Van Hiele Levels of Geometric Thought...

Van Hiele Levels of Geometric Thought RevisitedAuthor(s): ANNE TEPPOSource: The Mathematics Teacher, Vol. 84, No. 3 (MARCH 1991), pp. 210-221Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/27967094 .

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Van Hiele Levels of

Geometric Thought Revisited By ANNE TEPPO

The purpose of this article is to reexam

ine the van Hiele theory of levels of geometric thinking and to compare this the

ory with the geometry curriculum recom

mended by the NCTM's Curriculum and Evaluation Standards for School Mathemat

ics (1989). Examples of activities for stu

dents are included to illustrate the ways in

which van Hiele's theory can be translated

into classroom practice. The van Hiele theory postulates a learn

ing model that describes the different types of thinking that students pass through as they move from a global perception of geo

metric figures to, finally, an understanding of formal geometric proof. Van Hiele cur

rently characterizes his model in terms of

three rather than five levels of thought, which he labels as visual (level 1), descrip tive (level 2), and theoretical (level 3) (Fuys, Geddes, and Tischler 1988; Geddes 1987, 1988; van Hiele 1986).

In the current model, these levels are

achieved by passing through different learn

ing periods. During each period students in

vestigate appropriate objects of study, de

velop specific language related to these

objects, and engage in interactive learning activities designed to enable them to prog ress to the next higher level of thinking.

The inclusion of these learning periods reflects the crucial role that van Hiele as

signs to instruction in the development of a

student's geometric understanding. "The

transition from one level to the following is not a natural process; it takes place under the influence of a teaching-learning pro

Anne Teppo is a mathematics education consultant, 1611 Willow Way, Bozeman, MT 59715. She is involved

in qualitative research investigating the development of mathematical concepts within the classroom environ

ment.

gram" (van Hiele 1986, 50). The visual, descriptive, and theoretical

levels of thinking and the learning periods that lead to each of these levels are de

scribed here and summarized in figure 1

(van Hiele 1986). Examples of activities in

volving the concept of symmetry are in

cluded to illustrate the type of behavior that is expected at each level.

Level 1?Visual

Students recognize shapes globally. "It is

possible to see similar triangles, but it is

Fig. 1. Van Hiele's current model of instruction. Each

level of thinking is separated by a learning period in which instruction, using the five phases of learning, enables students to progress to the next higher level of thought.

Theoretical (Level 3)

Learning Period 2

Descriptive (Level 2)

Learning Period 1

Visual (Level 1)

Use deductive reasoning to prove geometric relationships.

Phases of Learning

Integration Free orientation

Explication Bound orientation Information

Recognize objects by their geometric properties.

Phases of Learning

Integration Free orientation

Explication Bound orientation Information

Recognize geometric objects globally.

210 Mathematics Teacher



senseless to ask why they are similar: There is no why, one just sees it" (p. 83).

Activity Students create shapes having a line of sym metry by folding paper and cutting out the shape.

Period 1

Students move from level-1 to level-2 think

ing. The objects of study during this period consist of properties of individual figures. For example, students begin to recognize that a square contains four congruent sides and has congruent diagonals that are per pendicular bisectors of each other.

Activity Students associate a line of symmetry with a

particular shape, fold cut-out shapes to line

up opposite sides, and draw a line of symme try along the fold. They discuss properties of

shape revealed by folding.

Level 2?Descriptive Students distinguish shapes on the basis of their properties.

Activity Students use the properties of line symme try of an isosceles trapezoid to construct such a figure when a diagonal and the base of the figure are given (fig. 2).

Period 2

Students move from level-2 to level-3 think

ing. The objects of study during this period are networks of relationships and the order

ing of properties of geometric figures. Stu dents investigate ways in which to order related properties so that each property is the outcome of a preceding property. Using informal deductive reasoning, students be come able to prove relationships.

Activity Students discover a set of conditions that are

needed to determine that line is a line of symmetry for the figure ABCD (fig. 3):

1. AD is perpendicular to line Z.

2. BC is perpendicular to line Z.

3. ?E is congruent to ED.

4. BF is congruent to FC.

5. AABE is congruent to ADCE.

6. Line is the perpendicular bisector of

segments AD and BC.

Does the set of items 1, 2, and 5 guarantee the result? Explain. Is item 5 sufficient? Ex

plain. Is item 6 sufficient? Explain.

Level 3?Theoretical

Students are able to devise a formal geomet ric proof and to understand the process em

ployed (p. 86):

The language of the theoretical level has a much more abstract character than that of the descriptive level because it is engaged with causal, logical, and other relations of a structure, which at the second level is not visual. Reasoning about logical relations between the orems in geometry takes place with the language of the third level.

Activity AA'B'C is the reflection of ABC about line Z. Students prove that the figure BB'C'C is a trapezoid (fig. 4).

Fig. 2. Construct an isosceles trapezoid given a diag- Fig. 3. Which properties of an isosceles trapezoid onal and one base. determine its line of symmetry?

March 1991 211



U a

Fig. 4. Use properties of reflection to prove that figure BB'C'C is a trapezoid.

Phases of Learning Van Hiele's model stresses the importance of the teaching-learning act. Students progress from one level to the next as the result of purposeful instruction organized into five phases of sequenced activities that empha size exploration, discussion, and integration. The van Hiele model postulates that these five phases of instruction are necessary to enable students in each learning period to

develop a higher level of geometric thinking (van Hiele 1986) (see fig. 1).

The five phases of instruction are de scribed and illustrated by examples dealing

with the concept of symmetry as treated dur ing th? first learning period. These activities are adapted from suggestions in Structure and Insight (van Hiele 1986).

First phase: Information

Material related to the current level of study is presented to the students.

Activity Students demonstrate the reflection of point A about line I using a Mira and show how this reflection can be drawn using graph paper (see fig. 5a).

Second phase: Bound orientation

The student explores the field of inquiry through carefully guided, structured activi ties.

Activity 1. Students reflect the given line segments

about line I (fig. 5b) and determine what shape results.

il

(c) Fig. 5. (a) Demonstration of reflection of point A about

line / using graph paper; (b) Reflect the given line seg ments about line I; (c) Given three vertices of an isosceles trapezoid, find the fourth.

2. After completing all three reflections about I, students make observations about the axes of symmetry of the figures created:

a) What properties must the rhombus have so as to exhibit these axes of

symmetry? 6) These axes are the diagonals of the

figure. On the basis of the symmetry, what observations can be made about the properties of the diagonals? Are the diagonals perpendicular bisectors?

Third phase: Explication

The students and teacher engage in discus sion about the objects of study. Language appropriate to the level is stressed (see the

preceding activity).

Fourth phase: Free orientation

The students engage in more open-ended activities that can be approached by several different types of solutions.

Activity Students are given three vertices of an isos




celes trapezoid (fig. 5c) and asked to find the fourth. They explain what they did and why their procedure worked.

Fifth phase: Integration

The teacher helps the students to gain an

overview of the field of study and to inte

grate the subject matter investigated. At

this stage rules may be composed and mem

orized.

Activity Students summarize the characteristics of

figures that have one or more axes of sym

metry. How can line symmetry be recog nized? They summarize and memorize the

properties of a rhombus.

Instructional Aspects of the Theory

According to the van Hiele model, each

learning period builds on, and extends the

thinking of, the preceding level. The instruc tion makes explicit that which was only im

plied at the preceding level of thought. Effective learning occurs as students ac

tively experience the objects of study in ap

propriate contexts of geometric thinking and as they engage in discussion and reflection

using the language of the learning period. Language is an important part of learn

ing. Each of the three levels in the van Hiele model is characterized by a vocabulary that is used to represent the concepts, structures, and networks within that level of geometric understanding. "Language is useful, be cause by the mention of a word parts of a

structure can be called up" (van Hiele 1986, 86). New language is introduced in each

learning period to make explicit and discuss new objects of study.

One important instructional aspect of van Hiele's theory is that students at a lower

level of thinking cannot be expected to un

derstand instruction presented to them at a

higher level of thought. According to van

Hiele (1986), "this is the most important cause of bad results in the education of

mathematics" (p. 66). Students must pass

through the learning periods leading to each level in succession to be able to develop an

appropriate understanding of the mathe

matical concepts expressed at each level. In this way students develop the ability to un

derstand and use geometric thinking and

insight. The passage of students through the vi

sual and descriptive levels and into the the oretical level occurs over a long time. The

subject of geometry is a rich, complex field whose topics must be effectively integrated during each learning period. A systemati cally developed field of knowledge must be

gained in all aspects of geometry before a

student is capable of reaching the theoreti cal level. To reach this level, van Hiele

(1986) comments, "It takes nearly two years of continual education to have pupils expe rience the intrinsic value of deduction, and still more time is necessary to understand the intrinsic meaning of this concept" (p. 64).

Validation off the van Hiele Theory

The Dutch educators Pierre van Hiele and Dina van Hiele-Geldof became interested in

developing an instructional theory involv

ing levels of geometric thinking as a result of their experiences in the 1950s teaching secondary school students in the Nether lands (Fuys, Geddes, and Tischler 1988; van

Hiele 1986). Classroom observations of stu dents' learning difficulties prompted the identification of these levels. As Pierre van

Hiele (1986, 39) explains,

When I began my career as a teacher of mathematics, I

very soon realized that it was a difficult profession. There were parts of the subject matter that I could

explain and explain, and still the pupils would not

understand_In the years that followed I changed my

explanation many times, but the difficulties remained. It always seemed as though I were speaking a different

language. And by considering this idea I discovered the

solution, the different levels of thinking.

Pierre van Hiele and Dina van Hiele

Geldof reported their theory of geometric levels of thinking in companion disserta tions at the University of Utrecht in 1957. In

1957 Pierre van Hiele also presented in

France a paper outlining their work. Soviet

educators became interested in these ideas

and during the 1960s conducted extensive research based on the van Hiele model that culminated in changes in their school geom

March 1991 213



etry curriculum (Fuys, Geddes, and Tischler

1988; Hoffer 1983). It was not until 1974,

however, that educators in the United

States became aware of the work of the van

Hieles when Izaak Wirszup presented a pa

per on the subject at the annual meeting of

the National Council of Teachers of Mathe

matics (van Hiele 1986). Since the late 1970s research has been

conducted in the United States investigat

ing this model using van Hiele's original 1957 classification of five levels of thinking.

Mayberry (1981) demonstrated that these

different levels form a learning hierarchy as

postulated by the van Hieles. Burger and

Shaughnessy (1986); Fuys, Geddes, and Tis

chler (1988); and Usiskin (1982) found that descriptions of students' behavior could be

used to classify students at different van

Hiele levels. Usiskin (1982) and Senk (1989) also found that this classification could be

used as a predictor of success in a traditional

high school geometry course. A study by Fuys, Geddes, and Tischler (1988) demon strated that appropriate instruction can be used to move students successfully from a

lower to a higher level of geometric think

ing.

Relationship of the Theory to School Instruction

Although van Hiele's theory has been

closely studied in the United States during the last decade, its ideas have yet to be

incorporated into daily instructional prac tices within the classroom setting. The pub lication of the NCTM's Curriculum and

Evaluation Standards in 1989, however, has

brought the van Hiele theory closer to actual

implementation. The Curriculum and Evaluation Stan

dards stresses the importance of sequential

learning as expressed by van Hiele's instruc

tional model (NCTM 1989, 48).

Evidence suggests that the development of geometric ideas progresses through a hierarchy of levels. Students first learn to recognize whole shapes and then to ana

lyze the relevant properties of a shape. Later they can see relationships between shapes and make simple de ductions. Curriculum development and instruction must consider this hierarchy.

The Curriculum and Evaluation Stan dards also advocates an active approach to

learning that incorporates many of the ideas

of van Hiele's phases of learning (NCTM

1989, 214).

The Curriculum Standards present a dynamic view of

the classroom environment. They demand a context in

which students are actively engaged in developing mathematical knowledge by exploring, discussing, de

scribing, and demonstrating. Integral to this social pro cess is communication. Ideas are discussed, discoveries

shared, conjectures confirmed, and knowledge acquired

through talking, writing, speaking, listening, and read

ing.

The following sections outline the K-12

geometry standards and compare their rec

ommendations to the van Hiele levels. The

specific language of the standards, with the

inclusion of examples of activities for stu

dents, serves as an excellent blueprint for

the incorporation of the van Hiele theory into American mathematics education. The

examples of activities included in the follow

ing sections are taken from NCTM's Curric

ulum and Evaluation Standards.

Grades K-4

Standard 9, "Geometry and Spatial Sense," for grades K-4 (p. 48) calls for students to be

able to?

describe, model, draw, and classify shapes; investigate and predict the results of combining, subdividing, and

changing shapes; develop spatial sense.

Standard 9 recommends that students learn to recognize geometric shapes through active exploration of the world around them

using a variety of "everyday objects and other physical materials." These activities

represent students' behavior at van Hiele's

visual level (see fig. 6). After students have developed a famil

iarity with various geometric objects, in

struction in the later grades begins to in

clude activities from van Hiele's first

learning period. The properties of objects become the focus of learning as students

manipulate shapes and discuss their find

ings (see fig. 7).

Geometry instruction is an important part of the K-4 curriculum. Such instruc tion develops a student's spatial sense and




Which shapes are triangles?

Fig. 6

furnishes the necessary foundation for the

development of higher levels of geometric thinking needed for modeling mathematical

problems and studying geometrical proofs. "Children who develop a strong sense of spa tial relationships and who master the con

cepts and language of geometry are better

prepared to learn number and measurement

ideas, as well as other advanced mathemat ical topics" (NCTM 1989, 48).

Grades 5-8

Standard 12, "Geometry," for grades 5-8 (p. 112) calls for students to?

identify, describe, compare, and classify geometric fig ures; visualize and represent geometric figures ... ; explore transformations of geometric figures; represent and solve problems using geometric models; understand and apply geometric properties and relationships.

The learning activities in the middle

grades continue the development of geomet ric thinking begun in grades K-4. Students discover geometric relationships and use them to make conjectures and test hypothe

Cut shapes and make new shapes from the parts.

ses. "Definitions become meaningful, rela

tionships among figures are understood, and students are prepared to use these ideas to

develop informal arguments" (NCTM 1989, 112).

According to van Hiele's theory, such

learning involves activities from both the first and second learning periods. Within the

span of the four middle grades, students

deepen their concepts of geometric objects by investing them with properties (see fig. 8) and then proceed to investigate relation

ships among these properties (see fig. 9). Such activities furnish essential preparation for the study of high school geometry and the use of more formalized geometric pro cesses.

A study by Usiskin (1982) linking stu dents' measured van Hiele levels to perfor mance in high school geometry concludes that systematic geometry instruction before

high school is necessary to insure students' later success in a traditional geometry course. Geometry instruction in grades K-8

must, therefore, be given an appropriate em

phasis; learning activities must proceed in correct sequence to bring students into at

"You are given a pile of toothpicks all the same size. First, take three toothpicks. Can you form a triangle using all three toothpicks placed end to end in the same plane? Can a different triangle be formed? What kinds of triangles are possible? Now take four tooth

picks and repeat the questions. Then repeat with five toothpicks, and so on" (NCTM 1989,

113).

z\

A

? Number toothpicks Is triangle possible?

Number of triangles 1 1

Fig. 7

Kind of triangle | Equilateral | | Isosceles | Equilateral | Isosceles

Fig. 8

March 1991 215



"Connect the midpoints of the sides of several

quadrilaterals. . . . How does the area of the new figure compare to that of the quadrilat eral? What quadrilateral would you start with so that the new figure is a rectangle? A

square?" (NCTM 1989, 114)

%

Fig. 9

least the second learning period by the time

they reach ninth grade. Without such prior instruction, students will not be at a level of

thinking necessary to understand the con

cepts of formal geometric proof. Van Hiele's theory of levels of geometric

thought has helped to focus attention on the

importance of geometry instruction in the

primary and middle grades. This theory's

emphasis on the sequential nature of geom

etry learning and the related assumption that a student at level cannot understand instruction presented at level + 1 mandate that students be given learning experiences through levels 1 and 2 and into period 2

before they are capable of successful perfor mance in high school geometry.

Grades 9-12

Standard 7, "Geometry from a Synthetic

Perspective" (p. 157), calls for students in

grades 9-12 to be able to?

represent problem situations with geometric models

and apply properties of figures; classify figures in terms

of congruence and similarity and apply these relation

ships; deduce properties of, and relationships between,

figures from given assumptions; . . . develop an under

standing of an axiomatic system through investigating and comparing various geometries.

Standard 8, "Geometry from an Alge braic Perspective" (p. 161), calls for students to be able to?

translate between synthetic and coordinate representa tions; deduce properties of figures using transforma tions and using coordinates; identify congruent and

similar figures using transformations; analyze proper

ties of Euclidean transformations and relate transla tions to vectors; deduce properties of figures using vec

tors; apply transformations, coordinates, and vectors in

problem solving.

Instruction in high school continues to use informal deductions that were begun in

the later middle grades (see fig. 10). Once

students have progressed through a thor

ough development of the ideas in the second

learning period, they are ready to apply this

knowledge within the theoretical level (see

fig. 11). The choice of language in which the 9-12 Curriculum and Evaluation Standards is written reflects the advanced level of geo

metric thinking that is expected of these

students; notice the use of such verbs as

translate, deduce, analyze, and apply.

Assessment of Students' van Hiele Level

Students can be assigned to a particular van

Hiele level or learning period by analyzing

Using only a ruler, find ways to determine that

figure ABCD is a rectangle (NCTM 1989, 158).

D

A 3? EF = 5 mzA= 90?, AC = BD -ABCD

so ABCD is a rectangle. is a rectangle.

Fig. 10

"In right triangle ABC with hypotenuse AB =

32, , , , Q, and R are midpoints of

segments AB, AC, CB, BM, and AM, re

spectively. Find the perimeter of NPQR"

(NCTM 1989, 160). (The diagram ... is not given to the students.)




their responses to specific geometric tasks. The Curriculum and Evaluation Standards recommends that this diagnostic assessment use teachers' observations or oral questions that ask students to explain their responses. This kind of assessment allows the teacher to measure students' use of appropriate lan

guage as well as to determine students' lev

els of concept development. An example of a diagnostic task that can

be used to assign van Hiele levels is the

following sorting exercise (Burger and

Shaughnessy 1986). The student is given a set of cutout quadrilaterals (fig. 12) and asked, "Can you put together some of these

that are alike in some way? How are they alike? Can you put together some that are

alike in a different way? How are they alike?"

This task can be used to measure stu

dents' concept development through level 2

and into the second learning period. Careful

questioning of students who are in the sec

ond learning period can probe their under

standing of class inclusion by asking them to order the shapes in terms of their properties, for example, a square is a special rectangle, which is a special parallelogram, which is a

special quadrilateral. Burger and Shaugh nessy (1986) and Fuys, Geddes, and Tischler (1988) have assembled excellent collections of geometric tasks that can be used for diag nostic purposes.

Usiskin (1982) and Mayberry (1981) have developed van Hiele tests that are

not as open-ended as the foregoing tasks. Usiskin's test consists of twenty-five

Fig. 12. Quadrilateral shapes used for diagnostic as sessment

multiple-choice items that are intended to

place students within van Hiele's original five-level classification scheme. This test

should be used with caution because of the

limited number of items included at each

level. Wilson (1990), Crowley (1990), and Usiskin and Senk (1990) discuss the uses and limitations of this test.

Mayberry's multiple-choice test includes

sixty-two items that are intended to place students within the first four of the original levels. The questions are developed specifi

cally to test students' knowledge of the geo metric concepts of square, right triangle, isosceles triangle, circle, parallel lines, sim

ilarity, and congruence. Both the Usiskin and Mayberry tests can serve as diagnostic tools if they are used in an open-ended format rather than as a simple paper-and

pencil test. Information on each student's

level of concept development can be obtained if students are observed individually and

asked to explain their responses to each

question. Until systematic geometry instruction is

included in grades K-8, as recommended by NCTM's Curriculum and Evaluation Stan

dards, students will continue to enter high school with low levels of geometric concept

development. Fuys, Geddes, and Tischler

(1988) demonstrated that effective activities can be given to ninth-grade students who are not yet ready for formal geometry learn

ing. Since these students cannot, according to the van Hiele model, understand the

higher level of high school geometry instruc

tion, any learning that they can accomplish will have to be at a lower level. It is impor tant, therefore, to assess students' levels of

geometric understanding as they enter

ninth grade and to furnish instruction at a

level that will benefit their concept develop ment.

Activities for Students

This section presents selected examples of

activities that characterize van Hiele's three

levels of thinking and two learning periods. These activities are based on the concepts of

reflection and line symmetry and represent activities suggested by Crowley (1987), Ged

March 1991 217



des and Tischler (1988), and van Hiele (1986). Other collections of van Hiele-based and general geometry activities can be found in articles by Burger and Shaugh nessy (1986), Shaughnessy and Burger (1985), Crowley (1987), and Fuys, Geddes, and Tischler (1988); in the NCTM's 1987 Yearbook Learning and Teaching Geometry, K-12; in the September 1985 issue of the Mathematics Teacher; and in Hoffer's (1979) van Hiele-based geometry textbook.

Visual level (early primary grades)

Students recognize symmetrical objects by their global appearance.

1. Students create shapes having a line of

symmetry.

a) Fold paper and cut out shapes or snow flakes.

6) Use grid paper to copy a design from one side of a given line to the other

(see fig. 13).

2. Students identify symmetrical objects.

a) Collect pictures of symmetrical ob

jects.

6) Sort objects according to whether they have a line of symmetry.

First period (late primary grades, early middle grades) The properties of line symmetry become the

objects of study.

1. Students learn to locate lines of symme try.

a) Fold shapes to line up the sides of a figure.

6) Use a Mira to locate lines of symme try; for example, use a Mira to draw a line that maps the image of side AB onto side CB (see fig. 14).

Fig. 14. Use a Mira to map the image of side AB onto side CB.

2. Students learn to identify properties of

figures on the basis of line symmetry.

a) Identify congruent sides of squares, rectangles, and so on, by paper folding.

6) Use a Mira to bisect the angles of an isosceles triangle.

3. Students learn to use the properties of

symmetry.

a) Use a Mira to draw the perpendicular bisector of a line segment, to bisect an

angle, and to draw parallel lines; for

example, in figure 15 use a Mira to draw a line perpendicular to line I

through point P. Draw a line through point A perpendicular to line Z.

) Locate coordinates of symmetrical points on grids when a point and the line of symmetry are given; for exam

ple, in figure 16 reflect each point about the diagonal line I and fill in the appropriate information.

4. Students discuss and reflect on the fore

going activities.

a) Ask students why a Mira can be used to draw a perpendicular bisector of a line segment; for instance, in figure 17 use a compass to locate the reflection of point about line 1. Confirm the location of this point with a Mira. Ex

plain.

6) Discuss how the symmetrical proper ties of a rhombus are used in construc tions with compass and straightedge;

Fig. 13. Copy the design from one side of the line segment to the other. Fig. 15. Use a Mira to draw parallel lines.




(1,4) -* (_, (-3,-1)- (_, (5,-3) - (_,

_) _)

Fig. 16. Reflect each point about line I and appropriate information.

in the

c)

for example, use the previous con struction to draw a rhombus. Discuss the symmetry of this figure. How did the construction with a compass use

symmetry? How do the line segment PP' and the line I relate to the rhom bus you have constructed?

Develop appropriate vocabulary for

objects of study.

Descriptive level (middle grades) Students can make use of the information

developed during the first learning period.

1. Students use properties of line symmetry to identify properties of shapes.

2. Students use properties of line symmetry to construct figures.

Second period (late middle grades, early high school) Networks of relationships involving line

symmetry become the objects of study.

P

Fig. 17. Use a compass to construct the reflection of

point about line I. Confirm the location of P' using a Mira.

1. Students learn to use properties of line

symmetry and reflection to investigate geometric relationships.

a) Demonstrate the congruence of figures on the basis of line symmetry; for ex

ample, for figure 18 use your knowl

edge about the axis of symmetry of the isosceles trapezoid ABCD to show that

?F is congruent to BG, that AD and BC are congruent, that E is the mid

point of AB, that F is the midpoint of DE, and that G is the midpoint of EC.

b) Develop informal proofs of geometric relationships.

Example. Let a line MN be given to

gether with two points A and on one side of it (see fig. 19a). Use the properties of reflection to locate a point X on MN such that A = . Explain in a se

quence of statements how point X can be located and why this location produces A congruent to (Yaglom 1962).

Solution. Reflect point A about MN to form A'. Draw a line segment connecting and A'. This line segment intersects at X. Points X and are reflections of them selves. Thus A' is the reflection of A . Then A = A' , since re

flections preserve angle measure. and A' are vertical angles of intersect

ing lines and are therefore congruent. Then A = (see fig. 19b).

2. Students discuss and reflect on these ac tivities.

a) Develop informal arguments for the selection of a particular ordering of

geometric relationships.

Fig. 18. Using properties of the axis of symmetry of the isosceles trapezoid ABCD, show that AF is congruent toBG.

March 1991 219



IN In

a a

a*

?

4M 4M

(a) (b) Fig. 19. (a) Use the properties of reflection to locate a

point X on ?? such that ?-AXN = LBXM. (b) Locate the reflection of point A about MN to find the solution to 19a.

?) Develop informal definitions of line symmetry and reflections.

c) Develop and use appropriate vocabu

lary.

Theoretical level (grades 9-12)

Students are able to use the concepts of line

symmetry and reflection as part of a formal

structure of geometric proof.

1. Students identify what is given and what

is to be proved in the following problem. They explain how the concept of line sym

metry can be used to prove the statement.

What other proof could be used?

Problem. The perpendicular bisector

of the base of an isosceles triangle passes

through the vertex of the triangle.

The foregoing examples using the ideas

of symmetry present only a limited aspect of

the total learning environment. The other

geometric transformations of rotation and

translation must be studied simultaneously with those of reflection. The complex net

works and relationships of this area of ge

ometry need to be addressed as students

progress toward the theoretical level of

thinking. Van Hiele no longer includes in his

model a level beyond his present theoretical

level. He is currently concerned with ad

dressing the levels of thinking that are usu

ally reached by the end of high school math

ematics. It is possible for students to attain a

more rigorous level of geometric thought

than the theoretical level through advanced

college courses. However, van Hiele's

present model does not address such think

ing (Geddes 1987; van Hiele 1986).

BIBLIOGRAPHY Burger, William F., and J. Michael Shaughnessy.

"Characterizing the van Hiele Levels of Development in Geometry." Journal for Research in Mathematics

Education 17 (January 1986):31-48.

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