As part of a suite of actions to address the teacher shortage in Australia, the Federal Government recently released its latest reform agenda for initial teacher education (ITE) titled ‘Strong Beginnings: Report of the Teacher Education Expert Panel’. Among other actions, the report identifies and mandates four areas of core content that must be included in every accredited ITE program. I read, with great interest, the section on ‘numeracy’ and tweeted my initial reaction. (Spoiler, I didn’t like it much.) While ‘the Panel acknowledges that the core content is not intended to cover everything a beginning teacher should learn’, days later I am still thinking about the unstated but implicit beliefs emanating from the report about what mathematics is and how it should be taught.
Researchers have explored beliefs about the nature of mathematics, including the ways in which they influence teachers’ practice. Some of the earliest work is from the late 1970s, when Alba Gonzalez Thompson (1984) used a case study approach to identify differences in specific elements of teachers’ conceptions of mathematics and mathematics teaching. Paul Ernest (1989) drew on this work to conceive three philosophies of mathematics, which he conjectured to be hierarchical:
- The instrumentalist view: “Mathematics is an accumulation of facts, rules and skills to be used in the pursuance of some external end.” Here, mathematics is viewed as a collection of disconnected, separate entities.
- The Platonist view: “Mathematics [is] a static but unified body of certain knowledge. Mathematics is discovered, not created.” Here, mathematics has a “consistent, connected and objective structure”.
- The problem-solving view: “Mathematics [is] a dynamic, continually expanding field of human creation and invention, a cultural product. Mathematics is a process of enquiry and coming to know, not a finished product, for its results remain open to revision.” Here, mathematics is “a dynamically organised structure located in a social and cultural context”.
However, our view can depend on the context. For example, one could argue that a ‘numeracy’ focus in school—accumulating facts, rules and skills that will be needed in everyday life—accords with an instrumentalist view of mathematics. In contrast, the creative and curiosity-driven pursuits of a research mathematician align to a problem-solving view of mathematics. These beliefs may then interact to influence ideas on how school mathematics should be taught. In her 2012 paper, Kim Beswick used the three categories above to consider different combinations of beliefs about school mathematics and the discipline of mathematics more broadly, and how they might impact on teacher practice. For example, someone holding the two beliefs described earlier in this paragraph might believe that while mathematics as a discipline is creative, you need to have a set of basic skills first and that mathematical creativity is not for school.
I propose to adapt this framework to contemplate beliefs about learning mathematics at school that may held by anyone, including teachers. To do this, I think a few changes need to be made. First, another dimension needs to be added to acknowledge that these beliefs may not interact. For some people, the school curriculum *is* the entirety of their view of mathematics and there is no conception of the broader discipline. Alternatively, for some people, there is no reason why there should be a relationship between the two contexts, that is, mathematics exists outside of but is separate from school mathematics. The figure below shows Beswick’s original table with these additions shown in italics. I have also made small alterations indicated with strike-outs to remove reference to ‘the gifted few’ and ‘some’ students so as to disconnect the meanings from beliefs about learners and learning. However, I have shown with a [T] or an [S] where I think a combination lends itself to a more teacher-led or student-centred approach to learning mathematics. (There are a lot of contestable ideas in this paragraph; I’m interested to hear what you think!)
Everyone who has encountered mathematics at school should be able to locate themselves on this modified grid. Can you? Perhaps different cells are relevant at different points in your mathematical journey.
That is certainly true for me when I map my maths autobiography in the grid below. I began [1] with no sense of what mathematics might be outside of the classroom. And, for much of my schooling in traditional teacher-led classrooms, I failed to see the deep connectedness of mathematical concepts. (The puzzle corner and other rich experiences existed outside of my conception of mathematics.) At some point [2] I realised that mathematics could be studied at university. I also started reading books in the genre of ‘recreational mathematics’, although I’m not sure I would consider that a shift to a problem-solving view of mathematics at this stage. It was probably during my undergraduate years at university [3] that I saw the connectedness of mathematical concepts but it wasn’t until I experienced a Mathematics-In-Industry Study Group [4], where I saw mathematicians creatively and collaboratively approach mathematical problem solving that I had a transformative moment in my conception of the discipline of mathematics. This marked my shift to a problem-solving view of the discipline. However, I still hadn’t reconciled those experiences with my implicitly held beliefs about school mathematics. It was only once my pedagogical repertoire developed and grew [5] that I began to conceptualise how school mathematics could reflect the creative and curiosity-driven pursuits of mathematicians. And, it was only once my beliefs about mathematics learning evolved that I believed all students were entitled to learn mathematics through ‘a process of enquiry and coming to know’.
This vignette shows how experiencing mathematics in different ways, at different times, in different settings, and with different people, shapes and moulds our beliefs about mathematics. And that’s what leads me back to thinking about politicians, policymakers and others in positions of power to influence mathematics education. If their mathematical experiences are narrow, if ideas of creative and curiosity-driven mathematics are completely alien to them, then how can we expect anything other than a narrow vision and policies for mathematics education? Of course, these people are being advised by experts. But depending on who is being consulted, it may just be a compounding of limited perspectives. If you don’t have a vision of what mathematics could be like, then how do you know what’s possible?
Paul Ernest wrote that ‘Teaching reforms cannot take place unless teachers’ deeply held beliefs about mathematics and its teaching and learning change.’ I would go further to say that teaching reforms cannot take place unless politicians’, policymakers’, and others’ deeply held beliefs about mathematics and its teaching and learning change. Richer mathematical experiences help shape beliefs. Maybe we should start asking those in positions of power about the shape of their mathematical experiences?
References
Beswick, K. (2012). Teachers’ beliefs about school mathematics and mathematicians’ mathematics and their relationship to practice. Educational Studies in Mathematics, 79, 127-147.
Ernest, P. (1989). The impact of beliefs on the teaching of mathematics. Mathematics Teaching: The State of the Art, 249, 254. https://education.exeter.ac.uk/research/centres/stem/publications/pmej/impact.htm
Thompson, A. G. (1984). The Relationship of Teachers’ Conceptions of Mathematics and Mathematics Teaching to Instructional Practice. Educational Studies in Mathematics, 15(2), 105–127. http://www.jstor.org/stable/3482244