The Sketchpad activities in this collection give students the opportunity to directly experience,... more The Sketchpad activities in this collection give students the opportunity to directly experience, through dynamic visualization and manipulation, the topics covered in precalculus. Students:
transform functions dynamically
- view and modify entire families of functions—polynomial, exponential, logistic, and more
- build parametric functions, and manipulate vectors and matrices
discover the relationship between rectangular and polar coordinates
- make complex numbers come alive on the coordinate plane
finish with a dynamic, visual introduction to the main concepts of calculus.
IN 2001–2002, THREE TEACHERS from Paul Robeson High School in Brooklyn, New York, joined an inter... more IN 2001–2002, THREE TEACHERS from Paul Robeson High School in Brooklyn, New York, joined an international collaboration to pilot U.S.-Russian Interactive Geometry (Armontrout et al. 2002) labs in their classrooms.
It starts as a thought experiment. You imagine an arbitrary quadrilateral with its four midpoints... more It starts as a thought experiment. You imagine an arbitrary quadrilateral with its four midpoints connected to form another quadrilateral nested inside. You wonder whether this inner quadrilateral possesses any special properties, and if these properties continue to hold when the vertices of the outer quadrilateral assume different locations. Almost instinctively, you move to your computer and launch a dynamic geometry software program such as The Geometer’s Sketchpad (Jackiw, 1995) or Cabri Geometry (Texas Instruments, 1994). With a few clicks and movements of your mouse, the construction is complete: there on your screen is a model of the quadrilaterals that can be freely manipulated, just as in your thought experiment. Some dragging of the outer quadrilateral’s vertices convinces you there is a behavior worth noticing; the inner quadrilateral appears to remain a parallelogram. The scenario described here is by now a common one. In an age when we become surprised if something on o...
This paper describes an innovative technology-based approach that enables students to learn funct... more This paper describes an innovative technology-based approach that enables students to learn function concepts by constructing and manipulating functions in the form of geometric transformations on the plane. Students’ direct sensorimotor experiences with variables, function rules, domain and range help them make sense of linear functions, Cartesian graphs, derivatives, multiplication of complex numbers, and Euler’s formula. Treating geometric transformations as functions is not a new idea in secondary mathematics, but few curricula take full advantage of the approach to develop students’ concept of function. Web Sketchpad, the technology described in this paper, supports a constructionist approach to students’ activities of creating, manipulating, and investigating mathematical objects, thus linking their sensorimotor activity to their conceptual understanding. The software provides a simple interface with no menus, based on dragging and on using a small set of tools designed by the...
Activities give users the opportunity to directly experience, through dynamic visualization and m... more Activities give users the opportunity to directly experience, through dynamic visualization and manipulation, the topics covered in precalculus, including: polynomial, exponential, and trigonometric functions; parametric functions; polar coordinates; and complex numbers. Also includes activities that are appropriate for use in the first few weeks of a calculus course. Chapters include: Function Transformations, Circular Functions, Trigonometric Properties, Other Functions, Data and Probability, Vectors and Matrices, Polar Coordinates and Complex Numbers, Sequences and Series, and Introduction to Calculus.
Sometimes beneath a simple problem statement lies unexpected mathematical depth. Consider this ch... more Sometimes beneath a simple problem statement lies unexpected mathematical depth. Consider this challenge from the Connected Geometry curriculum (Education Development Center 2000): “Find as many ways as you can to divide an arbitrary triangle ABC into four equal-area triangles.” Before reading further, try solving the problem yourself.
Students see that geometric transformations—dilation and translation—correspond to algebraic para... more Students see that geometric transformations—dilation and translation—correspond to algebraic parameters—m and b—in the familiar equation for a linear function.
Without Paul Goldenberg, this interview study would not exist. His thinking on the subject of dyn... more Without Paul Goldenberg, this interview study would not exist. His thinking on the subject of dynamic geometry influenced nearly every aspect of this dissertation. Together, we developed and implemented the interviews described within. Ken Goldberg kept his door wide open during the entire dissertation process and provided nearly instantaneous feedback on all of my drafts. His commitment to my progress was strong and reassuring. Fran Curcio's eagle-eyed editing skills greatly improved the text. Joe McDonald showed a keen understanding of mathematics education issues and has been an enthusiastic supporter of my work. Mary Anne Gallagher Landi contributed to the analysis and provided constant encouragement throughout the writing process. Special thanks to my parents, Joan and Michael.
Such software programs as The Geometer's Sketchpad (Jackiw 1995) and Cabri Geometry II (T... more Such software programs as The Geometer's Sketchpad (Jackiw 1995) and Cabri Geometry II (Texas Instruments 1994) allow students to turn the static images in their geometry textbooks into moving, deformable figures. With this ability comes the chance to pose challenges and approach theorems in ways that are not possible within a paper-and-pencil context. This article explores several dynamic-geometry constructions focusing on rectangles and shows how they foster fre h insights into traditional geometry theorems.
Some of the most lasting impressions that we retain from the study of circles come from our infor... more Some of the most lasting impressions that we retain from the study of circles come from our informal exposure to them in daily life. It may have been years since we learned, and perhaps forgot, the equation x2 + y2 = r2, but we can still summon our hands-on experiences to remember the essential features of a circle. The notion of a circle's having a center that is equidistant from all its points gets reinforced whenever we use a compass or twirl a rope or cord with our fingers. The symmetry of a circle becomes apparent each time we turn a knob or struggle to loosen a lid on a jar.
Experience with multiple representations fosters students' robust understanding of what funct... more Experience with multiple representations fosters students' robust understanding of what functions are, how they behave, and how they can be composed.
The Sketchpad activities in this collection give students the opportunity to directly experience,... more The Sketchpad activities in this collection give students the opportunity to directly experience, through dynamic visualization and manipulation, the topics covered in precalculus. Students:
transform functions dynamically
- view and modify entire families of functions—polynomial, exponential, logistic, and more
- build parametric functions, and manipulate vectors and matrices
discover the relationship between rectangular and polar coordinates
- make complex numbers come alive on the coordinate plane
finish with a dynamic, visual introduction to the main concepts of calculus.
IN 2001–2002, THREE TEACHERS from Paul Robeson High School in Brooklyn, New York, joined an inter... more IN 2001–2002, THREE TEACHERS from Paul Robeson High School in Brooklyn, New York, joined an international collaboration to pilot U.S.-Russian Interactive Geometry (Armontrout et al. 2002) labs in their classrooms.
It starts as a thought experiment. You imagine an arbitrary quadrilateral with its four midpoints... more It starts as a thought experiment. You imagine an arbitrary quadrilateral with its four midpoints connected to form another quadrilateral nested inside. You wonder whether this inner quadrilateral possesses any special properties, and if these properties continue to hold when the vertices of the outer quadrilateral assume different locations. Almost instinctively, you move to your computer and launch a dynamic geometry software program such as The Geometer’s Sketchpad (Jackiw, 1995) or Cabri Geometry (Texas Instruments, 1994). With a few clicks and movements of your mouse, the construction is complete: there on your screen is a model of the quadrilaterals that can be freely manipulated, just as in your thought experiment. Some dragging of the outer quadrilateral’s vertices convinces you there is a behavior worth noticing; the inner quadrilateral appears to remain a parallelogram. The scenario described here is by now a common one. In an age when we become surprised if something on o...
This paper describes an innovative technology-based approach that enables students to learn funct... more This paper describes an innovative technology-based approach that enables students to learn function concepts by constructing and manipulating functions in the form of geometric transformations on the plane. Students’ direct sensorimotor experiences with variables, function rules, domain and range help them make sense of linear functions, Cartesian graphs, derivatives, multiplication of complex numbers, and Euler’s formula. Treating geometric transformations as functions is not a new idea in secondary mathematics, but few curricula take full advantage of the approach to develop students’ concept of function. Web Sketchpad, the technology described in this paper, supports a constructionist approach to students’ activities of creating, manipulating, and investigating mathematical objects, thus linking their sensorimotor activity to their conceptual understanding. The software provides a simple interface with no menus, based on dragging and on using a small set of tools designed by the...
Activities give users the opportunity to directly experience, through dynamic visualization and m... more Activities give users the opportunity to directly experience, through dynamic visualization and manipulation, the topics covered in precalculus, including: polynomial, exponential, and trigonometric functions; parametric functions; polar coordinates; and complex numbers. Also includes activities that are appropriate for use in the first few weeks of a calculus course. Chapters include: Function Transformations, Circular Functions, Trigonometric Properties, Other Functions, Data and Probability, Vectors and Matrices, Polar Coordinates and Complex Numbers, Sequences and Series, and Introduction to Calculus.
Sometimes beneath a simple problem statement lies unexpected mathematical depth. Consider this ch... more Sometimes beneath a simple problem statement lies unexpected mathematical depth. Consider this challenge from the Connected Geometry curriculum (Education Development Center 2000): “Find as many ways as you can to divide an arbitrary triangle ABC into four equal-area triangles.” Before reading further, try solving the problem yourself.
Students see that geometric transformations—dilation and translation—correspond to algebraic para... more Students see that geometric transformations—dilation and translation—correspond to algebraic parameters—m and b—in the familiar equation for a linear function.
Without Paul Goldenberg, this interview study would not exist. His thinking on the subject of dyn... more Without Paul Goldenberg, this interview study would not exist. His thinking on the subject of dynamic geometry influenced nearly every aspect of this dissertation. Together, we developed and implemented the interviews described within. Ken Goldberg kept his door wide open during the entire dissertation process and provided nearly instantaneous feedback on all of my drafts. His commitment to my progress was strong and reassuring. Fran Curcio's eagle-eyed editing skills greatly improved the text. Joe McDonald showed a keen understanding of mathematics education issues and has been an enthusiastic supporter of my work. Mary Anne Gallagher Landi contributed to the analysis and provided constant encouragement throughout the writing process. Special thanks to my parents, Joan and Michael.
Such software programs as The Geometer's Sketchpad (Jackiw 1995) and Cabri Geometry II (T... more Such software programs as The Geometer's Sketchpad (Jackiw 1995) and Cabri Geometry II (Texas Instruments 1994) allow students to turn the static images in their geometry textbooks into moving, deformable figures. With this ability comes the chance to pose challenges and approach theorems in ways that are not possible within a paper-and-pencil context. This article explores several dynamic-geometry constructions focusing on rectangles and shows how they foster fre h insights into traditional geometry theorems.
Some of the most lasting impressions that we retain from the study of circles come from our infor... more Some of the most lasting impressions that we retain from the study of circles come from our informal exposure to them in daily life. It may have been years since we learned, and perhaps forgot, the equation x2 + y2 = r2, but we can still summon our hands-on experiences to remember the essential features of a circle. The notion of a circle's having a center that is equidistant from all its points gets reinforced whenever we use a compass or twirl a rope or cord with our fingers. The symmetry of a circle becomes apparent each time we turn a knob or struggle to loosen a lid on a jar.
Experience with multiple representations fosters students' robust understanding of what funct... more Experience with multiple representations fosters students' robust understanding of what functions are, how they behave, and how they can be composed.
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Books by Daniel Scher
transform functions dynamically
- view and modify entire families of functions—polynomial, exponential, logistic, and more
- build parametric functions, and manipulate vectors and matrices
discover the relationship between rectangular and polar coordinates
- make complex numbers come alive on the coordinate plane
finish with a dynamic, visual introduction to the main concepts of calculus.
Papers by Daniel Scher
transform functions dynamically
- view and modify entire families of functions—polynomial, exponential, logistic, and more
- build parametric functions, and manipulate vectors and matrices
discover the relationship between rectangular and polar coordinates
- make complex numbers come alive on the coordinate plane
finish with a dynamic, visual introduction to the main concepts of calculus.