Inquiry In Mathematics Education
DISSEMINATING INQUIRY-BASED SCIENCEAND MATHEMATICS EDUCATION IN EUROPEBackground Resources For ImplementingInquiry in Science and Mathematics at SchoolInquiry inMathematicsEducation112351382134WITH THE SUPPORT OF
Resources for Implementing Inquiry in Scienceand Mathematics at SchoolThe Fibonacci Project (2010-2013) aimed at a large dissemination of inquiry-based science education and inquiry-based mathematics education throughout the European Union. The projectpartners created and trialled a common approach to inquiry-based teaching and learning in scienceand mathematics and a dissemination process involving 12 Reference Centres and 24 Twin Centresthroughout Europe which took account of local contexts.This booklet is part of the Resources for Implementing Inquiry in Science and in Mathematics atSchool. These Resources include two sets of complementary booklets developed during the Fibonacci Project:1) Background ResourcesThe Background Resources were written by the members of the Fibonacci Scientific Committee.They define the general principles of inquiry-based science education and inquiry-based mathematics education and of their implementation. They include the following booklets: 1.1 Learning through Inquiry 1.2 Inquiry in Science Education 1.3 Inquiry in Mathematics Education2) Companion ResourcesThe Companion Resources provide practical information, instructional ideas and activities, andassessment tools for the effective implementation of an inquiry-based approach in science andmathematics at school. They are based on the three-year experiences of five groups of Fibonaccipartners who focused on different aspects of implementation. The Companion Resources summarise the lessons learned in the process and, where relevant, provide a number of recommendationsfor the different actors concerned with science and mathematics education (teachers, teachereducators, school directives, deciders, policy makers ). They include the following booklets: 2.1 Tools for Enhancing Inquiry in Science Education 2.2 Implementing Inquiry in Mathematics Education 2.3 Setting up, Developing and Expanding a Centre for Science and/or Mathematics Education 2.4 Integrating Science Inquiry across the Curriculum 2.5 Implementing Inquiry beyond the SchoolReference may be made within this booklet to the other Resource booklets. All the booklets areavailable, free of charge, on the Fibonacci website, within the Resources section.Editorial coordinator : Susana Borda CarullaFibonacci Scientific Committe : Michèle Artigue, Peter Baptist, Justin Dillon,Wynne Harlen, David Jasmin, Pierre LénaWWW.FIBONACCI-PROJECT.EUFibonacci Project, December 2012This project has received funding from the European Union’s Seventh Framework ProgrammeInquiry inMathematicsEducationMichèle ARTIGUE, Peter BAPTIST
3Table of contentsIntroduction. 31. What is Inquiry-Based MathematicsEducation (IBME)?. 3Michèle ARTIGUE1.1 Mathematical inquiry: internal andexternal questions. 41.2 Mathematical inquiry: some specificitiesof the process.71.3 The connected and cumulative dimensionof mathematics. 81.4 IBME and digital technologies. 101.5 IBME and other approaches in mathematics education. 111.6 Final comments. 122. Towards Teaching and LearningInquiry-Based Mathematics . 13Peter BAPTIST2.1 Towards new teaching in maths - why? . 132.2 What is special about maths teachingin the Fibonacci Project? . 142.3 IBME – Stimulating acts . 142.4 The role of the key features of inquiry pedagogyin the Fibonacci Project. 162.5 Problem solving – the journey is the goal .182.6 Problem-based teaching – Implementationin the classroom . 192.7 Stimulating acts – Surface of a golf ball . 202.8 Experiments on the computer screen – Recordsin a study journal . 222.9 Design of a computer-based lesson . 232.10 Guiding teaching concept – Reconsideringone’s own teaching . 23References. 25IntroductionIn recent decades, inquiry-based education and inquiry-based learning, whose roots can be traced at least toDewey’s approach towards education nearly one century ago (Dewey, 1926, 1933, 1938), have been gaininginfluence in education, but this terminology has remained essentially attached to science education. The authorsof the report known as Rocard’s report (Rocard & al., 2007), which has been very influential in promoting inquirybased science education (IBSE) at the European level, seem themselves hesitant to extend this terminology tomathematics. They write:“In mathematics teaching, the education community often refers to “Problem-Based Learning” (PBL)rather than to IBSE. In fact, mathematics education may easily use a problem based approach while, inmany cases, the use of experiments is more difficult. Problem-Based Learning describes a learning environment where problems drive the learning. That is, learning begins with a problem to be solved, andthe problem is posed in such a way that children need to gain new knowledge before they can solve theproblem. Rather than seeking a single correct answer, children interpret the problem, gather needed information, identify possible solutions, and evaluate options and present conclusions.” (p.9)In the Fibonacci Project, the same terminology was used both for mathematics and sciences, hence the expression ‘inquiry-based mathematics education’ (IBME). This makes it necessary to clarify exactly how we understand this expression, and how its characteristics compare with those of IBSE. The first part of this document,written by Michèle Artigue, is devoted to such clarifications, while in the second part Peter Baptist focuses onthe implementation of IBME, on the important changes that the move from traditional education to IBMErequires, and on how these changes can be progressively achieved. The view of IBME expressed in this bookletis coherent with the nine key features of inquiry pedagogy (basic patterns) that emerged from the modules ofthe German SINUS project (1998-2007) and were adopted by the Fibonacci project, and which are described indetail in the Fibonacci Background Resource Booklet Learning Through Inquiry1.1. What is Inquiry-Based Mathematics Education (IBME)?Michèle ARTIGUEUniversité Paris Diderot, FranceIn the introduction we pointed out that, contrary to what occurs in science education, the term inquiry-basedmathematics education (IBME) is of recent use in mathematics education. Nevertheless, several attempts atpromoting teaching practises with evident similarities to those associated with inquiry-based science education(IBSE) can be found in this field. Indeed, throughout the last fifty years, one of the main ambitions of innovationand research in the field of mathematics education has been to promote mathematical learning with understanding, to help pupils experience authentic mathematical activity from the early grades, and to elucidate theconditions for making this possible. It is thus necessary to consider IBME in relation to this heritage, and investigate how the scientific resources it provides can be used for developing and strengthening IBME2.The Fibonacci Background Resource Booklet Inquiry in Science Education3 points out that “inquiry is a term usedboth within education and in daily life to refer to seeking knowledge or information by asking questions”, andthat “what distinguishes scientific inquiry is that it leads to knowledge and understanding of the natural and1 Available at www.fibonacci-project.eu, in the Resources section.2 A more substantial reflection on this issue is developed in Artigue & Blomøj (in press).3 Available at www.fibonacci-project.eu, in the Resources section.
45made word around us”. The text also proposes a model of learning science through inquiry which is summarisedin the schema reproduced below (fig. 1) and described as “the process of building understanding throughcollecting evidence to test possible explanations and the ideas behind them in a scientific manner” (p. 9).Newexperience or ideaBiggerideaPredictionPlan and conductinvestigationInterpret dataConclusionthat mathematics as a science also creates its own objects and reality, and that the questions raised by theseobjects have always been an essential motor of its development. As stressed in the Fibonacci BackgroundResource Booklet Learning Through Inquiry:“As they become familiar, mathematical objects also become the terrain for mathematics experimentation.Numbers, for instance, have been used for centuries and are still an incredible context for mathematicsexperiments, and the same can be said of geometrical forms. Patterns play a great role in mathematics,whether they are suggested by the natural world or fully imagined by the mathematician’s mind. Digitaltechnologies also offer new and powerful tools for supporting investigation and experimentation in thesemathematical domains. IBME must, therefore, not just rely on situations and questions arising from realworld phenomena, even if the consideration of these is of course very important, but use the diversity ofcontexts which can nurture investigative practices in mathematics.”Thus the sources of mathematical inquiry in IBME and the associated questions may be very diverse. They canemerge from: natural phenomena (e.g.: how to understand and characterise changes in the shadow of an object cast bythe sun?), technical problems (e.g.: how to measure inaccessible magnitudes and objects?), human artefacts (e.g.: what is the effect of a pantograph on geometrical figures and why? How does a GPSwork?), art (e.g. what are the symmetries of an architectural object or piece of art? What are the minimal elementswhich can be used for generating a periodic tessellation?), daily life problems (e.g. how to choose between different offers for mobile telephony and internet?).Fig. 1: A model of learning science through inquiry.As pointed out in the Fibonacci Background Resource Booklet Learning Through Inquiry4, mathematical inquirypresents evident similarities with scientific inquiry as described above. Like scientific inquiry, mathematicalinquiry starts from a question or a problem, and answers are sought through observation and exploration;mental, material or virtual experiments are conducted; connections are made to questions offering interestingsimilarities with the one in hand and already answered; known mathematical techniques are brought into playand adapted when necessary. This inquiry process is led by, or leads to, hypothetical answers – often calledconjectures – that are subject to validation.This is rarely a linear process. Quite often, initial conjectures are found to be true only under specific conditions, which may lead to their modification, or even to questioning the definition of the mathematical objectsinvolved (as illustrated by Lakatos (1975) in his paradigmatic study on Euler’s formula for polyhedrons). Further,the process may lead to new questions and problems whose solution may affect the answers to the initial question, or even the formulation of the question itself.Nevertheless, despite the existence of similarities with scientific inquiry, mathematical inquiry has some distinctspecificities, both regarding the type of questions it addresses and the processes it relies on to answer them.But mathematical objects themselves from the early ages can be an essential source of mathematical inquiry. What is the greatest product that can be obtained by decomposing a positive integer into a sum of positiveintegers and multiplying the terms of the sum? Can all positive integers be obtained as the difference of two squares of integers? Are all positive integersthe sum of consecutive positive integers? What can it mean for two triangles, two rectangles, two polygons to have the same form? Given two triangles with the same area, can they be transformed one into the other by cutting and pasting?Does this extend to any pair of polygons? If two triangles have the same perimeter and the same area, are they necessarily isometric?The nature of the question obviously has an impact on the inquiry process. In the case of questions from anexternal source, such as in the first examples mentioned above, transforming these questions into questionsaccessible to mathematical work is an important part of the process of inquiry, engaging a modelling process5.In recent decades, research in mathematics education has paid more and more attention to these processes asshown, for instance, by the activities of the international group ICTMA or the extended literature devoted to thistheme6. This literature generally presents modelling as a cyclic process, which creates some similarity with themodel for IBSE in figure 1, at least at a surface level.1.1 Mathematical inquiry: internal and external questionsAs in scientific inquiry, mathematical inquiry is often motivated by questions arising from the natural world orthe made world around us. But if one main ambition of mathematics is to contribute to understanding of thenatural, social and cultural world, and to empower human beings to act on this world, it should not be forgotten4 Available at www.fibonacci-project.eu, in the Resources section.5 Models used in science are not necessarily mathematical models, and even when this is the case, models donot necessarily take the form of laws and equations as is mostly the case in physics. They can, for instance, begeometrical shapes (DNA, fullerenes, proteins), graphs or symbolic codes (as in chemistry and genetics), butinternal logical consistency and the ability to go beyond a simple heuristic description remain the absoluterequirements for any good model. In mathematics education and IBME, the term modelling is thus used in arestricted sense: it refers to a process engaging mathematisation and the construction of mathematical models.6 See the activities of the international group ICTMA (www.ictma.net) or (Kaiser & al., 2011).
67Figure 2 presents one classical vision of this modelling cycle reproduced (from Blomhøj & Højgaard Jensen,2003, p.127) with the corresponding description of the different phases:Perceivedreality(a) Formulationof task(f) ValidationTheoryAction/insightDomain of in quiry(b) Systematiziation(e) Interpretation/evaluationDataModel resultsSystem(c) Mathematization(d) Mathematical analysisMathematicalSystemFigure 2 : Modelling cycle“a) Formulation of a task (more or less explicit) that is related to a perceived reality and influenced by themodeller’s interests. Through this process the object of the modelling process is constructed. The object canbe reconstructed as a result of the modelling process. However, it is the object and the formulated task thatguide the identification and construction of a domain of inquiry.b) Selection and construction of the relevant objects, relations etc. from the domain of inquiry, and idealisation of these, in order to make a mathematical representation possible.c) Transformation and translation of selected objects and relations from their initial mode of appearance tomathematics by further abstraction and idealisation.d) Using mathematical methods to achieve mathematical results and conclusions.It should be added that inquiries motivated by mathematical questions (internal inquiries) may also requireor benefit from building interactions between different systems (between numerical and algebraic systems,algebraic and geometrical systems, deterministic and stochastic perspectives, etc.) since mathematics is ahighly connected field. Such interactions can be seen as specific forms of modelling processes, internal to mathematics7.1.2 Mathematical inquiry: some specificities of theprocessBox 1 presents an example of an inquiry process based on a mathematical question: What is the greatest productthat can be obtained by decomposing a positive integer into a sum of positive integers and multiplying theterms of the sum?This question has led to the development of many different activities at different levels of schooling, fromprimary education to senior high school (ERMEL, 1999; Aldon & al., 2010).Box 1. The greatest productWhat is the greatest product that can be obtained by decomposing a positive integer into a sum of positive integers and multiplying the terms of the sum?Let’s select for instance the number 10 and carry out some exploration.If 10 is decomposed into the sum of two numbers, an exhaustive exploration of possibilities is straightforward. We might quickly conclude that the greatest product is obtained through the balanced decomposition 10 5 5, leading to the product 5x5 25. This conjecture is often expressed at an early stage of the inquiryprocess, before any systematic work has been engaged.Nevertheless, the 5 5 conjecture does not survive when 10 is decomposed into the sum of three numbers(for instance 10 5 3 2, leading to the product 5x3x2 30). In that case, systematic exploration requires consideration of many different cases. Yet that does not solve the problem, since 10 can also be decomposed intomore than three numbers.e) Interpretation of these as results and conclusions regarding the system or the initiating domain of inquiry.f) Evaluating the validity of the model by comparison with data (observed or predicted) and/or with alreadyestablished knowledge (theoretically based or shared/personal experience based).”This modelling cycle thus organises the relationship and interaction between two systems: an extra-mathematical system and a mathematical system. Each obeys its own logic, and thus the inquiry process is not onlysubject to the rules of mathematical rationality. As is made clear by the description above, mathematical rationality regulates the work carried out in phase d, and this work can itself have an inquiry dimension, but validation of the answers obtained through the inquiry process as a whole is also submitted to the rationality of theextra-mathematical system. In IBME, it is certainly important to make students aware of this and give them theopportunity to experience the diversity of domains that are accessible to mathematical inquiry through modelling processes, beyond the sole natural world.Comparing with IBSE, beyond the fact that the modelling process does not necessarily use mathematics, itshould be noted that the loop may often involve more preconceived ideas than in mathematics, since thenatural world is more immediately perceived by the senses, and requires a crude model to be dealt with, even atearly ages. These pre-conceptions may have a strong impact on the process.Thus, we engage in a dialectic process involving trials and the progressive elaboration of partial results andconjectures. For instance, quite soon we find that optimal decompositions should not include the number1 nor the number 5 (because 3x2 5), nor the number 6 (because 3x3 6). This line of reasoning excludes allnumbers different from 2, 3 and 4. We then notice that 3x3 2x2x2, and that in any decomposition the number4 can be replaced by two number 2 (as in 4 2 2 2x2). Thus, we finall
The Fibonacci Project (2010-2013) aimed at a large dissemination of inquiry-based science educa-tion and inquiry-based mathematics education throughout the European Union. The project partners created and trialled a common approach to inquiry-based teaching and learning in science
Guided inquiry in the Chemistry lAborAtory exPerienCe 15 The degree of inquiry is based on the amount of self-direction by the student compared to directions provided by the teacher. This concept is summed up in the following table: essential Features of Classroom inquiry and Their Variations Essential Feature Open Inquiry Guided Inquiry Structured
Module 1: Introduction to Scientific Inquiry Lesson 1: Scientific Inquiry Activity 4: Misconceptions and Challenges of Scientific Inquiry Estimated Time: 10 minutes What aspect of scientific inquiry are you already doing or what appeals to you about scientific inquiry? Record your ideas below. Modu
John Dewey's Dual Theory of Inquiry and Its Value for the Creation of an Alternative Curriculum Fred Harris, independent scholar Abstract Dewey's theory of inquiry cannot be reduced to the pattern of inquiry common to both common-sense inquiry and scientific inquiry, which is grounded in the human life process, since such a reduction ignores
2. 3-4 Philosophy of Mathematics 1. Ontology of mathematics 2. Epistemology of mathematics 3. Axiology of mathematics 3. 5-6 The Foundation of Mathematics 1. Ontological foundation of mathematics 2. Epistemological foundation of mathematics 4. 7-8 Ideology of Mathematics Education 1. Industrial Trainer 2. Technological Pragmatics 3.
The Fibonacci Project (2010-2013) aimed at a large dissemination of inquiry-based science education and inquiry-based mathematics education throughout the European Union. The project partners created and trialled a common approach to inquiry-based teaching and learning in science
the brain module responsible for morality was shaped by millions of years of evolution. Based on this line of inquiry, many philosophers argue that, over time, as this line of descriptive inquiry unfolds, it will gradually replace prescriptive inquiry. That is to say, prescriptive inquiry will someday be absorbed by descriptive inquiry in the form of brain science. I don‟t believe it .
to the INQUIRY PROCESS Inquiry Model Refl ecting on the Process. 2 PRINCE EDWARD ISLAND SOCIAL STUDIES CURRICULUM: GEOGRAPHY 631A Guided Practice Selecting a Topic and Planning an Inquiry Outcome 2.1.1 . My biggest challenge with taking notes for this inquiry project will be .
Alex Rider was woken by the first chime. His eyes flickered open, but for a moment he stayed completely still in his bed, lying on his back with his head resting on the pillow. He heard a bedroom door open and a creak of wood as somebody went downstairs. The bell rang a second time, and he looked at the alarm clock glowing beside him. There was a rattle as someone slid the security chain off .
