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Punjab University \

Journal of Mathematics (ISSN 1016-2526) \

Vol. 51(7)(2019) pp. 00.00  vspace*{1pc}



begin{center}

{bf Creating an Environment for Learning Mathematics}

par noindent

vspace*{1pc}

parnoindentparnoindent

par noindent

Muhammad Ahmad \

Department of Mathematics, \

University of sargodha, Pakistan,\

Email: {m.ahmadpak@gmail.com}\

par noindent

end{center}

vspace*{0.5pc}Received: 07 April, 2018 / Accepted: 06 June, 2018 /

Published online: 20 December, 2018 vspace*{0.5pc}

begin{quote}

{bf Abstract.} It was cite{article-full} with the theoretical ideas about

constructivists' view of learning discussed in the preceding chapter

that we began our collaboration with the classroom teacher. Although

we communicated our intentions in discussions about the importance

of problem solving for learning and the necessity of social

interaction and class discussion, it was still the teacher's

obligation to enact these in the classroom. Admittedly we were well

aware that children actively discussing challenging problems in

primary grades was different from the way mathematics had been

taught in the past, but we had not yet realized the extent to which

these ideas would influence the practice of elementary school

mathematics. These aspects--challenging problems, collaborative

group work, and class discussion about students' solutions-were, for

the teacher, against tradition. It was accepted practice for her to

initiate grouped settings and discussions in social studies,

science, and reading, but she did not do this in mathematics. It was

against this background that the classroom teaching experiment

began.

end{quote}

vspace*{1.5pc} noindent {bf AMS (MOS) Subject Classification

Codes: 35S29; 40S70; 25U09} \

smallskip noindent

{bf Key Words:} -------------------------------------------------.



markboth{underline{hspace{3.7in} Muhammad Ahmad}}

{underline{hspace{0pt}Creating an Environment for Learning

Mathematicshspace{2.5in}}}pagestyle{myheadings}

{setcounter{section}{0}}

section{Introduction}

Typically a class session began with the teacher leading a brief

introduction intended to insure that the children understood what

they would be working on for the day. Once the teacher was satisfied

that the children understood the intent of the activities, she then

passed out the activity sheets and small-group work began. Children

worked in pairs on activities, which were on sheets of paper that

provided room for students to write. Each pair received one sheet to

share in completing the activity. Generally three to four sheets,

each containing four to six problems, were available for the

students to work on. Some children completed all the activity

sheets, whereas others only finished one. The problem solving as

pairs generally lasted 20 to 25 minutes.





section{Notations and Preliminaries}



The expectations for children's actions in the mathematics class

were quite different from their previous experiences in school.

However, in this mathematics class it was necessary for children to

express their thinking in order to create opportunities for learning

and so that their existing constructions could be investigated by

both the teacher and researchers

section{Discrete Evolution Semigroup}

Using these premises of children's learning as her guideline, the

teacher initiated the mutual construction of a different set of

norms for mathematics lessons as she acted to help the students

reconceptualize their role during mathematics instruction. Her

intention was for the children to figure things out for themselves

and to express their ideas in the public arena of whole-class

discussions. Additionally, during small-group work she expected them

to cooperate and work together to solve problems and to agree on an

answer. Her expectation that the children would express their

thoughts placed the students under the obligation of having to

recall their solutions and explain them to others during the

whole-class discussion.

section{Results}

The nature of the teacher and student interaction that occurred

within the whole-class discussion was crucial to establishing the

social norms that were necessary for developing a setting in which

the children would feel psychologically safe to express their

mathematical thinking.] The teacher's intention as she led class

discussion was to encourage children to verbalize their solution

attempts.

section{Applications}

Her comments were focused on talking about how in this class they

were going to talk about mathematics. In this example she told the

students that thinking was valued even more than right answers.

These mutual obligations and expectations were negotiated and

renegotiated by the teacher and students as they established an

interaction pattern that would form the basis for their activity.

These mutually constituted patterns of interaction were taken for

granted and made possible the smooth functioning of their collective



section{Conclusion}

It became evident that a psychological perspective alone could not

account for the complexity of the events occurring in the classroom.

Establishing social norms that provided the setting in which

children engaged in meaningful activity was an aspect of social

interaction not considered prior to the classroom teaching

experiment. As these norms became accepted, the students

participated in a type of discourse in which they were expected to

explain and justify their solutions and listen to others. The

teacher acted to initiate and guide students' learning by posing

questions and highlighting children's expectations. As students

engaged in this discourse, their personal meanings were negotiated

until an agreement was reached. The establishment of taken-as-shared

meanings between the participants enabled mathematical ideas to be

established by members of the class.

section{Acknowledgments}

I would like to thank my supervisor, Prof. Nicholas Young, for the

patient guidance, encouragement and advice he has provided

throughout my time as his student. I have been extremely lucky to

have a supervisor who cared so much about my work, and who responded

to my questions and queries so promptly. I would also like to thank

all the members of staff at Newcastle and Lancaster Universities who

%helped me in my supervisors absence.



newcommand{noopsort}[1]{} newcommand{printfirst}[2]{#1}

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begin{thebibliography}{99}



bibitem{article-minimal}

L[eslie]~A. Aamport.

newblock The gnats and gnus document preparation system.

newblock {em mbox{G-Animal's} Journal}, 1986.



bibitem{article-full}

L[eslie]~A. Aamport.

newblock The gnats and gnus document preparation system.

newblock {em mbox{G-Animal's} Journal}, 41(7):73+, July 1986.

newblock This is a full ARTICLE entry.



bibitem{article-crossref}

L[eslie]~A. Aamport.

newblock The gnats and gnus document preparation system.

newblock In {em mbox{G-Animal's} Journal/} cite{whole-journal}, pages 73+.

newblock This is a cross-referencing ARTICLE entry.



bibitem{whole-journal}

{em mbox{G-Animal's} Journal}, 41(7), July 1986.

newblock The entire issue is devoted to gnats and gnus (this entry is a

cross-referenced ARTICLE (journal)).



bibitem{whole-set}

Donald~E. Knuth.

newblock {em The Art of Computer Programming}.

newblock Four volumes. Addison-Wesley,

{noopsort{1973a}}{switchargs{--90}{1968}}.

newblock Seven volumes planned (this is a cross-referenced set of BOOKs).



bibitem{inbook-minimal}

Donald~E. Knuth.

newblock {em Fundamental Algorithms}, chapter 1.2.

newblock Addison-Wesley, {noopsort{1973b}}1973.



bibitem{inbook-full}

Donald~E. Knuth.

newblock {em Fundamental Algorithms}, volume~1 of {em The Art of Computer

    Programming}, section 1.2, pages 10--119.

newblock Addison-Wesley, Reading, Massachusetts, second edition, 10~January

{noopsort{1973b}}1973.

newblock This is a full INBOOK entry.



bibitem{inbook-crossref}

Donald~E. Knuth.

newblock {em Fundamental Algorithms}, section 1.2.

newblock Volume~1 of {em The Art of Computer Programming/} cite{whole-set},

second edition, {noopsort{1973b}}1973.

newblock This is a cross-referencing INBOOK entry.



bibitem{book-minimal}

Donald~E. Knuth.

newblock {em Seminumerical Algorithms}.

newblock Addison-Wesley, {noopsort{1973c}}1981.



bibitem{book-full}

Donald~E. Knuth.

newblock {em Seminumerical Algorithms}, volume~2 of {em The Art of Computer

    Programming}.

newblock Addison-Wesley, Reading, Massachusetts, second edition, 10~January

{noopsort{1973c}}1981.

newblock This is a full BOOK entry.



bibitem{book-crossref}

Donald~E. Knuth.

newblock {em Seminumerical Algorithms}.

newblock Volume~2 of {em The Art of Computer Programming/} cite{whole-set},

second edition, {noopsort{1973c}}1981.

newblock This is a cross-referencing BOOK entry.



bibitem{booklet-minimal}

The programming of computer art.



bibitem{booklet-full}

Jill~C. Knvth.

newblock The programming of computer art.

newblock Vernier Art Center, Stanford, California, February 1988.

newblock This is a full BOOKLET entry.



bibitem{incollection-minimal}

Daniel~D. Lincoll.

newblock Semigroups of recurrences.

newblock In {em High Speed Computer and Algorithm Organization}. Academic

Press, 1977.



bibitem{incollection-full}

Daniel~D. Lincoll.

newblock Semigroups of recurrences.

newblock In David~J. Lipcoll, D.~H. Lawrie, and A.~H. Sameh, editors, {em

    High Speed Computer and Algorithm Organization}, number~23 in Fast Computers,

part~3, pages 179--183. Academic Press, New York, third edition, September

1977.

newblock This is a full INCOLLECTION entry.



bibitem{incollection-crossref}

Daniel~D. Lincoll.

newblock Semigroups of recurrences.

newblock In Lipcoll et~al. cite{whole-collection}, pages 179--183.

newblock This is a cross-referencing INCOLLECTION entry.





bibitem{M.Ali,}  emph{A new capacity for plurisubharmonic functions}, Acta Math. textbf{170}, (1988) 1-21.

bibitem{H. Asad,} emph{Tranchage et prolongement des  courants positifs fermes}, Math. Ann. textbf{507}, (1991) 673-687.

bibitem{De}{J. P. Domnay,} emph{Complex Analytic and Differential Geometry}, http://www-fourier.ujf-grenoble.fr/demailly/books.html

bibitem{R. komar and B. Sharma,} emph{A characterization of Holomorphic chains}, Ann. Math.

textbf{18}, (1974) 253-287.

bibitem{H. Raheel,} emph{Geometric Measure Theory}, Berlin, New-York, Springer-Verlag, 1989.

bibitem{M.Waqas and Khuram Shahzad,} emph{Op'erateur de Monge Amp`ere, tranchage et extention de courants positifs ferm'es, Th`ese d''etat}, Fac. Sc. Tunis 1996.



end{thebibliography}

end{document}