Posing Problems with Interactive Images - I: Objects & Actions - This GeoGebra book is the principal online resource for a course on posing K-12 mathematics problems. The
course was designed for middle and secondary school teachers in the
"Mathematics for Teaching" graduate program at the Harvard Extension School. The
primary goal of this course is to introduce the use of interactive
images in the posing of problems in Algebra, Arithmetic and Geometry.
Interactive Images are computer applets the allow users to explore and
conjecture about mathematical objects and the actions that can be performed on them or by them. [The course was offered as a face-to-face course at Harvard in the fall of 2014 and in the fall of 2015. It was offered as an online course in the fall of 2016 and the fall of 2018. It may be offered again in an online format in the future.] Posing Problems with Interactive Images- II: Reflective Teaching - This GeoGebra book is the principal online resource for a follow-on course on posing K-12 mathematics
problems. The course was designed for middle and secondary school
teachers in the "Mathematics for Teaching" graduate program at the
Harvard Extension School. The primary goal of this course is to give
teachers a more systematic and theoretical framework for formulating the
mathematics problems they pose to their students. - Many (although not all) of the mathematics problems that we pose to K-12
students in order to stimulate invention and creativity can be
classified along several several pairs of perspectives. These pairs of
perspectives seem to overlap in some cases and in some cases to contrast
with one another. - In this course we explore some of these perspectives and the degree to
which a sensitivity to them can help teachers pose problems both in
class and for homework. The aim is to help teachers help their students
to develop deeper insight into, and appreciation for, mathematics. In
the course teachers work on problems that stimulate their inventiveness
and creativity and then analyze the approaches both they and their
colleagues have taken. The course relies heavily on interactive images, computer applets that use multiple graphical representations of the mathematical problems. [This course was offered as an online course in the spring of 2019. It may be offered again in an online format in the future.] These are smaller collections of applets, almost all drawn from the challenge archive, that are organized according to thinking strategies that cut across traditional mathematics subject matter. Formulating Measures - For background on this collection of applets, please read the essay entitled "Formulating Measures: toward modeling in the K-12 science and mathematics curriculum" . Word Problems & Multiple Graphical Representations - the applets in this collection analyze each of the most common types of word problems using two conceptually different graphical representations. For background please read the essay entitled "Hens & Rabbits, Feet & Animals; Some Thoughts on Representing Word Problems Graphically" - the applets in this collection offer opportunities to explore synthetic rather than analytic approaches to problems in both algebra and geometry Generalizing - a THEME under construction Scaling - provisional version Overlapping Issues - under construction 'Between-ness' - 'Between-ness' and ordering are two sides of the same coin. From a pedagogic perspective, 'between-ness' offers an opportunity for us to pose problems to our students that do not have unique correct answers while simultaneously being well-defined and open-ended. - In trying to understand the nature of motion we are confronted with the problem of reconciling two profoundly different representations of reality. One representation is the path (trajectory) taken by the object in motion as it moves. The eye perceives that path and the brain "records" it [image for the eye]. Unfortunately, the representation of the path traversed is inadequate for understanding the motion - it does not tell us where the object is at any given time nor does it tell us how fast the object is moving at any given time. For that purpose we need graphs of position and velocity as functions of time [images for the mind's eye]. The applets in this collection all deal with motion and are an attempt to help the user to become nimble is moving between images for the eye and images for the mind's eye. * * * * Polygonal linkages - applets in this collection approach the exploration of polygons by considering them to be closed chains of hinged links. - applets in this collection allow the graphs of functions of one variable to be manipulated and deformed. In each section of its domain a function is treated either as an uncooked noodle having a particular kind of shape or a flexible cooked noodle. On Rotating and Rolling - applets in this collection all focus on circular and/or rotational motion in different settings. Judah L. Schwartz  mathMINDhabits by Judah L. Schwartz is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License. |