Keywords

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

1 Introduction

Mathematical modelling and applications is a central theme in mathematics ­education, evidenced by the many publications in journals, conference proceedings, and programs of the International Community of Teachers of Modelling and Applications (ICTMA), International Congress on Mathematical Education (ICME), and the International Commission on Mathematical Instruction (ICMI). In teaching and learning, modelling is variously covered from elementary school to tertiary ­education (Greer et al. 2007). This study reports on modelling from the perspective of applied mathematicians actively involved in undergraduate teaching and research. We take the position that the learning of mathematics will help develop competencies for extra-mathematical purposes. Extra-mathematical worlds are other domains outside mathematics, but which are in many ways served by ­mathematical applications (Blum et al. 2007).

With respect to the application of mathematics in the extra-mathematical or real world, Burkhardt (2006) reflects:

…there is no point in educating human automata; they are losing their jobs all over the world. Society now needs thinkers, who can use their mathematics for their own and for their society’s purposes. Mathematics education needs to focus on developing these capabilities (p. 183).

Burkhardt’s reflection in essence points to the urgency for mathematics education to take its rightful place in society, and play the role of linking mathematics learning with application, especially in tackling problems encountered outside mathematics itself.

Much research has been done on what kinds of competencies students need in order to engage in modelling. (Blomhøj and Jensen 2007; De Bock et al. 2007; Greer and Verschaffel 2007; Henning and Keune 2007; Houston 2007; Singer 2007). These studies describe a range of modelling skills, which serve to guide the teaching and learning, and assessment of modelling as a discipline. For example, the studies propose that modelling should be properly incorporated into the curricula, and should start in the early years of school, taking into account the appropriate mathematical disposition of the students. This is a crucial point, given that in general, modelling is not taught on its own, but within mathematics, making its status in the curricula unclear. A challenge for the teacher is then to incorporate appropriate models that help the students relate what they are doing in a mathematics modelling class to extra-mathematical world problems, without over simplifying the mathematics (see for example Greer and Verschaffel 2007, p. 220).

We propose in this chapter that if mathematical modelling in the classroom is to link with the real world, then there has to be some enculturation process where students, teachers, researchers, and educators, share a language and practices, and develop knowledge through communication (Lerman 1996; Nardi 2008; Sierpinska 1994; Vygotsky 1962; Wenger 1998). In that respect, our study focuses on one group in this shared community: the applied mathematicians. As part of the enculturation process, students need to learn what applied mathematicians do, what tools and language they use in the modelling processes. For their part, applied mathematicians have to understand the needs of the students by putting together learning activities and programs that build their competencies. In this study, applied mathe­maticians also assume the roles of researchers and teachers.

We seek to find out what modelling experiences applied mathematicians would give to their undergraduate students, by addressing the following specific questions:

  1. 1.

    From the perspective of applied mathematicians, what mathematical modelling experiences are needed at undergraduate level?

  2. 2.

    What might we learn from those experiences to inform teaching?

The purpose of the study is to record the modelling experiences of the applied mathematicians that will inform the teaching and the practice at that level. Our data comprise the narrative gathered from applied mathematicians, paying attention to the use of language and practices that are taken for granted in that community.

2 Methodology

Applied mathematicians in a university department of mathematics were asked by email if they would agree to be interviewed. Initially, it was not easy to find convenient time as many had commitments. The interviews were scheduled on an individual basis, convenient to each member contacted; the resulting interview period stretched over 6 months. There were ten respondents, all were interviewed, but in this chapter, only four interviews with applied mathematicians are presented because of space. The interviews took place in the mathematicians’ offices, to cut down on their time of moving to another location, but this also gave the interviewers an opportunity to see the work space of the interviewees, for instance, the tools, equipment, materials, and resources they mostly use in their teaching and research.

The interviews followed a qualitative design (Creswell 2008), with open questions. The initial responses were often followed by probes to get further clarification on what the interviewees meant. At the beginning of the interview, the interviewees briefly described their areas of teaching and research, and the problems they were working on. Later on, they described their mode of work on the problems: For ins­tance, did they use computers, drawings, paper, and pencil, and if so, on what kinds of problems were these tools used? Did they use mathematical concepts differently from their colleagues in pure mathematics? In the final phase of the interview, they were introduced to dynamic conceptual models (dynamic number line, and matrix transformations) designed on Dynamic Geometry Sketchpad (DGS) software. We conjectured that applied mathematicians would support the use of dynamic models; such models help students focus on the behaviour of things that are moving, and hence the concepts involved. After interacting with these models for about 30 min, they gave their feedback, relating it to their own practice and experiences. In the analysis of the video transcript, we looked for themes that related to the practices and language that are taken for granted in the community of applied mathematicians.

3 Results

The original data comprised video recordings. After transcribing the data, four major themes stood out from the applied mathematicians with respect to the modelling competencies they would like their students to have:

  1. 1.

    Finding similar examples or phenomena.

  2. 2.

    Connecting physical phenomena with abstract concepts.

  3. 3.

    Building models from the ground up.

  4. 4.

    Communicating broader context of a modelling solution.

We briefly discuss each of the four themes in the following paragraphs. We note that Joan, Jeff, Bob, and John (all pseudonyms) are the applied mathematicians, and Nathalie is the interviewer. The response to questions does not follow the order of names.

3.1 Finding Similar Examples or Phenomena

  1. 01.

    Nathalie: Can you talk a little bit about the mathematics … what is needed to mobilize the math in modelling …, what is it that you are good at?

  2. 02.

    Jeff: Part of it is having an encyclopedic collection of things that are important. You know that this thing has been done for these sorts of problem so you could extract things that are similar.

Here, in Jeff’s response we observe that modelling requires an open approach to dealing with problems, but it also demands some degree of preparedness to handle the problems. On the other hand, transfer of experience from one area to another is essential. That implies that students can draw on examples from other subjects such as physics, chemistry, biology, and bring them to enhance their modelling competences.

  1. 03.

    Nathalie: What else does the “encyclopedic” collection include?

  2. 04.

    Jeff: Having a wider understanding. In this process they have to identify what the tools are, and then also they have to know how to use the tools once they have identified the problem.

Use of “tools” is important in modelling. The tools mentioned by Jeff could be physical such as pencil and paper, or use of diagrams, computer simulations, but they could also be nonphysical tools such as a procedure or algorithm that one uses to solve a modelling problem. All these imply some knowledge of the problem area one is working on, and related information, or experience that have been built over time.

In the next interview, Joan described a modelling problem she had to deal with in biology.

  1. 05.

    Nathalie: So where did you start … when you wanted to model that, did you think of an equation, did you think of something more geometric?

  2. 06.

    Joan: The data was very suggestive of things that we’d seen in other contexts, so we had to think about what types of mathematical objects would give rise to such pictures.

  3. 07.

    Joan: That was much harder and I have to confess that the model was cobbled with different terms that each individually described certain aspects.

What are those “mathematical objects” Joan refers to in [06]? Sfard (1994) brings in the theory of reification – a transition from an operational to a structural mode of thinking in the formation of a mathematical concept. Although reification is beyond the scope of this chapter, it is worth mentioning here, just in case there might be any parallels to what Joan went through in [06].What comes out clearly from Joan, though is that she draws from her experience to make connections to similar situations she had seen in other contexts.

3.2 Connecting Physical Phenomena with Abstract Concepts

  1. 08.

    Nathalie: About the concepts that you use in modelling do you feel that they differ at all from what somebody doing pure mathematics might use?

  2. 09.

    Jeff: I think the concepts are not any different; it’s just a question of what types of problem you are interested in. The basic tools are, how can I write something down which describes certain aspects of what I’m interested in, in some sense?

  1. 10.

    Jeff: In terms of someone from pure mathematics, the questions that they are interested in are slightly different but the basic ideas are being able to make some abstract representation of what it is that they are thinking about. That’s the starting point and that’s always the same.

Bob’s response to the same question does not differ very much from Jeff’s.

  1. 11.

    Bob: Well, in modelling you have to have some physical model of what the object is, its structure, shape. So you have to write down the equations that actually describe that.

Having a physical model and abstracting this to a mathematical model is a key component of modelling. Students should not ignore the structure and shape of objects in modelling situations, because these can provide some hints for formulating mathematical solutions.

3.3 Building Modelling from the Ground up

  1. 12.

    Nathalie: Ok, so all of these [dynamic models] that you have looked at … we think of them as ways that help students focus on the behavior of things that are moving, …

  2. 13.

    Nathalie: …and our hypothesis is that the dynamic models might help them with modelling or applied situation. What’s your reaction to that?

  3. 14.

    John: When I look at a tool like this one [dynamic models] my first question is what would I use it for? I’m always keen on anything that gets people to play, anything that brings a sense of discovery and wonder, the fun thing.

  4. 15.

    John: Well, there is certainly a lot of test cases and playing around, proto­typing small cases, “what if I had to fit a square into a circle, what would really happen?”

We observe that John does not object to using dynamic models. This also supports our original conjecture that dynamic models might help students develop competencies in applied situations.

Jeff, still referring to the same question [12.13] says:

  1. 16.

    Jeff: Ah, you think of adjusting a parameter and seeing the consequences, for example in a dynamical system, you have some function that has a parameter, and as you adjust the parameter, the function is going to change.

We observe unanimity among the applied mathematicians about the use of dynamic tools or technology for experimentation purposes (what if?), prototyping, play, wonder, fun, and discovery. Modelling should incorporate all these attributes. Evidence from research also supports use of technology for exploration purposes. Papert (1980) has very good examples.

3.4 Communicating Broader Context of a Modelling Solution

  1. 17.

    Nathalie: Is there a different culture of writing in applied mathematics?

  2. 18.

    John: In applied math, there is always a lot of communication between the people who are working on the problem together, so a lot of writing, starting right away with something broader, with a lot more background, in some sense.

  3. 19.

    John: [With respect to students’ modelling] after working their solutions, some will say, ok, that is the solution, even if they made a mistake. You look at the solution and it can’t be anything physical. The physical world doesn’t work that way.

  4. 20.

    John: But they are resistant that they might be able to apply their intuition about the real world to the solution of the model.

Communication in general is arguably the most challenging aspect of modelling. Students tend to ignore it as not the main part of modelling but indeed it is a very significant part of modelling. Clear communication is important because without it, the process is incomplete. So students need to develop the skill of communicating their results, and the best way is through practice, through individual and group projects.

Modelling solutions should be realistic and amenable to the extra-mathematical world or physical reality (Blum et al. 2007). In communicating this solution, some background information is necessary to inform nonexperts in the subject area what the solutions to the problems are and some implications of the solution.

4 Discussion and Summary

The study probed the modelling experiences of four applied mathematicians, who are teachers at undergraduate level as well as being researchers. Four major themes emerged from the interviews:

  • Finding similar examples or phenomena: the importance of drawing examples from one’s experiences and using them in the modelling situation.

  • Connecting physical phenomena with abstract concepts: Moving from a physical model to a mathematical model, and [after solving], interpreting and communicating the solution in the real setting outside mathematics.

  • Building a model from the ground up: starting with a simple idea about a problem and experimenting, using appropriate tools, until the problem becomes clearer.

  • Communicating a mathematical modelling solution: recognizing the importance of clear communication of results that accommodates a wider audience, other than the expert audience themselves.

These four themes address the question of competencies needed in undergraduate mathematical modelling, although their application extends outside education. We agree with Burkhardt that mathematics education should contribute to developing the capabilities of students for their own benefit, and for the benefit of society as a whole. The other insight we get from this study is the notion of “play” in modelling.

I’m always keen on anything that gets people to play, anything that brings a sense of ­discovery and wonder, the fun thing (John, line 13).

This remark is rather surprising because at undergraduate level, play is not usually considered part of serious learning. On the other hand, John is probably advo­cating a different game, perhaps a more serious game than just “play.” A game where students explore concepts through testing, guessing, estimating, simulating, checking and cross-checking, in a “playful” way using tools that they have.

Furthermore, modelling demands adaptive expertise in nature and is a social activity which should be properly supported by good curricula (Greer and Verschaffel 2007). In the context of our study, adaptive expertise means an ability to interpret the context, environment, of a modelling problem, and to apply the requisite mathematical tools in resolving the task.

  • Having a wider understanding. In this process they have to identify what the tools are, and then also [they] have to know how to use the tools once they have identified the problem [10].

We have argued in this chapter that modelling is also a multidisciplinary undertaking because one must draw from many areas to formulate a mathematical problem. “Wider understanding” [10], also implies that students are being challenged to look beyond their subject areas, draw from different areas of learning, and also from their personal experiences.

  • The data was very suggestive of things that we’d seen in other contexts, so we had to think about what types of mathematical objects would give rise to such pictures [06].

This clearly demonstrates a multidisciplinary approach to mathematical ­modelling and application. Overall, we believe that our conjectures have been well supported by the data that we collected.