Mathematics by Inquiry The Australian government has announced funding of $7.4 million for the ‘Mathematics by Inquiry’ project in an attempt to improve the teaching of maths from early years to year 10. The project, which runs from November 2015 to June 2018, will prepare and disseminate inquiry resources for use in classrooms across the country. Alongside the resources, the project will provide teacher training related to assessing higher order thinking and supporting inquiry in STEM contexts. The project represents a huge opportunity to develop ideas about mathematical inquiry and create a model of inquiry that can have a systemwide impact. However, in the spirit of critical inquiry, we should examine the project proposal more carefully. Firstly, we should heed the experience of two previous largescale European projects to promote inquirybased learning in maths. The PRIMAS and Fibonacci projects ran between 2010 and 2013 in 12 European countries. The €9 million spent on the projects went to universities, with nearly €1 million going to three universities in the UK. One problem with the projects was their aim to promote inquiry in mathematics and science. This led to confusion, particularly in the PRIMAS theoretical documents, about inquiry in the two subjects. The inquiry processes, which are different in maths and science, were conflated into one set of generic stages (see this article for an extended discussion). Another problem has been the low impact of the projects on classrooms. In the UK, the universities promoted materials they had already developed in a handful of oneday conferences with highlypaid marquee speakers and restricted audiences. The teaching community is left with some disparate webbased collections of resources and training materials that hardly anyone knows exist. Secondly, we should analyse the statements from the two organisations chosen to manage the project in order to understand their conceptions of mathematical inquiry. The Australian Academy of Science (AAS) and the Australian Association of Mathematics Teachers (AAMT) were invited to submit 'desktop reviews' of the current state of maths teaching in Australia before being confirmed as the managers of the new project. Their answers to question 2 are of most interest to us. The question was: What is the role of inquirybased pedagogy in the teaching of mathematics? The AAS used its answer as an opportunity to raise objections to inquiry as a legitimate pedagogy in maths: The term ‘inquirybased pedagogy’ is problematic for mathematics education and terms such as a problem solving approach or an investigative approach are more commonly used. The pedagogy of inquirybased learning is founded on the principle that students should be actively and socially engaged in the process of learning, constructing new concepts based on their current knowledge and understanding. Inquirybased learning, as described in the research literature, often refers to highly student driven approaches where the student decides the questions to ask, the research methods to use, and different learning occurs for different students. This very open studentled interpretation of inquirybased pedagogy has only a very small place in mathematics. 
Instead the best investigative pedagogies for mathematics use ‘well engineered’ mathematical problems, where engagement in the problem solving process individually and with others and supported by the teacher will assist in the development of targeted concepts, or strategic skills, or the ability to transfer knowledge. (p. 11) This response is disappointing. Yes, inquiry teachers structure and guide learning (see levels of inquiry), but their ultimate aim is to develop independent inquirers who leave school able and enthusiastic to engage in open inquiry. The AAS define inquiry as problem solving in which the teacher 'engineers' the process from start to finish. Even more concerning is the AAMT document, which repeats exactly the same mistake made in the European projects; the document states that "there are clear parallels between science inquirybased approaches and contemporary thinking about pedagogy in mathematics" (p. 6). It then lists six generic inquiry 'principles': articulating goals, making connections, fostering engagement, differentiating challenges, structuring lessons, and promoting fluency and transfer. According to the AAMT, mathematics is "a practical vehicle for implementing these principles" (p. 7). Where are the processes associated with mathematical inquiry? Questioning, noticing, conjecturing, generalising, deducing and proving get hardly a mention in the AAMT’s document or no mention at all. The most interesting aspect of the project is its aim to base the inquiry resources on realworld contexts. I have written here about concerns at using 'real life' in mathematics classrooms, particularly at secondary school level, because it can inhibit the development of abstract reasoning. However, the Australian project has the opportunity to evaluate the role of realworld contexts in the inquiry learning of mathematics across phases of education. The Mathematics by Inquiry project is a great opportunity. It should seek to develop a distinctive approach to inquiry that is appropriate to the discipline of mathematics. Furthermore, the project managers should utilise the expertise that already exists in Australian schools, rather than, as in the case of the European projects, allocate the resources solely to higher academic institutions. In this way they will not only have greater credibility, but they will also have a greater impact.
Andrew Blair November 2015
Postscript Kath Murdoch (author of The Power of Inquiry and a primary teacher, international speaker and consultant based in Australia) responded to this article on twitter by saying that, "The focus should be on inquiry specific to the discipline and on generic, shared inquiry skills and processes." This comment opens up new avenues for inquiry. How are the generic and specifically mathematical inquiry processes merged in classrooms? Is the balance the same in different phases of education? You can follow Kath on twitter @kjinquiry.
