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Over the past decade, mathematics education researchers have used the lens of participation to explore how student activity relates to mathematical learning. While this lens is useful in drawing connections between mathematical activity that occursMoreOver the past decade, mathematics education researchers have used the lens of participation to explore how student activity relates to mathematical learning.

While this lens is useful in drawing connections between mathematical activity that occurs during lessons and subsequent mathematical activity, it does not provide a specific conception of central concerns of mathematics education: student learning and thinking. Narrowing participation to communication and equating thinking with communication addresses this problem.

Using this frame, discourse becomes the object of study and learning is defined as a change in discourse. However, what remains unexplained are questions about why students in similar circumstances engage in different activity or why the same student engages in different activity on different occasions. Researchers have turned to identity to tie student activity to who students think they are.

By examining the interplay of discourse about who students think they are (identifying) and discourse about mathematical objects (mathematizing), it is possible to explore, elaborate, and clarify connections across mathematical activity, identifying activity, and learning.-One tool that supports the analysis of mathematizing, identifying, and learning is a theory of autonomous learning. Autonomy is a central goal of reform mathematics education, yet it has not been well defined or well studied in classroom settings.

Defining autonomous learning as a constellation of mathematizing and identifying activities reflecting curiosity creates a tool for evaluating students learning activities and suggesting how mathematics educators might support more mathematically desirable learning outcomes.-This study used the lenses of mathematizing, identifying, and autonomous learning to probe the discourse of three fourth grade students engaged in a mathematics lesson.

The analysis of discourse evaluated changes in each students mathematical discourse across the lesson, examined mathematizing and identifying activities might be grouped and described as a kind of learning, and made connections between those changes in discourse and kinds of learning.

Finally, the kinds of learning and changes in mathematical discourse were examined using the theoretical lens of autonomy to determine what advantages the activities of autonomous learning might have for the development of a mathematically desirable discourse.-The findings show five kinds of learning: engaged, directed, covert, watchful, and guided.

Each was characterized by a distinctive group of intertwined identifying and mathematizing activities that could be connected to changes in discourse. The kinds of learning also varied in their resemblance to the activities of autonomous learning, with the kind of learning which was the most autonomous exhibiting the development of a more mathematically desirable discourse.-The findings carry implications for how mathematics educators might construct learning situations to encourage identifying and mathematizing to best support student learning.

They also suggest that the lenses of mathematizing, identifying, and autonomous learning are useful tools for examining complexities of students actions and interactions as they learn mathematics.

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