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Geometric Thinking: Van Heile Applications in Grades K-5

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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

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 StandardsNavigating Through Geometry

- Analyzing characteristics and properties
- Alike and Different PreK - K
- Shapes from Shapes PreK - K

- Specifying locations and describing spatial relationships
- From here to there K - 1
- Map Maker 1 - 2

- Applying transformations and symmetry
- Design Tiles 1-2
- Rotating Geo-boards 2

- Using visualization, spatial reasoning, and geometric modeling
- Exploring Packages 3-4
- It’s All in the Packaging 4

Benchmark State

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

Measurement

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.

BackSample Problem

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?

- Compass
- Measuring Cup
- Scale
- Clock
- Yard Stick

Back

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.

BackSample Problem

Sample Problem for B3-I1

Which of the figures on the right represents a rotation and a reflection of the figure on the left?

Back

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).

BackSample Problem

Sample Problem B1-I2

Which shapes were used to create the drawing?

- Circle, hexagon, triangle, octagon
- Circle, rectangle, triangle, parallelogram
- Circle, square, triangle, ellipse
- Circle, rectangle, square, triangle Back

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

b) perimeter

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?

- 1-2
- 3-4
- 5-6
- 7-8 Back

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.”

- Students recognize and name figures based on the global, visual characteristics of the figure.
- Children at this level are able to make measurements and even talk about properties of shapes, but these properties are not abstracted from the shapes at hand.
- It is the appearance of the shape that defines it for the student.
- A square is a square because it looks like a square.

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.

- Students at this level are able to consider all shapes within a class rather than a single shape.
- At this level, students begin to appreciate that a collection of shapes goes together because of properties.
- Students operating at level 1 may be able to list all the properties of squares, rectangles, and parallelograms but not see that these are subclasses of one another, that all squares are rectangles and all rectangles are parallelograms.
- A square is a square because it looks like a square.

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.

- As students begin to be able to think about properties of geometric objects without the constraints of a particular object, they are able to develop relationships between and among these properties.
- If all four angles are right angles, the shape must be a rectangle. If it is a square, all angles are right angles. If it is a square, it must be a rectangle.
- With greater ability to engage in “if-then” reasoning, shapes can be classified using only minimum characteristics.
- Four congruent sides and one right angle can define a square.

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.

- Earlier thinking has produced in students conjectures concerning relationships among properties. Are these conjectures correct? Are they true?
- As this analysis takes place, a system complete with axioms, definitions, theorems, corollaries, and postulates begins to develop and can be appreciated as the necessary means of establishing truth.
- At this level, students begin to appreciate the need for a system of logic that rests on a minimum set of assumptions and from which other truths can be derived.
- This is the level of the traditional high school geometry course.

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.

- At the highest level of the van Hiele hierarchy, the objects of attention are axiomatic systems themselves, not just the deductions within a system. There is an appreciation of the distinctions and relationships between different axiomatic systems.
- Spherical geometry is based on lins drawn on a sphere rather than in a plane or ordinary space. This geometry has its own set of axioms and theorems.
- This is generally the level of a college mathematics major who is studying geometry as a branch of mathematical science.

The products of thought at level 4 are comparisons and contrasts among different axiomatic systems of geometry.

Instruction at Level 0

- Involving lots of sorting and classifying. Seeing how shapes are alike and different is the primary focus of level 0
- Include a variety of examples of shapes so that irrelevant features do not become important. Students need ample opportunity to draw, build, make, put together, and take apart shapes in both two and three dimensions.
- Combine and compare shapes. (Mosaic)

Comparing Shapes

- The 7-Piece Mosaic
- Cut apart the pieces of the Mosaic
- What can we do with these pieces?
- Allow students free exploration time.
- Find all the pieces that can be made from two others.
- Find a piece that can be made from three other pieces.
- How many shapes can be made with a pair of pieces?

Comparing Shapes

- The 7-Piece Mosaic
- With pieces 5 and 6, six shapes are possible.
- How many pieces are possible with 1 and 2?
- On a piece of paper make a house using two shapes.
- Trace around the house with a pencil and see if you can make the same shape with two other pieces.
- On a piece of paper make a tall house with piece 2 as the roof and one other piece. Trace around the house. Make the house with pieces 5 and 7.
- Can it be made with three pieces?
- Use any two, three, or four pieces. Make a shape. Trace around it on a large index card. Color it.
- Can you make this shape with other pieces?
- Write your name and a title for your shape on the index card.

Instruction at Level 1

- Focus more on the properties of figures rather than on simple identification.
- Apply ideas to entire classes figures (all rectangles, all prisms etc.) rather than on individual models. Analyze classes of figures to determine new properties. Find ways to sort all triangles into groups.

Instruction at Level 2

- Encourage the making and testing of hypothesis. “Do you think that will work all the time?”
- Examine properties of shapes to determine necessary and sufficient conditions for different shapes or concepts. “What properties of diagonals do you think will guarantee that you will have a square?”

Linking Children’s Literature

- The Greedy Triangle (Burns)
- Sir Cumference (Series by Neuschwander)
- Hello Math Reader (Series by Maccarone)
- Grandfather Tang’s Story (Tompert)

Important Concepts In Linear Measurement

- Partitioning is the mental activity of slicing up the length of an object into the same-sized units.
- The idea of partitioning a unit into smaller pieces is nontrivial for students and involves mentally seeing the length of the object as something that can be partitioned (cut up) before even physically measuring.
- Activity--Make your own ruler can reveal how well students understand partitioning.

Important Concepts In Linear Measurement

- Unit Iteration is the ability to think of the length of a small block as part of the length of the object being measured and to place the smaller block repeatedly along the length of the larger object.
- Students may iterate a unit leaving gaps between subsequent units or overlapping adjacent units. For these students, iterating is a physical activity of placing units end-to-end in some manner, not an activity of covering the space or length of the object without gaps.

Important Concepts In Linear Measurement

- Transitivity is the understanding that:
- if the length of object 1 is equal to the length of object 2 and object 2 is the same length as object 3, then object 1 is the same length as object 3.
- If the length of object 1 is greater than the length of object 2 and object 2 is longer than object 3, then object 1 is longer than object 3.
- If the length of object 1 is less than the length of object 2 and object 2 is shorter than object 3, then object 1 is shorter than object 3.

Important Concepts In Linear Measurement

- Conservation of length is the understanding that as an object is moved, its length does not change.
- Two strips of paper of equal length are laid down next to each other. Student identifies that they are the same length. One strip of paper is moved to the right two inches. Students who can not conserve length answer that the strips are no longer equal.

Important Concepts In Linear Measurement

- The accumulation of distance means that the result of iterating a unit signifies, for students, the distance from the beginning of the first iteration to the end of the last.
- Student paced off the length of a rug. The teacher stopped her on the 8th step and asked her what 8 meant.
- Some students claimed the 8 represented the distance covered by the 8th step.
- Others claimed the 8 represented the distance covered from the 1st step to the last.

Important Concepts In Linear Measurement

- Relation between number and measurement-Measuring is related to number in that measuring is simply a case of counting. However, measuring is conceptually more advanced since students must reorganize their understanding of the very objects they’re counting (discrete versus continuous units).
- Measuring with matches
- Starting measurement with 1 instead of 0

- Measurement Instruction Sequence (recommended by most math textbooks)
- Students compare lengths
- Measure with nonstandard units
- Incorporate the use of manipulative standard units
- Measure with a ruler

- Comparing lengths is at the heart of developing the notions of conservation, transitivity, and unit iteration but most textbooks do not include these types of tasks.
- Instead of “How many paper clips does the pencil measure?” the question “How much longer is the blue pencil than the red pencil?” gets at the relational aspect of measurement and thereby relational mathematics.

- Teachers should focus students on the mental activity of transitive reasoning and accumulating distances.
- One task involving indirect comparisons is to ask students if the doorway is wide enough for a table to go through. This involves an indirect comparison (and transitive reasoning) and therefore de-emphasizes physical measurement procedures.

Back

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.

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