# It's models all the way down

- Cambridge Mathematics
- Mathematical Salad
- It's models all the way down

## It's models all the way down

A system can be broadly defined as a set of elements and ways in which they interact with each other to form a complex whole (Bertalanffy, 1950). In that sense, everything in the universe as well as everything in our minds is a giant system. To make sense of things, we simplify and prioritise, making working models of (sub)systems appropriate to various different scales interest, ranging from cosmological to biogeophysical to social to physiological. The elements and rules that make up a system need not be purely physical; mathematics, for example, can be thought of as a system of concepts and practices that can be related to each other in defined ways.

Mathematics education, then, is a system of learning and teaching mathematics, involving not only the field of mathematics itself but also how students, teachers, and learning contexts are involved in building understanding. Everyone’s various roles and individual experiences in that system provide a certain perspective on it. What sort of tool or reference structure might help us all to build on what we hold in common? How can we make a useful simplification? Do we start and stay simple, or do we at first lay out more detail than we need so that the final version can benefit from good feedback and responsiveness to other perspectives? As we work on the Cambridge Mathematics framework, we keep these questions in mind. We have provisional answers, but since it is a simplification of something much more complex, we are mindful that the value of the structure we design will be determined by how we inform its priorities and tradeoffs.

It takes mathematicians many years to conceptualise and reconceptualise the field as they go, and many would not claim to understand it completely. Those of us who do not become professional mathematicians only ever experience parts of that system and that learning process, even though the mathematical knowledge of non-mathematicians may tie in more extensively with a variety of applications, e.g. research, the sciences, teaching, and critical citizenship.

Humans are good (sometimes too good) at imagining patterns in systems; even when the patterns we detect are imperfect or don’t really hold up when tested systematically. We typically cannot directly perceive or perfectly comprehend every piece and process of important systems; instead, we try to build and work with the best models we can. A model is a simplification that emphasises some features of a system at the expense of others so that the model can be used in a way that helps the user to act meaningfully in the real world (Norman, 1983).

Models and modelling are an integral part of mathematics teaching and learning. They’re also a significant part of curriculum design, assessment design, professional development, and research to develop and test theories in mathematics education.

** 1. 2. 3.**

(based on Norman, 1983)

Each set of actors involved, directly or indirectly, in teaching and learning mathematics in schools, has an incomplete perspective on the whole, and a set of experiences and assumptions underlying that perspective. In the picture above…

- People with expert understanding of the system from a variety of perspectives contribute to creating conceptual models designed for experts to teach to novices in order to help the novices bootstrap their understanding effectively.
- Pedagogy, an inseparable part of the conceptual model, ensues, and students build their own understanding as a mental model of the conceptual model which can be refined over time – and in time may help them extrapolate to insights about the real system.
- In order to improve conceptual models used for teaching, researchers develop studies that help them integrate evidence from classroom practice to consider what happens when learners engage with the conceptual model.

**Ideally, this research feeds back to curriculum development and teaching practice** along with more immediate-level feedback teachers receive as they work with students using a particular curriculum, and mid-level feedback curriculum developers receive from summative student performance data. But how?

Educational design research is one category of approaches for systematically integrating evidence from research into models for learning and evaluating the result. Researchers at the Shell Centre at the University of Nottingham call this an engineering approach, since it depends on evaluating and refining designs based on their performance. This can be applied to scales ranging from a fine-grained look at learning particular concepts to a system-wide examination of sets of concepts and processes.

In the midst of these chains of constructs, there is a great deal of room for biases, agendas, and assumptions to have defining effects, even before other constraints and interests at jurisdiction level come into play. At Cambridge Mathematics, the framework team is building a model that will function as a tool to help inform refinements to curriculum, teaching, and professional development. We certainly have perspectives, biases, agendas, and assumptions, and one of our strongest design considerations so far has been making these articulated and transparent so that they can become an essential part of how this tool would be used in curriculum development. We’re tracking a number of choices made along the way, subjecting the design of this model - and our review of the research which informs our choices - to feedback from representatives of the groups that would be using it, and adapting our structure as a result. As we proceed, you can follow some of the precursors to a full discussion of the design and the methodology on the Research Base page of this site.