Today’s mathematics classrooms should be dynamic places that involve and engage students in their own learning and promote a higher level of thinking. One way to achieve this is for teachers to use an investigative approach to mathematics by incorporating open-ended mathematical tasks and investigations into their mathematics programs.

An investigative approach engages students in open-ended tasks and investigations that:

• promote a higher level of thinking
• encourage students to think, learn, analyse, criticise, and to solve unfamiliar problems
• ask for more than recall of facts or the replication of processes

Open-ended mathematical tasks and investigations link the learning of mathematical concepts to a real-world context. They:

• involve solving a meaningful problem
• emphasise critical thinking and reasoning
• produce evidence of what students know and can do

Open-ended mathematical tasks and investigations can be varied. Some pose problems to be solved, others ask questions to be answered, and still others describe tasks or challenges to be completed or define issues to be explored in real-life or lifelike contexts. But all have one thing in common—they provide opportunities for students to use multiple pathways to arrive at a solution.

Some open-ended tasks and investigations can be completed within a lesson, while others take longer to complete. However, all require students to:

• identify the maths required to undertake the task or investigation
• describe what they have to do
• collect relevant information
• develop a suitable plan or strategy
• carry out the plan
• summarise their results

Teachers will gain valuable information about what students know and can do well by observing strategies and behaviours they use as they undertake these tasks or investigations. These might include:

Strategies	Behaviours
Draw diagrams and tables	Discussion with peers
Look for patterns	Check reasonableness of result
Make and test predictions	Self correct
Use trial and error	Reflect
Estimate	Persevere with task at hand
Work backwards

Questioning of students is an essential tool in the learning process. It not only helps to keep students on track but also helps them clarify and refine concepts, strategies and procedures. While students are undertaking mathematical tasks or investigations, teachers may find the following types of questions useful.

Questions that focus students’ thinking in a general direction

• What does this remind you of?
• How many ways can you find to…?
• How could you sort these…?

Questions that assist students to focus on particular strategies and to see patterns or relationships

• What is the same?
• What is different?
• How do you know?
• Can you find more examples?
• What are some things you could try?
• Can you see a pattern?

Questions that require students to explain what they are doing

• What have you discovered?
• Explain to me what you are thinking.
• How did you find that out?
• Why did you do it that way?
• Why do you think you are correct?

Reflective questions to draw together the efforts of all students

• Who has the same solution as this one?
• Who has a different solution?
• Have we found all the possibilities?
• Do you think we have found the best solution?

The way students go about completing open-ended mathematical tasks and investigations can give teachers valuable information to improve mathematics teaching and learning within the classroom. In addition, students experience real mathematical activities—activities that are not contrived—which allows them to experience success in mathematics.

References
This article is an excerpt from Ways of Working with Maths Plus for Queensland Investigations Edition © Oxford University Press 2009
To view the latest titles supporting the investigative approach, visit:
Maths Plus Queensland Investigations Edition