Our Fearless Leaders: Annie, Steve, and David
Dick Lesh
The session began with an introduction to assessment and math education, continued with a learning activity for all of us, and concluded with a discussion outlining how to write good problems. The introduction laid out the parts of the mission. Dick's main emphasis is on the middle school grades (6-8). One of his goals is to have math educators mathematize real situations: take something sensible, and then add in math. He wants students to be able to see math in the real world, not just in the pre-fabricated world of hypothetical textbook problems.
There were five main components of the argument:
(1) Psychology - Teaching students how to construct powerful mathematical models, stuctural metaphors to think about the world. Dick believes that the ability to develop models is essential to the future members of the workforce. The world is made up of systems, and we need to be able to create models to understand and make use of those systems.
(2) Technology - Basically, he says the world is becoming completely composed of complex systems as a result of technology: communication systems, economic systems, etc. We need students who can think in a system mode.
(3) Equity - The availability of powerful ideas for all students, not just for a handful of select students.
(4) Math Quality - The creation of good problems, activities, and means of assessment.
(5) Assessment - ASSESSMENT does not = TESTS: This myth he tries to dispel.
We next jumped head-first into an activity from the Packets Project entitled "Smart Shadows." We were given a sheet that described the background of the problem: a television program on which a mathematician claimed that you can make a square shadow from any convex quadrilateral. Our mission was to prove or disprove this claim, show how it could be done, and explain our reasoning. The convex quadrilaterals we were given were a square, kite, isosceles trapezoid, rhombus, irregular quadrilateral, parallelogram, and rectangle. We were also given two concave quadrilaterals for the fun of it.
Our Fearless Leaders: Annie, Steve, and David
The entire group was divided into several groups of three to work with the
cut-out shapes and tiny flashlights. Each group went through trials and
discoveries, made new theories, and discarded theories. We invented new
language ('fat angle', 'tilting') while working, judged our strengths and
weaknesses, and focused on geometry and proof. Some specific discoveries
and questions were mentioned:
(1) Can any angle look parallel?
(2) Shadow length has no bearing on length of object - it's the distance
between the light and the object that is important.
(3) Any angle can be a right angle.
Allison, with Forum Project Director Gene Klotz, and Research
Coordinator Ann Renninger
Everyone learned, discussed, argued, and had fun. I heard a physics professor say as he walked by in the hall, "Boy! Looks like they're having fun in there!"
There's me (otherwise known as 'the kid') trying to demonstrate my point to Howard
and Mary Denise
The last part of Dick's talk was an outline of the characteristics of good questions. He gave us six principles, and an example of how not to do it (a vague 'carpentry' problem that was a perfect example of a typically bad problem that is used frequently).
The 6 Principles
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1. Reality Principle - letting students know who wants to know and why. We need to give 'real life' situations, but ones that make sense based on extensions of students' own experience.
2. Model Construction Principle - Does the problem create the need for a model? Does it allow the students to constuct, explore, manipulate, predict, or control a structurally interesting system? Can they discover underlying patterns and regularities?
3. Self-Evaluation Principle - Are students enabled to judge for themselves when responses are good enough?
4. Model-Documentation Principle - The students need to reveal how they thought about the problem.
5. Model-Generalization Principle - Can the situation in a problem be generalized to encompass a general mathematical idea?
6. Simple-Prototype Principle - Even if the situation is simple, students still need to create a significant model. Also, is the problem a useful prototype for interpreting a variety of other structurally similar problems?
Dick's talk was a big hit. It was also a great change of pace from computer work. We hope he'll come back sometime!
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