teacher's Perspective on Problem-Solving

Claire Coolin, a maths specialist teacher, shared insights into the challenges and solutions for high-achieving Year 5 students regarding mathematical inquiry:

• Student Sentiment: Despite being high-achieving and generally enjoying mathematics, students often froze when faced with problem-solving. They reported disinterest, nervousness, and confusion, stating things like, ‖The information is hard to process,‖ and ‖I couldn’t understand the information as it was just a block of words.‖

• The Solution: Implementing Polya’s problem-solving steps provided a necessary framework for thinking in a logical way.

• Outcomes: Using a systematic framework stopped ‖manic scribbling‖ and allowed students to work through problems phase by phase.

• Inclusivity and Scaffolding: Using Polya’s steps with non-routine problems provides more able pupils with the scope and freedom to manipulate numbers differently, while providing critical scaffolding for less able students.

Polya’s Problem-Solving Process

George Polya’s approach guides students through a logical, four-step structure to navigate mathematical problems:

Step 1: Understanding the Problem

• Do you understand all the words used in stating the problem?

• What are you asked to find or show?

• Can you restate the problem in your own words?

• Can you think of a picture or diagram that might help you understand the problem?

• Is there enough information to enable you to find a solution?

Step 2: Devising a Plan to Solve the Problem

• Can you relate this problem to a previous problem that you’ve worked on before?

• Is there any basic formula that applies to this problem?

• Can you identify any patterns or relationships?

• Can you make up an analogous but simpler problem?

• Can you use objects to illustrate the problem or act out the problem?

• Will listing the information in a table help in solving the problem?

• Can you work backward to determine the correct procedure?

• Have you used a ‖trial and error‖ process?

Step 3: Carrying out the Plan

• Can you see clearly that the step is correct?

• Can you prove that it is correct?

Step 4: Checking the Results (Looking Back)

• Can you check the result?

• Can you check the argument?

• Can you derive the solution differently?

• Can you see it at a glance?

• Can you use the result, or the method, for some other problem?

Polya’s Process Example 1: Sum of Even Numbers (Year 2/3)

Problem Statement: Find the sum of all even numbers between $1$ and $10$.

Curriculum Context:

• Year 2 (VC2M2N04): Add and subtract one- and two-digit numbers, represent problems using number sentences, and solve using part-part-whole reasoning.

• Year 3 (VC2M3N01): Identify, explain, and use properties of odd and even numbers.

• Year 3 (VC2M3N09): Follow and create algorithms involving a sequence of steps; describe emerging patterns.

Solution Process:

1. Understand Problem: ‖Sum‖ means to add; ‖even numbers‖ are whole numbers (integers) divisible by $2$.

2. Devise Plan: List even numbers between $1$ and $10$; add them; state the answer.

3. Implement Plan: List includes $2$, $4$, $6$, $8$, and $10$. Calculate $2 + 4 + 6 + 8 + 10 = 30$. The answer is $30$.

4. Look Back: A straightforward approach was used. The arithmetic series formula could also be used: $S_n = n \frac{a_1 + a_n}{2}$.

Polya’s Process Example 2: Missing Values (Year 4)

Problem Statement: A number plus $9$ equals $17$. What is the number?

Curriculum Context:

• Year 4 (VC2M4A01): Find unknown values in numerical equations involving addition and subtraction, using properties of numbers and operations.

Solution Process:

1. Understand Problem: Find the missing number where $x + 9 = 17$.

2. Devise Plan: Write the equation; subtract $9$ from both sides; find the answer.

3. Implement Plan:

   • $x + 9 = 17$

   • $x + 9 - 9 = 17 - 9$

   • $x = 8$

4. Look Back: Check the result by substituting into the original equation (LHS: $8 + 9 = 17$; RHS: $17$). Alternate method: Trial-and-error.

Polya’s Process Example 3: Garden Measurement (Year 6)

Problem Statement: Therese has a rectangular garden that is $12\,m$ long and $8\,m$ wide. A walking path through the middle of the garden is $2\,m$ wide. How many square metres of the garden will be left for planting flowers after the path is built?

Visual Setup:

• Total garden: $12\,m \times 8\,m$.

• The path divides the garden into two sections. If the path is in the center, there are two remaining garden strips of $3\,m$ width each on either side of the $2\,m$ path ($3 + 2 + 3 = 8\,m$ width total).

Curriculum Context:

• Year 6 (VC2M6M02): Establish the formula for the area of a rectangle and use it to solve practical problems.

Questions to Stimulate Mathematical Thinking

Effective questioning is a vital tool for prompting student inquiry and assessment throughout the problem-solving journey.

Starter Questions

• How could you sort these…?

• How many ways can you find to…?

• What happens when we…?

• What can be made from…?

• How many different… can be found?

Questions to Stimulate Mathematical Thinking

• What is the same? What is different?

• Can you group these… in some way?

• Can you see a pattern?

• How can this pattern help you find an answer?

• What do you think comes next? Why?

• Is there a way to record what you've found that might help us see more patterns?

• What would happen if…?

Assessment Questions

• What have you discovered?

• How did you find that out?

• Why do you think that?

• What made you decide to do it that way?

Final Discussion Questions

• Who has the same answer, pattern, or grouping as this?

• Who has a different solution?

• Are everybody's results the same? Why or why not?

• Have we found all the possibilities? How do we know?

• Have you thought of another way this could be done?

• Do you think we have found the best solution?