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Bus conductors’ use of mental

computation in everyday settings

– Is it their ethnomathematics?

Nirmala Naresh

FOURTH INTERNATIONAL

CONFERENCE ON

ETHNOMATHEMATICS

(ICEM-4)

Session Agenda

!!Introduction

!!Background of the study

!!Research Methods

!!Data Analysis

!!Conclusion

!!Discussion

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The Study

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!! Focus on adults’ use of

mental mathematics in

everyday situations.

!! To observe, understand,

describe, and analyze the

mental mathematical

practices of the bus

conductors (in Chennai,

India) at their workplace.

Ethnomathematics

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!!Issues of social change, equity, and justice – Catalyst to the ethnomathematics movement.

!!Forerunners of ethnomathematics

•! Otto Raum – Arithmetic in Africa (1938),

•! Dirk Struik – Sociology of mathematics (1942),

•! Gay & Cole – The new mathematics and an old

culture (1967).

!!D’ Ambrosio – Father of ethnomathematics.

Four Strands of Ethnomathematics

!! A challenge to the traditionally told history of mathematics.

!! Analyses of mathematics associated with the traditional

cultures of indigenous people.

!! Explore the mathematics of different groups of people in

everyday settings.

!! Ethnomathematics and multicultural mathematics education.

(Vithal &Skovsmose, 1997)

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Everyday Mathematics

!! Investigation of everyday practices of groups of people.

!! Mathematical concepts that were described include mental

computation, arithmetic, proportional reasoning, and

geometry.

!! Grocery shoppers, Bankers (Lave, 1991).

!! Street vendors in Beirut (Jurdak & Shaheen,1999).

!! Candy sellers in Brazil (Carraher, Carraher, & Schliemann,

1987; Saxe, 1991).

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Workplace mathematics – A subset of research on

everyday mathematics

!! Insight into how people conceptualize the role of

mathematics in their work.

!! Is in its formative years.

!! What mathematics is used by workers? How is it used? Do

they use school-taught mathematics at work?

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Mental computation in everyday situations

!! Adults reason in problematic situations quite differently from the

way they did mathematics at school.

!! Problem solving in everyday situations is “characterized by a

pragmatic agenda and geared to solving particular

problems” (Noss, Hoyles & Pozzi, p. 17).

!! Further investigations needed to understand how people make

sense of mathematics outside school. (Carraher, 1991)

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Mental computation at workplaces

!! Work places varied mathematical demands upon workers.

(Carraher, p.195)

!! “Most of the knowledge used at the workplace remains implicit

and sometimes unconscious” (Vergnaud, 2000, p. xvii).

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Rationale for investigating everyday mathematics

!! How do people make sense of mathematics outside school ?

!! “Most of the knowledge used at the workplace remains implicit and sometimes unconscious” (Vergnaud, 2000, p. xvii).

!! Theoretical and empirical research base needed to develop research methodologies and also for curriculum development. (Millroy, 1992; Zevenbergen, 2000)

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Rationale for investigation in India

!!Desire to add to mathematics education research in

India.

!! Few contributions to the research area of everyday

mathematics from India. (e.g., Mukhopadhyay,

Resnick, & Schauble, 1990)

!! Several professions in India serve as rich sources for

investigation of mathematical practices of adults.

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Significance of the study

!! Draw out specific aspects of the bus conductors’ practice that contribute to their mental computational activities.

!! Contribution to research field of ethnomathematics, everyday mathematics, and workplace mathematics by extending it to a new social context.

!! Contribution to the line of research on mental computation.

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Theoretical fields

Adults’ use of mental math at

work

Ethnomathematics

Everyday Mathematics

Mental computation

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Theoretical Model – Saxe’s model

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!!Reformulation involved alteration of the numerical data

in the problem.

!!Translation strategy involved modifying the mathematical

structure of the problem.

!!Compensation technique was used to make adjustments

(to the problem) to compensate for the reformulation

and transformation techniques used.

Theoretical model: RTC framework

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Bus conductors are employees of the

Metropolitan Transport Corporation (MTC) of

Tamilnadu.

Study

Setting

Metropolitan

Transport

Corporation

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Fleet strength of 2,773 buses and 2,554

services. 486 routes 3.7 million passengers on

any single day.

Study

Setting

Metropolitan

Transport

Corporation

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The average daily collection per day is

estimated to be 10.9 million rupees.

Study Setting

Metropolitan

Transport

Corporation

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Research Purpose

Analyze, compare and characterize bus conductors’ use of

mental computation.

- Focus of this paper: One bus conductor’s mental

computational acts at work

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Overarching research methodology

!! Qualitative study

!! A non-experimental design

!! Case study approach

!! Instrumental case studies: Focus of this paper

"! Mr. Delta’s mental computational acts at work

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Research Methods: Data Sources

Mental computation: Work Setting

"!Archival records,

"!Official documents,

"!In situ observations.

"!Participant Responses to computational

tasks (Stimulated recall techniques)

"!Interviews – Formal, and semi structured.

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Research Methods: Data Analysis

Three levels of data analysis.

!!Level 1: Analyze data as the data are

collected.

!!Level 2: Creation of a case study database.

!!Level 3: Use Saxe’s model & RTC model to

analyze and interpret the data.

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Participant – Mr. Delta

!!Educational qualification – PUC fail (equivalent to higher secondary school).

!!Good recommendations from superiors.

!! 27 years of service.

!!Bitter school experiences; not proficient in maths.

!!High opinion about mathematics learning.

!! Sincere and dedicated – did not hold his job in high esteem.

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List of Terminologies

!!Fare – Depends on entry and exit point, type of bus.

!!Batta (allowance) – Allowance conductors get everyday based on total collection amount, type of bus operated.

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TICKET

Receipt for passenger paid fare

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Traffic Return

An official

document used to record ticket sales

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Waybill

Official

document used

to record total

ticket sales,

collection

amount.

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Analysis: What are Mr. Delta’s duties at work?

!! Commute several times from point A to point B & pick up and drop off passengers en route.

!! Issue ticket to a commuter based on his entry and exit point.

!! Tender exact change back if the commuter gives more money than the required amount.

!! Keep a record of the number of tickets issued from stage to stage.

!! Calculate the daily allowance based on the day’s collection.

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Analysis: Three-phase structure of Mr.Delta’s work

Sign on time (Before bus trips)

Collect imprest amount (Rs. 30.00), locate bus route, bus #, bus driver.

Spread over time (During bus trips)

Issue tickets to passengers, update travel records

Sign off time (End of bus trips)

Calculate daily batta, submit trip earnings

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Analysis: Evidence of mental computation at work

Sign on time

•!Little evidence of mental computational acts

Spread over time

•!Execution of ticket transactions

•!Completion of official records

Sign off time

•!Calculation of daily batta

•!Calculation of overall trip earnings

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Ticket transaction scenario: More terminology

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!! PPF: Passenger Paid Fare "! Fare a passenger hands over to the conductor

!! ATF: Actual Ticket Fare "! Fare pre determined to travel from point A to point B

!! BA: Balance Amount "! If the PPF > ATF, the balance amount the conductor hands

over to the passenger

Possible ticket transaction scenarios

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A snippet from workplace activity

!!A passenger boarded the bus at point P along with four members of his family. Mr. Delta approached the passenger. The passenger handed out a 50-rupee note

and requested the conductor for five tickets to point Q. Mr. Delta requested the passenger to give him an extra 50 paise coin and the passenger obliged. He then gave the passenger five tickets and handed 33 rupees back to

the passenger.

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Concomitant perceptions of goals associated with

the activity

Conductor’s perception

Carry out a ticket transaction - Issue 5 tickets to passengers travelling from P to Q.

Researcher's perception

Complete a mental computational problem

Solve 50 – (3.50 * 5)

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Overall work-related goals

!! Have an incident free day,

!! Serve passengers to the best of his ability to avoid

confrontations with the commuters

!! Authenticate every person’s travel in the bus;

!! Earn maximum collection and daily allowance.

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Emergent goals

!! Associated with work-related goal-directed

mathematical activities.

!! Emerge and fade with each activity.

!! Little “I need to do ….” things associated with work-

related activities.

!! Concomitant perceptions of goals

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Activity structures

Tasks in activity cycle (conductor’s interpretation) !! Approach passenger !! Gather information !! Issue tickets

Tasks in activity cycle (researcher’s interpretation) !! Determine actual ticket fare !! Evaluate passenger paid fare !! Compare passenger paid fare with the actual ticket fare; !! Determine the balance amount !! Match currency with the balance amount

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Prior understandings

!! Knowledge about the bus routes;

!! Names and number of bus stops and fare stages along their bus routes;

!! Stage fares and ticket denominations for different types of buses; and

!! School-learnt mathematical strategies.

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Conventions and artifacts

!! Routine activities – Approach passenger, gather

travel information, negotiate in reformulating the

problem.

!! Use of currency – helps do quick computation,

serves to cross check calculation.

!! Printed tickets help keep track of the figures

involved in the transaction.

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Social Interactions

!! Interacted with the passenger to initial ticket transaction

!! Crucial to completion of tasks set in activity structures.

!! Sought passenger’s cooperation in reformulation the

original problem.

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Evidence of mental computation: Mr. Delta’s method of

completing a waybill

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Evidence of Mental calculation: Completion of a waybill

Waybill contents

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Evidence of Mental calculation: Mr.Delta’s method of

calculating batta

Note: Total ticket collection amount for the day 7100.

"! Variable allowance: Rs. 2.00 (for every Rs. 100 collected)

"! For Rs. 1000, the allowance is Rs. 20.00.

"! For Rs. 7000 it is 140.

!! I added 7 thousands to get 7000. So I added 20 seven times to get

140. I added another two to this amount to account for the extra

100. The total variable allowance is 142.00.

!! The fixed allowance is Rs. 45. Penalty for lost trips: Rs. 10.

!! Actual allowance claimed: 142 + 35 = 177.

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Characterization of Conductors’ Mathematics

Three components of conductors’ mathematics:

"! Use of prior knowledge.

"! Use of work-specific knowledge.

"! Use of monetary units.

Acquire, appropriate, and specialize the above three

forms, that I term

“Bus conductors’ mathematics”.

Their ethnomathematics indeed!

Discussion

Mr. Delta’s reaction:

“You may be right. However, when I work, I don’t think about what mathematics I

am using. I see money, I hear words, my hands deal with tickets and

money and I calculate in my heart. There is little time to think about anything else. I don’t gloat about the mathematics that I do everyday. I cannot afford to make mistakes in my calculations because I

may end up losing government money. So I only concentrate on doing my job quickly and correctly.”

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Thank You!

Questions?

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