Imagine a young girl sitting at her desk, having just been given a maths task that seems a bit more challenging than usual. The task looks intriguing, but she’s not sure how to tackle it. Around her, a hum of focused activity fills the classroom. The walls are adorned with reminders that mistakes are a natural part of learning, and that maths isn’t about memorising but about making sense. Despite being unsure where to start, the problem piques her curiosity. She takes a deep breath and whispers softly to herself “This is tough, but so am I”. She looks up and sees her teacher, smiling encouragingly. With determination lighting her eyes, she picks up her pencil and gets to work.
What’s required for students to succeed in mathematical thinking and problem solving? Alan Schoenfeld has argued in various places that there are five key factors: the mathematical resources available to them, including content knowledge, mathematical processes, and practices; their facility with problem-solving strategies; effective metacognitive tools such as monitoring and self-regulation; their beliefs about themselves and mathematics; and the environments and contexts in which they engage with mathematics.
Today I want to focus on students’ beliefs about themselves and mathematics — what’s often referred to as a productive disposition — and illustrated in the vignette above. Productive disposition is one of the five strands of mathematical proficiency described in the seminal ‘Adding It Up’ report by Kilpatrick et al. (2001), who defined it as “habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy” (p. 26). The inclusion of disposition as a strand recognises that success in mathematics is influenced by affective factors, as well as cognitive ones. In other words, for students to successfully develop and attain mathematical understanding, skills and competencies, they also need to believe that mathematics is understandable, can be learned and used, and that with effort, they are capable of figuring it out for themselves.
A range of affective factors has been shown to be correlated with achievement in mathematics. The 2012 Programme for International Student Assessment (PISA) results for mathematical literacy indicated that each of the following was significant for students:
intrinsic motivation (i.e. being interested in and enjoying mathematics)
instrumental motivation (i.e. believing that mathematics is useful for their future)
self-concept (i.e. believing that they are good at mathematics)
self-efficacy (i.e. believing they can succeed with a particular mathematics task)
responsibility for one’s own success in mathematics.
Thomson et al. (2013) reported that the relationship between self-concept and achievement is much stronger than that between instrumental motivation and achievement. They also found the relationship between self-efficacy and achievement to be even more pronounced. While bearing in mind that correlation is not the same as causation, these findings appear to support Kilpatrick’s position that the affective and cognitive aspects of learning mathematics, as articulated through the five intertwined proficiencies, develop together and bolster each other. (It will be interesting to see whether these relationships have changed when the results of Australia’s student characteristics for PISA 2022 are released in May 2024.)
Those familiar with the Australian Curriculum: Mathematics will recognise that four of Kilpatrick et al.’s (2021) five proficiencies, renamed for Australian contexts, were adopted: understanding (termed conceptual understanding by Kilpatrick et al.), fluency (procedural fluency), problem solving (strategic competence), and reasoning (adaptive reasoning). But what happened to productive disposition?
Peter Sullivan, one of the architects of the national curriculum for mathematics in Australia, explains that “disposition was taken to refer to pedagogical approaches” (Sullivan 2012, p. 182) and that the Australian Curriculum: Mathematics was not to prescribe approaches to teaching. Another reason was that “the proficiencies [were] intended to inform assessment and particularly the articulation of standards, and it is neither meaningful nor appropriate to set standards for disposition”. Sullivan goes on to say that “the omission of disposition is not intended to infer that disposition is less important than other proficiencies”. This, however, causes some problems. If we don’t have clear descriptions of what is meant by ‘productive disposition’ then how do we know how to foster them? As one example, Ollie Lovell describes his difficulties with identifying and ‘teaching’ them here.
To help conceptualise productive disposition, I propose a framework comprising four dimensions:
Does the student have productive thoughts about themselves mathematically? Does the student believe they are capable in mathematics? Do they have a growth mindset? Do they view mistakes as a natural part of learning mathematics? This dimension focuses primarily on self-concept.
Does the student have productive thoughts about the mathematics they are learning? Do they value learning it? Are they interested and curious? Do they believe that it should make sense? This dimension brings together intrinsic and instrumental motivation.
Does the student have productive thoughts about how they engage with the mathematics they are learning? Are they willing to engage with the task? Do they believe in their ability to succeed? Are they confident and resourceful? Do they know how to recruit support? Do they demonstrate resilience and perseverance? Are they taking responsibility for their own learning? Are they reflective, and able to identify their strengths and areas for growth (i.e. make an assessment of the mathematical resources available to them)? This dimension brings together self-efficacy and responsibility for one’s own success, and includes the metacognitive aspects described by Schoenfeld.
Is the student learning in a productive environment? Do the people around them see them as mathematically capable? Are they provided opportunities to engage in cognitively demanding tasks and in meaningful ways? To use their own mathematical ideas? To participate in mathematically rich discussions with teachers, peers, and others? To collaborative with others? What messages are they receiving about how to be good at mathematics and who can be good at mathematics? This dimension is pivotal in recognising that factors beyond students’ control can influence their disposition toward mathematics, including classroom, home, and societal attitudes about mathematics.
The framework demonstrates that productive dispositions can be situational, influenced by the type of mathematics being learned, the tasks being undertaken, and the environment in which learning occurs. For example, I have a positive attitude towards mathematics and my own ability to engage with it, but certain types of mathematics and specific tasks can influence my disposition. Evaluative or competitive environments turn me off completely; I’m much more engaged in collaborative, curiosity-driven settings1. As shown through this example, the framework suggests specific ways that teachers can help foster productive dispositions in their students. This includes acknowledging how their own beliefs about mathematics, teaching, and learning affect not only what students learn but also how they perceive themselves as mathematics learners. (I wrote more about that in this blog post.)
Like the proficiencies, these dimensions are intertwined, with each developing and reinforcing the others. Success in solving a challenging mathematical task can increase a student’s confidence, which in turn may make them more willing to persevere with challenging tasks in the future. Similarly, the absence of one of the dimensions, such as a supportive learning environment, can diminish one’s disposition towards mathematics. I believe that all four dimensions must be present to a certain degree for a student to have a productive disposition. I’m not yet certain what those levels might be, but I’m interested in exploring further.
What are your thoughts on my framework? I’m interested to hear what you think might be missing, where it could be improved, and whether it could help teachers conceptualise productive dispositions differently. Do you focus on productive dispositions in your teaching? If so, how? Please, get in touch!
References
Kilpatrick, J., Swafford, J., Findell, B., & National Research Council (U.S.) (Eds.). (2001). Adding it up: Helping children learn mathematics. National Academy Press.
Schoenfeld, Alan H. (2022). “Why Are Learning and Teaching Mathematics So Difficult?” In Handbook of Cognitive Mathematics, edited by Marcel Danesi, 1–35. Cham: Springer International Publishing, 2022. https://doi.org/10.1007/978-3-030-44982-7_10-1.
Sullivan, P. (2012) The Australian Curriculum: Mathematics as an Opportunity to Support Teachers and Improve Student Learning, In B. Atweh, M. Goos, R. Jorgensen & D. Siemon, (Eds.). (2012). Engaging the Australian National Curriculum: Mathematics – Perspectives from the Field. Online Publication: Mathematics Education Research Group of Australasia pp. 175-189. Access for free on the MERGA website: https://merga.net.au/Public/Public/Publications/Publications_Info_Other_Pubs.aspx?hkey=cdd19cd7-b1f4-4f42-9d93-0b559d91c333
Thomson, S., de Bortoli, L., & Buckley, S. (2013). PISA 2012: How Australia measures up. Camberwell, VIC: Australian Council for Educational Research. https://research.acer.edu.au/ozpisa/15/
[1] Eugenia Cheng terms these as ‘ingressive’ and ‘congressive’. Ingressive behaviour includes focussing on oneself, valuing individualism and taking an aggressive or combative approach to tasks. Congressive behaviour is concerned with community and connectedness and taking a collaborative and cooperative approach to tasks. Read more here: https://www.protectpuremaths.uk/blog/mathematical-thinking-for-a-new-approach-to-gender
