International Group for the Psychology of Mathematics Education, 2016
This study examined discussions centered on precise mathematical language use in two fifth grade ... more This study examined discussions centered on precise mathematical language use in two fifth grade classrooms. Drawing on episodes from lessons in which teachers focused on encouraging mathematics reasoning, our analysis examines the relationship between precise language use and mathematical justifying. We present three classroom episodes that illustrate facets of the relationship between precise language use and mathematical argument, and explore how blurring the borders of two types of pedagogical tools (attention to precise mathematical language and engagement of students in justifying) promotes enculturation of students into mathematics.
International Group for the Psychology of Mathematics Education, 2018
Middle school is a critical time when students begin formal study of functional relationships in ... more Middle school is a critical time when students begin formal study of functional relationships in algebra. However, many students struggle in understanding functions as relationships between quantities that change according to a dependency relationship. We report on the influence of Scaling Continuous Covariation in fostering productive ideas about graphical relationships and rates of change. Scaling Continuous Covariation entails the ability to imagine a re-scaling to any increment for x and coordinate that scaling with associated values for y. We present findings from two students, one who reasoned with Scaling Continuous Covariation and one who did not, and report on how Scaling Continuous Covariation supported students' reasoning in three ways: (a) sense making about graphs, (b) forming constant rates of change, and (c) understanding constantly-changing rates of change.
Given a line and points B and C, we construct the two points on that maximize and minimize the ra... more Given a line and points B and C, we construct the two points on that maximize and minimize the ratio XB XC for X on. In this note we solve a problem that generalizes the main result of [1]. Given triangle ABC with the line bisecting angle A, in [1] we asked to find the two points X 1 and X 2 that maximize and minimize BX CX. It was proved that these two points are the incenter and excenter corresponding to angle A. It is worthwhile to notice that it is not difficult to prove a similar result where is the external bisector of A. In this case the two extremal points are the excenters which correspond to angles B and C. In this note we consider a more general problem where is an arbitrary line which does not contain the points B and C and find the two points X 1 and X 2 on which give the minimum and maximum of BX CX. x y X A O B C θ α α R R Figure 1 If is not perpendicular to BC, it intersects the perpendicular bisector of BC at a point O. Let R be the radius of the circle with center ...
This paper describes a theoretical model for systemic change as it concerns the learning and teac... more This paper describes a theoretical model for systemic change as it concerns the learning and teaching of mathematics in K – 12 schools, with particular attention being paid to the rural context. Systemic change is the active process of establishing change in the community through lasting, long-term relationships, practices, and procedures (Adelman & Taylor, 2003). Our purpose is to describe the mechanics of such change provided by the strategic, continuous, and monitored support of all three of the constituents: Teachers, administrators and community, and externally supported by a temporary catalyst. Systemic change is achieved when the removal of the external catalyst does not affect the rest of the model. Evidence to support this claim has been derived from our case studies.
The Editorial Panel of MTMS outlines its thoughts about the choice of the Focus Issue topic, Rati... more The Editorial Panel of MTMS outlines its thoughts about the choice of the Focus Issue topic, Rational Number Sense.
International Group for the Psychology of Mathematics Education, 2016
This study examined discussions centered on precise mathematical language use in two fifth grade ... more This study examined discussions centered on precise mathematical language use in two fifth grade classrooms. Drawing on episodes from lessons in which teachers focused on encouraging mathematics reasoning, our analysis examines the relationship between precise language use and mathematical justifying. We present three classroom episodes that illustrate facets of the relationship between precise language use and mathematical argument, and explore how blurring the borders of two types of pedagogical tools (attention to precise mathematical language and engagement of students in justifying) promotes enculturation of students into mathematics.
International Group for the Psychology of Mathematics Education, 2018
Middle school is a critical time when students begin formal study of functional relationships in ... more Middle school is a critical time when students begin formal study of functional relationships in algebra. However, many students struggle in understanding functions as relationships between quantities that change according to a dependency relationship. We report on the influence of Scaling Continuous Covariation in fostering productive ideas about graphical relationships and rates of change. Scaling Continuous Covariation entails the ability to imagine a re-scaling to any increment for x and coordinate that scaling with associated values for y. We present findings from two students, one who reasoned with Scaling Continuous Covariation and one who did not, and report on how Scaling Continuous Covariation supported students' reasoning in three ways: (a) sense making about graphs, (b) forming constant rates of change, and (c) understanding constantly-changing rates of change.
Given a line and points B and C, we construct the two points on that maximize and minimize the ra... more Given a line and points B and C, we construct the two points on that maximize and minimize the ratio XB XC for X on. In this note we solve a problem that generalizes the main result of [1]. Given triangle ABC with the line bisecting angle A, in [1] we asked to find the two points X 1 and X 2 that maximize and minimize BX CX. It was proved that these two points are the incenter and excenter corresponding to angle A. It is worthwhile to notice that it is not difficult to prove a similar result where is the external bisector of A. In this case the two extremal points are the excenters which correspond to angles B and C. In this note we consider a more general problem where is an arbitrary line which does not contain the points B and C and find the two points X 1 and X 2 on which give the minimum and maximum of BX CX. x y X A O B C θ α α R R Figure 1 If is not perpendicular to BC, it intersects the perpendicular bisector of BC at a point O. Let R be the radius of the circle with center ...
This paper describes a theoretical model for systemic change as it concerns the learning and teac... more This paper describes a theoretical model for systemic change as it concerns the learning and teaching of mathematics in K – 12 schools, with particular attention being paid to the rural context. Systemic change is the active process of establishing change in the community through lasting, long-term relationships, practices, and procedures (Adelman & Taylor, 2003). Our purpose is to describe the mechanics of such change provided by the strategic, continuous, and monitored support of all three of the constituents: Teachers, administrators and community, and externally supported by a temporary catalyst. Systemic change is achieved when the removal of the external catalyst does not affect the rest of the model. Evidence to support this claim has been derived from our case studies.
The Editorial Panel of MTMS outlines its thoughts about the choice of the Focus Issue topic, Rati... more The Editorial Panel of MTMS outlines its thoughts about the choice of the Focus Issue topic, Rational Number Sense.
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