Choices and Challenges: Redefining Mathematics Teaching for the 21st Century

1
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.
2
National Standards

• 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.

3
Navigating 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

4
Navigating Through Geometry

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

• Using visualization, spatial reasoning, and
  geometric modeling
• Exploring Packages 3-4
• Its All in the Packaging 4

5
Kansas State Standards
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
6
Kansas State Standards
Benchmark 2The 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. Back Sample Problem

7
Kansas State Standards

• Sample Problem for B2-I4
• You need to buy carpet to cover the floor of
  your dogs house. Which tool would you use to
  help you decide how much carpet to buy?
• Compass
• Measuring Cup
• Scale
• Clock
• Yard Stick
• Back

8
Kansas State Standards
Benchmark 3The 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. Back Sample Problem

9
Kansas State Standards
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
10
Kansas State Standards
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). Back Sample
Problem

11
Kansas State Standards
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

12
Kansas State Standards
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 Back Sample
Problem

13
Kansas State Standards

• 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

14
Kansas State Standards
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 Sampl
e Problem

15
Pierre van Hiele
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)
16
Pierre van Hiele

• 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.

17
Pierre van Hiele

• 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.

18
Pierre van Hiele

• 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.

19
Pierre van Hiele

• 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.

20
Pierre van Hiele

• 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.

21
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)

22
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?

23
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.

24
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.

25
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?

26
Linking Childrens Literature

• The Greedy Triangle (Burns)
• Sir Cumference (Series by Neuschwander)
• Hello Math Reader (Series by Maccarone)
• Grandfather Tangs Story (Tompert)

27
Tangram Activities
28
Pentominoes
29
Measurement Concepts
30
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.

31
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.

32
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.

33
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.

34
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.

35
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 theyre counting (discrete versus
  continuous units).
• Measuring with matches
• Starting measurement with 1 instead of 0

36
Measurement Concepts

• 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

37
Measurement Concepts

• 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.

38
Measurement Concepts

• 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

39
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.