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Agenda
Click a link below to connect to the presentation summary or abstract, and to review links, sketches, Powerpoint presentations, and other contributed material relating to each presentation.
Friday pm, 7 February 2003
Saturday am, 8 February 2003
- Plenary
- Reports from the Classroom
- Reports from Curriculum Developers
Saturday pm, 8 February 2003
- Working Groups on Activity Potentials
- In Elementary School (geometry strands)
- In Elementary school (non-geometry strands)
- In middle school
- Research reports
Sunday am, 9 February 2003
Other Contributions
Abstracts
"Beyond Geometry: From Algebra to Calculus with Sketchpad Version 4"
Nicholas Jackiw, KCP Technologies
Steven Rasmussen, KCP Technologies / Key Curriculum Press
With powerful new capabilities for defining, evaluating, graphing, and iterating functions, Sketchpad 4 brings its full dynamic power to the study of mathematics from algebra to calculus as well as geometry. Examples will range from sketches showing why a negative times a positive is a negative to the summation of sine waves to slope fields in integral calculus.
Presentation Document: MMC_Feb2003_Complete.gsp (Sketchpad document)
Related Links: Sketchpad Resource Center
"Dynamic Geometry Situations: Design Dimensions & Opportunities"
Nicholas Jackiw, KCP Technologies
Dynamic Geometry software is widely held to be a powerful and useful tool for mathematics visualization and investigation. But the dominant conception of Dynamic Geometry held outside of the classrooms in which it is actually practiced is that its purpose is "to drag triangles"---that is, to investigate an (important, yet local) family of properties on an (important, yet small) class of figures from elementary geometry. On inspection, however, the central idea of Dynamic Geometry---of maintaining automatically the integrity of a given system of mathematical relationships while students explore degrees and dimensions of its variation--reaches far beyond Geometry, as a strand formally constituted by our curricula, and even beyond the realm of continuous mathematics. At the elementary level, measurement, proportion, and number pattern are as fit topics as "shape" for Dynamic Geometry investigation. Confronting the daunting challenge of integrating effective technology use into our existing mathematics curricula, in this talk I urge us to start on an approach which is simultaneously technologically conservative, with respect to "how much" or how little technology investment we must initially make to find benefit; and pedagogically ambitious, with respect to how broadly applicable we find Dynamic Geometry software in our vision of meaningful student mathematics.
Presentation Document: DGSituations.gsp (Sketchpad document); Flatland (JavaSketchpad website)
"Geometry & Its Connections Through Sketchpad"
Cathi Sanders, Punahoe School
There is such a variety of interesting concepts in geometry, at all grade levels, and so many opportunities to enrich my students' experiences with hands-on projects that are directly related to the curriculum. In this presentation, I would like to show you some of the projects we do in my geometry classes, and in my online class for teachers at all grade levels. I've listed some of the projects below:
Transformations: "Geometric Graphics"
Congruent and Similar Triangles, Quadrilaterals: "Tangrams"
Parallel Lines, Parallelograms: "Tessellations", "Quads Symmetry", and "Op Art"
Polygons: "Mathematical Origami"
Circles: "Cardioid Construction" and "Eggstra Credit"
Surface Area and Volume: "3D Drawing" and "Package Design"
In addition to teaching at Punahou, I also have developed and taught Distance Learning classes for ESchool. ESchool is a U.S. Challenge Grant funded project of the Hawaii Department of Education. Although these classes are intended for high school students, 7th and 8th grade students have taken the classes and done well on them. The students are use The Geometer's Sketchpad in these classes, and we communicate via email and attached GSP files.
Presentation Documents: How to Star Card.gsp (Sketchpad document); How To Tessellate.gsp (Sketchpad document)
Related Links:
- I-MATH online class for teachers: http://www.k12.hi.us/~mathappl/imwelcome.htm
- Connecting Geometry online class for high school students: http://www.k12.hi.us/~csanders/
- MathApplications online class for high school students: http://www.k12.hi.us/~mathappl/
- Publication of student work: http://forum.swarthmore.edu/sum95/suzanne/hawaii.html
- Student work--ornamental tiles: http://www.punahou.edu/acad/sanders/SpanishGeometry.html
- 3-D drawing: http://forum.swarthmore.edu/workshops/sum98/participants/sanders/
- Math & Art connections: http://www.punahou.edu/acad/sanders/MathArt/
"Discovering Geometry using GSP with second graders"
Beth Lackey, Sims Elementary School
I believe that young children can use Geometer's Sketchpad in a variety of ways. Students can explore concepts, such as ray, line and segment and develop their own definitions, by experimenting with these concepts in a dynamic way.
Students can also use GSP to explore the concepts of shape and measurement. My second graders enjoyed constructing pictures using line segments and animating their designs.
Teachers can use GSP to prepare activities for students, such as sorting shapes in a Venn Diagram, as a means of studying relationship, which is an important principle of algebraic thinking.
Presentation Documents: BethLackey.gsp (Sketchpad document)
"Creative Explorations of Similarity, Transformations, and Fractals with Sketchpad"
Karen Wyatt, Mountain Brook Junior High School
I showed student examples of two Sketchpad projects I do with my students.
One was called Ode to the Pythagorean Theorem. The students used a grid of points to create similar figures they could save as custom tools. They used these tools to construct similar figures on the sides of a right triangle. They then measured the areas of the figures (or similar parts of their figures) to confirm that the sum of the areas constructed on the legs was equal to the area of the figure constructed on the hypotenuse. They dragged the vertices of their right triangle to confirm that this was always true.
The second project was a transformations project. Students learned about transformations and had to create a sketch containing, reflections, translations, rotations, dilations, and animation. They worked with a partner on this project.
Lastly, my fellow geometry teacher, Mrs. Peerson, designed a fractal web quest for our geometry students. After the students complete the web quest, they create three fractal designs from instructions Ann Lawrence and I wrote that help them create a Binary Tree, Sierpinski Gasket, and a Dragon fractal. The students then experiment and construct a fractal design of their own. The students and I are not always sure of what is the zero stage and first stage of their fractal. Any suggestions or clarifications on the stages are welcome!
Presentation Documents: PythagoreanSamples.gsp (Sketchpad document); PythagoreanGrids.gsp (Sketchpad document); ExploringPythagoras.doc (Word file); Fractal Project.gsp (Sketchpad document); Trans Project.gsp (Sketchpad document)
"Dynamic Geometry in the Primary School through Mentoring"
Ann Lawrence, Capitol Hill Day School
At Capitol Hill Day School in Washington, DC, middle school students have been paired with younger students for various projects involving the use of Sketchpad. The older students act as "expert advisors" to the younger ones; eighth graders are paired with third graders, seventh graders with fifth graders, and sixth graders with fourth graders. This school year is the second year of this endeavor, and some of the teachers in the lower grades have begun to have their students doing some activities on their own as well. The presentation included explanations and showed some of the sketches from a sampling of cross grade projects and single grade level activities and projects. The presenter welcomes inquires (see Participants).
Presentation Documents: Ann1.gsp (Sketchpad document); Ann2.gsp (Sketchpad document); Ann3.gsp (Sketchpad document)
"Measurement Units and Logical Relations in a Dynamic Geometry Environment"
Jeff Barrett, Illinois State University
In classroom trials over the past three years, the sketchpad environment afforded children in grades 2-7 clear access to ideas about precision of measures, the relation of units to segment and perimeter length and the shortest distance postulate. In work with children in grades 3 and 4, the concepts of distance and geometric shape were related through activities involving rule-making for playground games. These activities promoted gains in children's thinking about units and precision. The tracing facility in sketchpad promoted ways of reflecting on close sequences of positions for a unit object along a measured object.
The sketchpad environment afforded teachers a tangible window to their students' intuitions and to their students' internal models for path length and operations with units. Sketchpad also provided teachers a way of prompting their students to examine the logical inter-relation between various cases of a mathematical model, such as the compensatory relation between side lengths of a triangle with a fixed perimeter. Sketchpad provided a unique and beneficial micro-world for examining the implications of varying levels of unit precision. Teachers and researchers found that the sketchpad environment provides tangible evidence of students' varying abilities to appropriate the arrow tool as a means of attending to critical vs non-critical characteristics of a geometric situation, examining logical interdependence among parts of sketches, or reasoning across close, systematic sequences of cases.
Presentation Documents: Jeff.ppt (Powerpoint presentation); hopping.gsp (Sketchpad document); msmathsketches.gsp (Sketchpad document)
"Research and Development of Geometry in Grades K-9 on Three Continents"
Retha Van Niekerk, Rice University School Mathematics Project
The black doubt that lurks in every pedagogue's heart is not so much whether s/he is teaching correctly as whether WHAT s/he is teaching is worth teaching at all. The real danger is not that we shall teach the right things inefficiently, but that we shall teach the wrong things more and more efficiently. In this presentation, I will share thoughts about my design research research model for developing a k-9 geometry curricula--one that teaches the right things efficiently!
Presentation Documents: VanNiekerk_Chicago.ppt (Powerpoint presentation)
"Math Trailblazers"
Philip Wagreich, University of Illinois at Chicago
Math Trailblazers is a K-5 mathematics curriculum developed by the Teaching Integrated mathematics and Science (TIMS) Project at the University of Illinois at Chicago. It is one of three comprehensive elementary mathematics curricula developed with NSF funding in the 1990s. The curriculum has significant connections to science and language arts. In this talk we gave some "off the top of our heads" thoughts about ways in which Sketchpad might be used to enhance student learning. The thoughts centered on a) the concept of a variable, in particular looking at length, area, and volume as scientific variables; b) 2-D and 3-D visualization. Potential applications to many other parts of the curriculum became apparent in the course of the meeting.
Presentation Document: PP_Wagreich.ppt (Powerpoint presentation)
"Opportunities for Dynamic Visualisation in The Connected Math Project"
Glenda Lappan, Michigan State University
Betty Phillips, Michigan State University
We are committed to finding ways of effectively integrating technology into our curriculum. By effective learning we intend the deep understanding of basic mathematical concepts and ideas. As examples, we identified a few "tools" that would be useful within the context of our curriculum (and probably beyond), and that would be well-suited to Dynamic Visualisation. The first is a simple scaling device that could be used for the teaching and learning of percent. [This sketch has since been implemented; see below.] The second tool could be used for our "Licorice problem," where students have to re-partition a piece of licorice into fractional parts to accommodate more licorice-eaters.
Related Links: DGYL_Prototypes.gsp (Sketchpad prototype of the percentage scaling device)
"The Shape Makers Dynamic Geometry Microworld"
Michael Battista, Kent State University
When students are studying types of geometric shapes, learning definitions and lists of properties is not nearly as important as being involved in the processes of developing and using a property-based conceptual system for reasoning about shapes. This system utilizes concepts such as angles, sides, angle measure, length, congruence, and parallelism to describe and analyze spatial relationships within and among shapes. This conceptual system specifies precisely the geometric essences of spatial configurations, enabling us not only to accurately identify shapes but to reason analytically about them. To help students genuinely understand, appreciate, and use the property-based conceptual system utilized in geometry, students should actively participate in developing and using the system, not in memorizing facts that others have established with the system.
The Shape Makers microworld, a special add-on to The Geometer's Sketchpad, consists of screen manipulable shape-making objects that can be used instructionally to encourage and support students in developing this property-based conceptual system. For instance, the computer Parallelogram Maker makes any parallelogram that fits on the computer screen, no matter its shape, size, or orientation-but only parallelograms. Investigating the Parallelogram Maker enables students not only to see the continuous transformations of one parallelogram into another, but to feel the constraints that maintain it as a parallelogram. In my presentation, I argued that the visual and kinesthetic abstractions that students develop in manipulating such computer Shape Makers are critical cognitive input for forming increasingly sophisticated conceptualizations of the structure of shapes. Indeed, in my presentation, I gave examples of students' work that illustrated how the Shape Makers environment moved students from visual reasoning that is holistic, vague, and imprecise to reasoning that is analytic and precise because it uses measurement-based properties. That is, in a proper classroom culture, and with appropriate instructional activities, the Shape Makers encourage and support students' development of increasingly sophisticated conceptualizations of shapes, conceptualizations that eventually become the formal geometric concepts that are used by mathematicians to characterize the class of shapes made by each Shape Maker.
"Constructing Improper Fractions Using Tools for Interactive Mathematical Activity (TIMA) Software"
John Olive, University of Georgia
The TIMA software were developed as part of a three-year constructivist teaching experiment in which our research project worked with 12 children, starting in their third grade and following them through their fifth grade in elementary school. We collected approximately 600 videotaped episodes over the 3 years of the project. These videotapes have become the data for a retrospective analysis of the teaching experiment, through which we are building models of children’s fractional knowledge. In the talk I presented some of the results found by our research that describe the emerging models of children’s fractional schemes and the role that the computer tools played in their construction; shared two teaching episodes from the Fractions Project that illustrate the potential of two of the computer tools; and discussed how similar tools might be considered in a Dynamic Geometry environment.
Presentation Documents: FractionManipulatives.gsp (Sketchpad document); DGYL_Paper_Olive.doc (Word document)
"Numberline Models for Basic Operations"
Steven Rasmussen, KCP Technologies
Very young children are usually introduced to basic number operations using discrete numberline models. There are two aspects to this discrete behavior. First, operations are usually introduced using integers, so issues of intermediate value (operations with values between integer values) are ignored. Second, operations are modeled one problem at a time on number lines. A contrasting approach is illustrated by using Sketchpad numberlines to model continuous behavior of binary operations as one or the other operands in the binary operation is continuously varied. This leads to a focus on the mathematical "behavior" of the operation, rather than on the specific numerical result. By dragging an addend in the Add Machine sketch, for example, students can see that the sum varies at the same rate as the addend and remains at a fixed distance from the addend that's moved. This is fundamental additive behavior and contrasts markedly from the behavior of the product when one moves a factor in the Multiply Machine. Subtraction and division exhibit yet different behaviors.
Presentation Documents: Calculating_Machines.gsp (Sketchpad document); NumberlineModels.doc (Word file)
"A Student-Engagement Perspective on Dynamic Geometry"
Nathalie Sinclair, Michigan State University
Rather than focus on the curriculum or the software, my presentation considered the possibilities of integrating dynamic geometry into the young learner's mathematical experience from the point of view of student engagement. In particular, I identified the range of feelings and values involved in learning and problem solving. It is well accepted that negative feelings such as fear and anxiety can essentially block thinking and learning. However, it is less well understood how more positive feelings and values can foster mathematical thinking and learning.
In this presentation, I proposed that the visual, concrete nature of dynamic geometry can be used to evoke feelings of surprise, mystery and visual attraction, which in turn makes it possible to both "catch" and "hold" student engagement. I also showed how values that are pervasive in the mathematical community, and in mathematics itself, such as efficiency and perspicuity, can be made accessible and meaningful in a dynamic geometry environment. Finally, I suggested that well-designed interactions between students and technology can help learners focus their attention on the act of their own problem solving and learning.
Presentation Document: Nathalie_DGYL.gsp (Sketchpad document)
"Activity Prototypes from First Working Groups"
Nathalie Sinclair, Michigan State University
Nicholas Jackiw, KCP Technologies
Several activity ideas were generated in the first working groups; these were presented to the entire group on Saturday afternoon. We presented some rough sketches that would give everyone a better sense of the way the ideas would work in Sketchpad.
Presentation Document: DGYL_Prototypes.gsp
"Euclid's Kitchen: Tiling and the GCD"
Daniel Scher
Most fifth graders can find the greatest common divisor (g.c.d.) of 48 and 18, but few teachers prior to college introduce the Euclidean algorithm as a generalized method for calculating the g.c.d. The accompanying sketch dispalys an outline of a rectangular kitchen floor. Using Sketchpad's square tool, young students are challenged to recreate a square tiling pattern that covers the floor. The method, which relies heavily on geometric intuition, corresponds to a visual, "hands-on" model of Euclid's algorithm. In the early grades, this tile-laying scenario serves mostly to introduce a visual representation of divisor. In later grades, students can use the models on the remaining pages of the sketch to connect their visual understanding with the standard arithmetic representation of Euclid's algorithm.
Presentation Document: tile laying.gsp
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