DG 19: Current problems and challenges in lower
secondary mathematics education
Team Chairs
Maryvonne Le Berre, IREM of Lyon
Address: 43 Bd. du 11 novembre 1918, F-69 622 Villeurbanne Cedex, France
leberre.maryvonne@free.fr
Gard Brekke, HiT Telemark Research Notodden, College of Telemark
Address: LÊrerskoleveien 35, N-3670 Notodden, Norway
gard.brekke@hit.no
Team Members
Suwattana Eamoraphan, Faculty of Education, Chulalongkorn
University, Thailand
suwattana_u@yahoo.com
Merrilyn Goos, Social Sciences Building, School of Education, University
of Queensland, Australia
m.goos@mailbox.uq.edu.au
Keiichi Shigematsu, Department of Mathematics Education, Nara University
of Education, Japan
shigek@nara-edu.ac.jp
Aims and Focus
The general aim of Discussion Group 19 is to encourage participants to engage in discussions related to current problems and challenges pertaining to the teaching and learning of mathematics at the lower secondary level. The discussions in DG19 should shed light on difficult or controversial issues of the mathematic curricula. One example could be the role of algebra in lower secondary school. Are such issues or dilemmas of a controversial nature? How should these problems, issues and challenges be dealt with?
Examples of issues
1. Mathematical literacy and "mathematics for everybody""The aim of the OECD/PISA assessment (http://www.pisa.oecd.org/index.htm) is to develop indicators of the extent to which the educational systems in participating countries have prepared 15-year-old to play constructive roles as citizens in society...
The OECD/PISA assessment focuses on real-world problems, moving beyond the kinds of situations and problems typically encountered in school class-rooms."
- How do we define "mathematics for everybody"? Is this what we could name a "minimum curriculum"?
- Does it just include applied mathematics?
- How can the teachers develop ability to apply math skills to different contexts?
"Such uses of mathematics are based on the skills learned and practised through the kind of problems that typically appear in school textbooks and classrooms. However, they demand the ability to apply those skills in a less structured context, where the directions are not so clear, and where the students must make decisions about what knowledge may be relevant, and how it might usefully be applied."
- This is a very difficult, but valuable challenge for most teachers. How could we organise this kind of teaching to cater for all students in the class? How do we organise the teaching to include those who have acquired very few elements of mathematical skills and conceptual understanding?
2. Relationships between different levels of knowledge
In the constructivist approach, mathematics should be taught through activities that invite pupils to reason, explain and justify rather than simply to memorize and imitate, in order to construct mathematical understanding. Nevertheless, memorizing and imitating are parts of the learning process.- Is it possible to find a "right balance"?
- Which are the relationships between computational skills and reasoning or understanding?
3. Different approaches of geometry
- What kinds of geometrical reasoning do 11-15 -year- old pupils develop?
- How will the dynamic geometry software (for example Cabri) change the teaching of geometry?
- Are there important differences between countries in the way of teaching geometry? (Inductive and deductive reasoning, modelling, application of software)
4. The role of technology and electronic tools
- How can the use of calculators and different software make easier or - on the contrary - disturb the mathematical learning? In which way should they be used?
- Do the computers use induce changes in the curricula?
5. What is the role of algebra in lower secondary school?
The twelfth ICMI study, on The Future of the Teaching and Learning of Algebra, was held at the University of Melbourne (Australia) in December 2001. One issue is described below:The technological future of a modern society depends in large part on the mathematical literacy of its citizens and this is reflected in the worldwide trend towards mass secondary education. For an individual, algebra is a gateway to much of higher education, and therefore to many fields of employment. Educators also argue that algebra is part of cultural heritage and is needed for informed and critical citizenship. However, for many, algebra acts more like a wall than a gateway, presenting an obstacle that they find too difficult to cross.
DG 19 will amongst other discuss issues such as:
- Should algebra be taught to all? In recent years we have seen a call for teaching algebra to all lower secondary students. The question above may raise several educational questions to the educators, such as: What aspects of algebra are of value to all? What should a minimal curriculum be consisted of? How do answers to these questions relate to regional or cultural differences?
- What do we expect of an algebra-literate individual? What are the values of algebra learning for the individual, especially in view of increasingly powerful computing capabilities? Access to higher learning and employment are two values, but what are the more immediate values and how can they be achieved?
- How can we reshape the algebra curriculum so that it has more immediate value to individuals? Can we identify explicit examples in contexts meaningful to students in which algebraic ideas have a clear, unambiguous value? Are there undesirable consequences of such orientations to algebra?
- How can we reshape the algebra curriculum so that specific difficult ideas are more easily accessed?
Call for Papers
The Discussion Groups at ICME-10 are designed to gather congress participants who are interested in discussing and exchanging ideas concerning the theme of the group. Therefore there will be no oral paper presentations by individual congress participants, but papers submitted by individuals will form the basis for discussion.
The Organizing Team of DG 19 is calling for papers that will structure and allow informed discussion. Each paper submitted will be reviewed by at least two members of the organizing team. Accepted papers will be posted on the DG19 web site.
Papers should be written in English and should be directed towards stimulating discussion and debate. Papers should not just report activities or describe programmes, but discuss and illustrate principles and ways of teaching in classrooms. The maximum length should be 10 pages, written in 12 points TIMES with 16 points spacing, and fit into an outline of 16 x 25 cm.
Relevant deadlines:
02.02.04: Deadline for submission of the first draft of papers
20.03.04: Deadline for group leaders to decide on acceptance or revision
20.04.04: Deadline for submission of revised papers
Submission of papers should be sent by email to both chairs of DG 19.
Practical
Information
Sessions at ICME:
- Session 1: Monday 5 July; 16:30 to 18:30
- Session 2: Wednesday 7 July; 16:30 to 18:30
- Session 3: Saturday 10 July; 15:00 to 16:00
Programme
First 2-hours session: Monday 5th, 16h30-18h30
- Introduction/explanation of programme
- Overview of issues 1, Math literacy, maths for everybody 2, relationship between technical skills and understanding 5 the role of algebra, by team members
- Feedback from participants
- Working in small groups (4-5) on the 2 topics
- Different ways and contexts in which letters can be first introduced
- Different meanings of letters and equalities
- Discussion on algebra in the whole group
- Overview of issues 3, the role of technology 4, geometry , by team members
- The whole group breaks into several sub-groups for discussion Geometry
- Comparing contents,
- Problems and methods
- The role of the dynamic geometry software
- The role of other software (such as spreadsheets...)
Group leaders report back to whole group, summarising discussion on all issues and identifying new questions/research directions
Papers
and Discussion Documents
There will be no oral presentation of any paper. Participants are expected to read them before the congress.
Discussion documents will be distributed during the sessions.
Learning from comparing
A review and reflection on qualitative oriented comparisons of teaching and learning mathematics in different countries
Christine Knipping, Hamburg (Germany)
The article reviews five qualitative comparative studies in order to reflect on methods and methodology of international comparative research within a qualitative paradigm. The goals, methodologies and theoretical perspectives of these studies are described and compared. Issues such as the different uses of the notion ‘culture’, and the purpose of comparison, are raised and discussed. The question of how comparative research can contribute to better a understanding of the ‘covert culture’ that shapes the mathematics education in different contexts is raised. (download)
Mathematics for Everybody – implications for the lower secondary school
Steve Thornton
University of Canberra
Australia
John Hogan
Redgum Consulting Pty Ltd
Australia
If we take as our starting point the quite reasonable proposition that numeracy is “having the competence and disposition to use mathematics to meet the general demands of life at home, in paid work, and for participation in community and civic life” (Willis 1992) then the interaction between school mathematics and numeracy becomes critical. The practice of school mathematics focuses on developing student knowledge and understanding of mathematics per se; it also emphasizes the applications of mathematics to other contexts. In both of these practices the primary purpose is, appropriately, the development of mathematical understanding. However we would argue that, in the school setting, the context for numeracy lies beyond the mathematics classroom, most obviously as an integral part of students’ learning across the curriculum. Here the focus is not on the mathematics; it is on the context. Yet understanding the mathematics is crucial in working within that context. In this paper we propose a Numeracy Framework as a way of describing numeracy, diagnosing learning issues, supporting teacher planning and for teaching to students so that they can choose to act numerately beyond the mathematics classroom. We use the results of an Australian research project in numeracy across the curriculum in the middle years of schooling, and examine the implications for teachers of mathematics.download
The differences between design intentions and implementation: The implementation of the Malaysian mathematics curriculum
Noor Azlan Ahmad Zanzali, Faculty of Education, Universiti Teknologi, Malaysia
The Malaysian secondary school mathematics curriculum is based on the constructivist approach. For effective implementation it is indispensable that teachers are not only aware of what the goals of the curriculum are, but at the same time cognizant of the underlying theoretical assumptions embedded in the curriculum. Recent surveys have however revealed that the most significant factors that tend to impede successful implementation are those related to teachers’ beliefs about teaching and learning. Thus the question of how their beliefs differ from those of the curriculum developers is an important one. The study attempts to discover the differences between teachers’ assumptions about teaching and learning, particularly related to problem solving in mathematics, as compared to those espoused by the curriculum. Data were collected from interviews, classroom observations and content analyses. The nature of mathematical knowledge transmitted, students and teachers activities were used as the orienteering constructs. These were then analyzed to identify teacher’s assumptions about teaching and learning, particularly those related to problem solving. Initial findings indicate that majority teacher’s beliefs are still traditional in nature despite various attempts by the Curriculum Development Center of the Ministry of Education to change teacher’s beliefs about teaching and learning. (download)
Towards a comparative analysis of proof
Christine Knipping, Department of Education of the University of Hamburg
In this paper, the first results of a comparative study on proof and proving in geometry teaching are represented. Twelve eighth grade classes (approximately 14 year-old students) were observed in France and Germany, in order to analyse the impact of culturally embedded classroom practices on the teaching and learning of proof. Also, the differences in functions of proofs based on the observed French and German teaching practices are presented here. In particular two ideal types of mathematical cultures in classroom practice have been singled out. (download)
Argumentations in proving discourses in mathematics classrooms
Christine Knipping
University of Hamburg
This paper focuses on argumentations in proving discourses in mathematics classrooms.
Examples of different types of argumentations in proving discourses that were observed in
a comparative study of French and German lessons on the Pythagorean Theorem are
presented. These illustrate different argumentations within two kinds of proving discourses.
In one case the argumentation is characterised as intuitive-visual, and in the other as
conceptual. It is suggested that there is an epistemological basis for the differences between
these discourses and that comparative studies like this one provide a way to investigate
these further.(download)
How to deal with Algebraic Skills in Realistic Mathematics Education?
Monica M. Wijers
Freudenthal Institute, The Netherlands
In the Netherlands, the mathematics curriculum for lower secondary has been modernised since 1992 and reflects a realistic approach to mathematics. For algebra this means that the focus is more on using algebra than on studying algebra as an abstract subject. Some years after the implementation of the new curriculum a survey among teachers and researchers revealed difficulties with the teaching and learning of algebra Tension was felt between: skills and understanding; tools and structure etc. Difficulties surfaced with the transition from lower to upper secondary. The Freudenthal Institute is investigating these difficulties and is experimenting with possible solutions. (download)
Literal calculation and equations during French collège time
Maryvonne Le Berre, IREM de Lyon, France
During the first years of secondary school, the introduction of letters makes it necessary to consider different status for equalities. Difficulty is found in the dichotomy true/false, very evident in French teaching, which is an obstacle to understanding equations as open formulas. A few examples of what can be done in classroom in order to construct the concept of variable are provided. (download)
Misconceptions in Mathematics: Solving the Equation
Suwattana Eamoraphan, Faculty of Education, Ghulalongkom University,Bangkok, Thailand
In the usual classroom context, teachers think that their students understand the lesson very well after they have explained the content clearly. When teachers assign exercises to check their students’ comprehension, however, they find that the results are not what they expected. Several problems lie in the discontinuity of content between the upper primary school level and the lower secondary school level, especially in solving equation problems. The student cannot link the symbol
(©or ® ) with the unknown or variable (x or y). In this case, no matter how effective the teaching methods employed, one sees the same kind of errors. In the article, the method of teaching was introduced.
(download)
Affects and beliefs in school mathematics : gender differences
Gard Brekke, Åse Streitlien, Lise Wiik, Notodden (Norway)
The TIMSS study found significant gender differences in achievement only in a few of the participating countries (grades 7 and 8). In general attitudes towards mathematics were positive for most countries, however gender differences in favour of boys were found in 31 out of the 40 participating countries. Norway was one of countries with the largest gender difference (Lie et al 1997). Leder and Forgasz (2000) have presented a rational and methods, which they used to develop a scale for investigating to what extent mathematics continues to be considered as a gender domain. Our study concerns issues from a large-scale project. A questionnaire containing a wide range of issues related to the teaching and learning of mathematics (125 items) was administered to 1482 students in grad 6 (11.5 y) and 1183 students in grade 9 (14.5 y). Forty-two of the items were related to students’ beliefs about mathematics, mathematics teaching and self, as well as attitudes towards the subject. Factor analysis showed that we could form aggregated variables, named Interest, Usefulness, Self-confidence, Diligence and Security. Relatively large differences between genders were found for these variables. The relation between attitudes and performances in mathematics were analysed.
(download)