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Geometric Thinking: Van Heile Applications in Grades K-5 Professional Development Workshop KATM 2003 Annual Conference October 24, 2003 David S. Allen, Ed.D. and Jennifer Bay-Williams Ph.D. National Standard: Geometry
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Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships.
Specify locations and describe spatial relationships using coordinate geometry and other representational systems
Apply transformations and use symmetry to analyze mathematical situations
Use visualization, spatial reasoning, and geometric modeling to solve problems.National Standards
Indicator# of Items Means
Knowledge 2.4 12 68.4
Knowledge 3.1 8 46.4
Application 1.2 4 67.9
Application 2.1 12 45.4
Application 4.1 4 60.3
Pierre van Hiele
Benchmark 2:The student estimates and measures using standard and nonstandard units in a variety of situations.
Indicator 4: The student selects, explains the selection of, and uses measurement tools, units of measure, and degrees of accuracy appropriate to the given situation to measure length to the nearest fourth of an inch, nearest centimeter; volume to the nearest pint, cup, quart, gallon or liter and nonstandard units of measure to the nearest whole unit; weight to the nearest pound or ounce and nonstandard units of measure to the nearest whole unit; and temperature to the nearest degree; and units of time.
Sample Problem for B2-I4
You need to buy carpet to cover the floor of your dog’s house. Which tool would you use to help you decide how much carpet to buy?
Benchmark 3:The student recognizes up to two transformations of basic geometric figures in a variety of situations.
Indicator 1: The student recognizes and performs up to two transformations (rotation/turn, reflection/flip, translation/slide) on simple two-dimensional shapes and uses cardinal or positional directions to describe translations such as move the triangle three units to the right and two units up.
Sample Problem for B3-I1
Which of the figures on the right represents a rotation and a reflection of the figure on the left?
Benchmark 1: The student recognizes or investigates properties of simple geometric figures in a variety of situations.
Indicator 2: The student categorizes a composite figure into the shapes used to form it. For the purpose of assessing this indicator on the Kansas assessment the student should be able to recognize the following figures which were used to form a composite shape: square, rhombus, octagon, pentagon, circle, square, rectangle, triangle, and ellipse (oval).
Sample Problem B1-I2
Which shapes were used to create the drawing?
Benchmark 2: The student estimates and measures using standard and nonstandard units in a variety of situations.
Indicator 1: The student formulates and solves real-world problems by applying measurements and measurement formulas. For the purpose of assessing this indicator the student should be able to work with the following measurements and conversions:
a) area of rectangle
c) length to the nearest fourth of an inch, nearest cenimeter and nonstandard units of measure to the nearest whole unit; volume to the nearest pint, cup, quart, gallon or liter; temperature to the nearest degree; and weight to the nearest pound or ounce.
d) conversions within the same measurement systems (inches and feet, cups and pints etc.)
e) units of time BackSample Problem
About how many candy bars touching each other could be laid in a row to equal the length of 2 feet?
Benchmark 4: The student relates geometric concepts to the number line and the first quadrant of the coordinate plane in a variety of situations.
Indicator 1: The student uses coordinate grids and maps to formulate and solve real world problems involving distance and location such as identifying locations and giving or following directions to move from one location to another. For the purpose of assessing this indicator on the Kansas Assessment the student should be able to use maps and grids which have positive number or letter coordinates.
Back Sample Problem
What generalizations can you make related to a persons ability to engage in geometric related tasks?
What geometric understandings (if any) do children bring to school with them when entering kindergarten?
“Not all people think about geometric ideas in the same manner. Certainly, we are not all alike, but we are all capable of growing and developing in our ability to think and reason in geometric context.”
(Van de Walle 2003)
Level 0: Visualization
The objects of thought at level 0 are shapes and what they “look like.”
The products of thought at level 0 are classes or groupings of shapes that seem to be “alike.”
Level 1: Analysis
The objects of thought at level 1 are classes of shapes rather than individual shapes.
The products of thought at level 1 are the properties of shapes.
Level 2: Informal Deduction
The objects of thought at level 2 are the properties of shapes.
The products of thought at level 2 are relationships among properties geometric objects.
Level 3: Deduction
The objects of thought at level 3 are relationships among properties of geometric objects.
The products of thought at level 3 are deductive axiomatic systems for geometry.
Level 4: Rigor
The objects of thought at level 4 are deductive axiomatic systems for geometry.
The products of thought at level 4 are comparisons and contrasts among different axiomatic systems of geometry.