## Introduction

I was surprised to see no reference to developing modelling as a mathematical practice? Why is that?

We agreed to have an email conversation about modelling in the (my) maths classroom - this post contains an edited version of this conversation.

## Time

Me: I think modelling is an incredibly powerful teaching approach, particularly for the applied (statistics and mechanics) A-level modules which I currently teach - for which it is the whole point! I worked as a financial 'modeller' for a number of years (and have an MSc in Applied Statistics), so I do understand how mathematics can be applied in the 'real-world'..

Geoff: I am a little surprised that you have a background in modelling yourself. My experience is that a lack of any recognition of modelling is often precipitated by a teacher being a 'pure' mathematician who finds engagement with the puzzles that school mathematics can provide satisfying enough - and I guess a lack of experience of modelling themselves. So good news that you understand the importance of modelling and applications.

Me: Yes, I definitely do, but… there are important constraints that are stopping me from creating a 'modelling classroom’, namely the

*time*needed to (1) create, and (2) explore, authentic modelling situations.

Although I have a reasonable amount of experience-of and motivation-towards modelling, I still lack the time, and perhaps the expertise, required to create and present (units of) modelling lessons whilst ensuring the students come out the other end having learned what needs to be learned.

As a teacher with limited resources I make decisions on how I can meet conflicting demands, and I often adopt a 'simple' approach that circumvents 'real-life'. This of course creates a conflict; I know that teaching statistics and mechanics (and to a lesser extent pure) only really makes sense in context, that mathematics should be meaningful and relevant to students, and I admire those very few (if any) full-time classroom teachers that manage to pull it off!

Geoff: I note your honesty in suggesting that you don't have the experience and expertise to develop modelling lessons that would give you confidence that you are both covering content as well as providing students with insight into an important mathematical practice. But it is this deficit for students that underpins my concern.

If we look at the data of student progression from A Level to degree programmes we can get a sense of the scale of the problem. In any one cohort approximately some 9 thousand students go on to study mathematics degrees and something approaching 52 thousand study some form of engineering or physical science degree. These engineering and physical science students will all engage with mathematical models and applications at university with many having to engage in mathematical modelling themselves. This means that they will have little or no experience of what this entails. And of course these students are just a very obvious tip of an iceberg - many other undergraduates will also engage with models and applications. From this perspective alone I would argue that we need to include some experience of models and applications.

Me: Yes, I agree, the modelling process is fundamentally important, although I also enjoy, and think there is value in, solving 'pure' mathematics problems...

Geoff: I’m in the same boat with this. I particularly like working with algebra, functions, graphs and geometry. But my experience is that we have plenty of supporters of mathematical practices associated with maths in these areas so I’m concerned to support modelling and applications above all.

## Meaningful modelling

Me: I totally agree. My entire reason for teaching (maths) is to try to give students opportunities to experience wider mathematical and social elements, but as I say: it is difficult to create truly

*meaningful*modelling situations.

As an example, I spent quite a long time last weekend trying to find some interesting/relevant datasets that exhibited exponential growth, but was largely unsuccessful. I could have explored some bacteria data, but this is not the sort of modelling I have in mind.

If I am going to spend the time planning and exploring modelling situations in my classroom, I really want to model something *important*, in the sense that it *matters*, perhaps politically, or socially, perhaps having a *local* relevance for the students in the classroom - this paper has been an inspiration (one of the authors, @LaurieRubel also suggested citydigits.org, which looks exciting). There are also some interesting ideas in this book, and this new ATM book (written by @PeteWrightIOE) looks really interesting. The idea that mathematics can be used to explore social justice, making decisions about important issues, is the kind of thing I am interested in bringing to my students.

In the past, I have created some resources of my own (for KS3/4), such as this on the UK 2010 election, but this quality of modelling experience takes time to plan and explore, and is also difficult to align with the curriculum.

Some of the other online modelling resources, such as Dan Meyer's blog, don't really fit what I would class as meaningful modelling, and can feel contrived, or of little relevance/import to students' lives; if I am going to spend the time bringing modelling situations to my classroom, I want to address matters of importance, socially or politically.

Geoff: Yes, I’m interested see how Dan Meyer promotes a sort of pseudo-modelling that seems to be quite popular among certain teachers. I think one aspect that appeals is that he suggests a narrative that is immediately accessible. On the other hand some of the questions are not particularly meaningfully tackled using mathematics seriously. The same is true of some of the Fermi problems that are promoted by the likes of Tim Gowers – although I entirely agree with his recent comments on the lack of imagination in maths teaching.

Me: Yes, I am inclined to agree with him. But it comes back to the constraints, and the ability, to bring meaningful modelling situations into the classroom. The lottery article I mentioned above is a brilliant example: it is authentic, full of local and social relevance, but was created by people with a vast amount of expertise and time. This kind of meaningful project is not easy to produce, or re-produce, in the classroom. Are there any resources that you might recommend that fit these criteria, what I would call meaningful modelling, that are readily available?

Geoff: Interestingly there was a time when modelling was an essential part of the A Level curriculum and there are resources that were developed at that time. An important set of resources that did this really well was the SMP 16-19 course materials.

My own career move to work in maths education in a university was precipitated by my interest in modelling in mechanics and I joined the Mechanics in Action Project at the University of Manchester. Our group contributed to the SMP 16-19 materials and also produced a wide range of other resources that supported the teaching of mechanics more generally. As part of that work we produced a number of practical kits to support practical work and teacher-led demonstrations in the classroom. These helped teachers promote mathematical models and modelling at different scales whilst also 'covering' content. I'm going to suggest at this point you take a look at the Mechanics in Action resources and let me know what you think about the suggested classroom activities. Would these provide you with a starting point to try something in your classroom?

Me: Thanks for this. Yes, they are simple enough to be do-able, although I would have to talk to the Physics department (or perhaps the premises staff!) to get the equipment; I will definitely consider using them when teaching mechanics next year.

But again, I come back to my concern about coverage of content: it's all well and good experimenting with pulleys on strings and so on, but is this really a much more valuable learning experience than solving a few mechanics problems?

Also, I might ask: are these activities really meaningful, or motivational, for students? Perhaps they will give students a better 'feel' for the kinds of problems they will ultimately face, but are they really motivated by a few pulleys, or a car rolling down a ramp? Perhaps they might be, but if I am to devote the time to modelling, I want it to have real

*meaning*for the students, to

*transform the way they view the world...*

Geoff: Well, I’m with you all the way on this. I want to see modelling tasks that really do focus on meaningful applications and I also want to work on developing tasks and lessons in which reality and mathematics are mutually supportive, indeed preferably where understanding the ‘reality’ being modelled leads to understanding of the mathematics. I’ve done some writing about this recently in a journal article - you can find the final version that is not copyrighted here - but first of all try out this Four Card Problem. Maybe this gives some insight into how I believe mathematical thinking can be facilitated by context.

Core maths seems the ideal space in which modelling and applications might be developed – although I hope it’s not too late by then and we may have already lost some of students. The qualifications that the awarding bodies have developed all try to address the issue in different ways. Are you likely to get involved in teaching in this post-16 area?

Me: I have no plans to, but will definitely consider it. Thanks for this conversation, it has made me think very carefully about how I can introduce meaningful modelling into my teaching next year.