Across the Curriculum
at Albuquerque TVI Community College
Archive of Teaching Ideas: Teaching tip for April 12-18, 1999
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Joe Krzyzanowski's tip
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A collaborative group technique for involving students in evaluating each other's writing and also in evaluating the evaluations they receive:
This idea comes from Joe Krzyzanowski, Developmental Math instructor, who uses these activities "to show [students] that they are perfectly capable of coming up with formulas and solutions on their own--if they observe, think, guess, and check."
I use Geoboards to discover the formulas for Perimeter and Area of rectangles. Students create rectangles on the geoboard with rubber bands. They count the squares inside to determine area, and the sections around the outside to determine the perimeter. The students fill in a table with the length, width, area, and perimeter of the rectangles they create, and then try to project the results into formulas which can be used on much larger rectangles. The students who finish early can be challenged to determine the formula of a triangle (first a right triangle which is 1/2 the rectangle, then any triangle).
I will use various circles to have students discover "pi." Have them measure the circumference and the diameter and then divide c/d. This forces the students to figure out how to measure non-linear dimensions.
I have students investigate Volume by giving them a can of beans and the formula for volume of a cylinder. I ask them to construct a box which will have the same volume. The test is to pour the beans into the box and assess the outcomes. I use this activity with my algebra students to set up a test question wherein they speculate the formula for volume of a cone, justify their speculations, and develop a means of testing their speculation.
Analysis: The students are asked to analyze basic geometric shapes for their components and the interrelation of those components.
Synthesis: By studying many specific shapes, the students are expected to generate formulas which allow them to determine perimeter, area, etc., of any size figure.
Evaluation: Ongoing evaluation during the project lets any error stand out and forces rethinking. The formulas derived can then be tested on other samples. Some formulas can be used to derive others (area of a rectangle to area of triangle, volume of a cylinder/box to volume of a cone, etc.).
Creativity: The students are free to use any materials to measure, to construct, etc.
Decision Making: Students develop a plan, decide which units of measure to use, how to test their results, etc.
Hypothesizing/Predicting: Students use the formulas discovered to derive other formulas and analyze other geometric shapes.
Applications: Besides deriving other formulas, students can apply the formulas to remodeling-type activities---carpet a floor, paint walls, cover tables, heat/cool a room, etc.
Assumptions: Since these are basically discovery activities for students at this level, there a few assumptions. They are usually very surprised to see that such regularity (patterns/formulas) actually exist.
Relevance: Students generally feel excited about thinking (in a puzzle-solving sense). They see the need to be able to work with fractions, decimals, calculators, etc. They see that these geometric ideas have a lot of practical value.
Learning Style: The activities are obviously kinesthetic and visual. The dialogue within the group benefits the more auditory learner.
Creativity: There are usually many approaches to solving the problem. This provides an excellent opportunity to show the flexibility inherent in mathematics.
Real-life: The geometry involved in these activities is readily useable in any decorating activities that these students may want/need around their own homes/apartments.
Writing explanations can be incorporated into these activities. Research into the history of mathematics can be incorporated. Extensions into the world of science--formulas, patterns like the Periodic Table, etc.Adaptability:
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If you have a tip, technique or assignment that has worked successfully in your classroom to elicit active thinking, please share it with us. Send them--one per message, please--to the Critical Thinking Across the Curriculum Initiative webmaster, making sure to include an explanation of how the technique or assignment promotes active, critical thinking.
| "Ongoing evaluation during the project lets any error stand out and forces rethinking." |
| . |
| "There are usually many approaches to solving the problem. This provides an excellent opportunity to show the flexibility inherent in mathematics." |