Title: Choices and Challenges: Redefining Mathematics Teaching for the 21st Century
1Geometric 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.
2National 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.
3Navigating 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
4Navigating 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
5Kansas 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
6Kansas 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
7Kansas 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
8Kansas 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
9Kansas 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
10Kansas 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
11Kansas 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
12Kansas 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
13Kansas 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
14Kansas 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
15Pierre 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)
16Pierre 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.
17Pierre 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.
18Pierre 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.
19Pierre 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.
20Pierre 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.
21Instruction 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)
22Comparing 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?
23Comparing 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.
24Instruction 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.
25Instruction 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?
26Linking Childrens Literature
- The Greedy Triangle (Burns)
- Sir Cumference (Series by Neuschwander)
- Hello Math Reader (Series by Maccarone)
- Grandfather Tangs Story (Tompert)
27Tangram Activities
28Pentominoes
29Measurement Concepts
30Important 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.
31Important 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.
32Important 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.
33Important 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.
34Important 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.
35Important 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
36Measurement 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
37Measurement 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.
38Measurement 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
39Geometric 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.