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232 MatheMatics teaching in the Middle school ● Vol. 16, No. 4, November 2010

wWhen you refl ect on assessment, do you think about it as an evaluation of what a student knows? As a helpful tool for students while learning some specifi c content? Or do you think it is helpful in learning about your instruc-tional practice?

One of the precepts of NCTM’s Assessment Principle is this: “Assess-ment should support the learning of important mathematics and furnish

To analyze students’ geometric thinking, use both formative and summative assessments and move students along the van Hiele model of thought.

M. Lynn Breyfogle and Courtney M. Lynch

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useful information to both teachers and students” (italics added, NCTM 2000, p. 22). As we understand assessment, we think of it as contributing to all three questions above. It is a tool to be used in the classroom as a way to deepen students’ learning and to allow the educator to make informed deci-sions regarding instruction. The focus of this article will be on the role of as-sessment, both in terms of teachers and

van Hielevan HieleRevisited

Copyright © 2010 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved.This material may not be copied or distributed electronically or in any other format without written permission from NCTM.

Vol. 16, No. 4, November 2010 ● MatheMatics teaching in the Middle school 233

van HieleRevisited


students, while developing students’ understanding of geometry. In particu-lar, we are interested in using authen-tic assessment to develop students’ geometric thought using the van Hiele model (Crowley 1987).

authentic assessMentAuthentic assessment allows students to demonstrate their acquired knowl-edge in a meaningful way and helps them continue to learn as they go through the process. Wiggins (1993) defines this term as assessments that are “engaging and worthy problems or questions of importance, in which stu-dents must use knowledge to fashion performances effectively and creative-ly” (p. 229). These assessments might

be projects completed at the end of a unit (summative assessment) or smaller activities peppered throughout the unit where the teacher is checking for understanding (formative assessment).

The van Hiele model of the de-velopment of geometric thought was created in the 1980s by two Dutch middle school teachers and research-ers, Dina van Hiele-Geldhof and Pierre van Hiele. The model described levels of understanding through which students progress in relation to geom-etry (Crowley 1987). The van Hieles outlined five levels that begin with the most basic level—visualization—and continue to the most advanced level—rigor (see table 1).

The van Hieles (and subsequent

researchers) agreed that, unlike Piagetian models, this is not a develop- mental model where students must reach a certain age to progress through the levels; rather, it is dependent on the experiences and activities in which students are engaged (van Hiele 1999). In other words, students progress through levels on the basis of their experiences rather than age, and it is imperative that teachers provide experiences and tasks so that students can develop along this continuum.

Consider formal deduction (level 3). An example of a question that students should be able to answer us-ing formal deduction can be found in figure 1. In this case, students would use their knowledge of quadrilateral

level name description example teacher activity

0 Visualization See geometric shapes as a whole; do not focus on their particular attributes.

A student would identify a square but would be unable to articulate that it has four con-gruent sides with right angles.

Reinforce this level by encouraging students to group shapes according to their similarities.

1 Analysis Recognize that each shape has different properties; identify the shape by that property.

A student is able to identify that a parallelogram has two pairs of parallel sides, and that if a quadrilateral has two pairs of parallel sides it is identified as a parallelogram.

Play the game “guess my rule,” in which shapes that “fit” the rule are placed inside the circle and those that do not are outside the circle (see Russell and Economopoulos 2008).

2 Informal deduction

See the interrelationships between figures.

Given the definition of a rectangle as a quadrilateral with right angles, a student could identify a square as a rectangle.

Create hierarchies (i.e., organiza-tional charts of the relationships) or Venn diagrams of quadrilaterals to show how the attributes of one shape imply or are related to the attributes of others.

3 Formal deduction

Construct proofs rather than just memorize them; see the possibility of developing a proof in more than one way.

Given three properties about a quadrilateral, a student could logically deduce which state-ment implies which about the quadrilateral (see fig. 1).

Provide situations in which students could use a variety of different angles depending on what was given (e.g., alternate interior or corresponding angles being congruent, or same-side interior angles being supplementary).

4 Rigor Learn that geometry needs to be under-stood in the abstract; see the “construction” of geometric systems.

Students should understand that other geometries exist and that what is important is the structure of axioms, postulates, and theorems.

Study non-Euclidean geometries such as Taxi Cab geometry (Krause 1987).

table 1 The van Hiele model of geometric understanding describes a progression that is independent of age or grade level.


properties to determine the sequence of implications. When they are able to demonstrate the correct application and understanding of these proper-ties, they will be at the formal deduc-tion level, regardless of their age. They will get to this point through experiences.

The van Hieles developed a frame-work for organizing classroom instruc-tion to help teachers structure activities that cultivate their students’ geometric thinking (Clements 2003; Geddes 1992). This framework includes a sequence of five phases of learning: information/inquiry, directed orienta-tion, explication, free orientation, and integration (see table 2). The idea is that students need to cycle (and, for many, re-cycle) through phases of learning while developing their under-standing at each level. We focus on the

last two phases, free orientation and integration, in this article.

• Free orientation is the phase in which students are challenged to make connections among geo-metric concepts as well as solve problems related to them.

• Integration is the process whereby students reflect on their observa-tions and how these observations fit into the overall structure of the concepts.

Because the phases of learning are cyclical, they do not necessarily correspond to certain types of as-sessment. In fact, for this article we have coupled a formative assessment

with an integration-phase activity and a summative assessment with a free-orientation-phase activity. The remainder of this article is devoted to examining authentic assessment and its use in encouraging students to progress along the van Hiele levels.

exaMples of authentic assessMentsAlthough the level of geometric thought for a student is dependent on his or her experiences, according to NCTM’s Curriculum Focal Points for Prekindergarten through Grade 8 Math-ematics (2006), students are expected, by the end of eighth grade, to use informal deduction. It is important that students are given opportunities

phase description

Information/inquiry Teacher: Assess students’ prior knowledge through discussion and allow questions to prompt topics to be explored

Directed orientation Teacher and students: Explore sets of carefully sequenced activities

Explication Students: Share explicit views and understandings about the activities

Free orientation Teacher: Challenge students to solve problems related to the geometric concepts and make connections among them

Integration Students: Reflect on observations and how they fit into the overall structure of the concepts

Source: Adapted from Geddes (1992)

table 2 The van Hiele sequence of phases of learning provides a framework for teachers to guide students through the levels of understanding.

three properties of a QuadrilateralProperty D: It has diagonals of

equal length.Property S: It is a square.Property R: It is a rectangle.

Which is true?a. D implies S, which implies R.b. D implies R, which implies S.c. S implies R, which implies D.d. R implies D, which implies S.e. R implies S, which implies D.

Source: Usiskin (1992)

fig. 1 Students should be able to answer this question using formal deduction.

Students progress through levels on the basis of experiences rather than age. It is imperative that teachers provide experiences and tasks so that students can develop along this continuum.


and experiences to develop the skills that are necessary to demonstrate this expectation.

The following examples show work toward van Hiele’s level 2, informal deduction. Given this, the teacher is looking for formative and summative assessments that engage students in reasoning activities related to the role of definition in geometry. In other words, assessments are needed that provide insight into how students understand definitions and the role of definitions in making and supporting their claims.

Formative Assessment with an Integration-Phase ActivityOne formative-assessment example is found in the Connected Mathematics Project (CMP) in its seventh-grade unit Stretching and Shrinking (Lappan et al. 2002). In this unit, students explore the concept of

similarity in polygons. The CMP curriculum materials use a variety of formative assessments, such as end-of-investigation questions, checkpoints, notebook reflections, and a set of end-of-unit questions called Looking Back and Looking Ahead.

In this particular unit, the Looking Back and Looking Ahead assessments include a set of questions that could provide both valuable feedback to a teacher about a students’ understanding and an experience that could help move students along the van Hiele model.

One such question asks students to examine four statements to determine if they are true or false (see fig. 2). The combination of the four separate statements is powerful in helping students think about what it means for figures to be similar. Asking students to consider similarity for both triangles and quadrilaterals and including one “true” and one “false” statement for each type of shape can help them think critically about the concept. In addi-tion, figure 2’s statement 4d, unlike the previous three statements, requires that students consider the angles.

Let us look at the instance in which Jerome answered statement 4b. Requiring him to justify his selec-tion and include a written rationale engaged him in the integration phase of instruction. He responded in this way to the true or false question: “Any two rectangles are similar”:

True, because all of the angles of rectangles are right so all of the angles

are congruent, and since the opposite sides of all rectangles are congruent their sides would be in proportion.

As teachers, we might note that although Jerome knew that angles of corresponding figures must be congru-ent and sides must be in proportion, he clearly needed help in understand-ing what “sides must be in proportion” meant. Jerome also marked 4a and 4c as true, which raised a further ques-tion regarding his reasoning. These answers gave the teacher an opportu-nity to ask about his understanding, “Did he understand that similarity is a comparison of corresponding sides across two figures—or is he thinking within the figure?” When asked this question, Jerome clarified his under-standing of the definition, which moved him along the levels.

Since Looking Back and Looking Ahead were intended to be formative assessment tools, teachers can use students’ responses to determine what is important to revisit with that par-ticular student. In this case, students were solidifying their understanding of similarity as a definition and using the components of the definition (i.e., corresponding angles congruent, corresponding side lengths in propor-tion) to justify a generalized state-ment. This enhanced their deductive-reasoning skills.

Summative Assessment with a Free-Orientation-Phase ActivityConsider again that an eighth-grade teacher is helping his or her class

unit Reflection Question4. Which of the following

statements about similarity are true and which are false?

a. Any two equilateral triangles are similar.

b. Any two rectangles are similar.c. Any two squares are similar.d. Any two isosceles triangles

are similar.

Source: Lappan et al. 2002, p. 87

fig. 2 This question can be used as a formative assessment at the integration phase of understanding.

Assessments are needed that provide insight into how students understand definitions and the role of definitions in making and supporting students’ claims.


fig. 3 This example of summative assessment explores what students can extrapolate from the activity at hand.

Source: Geddes 1992, p. 54

fig. 4 This student’s family tree for the Pythagorean theorem reveals his understand-ings about the concept.

secure their notion of the importance of properties and the role of definition to develop logical reasoning. In this case, however, the context has changed and the class has completed a unit focused on the relationships among properties of shapes including angle sums. The activities included investiga-tions of parallel lines with transversals and the angles, quadrilaterals, triangles, and tessellations that were created (see Geddes 1992 for sample activities).

Summative authentic assessments provide insight into students’ under-standing of the concepts from the unit. For example, figure 3 pre sents a series of questions to acquaint students with the use of a family-tree diagram (i.e., a flow chart), which shows relationships among concepts. After students have an understanding of how the family trees are used and what information is gleaned from them, a summative assessment could involve this fifth question (Geddes 1992, p. 54):

Try to build some other geometry family trees. For example: “The measure of an exterior angle of a triangle is equal to the sum of the two nonadjacent interior angles.” What are its ancestors?

One student who was given this task created figure 4, a family tree for the Pythagorean theorem. He saw two important aspects of the Pythagorean theorem, namely, the equation and the characteristics of the figure necessary to employ the formula.

In this tree, he clearly identified that lines and angles make up poly-gons, that some polygons are right triangles, and that this will lead to having a hypotenuse in the figure. It is also interesting to note that he identi-fied the property of substitution when using “equations” to solve the prob-lem. Although it is not obvious in this figure, erasure marks below this pencil


drawing indicated that several differ-ent trees were drawn and reconsid-ered. It is obvious that as the student participated in this assessment, he engaged in a variety of logical thought and was working toward the level of formal deduction. This family-tree structure helped students visualize connections, but it also forced them to consider the definitions of the words and how one implied another.

shaping instRuctionTeachers need to consider the out-come of authentic assessments when reflecting on and planning instruc-tion. For example, after completing the aforementioned assessments, the teacher would consider the students’ level of progression through the van Hiele model. As the students achieve the informal deduction level, the teacher would begin to lay the foun-dation for those particular students to reach level 3: formal deduction. Since the students are able to see the interrelationships between figures, the teacher would give students the op-portunity to discover a proof through activities and practice constructing proofs rather than memorizing them.

Teachers must carefully moni-tor the progression of students through the van Hiele model and tailor instruction accordingly so that they receive the experiences neces-sary to move on. Consider Jerome, who completed the formative assess-ment in figure 2 about similarity. The teacher was left wondering about his full understanding of the true defini-tion of the geometric term. Using the information from the assessment, the teacher prompted Jerome to justify his rationale, which in doing so moved him along the van Hiele levels be-cause of the experience he had gained.

concluding thoughts We wanted to introduce teachers to (or remind them of ) the van Hiele

model of geometric thought and how movement through this model depends on experiences of the learner. Well-devised tasks help move stu-dents through the levels. The second purpose was to make the case for using assessments (both formative and summative) as an avenue for moving students along the van Hiele model.

RefeRencesClements, Douglas. “Teaching and Learn-

ing Geometry.” In A Research Com-panion to Principles and Standards for School Mathematics, edited by Jeremy Kilpatrick, W. Gary Martin, and Deborah Schifter, pp. 151−78. Reston, VA: National Council of Teachers of Mathematics, 2003.

Crowley, Mary L. “The van Hiele Model of the Development of Geometric Thought.” In Learning and Teach-ing Geometry, K-12, edited by Mary Montgomery Lindquist, pp. 1−16. Reston, VA: National Council of Teachers of Mathematics, 1987.

Geddes, Dorothy. Geometry in the Middle Grades. Curriculum and Evaluation Standards for School Mathematics Addenda Series, Grades 5−8, edited by Frances R. Curcio. Reston, VA:

National Council of Teachers of Mathematics, 1992.

Krause, Eugene F. Taxicab Geometry: An Adventure in Non-Euclidean Geometry. New York: Dover Publications, 1987.

Lappan, Glenda, James T. Fey, William M. Fitzgerald, Susan N. Friel, and Elizabeth Difanis Phillips. Stretching and Shrinking: Connected Mathematics. Glenview, IL: Prentice-Hall, 2002.

National Council of Teachers of Math-ematics (NCTM). Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics: A Quest for Coherence. Reston, VA: NCTM, 2006.

________. Principles and Standards for School Mathematics. Reston, VA: NCTM, 2000.

Russel, Susan Jo, and Karen Economo-poulos. Investigations in Number, Data, and Space. 2nd ed. Glenview, IL: Scott Foresman, 2008.

Usiskin, Zalman. “Van Hiele Levels and Achievement in Secondary School Geometry.” http://ucsmp.uchicago .edu/van_Hiele.html.

van Hiele, Pierre M. “Developing Geo-metric Thinking through Activities That Begin with Play.” Teaching Chil-dren Mathematics 5 ( January 1999): 310−16.

Wiggins, G. P. Assessing Student Perfor-mance. San Francisco: Jossey-Bass Publishers, 1993.

M. lynn Breyfogle, lynn [email protected], is an associate professor in the mathematics depart-ment at Bucknell Universi-ty in Lewisburg, Pennsyl-vania. She is interested in teachers’ professional development in the teach-

ing and learning of mathematics. courtney M. lynch, [email protected], teaches at the Oxford Area High School in Oxford, Pennsylvania. She is a graduate of Bucknell University and is pursuing her master’s degree in instructional technology at Saint Joseph’s University.

Teachers must carefully monitor the progression of students through

the van Hiele model and tailor

instruction accordingly.

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