ETNOMATEMÁTICAS. Matemáticas y cultura.

Introduction to Mathematical Knowledge in Different Cultures

This section introduces the existence and importance of mathematical knowledge in various cultures throughout history. It highlights the empirical nature of this knowledge, passed down through generations and utilized in different professions.

Mathematical Knowledge as Empirical and Cultural

• Mathematical knowledge has been present in all cultures throughout history.

• It is an empirical knowledge used in various professions and passed down through generations.

• In the past, this knowledge derived from social practices was often ignored in education.

• In recent years, non-mathematics has emerged as a new branch of mathematical knowledge that aims to incorporate cultural values, customs, and traditions into education.

Ethnomathematics: Connecting Mathematics and Culture

This section explores the concept of ethnomathematics coined by D'Ambrosio. It focuses on studying the relationship between mathematics and culture, aiming to understand both aspects better. The existence of six universal mathematical activities across societies is discussed.

Ethnomathematics Defined

• Ethnomathematics refers to the study of connections between mathematics and culture.

• Its goal is to contribute to understanding both culture and mathematics.

• Alan Bishop's research confirms six universal mathematical activities found in all societies.

Origins of Mathematics in Society

This section delves into the origins of mathematics within society. It suggests that mathematics evolved from human needs for measurement, determination, and understanding their surroundings. The development of basic concepts like shape, number, etc., is explored.

Evolutionary Nature of Mathematics

• Mathematics is an evolving body of knowledge closely related to other procedures.

• It originated from human needs for measuring, determining shapes, etc., in their surroundings.

• Primitive humans likely conceived the idea of numbers before the development of language.

• Early societies, such as hunter-gatherer communities, developed primary ideas of shape, form, and number.

• Over centuries, these mathematical concepts were refined and expanded upon by ancient civilizations like the Babylonians and Egyptians.

Theories on Development of Mathematical Knowledge

This section discusses two psychological theories on how individuals develop mathematical knowledge. The modular theory suggests that the mind functions with independent modules for different tasks. The constructivist and social cognitive theory emphasize the role of endogenous and social-cultural activities in generating logical-mathematical knowledge.

Modular Theory vs. Constructivist and Social Cognitive Theory

• Modular Theory: The mind operates using independent modules or computational processors for various tasks.

• Each module is functionally distinct and processes specific data inputs.

• These modules are pre-established or innate rather than constructed from previous processes.

• Constructivist and Social Cognitive Theory:

• Knowledge is generated through endogenous activity (Piaget) and social-cultural interaction (Vygotsky).

• Individuals assimilate new situations using existing cognitive schemas.

• As a result, cognitive schemas are restructured to accommodate new information.

Piaget's Solipsist Theory vs. Vygotsky's Socio-Cognitive Development

This section compares Piaget's solipsist theory with Vygotsky's socio-cognitive development theory regarding the origin of knowledge. Piaget emphasizes individual exploration as the driving force behind cognitive development, while Vygotsky highlights the social origins of cognition.

Piaget's Solipsist Theory

• Knowledge originates from subject-object interaction through exploration.

• Piaget uses a biological metaphor where the mind acts as a stomach that assimilates knowledge to create intelligence.

• Factors influencing cognitive development include nervous and endocrine system maturation, physical experimentation, sociocultural transmission, and equilibrium with assimilation and accommodation.

Vygotsky's Socio-Cognitive Development Theory

• Human mind has a social origin, and cognitive development is both endogenous and exogenous.

• Social interaction, including mundane and educational contexts, drives the development of mathematical knowledge.

• Tools and artifacts play a crucial role in cognitive development through action.

Piaget's Sensorimotor Schemes vs. Vygotsky's Tool Use

This section explores Piaget's sensorimotor schemes and Vygotsky's emphasis on tool use as they relate to cognitive development. Both theories highlight the importance of these processes in developing mathematical concepts.

Piaget's Sensorimotor Schemes

• Infants develop sensorimotor schemes related to quantity recognition.

• For example, a 9-month-old baby can distinguish between "much" and "little."

• These schemes form the basis for the notion of numbers.

Vygotsky's Tool Use

• Cognitive development is influenced by using tools or artifacts.

• The motor skills involved in using tools contribute to the formation of mathematical concepts.

• An example is children selling candy, which involves counting money and understanding basic arithmetic operations.

The Role of Culture in Mathematical Thinking

This section discusses how the mind of a mathematician develops within a cultural context. Mathematics is seen as a set of procedures used to investigate small changes in psychological processes. The sociocultural perspective and its theoretical assumptions are also mentioned.

Cultural Context and Mathematical Thinking

• Mathematics is developed within a cultural context.

• Mathematics involves procedures to investigate small changes in psychological processes.

• The sociocultural perspective plays a key role in understanding mathematical thinking.

Components of Working Memory

• Working memory consists of several components, including the phonological loop, visuospatial sketchpad, and episodic buffer.

• The phonological loop manipulates verbal information for short-term storage.

• The visuospatial sketchpad creates and manipulates visual images.

• The episodic buffer integrates contextual information into a multimodal representation.

• These components are supervised and controlled by the central executive.

Functions of Working Memory Components

• The visuospatial sketchpad stores observed information related to the problem at hand.

• The phonological loop stores verbal information, such as numerical instructions.

• The central executive manages the stored information from both systems to coordinate problem-solving.

• The episodic buffer temporarily stores multidimensional episodes related to problem-solving.

Problem-Solving Components

• Four components are necessary for solving mathematical problems: translation, integration, planning, and execution.

• Translation involves understanding and translating linguistic and semantic aspects of the problem statement.

• Integration requires knowledge about existing types of mathematical problems relevant to the given task.

• Planning involves strategic knowledge to follow steps towards achieving the solution goal.

• Execution relies on procedural knowledge for performing mathematical operations.

Evolution of Numerical Notation Systems

This section explores the historical development of numerical notation systems. Various ancient systems, such as Greek, Roman, and Egyptian numerals, are discussed. The invention of the Hindu-Arabic numeral system is highlighted.

Ancient Numerical Notation Systems

• Early numerical notations consisted of groups of vertical or horizontal lines to represent numbers.

• These notations were challenging for handling large quantities effectively.

• Ancient civilizations developed their own systems, including Greek Ionic, Old Slavic, Cyrillic, Hebrew, and Arabic numerals.

Limitations of Early Notation Systems

• While these notations were concise in writing, they were impractical for performing operations with large numbers.

• Significant effort was required to perform calculations using these systems.

Hindu-Arabic Numeral System

• Approximately 2,200 years ago, a Hindu mathematician invented the Hindu-Arabic numeral system.

• This system introduced positional notation where each digit's position represents its value.

• The base 10 system allowed for easy representation and manipulation of large numbers by adding digits to the left.

Mathematical Skills and Cultural Context

This section discusses the various mathematical skills generated by the human brain and emphasizes that mathematical abilities can only exist within a culturally mediated activity. It also mentions the importance of recognizing ethnomathematics or living mathematics in different cultures.

Mathematical Skills Generated by the Brain

• The human brain generates various mathematical skills such as numerical reasoning, mental calculation, quantitative reasoning, algebraic reasoning,

spatial orientation,

spatial coordination,

visual imagery,

aesthetic sense,

numerical skills,

estimation,

visualization,

figurative interpretation,

drawing representation,

visual memory,

strategic thinking,

planning,

social-interpersonal skills,

logical reasoning,

and verbal reasoning.

Mathematics as Culturally Mediated Activity

• Mathematical abilities can only exist within activities mediated by cultural instruments.

• Different cultures have their own mathematical practices and elements of ethnomathematics or living mathematics.

• Some cultures and practices are often overlooked or invisible, leading to the concept of "frozen" mathematics.

Conclusion

Mathematical thinking is influenced by cultural context, and understanding the role of culture in mathematical development is crucial. Working memory components play a significant role in problem-solving, and different numerical notation systems have evolved over time. Mathematical skills are generated by the brain within culturally mediated activities, emphasizing the importance of recognizing diverse mathematical practices across cultures.

Mathematics in Production Techniques

This section discusses how mathematics is involved in the invention and learning of production techniques. It focuses on the organization and application of mathematical knowledge in artisanal cultures.

Mathematics in Cultural Activities

• Artisan cultures engage in activities such as counting, locating, and calculating.

• They develop their own systems of numeration and ways to represent numbers.

• The Inca culture had the yupana and the quipu as their numerical representation tools.

Yupana and Quipu

• The yupana is a board with squares or compartments used for arithmetic calculations.

• The quipu represents quantities and was used for storytelling, statistics, calendars, etc.

• Researchers have attempted to decipher the quipu but there is still much unknown about its exact representation.

Numerical Representation in Quipu

• Quipus are recognized by groups of knots on strings, with each group representing a number.

• Symmetrical spacing helps detect the number zero when there are no knots in a specific space.

Problem-Solving Exercise

This section presents a problem-solving exercise involving counting cars. It highlights the importance of understanding key words and following problem-solving steps.

Problem-Solving Steps

1. Translation: Understand the problem statement and identify key information.

2. Integration: Recognize that this is a compound problem that requires multiple steps to solve.

3. Planning: Determine the solution approach, considering intermediate operations if necessary.

4. Execution: Perform the calculations based on the plan.

Analysis of Responses

• Two subjects attempted to solve the problem without fully understanding all problem-solving steps.

• Both subjects focused only on executing the solution without considering translation, integration, planning, or supervision stages.

• Lack of cognitive development in problem-solving stages led to incorrect answers.

Mathematics as a Cultural and Cognitive Product

This section explores the cultural and cognitive aspects of mathematical knowledge. It emphasizes that mathematical understanding is not solely innate but also influenced by cultural interactions.

Mathematical Knowledge Origins

• Mathematical knowledge is a product of both biological and cultural interactions.

• The human mind cannot be understood outside of its cultural context.

• Six universal activities form the basis for mathematical skills.

Importance of Cultural Artifacts

• Mathematical concepts are discovered in various artifacts, tools, toys, architecture, and artwork.

• These artifacts provide insights into human cognitive development.

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