Studying mathematics in science: blind spots
To integrate mathematics into classroom science requires teaching practices that appropriately select ideas from one body of knowledge (mathematics) and recontextualize that selection within a second selection of ideas from another body of knowledge (science). A key issue is thus how different teaching practices shape the forms taken by ideas from those bodies of knowledge when they are brought together. Put simply, the question is: what teaching practices enable or constrain the integration of mathematical knowledge into scientific knowledge? These statements may seem unnecessary. However, studies of science education typically sideline both teaching practices and changing forms of knowledge. These blind spots arise from three assumptions that pervade the field: that knowledge equates to knowing, that education equates to learning, and that ‘science’ and ‘mathematics’ are self-evident.
Knowing and learning
The first assumption is that ‘knowledge’ comprises mental processes of understanding that reside ‘in the heads of persons’ (von Glasersfeld 1995: 1). Reflecting this ‘subjectivist doxa’ (Maton 2014: 3-14), research focuses almost exclusively on cognitive and affective ways of knowing. Knowledge as an object of study in its own right - one taking particular forms which have effects for bringing that knowledge together with other forms - is left out of the picture. Put another way, the assumption is that to analyze knowledge one must analyze ways of knowing. Rather than distinguish between students’ dispositions and what they are learning, as a precursor to exploring relations between knowing and knowledge, the only concern is the former. This assumption that knowledge is nothing but knowing is typically accompanied by a second assumption: that education is nothing but learning. When studying ‘ways of knowing’, research overwhelming focuses on student interactions, such as when solving a scientific problem. Teaching is rarely centre stage, if considered at all. Each of these assumptions thereby takes part of the picture for the whole.
This focus on learning and ways of knowing is illustrated by studies using the ‘resources framework’ (e.g. di Sessa 1993, Hammer 2000, Redish 2014, 2017), an influential approach to physics education research. The framework explores ‘how our students think’ (Redish 2014), such as ‘the student’s perception or judgement (unconscious or conscious) as to what class of tools and skills is appropriate to bring to bear in a particular context’ (Bing and Redish 2009: 1). The concern is how ‘cognitive resources’ are ‘activated in response to a perception and interpretation of both external and internal contexts’ (Redish and Kuo 2015: 573). Thus student perceptions are central - what their perceptions may be about, the forms taken by knowledge itself, is not analyzed. This subjectivism is thoroughgoing - everything is psychological. For example, the term ‘epistemology’ is used to refer not to inter- subjectively shared field-level knowledge practices but rather to personal frames of individual understanding (e.g. di Sessa 1993, Hammer and Elby 2002). Accordingly, disciplines such as mathematics are viewed as comprising‘ways of knowing’ (Redish 2017) and studies of mathematics in science explore how students solve problems in order‘to model their thinking’ (Bing and Redish 2009: 2).
Research using the ‘resources framework’ offers valuable insights into how students learn ways of knowing. However, learning is not the sum of education, and ways of knowing are not the sum of disciplines. Much remains missing from the equation. Such studies could provide a powerful basis for understanding the integration of mathematics into science lessons if they were complemented by analyses of teaching and analyses of knowledge. However, frameworks for bringing these issues into the picture are lacking in the wider field. Currently, whatever approach they are using, studies of science education tend to reduce knowledge to knowing and education to learning, generating blind spots in the overall field of vision.2 For example, studies of mathematics in science education draw on such frameworks as ‘thinking dispositions’ to suggest that attributes such as curiosity can help students shift from ‘rigidity of mind’ to ‘fluid thinking’ that ostensibly supports successful integration (Quinnell et al. 2013). Similarly, studies adopting a ‘cognitive blending framework’ analyze how students draw on ‘mental spaces’ when combining physical and mathematical knowledge (Bing and Redish 2007). Thus, everything lies in the mind of the beholder. Similarly, teaching is sidelined. The implications of studies of student learning for how mathematics should be taught in science often take the form of afterthoughts, as if teaching is merely an epiphenomenon of learning. Typically, such implications simply comprise calls for the integration of mathematics into science in teaching (e.g. Meli et al. 2016; Planinic et al. 2012), leaving unsaid what teaching practices would support that integration. In short, the widely shared focus on ways of knowing (rather than also knowledge) and on learning (rather than also teaching) limits current understanding of how pedagogic practices enable or constrain the integration of mathematics into science within classrooms.