At its most fundamental level, an investigative approach to mathematics involves teachers beginning with students’ questions (Department of Education and Science [DES], 1982). Students are encouraged to think enquiringly, and teachers must be responsive to the student’s propositions by pursuing their curiosities. The Cockcroft Report (1982) underscores the importance of discussing the outcome of the investigation, even when it reveals a false trail because the analysis highlights the underlying value of the experience. Jaworski (2003a) supports the notion of investigations being student-centred by adding that investigations involve students in loosely-defined situations asking their own questions and following their own interests whilst setting their own goals and, importantly, having fun. Such a constructivist approach, allows its users to mirror the processes of professional mathematicians through exploration, enquiry and discovery (Bruner, 1960; Jaworski, 2003b).

In contrast, Greenes (1996) identifies that investigations present curiosity provoking situations, problems and questions that are intriguing and captivate the student’s attention and interest. This definition places the design onus squarely with the teacher to construct such an experience. However, Greenes (1996) substantiates the Cockcroft Report’s (1982) claim that identifying and evaluating different solution paths is crucial to the process.

AusVELS (2016) embodies the discussed literature above as it mentions choice making; communicating solutions effectively; students designing investigations; using mathematics to represent unfamiliar or meaningful situations; and using existing strategies to verify that responses are reasonable. It is only the last point that silences the discovery aspect of investigations, as strategies can and should be acquired along the way.

## Principles of an Investigative Approach

**• **PRINCIPLE 1: START WITH THE STUDENTS

Largely identified by DES (1982) and Jaworski (2003a), and embodying student voice (Manefield, Collins, Moore, Mahar, & Warne, 2007), starting with the students is the foundation upon which to construct a mathematical investigation. This principle can be made visible, for example, during teacher exposition when a teacher pursues a student’s question.

**• **PRINCIPLE 2: NEED BE NEITHER LENGTHY NOR DIFFICULT. PROCESS COUNTS!

There is a perception that mathematical investigations encompass an extensive piece of work that consumes much time to complete (Department of Education and Science [DES], 1982). Mathematical investigations are defined by their analytical and evaluative processes, which encompass higher order thinking, rather than their timespan. They could entail mathematical paradoxes and puzzles, real-world scenarios, and yes, long drawn out pieces of work.

**• **PRINCIPLE 3: TEACHERS PLAY A GUIDING AND SUPPORTING ROLE

This principle can be broken down further into two ideas: the teacher should pursue the interests and curiosities of the student when they arise (Department of Education and Science [DES], 1982); and the teacher’s own knowledge, experience and philosophy of mathematics supports student learning (Jaworski, 2003a). The essence of this principle is teacher’s providing timely feedback (Hattie, 2012).

**• **PRINCIPLE 4: EXPLORATIVE AND FUN

If we start with the student’s interests, then the elements of exploration, discovery and fun should be a natural extension. Investigations should rouse curiosity and wonder from the students, and should also encourage the student to ask question upon question (Greenes, 1996). Investigations invoke probing discussions and structured arguments that lead to students evaluating their own and each other’s approaches (Baroody & Coslick, 1998).

#### REFERENCES

AusVELS. (2016). Content Structure: Problem Solving In. *Mathematics.* Retrieved from http://ausvels.vcaa.vic.edu.au/Mathematics/Overview/Content-structure

Baroody, A., & Coslick, R. T. (1998). Fostering children’s mathematical power: An investigative approach to K-8 mathematics instruction: Routledge.

Bruner, J. S. (1960). *The process of education*. Cambridge, Mass: Harvard University Press.

Day, L. (2014). Purposeful Statistical Investigations. *Australian Primary Mathematics Classroom, 19*(3), 20-26.

Department of Education and Science [DES]. (1982). Mathematics in Schools. InMathematics Counts (The Cockcroft Report) (pp. 56-82). Retrieved from http://www.educationengland.org.uk/documents/cockcroft/cockcroft1982.html.

Greenes, C. (1996). Investigations: Vehicles for learning and doing mathematics. *Journal of Education*, 35-49.

Hattie, J. (2012). Visible learning for teachers. [electronic resource] : maximizing impact on learning: London ; New York : Routledge, 2012.

Jaworski, B. (2003). An Investigative Approach: Why and How? In. *Investigating Mathematics Teaching: A Constructivist Enquiry* (e-Library ed., pp. 1-13): Taylor & Francis.

Manefield, J., Collins, R., Moore, J., Mahar, S., & Warne, C. (2007). *Student Voice: A historical perspective and new directions*. Retrieved from Melbourne, Australia: www.education.vic.edu.au

Taylor, M., & Hawera, H. (2016). Work show?: From playing a game to exploring probability theory what can student. *Australian Primary Mathematics Classroom, 21*(2), 32.

Thomas, T. A., & Wiest, L. R. (2013). Video games as a context for numeracy development. *Australian Primary Mathematics Classroom, 18*(3), 29.