Teaching mathematical modelling: a framework to support teachers’ choice of resources

Abstract

Mathematics teachers are often keen to find ways of connecting mathematics with the real world. One way to do so is to teach mathematical modelling using real data. Mathematical models have two components: a model structure and parameters within that structure. Real data can be used in one of two ways for each component: (a) to validate what theory or context suggests or (b) to estimate from the data. It is crucial to understand the following: the implications of using data in these different ways, the differences between them, the implications for teaching and how this can influence students’ perceptions of the real-world relevance of mathematics. Inappropriately validating or estimating with data may unintentionally promote poor practice and (paradoxically) reinforce in students the incorrect idea that mathematics has no relevance to the real world. We recommend that teachers approach mathematical modelling through mathematizing the context. We suggest a framework to support teachers’ choice of modelling activities and demonstrate these using examples.

1 Introduction

Connecting classroom learning to the real world is seen as important in teaching generally. This is especially true for mathematics (Gainsburg, 2008; Smith & Morgan, 2016) where the connection increases students’ understanding of mathematics, increases motivation to study mathematics and helps students to apply mathematics (Gainsburg, 2008). For example, one of the aims of the Australian Curriculum: Mathematics is to ensure that

students are confident, creative users and communicators of mathematics, able to investigate, represent and interpret situations in their personal and work lives and as active citizens. (Australian Curriculum, Assessment and Reporting Authority, n.d., p. 1)

That is, numeracy is inseparable from real-world contexts and so students must be exposed to mathematics in a wide range of real-world contexts (Department of Education, Training and Youth Affairs, 2000).

In mathematics, many means are possible for making connections with the real-world (Gainsburg, 2008), including using analogies, using word problems, facilitating discussions, using physical models, using simulations, analysing real data and developing mathematical models. One way to connect mathematics and the real world is to set the mathematics in real-world contexts:

Problem-solving in mathematics can be set in purely mathematical contexts or real-world contexts. When set in the real world, problem-solving in mathematics involves mathematical modelling. (The State of Queensland (Queensland Curriculum and Assessment Authority), 2017, p. 12)

The Common Core State Standards for Mathematics (Common Core State Standards Initiative, 2010) from the USA identifies the steps involved in modelling as follows: specify: identifying variables, assumptions and approximations; formulate: formulating the mathematical model to identify relationships; analyse: analysing these relationships to draw conclusions; interpret: interpreting the solution to ensure it makes sense in the context of the original problem; and refine: refining the model if necessary. This is often interpreted as a cyclical process (Fig. 1; Stillman et al., 2007). One way to teach mathematical models with real-world applicability (Kaiser, 2014) using this process is through using data.

Fig. 1.

Modelling as a cyclical process (adapted from Stillman et al., 2007).

Fig. 2.

A framework to support teachers’ choice of resources for mathematical modelling. ST and SE means that the model structure is based on theoretical or empirical methods; PT and PE means that parameters are estimated using theoretical or empirical methods.

Fig. 3.

The marble sifter. (a) The `sifter’ made from a cardboard box lid; (b) students sifting marbles using the sifter.

This paper was motivated by seeing teachers struggle to teach mathematical modelling with the unfortunate outcome that their use of real data convinced students that mathematics had no practical purpose. In this paper, we first give some background and discuss the role and importance of real data and then describe mathematical modelling. We then present a framework with some examples for teachers to use when deciding how to incorporate real-world contexts and/or data in their classrooms.

2 Background

Much has been written about teaching mathematical modelling. Galbraith (2011) describes six different meanings associated with mathematical modelling in schools: using real problem situations as a preliminary basis for abstraction, emergent modelling, modelling as curve fitting, word problems, modelling as a vehicle for teaching other mathematical material and modelling as real-world problem solving (pp. 280–282). In this paper, we are mainly focussing on the `modelling as a vehicle for teaching other mathematical material’ (p. 282). In this case, the curriculum determines the models chosen. This aligns with what Kaiser & Sriraman (2006) call an `educational modelling’ perspective (p. 301) where the purpose of the modelling is introducing and developing mathematical concepts and `the real world examples and their interrelations with mathematics become a central element for structuring of teaching and learning mathematics’ (p. 306).

Technology, including graphing calculators and software such as Excel, can be a useful tool for students to explore these mathematical models (da Silva Soares, 2015) without the need to restrict examples to artificial data that is easy to manage computationally. Modelling as a vehicle for teaching other mathematical material can entail difficulties. One of these difficulties is that despite the intention to make real-world connections, students and pre-service teachers can `exclude real knowledge from their solutions’ (Verschaffel et al., 1997, p. 339). For example,

[...] students demonstrated a very strong overall tendency to exclude real-world knowledge and realistic considerations [...] students’ lack of sense making when solving arithmetic word problems in a typical school setting is not caused by a strange cognitive deficit. Rather, the cause seems to be students’ beliefs about the role of mathematical word problems and how they should be treated and solved in the mathematics class. (Dewolf et al., 2014, p. 105)

In other words, students often think that mathematics is not useful for realistically describing the real world (Greer, 1997), and this may be reinforced by how mathematics is taught. Mathematical modelling includes `the connection between mathematical and non-mathematical contexts in this process’ (Falsetti & Rodriguez, 2005, p.15). Using mathematical modelling of real situations with real data can help students connect mathematics with the real world. In our experience though, modelling with real data can also have the unintended consequence of reinforcing that mathematics is disconnected from reality. The actual modelling process is vital for students’ learning and understanding:

The students’ responsibility is figuring out how to solve the problem as well as finding the solution. It is the strategies used for figuring out, rather than the answers, that are the site of the mathematical argument... (Lampert, 1990, p. 40)

[...] the demand that learners make connections between mathematics and the real world has been, and will continue to be, at the forefront of most major reform efforts (Carrejo & Marshall, 2007, p. 71)

Carreira & Baioa (2018) believe it is important to design tasks that `ascribe credibility to a modelling situation deliberately designed to stimulate what happens in a real situation’ (p. 202). Teachers must be honest with students when engaging them in experimental work to generate data or prototypes of actual objects. Students need to be aware of the authenticity of activities (Jablonka, 2007).

This paper focuses on how to use real data and how to approach the modelling: both are critical to promoting that mathematics describes the real world. We develop a framework to support teachers make these choices.

3 Teaching mathematics with real data

The use of real data and context in teaching has many advantages. They can emphasize solving real (rather than artificial) problems rather than just `theoretical’ and computational problems (Hand et al., 1996); increase student interest and engagement (Aliaga et al., 2005); counter the idea that mathematics is dull and dry; enable students to learn to good quantitative thinking habits; help students to understand why we do mathematics, not just how (Willett & Singer, 1992); ensure students remember the analysis as solving a real problem (Bradstreet, 1996); and become a trigger for later recall of the techniques (Singer & Willett, 1990). Real data can be found in books (Hand et al., 1993), online (Smyth, 2011; JSE Data Archive: http://www.amstat.org/publications/jse/jse_data_archive.htm) or in journal articles. Alternatively, students can collect their own data (for example, Carrejo & Marshall, 2007) or even obtain simulated data from more challenging and interesting situations based on realistic computer simulation (Huynh et al., 2016).

For these reasons, real data have a role to play in teaching mathematics, especially the mathematical modelling process. However, the role that data are allowed to play in a classroom impacts how students perceive the relevance of mathematics.

4 Models

4.1. Mathematical models

Models are used in a variety of human endeavours, and so many definitions exist for a `model’. All models compromise two features:

1. Accuracy: A model agrees with reality in important (but not necessarily all) ways.

2. Simplicity: A model is generally simpler than reality (i.e. is an approximation).

Rodgers (2010, p. 5) notes that a `mathematical model is one that captures these two features [accuracy; simplicity] within one or more mathematical equations’, while Blum & Ferri (2009, p. 45) define mathematical modelling as `the process of translating between the real world and mathematics in both directions’. Kaiser (2014) notes that `to create a mathematical model, the real-world model has to be translated into mathematics’ (p. 399) or `mathematised’ (Stillman et al., 2007). Mathematical models are usually built on theoretical or scientific means of relating quantities using mathematical functions (for example, differential equations). A consequence of `mathematising’ is that mathematical functions are allowed to naturally emerge. For this reason, mathematical modelling is an excellent way of showing students the relevance of mathematics in, and emergence of mathematics from, the real world.

Creating a mathematical model may be difficult but is an important conduit to learning:

...our continuing efforts to bring the discovery method to the classroom naturally go hand-in-hand with attempts to bring genuine applications to the classroom. The two efforts reinforce each other, and both are essential for a complete and honest presentation of mathematics in our schools. (Pollak, 1969, p. 403)

To be useful and relevant for describing the real world, mathematical models need to have meaningful interpretations, allowing students to sensibly `interpret’ their mathematics in the real world (Australian Curriculum, Assessment and Reporting Authority, n.d., p. 1). Dewolf et al. (2014) discuss examples of students dismissing the real world when solving mathematical problems and not even expecting mathematical solutions to be sensible in the real world.

4.2. Mathematical models have a structure and parameters

Mathematical models are useful for describing the relationships between variables; in this paper, we focus on the case of two variables only (a response or dependent variable and an explanatory or independent variable), though the ideas extend to more than two. Mathematical models have two components: the structure of the model (for example, is the relationship between the variables best described by a linear or exponential function?) and the parameters in that model (for example, what is the slope of the linear function?). The structure of a mathematical model can often be developed purely from a theoretical understanding of a situation. Sometimes the structure of the model has no underlying theory and is established based on the data (for example, a graph of the data may appear linear). In either case, the values of the parameters in that model usually need to be assigned values, which can come from a theoretical description of the situation or from the data.

Table 1

Different ways to use data when fitting models. Using data to validate or estimate can be done formally (using statistical methods) or informally

	Theoretical parameter values (PT)	Empirical parameter values (PE)	Role of data
Theoretical model structure (ST)	Useful for mathematical modelling in schools (ST, PT)	Suitable for mathematical modelling in schools in many cases (ST, PE)	Validation of model structure
Empirical model structure (SE)	(Impossible) (SE, PT)	Not recommended for mathematical modelling in schools (SE, PE)	Estimation of model structure
Role of data	Validation of parameter values	Estimation of parameter values

Establishing the structure of a model based on theory allows students to see the relevance of mathematics, where mathematical functions emerge naturally from the real-world development, understanding and description of the situation. Sometimes, the values of the parameters in the model also emerge through the same process. In contrast, developing a model (especially the model structure) purely based on the data presents numerous challenges, both for mathematicians in practice and especially for students in schools, as the suite of tools necessary to do this well are statistical, often complicated, not in the curriculum and perhaps produce a model that was not interpretable in the real-world context (Galbraith’s modelling as curve fitting).

4.3. Data can play two roles in model development

In model development, data can be used to validate or to estimate. Validation refers to the process of using theory and/or context to guide development of the model structure and/or the values of the parameters, then using the data confirms that the result is sensible and sufficiently accurate. That is

… the objective is not to confirm or deny the model (we already know it is not precisely correct because of the simplifying assumptions we have made), but rather to test its reasonableness [i.e. validity]. We may decide that the model is quite satisfactory and useful, and elect to accept it. Or we may decide that the model needs to be refined or simplified. In extreme cases, we may even need to redefine the problem, in a sense rejecting the model altogether. [...] this decision process really constitutes the heart of mathematical modelling. (Giordano & Weir, 1997, p. 38).

Estimation refers to using the data to determine the model structure or values of the parameters. If the structure is determined based on theory, then it can be validated by comparing to data; if no theory underpins the structure, then the data are used to estimate the structure. Similarly, if the values of the parameters are based on theory, then these can be validated using the data; if no guiding theory exists, then the data can be used to estimate the parameter values.

Validation and estimation can be performed using formal or informal tools. A crucial observation is that formal approaches using data require the use of statistical tools (such as residual analysis), which require experience to use well and appropriately and are usually not covered in the school curriculum. Even those few techniques that are covered in the school curriculum (such as R²) are often used inappropriately even by experienced researchers (Kvålseth, 1985). In many cases, informal processes might be suitable in the classroom (`there are no obvious departures from the proposed model’) and are often better didactically as they build conceptual understanding of the mathematics and ability to interpret the model to describe the real world. In Section 5, we discuss the challenges when teachers attempt to use statistical tools.

Hence, we see that real data have a role to play in the mathematical modelling process, but ensuring that the role is appropriate remains crucial. This has implications for classroom teaching.

4.4. Teaching implications

In general, mathematical models can be developed in one of three ways (Table 1).

1. Theoretical structure and parameter values: Both the model structure and the parameters values are determined from underlying theory. Data are then used for validating the structure and the parameter values. This approach is an excellent choice for teachers (see Cramer, 2001 for examples) as the mathematics emerges naturally in the description of the real world, and real-world data are used for validating the model. Hence, the modelling is a `vehicle for teaching other mathematical material’ (Galbraith, 2011, p. 282).

2. Theoretical structure and empirical parameter values: The model structure is developed from an understanding or description of the context, and data are used to validate the structure. The unknown parameter values are then estimated using the data. This approach is also a good choice for teaching as students can see the mathematics emerging from the real-world situation and once again the modelling is a `vehicle for teaching other mathematical material’ (Galbraith, 2011, p. 282). Estimation of the parameters can be as simple or as complex as necessary and may not even require statistical tools for parameter estimation. Formally, a statistical approach would be used for model validation and parameter estimation, but in a school situation, this may be unnecessary and even counterproductive. In fact, a non-formal approach to parameter estimation is often preferred as this clearly shows how the mathematics describes the real world. Without a formal approach, students may develop many varied ways to estimate parameters based on the meaning of the parameters (Section 6.3).

3. Empirical structure and parameter estimation: Both the structure and parameter values are estimated from the data. Situations often exist where the scientific or physical understanding of the relationship between two quantities is either difficult or impossible to determine. In these cases, the selected structure of the model is the one that `best’ fits the real data (for some definition of `best’). This approach is, in general, not recommended for use in schools, as the necessary statistical techniques are not available to students, and it encourages students to fit models (using a graphics calculator or Excel) without thinking about the real-world context. Galbraith (2011) calls this `modelling as curve fitting’ and states that such models

   [...] can be generated in complete ignorance of the principles underlying the real situation […] when used mindlessly it creates a dangerous aberration of the modelling concept. (p. 271).

4. Many reasons exist for why this approach should be avoided in schools, which we discuss in the next section.

The fourth combination—empirical model structure and theoretical parameter values—is unusual, if not impossible. If there is no guiding theory to suggest a model structure, then it is unlikely that theory will exist to determine the parameters of the structure with no theoretical basis. These approaches have been summarized in Table 1.

Validation and estimation use data very differently, and the tools necessary are different for both. Formal (statistical) methods of validation and estimation require appropriate knowledge to do well, while informal methods are simple and retain connections with the real-world context.

5 Empirical model structures present challenges for teaching

As noted above, data can be used to validate or estimate the structure of a model and/or to validate or estimate the parameter values. We argue that adopting a theoretical approach to developing model structure is the best way for students to see how mathematical functions emerge naturally to describe the real world and hence are useful for showing the relevance of mathematics in the real world (Galbraith, 2011). Using an empirical approach presents many challenges in the classroom. (Using a theoretical approach is also preferred in the real world, but sometimes an empirical approach is the only option.)

Firstly, an empirical approach to model structure is not usually required by the curriculum. The curricula require students to develop models by describing the real world but rarely cover formal empirical validation or estimation techniques for modelling with non-linear functions. The empirical approach imposes functions on a context, rather than having functions emerge from an understanding of the context. Commonly in Australian schools, R² is used to help with decision making as it is easily accessible (using Excel and graphing calculators), even when this is not appropriate. Without guidance from the real-world context, the resulting model may be a poor approximation to reality if the decision is made solely on which model has the `best’ fit. Furthermore, different students with different data from the same context may select different model structures. `Fitting the data well’ is not the only criterion for deciding on a model (see Dunn & Smyth, 2018, Section 1.10).

Secondly, the resulting models may have no real-world interpretation if the model structure is not based on theory. When using mathematics to describe the real world, mathematical ideas and functions emerge naturally, and students need to think about the context to `mathematise’ it (`mathematising’ is just as important as `interpreting’: Wake, 2016). In contrast, when models are imposed upon data, no context is necessary, and the mathematics becomes subservient to the data (modelling as curve fitting Galbraith, 2011).

Thirdly, teachers are often not trained to teach empirical modelling. Determining model structures empirically requires statistical techniques, yet mathematics teachers are often not trained in the formal statistical techniques used to properly fit models. Furthermore, many teachers may not understand many basic statistical concepts (Engel & Sedlmeier 2005) so that empirical modelling presents significant challenges. For example, one study (Dunn et al., 2015) asked teachers to define `regression’ and `correlation’. Although a small sample, the sample was a voluntary, self-selected group of keener mathematics teachers, so the results are optimistic. The results indicate that the teachers’ understanding of regression and correlation was poor. Only 32% gave a correct definition of regression and 16% for correlation (see Dunn et al., 2015 for details of the study), yet an almost identical group of teachers expressed confidence in their knowledge of regression and correlation (Marshman et al., 2015). This aligns with other research showing that mathematics teachers may be proficient with the formulae and computations used with regression but do not necessarily understand the statistical concepts or the meaning of the formulae (Engel & Sedlmeier, 2005) and hence the connections between the model and the real world.

Fourthly, teachers may not understand the difference between statistics and mathematics. Formal attempts to fit a model empirically (that is, using statistical approaches) require the use of statistical tools. However, mathematics and statistics have many differences. For example, teachers and researchers alike are often surprised to learn that the language and notation of statistics and mathematics may contradict each other (Dunn et al., 2016). Even the concept of `linear’ is different in mathematics and statistics (Dunn et al., 2016). Furthermore, a straight line has different presentations in mathematics and statistics. Straight lines in mathematics are presented deterministically as (say) |$y= mx+c$|⁠, implying that every value of |$x$| is associated with a single possible value |$y$|⁠. In statistics, the structure of the model is written as (say) |$\hat{y}=a+ bx$|⁠, where the left-hand side shows that the model predicts a mean value |$\hat{y}$| for a given value of |$x$|⁠, and the actual (observed) values of |$y$| for that value of |$x$| are generated randomly from the given (often normal) distribution of possible values around the given mean. Failure to understand these differences produces a clash of notation. Other examples of language inconsistencies between mathematics and statistics include the words `graph’, `estimate’, `significant’, `variable’ and the symbol |$\pm$| (Dunn et al., 2016).

Finally, teachers are not adequately supported to teach a statistical approach. Because many teachers lack an understanding of the empirical modelling approach (i.e. statistical techniques), teachers may reasonably be expected to turn to textbooks. However, many textbooks used in teaching mathematics contain errors and inconsistencies when discussing statistics (Dunn et al., 2015) or lack useful features to enhance learning (Dunn et al., 2017). Problems with textbooks include the following: errors in formulae (Bland & Altman, 1988; Dunn et al., 2015), cumbersome and misleading language, confusing or incorrect notation, unexplained language, misusing notation or not explaining what the model means. Furthermore, very few textbooks use real data, and many of those that do use real data (for example, Barnes et al., 2016) use contrived examples such as the relationship between student height and hair length. Many examples and exercises are clearly made up or simply provide a list of |$x$| and |$y$| values and ask students to compute a regression line with no attempts at a context or to attach meaning (for example, Morris, 2016, p. 598, 599). These approaches do not even try to help students see a connection between mathematics and the real world and may reinforce students’ belief that there is no connection (`these data are made up, because there are no real examples’). Other easily accessed resources, such as Wikipedia, also have significant problems explaining simple statistical concepts (Dunn et al., 2018).

6 Teaching functions with data: examples of modelling in schools

6.1. Recommended approach for teaching modelling

We recommend approaching mathematical model by using data for validation of the model structure, for reasons described above, and suggest a framework to support teachers’ choice of resources for modelling activities (Fig. 2). We then describe some examples of the modelling process (using Fig. 1) of how to do this in the classroom and some examples of how a different approach leads to poor experience of mathematics being useful in the real world.

6.2. Good practice: theoretical structure and parameter values (ST, PT)

A simple example that could be used with Year 11 and 12 (aged 16 and 17) students in their final 2 years of school is to simulate radioactive decay to introduce power functions. For example, consider repeatedly tossing coins on a table, where coins that turn up as `heads’ are considered to have radioactively decayed. A model is sought for the number of coins remaining after n tosses. After each toss, the expected number of coins removed at each toss is about one-half. Hence, the `half-life’ is one toss. Using dice and removing dice that land with a (say) six uppermost is similar but with a longer `half-life’ (about one-sixth of the dice are removed with each throw, and the `half-life’ is between three and four rolls). In both cases, a mathematical model can be developed with known parameters. Students can then validate the model by tossing coins (or rolling dice) to see how well the model fits the data. Students only need to be familiar with index notation (including fractional indices) and some guiding questions from the teacher to introduce power functions. This may also give a reason to introduce logarithms.

6.3. Good practice: theoretical structure and empirical parameter values (ST, PE)

An example that was used by one of the authors with Year 11 and 12 students both as an in-class activity and a take-home assessment task uses a `marble sifter’ in which holes just large enough for marbles to comfortably pass through are cut in the lid of a cardboard box (for example, printer paper box lid; Fig. 3a). Students begin with 100 marbles and `sift’ them through the sifter by tilting the box (Fig. 3b) and counting how many remain unsifted after each tilt. The process is repeated for 11 to 20 tilts. We can define |$t$| as the number of tilts and |$N$| as the number of marbles (out of 100) that remain unsifted (specify: identifying variables). Some (real) example data are shown in Table 2. Again, students could mindlessly fit many functional relationships to the data. However, this activity lends itself to a useful discussion of mathematical modelling to establish the model structure. The initial assumption is that we can model a discrete situation with a continuous model (specify: identifying initial assumptions).

To determine a suitable model structure for the marble sifter, students can graph the number of marbles remaining against the tilt number, so that they can visualize the data to help them understand the data (formulate). Then various model structures can be critiqued (analyse; interpret). For example, the model cannot be linear as the graph does not look linear. In addition, a linear model would imply that approximately the same number of marbles would fall through the sifter; however, many marbles began the sift at any iteration, which makes no sense. A linear model also implies that more and more sifting would eventually end up producing a negative number of remaining marbles.

The model cannot be quadratic, as the number of marbles remaining cannot start increasing as a quadratic would permit (marbles cannot jump back up through the holes). Furthermore, if students attempt to fit a quadratic, then they usually see a turning point occurring and the function increasing within the range of the data.

Attempting to fit a logarithmic model of the form |$N=a\ \log (t)+b$| will cause difficulties as the data begins at |$t=0$| and |${N}_0=100$| (since |$\log\ 0$| is undefined). A similar problem exists when trying to model with a power function (the initial conditions are undefined; |$N=a{t}^b$| can never satisfy the initial conditions).

An exponential relationship, however, makes sense. From Table 2, the fraction remaining (say |$a$|⁠) at each tilt is approximately constant. For example, if we start with |${N}_0$| marbles, then after one tilt, we would have an average of |${N}_0a$| marbles remaining. After two tilts, an average of |${N}_0\ {a}^2$| would remain. Continuing, after |$t$| tilts, the predicted number of marbles remaining would be |$\hat{N}={N}_0\ {a}^t$|⁠, for some value of |$a$|⁠, and where |$\hat{N}$| is the average number of marbles left after |$t$| tilts. This is the process of mathematical modelling describing the real world.

Table 2

Data from one example of conducting the marble-sifting study

Sift #	Trial 1	Trial 2	Trial 3	Mean number remaining	Fraction remaining compared to previous sift
0	100	100	100	100.0
1	85	87	89	87.0	0.8700
2	72	78	82	77.3	0.8889
3	67	66	74	69.0	0.8922
4	57	58	63	59.3	0.8599
5	51	51	60	54.0	0.9101
6	45	49	49	47.7	0.8827
7	43	43	43	43.0	0.9021
8	39	37	34	36.7	0.8527
9	37	30	32	33.0	0.9000
10	32	28	30	30.0	0.9091
11	28	25	29	27.3	0.9111
				Mean, |$a$|⁠:	0.8890

Opportunities emerge to discuss the assumptions made in developing the model (interpret) (for example, is it reasonable to assume that decay rate is constant?) and strengths and limitations of the model (for example, the data are discrete, so does it matter that |$N$| is continuous as we develop the model?) (The State of Queensland (Queensland Curriculum & Assessment Authority), 2017). A discussion of half-life also fits with the discussion.

To determine values for the parameters, first observe that the model has two unknown parameters to be estimated: |${N}_0$| and |$a$|⁠, though |${N}_0$| could be constrained to 100 as it is the only known value. To estimate these values, students could use technology (such as graphics calculators or Excel) as a `black box’ or formal statistical tools. Technology allows students to determine equations and coefficients of determination, R², for various regression lines from the scatterplot. However, if students are not encouraged to think about each model as it is generated, then the technology is a black box and opportunities for strengthening students’ understanding of modelling have been missed. Of concern is that the students (and most of their teachers) do not understand how the computer creates the estimates and the reason why it is important to refer each model generated to the real world for validation. Secondary school students were expected to evaluate every possible model before deciding on their final choice as part of the written assessment or so that they could participate in a whole class discussion.

An alternative approach to using a black-box approach, which has arisen when the activity was used with pre-service mathematics teachers, is for pre-service teachers to understand the meaning of the parameters, helping them connect the mathematics to the real world. For example, the value of |${N}_0$| is the initial number of marbles, so using |${N}_0=100$| seems sensible. An estimate of |$a$| can be found by combining results from all groups and estimating |$a$| as the fraction of marbles retained at each sift (Table 2). Of course, the modelling can be rearranged to lead to the exponential function also.

For most students, this approach allows them to see explicitly how mathematics is used to describe the real world, how mathematics emerges naturally and that the parameters have physical meaning. However, students could also use some of the basic statistical modelling ideas that are in the curriculum to estimate the parameters, by first transforming to a linear model (taking logarithms of the modelling equation): |$\textrm{log}\ \hat{N}=\textrm{log}\{N}_0+t\ \textrm{log}\ a$|⁠. This equation is now linear in |$\textrm{log}\ \hat{N}$| and |$t$| (a plot, not shown, of the data confirms this is approximately true), and the slope of the line is expected to be about |$\textrm{log}\ a$|⁠. The slope can be estimated formally using technology or informally using a ruler (⁠|$a$| may be estimated using a graph and fixing the intercept at |$\textrm{log}\{N}_0$|⁠, then a ruler used to estimate the slope |$\textrm{log}\ a$|⁠).

Furthermore, different approaches can be used to estimate the parameters, allowing students to be creative and demonstrating to students that there are different approaches, which produce slightly different estimates for the data in Table 2:

• Estimating a by averaging the three values of a that emerge from each individual trial: |$\hat{N}=100\ {t}^{0.8890}$|

• Fitting the model |$\hat{N}={N}_0{a}^t$|directly using a statistical approach: |$\hat{N}=98.892\ {t}^{0.8858}$|

• Log-transform and the fitting a linear model statistically, fixing |${N}_0:\hat{N}=100\ {t}^{0.8854}$|

• Fitting an exponential model: |$\hat{N}=98.0\ \exp (-0.119t)$|⁠.

• Fitting an exponential model, fixing |${N}_0$| as 100: |$\hat{N}=100\ \exp (-0.122t)$|

All produce similar, but not identical, values for |$a$|⁠. This leads to useful discussions of why they are different, which are the `best’ estimates and what is meant by `best’. It is also useful and helpful for students to see that there is not just one possible answer (Carrejo & Marshall, 2007). A discussion of these different methods would be dependent on whether students used these methods themselves and whether time was available.

6.4. Poor practice: empirical structure and empirical parameter values (SE, PE)

A child of one author was given a mathematics assignment in Year 11 where students were asked to generate data as follows. Stand 30 cm from a wastepaper basket, take 10 shots at landing a wad of paper in the bin and record the number of successful shots. Students were to repeat at other distances from the wastepaper basket (60 cm, 90 cm, 120 cm and up to 330 cm). Students were then asked to plot the data and fit linear, quadratic, cubic and reciprocal functions to the data using software. Students were told to vary the parameters of the model and then select a `best fitting’ model by eye (rather than by computer optimization, say). Students used their chosen model to predict what would happen at a different distance (say 105 cm). The data collected by the author’s child are plotted in Fig. 4. Students can identify the variables as the distance from the bin |$x$| and the number of successful shots |$y$| out of 10 (step 1: specify; Fig. 1). They then defer to technology to fit a model (step 2: formulate), which they chose `by eye’ as the best (step 3: analyse).

Fig. 4.

Data collected from the paper toss study, with various fitted models shown over the data.

This task uses the empirical approach entirely: no attempt is made to mathematize the situation or even to understand what the fitted parameters mean. For example, none of the candidate functions restrict the values of |$y$| to between 0 and 10 even within the domain of the data. None of these proposed models sensibly describe reality, and none of the models or their parameters can be sensibly interpreted or have any physical meaning relating to the actual task. The mathematical functions have been imposed onto the data (the instructions tell students to fit specific functions), and none actually make sense. How can a student be expected to see mathematics being useful in this context, when the models are all clearly ridiculous?

This is an example of a task that `exclude[s] real knowledge from [the] solutions’ (Verschaffel et al., 1997, p. 339). Despite intentions, this activity then could `actually teach students to suspend real-world sense making (Greer, 1997)’ (Gainsburg, 2008, p. 200; citation in the original) and reinforce the idea that mathematics has no connection to the real world (as it did for the author’s child):

Secondary mathematics teachers count a wide range of practices as real-world connections. Teachers make connections frequently, but most are brief and many appear to require no action or thinking on the students’ part. (Gainsburg, 2008, p. 215)

A sensible model would be a logistic regression model (Fig. 1, left panel), which is outside the scope of many curricula (Morrell & Auer, 2007).

6.5. A more involved example: theoretical structure and empirical parameter values (ST, PE)

This example (from Dunn & Smyth, 2018) considers real data from small-leaved lime trees Tilia cordata grown in Russia (Fig. 6, left panel; Schepaschenko et al., 2017). It has been used with university students but could also be used with Year 11 and 12 students (16 and 17 year olds). The interest is in estimating the oven-dried foliage biomass (hard to measure) from just the diameter of the tree (easy to measure). A data-based approach would require students to fit quadratic models, exponential models and other functional forms to model the structure of the data and then decide which fits the data best (Section 6.4). However, this approach is poor mathematical modelling and misses rich learning opportunities.

Fig. 5.

A small-leaved lime tree (T. cordata). (Used under the Creative Commons Attribution-Share Alike 3.0 Austria licence. Photo by Haeferi, downloaded from https://commons.wikimedia.org/wiki/File:Feldkirchen_an_der_Donau_-_Naturdenkmal_nd325_-_Linde_in_Feldkirchen_-_Winter-Linde_(Tilia_cordata).jpg on 07 November 2017).

A better approach is to understand the real problem context. As Fig. 5 shows, the tree canopy is roughly spherical. Therefore, it is reasonable, as a first approximation, that the foliage biomass canopy be modelled (following the modelling steps from The Common Core State Standards for Mathematics (Common Core State Standards Initiative, 2010)) as a sphere (specify: identifying assumptions and approximations). Furthermore, since most foliage grows on the outside of the approximately spherical canopy (where sunlight is available), the foliage biomass may be related to the surface area of that sphere (formulate). It also seems reasonable that the diameter of that canopy sphere is proportional to the size (say, diameter) of the tree; that is, trees with larger diameter trunks generally have larger canopies (specify: identifying assumptions). The variables can be defined by writing |$y$| as the foliage biomass and |$x$| as the trunk diameter (specify: identifying variables). This suggests that |$y$| is proportional to |$\pi \ {x}^2$|⁠. That is, |$\hat{y}$| is proportional to |${x}^2$| (recalling |$x$| is the diameter of the trunk) so that a quadratic relationship between |$x$| and |$y$| may be reasonable:

$$\hat{y}\propto{x}^2, \textrm{ or alternatively, } \hat{y}=b{x}^2,$$

for some proportionality constant |$b$|⁠, where the value of |$b$| is to be estimated. This development can occur in small groups of students, with teacher guidance where necessary, or as a class discussion.

Of course, the value of the proportionality constant |$b$| is unknown, but can be estimated informally, as the modelling implies that |$y/{x}^2$| should be approximately constant (analyse). This estimation could be done formally using the data using statistical methods, but the complications this entails are easily avoided and teaching opportunities increased. For each tree, we can compute |$y/{x}^2$| and then find the mean of all these values; this produces an estimate of the proportionality constant of |$0.005309$|⁠, so that |$y=0.005309\ {x}^2$|⁠. This model can be plotted on the data (Fig. 6, right panel), and the fit appears to be reasonable, and so the mathematical modelling seems reasonable; this is informal validation (interpret). In addition, the students can see directly where the estimate of the proportionality constant comes from, and the parameter has meaning.

Fig. 6.

The oven-dried foliage biomass from 185 naturally growing lime trees in Russia. Left panel: the original data; right panel: some mathematical models plotted on the data.

Note that rather than suggesting using the mean of |$y/{x}^2$|as an estimate of the proportionality constant (Method 1), students may also suggest the mean of |$y$| divided by the mean of |${x}^2$|(Method 2; giving 0.004742) or the mean of |$y$| divided by the square of the mean of |$x$| (Method 3; giving 0.005766). These can all be drawn on the graph of the data and all seem reasonable. All these approaches—while not specifically statistical—demonstrate an understanding of the mathematics and the meaning of the parameters and connect the mathematics to the real world. Furthermore, all models are reasonable so there is not a single correct answer (which may surprise students in mathematics classes; Grootenboer & Marshman, 2016), and there is no obvious answer to which estimate is the `best’, or even an answer to what `best’ means in this context.

An alternative approach to fitting a quadratic model is to use Excel:

$$\hat{y}=0.0072{x}^2-0.0888x+0.6412 $$

(1)

with |${R}^2=0.637$|⁠. This model may be a quadratic as the development suggests, but the three parameters estimated in the model are hard to interpret, as a term involving |$x$| and a constant term is fitted, which did not emerge as part of the earlier mathematical development. In Excel at least, fitting a quadratic model without these two terms is difficult. Furthermore, this model is not monotonic (the minimum occurs at x = 0.16): the value of |$y$| is predicted to increase as the value of |$x$| decreases for small values of |$x$| (that is, the model predicts that the foliage biomass is larger for smaller diameter trees). All connection to the real world has now been lost. In addition, the value of |${R}^2$|must be larger (that is, `better’) for the inferior Excel-derived model since it has three parameters compared to the original one-parameter model based on theory.

Numerous discussion questions naturally emerge, from which rich learning opportunities arise:

• What else might impact foliage biomass, and how might these be incorporated into a model? (Soil type, hours of sunlight, amount of water received, genetics, planting density, competitor species present etc.)

• In what ways is the spherical assumption for the canopy likely to break down? What might be the implication of this? (The canopy sphere may be not quite accurate; there is a `gap’ in the sphere where the trunk is; some foliage is inside the canopy. So maybe the `power’ may not actually be two, but the general structure of the relationship is probably sound.) This shows the balance of simplicity and accuracy, and the level of accuracy needed may depend on what the model will be used for.

• What are the units of measurement for the proportionality constant, and what does it mean? (kg/cm²; the average mass of foliage per unit surface area.)

• Why are the estimates of the proportionality constant different? Which is the best estimate and why? What is meant by `best’?

• Why is the power not two (using one of the approaches), when the modelling led us to expect a power of two? (The model is just an approximation.)

7 Discussion and conclusion

Connecting mathematics to the real world is important (Smith & Morgan, 2016) for increasing students’ understanding of and motivation to study mathematics (Gainsburg, 2008). Using real data and real contexts is a useful way of showing students the applicability of mathematics, and this can be done through mathematical modelling (Jablonka, 2007). In this article, we considered mathematics modelling as a vehicle for teaching mathematical material (Galbraith, 2011) to develop their students’ understanding of particular mathematical functions. We have provided a framework to support teachers as they choose resources and data sets for mathematical modelling.

Using examples where the model structure can be described theoretically (ST; Fig. 2) is a better approach for teaching (and in practice), as the mathematics is being used to describe the real world. Parameter estimates then are usually determined from the data (PE), and a theoretical basis for the model structure often leads to sensible ways to estimate the parameters that retain meaning in the parameters.

In contrast, asking students to fit many functional relationships to real (or artificial) data without any meaning behind the resulting models or their parameters (SE) can reinforce the misconception that mathematics is irrelevant in the real world (Section 6.4). Finding suitable real data sets—or having students generate suitable real datasets—is not always easy or straightforward. We have given some suggestions here for students to potentially generate different data sets (Sections 6.2 and 6.3).

We do not recommend fitting mathematical models by determining the model structure through empirical means. When teachers are trying to teach mathematical modelling this way, by using `modelling as curve fitting’ (Galbraith, 2011, p. 281), the applicability of mathematics is lost in the procedure and students forget about the real-world knowledge. Generally, students and teachers do not have the skills in statistical modelling to do this effectively as it is not generally included in the curriculum. As a consequence, we believe that if data are used in the teaching of mathematical modelling, then it should be used in a confirmatory (for validation) and not exploratory (for estimation) manner when applied to determining the model structure.

Teaching functions using real data is possible and sensible and can be abounding with meaning. Our framework is useful to remind teachers that the model should be derived from the real world rather than the data and the data then used for validation. If done well, using real data can encourage good mathematical modelling skills in students and help them see the utility of mathematics in describing the real world.

Dr Peter Dunn has broad expertise in the application of statistics in a variety of fields, with publications in diverse areas that includes the teaching of statistics, health, ecology, climatology, agriculture and mathematical statistics.

Dr Dunn has developed a unique R package (called tweedie) for understanding the Tweedie class of distributions. He has presented numerous conference papers (winning a prize at the 16th International Workshop on Statistical Modelling in Odense, Denmark; and the EJ Pitman Prize at the Australian Statistics Conference in 2002).

Dr Dunn is the co-author (with Prof. Gordon Smyth) of Generalized Linear Models With Examples in R, and developed the Dunn-Smyth (quantile) residuals used in statistical modelling. He has been invited to give presentations on the R statistical environment at conferences and workshops.

Dr Dunn was awarded a national Office of Learning and Teaching Citation in 2012, has worked as a climate scientist, and has over 20 years' experience as a lecturer in statistics and mathematics at university level.

Dr Margaret Marshman worked as a physicist in a hospital, laser physics and magnetic resonance imaging before becoming a secondary mathematics and science teacher and mathematics Head of Department. She is involved in teacher education in the undergraduate and Master of Teaching programs.

Dr Marshman supervises a range of doctoral and masters students in Education. Her research interests include Mathematics Education and Science Education and she is particularly interested in how students formulate and solve mathematical and scientific problems and peoples' beliefs about mathematics, its teaching and learning. Margaret also researches areas of collaborative learning, middle schooling, statistics education and first year in higher education.