# Two types of prompt

In mathematics education, inquiry is commonly divided into two types, which mirror the division in the subject between pure and applied, and each type of inquiry starts with a different type of prompt.

Staples (page 4), for example, distinguishes between, on the one hand, inquiry into mathematics when students delve into ideas and concepts and come to an understanding of the structures and power of the subject; and, on the other hand, inquiry with mathematics when students apply mathematical tools to solve problems and model situations that arise in contexts outside the discipline.

We characterise the two types of inquiry as, respectively, disciplinary inquiry and modelling.

Each type of inquiry is, in turn, generated from different types of prompts. For Artigue and Baptist (page 5), the prompt is either an internal question about the objects and ideas that mathematics itself creates or an external question that arises from the natural or made world, including artefacts, art and 'daily life' problems. (See here for an example of the latter in trans-disciplinary inquiry.)

Limitations

One limitation of disciplinary inquiry is that students might feel that the inquiry, in arising outside of their immediate experience, lacks authenticity. Another limitation is that, in focusing on mathematical objects, inquiry might reinforce a 'silo' mentality about school subjects that ignores the connections between them.

A limitation of modelling is that, in focusing on using mathematics as a tool in a context outside the subject, students apply only the concepts they already know rather than learn new ones. There is also the danger that students, in being drawn into practical and concrete issues, do not learn about the abstract nature of the subject and, in particular, algebra.

Connections

Rather than reinforce the dichotomy between pure and applied, inquiry teachers benefit from emphasising the connections between the two types.

Mathematics develops through inquiry into external questions. As Vygotsky noted, even the most abstract mathematical concept "contains a clot, a sediment of the concrete, real and scientifically known reality, albeit in a very weak solution."* That echo of real relations is evident in, for example, the origins of algebra in problems related to land distribution, inheritance and salaries, and the roots of calculus in the area shapes and gradients of curves.

The reverse is also true. While mathematics develops through the "essential motor" of internal questions (Artigue and Baptist), the answers to those questions often have applications outside the subject. Even the most abstract algebra has contemporary applications to climate modelling, computer technology, and interpretations of medical data amongst many others.

Not only does an appreciation that mathematics originated in human activity and continues to fuel developments in that activity broaden students' perspectives, but it also combats the complaint that learning mathematics is irrelevant to life outside school.

Inquiry Maths prompts

So why are Inquiry Maths prompts exclusively of the first type? That is, why are they internal to the subject and predominantly generate inquiry into mathematics?

One answer relates to constraints that schools and curricula place on the inquiry teacher. School timetables that break the day into separate subjects militate against considering external questions. Moreover, a mathematics curriculum, built, by definition, on concepts internal to the subject, further circumscribes the field of inquiry. Both factors lead teachers to using the first type of prompt.

However, even without those constraints, Inquiry Maths prompts would be internal to the subject. That is because disciplinary inquiry has the potential to change students' attitudes towards mathematics and increase their sense of agency in classrooms. Students realise that inquiring into mathematics is itself a human activity over which they can exercise control.

Rather than seeming like an external (and imposed) body of knowledge, mathematics becomes both an object to question and explore and a tool to use in questioning and exploring.

Andrew Blair, March 2023

* Vygotsky, L. S. (1997). The Historical Meaning of the Crisis in Psychology: A Methodological Investigation. In Rieber, R. W. and Wollock, J. (Eds.) The Collected Works of Lev Vygotsky, Volume 3: Problems of the Theory and History of Psychology. New York: Plenum Press, p. 248.