A study of the developing procedure of the van Hiele geometry test for
elementary school students
National Taichung Teachers College
This research was undertaken to develop the van Hiele geometry test for elementary school students. This research has the following findings:
1. There are five van Hiele levels. The Wu-Ma‘s van Hiele geometry test (WMVHGT) in this study attempts to
develop a test to measure the students at the first three van Hiele levels (visual, descriptive, and theoretical level). 2. There are three basic geometrical concepts (triangle, quadrangle, and circle) including in the Wu-Ma‘s van
Hiele geometry test.
3. There are nine types of geometry shapes (i.e. open and close images, concave and convex images, straight and curved images, rotate images, large and small images, big obtuse angle images, narrow and broad images, thin-lined and thick-lined images, filled and hollow images) in the first van Hiele level (visual level). 4. There are 25 questions for the first van Hiele level, 20 questions for the second van Hiele level, and 25 questions for the third van Hiele level.
Keywords: van Hiele, geometry, elementary school students
In the late 1950s, two Dutch school teachers, Dina van Hiele-Geldof, and her husband, Pierre M. van Hiele, devised a model of geometric thought for helping students to learn geometry. They interested "in improving teaching outcomes" (van Hiele, 1986, P. vii). The van Hieles' doctoral dissertations studied complementary aspects of developing insight in geometry.
Adapted from Gestalt psychology, many of the ideas in the van Hiele model were centered around the idea of a structure (Molina, 1990; van Hiele, 1986). P. M. van Hiele (1986) pointed out, "Most of the ideas of structure I have developed . . . are borrowed from Gestalt theory" (p. 5). P. M. van Hiele adapted the concepts of levels, derived from Jean Piaget, although he disagreed with Piaget at several points (Molina, 1990; van Hiele, 1986). P. M. van Hiele (1986) claimed, "In any case, an important part of the roots of my work can be found in the theories of Piaget. It is important then, too, to emphasize the differences . . ." (p. 5). He pointed out six main differ-ences between his theories and Piaget's theories. Pierre M. van Hiele "formulated the scheme and psychological principles; D. van Hiele-Geldof focused on the didactic experiment to raise students' thought levels" (Hoffer, 1983, p. 207).
The instruments used to assess students' van Hiele levels, whether interview (Burger & Shaughnessy, 1986a) or written (Usiskin, 1982), have consistently used triangle and quadrilateral definitions and classifications. Similarly, the topics of instruction, except for a few isolated topics in the Fuys, Geddes, and Tischler (1988) study, have been to promote an understanding of triangle and quadrilateral classification. It appears that, in general, all American research on the van Hiele theory has centered around using triangles and quadrilaterals to classify students by levels and to describe the levels, and has focused around using instruments to assign a student into the first four van Hiele levels.
This study attempt to develop a test to measure the students at the first three van Hiele levels; the last two levels of learning were not considered.
The structure of developing the test
The structure of developing the ―Wu-Ma‘s van Hiele geometric test for elementary school students‖ is shown as
collection of the relative Tool to assess elementary school students‘ geometric information and references thinking level
Editing the test paper
Opinions The first from experts version of
the test paper
The second version of
the test paper
The third version of the test paper
The last modification on the
test paper Confirming the test paper
Figure 1: the procedure of developing the test
Wu’s van Hiele geometry test
To remain the reliability and the validity of the research tool, researchers will compare the relative validity with ―Wu‘s van Hiele geometry test (WVHGT)‖ developed by Wu (2000).
Because of the need of the research to designing and the research objects, before designing the research tool need to widely collect the references and then design the test tool based on concepts and relative operation contents that students contact in their daily life. Then, according to expert‘s opinions to modify the test tool and then test again. Analyzing the development of van Hiele level of geometric thinking in the basic figures~triangle,
quadrangle, and circle based on testers‘ scores in the paper-pencil-test.
The research tool is a multiple-choice test. Each concept is chosen by the first three level of van Hiele level of geometric thinking. Each level has five questions, four concepts (square, circle, isosceles triangle, and right triangle), 60 questions in total. The distribution level is as followed~
Table 1: Distribution of the question number of WVHGT
Level one Level two Level three
square 1-5 6-10 11-15
circle 16-20 21-25 26-30
isosceles triangle 31-35 36-40 41-45
right triangle 46-50 51-55 56-60
The reliability of the whole test is 0.890，p<0.01：. The reliabilities of each geometry figure are: square 0.655,
circle 0.758, isosceles triangle 0.560, right triangle 0.696. This show the reliability is pretty high. The validity of this test and the van Hiele‘s level test edited by Lin (1992) has a correlation coefficient 0.872. It is
obvious known that the validity is pretty high.
Procedure of developing the Wu-Ma’s van Hiele geometry test
The Based Theory to Edit the Test
The test tool was designed according to ―theoretic model of van Hiele‘s geometric thinking ‖ and the description of van Hiele level and the examples of students‘ behaviors that were raised by Fuys (1983). Because the research objects were students, the test were edited based on the first three levels (from level one to level three) of van Hiele‘s geometric thinking level. Meanwhile, we design this tool by taking references from 「Grade 1-9
Curriculum Provisional Guidelines—Mathematics」，Ministry of Education of Taiwan, 2000：, the versions of
mathematics textbooks in every book store from the first grade to the sixth grades (textbooks, activity books and teacher‘s manual) and relative research in geometry ，Fuys,Geddes & Tischler, 1988; Usiskin；1982; van Hiele,
. 1986; Wu, 1994, 1995, 2003：
Process of Edition
1. The first edition of test
According to the references and the tool that was mentioned above, we develop the test of van Hiele‘s geometric thinking level, which include the concepts of three basic figures~triangle, quadrangle, and circle. The
edition of the questions could let students to answer them in a quick and simple way.
2.The modification of the test~
The test became a formal one after several modification, the procedure of the modification were as followed~
(1) The first modification~
After the first edition of the questions, we pre-tested two six graders (one boy and one girl students) at random who were studied in the research‘s school. After the pre-test, we will have conversation with each of the two
students to talk about the content of the test. From the conversation, we will modify the questions (the records of the conversation will be found in attachment 3). The modification of the contents was as follow~
;Students can understand the symbol of angle (),so we modify 「A」 into 「angle A」. ;;
AB....；Students can NOT understand the symbol of edge ，：, so we modify 「」into 「edge AB」.
(2)The second modification~
After the first modification of the test paper in January in 2003, the researchers asked for suggestions from elementary teachers who are in other counties and teachers who major in mathematic education of graduate school in National Tai-Chung Teachers College (NTCTC). The modification is as followed~
;Each question has five options, which is not the same as the four options that elementary students were familiar with. Therefore, we modified ―five options‖ into ―four options‖.
；The options in the test paper ―all of above are true‖ are not quite well, so we modified them.
？The symbols of the figures such as figure A or angle ABC are not suitable of students, so we modify angle ABC into Chinese ㄅㄆㄇ.
？The original symbols, such as 甲、乙、丙、丁 in the figures are not clear enough, we are suggested move them
to the outer figures.
(3) The third modification~
Except asking opinions from elementary students and elementary teachers, we also ask suggestions and modification from several professors who teach in Normal Universities or Teachers colleges in January and February in 2003. The modified opinions will be stated in the following statement.
In 2003, opinions from three professors from Taichung teachers college in January~
;question 23~how many degrees in a circle? Opinions from scholarship~the sentence is not stated suitably,
which should change into~how many degrees does a angle in circular segment have?
；question 39~‖correspond side‖ should change into ‖opposite side‖。
？we were suggested to add different bold figures to identify.
？We will change the athwart ship arrangement, such as option，;；？？： into vertical arrangement.
In February, 2003, a professor from Chai-Yi University and a professor in mathematic department from Normal University suggest us~
;Modify the different bold lines and parts which are not smooth enough in the original figures. We will modify the zigzag circle in to a smooth circle, which means that the original matrix graph，BMP file： into vector figure
；We consider the factors of time and reading comprehension of the testers, so we part the test into two parts to test the higher graders. The first and second parts will be tested for 40 minutes, and the third parts will be tested for 40 minutes.
(3) Rules of the edition~
The test tool is mainly design for the elementary school students, so its edition only focuses on van Hiele‗s geometric thinking level from level one to level three (according to what the experts said that students‘ geometric thinking levels are limited only the first three level. Liu，1993：analyze the materials of elementary school, which
accounts for our geometric classes are edited by van Hiele‘s geometric thinking level~materials of the lower
graders are level one – visual level, materials of the medium graders are level two – describable level, and
materials of the higher graders are level three – theoretical level.) The way to answer the questions was designed
as simple and clear as we can. The description of the questions will be verbalization as possible. Ranges of the questions will cover three basic concepts~triangle, quadrangle and circle, so we will try our best to make a figure which contains three basic geometric figures, so that the testees could have a basic comprehension on the figural identification.
The test tool will put into action in March, 2003. The samples of the pre-test are chosen from two elementary schools (school A and school B). We pre-test each class from each grade, and the returnable ratio is 95%. After removing the fragment information and test papers that were not completely answered, we have out pre-test sample such as table 2~
Table 2: the distribution of the test sample
level Lower grade Medium grade Higher grade
total Second Fourth First grade Third grade Fifth grade Sixth grade grade grade school
School A 19 17 18 13 13 18 98
School B 30 30 30 32 30 39 191
total 49 47 48 45 43 57 289
From the result of the pre-test, we notice that in question 22 【identification on the bold lines of outer figure】 the
factors of the bold line make circumference of a circle zigzag in the process during printing and affect its identification to students. Therefore, we will modify this question.
Reliability and Validity
Reliability and validity are important indexes, which should depending on the quality of the questions. The
quality of the test questions could be analyzed through the questions ，Kao, 1989：.Now, we will analyze in
reliability and validity.
It had been tested in March, 2003. The pretest students random selected from two counties. The statistics data, after analyzing by spss10.0 for windows, we got alpha reliability~
Test A is 0.871698，p?.01：, Test B is 0.881643，p?.01：, Test C is 0.944702，p?.01：, which showed the
reliability of this tool is quite high.
The test tool not only invites experts to examine but also take reference to ―Wu‘s van Hiele geometric thinking
level test‖ (Wu, 2000) and get the validity of the test. The correlation coefficient is 0.5338. There are 60 questions in total in tool ―Wu‘s van Hiele‘s geometric thinking level test‖, which is in three different levels. Each level has 20 questions, the validity of the whole test is 0.8105. The co-identity of sub-test of this tool is 0.5492, 0.7619 and 0.7222 (p<.01), which shows high in co-identity.
There are 70 questions in the WMVHGT, and each of them has four options. Only an answer is correct. All the statements were edited by van Hiele level of geometric thinking level, each level include three concepts of basic geometry.
Owing to up 70 questions in this test tool, students from subway or country side, who has poorer language ability can‘t answer all of them in 40 minutes. Time to think about the questions is less than 30 seconds after deducting reading time. Therefore, it is hard to test the students‘ real ability. The researchers changed the test papers into A, B, C based on others‘ opinions. There are 25 questions in Test A, which belongs to the van Hiele level one; 20 questions in Test B, which belongs to level two, and 25 questions in Test C, which belongs to level three . Tests were given to: test A was given to lower grades (the first and second graders); test A and B were given to middle grades (the third and the fourth graders); test A, B, and C were given to higher grades (the fifth and the sixth graders). The distribution of the test papers and the questions are in table 3.
Table 3: the distribution of the questions
Test part Number of the questions
TEST A Q1 to Q25 (van Hiele level 1)
TEST B Q26 to Q45 (van Hiele level 2)
TEST C Q46 to Q70 (van Hiele level 3)
The research reported in this paper was supported by the National Science Council under Grand No. NSC91-2521-S-142-004 and NSC92-2521-S-142-004. Any opinions, viewpoints, findings, conclusions, suggests, or recommendations expressed are the authors and do not necessarily reflect the views of the National Science Council, Taiwan.
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