Motivation

General educational character of mathematics class

What is it about? What is supposed to be learned? Why this exactly?

Three experiences characterize the general educational mathematical class according to H.Winter:

• Perceiving and understanding appearances of the world around us ... in a specific way
• Get to know and understand mathematical subjects ... as a deductive peculiarly ordered world.
• Acquiring problem solving skills by discussing problems (heuristic skills) beyond math.

According to H. Winter math class should for all students be geared towards:

• creativity
• rational argumentation
• making it mathematical/ applicable
• formal proficiencies
• ability to visualize
• ability and willingness for dialogue
• creating and mastering new situations.

Creativity of students in math class becomes apparent:

• by prompting questions and questioning issues or presentations about mathematical topics
• in discovering or developing (subjective new) mathematical coherences, problem solving methods or applications of mathematics - especially in recognizing possibilities to describe coherences of every day life.
• in discovering and trying a new (unconventional) option to solve a problem posed
• in varying and creating problems with inventive presentation, reasoning and valuing of working results (projects)

All listed expressions of creativity demand a math class that is actionoriented, allowing such actions and opportunities in the first place.

Historical believes about the nature of mathematics:

"Mathematics is the prototype of science and a suppliant of methods transferable to daily life." (Bigalke 1976)

Diligent pursuit of mathematics promotes:

• objective thinking
• denial of illegitimate generalizations
• precision of language (Meschkowski 1980)

Aristotele recommends mathematics (only) for more mature adolescents of 15 years of age. According to Plato mathematics is the "call of knowledge". The intellectual character of mathematics is reflected in the following four points if you

1. are silent, if you have nothing to say
2. only claim things that can be proven
3. make a difference between axioms and things that can be proven
4. protest in the strongest terms against the mixing of acknowledgements and insight or inverifiable statements

... and that is part of the understanding of general education

A knowledge-based society implies the acquisition and processing of knowledge by people.

The area of general education was divided in four areas, that are

1. personal competency
2. social competency
3. instrumental competency and
4. content competency or basic knowledge.

"Personal expertise is defined by practical knowledge and the ability to deal with knowledge, and social competency is defined as communicative skills and social responsibility. Personal and social competency are renowned as the most important competencies for the individuals orientation in knowledge-based society!"

"Urgent needs for reformation - according to experts - are among the learning arrangements and the learning methods."

_{(1) Source: Horster, L. und Rolff, H.-G.: Unterrichtsentwicklung. Grundlagen, Praxis, Steuerungsprozesse. Beltz Weinheim und Basel 2001}

Different learning objectives for math class (according to ZECH):

A) Interdisciplinary objectives

1^st fundamental thinking activities and - attitudes

• Ability to visualize (visual thinking, use of presentiveness),
• logical thinking (inner-mathematical, outer-mathematical, rules of formal logic),
• ability to communicate and cooperate (rational dialogue, taking statements in and examining them),
• promotion of speech and critical faculties (verbalization of issues, specification of language),
• supporting actions of problem solving and creativity (combinatory thinking, ability to combine, acquisition of new thinking strategies)
• independence and ability to work on your own.

2^nd Basic mental techniques

• comparing, order, abstract, generalize, classify, specify, formalize, analogize

3^rd General "educational objectives" of math class

• virtues of working like: carefulness, exactness, preciseness, clarity and order.

B) General subject related objectives of math class

1^st Skills and attitudes related to single contents

• arithmetic proficiency ("culture techniques"),
• arithmetic ability (situations in our environment are mathematized),
• Understanding and assessing environmental appearances of mathematical kind.

2^nd "Insights and attitudes that are related to the whole subject

• acknowledging skills, opportunities and limits of mathematics,
  
• enjoyment of "esthetic and play"