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  4. Teaching Mathematics: Toward a Sound Alternative

In our review we also included proceedings of some international conferences as well as selected books on mathematics education research. These literature sources were organised into three layers, described in more detail below. Layer 1 — Journals In this layer we included international and regional journals.

In each case we selected the journals that we consider to be the most influential and representative. It is important to clarify that, although these journals are produced in particular regions of the world, it is likely that at least some of them may not represent the thinking of the researchers within the geographical areas where they are produced, simply because they may contain articles written by authors from outside that region. This is particularly true for some of the journals included from Europe and North America.

The Australasian region was represented in this review by the journal Mathematics Education Research Journal. Finally, we included the journal Pythagoras to represent the African region. Layer 2 — Conference proceedings In this layer we included proceedings of international conferences that were freely available on the Internet. The proceedings of the conference MES 4 were not included in the review because they were not freely available on the Internet. Due to its importance and influence, we would have liked to include in our review the proceedings of the conferences organised by the International Group for the Psychology of Mathematics Education PME.

However, these proceedings are not available online and they were not accessible through the library of our university; hence, they could not be included in this review. Layer 3 — Books In this layer we included books on mathematics education research. These books were identified though a study of the bibliographies of the selected articles contained in the first layer.

As we went through the lists of references used in these articles, we noticed that certain books were cited frequently; after examining them directly, we decided which ones to include in this third layer of this review. What should we look for? The layers just described illustrate where we looked when searching the materials used in this review. What we want to do now is to clarify how we selected the sources used in this review.

We set this condition to try to ensure that the selected texts would address the relationship between mathematics education and democracy. This should be considered as another limitation of our review. We used search engines to find key terms within the documents.

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For example, for the articles contained in Layer 1 we used the Web-based search engines included in the Web pages of the journals. These tools allowed us to quickly locate relevant articles contained in large collections of documents. For some documents in Layer 3 it was necessary to locate the key terms manually. What was excluded from the review? Not all the materials identified in Layer 1, Layer 2 and Layer 3 were included in our literature review.

Next we explain the various reasons for exclusion. Within the article, however, the phrase is used to refer to the title of an article written by Skovsmose In fact, Hanna and Sidoli do not address the relationship between democracy and mathematics education at all, but rather focus on providing a statistical profile of articles published in the journal Educational Studies in Mathematics. Another category of articles that was not considered in this review was those that mentioned a possible relationship between mathematics education and democracy, but only superficially.

For example, there were texts claiming that mathematics is important for the education of citizens, but the reasons why mathematics is important were not clarified. We considered only materials written either in Spanish, Portuguese or English; as a result, articles written in other languages were excluded, as was the case for three articles from the special issues on mathematics education and democracy in the journal ZDM: The International Journal on Mathematics Education , issues 30 6 and 31 1 , which were written in German. Definitions of democracy used in these texts. Skovsmose and Valero affirm that this richness of definitions suggests that the open nature of democracy is such that a precise definition of the concept is not possible.

However, we think is important to try to identify the interpretation of democracy that each author adopts since the ideas, concepts and proposals that they present are usually related to their own interpretation of democracy. One of the most elaborate definitions of democracy is that of Murillo and Valero , cited in Valero , in which democracy is interpreted as an ideal form of social organisation with four dimensions: Democracy can be defined as an ideal way social organization establishes a series of political, juridical, economic and cultural values, norms and behaviors aiming at providing a better living for the whole population of a given state.

This definition highlights a conception of democracy not as an actual reality, but as a goal to reach. The political dimension includes the series of procedures to form governments by means of regular, free elections as the corner stone of representative democracy. The juridical dimension sets and protects the different basic legal human rights and duties. The economic dimension deals with the material conditions of living and the organization of the economy by the state.

We decided to use this definition because it offers an overarching characterisation of democracy covering the different dimensions of democracy that are identified in other definitions. The political dimension of democracy Authors such as Woodrow and Almeida refer to the political dimension of democracy; that is, the type of democracy where citizens elect their representatives to participate in discussions about public affairs and make decisions related to those public affairs.

Representatives are elected through free elections in which citizens exercise their right to vote. This interpretation of democracy assumes that citizens do not directly participate in the discussion of public affairs, but they do so through the representatives of their choice. Skovsmose criticises this interpretation of democracy because it puts the election of the government at the centre of the discussion, and makes other conditions or dimensions of democracy irrelevant. Skovsmose proposes an alternative interpretation of democracy, inspired by the concept of direct democracy.

Here democracy is conceptualised as a form of political democracy in which citizens participate directly in the discussion of public affairs. This position may seem impractical if we think of a state, but Skovsmose conceives the application of this type of democracy in all types of institutions, such as workplaces, schools and classrooms. Furthermore, this conception of democracy puts the type of skills that a citizen must possess in order to fully participate in the public discourse at the centre of the discussion. This point will be addressed later in the review, when we discuss the links between mathematics education and democracy.

The juridical dimension of democracy Respect for the rights and freedoms of individuals is another element included in some definitions of democracy. For example, Harris mentions that a characteristic of democracies is that they are social formations where people have largely equal rights as citizens. Such freedoms include freedom of speech, freedom to work, freedom from hunger, freedom from oppression and freedom to worship.

The economic dimension of democracy Skovsmose points out that democracy is subject to the fulfilment of certain conditions. One of these conditions is the fair distribution of goods; in other words, democracy is not possible in a context where material goods are unevenly distributed. The socio-cultural dimension of democracy Democracy not only refers to the fair distribution of goods, equal rights for citizens and the free election of representatives.

It also refers to a type of social organisation that can accommodate different views and ways of thinking. Hannaford pays special attention to this aspect of democracy, referring to two types of democracy.

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In this kind of democracy there is no room for people with alternative ways of thinking. The second kind of democracy is that in which there are almost as many ways of thinking as there are people. Hannaford claims that the latter type of democracy is slower and seems less efficient; however, history has shown that in the long term it is more efficient than depending on only one idea. As the above discussion shows, the concept of democracy is multidimensional; that is, it is a concept that refers to freedoms, rights, obligations, the distribution of material and cultural goods, and respect for diversity of ideas and ways of thinking.

The question now is: What are the links between mathematics education and democracy? In the following section of the article we will present the links that we have identified through the literature review. Links between mathematics education and democracy. The first one refers to mathematics education as a field of research, whilst the second one refers to mathematics education as a set of practices associated with the teaching and learning of mathematics.

Such practices are not confined to the classroom.

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As noted by Valero , they include external educational practices that affect the learning and teaching of mathematics, such as curricular policymaking, mathematics textbook writing and pre-service and in-service education. In this section, where we address the links between mathematics education and democracy, we have adopted the second connotation. We have identified three links between mathematics education and democracy in our literature review. Firstly, mathematics education can provide students with mathematical skills to critically analyse their social environment, and also to identify and evaluate the uses and misuses of mathematics in society.

The second link relates to the fact that the mathematical education that students receive in a classroom can promote or inhibit values and attitudes that are essential to build and sustain democratic societies. The third link is the acknowledgment that mathematics education can function as a sort of social filter that restricts the opportunities for development and civic participation of some students. Mathematics is an integral component of society.

In fact, society is largely shaped by mathematics. Thus many decisions that are socially relevant may be strongly influenced by mathematical models and applications, for example which municipalities in a country are considered poor enough to receive additional financial aid from the state, how much an employee should produce in order to maintain their position within a company, or what level of pollution levels in a city should lead to a recommendation that the inhabitants avoid exercising outdoors.

The important point here is that it would be difficult for citizens to assess whether these decisions are fair or appropriate if they have not received a proper mathematical education. In sum, a mathematical education helps citizens to identify how mathematics is being applied to support such decisions, and to reflect on the consequences, positive and negative, that this application can produce.

In order to maintain a democratic society, it is important that citizens are capable of critically analysing such questions and their answers because if they are to understand the economic and juridical dimensions of democracy, for example how the economic resources are distributed in a country or the defence of labour and environmental rights, it is vital that they understand the mathematics underlying those decisions. When we refer to the particular case of the application of mathematics in politics, we are addressing the connection between mathematics education and the political dimension of democracy.

For example, Almeida remarks: One of the ways that the government or elected representatives convince the citizens that their policies are the correct ones is by producing reports which include a mass of numerical and statistical data. There are many instances where this data is misleadingly summarised. Children and adults need their number sense to be part of their critical sense. A democracy without this kind of citizenry is a fragile democracy. Link 2: Mathematics education as a source of values and attitudes In the previous section we emphasised the importance of having mathematically educated citizens, able to critically analyse how mathematics is applied in their societies.

However, an adequate mathematical education is not sufficient to produce critical citizens. A critical citizenry also requires the promulgation of democratic values and attitudes. Values like tolerance and respect for diversity, and attitudes about truth that demand the critical analysis of information. The second link between mathematics education and democracy identified in our review is the claim made by several authors that the mathematics classroom can be any place where, alongside mathematics learning, it is possible to transmit and acquire perhaps subconsciously both democratic and undemocratic values and attitudes.

This link is closely related to the socio-cultural dimension of democracy, which refers to the social space where democratic values are produced. This is problematic at the present time because we neither know what currently happens with values teaching in mathematics classrooms, or why, nor do we have any idea how potentially controllable such values teaching is by teachers. In addition, many mathematics teachers are not even aware that they are teaching any values when they teach mathematics. Changing that perception may prove to be one of the biggest hurdles to be overcome if we are to move to a more just mathematics education.

Skovsmose argues that the nature of mathematics classroom interactions can teach the students to follow explicitly stated prescriptions. Hence, Skovsmose suggests that this sort of mathematical education, more than producing critical citizens, prepares students to perform routine work and become part of the workforce.

In turn, Hannaford suggests that the teaching of mathematics in which students are taught that there is no room for mistakes and there is only one correct answer does not promote plurality and respect for the diversity of ideas. Link 3: Mathematics education as a social gatekeeper The third link that we have identified relates to the fact that mathematics education can function as a kind of social filter.

Several researchers acknowledge this situation e. For instance, Thomas , referring to the Australian situation, states: Australia and some other nations risk becoming societies divided by access to mathematical knowledge. A minority will have access to high levels of mathematics and will be the highly paid professionals and leaders.

This is not the basis for either a clever country or a democracy but it is the basis for a divided society. In short, it decreases their chances of economic and social success. Skovsmose has even suggested that lack of a mathematics education may contribute to the growth in modern societies of a new lower class. For example, when mathematics is used in political discussions of social problems, only those who understand the mathematics being used can criticise its use and participate in the discussion, effectively leaving citizens who lack such knowledge out of any deliberations. Orrill and Skovsmose , go so far as to argue that the lack of such knowledge is a threat to democracy because people who are not mathematically literate cannot fully participate in civic life.

In the words of Skovsmose : Democracy may be destroyed by a dictatorship which obstructs formal democratic procedures. Democracy refers not only to formal, but also to material and ethical conditions and to possibilities for participation and reaction. In particular, democracy can be destroyed if a critical citizenship cannot brought into life. How to promote such democratic competence in mathematics students, however, is the subject of much debate. Fostering critical mathematical skills The term critical mathematical skills refers to the mathematical knowledge that allows students to use mathematics to analyse social problems or to address issues relevant in their personal lives.

Such critical mathematical skills enable students to identify and judge how mathematics is applied to address socially relevant issues, as well as to reflect on the consequences of their application. These activities can be used both to assist students in developing an understanding of mathematical tools and ideas, and to analyse social problems. For example, Christiansen asked pre-service teachers to represent in different ways the share of land that Black people and White people in South Africa owned in , and then to reflect on the impression given by each representation.

Besides promoting a reflection on the different ways quantitative information may be represented, this activity also made these future teachers aware of the racial problems in South Africa. Moreira used similar methods to introduce students to mathematical applications that allow them to analyse various economic, political and social problems, such as trends in the number of people with AIDS, the impact of fishing policies on endangered species, and the advantages and disadvantages of adopting nuclear power as a source of energy.

Malloy in turn suggests that students should be confronted with moral issues that surround the uses mathematics: We must present them with problems that not only tackle issues that affect their communities, but also reveal the motivations and the hidden agenda curriculum in their world. When students use and apply mathematical knowledge in such situations, they are learning to think critically about world issues and their environment through mathematics.

Through this process students will have an understanding of inequities in society, and will be able to critique the mathematical foundations of social situations. However, as Skovsmose also points out: It is not possible to develop a critical attitude towards the application of mathematics solely by improving the modelling capability of students. We must be able to point out which economical ideas are hiding behind the curtain of mathematical formulas. Orrill goes even further and argues that we should avoid the compartmentalisation of the mathematical knowledge in the school curriculum.

In other words, he argues that the teaching of mathematics should be spread across the curriculum. The logic behind this idea is that in real life mathematics is everywhere; it should not be isolated into a single subject. Skovsmose and Valero express the same idea this way: There is also a need to consider that mathematical competencies do not operate in isolation outside school but as part of integrated units assembled in schooling. This implies interdisciplinarity among the school subjects as an important research issue.

Such proposals aim both at modifying the kinds of interactions that occur between the teacher and the students, and at changing the mathematical activities that mediate such interactions. A basic idea behind the promotion of democratic values and attitudes in students is the one proposed by Vithal : that within the mathematics classroom it is possible for students to experience democratic life. Ernest also makes this point: Teaching approaches should include discussions, permit conflict of opinions and views but with justifications offered, the challenging of the teacher as an ultimate source of knowledge not in their role as classroom authority , the questioning of content and the negotiation of shared goals.

He argues that the use of open materials see description above is compatible with the kind of investigative activities known as project work. He further posits that these kinds of mathematical activities give students more power to make decisions about what to study and how to study it. In other words, this approach fundamentally changes the roles and the power relations between teachers and students.

Hannaford makes another important proposal to promote dialogue and negotiation in the mathematics classroom. He argues that students should be taught to listen, to think, to argue effectively, and to respect others because democracy depends on those values. In a similar vein, Almeida recommends the use of informal mathematical proof as a means to introduce students to a culture of interrogating explanations. Students should be invited to consider the explanations provided to them, and to question their level of plausibility.

It is especially important for them to learn to look critically at the information and explanations provided by the teacher. According to Almeida , if students uncritically accept the information that teachers provide simply because they are authority figures, then it is likely that as citizens they will tend to accept uncritically the information and proposals politicians provide. Vithal similarly envisages the mathematics classroom as a democratic microsociety where the students can learn to both live together with and talk back to authority figures. A tension between open and empowering materials It is difficult to design activities that are both open and empowering at the same time.

For example, what do you do when a student is truly interested in an activity but it does not address any socially relevant problem? Similarly, because teachers want students to understand the functions and assumptions behind a real mathematical model, it is difficult for them to avoid proposing activities that are too structured and guided. Skovsmose sums up this problem: Open material could result in open and democratic educational situations — but no empowerment is guaranteed; and empowering material could result in critical understanding — but no openness is guaranteed.

For example, to protect human rights, an authority must exist to defend and guarantee them. Vithal illustrates very clearly how this tension between democracy and authority may occur at different levels within the mathematics classroom: at the classroom level, within working groups, and even in the teacher—student, teacher—researcher domain.

This tension brings two important points into focus: firstly, in order to promote democracy sometimes it is necessary to engage in non-democratic practices; and secondly, it is important to recognise that democracy requires some kind of authority, but it is also important to be aware that authority can turn into authoritarianism. The paradox of empowerment vs disempowerment Attempts to introduce mathematical activities into school curricula aimed at promoting democratic competencies in the students often face obstacles and resistance.

For example, Almeida notes how such activities may cause delays in delivering the traditional curriculum. Such situations raise a paradox, as noted by Christiansen , that the intention to empower students can be transformed into their actual disempowerment. For instance, we may propose activities that use mathematics to help the students to analyse social problems in their own communities.

Behind this type of instruction is an assumption that these activities will empower students. However, whilst students may develop mathematical skills through such empowering activities, there is no guarantee that these skills will be those assessed on their exams. This could cause students to obtain poor marks or even to fail, ultimately disempowering them. The paradox is that, despite the vital importance of the UNESCO declaration, many mathematics educators are indifferent to this right, and to other dimensions of democracy. This paradox is similar to the issue raised by Ernest concerning the status of social justice within the mathematics education community: Why do some individuals believe in social justice?

There is great divergence in interest and commitment to social justice among mathematics educators. Some view it as central to their professional concerns, whereas others take no personal or professional interest in pursuing social justice issues. Why this divergence? Consequently, the competencies that these teachers require to implement this educational approach in their classroom are the subject of much debate. Skills needed The specific skills that teachers need to possess can be divided into two groups: mathematical skills and pedagogical skills.

With respect to the former, Christiansen holds that such teachers should be mathematically creative. This statement makes sense if they must select and even design empowering activities for their students. Christiansen also states that this requires that teachers understand the mathematical potential that such activities entail. They must understand why algorithms are applicable, must know the different ways a problem may be solved and must have a good understanding of how concepts and mathematical structures are interconnected.

Mathematics can be used either as a tool to improve the welfare of humanity, or applied to increase inequality and injustice. Awareness of its dual role is especially important when teaching students to identify the uses and abuses of mathematical applications in society.

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A mathematics classroom that aims to promote democratic values and attitudes should model deliberative interaction, argumentation, critical analysis of the information, and respect for the ideas of others. These features require that teachers possess the pedagogical skills to manage and promote such dynamics in the classroom. As Almeida points out, this requires that teachers use effective questioning techniques and appropriately manage class discussions.

Christiansen argues that teachers should be aware of the pedagogical potential of activities both for the individual student and for the class as a whole. This requires, amongst other things, specific pedagogical knowledge about how students learn mathematics, and about concept development. Attitudes needed Besides having specific mathematical and pedagogical knowledge, teachers need to possess particular attitudes in order to promote democratic competencies in students.

For instance, as Harris states, one of the important qualities that teachers must possess is commitment, particularly to social renewal along rational democratic lines. Christiansen stresses the importance of having critical teachers who are willing to speak up when they detect that a potentially empowering curriculum is being blocked or mathematical creativity is being hindered by limited assessment criteria representing traditional values or by recipe-like instructions about how to teach.

Another necessary attitude that Almeida highlights is the egalitarian treatment of students in the classroom. He claims that it is the responsibility of teachers to treat students as equal partners in the teaching—learning process. Along similar lines, Vithal suggests that teachers can be useful models of authority for students not only to learn about their individual limits, but also to learn that it is possible to raise their voice against authorities. But for this to happen, teachers must be willing to give up part of the authority they traditionally have enjoyed in the classroom and understand that challenges to their authority are part of what constitutes a democratic education for their students.

Finally, another of the necessary attitudes that some texts address is a willingness of teachers to work in academic environments that are full of uncertainty.


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This is because, as Skovsmose observes, the evolution of a lesson based on open activities may be unpredictable, so the control that teachers usually have over the mathematical knowledge that is assigned and discussed in the classroom can be diluted and replaced with uncertainty. Rights and obligations Obligations, as conceptualised by Christiansen , are those qualities that teachers responsible for implementing a mathematical education for democracy should possess.

Framed in a juridical sort of rights and obligations discourse that is one dimension of democracy, Christiansen also refers to the rights of teachers: In extension of these obligations, it must also be a democratic right for teachers to have a say in how curricula, guidelines and recommended teaching materials are put together; a right to have the many years of experience from the teaching profession being put to use.

A right to be taken serious if they choose to criticise curricula and required teaching methods for being too idealistic and too demanding to realise in practice. Do we secure these rights? Selected criticisms.


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  5. INTRODUCTION;
  6. Some of these criticisms are aimed at applications of this approach, whilst others refer to unwanted results that it could produce. We reviewed enough works for it to be evident to us that most of this literature consists of programmatic theoretical and rhetorical statements rather than careful empirical research. There is a clear need for empirical studies to test and expand these theoretical ideas.

    Vithal explicitly addresses this issue: There is now a considerable literature exploring the connections between mathematics education and democratic society, much of it theoretical about what could or should occur. The question is what happens when an attempt is made to deliberately realise such a link in a mathematics classroom. Woodrow points to some studies that show that this approach may produce discriminatory results. For example, he cites a study where some pre-service teachers were empowered to create new curricula based on exploratory and investigative work.

    However, when faced with the school reality and having found out that school mathematics can be quite ritualistic and confirmatory, one of these empowered teachers became so disillusioned that he left teaching. An imposition of emancipation Mathematics education for democracy consists of a series of mathematical activities and modes of interaction that are considered to be beneficial and empowering.

    They are based on the assumption that teachers with the appropriate training can tell what kinds of education will further the civic development of students. Yet entailed in this proposition is the assumption that their superior position gives them the right to modify the curriculum, and to decide what is beneficial for their students. How would the historically advantaged feel if the educational system really came to function on the premises of the historically disadvantaged?

    If our cultural capital … was depreciated overnight? The applicability and relevance of a critical ideology Valero makes a criticism of the critical ideology that underlies the link between mathematics education and democracy that seems particularly relevant to us as Latin Americans. We refer to the theoretical position that holds that mathematics is ubiquitous in modern societies, and that mathematical models and applications influence many of the decisions that affect and shape modern societies.

    Valero analyses this ideology from a Latin American perspective: Critical ideology overemphasizes the role of mathematics in society. In Latin America, the power structure has lead to a clientelist political system where decisions are made based on personal loyalty of clients to patrons, political convenience, power of conviction through the use of language or violent and physical imposition. We live in Mexico, a country with a developing but still fragile democracy. When we discovered that there was a research area focused on studying the relationship between mathematics education and democracy, we were very interested in studying it.

    We wanted to know what the links between mathematics education and democracy are, but we also wanted to know what knowledge and strategies our field has produced that may be used to promote democracy. What are the main ideas that we found in the literature? Dispatched from the UK in 4 business days When will my order arrive? Arthur H. Robert A. Brent Davis. William F. Margery Osborne.

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    Teaching Mathematics: Toward a Sound Alternative

    Free delivery worldwide. Bestselling Series. Harry Potter. Popular Features. New Releases. Teaching Mathematics : Toward a Sound Alternative. Description This book presents an approach to the teaching of mathematics that departs radically from conventional prescription-oriented and management-based methods.

    It brings together recent developments in such diverse fields as continental and pragmatist philosophy, enactivist thought, critical discourses, cognitive theory, evolution, ecology, and mathematics, and challenges the assumptions that permeate much of mathematics teaching. The discussion focuses on the language used to frame the role of the teacher and is developed around the commonsense distinctions drawn between thought and action, subject and object, individual and collective, fact and fiction, teacher and student, and classroom tasks and real life.

    The discussion also addresses the question of how mathematics teaching can be reformed to better suit current academic and social climates. Making use of the theoretical framework of enactivism, the book explores the subject through an account of a middle school teacher's appreciation and understanding of her role. Teaching mathematics, as both the report of this teacher's experience and the discussion make clear, demands an embracing of ambiguity, uncertainty, complexity, and moral responsibility. Employs reflective teaching techniques to focus students on their own learning, knowledge, and understanding of mathematics.