COLLEGE OF EDUCATION,
BENUE STATE – NIGERIA.
It is an abstraction to think that mathematics is only confined to the classroom. This paper attempts to analyse how mathematics can be used in our society today through mathematical models. Attempt is also made to justify how these models can be used in mathematics teaching and teacher education. Arguments for considering the use of mathematics in politics as a useful resource for mathematics in the classroom have also been advanced.
At Roskilde University in Denmark, the regulation for mathematics teacher education for upper secondary school level included as one of its main aims and objectives that the teachers should be able to teach mathematics modeling to students in such a way that the roles and functions of mathematics in the society should be demystified (Roskilde University, 1975, p. 1). Mogen Niss, who was the main architect behind the programme, explains in Niss. M (1977) how this programme should be seen as a cure against what he described as the crisis of mathematics instruction. The crisis, which according to him, arose because of an imbalance between the societal developments towards a highly technological society with a need for mathematical competency in the wide population as well as a quest for critical citizenship in relation to the use of mathematics in society and on the otherhand, an abstract and isolated mathematics instruction, especially at upper secondary level skewed towards a mathematical elite and without connections to the use of mathematics in the society. In Denmark, this discussion together with other developments, actually led to the inclusion of models and applications in the upper secondary mathematics curriculum (the equivalence of senior secondary mathematics curriculum in Nigeria). Facilitated by a Danish research initiative; called mathematics education and Democracy (Nissen, 1993), this period also marked the beginning of a strong trend in Danish mathematics education research focusing on the roles and functions of mathematics in society. The overall objective was to understand how mathematics teaching can contribute to the development of a democratic society (Niss, 1994). Mogen Niss followed this path in some of his research (Blum and Niss, 1991, Niss 1994 and 1996) and another internationally well-known Danish Professor, Ole Skovmose, focused his research on the roles and functions of mathematics in society and on the related issue of investigating the conditions; actual and possible for a critical mathematics education (Skovsmose, 1994, 2000 and 2004, Alro and Skovsmose, 1998). Description is to be made of some of the ideas and concepts developed by Moges Niss and ole Skovsmose and others, and illustrations with some examples of the current use of mathematics in politics in society, how their theoretical ideas can be used to analyse cases of societal use of mathematical models and how they can be transformed into teaching.
The Nigerian Situation
Some examples will be considered from the Nigerian socio-political context. Nigeria is a developing country trying to establish a democratic, just and inclusive society. The Nigerian educational institutions play a determining role in the establishment of this kind of society. So, these institutions should prepare the future citizens to actively participate in the decision-making that affects and defines the social reality of Nigeria.
In particular, the mathematical education that is being provided in the Nigerian educational institutions should pay special attention to the socio-political uses of mathematics. There is evidence showing that mathematics plays an important role in shaping the social reality of Nigeria. An example of this is the “marginalization index”, (Sanchez, 2009 and 2010) applied in Mexico, which is a measure based on a mathematical model that is used by governments to define the municipalities or local government areas that are in need of resources to promote social development such as, buildings, hospitals, roads, schools etc. Although mathematics plays an important role in shaping the social reality, the mathematics education that is provided in Nigeria does not seem to acknowledge this role. The perception of the role of mathematics within the Nigerian educational system has been discussed and was intended to provide a modest answer to the justification for teaching mathematics for social reality in Nigeria. The problem of justification, if we can underscore the Mexican example, the Ministry of Public Education in one of the documents, to some extent, provides an answer to this question in “Foundation of the curriculum for the reform of the lower secondary education”. The document states:
[mathematics is useful] to cope with fractions, to plot functions, to conclude angles, probabilities and perimeters. But also to encourage abstraction in order to facilitate reasoning, develop the argumentation and introduction to the proof (Secretaria de Education Public, 2006, p. 9) as translated.
Other official documents from the Ministry of Public Education portray mathematics as a tool that helps students to understand the physical phenomena around them (Sanchez, 2007). Thus the official justification for teaching mathematics is to provide students with mathematical understanding and the understanding is relevant to the society in general. Besides, mathematics is presented to the students as a topic that is important to study and to understand because it will help them to learn more about mathematics. At best, mathematics is presented to the students as a topic that will help them to understand other school topics or subjects as Physics and Chemistry. Mathematics is not presented to the students as a tool that can be used for prescribing our political and economic reality. It is believed that this perception of the role of mathematics is not specific to the educational institutions. It is common to find mathematics teachers who are not aware of the connections between mathematics and the configuration of social reality and therefore, do not include them in their teaching.
The lack of connections between mathematics and society has consequences. For example, there is a risk of making students to interprete mathematics as a school subject that exists and is only relevant within the four walls of the school. This tends to produce a poor image of mathematics in the students. Another consequence is that the school curriculum does not encourage teachers to relate mathematics to other subjects. This also contributes to create a restricted image of mathematics in the students.
If it is agreed that the idea presented in Skovmose (2000) about the need for educating the youth and our citizens, so that they begin to understand and critique the formatting power of mathematics in society, then more work is needed in order to change the general perception about the role and nature of mathematics. Part of the contribution of mathematics educators to support such a change may consist of designing mathematical activities for the classroom, aim at explicitly illustrating and analyzing such formatting power. It is thought that, in order to prepare students to identify and evaluate socio-political applications of mathematics, it is essential to show them and discuss with them real instances of such applications. Such activities could be based on authentic applications of mathematics within the political systems of our societies.
Reasons for the Use of Mathematics in Politics
The arguments for advocating the use of mathematics in politics as a resource for mathematics teaching are mainly two. Firstly, to show how mathematics is applied within the political context to give an important motivational value. There, is a research indicating that the study of authentic and contemporary applications of mathematics can arouse a great interest among students. An example of this can be found in the empirical research reported in Jankvist (2009), where modern histories of applications of mathematics are used to motivate and change student’s conception about mathematics. In this regards, the author states:-
The fact that history is a newer and fairly recent, history of mathematics seems to make it easier for the students to relate […] concerning the history of modern applications of mathematics. Some students may find it more interesting to work with such a history and possibly even more so, if they recognize elements from everyday life (Jankvist, 2009, p. 11).
Although, the study refers to the use of history of mathematics in the classroom, it is a case that can be used as an analogy to illustrate the motivational value that the use of mathematics in contemporary politics could have for the students. Such examples might be easy to relate to aspects of students’ daily life. However, there are other reasons to believe that the use of mathematics in politics would be interesting and motivating for mathematics students and their teachers. A specific example of this is the video, “Government and Mathematics” which was published on “YouTube” (Sanchez, 2010). Several comments from people who would have otherwise ignore mathematics at a glance, were received and many of them expressed their feelings that the topic addressed in the video was interesting and they even recommended it to other people. A very remarkable comment was the one on realization of how the politicians govern the people with limited mathematical models based on the expositions by the video. This comment actually promoted the video among his contacts in the social network, but he in turn received commendation from the users of the social network. It can be seen that this process serves as a manifestation of the interest that certain people had in mathematics based on the contents of the video, because it was related to the social reality and experience in society. Consequently, it can be argued that, the mathematics teaching that is detached from the social reality experienced by any society or country can appear demotivating for the students. Conversely, the mathematics teaching that has strong links with the social reality may be more attractive and motivating for the mathematics students. This may be perhaps, true in particular for developing countries like Nigeria.
The second argument for considering the application of mathematics in politics as a resource for mathematics teaching is that, politics is a context that can be useful to explicitly illustrate the formatting power of mathematics. In other words, it is a context that can help to make evident how mathematics can be used to legitimize and justify political decisions that directly and significantly affect the social dynamics of communities and the lives of the inhabitants. It is important to explicitly illustrate and study these kinds of applications of mathematics, because they can serve to nourish a sense of civic awareness in the students and teachers towards the use of mathematics by politician and government institutions. We think that such kind of activities would contribute to prepare students and teachers to identify, evaluate and respond critically to the consequence, of such use of mathematics. Furthermore, we believe that the discussion of such mathematical applications can enrich students’ perceptions about the nature and role of mathematics. Nevertheless, we are aware of the fact that locating instances of applications of mathematics in politics is not enough for using them into the classroom. Teachers need to have some sort of guide regarding the aspects they should focus on when discussing this kind of applications of mathematics in the classroom. We understand that mathematics education research can guide us on the aspects that could be the focus of attention when using the applications of mathematics in politics as an aid for the development of mathematics teaching.
The Integration of Mathematics in Politics in Mathematics Teaching
Whenever mathematics is used in politics to describe, predict or even prescribe reality, there is always some type of mathematical model involved (Niss, 1994, p. 369). Even the most simple statistics presupposes assumptions and choices about what to count and how to represent the results. Mathematical models used in politics or societal administration vary a lot in function, complexity, mathematical content and representation. Therefore, there is a need for theoretical ideas that can guide and structure the analysis of the use of mathematics in politics and society, but we are in need of didactical ideas that can support the transportation of such analyses into mathematics teaching practices. Some examples of these ideas as already tried out in practices of mathematics teaching in other societies, especially Denmark, are presented in this part. This is by no means a claim to have covered the areas of mathematics education research relevant for including the use of mathematics in politics in the teaching of mathematics.
Despite the great variation of mathematical models used in politics, they can all be discussed according to their role(s) in the context or contexts where they are to be applied. One very general categorization of models use in politics is the division in descriptive, predictive and prescriptive models (Davis and Hersh, 1986, p. 120). A mathematical model is descriptive when it is used to represent and communicate the current state of a situation. A predictive mathematical model is the one that is used to anticipate or predict what the future state of a situation or problem will be based on the current state. Predictive use of mathematical models is common in relation to societal decision-making, where it is important to be able to predict the effects of possible political regulations in often very complex societal systems. Models in economic planning, traffic planning, environmental planning, planning of energy supply and production are all examples for mathematical models that are used by the political or administrative systems to predict the possible effects of changes or regulations in these systems. Mathematical models are also used in society to define systems that actually shape the political or social reality in which we are living. Such use of mathematical models is characterized as prescriptive.
[….] there are plenty examples of recent and recently reinstated prescriptive mathematizations: examination grades, IQs, life insurance telephone switching systems, credit cards, zip codes, proportional representative voting… we have prescribed these systems often for reasons known only to a few, they regulate and alter our lives and characterize our civilization. They create a description before the pattern itself exits (Davish & Hersh, 1986, p. 120-121).
Of course, not all the examples mentioned in the quote belong to the domain of politics and many of them are integrated in technological systems, which are, infact mathematical models, have political and societal impacts. This is just to make sure it is the functions of the models that are characterized here not the models themselves. The same mathematical model can have different functions in different contexts.
If we want to analyse and discuss applications of mathematics in politics with students, one possible and first approach is to let the students experience and discuss concrete examples of mathematical models used in the society in relation to these three categories, viz, descriptive, predictive and prescriptive models. From experiences, it is possible and motivating for students from secondary school level and above as well as for mathematics teachers to work these categories and even find examples within each category themselves. The students do not need to understand completely the mathematical structure of the models in order to work with them in relation to these categorization. The function(s) that a model plays can be organized at different educational levels. Even though these sort of discussions are somehow general, and not strongly related to the internal mathematical structure of the model, it can be very useful for enriching the students image of mathematics and its applications. For most students and even some teachers of mathematics, it appears as a surprise for them that mathematical models can play different roles and even sometimes prescribe parts of the political and economic reality we are living in.
Skovsmose has analysed deeply the roles and functions being played by mathematical models in society both from a philosophical point of view and analysis of concrete cases of societal applications of mathematical models. Skovsmose, (1990, 1994) and Skovsmose and Yasukawa (2004). In this research, it is established that mathematics through modeling and models exerts a formatting power in modern societies. Furthermore, this formatting power of mathematics constitutes a major challenge for mathematics education research and for the practice of mathematics teaching. In order for mathematics teaching to contribute to general education in favour of democratic societal developments and if its teaching needs to have the desired role in society and politics, it must be taken seriously.
During a Danish research initiative in the years 1998-2004, a number of research and developmental projects with experimental teaching were carried out in order to investigate how this challenge can be met in mathematics teaching already at lower secondary level (Skovsmose & Blomhaj, 2003 & 2006). Findings from analyses of authentic application of mathematical models were used to structure courses of lessons which aimed at including a political dimension. One example of such findings, which is relevant for mathematics teacher education, is the following four types of general side-effects in relation to the use of mathematical models in technical or societal investigation or decision-making process (Skovmose, 1990, p. 128-133), which include the following:-
- A reformation of the problem in hand in order for it to be suitable for analysis by means of a mathematical model.
- A delimitation of the group of people engaging in the public discussion about the problem in hand, to those who are able to understand the model and its role in the decision-making process. This group is in this case referred to as the base of critique.
- A shift away in the discourse from the political and societal reality toward the political and societal reality towards quantitative claims and arguments related to the model. The model now becomes the object for the discourse.
- A delimitation of the possible solutions or the alternative political actions taken under consideration to those that can be evaluated in the model.
The point here is of course not that the use of mathematical models should be avoided as a tool in political and societal decision-making because of the side effects. Mathematical modeling is an indispensable part of a modern technological society. The point here is that the use of mathematical models in society is neither good nor bad nor neutral by any means. Therefore, it is important that mathematics teaching in general education contributes to the development of a critical awareness of, and a competence to analyse the possible effects of the use of a mathematical model in a decision-making process. That the detection of such effects and the related reflections do not need to be closely related to the mathematical structure of the model involved, mathematical modeling competency is a prerequisite for conducting such analyses. Hence, it is mathematics as a subject that has to address the educational challenge related to the formatting power of mathematical models in society, and therefore, these issues must be included in mathematics teacher education, else nobody will to it.
General discussions about the types of mathematical models and their functions in politics could serve to nurture and broaden teachers’ and students’ images of mathematics and its applications. However, in mathematics curricula, say, at secondary level, it might be difficult to find space for such extra mathematical aspects. Analyses of the inner mathematical structure of an authentic model might be a way of approaching the study of mathematical applications in politics, which is easier to integrate in the practice of mathematics teaching. Such analyses can be structured according to a general model of a mathematical modeling process (fig. 1). A concrete case of modeling can be analysed with respect to one or more of the six sub-processes included in a modeling process, namely; problem formulation, systematization, mathematization, mathematical analysis, interpretation/evaluation and validation. For example, we can discuss with the students the assumptions underlying a model and what their implications are. Such kind of discussion refers to the systematization sub-process. It is also possible to discuss the role of the variables and parameters within a mathematical model; like what do they represent?, how are the values of the parameters? And what are their effects on the models’ results? This is one way of addressing the sub-process of mathematical model analysis. It is also important to discuss with students the sub-process of interpretation/validation. Questions like; on what ground(s) can the model be validated? Is it possible to obtain an alternative interpretation of the modeled situation? Or, does the model adequately capture the situation in hand? These questions are relevant within the discussion of the interpretation/validation sub-process. A typical figure of a mathematical modeling process is shown is figure 1 below:-
The point here is that the students through their work with concrete case, get acquainted with a model of a modeling cycle as a tool for analyzing the process behind models used in relation to particular political decisions. Examples from the project work at Roskilde University are found in Blomhoj and Kjeldsen (2010). Such approach is of course, relevant only for students that have previous experiences with modeling cycle as a tool for supporting their modeling activities. Last but not the least, students’ work with authentic societal and political problems can serve as a motivation for and as a means for the learning of important mathematical concepts and methods. As illustrated in examples before, it is possible for students already from lower secondary level to activate their mathematical competencies in relation to societal and political issues.
Politicians and government institutions use mathematics to underpin their proposals and arguments and in decision-making related to social problems. Skovsmose (1990) has warned us about some of the effects that are produced when mathematics is incorporated into the discussion of non-mathematical problems. For instance, if the original problem is reformulated into a different kind of discourse (a mathematical discourse). As a consequence, the group of people who could have participated in the discussion of the problem and its possible solution becomes smaller or reduced because of its very specific composition. It would be integrated only by those citizens with certain mathematical knowledge, who could discern and criticize the use of mathematics in the discussion.
It is the responsibility of mathematics educators to provide the students with a mathematical education which could enable them to take an active and critical participation in the society. A mathematical education that prepares them to identify and analyse the applications of mathematics in addressing social problems. We believe it is worth discussing with the students, examples as those that have been shown in this article. This is because, these examples clearly and explicitly illustrate the kinds of consequences that may result from the application of mathematics in such contexts. This type of mathematics education, it is believed, would positively affect their education as citizens. It is however, necessary to promote these ideas in the classroom. We need more enthusiastic teachers and researchers to conduct experiments and empirical research, reporting the type of results that this approach of mathematical education can produce. Besides, authors of mathematics books are encouraged to try to relate topics in mathematics and their applications to other subjects and also give examples in these books that are applicable to societal issues, so that mathematics cannot be seen as being an abstract subject that is only useful and discussed in the classroom.
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