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\pagestyle{myheadings} \pagenumbering{arabic} \setcounter{page}{1}
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\par \noindent
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}
\par\noindent\par\noindent
\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
Mathematics\hspace{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.

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  • 2
    If you add references manually to thebibliography it is your job to sort them correctly. LaTeX has no provision for doing so and since LaTeX itself is not good at sorting strings (that's one of the reasons for BibTeX to exists in the first place) it would be quite challenging to implement (though not impossible, I guess, it is already possible with a bit of trickery to sort by cite order: tex.stackexchange.com/q/458422/35864, but you need to change the syntax for that slightly). – moewe Mar 25 '19 at 5:14
  • 1
    Roughly the same holds for the bibliography format: If you use a manual thebibliography, you have to take care of formatting yourself. Changes to the style mean that you have to go through all references manually and reformat them according to the desired style. BibTeX of course would allow you to automate this. – moewe Mar 25 '19 at 5:15
  • 1
    Off-topic (but since you posted the entire preamble): Note that it is strongly recommended to use \newcommand instead of \def or \let to define commands. \newcommand checks if the name you want to use is already taken. Especially with one-letter names (that people seem to like) there is a danger that you overwrite an important internal command. Two-letter font commands like \bf have been deprecated for more than twenty years now (texfaq.org/FAQ-2letterfontcmd), they should be avoided and replaced by their modern counterparts: \bfseries for \bf ... – moewe Mar 25 '19 at 5:29

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