ICME 9ICME 9WORKING GROUP 2WORKING GROUP 2MATHEMATICS EDUCATION IN JUNIOR MATHEMATICS EDUCATION IN JUNIOR SECONDARY SCHOOLSECONDARY SCHOOL

Ferdinando Arzarello (Italy), CO

Alwyn Olivier (South Africa), CO

Rick Billstein [USA], AOKeiichi Shigematsu [Japan],

AOSuwattana Utairat

[Thailand], AONanae Matsuo , LAO

Ferdinando ArzarelloFerdinando ArzarelloCritical issues for Math Critical issues for Math

Education Education in the Junior Secondary in the Junior Secondary

School School

1. Some approaches to the problem 1. Some approaches to the problem

2. Some examples to give fuel to the 2. Some examples to give fuel to the discussiondiscussion

3. Consequent subjects for the discussion3. Consequent subjects for the discussion

1. Some approaches to the problem

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In most countries of the world, school education from grade VI to grade X is characterised by transitionstransitions:

from compulsory education to jobs and/or vocational school education;

from school-for-all to different schools (or curricula).

Furthermore, there is the transition from child to adult behaviours of learners within the 11-16 age range.

ICME ICME 99Mathematics education plays a special

function in these transitions:

it must ensure the acquisition of crucial tools for life and many professions;

it must prepare students for further studies in scientific and technological areas;

it is supposed to pass over relevant components of modern scientific culture.

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The problem of transition must be investigated and discussed from different points of view, e.g.:•MATHEMATICAL•COGNITIVE•CULTURAL•POLITICAL

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From the MATHEMATICAL point of view, the main problem of transition is characterised by the necessity of introducing important and challenging issues, which appear as unifying and unifying and crucialcrucial in the present historical moment (which is itself also featured by deep transition aspects -social and political).

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Which must be the theoretical status of mathematical knowledge in JSS? • On the one hand, it has not any longer the empirical and episodical features of the elementary school. • On the other hand it cannot yet have the character of a structured theory, like possibly at the senior level.

1. BUT WHAT MUST IT BE IN THE POSITIVE?

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For ex., the transition to the new status of mathematical statements compels teachers both to introduce new topics (e.g. algebra, functions, etc.) and to approach old ones in new ways (e.g. geometry, number systems).

This transition involves the way all mathematical topics (from numbers to geometry, through algebra, statistics, probability, etc.) can be approached.

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From a COGNITIVE point of view, the transition deals with the change from child to adult behaviors within the perspectives of continuing school at a higher level or of entering the working world.

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This transition poses various domains of problems:

a) balancing the math with the new interests of young people

b) situated learning

c) the jump from the elementary school

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a) How to balance the learning of mathematics with the developing knowledge and interests of students, particularly when their attitude towards mathematics has been negative because of different complex reasons (previous school experiences, mass media images, etc.)?

a) balancing the math with the new interests of young peopleb) situated learningc) the jump from the elementary school

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b) How to develop a situated teaching of mathematics within suitable fields of experience that connect a deep real level of mathematical knowledge with that of students and teachers in a positive interaction framework?

a) balancing the math with the new interests of young peopleb) situated learningc) the jump from the elementary school

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c) In many countries there is a big jump between the way maths are framed from elementary to secondary school: how can the different types of interactions in the class [ student(s)-student(s), teacher-student(s)] be organized within different pedagogical frameworks?

a) balancing the math with the new interests of young peopleb) situated learningc) the jump from the elementary school

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From a CULTURAL point of view:

• the learning of mathematics evolves differently in different countries, because of CULTURAL reasons.

• one must distinguish between general and specific issues.

ICME ICME 99Typical general issues:

* How the technological evolution changes the ways mathematics can be taught; for example, is it worthwhile to investigate how the new technologies allow concepts and images to interact at a new level?

* The stereotyped ways of looking at maths inside and outside school influence its teaching: how do these change in the passage from elementary to secondary teaching?

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Examples of specific issues:

* How is the teaching of maths influenced by the organisation of the school, particularly by the changes which learners encounter in their own country passing from the elementary to the secondary level?

* How is the teaching of maths determined by that of other disciplines like natural sciences, first language, etc.?

ICME ICME 99From a POLITICAL point of view, the transition

concerns:* The evolution of the school organization. Which the specific tendencies and the general features of the teaching of mathematics in JSS in different countries?

* Which suggestions mathedu research can give to politicians in order to improve mathematics teaching in the school; in particular: what suggestions can be made to help those responsible for school organization in the various countries?

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In general, we can remark that mathematical theories and theoretical and cultural aspects of mathematics (like the systemic character of mathematical knowledge and the role of mathematical models in shaping scientific knowledge of the physical world) represent a challenge for mathematics educators all over the world.

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Neither abandoning them in curricula designed for most students, nor insisting in traditional teaching of them seem to be good solutions.

The former means dismissing the school in its task of passing over scientific culture and models of rationality to new generations.

The latter is scarcely productive and impossible to keep in today's school systems.

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An interesting and important area of investigation is opened to mathematics education research:

• on what theoretical and cultural aspects of mathematics should the efforts be concentrated?

• how must they be implemented in curricula?

• how to ensure a reasonable success in classroom activities about them?

ICME ICME 99In relationship with these "transitional"

characters of mathematics education in junior high school, we need to consider:• the epistemological questions (what does it mean "theoretical status of mathematical knowledge"? How can we distinguish mathematical theories from other scientific theories?)• the related cognitive and didactical questions (how can students approach theoretical knowledge? How can they arrive to distinguish some features of mathematical theories and other scientific theories?).

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2. SOME EXAMPLES TO GIVE FUEL TO THE DISCUSSION.

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1. The mathematical war

2. Which competencies for the XXI century?

3. New Standards and Testing

4. Methodologies

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1. The mathematical war

From the one side: those who support programs to make children think critically and solve problems.

From the other: those who support programs that stress basic skills.

2. How does the math war concern the JSS as a compulsory school?

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The revision of 1989 Standards by NCTMThe old emphasis on the mechanics of math -memorising tables, rules and equations- has gradually been replaced in many schools by a curriculum that encourages kids to explore math through estimation and real-life experience, such as the stock market.

Scores in some states are falling: students who never memorised rules are finding it can cripple them later.

The problem of getting (again) comfortable with numbers.

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JSS is crucial with this respect:

3. How to nurture creativity in students, giving them also the tools to get it?

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2. Which competencies for the XXI century?

(from the book: How people learn: brain, mind experience, Wash. D.C., Nat.Sci. Acad. Press, 1999)

An educational system for the 21st cent. must produce learners who read and think critically, express themselves clearly and persuasively, are able to solve complex problems, and become self-sustaining, lifelong learners.

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Some of the conclusions seem important for JSS:• focus on ‘conditionalized’ knowledge (not just content but also contexts)

• competencies that build foundations for later learning

• learner environments: learner-, knowledge-, community- centered

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3. New Standards and Testing

An article on the NYTimes (Jul 11) says that success in the new jobs doesn’t depend on mastery of one uniform body of knowledge as measured by standardized tests. Instead, many of them require an ability to learn on the job. Some of the required abilities depend on creativity, out-of-the-box thinking, …. Others depend on the ability to listen and understanding what other people are feeling and needing. Most require soft skills like punctuality and courtesy….

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Yes, people need to be able to read, write and speak clearly. And they have to know how to add, subtract, multiply and divide. But given the widening array of possibilities, there is no reason that every child must master the sciences, algebra, geometry, biology or any of the standard curriculum.

…..Our challenge is to find different measures of the various skills relevant to the jobs of the new economy. It’s our job not to discourage our children but to help them to find their way

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4. What about the Standards in the transitional period that features JSS?

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interacte in a complex and often contradictory way.In such cases, quite often results got by pupils in some specific objective tests seem to contradict their general performances: namely, clever pupils get less marks than good ones.An explanation of this countertendency points out the more complex level at which clever pupils look at some problems, which in fact become more difficult for them than for lower pupils, who do not even percieve such deeper aspects. This is not an exceptional in developing age.

Testing in transitional ages is very delicate. Typically when one tries monitoring the acquiring of such new delicate mathematical objects, where symbols, concepts, everyday language

LEVEL

SCORE

E D C B A

LEVEL

A

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4. Methodologies: the new, the old and other

The debate of the new Math War reduces everything to a contrast between the old methods and the new constructivistic ones.

But there are elements which are difficult to construct in a constructivist approach to theoretical knowledge and difficult to mediate through a traditional approach.

They concern concepts and contents which are typical of JSS.

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EXAMPLES

• contents (especially, counter-intuitive conceptions) which are difficult to construct individually or socially;• methods (for instance, mental experiments) far beyond the students' cultural horizon;• kinds of organization of scientific discourse (for instance, scientific dialogue; argumentation structured into a deductive chain) which are not a natural part of students' speech.

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3. CONSEQUENT SUBJECTS FOR THE DISCUSSION.

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A) Unifying concepts and ideas

B) Curricula

C) Goals and methods of math teaching in Junior Sec. School

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A) Unifying concepts and ideas

According to Billstein they are those that give depth (contrasted with breadth) to the curriculum.

5. WHICH ONES FOR JSS?

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Some issues to be focused specifically for JSS:

• The transition from Arithmetic to Algebra: why and how? Variables, functions, patterns, modeling (see Standards of NCTM)

•The transition to the theorical thought from episodic and empirical knowledge

• New technologies: which support to - visualisation (e.g. DGS) - symbolic languages (e.g. Spreadsh.)?

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B) Curricula

- 6 - Comparisons:

- in the space (different countries)

- in time (changes, new tendencies)

Recommendations

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C) Goals and methods of math teaching in Junior Sec. School

- 7 -•Which the most critical?

•Which the most robust?

•Which the major novelties at the beginning of the 21st century?

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SOME MAIN QUESTIONSSOME MAIN QUESTIONS

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1. What must the theoretical status of math be in the positive?2. How does the math war concern the jss as a compulsory school?3. How to nurture creativity in students, giving them also the tools to get it?4. What about the Standards in the transitional period that features JSS?5. Which unifying concepts for JSS?6. Curricula: comparisons and recommendations7. Goals and methods: the most critical, the most robust? Which the major last novelties?