and then there was luke -...

What might students say about the

angles of the pentagonal block

shown in figure 1? Children

might respond in different ways, depending on their

abilities and experiences with angles. Some might

say that the block “has five angles” after touching

each of the corners. Others might observe that “it

looks like all the angles are the same size.” Perhaps

a few would respond as did Luke, a bright fifth

grader who is featured in this article. Luke

explained that each angle was 108 degrees.

You might wonder, How did Luke come to this con-

clusion? What else does Luke know about angles

and shapes, and how could we assess what he

knows? This article addresses these questions by

telling the story of how Luke’s teacher used a

specifically designed activity to assess his thinking

about angles and shapes. The van Hiele levels of

thinking provide a theoretical framework for ana-

lyzing Luke’s thinking and for offering suggestions

for instruction in geometry. The article concludes

with a discussion of how Luke and his teacher

became partners in studying geometry. This exami-

nation is grounded in the student’s words, writings,

and drawings as he worked to resolve geometric

tasks involving angle measurement. Although this

article discusses a gifted fifth grader to illustrate the

use of the van Hiele theory, the theory can be used

to analyze the thinking of, and plan instruction for,

any student in any grade.

432 TEACHING CHILDREN MATHEMATICS

Janet M. Sharp and Karen Bush Hoiberg

Janet Sharp, [email protected], teaches at Iowa State University, Ames, IA 50011. She isinterested in developing innovative and effective approaches for teaching mathematics. KarenHoiberg, [email protected], teaches fifth grade at Fellows Elementary School,Ames, IA 50014. She has received the Presidential Award for Excellence in Science and Math-ematics Teaching.

And Then ThereWas Luke:

The Geometric Thinking of a Young Mathematician

Copyright © 2001 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.

The van Hiele Theoryand Angle MeasurementThe van Hiele theory (1986) stipulates that withappropriate instruction, learners can pass throughfive distinct levels of geometric thinking. A student’sgeometric thinking may be at any level of vanHiele’s model for any given concept and is not agedependent. Here, we consider a student’s responseto a task about angle measurement, during which hedemonstrates thinking and reasoning at each of thefirst three levels. The first three levels of the vanHeile theory are briefly described in figure 2.

The activity in sheet 1is designed to stimulatelevel 2 thinking about angle measurement. Stu-dents determine the interior angle measurement indegrees for six polygons, without using a protrac-tor but with access to Pentablocks (see fig. 1).Pentablocks come in six shapes and can be used tohelp students develop geometry concepts (Berman,Plummer, and Scheuer 1998).

Luke’s teacher, Mrs. Hoiberg, used this specifi-cally designed task in a clinical assessment as partof an assignment for a course that she was takingon teaching geometry. Her task was to select atopic and choose a task to assess the thinking ofone of her students on that topic. Mrs. Hoibergused probes, comments, and questions to draw outLuke’s reasoning and to develop a clear picture ofhis thinking in terms of the van Hiele levels.

Luke, theMathematicianLuke is an exceptional, well-rounded fifth-gradestudent and a gifted child mathematician.Hoiberg’s characterization of Luke resembles thatof a budding Renaissance man:

Luke is a writer. Luke is a mathematician. Luke is a reader.Luke is an athlete. Luke is a musician. Luke is well liked.Luke is funny. Luke loves learning. Luke is self-motivated.Luke challenges himself in every area every day! Luke isone in a million. I was fortunate to be his fifth-grade teacher.

Before his fifth-grade year, Luke attended aSuper Summer mathematics program. There, hestudied a variety of mathematical topics with Dr.Leslie Hogben, a mathematician who taught in theprogram. With her, Luke investigated geometricideas, such as tessellations, Platonic solids, andconstructions in the plane. In so doing, he discov-ered ideas about angle measurement, such as justi-fying triangle congruence, comparing angles in tes-sellating figures, and explaining why the angle sumof a triangle is 180 degrees. During fifth grade,Luke and his classmates applied angle-measure-ment ideas in map contexts in which they esti-

mated, compared, measured, added, and subtractedangle measures. The analysis in the following para-graphs notes the instances when Luke is buildingon these past mathematical experiences, formulat-ing his own arguments, or engaging in some com-bination of the two.

Analysis ofLuke’s ThinkingAt first, Luke was challenged when Hoiberg pre-sented the activity in sheet 1:“You can’t do thatwithout a protractor!” Then Luke recalled an ideathat he had discovered while working with Hog-ben: “I think if you take a polygon and add up thesides and subtract 2, then multiply by 180, you getthe amount of all the angles inside that polygon.”For the first polygon, which was a regular penta-gon, he wrote these equations:

5 – 2 = 3 180 × 3 = 540 540 ÷ 5 = 108

He then carefully recorded 108 degrees in each ofthe angles of the pentagon. When asked how heknew that all interior angles were congruent, at firsthe said, “They all look the same.” Hoiberg urgedhim to reconsider this level 0 answer: “But how canyou be certain?” She suspected that Luke couldgive a more sophisticated explanation than that“they look the same.” For children who are at level0, the level of visual thinking, “looking the same”is sufficient reasoning for congruence.

Luke recognized his teacher’s question as anindication that “looking the same” was an insuffi-cient rationale. He physically placed a pentagonblock on the drawing and rotated it five times to

433MARCH 2001

FIG

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E 1 The Pentablock set

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empirically justify his claim that the shape wasequiangular. Luke explained this answer at ahigher van Hiele level than before, level 1, afterpurposeful prodding from his teacher.

Then Hoiberg steered the conversation back tothe rule that Luke had used, (n – 2) × 180°, andasked him, “Why does that property work?” Tojustify this rule, Luke gave an explanation for theangle sum of a pentagon. He carefully tracedaround a pentagon block, completed the pictureshown in figure 3, and stated that the sum of thecentral angles was 360 degrees. He had learnedwhile studying with Hogben that the sum of anglesabout a point is 360 degrees. Then he found that

360° ÷ 5 = 72°, which he recorded as the measureof each central angle. Next he decided that thebase angles of each of the five triangles were con-gruent. Luke applied a known Euclidean notion,that the angle sum for a triangle is 180 degrees, toground his subsequent deductive argument that thesum of the two base angles in each triangle was108 degrees, or 180° – 72° = 108°. Next he con-cluded that because each of the base angles of thefive triangles was congruent, the measure of anangle of the pentagon would be equal to the sumof the measures of any two base angles from thetriangles. That sum also equaled 108 degrees.Then he found that the sum of the interior anglesof the pentagon is 180 × 5, or 540, degrees. He didnot offer a generalization of this result for all pen-tagons. Perhaps an appropriate question wouldhave stimulated him to give a level 2 justificationfor the generalization.

As Luke talked through, and wrote about, thework shown in figure 3, he made a generalizationthat was an example of level 2 thinking: “Theangle sum of any polygon would be equal to 360degrees less [than 180 degrees times the numberof sides of the polygon].” Luke’s thinking can becharacterized by a formula: (n × 180°) – 360°,where n is the number of sides of the polygon.Note that Luke did not give this exact formula; itis offered to clarify his words. After Luke madehis comment, his eyes lit up. He realized that hehad not explained the formula he originallyrecalled: (n – 2) × 180°.

Because the two formulas are algebraically

FIG

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E 2 Characteristics of the first three levels of thinking in the van Hiele model

Level General Description Angle Measurement Examples

0 “Student identifies, names, Decides an angle’s sizeVisual compares, and operates on on the basis of “how it looks.”

geometric figures according Compares angles visually toto their appearance.” determine size.

1 “Student analyzes figures in Uses protractors to measure;Descriptive terms of their components and discovers angle relationships,

relationships among components such as that “vertical angles and discovers properties/rules have equal measure.”of a class of shapes empirically.”

2 “Student logically interrelates Uses properties of parallel linesInformal previously discovered to explain “why the angle sum deductive properties/rules by giving or of a quadrilateral is 360

following informal arguments.” degrees” or “why oppositeangles of a parallelogram are congruent.”

(Adapted from Fuys, Geddes, and Tischler [1988, p. 5])

FIG

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E 3 Luke’s first picture to demonstrate the

formula

435MARCH 2001

From Teaching Children Mathematics, March 2001

Pentablock Activity Sheet 1

Find the measures of the interior angles of each polygon without using a protractor.

Name of yellow shape ______________

Name of green shape ______________

Name of pink shape _______________

Name of purple shape ______________

Name of black shape _______________

Name of white shape ______________

(Adapted from Berman, Plummer, and Scheuer [1998])


equivalent, they give the same result. The explana-tions for the formulas, however, are significantlydifferent because each equation requires differentprocedures and different figures to accompanythose explanations. Luke’s attention to the mis-match between the explanation describing hiswork with the pentagon in figure 3 and his for-mula, (n – 2) × 180°, is evidence of level 2 think-ing. Luke’s perceptive observation that his expla-nation did not match his rule exemplifies thecharacteristics of a mathematician.

Undaunted by the lack of fit between the for-mula and his drawing, Luke drew another picture,shown in figure 4. He placed the three appropriatetriangle Pentablocks in the corresponding spacesof the pentagon shape in his drawing and wrote180 × 3 = 540. He stated that the polygon wouldalways have two fewer triangles than the number

of its sides. Luke drew and referred to a regularnine-sided polygon, shown in figure 5 (which isnot in the Pentablock set), to verify his claim.Luke explained:

I did a nine-gon. This is how I did it: 9 – 2 = 7, 7 times180 is 1260; 1260 divided by 9 is 140. So when I made mynine-gon, I measured each angle to be 140 degrees and itworked out perfectly. You can use this theory with anyshape.

Luke then returned to his explanation of the for-mula that he had recalled at the beginning of theassessment:

To do the thing to find the total of the . . . angles, there is areason for having to subtract 2. It happens when you do thetriangle way to find it. [See fig. 4.] There are the same num-ber of vertices as sides, so I’ll talk about vertices. There isone [vertex] that you start from. Then there are the two thatare next to it (that can’t be used). This would mean that youneed to subtract 3, but when you make the last line, youmake two triangles, so you only need to subtract 2.

Here, Luke demonstrated correct if-then reason-ing, further demonstrating his level 2 thinkingwhen he concluded, “So you only need to sub-tract 2.” Luke continued, using his nine-sidedpolygon to demonstrate:

You can only divide the [angle sum] total to find the angleof each side if the shape is regular. You can find the totalof the . . . angles of an irregular polygon, but all the anglesprobably won’t be the same.

Luke paid careful attention to whether his state-ments could be generalized. He correctly did nottry to find the measure of each angle of irregularshapes by dividing the angle sum by the numberof sides. This recognition of what notions can begeneralized and when those generalizationsapply is clearly level 2 thinking.

FIG

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E 4 (a) Luke’s second picture to demonstrate the formula and (b) the Pentablocks

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(a) (b)

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E 5 Luke’s nine-sided polygon

In general, Luke seemed to move easilybetween thinking with diagrams and recordingwith symbols. He distinguished between the twomethods as verification strategies. Rather thandraw n – 2 triangles on a nonconvex figure, Lukeopted instead to accept the known formula:“[Drawing] the triangle thing won’t work with anonconvex figure, but the side-minus-2-times-180will.” He knew the shortcomings of using a dia-gram for showing the n – 2 triangles on nonconvexpolygons. He completely abandoned the first dia-gram (fig. 3), in which he drew n triangles meetingat a central point, because it did not match hisexplanation for the formula (n – 2) × 180°.

Luke explained that his n – 2 triangle diagramwould not work for a nonconvex figure, becausewhen all vertices are joined to a given vertex,some of the segments fall outside the shape. Lukepointed to the two vertices of the decagon onsheet 1,which, when joined, would produce aline segment falling outside the shape. He wasreluctant to record this effort on his drawing. Hehad already stated that drawing the segmentswould not work, and he appeared unwilling tospend any more time with the task. He was con-fident in his use of the formula and clearly under-stood the difference between an example of, andan explanation for, the formula. Perhaps the abil-ity to recognize that examples are insufficient asexplanations is one of Luke’s greatest skills as agifted young mathematician.

After explaining his work with the formula,Luke returned to the task of finding the angle mea-surement of the remaining shapes on sheet 1.Hiswork with the familiar quadrilaterals and triangleswas fairly uneventful. For the green rhombus, hereturned to the pentagon block. He found that eachof the pentagon’s angles was congruent to thelarger angle of the rhombus; therefore, he recorded108 degrees for that angle and the opposite angle.Then he subtracted 360° – 216° = 144° andrecorded 72 degrees for the other two angles. Toarrive at his answers for the smaller angle mea-surement, Luke used the ideas that (a) the anglesum for a quadrilateral was 360 degrees and (b) ina parallelogram, opposite angles are congruent. Heknew these facts from his previous work with Hog-ben. His fluent use of these two ideas to deduce theangle measurements of the two smaller anglesagain represents level 2 thinking.

Luke continued using known angles, alongwith the facts that the angle sum of a triangle is180 degrees and that the angle sum of a quadri-lateral is 360 degrees, to find the remaining anglemeasurements for the pink rhombus and the twotriangles. Hoiberg asked, “Why is the angle sumfor a quadrilateral 360 degrees?” Luke’s some-

what amused facial expression (see fig. 6)accompanied his answer: “There are two trian-gles inside a quadrilateral, so there are two times180 degrees inside a quadrilateral.”

When he encountered the nonconvex decagon(see sheet 1), Luke was momentarily challenged.Because he had already stated that he did not wantto draw his “triangle thing” on a nonconvex figure,he used another approach. He stacked blocks toshow that the acute angle of the decagon was con-gruent to the acute angle of the pink rhombus,which he had found to measure 36 degrees. Hethen said that he knew the decagon’s angle wasalso 36 degrees. He tried to determine the anglemeasurement of the decagon’s larger interior angleby using symbolic numeral manipulation; he sub-tracted 180° – 36° = 144°, but he could not decide

437MARCH 2001

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E 6 Luke explains the angle sum of a quadrilateral.

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E 7 Luke’s strategy to determine the exteriorangle of the decagon

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360˚– 180˚ 180˚


what to do with the result. Eventually, he placedthe pentagon block flush against the exterior angleof the decagon, as shown in fig. 7. When Luke’ssymbolic manipulation was not meaningful tohim, he readily used the blocks to focus his think-ing or to verify his measurements or claims ofcongruence.

Luke then calculated that the large interiorangle was 360° – 108°, or 252°. His use of thephysical blocks to measure the angles markedLuke’s voluntary return to level 1 activity whenhe needed to verify his claims before he couldmove forward. The van Heile theory claims that alearner must experience visual recording (level 0)before he or she is able to formulate descriptionsat level 1 or engage in informal deduction at level2 in completing geometric tasks. Luke’s thinkingprocesses outlined in this article seem to supportthis theory.

Teacher and Student:Partners in LearningHoiberg is locally and nationally recognized for herexcellence in teaching mathematics and for regularlyproviding challenges for all her fifth-grade students.In Luke, she recognized an excellent learner. Sheworked diligently to give him opportunities thatwould challenge him to higher levels of thinking andpush him toward becoming a mathematician. Attimes, Hoiberg worried about her ability to presentLuke with situations or ask him questions that wouldlead to deeper mathematical thinking. Her concernsare elaborated in this description:

Before he entered my classroom, I knew Luke. I knew of

Luke’s family. His sister had been one of my students. I knewhe was extremely bright. I worried about how I would chal-lenge him. I needn’t have worried. Luke loves learning andmakes challenges for himself.

Later in the school year, Luke’s teacher recog-nized an unusual opportunity to further nurtureLuke’s ability to think at higher levels.

I found another challenge when I signed up for a graduateclass in geometry learning and teaching at the local univer-sity. I asked Luke if he’d like to look over my work with me.He loved it! He often helped me with assignments. He lovedbeing my teacher! When it fit into his busy schedule, hecame to class with me. He enjoyed sharing his theories withmy fellow classmates.

One problem presented in the graduate classwas to use the computer program The Geometer’sSketchpad (1995) to find the areas of successivemidpoint quadrilaterals. Figure 8 shows an exam-ple of this process. When several teachers in theclass struggled with the problem, Luke clarified thewording of the problem, then solved it. He foundthat each of the successive midpoint quadrilateralswould be parallelograms and used the area tool toconsider the areas of successive midpoint quadri-laterals. He summarized the investigation byobserving, “I think that the area is cut in half eachtime that you make another polygon inside the oldone. It’s like shrinking by the powers of 2.”

Luke’s empirical discovery of this area relation-ship is an example of his level 1 thinking about thenew topic. He did not try to explain why his obser-vation was true (in part, because he was late forsoccer practice). If we had examined simplershapes, such as squares or rectangles, Luke mighthave been able to develop an informal explanationfor why the area of the midpoint figure is half thearea of the original (Welchman 2000).

Hoiberg was willing to risk allowing Luke to beher teacher, and Luke responded to the challenge.Conventional wisdom says that the best mathemat-ics teachers are those who can push their studentsto gain knowledge beyond even that which is heldby the teacher. This statement would seem particu-larly pertinent to teachers who work with giftedstudents. Luke’s teacher pushed Luke toward thehighest van Hiele levels, even though she wasachieving some of those levels at the same time.

Luke’s teacher used the van Hiele learning the-ory to pose questions and to interpret Luke’sanswers. She used this information to provide Lukewith meaningful learning activities. Recognizinglower-level answers, she continued to ask ques-tions that pushed Luke toward higher-level reason-ing. She asked him to work alongside her as shecompleted her geometry assignments, whichincluded writing proofs. In turn, Luke offeredsound reasoning about his geometric ideas and

FIG

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E 8 An example of the midpoint-quadrilateral problem

taught his teacher about geometry; he also showedher how to animate the mathematician waitinginside a gifted child.

Implications forTeachingAlthough we have focused on using the van Hieletheory to stimulate the mathematical developmentof a gifted fifth grader, we want to stress that thevan Hiele theory can be used with students of allages and all ability levels. The kinds of tasks andquestioning techniques used in assessing Luke canbe incorporated into instruction to foster students’advancement through the levels of thinking. Thetask of finding the measurements of the interiorangles of the Pentablocks releases the learner fromreliance on protractors. In turn, the task gives theteacher deeper insight into the learner’s reasoningabout angle measurements, and this insight can beused to plan subsequent instruction.

The van Hiele theory provides the teacher witha tool to recognize the level of a student’s thinkingand to facilitate a student’s advancement to succes-sive levels by asking questions at the next higherlevel. For instance, when Luke said that he knewthe interior angles of the pentagon were congruentbecause “they all look the same,” the teacher prod-ded Luke forward. She continued to ask questionsthat encouraged Luke to use level 1, descriptive,thinking. She used his words or numbers to poseher questions, for example, “How did you knowthis angle was 108 degrees?” or “Why did you sub-tract 216 from 360?” At times, she asked him toexplain how he knew that mathematics allowedhim to make certain claims. At other times, sheurged him to make generalizations or asked him todevelop counterexamples: “Does this work for anypolygon? What about an n-sided polygon? Whatabout a nonconvex polygon?”

Encouraging children to answer questions atthe higher levels of the van Hiele model requirescareful thought. While some classmates aredescribing properties of shapes, using level 1thinking, gifted children might be pressed toclassify a set of triangles on the basis of theseproperties or to deduce ideas and relationshipsabout the triangles. These children may be fur-ther pressed to develop their own definitions of atriangle or to deduce relationships between a tri-angle and a quadrilateral.

Teachers must select geometrically worth-while tasks and problems, such as those on sheet1, that have the potential to foster higher levels ofthinking. Teachers can develop collections ofworthwhile tasks for exceptional children, per-haps using them for special individual or small-

group explorations. Another possibility is toassemble a learning center that includes somebasic tasks, along with challenges. Resources forcenter activities include Geometry and SpatialSense(Leiva 1993); Geometry in the MiddleGrades (Geddes 1992); Shapes Alive!(Leeson1993); and Geometry and Geometric Thinking,afocus issue ofTeaching Children Mathematics(NCTM 1999). From the moment that each childenters a classroom, the teacher should begin lis-tening for, and recording clues to, the child’s vanHiele level. The teacher can then design geome-try tasks to match the student’s level and eventu-ally raise the child’s thinking to a higher level.

Working with an exceptional child such as Lukecan make a teacher aware of the special mathemat-ical potential in certain children and can help theteacher observe more carefully the mathematicalpotential in all students. Readers are encouraged tointerview several of their students to determinetheir van Hiele levels of thinking about a certainconcept, then to use the students’ answers to designand develop questions at or above those levels.

ReferencesBerman, Sheldon, Gary A. Plummer, and Don Scheuer. Investi-

gating Mathematics with Pentablocks.White Plains, N.Y.:Cuisenaire Co. of America, 1998.

Fuys, David, Dorothy Geddes, and Rosamond Tischler. The vanHiele Model of Thinking in Geometry among Adolescents.Journal for Research in Mathematics Education MonographSeries No. 3. Reston, Va.: National Council of Teachers ofMathematics, 1988.

Geddes, Dorothy. Geometry in the Middle Grades. Curriculumand Evaluation Standards Addenda Series, Grades 5–8.Reston, Va.: National Council of Teachers of Mathematics,1992.

The Geometer’s Sketchpad. Berkeley, Calif.: Key CurriculumPress, 1995. Software.

Leeson, Neville. Shapes Alive! Lincolnshire, Ill.: LearningResources, 1993.

Leiva, Miriam A., ed. Geometry and Spatial Sense. Curriculumand Evaluation Standards Addenda Series, Grades K–6.Reston, Va.: National Council of Teachers of Mathematics,1993.

National Council of Teachers of Mathematics. “Geometry andGeometric Thinking” (Focus Issue). Teaching ChildrenMathematics5 (February 1999).

van Hiele, Pierre. Structure and Insight: A Theory of Mathe-matics Education.Orlando, Fla.: Academic Press, 1986.

Welchman, Rosamond. “Investigations: Midpoint Shapes.”Teaching Children Mathematics6 (April 2000): 506–9.

This manuscript was developed, in part, as a resultof support from the Exxon Education Foundation.The authors would like to thank Gary Talsma forhis tremendously helpful comments in the manu-script’s completion. His careful editorial remarkswere extraordinarily perceptive and immeasurablybeneficial. ▲

439MARCH 2001

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