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

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1 van Hiele Rev 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 When you reflect on assessment, do you think about it as an evaluation of what a student knows? As a helpful tool for students while learning some specific content? Or do you think it is helpful in learning about your instructional practice? One of the precepts of NCTM s Assessment Principle is this: Assessment should support the learning of important mathematics and furnish 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 decisions regarding instruction. The focus of this article will be on the role of assessment, both in terms of teachers and ALEAIMAGE/ISTOCKPHOTO.COM 232 MatheMatics teaching in the Middle school Vol. 16, No. 4, November 2010 Copyright 2010 The National Council of Teachers of Mathematics, Inc. All rights reserved. This material may not be copied or distributed electronically or in any other format without written permission from NCTM.

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3 students, while developing students understanding of geometry. In particular, we are interested in using authentic assessment to develop students geometric thought using the van Hiele model (Crowley 1987). Authentic Assessment Authentic assessment allows students to demonstrate their acquired knowledge 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 students must use knowledge to fashion performances effectively and creatively (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 development of geometric thought was created in the 1980s by two Dutch middle school teachers and researchers, Dina van Hiele-Geldhof and Pierre van Hiele. The model described levels of understanding through which students progress in relation to geometry (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 developmental 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 using formal deduction can be found in figure 1. In this case, students would use their knowledge of quadrilateral Table 1 The van Hiele model of geometric understanding describes a progression that is independent of age or grade level. Level Name Description Example Teacher Activity 0 Visualization See geometric shapes as a whole; do not focus on their particular attributes. 1 Analysis Recognize that each shape has different properties; identify the shape by that property. 2 Informal deduction 3 Formal deduction See the interrelationships between figures. Construct proofs rather than just memorize them; see the possibility of developing a proof in more than one way. 4 Rigor Learn that geometry needs to be understood in the abstract; see the construction of geometric systems. A student would identify a square but would be unable to articulate that it has four congruent sides with right angles. 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. Given the definition of a rectangle as a quadrilateral with right angles, a student could identify a square as a rectangle. Given three properties about a quadrilateral, a student could logically deduce which statement implies which about the quadrilateral (see fig. 1). Students should understand that other geometries exist and that what is important is the structure of axioms, postulates, and theorems. Reinforce this level by encouraging students to group shapes according to their similarities. 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). Create hierarchies (i.e., organizational 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. 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). Study non-euclidean geometries such as Taxi Cab geometry (Krause 1987). 234 Mathematics Teaching in the Middle School Vol. 16, No. 4, November 2010

4 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. properties to determine the sequence of implications. When they are able to demonstrate the correct application and understanding of these properties, they will be at the formal deduction level, regardless of their age. They will get to this point through experiences. The van Hieles developed a framework for organizing classroom instruction 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 orientation, 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 understanding at each level. We focus on the Fig. 1 Students should be able to answer this question using formal deduction. Three Properties of a Quadrilateral Property 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) last two phases, free orientation and integration, in this article. Free orientation is the phase in which students are challenged to make connections among geometric concepts as well as solve problems related to them. Integration is the process whereby students reflect on their observations 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 assessment. In fact, for this article we have coupled a formative assessment Phase 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 Assessments Although 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 Mathematics (2006), students are expected, by the end of eighth grade, to use informal deduction. It is important that students are given opportunities Table 2 The van Hiele sequence of phases of learning provides a framework for teachers to guide students through the levels of understanding. 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) Vol. 16, No. 4, November 2010 Mathematics Teaching in the Middle School 235

5 Assessments are needed that provide insight into how students understand definitions and the role of definitions in making and supporting students claims. 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 Activity One 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 Fig. 2 This question can be used as a formative assessment at the integration phase of understanding. Unit Reflection Question 4. 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 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 addition, 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 selection 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 congruent and sides must be in proportion, he clearly needed help in understanding what sides must be in proportion meant. Jerome also marked 4a and 4c as true, which raised a further question regarding his reasoning. These answers gave the teacher an opportunity 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 understanding 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 particular 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 proportion) to justify a generalized statement. This enhanced their deductivereasoning skills. Summative Assessment with a Free-Orientation-Phase Activity Consider again that an eighth-grade teacher is helping his or her class 236 Mathematics Teaching in the Middle School Vol. 16, No. 4, November 2010

6 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 investigations 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 understanding 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 polygons, 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 identified the property of substitution when using equations to solve the problem. Although it is not obvious in this figure, erasure marks below this pencil 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 understandings about the concept. Vol. 16, No. 4, November 2010 Mathematics Teaching in the Middle School 237

7 drawing indicated that several different trees were drawn and reconsidered. 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 Instruction Teachers need to consider the outcome of authentic assessments when reflecting on and planning instruction. 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 foundation 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 opportunity to discover a proof through activities and practice constructing proofs rather than memorizing them. Teachers must carefully monitor the progression of students through the van Hiele model and tailor instruction accordingly so that they receive the experiences necessary to move on. Consider Jerome, who completed the formative assessment in figure 2 about similarity. The teacher was left wondering about his full understanding of the true definition 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 because of the experience he had gained. Concluding Thoughts We wanted to introduce teachers to (or remind them of ) the van Hiele Teachers must carefully monitor the progression of students through the van Hiele model and tailor instruction accordingly. model of geometric thought and how movement through this model depends on experiences of the learner. Well-devised tasks help move students 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. References Clements, Douglas. Teaching and Learning Geometry. In A Research Companion to Principles and Standards for School Mathematics, edited by Jeremy Kilpatrick, W. Gary Martin, and Deborah Schifter, pp Reston, VA: National Council of Teachers of Mathematics, Crowley, Mary L. The van Hiele Model of the Development of Geometric Thought. In Learning and Teaching Geometry, K-12, edited by Mary Montgomery Lindquist, pp Reston, VA: National Council of Teachers of Mathematics, 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, Krause, Eugene F. Taxicab Geometry: An Adventure in Non-Euclidean Geometry. New York: Dover Publications, Lappan, Glenda, James T. Fey, William M. Fitzgerald, Susan N. Friel, and Elizabeth Difanis Phillips. Stretching and Shrinking: Connected Mathematics. Glenview, IL: Prentice-Hall, National Council of Teachers of Mathematics (NCTM). Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics: A Quest for Coherence. Reston, VA: NCTM, Principles and Standards for School Mathematics. Reston, VA: NCTM, Russel, Susan Jo, and Karen Economopoulos. Investigations in Number, Data, and Space. 2nd ed. Glenview, IL: Scott Foresman, Usiskin, Zalman. Van Hiele Levels and Achievement in Secondary School Geometry. van Hiele, Pierre M. Developing Geometric Thinking through Activities That Begin with Play. Teaching Children Mathematics 5 ( January 1999): Wiggins, G. P. Assessing Student Performance. San Francisco: Jossey-Bass Publishers, M. Lynn Breyfogle, lynn.breyfogle@bucknell.edu, is an associate professor in the mathematics department at Bucknell University in Lewisburg, Pennsylvania. She is interested in teachers professional development in the teaching and learning of mathematics. Courtney M. Lynch, lynch.courtneym@gmail.com, 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. 238 Mathematics Teaching in the Middle School Vol. 16, No. 4, November 2010

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