references adams, r. j. & khoo, s. (1993). quest - the interactive test analysis ... · 2019. 3....

REFERENCES

Adams, R. J. & Khoo, S. (1993). Quest - The Interactive Test Analysis System. Hawthorn,

Victoria: ACER.

Biggs, J. & Collis, K. (1991). Multimodal learning and the quality of intelligent behaviour. In

H. Rowe (Ed.), Intelligence, Reconcertualization and measurement(pp.57-76).New

Jersey: Erlbaum.

Board of Secondary Education NSW (1988). Syllabus Years 7-8. Sydney: Board of

Secondary Education.

Bobango, J. C. (1987). Van Hiele Levels of Geometric Thought and Student Achievement in

Standard Content and Proof Writing: The Effect of Phase-based Instruction. Doctoral

Thesis, The Pennsylvania State University.

Burger, W. F. & Shaughnessy, J. M. (1986). Characterizing the van Hiele levels of

development in geometry. Journal for Research in Mathematics Education, 17(1),

31-48.

Chaiyasang, S. (1988). An Investigation into Level of Geometric Thinking and Ability to

Construct Proof of Students in Thailand. Doctoral Thesis, Graduate College of The

University of Iowa.

Clements, D. H. & Battista, M. T. (1989). Learning of geometric concepts in a Logo

environment. Journal for Research in Mathematics Education, 20, 450-467.

Clements, D. H. & Battista, M. T. (1991). Van Hiele levels of learning geometry. In F.

Furinghetti (Ed.), Proceedings of the fifteenth annual conference of The International

Group for the Psychology of Mathematics Education, 2, (pp.223-230). Assisi:

Program Committee.

Clements, D. H. & Battista, M. T. (1992). Geometry and spatial reasoning. In D. Grouws

(Ed.), Handbook of Research on Mathematics Teaching and Learning: A project of the

National Council of Teachers of Mathematics, (pp.420-464). New York: Macmillan.

216

Coxford, A. F. (1978). Research directions in geometry. In R. Lesh & D. Mierkiewicz

(Eds.), Recent research concerning the development of spatial and geometric concepts

(323-331). Colombus, Ohio: ERIC/SMEAC.

Crowley, M. L. (1987). The van Hiele model of the development of geometric thought. In M.

M. Lindquist & A. P Shulte (Eds.), Leirning and Teaching Geometry, K-12 (1-16).

Virginia: NCTM.

Crowley, M. L. (1990). Criterion referenced reliability indices associated with the van Hiele

geometry test. Journal for Research ir, Mathematics Education, 21, 238-241.

Davey, 0. & Pegg, J. (1989). Relating of common 2-D shapes to underlying geometric

concepts. Paper presented at the twelfth annual conference of the Mathematics

Education Research Group of Australasia, Bathurst, NSW.

Davey, 0. & Pegg, J. (1990). Students' understanding of parallelism. Paper presented at the

thirteenth annual conference of the Mathematics Education Research Group of

Australasia, University of Hobart, Tasmania.

Davey, G. & Pegg, J. (1991). Angles on angl es: students' perceptions. Paper presented at the

fourteenth annual conference of the Mahematics Education Research Group of

Australasia, University of Western Auitralia, Perth.

Davey, G. & Pegg, J. (1992). Research in geometry and measurement. In B. Atwell & J.

Watson (Eds.), Research in Mathemati Education in Australasia 1988-1991.

Brisbane: MERGA.

De Block-Docq, C. (1992). Analyse tpistemologique Comparative de Deux Enseignements de

la Geometrie Plane Vers l'Age de Douze Ans. Doctoral Thesis, UCL, Louvain-la-

Neuve..

De Block-Docq, C. (1994). Modalites de la Nnsee mathematique d'eleves de douze ans devant

des problemes de pavages (Forms of mathematical thought of twelve years old students

at tiling problems). Educational Studir?s in Mathematics, 27, 165-189.

217

Denis, L. P. (1987). Relationships Between Stage of Cognitive Development and van Hiele

Level of Geometric Thought Among Puerto Rican Adolescents. Doctoral Dissertation,

Fordham University, New York.

De Villiers, M. D. (1987). Research evidence on hierarchical thinking teaching strategies and

the van. Hiele theory: some critical comments. Document presented to the Working

Conference on Geometry, Syracuse University, New York.

De Villiers, M. D. & Njisane, R. M. (1987). The development of geometric thinking among

black high school pupils in Ksazulu (Republic of South Africa). In J. Bergeron, N.

Herscovics & C. Kieran (Eds.), Proceedings of the eleventh annual conference of The

International Group for the Psycholog y of Mathematics Education, 2 (pp.117-123).

Montreal: Program Committee.

Dick, 0. L. (Ed.) (1962/1949). Aubrey's brief lives. Mitcham, Victoria: Penguin Books.

Freudenthal, H. (1973). Mathematics as an Educational Task. Dordrecht, Holland: Reidel.

Fuys, D. J. (1982). Van Hiele levels of thinking of sixth graders. In A. Vermandel (Ed.),

Proceedings of the sixth annual conference of The International Group for the

Psychology of Mathematics Education, (pp.113-119). Antwerp: Program Committee.

Fuys, D. J. (1985). Van Hiele levels of thinking in geometry. Education and Urban Society,

17, 447-462.

Fuys, D., Geddes, D. & Tischler, R. (1984). English Translation of Selected Writings of Dina

van Hiele-Geldof and Pierre M. van Hiele. New York: Brooklyn College.

Fuys, D., Geddes, D. & Tischler, R. (1985). An Investigation of the van Hiele Model of

Thinking in Geometry among Adolescents. New York: Brooklyn College.

Fuys, D., Geddes, D. & Tischler, R. (1988). The van Hiele model of thinking in geometry

among adolescents. Journal for Resea,-ch in Mathematics Education Monograph, 3.

218

Gutierrez, A. & Jaime, A. (1987). Study of the characteristics of the van Hide levels. In J.

Bergeron, N. Herscovics & C. Kieran (Eds.), Proceedings of the eleventh annual

conference of The International Group for the Psychology of Mathematics Education, 3

(pp.131-137). Montreal: Program Committee.

Gutierrez, A., Jaime, A. & Fortuny, J. (1991) An alternative paradigm to evaluate the

acquisition of the van Hide levels. Jot'rnal for Research in Mathematics Education,

22(3), 237-251.

Gutierrez, A., Jaime, A., Shaughnessy, J. M. & Burger, W. F. (1991). A comparative

analysis of two ways of assessing the van Hiele levels of thinking. In F. Furinghetti

(Ed.), Proceedings of the fifteenth annual conference of The International Group for the

Psychology of Mathematics Education, 2, (pp.109-116). Assisi: Program Committee.

Hoffer, A. (1981). Geometry is more than proof. Mathematics Teacher, 74, 11-18.

Hoffer, A. (1983). Van Hiele based research. In R. Lesh & M. Landau (Eds.), Acquisition of

mathematics concepts and processes, (pp.205-227). New York: Academic Press.

Lawrie, C. J. (1993). Some problems identified with Mayberry test items in assessing

students' van Hiele levels. In B. Atweh, C. Kanes, M. Carss & G. Booker (Eds.),

Proceedings of the sixteenth annual conference of the Mathematics Education Research

Group of Australasia, (pp.381-386). Brisbane: MERGA.

Lawrie, C. J. (1996). First year teacher trainees' understanding of geometry. In P. Clarkson

(Ed.), Proceedings of the nineteenth annual conference of the Mathematics Education

Research Group of Australasia, (pp.345-351). Melbourne: MERGA.

Lawrie, C. J. (1997). An evaluation of two coding systems in determining van Hiele levels.

In F. Biddulf & K. Carr (Eds.), Proceedings of the twentieth annual conference of the

Mathematics Education Research Group of Australasia, (pp.294-301). Rotorua:

MERGA.

Lawrie, C. J. (in press). An alternative assessment: the Gutierrez, Jaime and Fortuny

technique. Proceedings of the twenty-second annual conference of The International

Group for the Psychology of Mathematics Education . Stellenbosch: Program

Committee.

219

Lawrie, C. J. & Pegg, J. (1997). Some issues in using Mayberry's test to identify van Hiele

levels. In E. Pehkonen (Ed.), Proceedings of the twenty-first annual conference of The

International Group for the Psycholog y of Mathematics Education 3, (pp.184-191).

Helsinki: Program Committee.

Malan, F. R. P. (1986). Teaching strategies for the transition of partition thinking to

hierarchical thinking in the classification of quadrilaterals: some case studies. (Internal

Report No. 3). Stellenbosch: University of Stellenbosch, Research Unit for

Mathematics Education.

Mason, M. (1989). Geometric understanding and misconceptions among gifted fourth-eighth

graders. Paper presented at the annual meting of the American Educational. Research

Association, San Francisco.

Masters, G. N. (1982). A Rasch model for partial credit scoring. Psychometrika, 47(2),

149-174.

Mayberry, J. W. (1981). An investigation of the van Hiele levels of geometric thought in

undergraduate preservice teachers. Doctoral Thesis, University of Georgia.

Mayberry, J. W. (1983). The van Hiele levels of geometric thought in undergraduate

preservice teachers. Journal for Research in Mathematics Education, 14(1)., 58-69.

Mitchelmore, M. C. (1993). Abstracting the angle concept. In B. Atweh, C. Kanes, M. Carss

& G. Booker (Eds.), Proceedings of the sixteenth annual conference of the Mathematics

Education Research Group of Australasia, (pp.401-406). Brisbane: MERGA.

Nasser, L. (1990). Children's understanding of congruence according to the van Hiele model

of thinking. In G. Booker, P. Cobb & T. N. de Mendicuti (Eds.), Proceedings of the

fourteenth annual conference of The International Group for the Psychology of

Mathematics Education, 2 (pp.297-303). Mexico: Program Committee.

Newman, M. A. (1977). An analysis of sixth-grade pupils' errors on written mathematical

tasks. In M. A. Clements & J. Foyster (Eds.), Research in Mathematics Education in

Australia, Melbourne, 1, 239-258.

220

NSW Department of Education (1989). Mathematics K-6. Sydney: NSW Department of

Education.

Olive, J. & Scally, S. P. (1982). Teaching and understanding geometric relationships through

Logo. In A. Vermandel (Ed.), Proceedings of the sixth annual conference of The

International Group for the Psycholog y of Mathematics Education, (pp.120-126).

Antwerp: Program Committee.

Pegg, J. (1992). Students' understanding of geometry: theoretical perspectives. In B.

Southwell, B. Perry & K Owens (Eds.), Proceedings of the fifteenth annual conference

of the Mathematics Education Research Group of Australasia, (pp.18-35). University

of Western Sydney, Sydney: MERGA.

Pegg, J. & Davey, G. (1989). Clarifying level descriptions for children's understanding of

some basic 2D geometric shapes. Mathematics Education Research Journal, 1(1),

16-27.

Pegg, J. & Davey, G. (in press). A synthesis of two models: Interpreting student

understanding in geometry. In R. Lehrer & C. Chazan, (Eds), New Directions for

Teaching and Learning Geometry. New Jersey: L. Earlbum.

Piaget, J. (1955). Les structures mathematique et les structures operatoires de L'intelligence.

L'enseignement des mathematiques. Neuchatel: Delachaux et Niestld, 11-34.

Piaget, J. & Inhelder, B. (1967). The child's conception of space. New York: Norton.

Pyshkalo, A. M. (1968). Geometriya v I-IV klassakh (Geometry in grades I-IV). In A.

Hoffer (Ed.) and I. Wirszup (Trans.), Problems in the formation of geometric ideas in

primary-grade school children (2nd ed.). Moscow: Prosveschchenle.

Schoenfeld, A. H. (1986). On having and using geometric knowledge. In J. Hiebert (Ed.),

Conceptual and procedural knowledge: the case of mathematics (pp.225-264).

Hillsdale, New Jersey: Erlbaum.

Secondary School Board (1982). Notes on the Syllabus in Mathematics, Years 9 and 10.

North Sydney NSW: Secondary Schools Board.

221

Secondary School Board (1983). Syllabus in Mathematics, Years 9 and 10. North Sydney

NSW: Secondary Schools Board.

Senk, S. L. (1982). Achievement in writing geometry proofs. Paper presented at the annual

meeting of the American Educational Research Association, New York.

Senk, S. L. (1989). Van Hiele levels and achievement in writing geometry proofs. Journal for

Research in Mathematics Education, 20(3), 309-321.

Shaughnessy, J. M. & Burger, W. F. (1985). Spadework prior to deduction in geometry.

Mathematics Teacher, 78(6), 419-428.

Silfverberg, H. (1984). Study of learning concepts related to triangles and quadrangles on the

basis van Hiele's theory. Doctoral Thesis, University of Tampere.

Smith, E. C. (1989). A comparative study of two van Hiele testing instruments. Poster

presented at the thirteenth annual conference of The International Group for the

Psychology of Mathematics Education, Paris.

Steffe, L. & Cobb, P. (1988). Construction of arithmetical meanings and strategies. New

York: Springer-Verlag. In Clements, D. H. & Battista, M. T. (1992). Geometry and

spatial reasoning. In D. Grouws (Ed.), Handbook of Research on Mathematics

Teaching and Learning: A project of the National Council of Teachers of Mathematics,

(pp.420-464). New York: Macmillan.

Teppo, A. (1991). Van Hiele levels of geometric thought revisited. Mathematics Teacher,

March 1991, 210-221.

Torgenson, W. S. (1967). Theory and methods of scaling. New York: Wiley.

University of New England Handbook (1992',. Armidale, NSW: Publications Office,

University of New England.

Usiskin, Z. (1982). Van Hiele levels and ach ievement in secondary school geometry.

Chicago: University of Chicago.

222

Usiskin, Z.& Senk, S. L. (1990). Evaluating a test of van Hiele levels: a response to Crowley

and Wilson. Journal for Research in Mathematics Education, 21, 242-245.

van Hiele, P. M. (1955). De niveau's in het denken, welke van belang zijn bij het onderwijs in

de meetkunde in de eerste klasse van Let V.H.M.O. In Pedagogische Studien, XXXII

(pp.289-297). Groningen: J. B. Wolters. In van Hiele, P. M., Structure and insight: a

theory of mathematics education. Orlando, Florida: Academic Press.

van Hiele, P. M. (1957). De problematick van het inzicht, gedemonstreed aan het inzicht van

schoolkinderen in metkunde-leerstof (The problem of insight in connection with school

children's insight into the subject matter of geometry). Summary of Doctoral 'Thesis,

Utrecht. In Fuys, D., Geddes, D. & Tischler, R. (Eds.) (1984), English Translation of

Selected Writings of Dina van Hiele-Geldof and Pierre M. van Hiele (pp.237-241).

New York: Brooklyn College.

van Hiele. P. M. (1959). La pens& de l'enfant et la geometrie (The child's thought and

geometry). In Fuys, D., Geddes, D. 8: Tischler, R. (Eds.) (1984), English Translation

of Selected Writings of Dina van Hiele-Geldof and Pierre M. van Hiele (pp.243-252).

New York: Brooklyn College.

van Hiele. P. M. (1986). Structure and insight: a theory of mathematics education. Orlando,

Florida: Academic Press.

van Hiele. P. M., & van Hiele-Geldof, D. (1958). A method of initiation into geometry. In

H. Freudenthal (Ed.), Report on methods of initiation into geometry. Groningen:

Wolters.

van Hiele-Geldof, D. (1957). De didaktiek van de meetkunde in de eerste klas van het

V.H.M.O (The didactics of geometry in the lowest class of secondary school). Doctoral

Thesis., Utrecht. In Fuys, D., Geddes D. & Tischler, R. (Eds.) (1984), English

Translation of Selected Writings of Dira van Hiele-Geldof and Pierre M. van Hiele

(pp.1-214). New York: Brooklyn College.

van Hiele-Geldof, D. (1958). Didactics of geometry as learning process for adults. In Fuys,

D., Geddes, D. & Tischler, R. (Eds.) 4 1984)., English Translation of Selected Writings

of Dina van Hiele-Geldof and Pierre M. van Hiele (pp.215-233). New York: Brooklyn

College.

223

Vygotsky, L. S. (1934/1986). Thought and language. Cambridge, MA: MIT Press.

Wilson, M. (1990). Measuring a van Hiele geometry sequence: a reanalysis. Journal for

Research in Mathematics Education, 21, 230-237.

Wilson, P. S. (1990). Inconsistent ideas related to definitions and examples. Focus on

Learning Problems in Mathematics, 12(3,4), 31-47.

Wirszup, I. (1976). Breakthroughs in the psychology of learning and teaching geometry. In

J. L. Martin & D. A. Bradbard (Eds.), Space and geometry: papers from a research

workshop, (pp.75-97). Athens, GA: University of Georgia, Georgia Centre for the

Study of Learning and Teaching Mathematics.

Appendix A

TEST ITEMS FOR PRELIMINARY AND MAIN STUDIES

Paper I contained Items 1, 2, 3, 5, 7, 9, 10, 12, 15, 16, 17, 19, 22, 23, 24, 25, 26, 27, 32, 33,34, 35, 42, 43, 44, 45, 46, 47, 48, 51, 52, 55, 56, and 57.

Paper II contained Items 1, 2, 4, 6, 8, 9,32, 36, 37, 38, 39, 40, 41, 42, 44, 45, 46,

11,48,

13,49,

14,50,

16,52,

18, 20, 21, 23,53, 54 and 58.

24, 25, 28, 29, 30, 31,

1.

These pairs of lines appear to meet at what kind of angle?

What is the word used to describe this relationship?

2.

This figure is which of the following?

a) triangle

b) quadrilateral

c) square

d) parallelogram

e) rectangle

3.

Are all of these triangles? YES NOExplain:

Do they appear to be a special kind of triangle.;?If so what kind?

225

4.

These appear to be what kind of triangles?

5.

Name this figure.

6.

Suppose these two lines will never meet no matter how far we draw them.

What word describes this?

7.

A

What is true of A and B? What is true of C a id D?


/7r

a

E

.7 7z

8.

VeAre these figures alike in any way? YES NO


9.

d

Which of these figures are squares?

List all of these figures which are rectangles.

e

b

10.

b

ad f

Which of these appear to be right triangles?

11.

cWhich of these figures appear to be isosceles triangles?

12.

a bWhich of these are circles?

7a b

14.

Which figure appears to be similar to a?

13.

a d

Which pair(s) of lines appear to be parallel?

15.*

< C

16.

Draw a square.What must be true about the sides? What must be true about the angles?

17.

Does a right triangle always have a longest side?If so, which one?

Does a right triangle always have a largest angle?If so, which one?

18.What can you tell me about the sides of an isosceles triangle?

What can you tell me about the angles of an isosceles triangle?

19.

This figure is a circle. 0 is the centre.

Name the following line segments.OB is a of the circle.OC is a of the circle.AC is a of the circle.

di d 2

20.

If d 1 = d 2, what if true about lines 11 and 12?

If d 1 � d 2, what is true about lines 1 1 and 12?

21.

Triangle ABC is similar to triangle DEF.

How long is ED? How do you know?

What is the size of L EDF? How do you know?

22.

1 24--

F

These are congruent figures.

What is true about their sides? AD =

What is true about their angles? LB =

231

23.

ABCD is a square, BD is a diagonal.

Name an angle congruent to Z ABD How do you know?

24.Circle the smallest combination of the following which guarantees a figure to be a square.

a. It is a parallelogram.b. It is a rectangle.c. It has right angles.d. Opposite sides are parallel.e. Adjacent sides are equal in length.f. Opposite sides are equal in length.

25.A. Name some ways in which squares an rectangles are alike?

B. Are all squares also rectangles? Why"

C. Are all rectangles also squares? Why"

26.Circle the smallest combination of the following which guarantees a figure to be a righttriangle?

a. It is a triangle.b. It has two acute angles.c. The measures of the angles add up to 180°.d. An altitude is also a side.e. The measures of two angles add up to 90°.

232

27.QAB is a triangle.

a) Suppose angle Q is a right angle. Does that tell you anything about angles A and B?If so, what?

b) Suppose angle Q is less than 90°. Could the triangle be a right triangle? Why?

c) Suppose angle Q is mote than 90°. Could the triangle be a right triangle? 'Why?

28.Circle the smallest combination of the following which guarantees a figure to be an isoscelestriangle?

a. It has two congruent angles.b. It is a triangle.c. It has two congruent sides.d. An altitude bisects the opposite side.e. The measure of the angles add up to 1

29.Give a definition of an isosceles triangle.

30.Suppose all we know about A MNP is that L M is the same size as Z N. What do you knowabout sides MP and NP? Suppose Z Mis larger than ZN.a) What do you know about MP and NP" b) Could A MNP be isosceles?

31.Triangle .DEF has three congruent sides. It is an isosceles triangle? Why or why not? Are the following true of false?a) All isosceles triangles are equilateral. b) All equilateral triangles are isosceles.

32.Which are true? Give reasons.a) All isosceles triangles are right triangles.b) Some right triangles are isosceles triangles.

233

33.

C) DC)

a

e

dTell why each of these figures is or is not a circle.a) b) c) d) e) Can you give a general rule to fit all the above answers?

34.

Figure A is a simple closed curve. Figure B is a circle.

Is figure B a simple closed curve? How are these figures alike? How are they di Terent?

(T--F) All simple closed curves are circles. (T—F) All circles are simple closed curves.

35.

This figure is a circle with centre 0.Would the following be: a) certain b) possible c) impossible.Give reasons for your answer.

1) OB =OA 2) OD = OA 3) 20B = AD 4) AD = EC

36.

Suppose Z1 and Z2 are congruent. What does that tell you about / 1 and / 2?

Suppose Z1 is larger than Z2. What does that tell you about / land / 2?

37.How do you recognise lines that are parallel?

38.Are these lines or line segments parallel?

a) always b) sometimes c: neverGive reasons for your answers.

A) Two lines which do not intersect.B) Two lines which are perpendicular to the same line. C) Two line segments in a square. D) Two line segments in a triangle. E) Two line segments which do not intersect.

235

39.Circle the smallest combination of the following which guarantee that two lines are parallel?

a) They are everywhere the same distance apart.b) They have no points in common.c) They are in the same plane.d) They never meet.

40.What does it mean to say that two figures are similar?

41.Triangle ABC is similar to triangle DEF (in that order).Are the following a) certain b) possible, orGive reasons for your answers.

a) AB =DE b) AB > DE c) LA= LE d) LA > LE e) AB = EF f) LA >

c) impossible?

42.Will figures A and B be similarGive reasons for your answers.

Aa) a squareb) an isosceles trianglec) a A congruent to Bd) a rectanglee) a rectangle

I-always II-sometimes or III-never?

B ANSWERa) a square b) an isosceles triangle c) a A congruent to A d) a square e) a ,riangle

43.A ABC is congruent to A DEF (in that order).Are the following a) certain b) possible, orGive reasons for your answers.

a) AB = DE b) LA= LE c) LA < LD d) AB = EF

c) impossible?

236

QB

44.Will figures A and B be congruent I-always II-sometimes or

III-never?

Give reasons for your answers.

a)b)c)

d)e)

Aa squarea square with a 10cm sidea right triangle with a10cm hypotenusea circle with 10cm chorda A similar to B

B ANSWERa triangle a square with a 10cm side a right triangle with a10,,;m hypotenuse a circle with 10cm chord a A similar to A

b)b)c)

d)e)

45.ABCD is a four sided figure. Suppose we know that opposite sides are parallel. What are thefewest facts necessary to prove that ABCD is a square?

46.Figure ABCD is a parallelogram, AB F.- BC alid /ABC is a right angle. Is ABCD a square?Prove your answer.

47.

D

CD is perpendicular to AB. LC is a right angle.If you measure ZACD and LB, you find that they have the same measure.Would this equality be true for all right triangles? Why or why not?

48.

Figures ABC and PQR are right isosceles triengles with angles B and Q being right angles.Prove that LA = LP and LC = L R.

237

49.

AZD

ABC is a triangle. A ADC A BDC.

a) What kind of triangle is A ABC? Why ?b) AD = BD. Why? c) CI) is perpendicular to AB. Why?

50.

AB is the line segment with A and B the midpoints of the equal sides of the isosceles triangleXYZ.AY = BY and A AYB is similar to AXYZ so L A = ZX and AB is parallel to XZ.

What have we proved?

51.

Figure 0 is a circle. 0 is the centre.AOB = LCOD, so AB = CD.


238

52.

Figure C is a circle. 0 is the centre.

Prove that A AOB is isosceles.

53.

Line / is parallel to AB.

Because of properties of parallel lines we can prove that L 1 = LA and L3 = LB.

Now, 1 is a straight angle (180°).


54.

Prove: If / 1 is parallel to 1 2 and 1 2 is parallel to 1 3, then / 1 is parallel to 1 3.

239

55.

In this figure AB and CB are the same length. AD and CD are the same length.

Will L A and LC be the same size? Why or why not?

56.

These circles with centres 0 and P intersect at M and N.

Prove: A OMP E A ONP.

57.Prove that the perpendicular from a point not on the line to the line is the shortest linesegment that can be drawn from the point to theline.

58. (This item was created after the preliminary study for use in the main study.)

Figures NINO and PQR are right triangles with LN and L Q being right angles. MO and PRare in the ratio a:b. What is the least addition al information needed to ensure that trianglesMNO and PQR are similar?

240

Appendix B

MARKING FORMATS, PRELIMINARY AND MAIN STUDIES(adapted from originals in Mayberry 1981, pp128-131)

PAPER I(preliminary and main studies)

Concept Level Question type Questionnumber

Score Questncriteria

Mark Levelcriteria

Totalscore

Square 1 Name 2 1Discriminate 9a 2 2/2=1 1 of 2

2 Properties 16 223a 1 2 of 3

3 Definition 24 1Class Inclusn 9b 4 3/4=1

25 3Relationships 42a 1

42d 144b 1

Implications 23b 1 6 of 94 Proof 45 1

46 1 1 of 2Right 1 Name 3 1

Triangle Discriminate 10 4 3/4=1 1 of 22 Properties 17 4 3 of 43 Definition 26 1

Implications 27 3Class Inclusn 32 2Relationships 44c 1 5 of 7

4 Proof 47 148 157 1 2 of 3

Circle 1 Name 5 1-- 1 of 2Discriminate 12 2 2/2=1 -

2 Properties 19 3 2/3=133 5 5 of 6

3 Definition 33 135 4

Relationships 44d 1Class Inclusn 34 4 6 of 10

4 Proof 51 152 1 loft

Congruency 1 Name 7 1Discriminate 15 1 1 of 2Pro • erties 22 4 ' 3 of 4

3 Relationships 43 4 deduct1 pointif missa and c

Implications 44 5Class Inclusn 42c 1 6 of 10

4 Proof 55 156 1 1 of 2 -

241

PAPER II(preliminary and main studies



Mark Levelcriteria

Totalscore

Square 1 Name 2 1Discriminate 9a 2 2/2=1 1 of 2

2 Properties 16 223a 1 2 of 3

3 Definition 24 1Class Inclusn 9b 4 3/4=1

25 3Relationships 42a 1

42d 144b 1

Implications 23b 1 6 of 94 Proof 45 1

46 1 1 of 2Isosceles 1 Name 4 1Triangle Discriminate 11 2 2/2=1 1 of 2

2 Properties 18 2 a 2 of 23 Definition 28 1

29 1Implications 30 3

42b 149 1

Class Inclusn 31 332 2 8 of 12

4 Proof 48 150 152 1 - 2 of 3

Parallel 1 Name 6 1Lines Discriminate 13 1 1 of 2

2 Properties 20 2 2 of 23 Definition 37 1

39 1Relationships 38 5Implications 36 2 6 of 9

4 Proof 53 154 1 1 of 2

Similarity 1 Name 8 1Discriminate 14 1 1 of 2


Relationships 41 642 5

Class Inclusn 44e 1 8 of 134 Proof *58 1

#48_

1 only iffitting

-1 (of 2)

_

* Question 58 was developed specifically to test Level 4 for similarity.# Similarity need not be used in solving Item 48.

242

Appendix C

RESULTS FOR PRELIMINARY STUDY

Achievement levels for each subject

PAPER I

Subject\conce )t Square Right trianglee Circle Congruency_

P01 (1,1,1,1) (1,1,1,1) (1,1,1,1) (1,1,1 1)

P06 (1,1,1,1) (1,1,1,0) ((1,1,1,1) (1,1,1,1)

P08 (1,1,1,1) (1,1,1,0) (1,1,1,1) (1,1,1,1)

P09 (1,1,0,0) (1,1,0,0) (1,1,1,0) (1,1,0,0)

P11 (1,1,0,0) (1,1,0,0) (1,1,1,0) (1,0,1,0)*

P12 (1,1,1,0) (1,0,0,0) (1,1,1,0) (1,1,0,1)*

*Response Pattern Error

PAPER II

Subject\concept Square Isosceles

triangle

Parallel lines Similarity

P02 (1,1,0,0) (1,0,1,1)* (1,1,1,0) (1,1,1,#)

P03 (1,1,0,0) (1,0,1,0)* (1,1,1,0) (1,1,0,#)

PO4 (1,1,0,0) (1,1,0,0) (1,1,0,0) (1,1,0,#)

P05 (1,1,1,0) (1,1, ,0) (1,1,0,0) (1,1,0,#)

P07 (1,1,1,1) (1,1, ,1) (1,1,1,1) (1,1,1,#)

P10 (1,1,1,0) (1,1, r,0) (1,1,0,0) (1,1,1,#)

P13 (1,1,1,0) (1,1,1,0) (1,1,1,0) (1,1,1,0)

*Response Pattern Error

#P13 was the only subject attempting Paper 2 to use similar triangles in the proof of Item 48.

All other subjects used properties of triangles in attempting the item, hence were not able to

be assessed for Level 4 reasoning for the concept similarity.

243

Scoring for each subject - preliminary test

Scoring for Square

Level 1

1 of 2

Level 2

2 of 3

Level 3

6 of 9

Level 4

1 of 2

Results

2 9a 16 23a 24 9b 25 42a 42d 44b 23b 45 46

P01 1 1 2 1 0 1 1,1,1 1 1 1 1 1 0 1,1,1,1

P02 1 1 2 1 0 0 1,0,0 1 1 1 1 0 0 1,1,0,0

P03 1 1 2 1 0 0 1,0,0 1 0 0 0 0 0 1,1,0,0

PO4 1 1 2 1 0 0 1,0,0 0 1 0 0 0 0 1,1,0,0

P05 1 1 2 1 0 0 1,1,1 1 1 1 0 0 0 1„1,1,0

P06 1 1 2 1 1 1 1,1,1 1 1 1 0 1 0 1,1,1,1

P07 1 1 2 1 1 1 1,1,1 1 1 1 1 1 0 1,1,1,1

P08 1 1 2 1 1 0 1,1,1 1 0 1 1 0 1,1,1,1

P09 1 1 1 0 0 1,0,0 1 1 0 0 0 1,1,0,0

P10 1 1 2 1 0 1 1,1,1 1 1 1 1 0 0 1,1,1,0

P11 1 1 2 1 0 0 1,0,0 1 1 0 0 0 0 1,1,0,0

P12 1 0 2 1 0 0 1,1,1 1 1 1 1 0 0 1,1,1,0

P13 1 1 2 1 0 1 1,1,1 1 1 1 1 0 0 1,1,1,0

Scoring for Right Triangle

Level 1

1 of :2

Level 2

3 of 4

.Le el 3

5 of '7

Level 4

2 of 3

_Results

3 10 17 26 27 32 44c 47 48 57

P01 1 1 1,1,1,1 1 1,1,1 1,1 1 1 1 1 1,1,1,1

P06 1 1 1,1,1,1 1 1,1,1 1,1 1 0 0 1 1,1,1,()

P08 1 1 1,1,1,1 1 1,1,1 1,1 1 0 0 0 1,1,1,()

P09 1 1 1,1,1,1 0 1,1,1 0,0 0 0 0 0 1,1,0,()

P11 1 0 1,1,1,1 0 1,0,0 1,1 0 0 0 0 1,1,0,()

P12 1 0 1,1,0,0 0 0,1,1 1,1 0 0 0 0 1,0,0,0

244

Scoring for Isosceles Triangle

Level 1

lof 2

Level 2

2 of 2

Level 3

8 of 12

Level 4

2 of 3

Results

4 11 18 28 29 30 49a 42b 31 32 48 50 52

P02 1 1 0,0 1 1 1,1,1 1 1 1,1,1 1,1 1 0 1 1,0,1,1*

P03 1 1 1,0 1 1 1,1,0 1 1 1,1,1 1,1 0 0 0 1,0,1,0*

PO4 1 1 1,1 1 1 1,1,0 0 0 0,1,0 0,1 0 0 1 1,1,0,0

P05 1 1 1,1 0 1 1,1,1 1 0 0,1,1 1,0 0 0 1 1,1,1,0

P07 1 1 1,1 1 1 1,1,0 1 1 0,1,1 1,1 1 1 1 1,1,1,1

P10 1 1 1,1 1 1 1,1,1 1 0 1,1,1 1,1 0 0 0 1,1,1,0

P13 1 I 1 1,1 1 1 1,1,1 1 0 1,1,1 1,1 0 0 0 1,1,1,0

* Response Pattern Error

Scoring for Circle

Level 1

1 of 2

Level 2

5 of 6

Le '213

6 of 10

Level 4

! of 2

Results

5 12 19 33 33 35 44d 34 51 52

P01 1 0 1 1,1,1,1,1 1 1,1,1,1 1 1,1,0,1 1 1 1,1,1,1

P06 1 1 1 1,1,1,1,1 0 1,1,1,1 0 1,1,1,1 0 1 1,1,1,1

P08 1 1 1 1,1,1,1,1 1 1,1,1,1 1 1,1,1,1 1 1 1,1,1,1

P09 1 1 1 1,1,1,1,1 0 0,1,1,1 0 1,1,1,1 0 0 1,1,1,0

P11 1 1 1 1,1,1,1,1 1 1,1,1,0 1 1,1,1,1 0 0 1,1,1,0

P12 1 1 1 1,1,1,1,1 0 1,1,1,0 1 1,0,1,1 0 0 1,1,1,0

Scoring for Parallel Lines

Level 1

1 of 2

Level 2

2 of 2

Level 3

6 of 9

Level.4

1 of 2

Results

6 13 20 37 39 38 36 53 54

P02 1 0 1,1 1 0 1,1,1,1,1 1,1 0 0 1,1,1,0

P03 1 1 1,1 1 0 1,0,1,1,1 1,1 0 0 1,1,1,0

PO4 1 1 1,1 0 0 0,0,0,0,0 1,1 0 0 1,1,0,0

P05 1 1 1,1 1 0 0,0,0,1,1 1,1 0 0 1,1,0,0

P07 1 1 1,1 1 0 0,0,1,1,1 1,1 1 1 1,1,1,1

P10 1 1 1,1 1 0 1,0,0,1,0 1,1 0 0 1,1,0,0

P13 1 1 1,1 1 0 1,0,0,1,1 1,1 0 0 1,1,1,0

245

Scoring for Congruency

Level 1

I of 2

Level 2

3 of 4

Level 3

6 of 10

Level 4

1 of 2

Results

7 15 22 43# 44 42C 55 56

P01 1 1 1,1,1,1 1,1,0,1 1,1,1,1,1 0 1 1 1,1,1,1

P06 1 0 1,1,1,1 1,1,1,1 1,1,1,0,0 1 0 1 1,1,1,1

P08 1 1 1,1,1,1 1,1,1,1 1,1,1,1,1 1 1 1 1,1,1,1

P09 1 1 1,1,1,1 1,0,1,0 1,1,0,0,0 0 0 0 1,1,0,0

Pll 1 1 0,1,0,1 1,1,0,1 1,1,0,1,0 0 0 0 1,0,1,0*

P12 0 1 1,1,1,1 1,0,1,0 1,1,0,1,0 0 0 1 1,1,0,1*

# Deduct one point if both (a) and (c) of 43 incorrect.


Scoring for Similarity

Level.1

1 of 2

Level 2

3 of 4

Lev ;13

8 of 13

Level 4

1 of 1

Results

8 14 21 40 41 42 44e 48#

P02 () 1 1,1,1,1 1 1,1,1,0,0,0 1,1,1,1,1 0 # 1,1,1,#

P03 0 1 1,1,1,1 0 1,0,1,0,0,1 1,1,0,1,0 1 # 1,1,0,#

PO4 1 1 1,1,1,1 0 0,0,0,0,0,1 0,0,0,0,1 0 # 1,1,0,#

P05 0 1 1,1,1,1 0 0,1,0,0,1,0 1,0,1,1,1 0 # 1,1,0,#

P07 1 1 1,1,1,1 0 1,1,1,1,1,1 1,1,1,1,1 1 # 1,1,1,#

P10 1 1 1,0,1,1 1 1,1,0,1,0,1 1,1,1,1,1 0 # 1,1,1,#

P13 0 1 1,1,1 1 1,1,0,0,1,0 1,0,1,0,1 1 0 1,1,1,0

# P13 Was the only student to use similar triangles in the proof of Item 48. Hence, the other

students 'were not able to be assesed for Level 4 for the concept similarity.

246

Appendix I)

SCHEDULES FOR INTERVIEW ITEMS

The same first question was asked for each interview item:

1 a Please read the question aloud.

lb Do the question, telling me what you are thinking and doing as you go.

Write clown your answer on the question sheet.

The prompting and probing questions are listed below each item.

Item 23

B C

A


Name an angle congruent to LABD

How do you know?

VERIFICATION

2 Is there any way you can check to make sure your answer is correct?

GENERALISATION

3 What if ABCD is a rectangle? Does this change your answer?

Item 27QAB is a triangle.

a) Suppose angle Q is a right angle. Does that tell you anything about angles A andB? If so, what?

b) Suppose angle Q is less than 90°. Could the triangle be a right triangle? Why?

c) Suppose angle Q is mote than 90°. Could the triangle be a right triangle? Why?

VERIFICATION

2 How do you know your answer is correct?

CONFLICT was taken to be inappropriate fo] . this item.

Item 35

This figure is a circle with centre 0.Would the following be: a) certain b) possible c) impossible.Give reasons for your answer.

1) OB =OA 2) OD = OA 3) 20B = AD

4) AD = EC

VERIFICATION

2 Is there any way you can check you answer?

CONFLICT

3 Suppose I tell you that AD is the same length as EC, what are your comments?

248

Item 39Circle the smallest combination of the following which guarantee that two lines are parallel?

a) They are everywhere the same distance apart.b) They have no points in common.c) They are in the same plane.d) They never meet.

PROMPTING2a Can you do with less?2b Is there another answer? (prompting for 3-D recognition)

VERIFICATION3 Is there any way you can check your z nswer'?

CONFLICT4a Suppose you consider the two lines in this room and

(interviewer indicated a pair of skew lines).What can you tell me about them?

4b (if needed)Do you now wish to change your original answer?

Item 41Triangle ABC is similar to triangle DEF (in tiat order).Are the following a) certain b) possible, or c) impossible?Give reasons for your answers.

a) AB = DEb) AB > DEc) ZA = LEd) ZA > LEe) AB = EF

f) Z A >

CONFLICT has already been raised in parts (c) and (e).

Item 45

ABCD is a four sided figure. Suppose we know that opposite sides are parallel. What are thefewest facts necessary to prove that ABCD is a square?

VERIFICATION2 Can you give me another answer?

CONFLICT3 What if ABCD is a rectangle? Is then; another answer?

(for weaker students: Is ABCD a remingle?)

249

Item 50

AB is the line segment with A and B the midpoints of the equal sides of the isosceles triangleXYZ.AY = BY and A AYB is similar to A XYZ so LA = ZX and. AB is parallel to XZ.


CONSOLIDATION

2a What has the question asked you to do

2b What are the key features in th proof?

CONFLICT

3 Can you answer the question without mentioning (here the interviewer should select

the appropriate condition)

a) isosceles triangles

b) similar triangles or

c) corresponding angles.

FURTHER PROBE (for higher levels)

4 If instead of A and B being the mid-points of XY and ZY, they divide the sides

XY and ZY in the same ratio, what changes does this make to your answer?

250

Item 55


Will Z A and LC be the same size? Why or why not?

PROMPT

2a (if first answer is very informal)

Can you phrase your answer in another way?

2b (if not really knowing how to tackle the question)

If BD is joined, can you now give an answer?

VERIFICATION

3 Can you justify that your answer is correct?

CONFLICT

4 Suppose a student gave the following answer:

"BD bisects and ABC, therefore triangles ABD and CBD are congruent (SAS)."

Can you justify such an answer?

251

Appendix E

INVESTIGATOR-OBSERVER AGREEMENT FOR THE CODING OF

RESPONSES

In order to demonstrate consistency of the use of the Mayberry marking scheme, a marker who

was experienced in coding students' response3 with the van Hiele levels agreed to act as a co-

marker with the investigator. A preliminary coding of responses was first undertaken,

resulting in the formulation of guideline rules which would then be used in the actual marking.

Preliminary coding

The investigator and the co-marker took ten papers from each of test Papers I and II and coded

them independently. The results were compared for the four van Hiele levels across four

concepts for each of the ten students randomly selected from Papers I and II, i.e., 160 results

were compared for each paper. There was found to be a high percentage agreement, 92.5%

(148 out of 160) for Paper I and 90% (144 oui of 160) for Paper II.

Almost all discrepancies occurred in the Levels 3 and 4 codings. For Level 3, it was found that

there were differences in the expectations of the two markers for the reasons given by the

students for responses, while for Level 4, the discrepancies concerned the markers'

expectations of the degree of completeness of the proofs. Additionally, markers showed a

slightly higher degree of discrepancy in the concepts parallel lines and similarity, leading to the

slightly lower percentage agreement for Paper II which contained the questions for these two

concepts.

As a consequence of this, a short list of guideline rules was formulated. These were used in

conjunction with the original behavioural definitions and the Mayberry guidelines for the actual

coding. This was done to clarify and fine-tuns; the coding scheme.

Actual coding

Both the investigator and the co-marker, working independently, then coded a different set of

ten papers selected from each of Papers I and II using the formulated guidelines as well as the

behavioural definitions and the Mayberry guidelines. This meant that altogether, sets of results

were compared for forty of the sixty-one students. On completion of the actual coding for the

second set of twenty students, the agreements for the four levels across all concepts was 98.8%

252

(158 out of 160) for Paper I and 98.1% (157 out of 160) for Paper II, i.e. there was agreement

for all except five of the 320 codings. These five discrepancies were discussed between

markers and the differences reconciled.

To ensure consistency of coding throughout all responses, a safety net was established. This

entailed, on the rare occasions when a response was difficult to code, a consultation between

the investigator and the co-marker, resulting in a joint decision. Coding reliability was,

therefore, established in the Mayberry coding.

253

Appendix F

RESULTS FOR MAIN STUDY

Achievement levcis for each concept

PAPER I

Concept Square Right triangle Circle Congruency

(1,0,0,0)SO1 (1,0,0,0) (1,0,0,0) (1,0,0,0)

S02 (1,1,0,0) (1,1,0,0) (1,0,1,0)* (1,1,0,1)*

S03 (1,1,0,0) (1,0,0,0) (1,1,0,0) (1,0,0,0)

SO4 (1,1,0,0) (1,1,0,0) (1,0,0,0) (1,0,0,0)

S05 (1,1,0,0) (1,1,0,0) (1,1,1,0) (1,1,0,0)

S06 (1,1,1,0) (1,1,1,0) (1,1,1,0) (1,0,1,0)*

S07 (1,1,0,0) (1,1,0,0) (1,1,0,0) (1,0,0,0)

S08+ (1,1,0,0) (1,1,0,0) (1,1,1,0) (1,1,0,0)

S09 (1,1,0,0) (1,1,0,0) (1,1,1,0) (1,1,0,0)

S10 (1,1,1,0) (1,1,0,0) (1,1,1,0) (1,0,0,0)

S 11 (1,1,0,0) (1,1,0,0) (1,1,1,0) (1,1,0,0)

S12 (1,1,0,0) (1,1,0,0) (1,1,0,0) (1,1,0,0)

S13 (1,1,0,0) (0,0,0,0) (1,0,0,0) (1,0,0,0)

S14+ (1,1,1,1) (1,0,1,0)* (1,1,1,1) (1,1,1,1)

S15 (1,1,0,0) (1,0,0,0) (1,1,1,0) (1,1,0,1)*

S16 (1,1,0,0) (1,1,1,0) (1,1,1,1) (1,1,0,1)*

S17 (1,1,0,0) (1,1,0,0) (1,1,1,0) (1,0,0,0)

S18 (1,1,0,0) (1,0,0,1) (1,1,0,0) (1,1,0,0)

S19 (1,1,0,0) (1,1,1,3) (1,1,1,1) (1,1,0,1)*

S20 (1,1,1,0) (1,1,0,3) (1,1,1,0) (1,1,0,0)

S21 (1,1,0,0) (1,1,0,0) (1,1,1,0) (1,1,0,0)

S22 (1,1,0,0) (1,0,0,0) (1,0,1,0)* (1,1,0,0)

S23 (1,1,0,0) (1,0,0,0) (1,1,1,0) (1,1,0,0)

S24 (1,1,0,0) (1,1,0,0) (1,0,0,0) (1,0,0,0)

S25 (1,1,0,0) (1,1,1,0) (1,1,1,1) (1,1,0,1)*

S26 (1,1,0,0) (1,1,0,0) (1,1,1,0) (1,1,1,1)

S27 (1,1,0,0) (1,1,1,0) (1,1,1,0) (1,1,0,1)*

S28 (1,1,0,0) (1,1,0.0) (1,1,0,0) (1,0,0,0)

S29 (1,1,0,1)* (1,1,1.1) (1,1,1,1) (1,1,0,1)*

S30 (1,1,0,0) (1,1,0.0) (1,01,0)* (1,0,0,0)

S31+ (1,1,0,0) (1,1,0 0) (1,1,0,0) (1,1,0,0)

*Response Pattern Error +Intervu'w subject

254

PAPER II

Concept Square Isosceles

triangle


S32 (1,1,0,0) (1,1,1,0) (1,1,0,0) (1,0,0,0)

S33+ (1,1,0,0) (1,1,1,0) (1,1,0,0) (1,0,0,0)

S34 (1,1,0,0) (1,1,0,0) (1,1,0,0) (1,1,0,0)

S35 (1,1,0,0) (1,0,0,0) (1,0,0,0) (1,0,0,0)

S36 (1,1,0,0) (1,1,0,0) (1,1,0,0) (1,1,0,0)

S37 (1,1,0,0) (1,0,0,0) (1,0,0,0) (1,0,0,0)

S38 (1,1,0,0) (1,1,0,0) (1,1,0,0) (1,0,0,0)

S39 (1,1,0,0) (1,1,1,0) (1,1,0,0) (1,1,0,0)

S40 (1,1,0,0) (1,0,0,0) (1,1,0,0) (1,1,0,0)

S41+ (1,1,0,1)* (1,1,1,1) (1,1,0,0) (1,1,1,0)

S42 (1,1,0,0) (1,1,0,1) (1,1,0,0) (1,1,0,0)

S43 (1,1,0,0) (1,1,1,0) (1,1,0,0) (1,1,0,0)

S44 (1,1,0,0) (1,1,0,0) (1,1,0,0) (1,1,1,0)

S45 (1,1,0,0) (1,1,0,0) (1,1,0,0) (1,1,0,0)

S46 (1,1,0,0) (1,0,0,0) (1,0,0,0) (1,0,0,0)

S47 (1,1,0,0) (1,1,0,0) (1,0,0,0) (1,1,0,0)

S48 (1,1,0,0) (1,0,0,0) (1,1,0,0) (1,0,0,0)

S49 (1,1,0,0) (1,0,0,0) (1,1,0,0) (1,1,0,0)

S50 (1,1,0,0) (1,1,0,0) (1,1,0,0) (1,0,0,0)

S51 (1,1,0,0) (1,1,0,0) (1,1,0,0) (1,1,0,0)

S52 (1,1,0,0) (0,0,0,0) (1,1,0,0) (1,0,0,0)

S53 (1,1,0,0) (1,1,1,0) (1,1,0,1)* (1,1,0,1)*

S54 (1,1,0,0) (1,0,0,0) (1,0,0,0) (1,0,0,0)

S55 (1,0,0,0) (1,0,0,0) (1,1,0,0) (1,0,0,0)

S56 (1,1,0,0) (1,1,0,0) (1,1,0,0) (1,0,0,0)

S57 (1,1,1,0) (1,0,1,0)* (1,1,0,0) (1,0,1,0)*

S58 (1,1,0,1)* (1,1,0,0) (1,1,0,0) (1,1,1, l )S59+ (1,1,0,0) (0,0,0.0) (1,1,0,0) (1,0,0,0)

S60 (1,1,0,0) (1,1,0.0) (1,1,0,0) (1,1,0,0)

S61 (1,1,0,0) (1,1,0.0) (1,1,0,0) (1,1,0,0)

*Response Pattern Error +Interview subject

255

Subject Scores for each Concept

Scorini for Square

Level 1

1 of 2

Level 2

2 of 3

Level 3

6 of 9

Level 4

1 of 2

Results

Q 2 9a 16 23a 24 9b 25 42a 42d 44b 23b 45 46

SO1 0 1 1,0 0 0 0 1,0,0 0 0 0 0 0 0 1,0,0,0

S02 1 1 1,1 1 0 0 1,1,1 1 0 0 1 0 0 1,1,0,0

S03 1 1 1,1 1 0 0 1,0,0 0 0 0 0 0 0 1,1,0,0

SO4 1 1 1,1 1 0 0 1,0,0 0 0 0 1 0 0 1,1,0,0

S05 1 1 1,1 1 0 0 1,0,0 0 0 0 0 0 0 1, 1,0,0

S06 1 1 1,1 1 0 0 1,1,1 1 1 1 1 0 0 1,1,1,0

S07 1 1 1,1 1 0 0 1,0,0 0 0 0 0 0 0 1.1,0,0

S08 1 0 1,1 0 0 0 1,0,0 0 0 0 0 0 0 1,1,0,0

S09 1 1 1,1 1 0 0 1,1,1 1 1 0 0 0 1,1,0,0

S10 1 0 1,1 0 0 0 1,1,1 1 1 1 0 0 0 1,1,1,0

S 1 1 1 1 1,1 1 0 0 1,0,0 0 0 0 1 0 0 1,1,0,0

S12 1 0 1,1 1 0 0 1,0,0 0 0 0 0 0 0 1,1,0,0

S13 1 0 1,1 0 0 0 1,0,0 0 0 0 0 0 0 1,1,0,0

S14 1 1 1,1 1 0 0 1,1 1 1 1 1 0 1 0 1,1,1,1

S15 1 0 1,1 1 0 0 1,0,0 0 0 1 1 0 0 1,1,0,0

S16 1 1 1,1 1 0 0 1,1,1 0 0 0 0 0 0 1,1,0,0

S17 1 1 1,1 1 0 0 1,0,0 0 0 0 1 0 0 1,1,0,0

S18 1 1 1,1 0 0 0 1,0,0 0 0 0 0 0 0 1,1,0,0

S19 1 1 1,1 1 0 0 1,0,0 1 1 0 0 0 0 1,1,0,0

S20 1 1 1,1 1 0 1 1,1,1 0 1 0 1 0 0 1,1,1,0

S21 1 0 1,1 1 0 0 1,0,0 0 0 0 0 0 0 1,1,0,0

S22 1 0 1,1 1 0 0 1,0,0 0 0 0 0 0 0 1,1,0,0

S23 1 0 1,1 1 0 0 1,0,0 0 0 0 0 0 0 1,1,0,0

S24 1 1 1,1 1 0 0 1,0,0 0 1 0 0 0 0 1,1,0,0

S25 1 1 1,1 1 0 0 1,1,1 0 0 0 1 0 0 1,1,0,0

S26 1 1 1,1 1 0 0 1,0,0 1 1 0 1 0 0 1,1,0,0

S27 1 1 1,1 1 0 0 1,0,0 0 0 0 0 0 0 1,1,0,0

S28 1 0 1,1 1 0 0 1,0,0 1 1 0 0 0 0 1,1,0,0_

1,1,0,1*S29 1 1 1,1 1 0 1 1,1,1 0 0 0 0 1 0

S30 1 0 1,1 1 0 0 1,0,0 1 0 0 1 0 0 1,1,0,0

S31 1 0 1,1 1 0 0 1,0,0 0 1 0 0 0 0 1,1,0,0_* Response Pattern Error

256

Level 1

1 of 2

Level 2

2 of 3

L:vel 3

6 of 9

Level 4

1 of 2

Q 2 9a 16 23a 24 9b 25 42a 42d 44b 23b 45 46 Results

S32 1 1 1,1 1 1 0 1,0,0 1 1 0 1 0 0 1,1,0,0

S33 1 1 1,1 1 0 0 1,1,1 1 0 1 0 0 0 1,1,0,0

S34 1 0 1,1 1 0 0 1,0,0 0 0 0 0 0 0 1,1,0,0

S35 1 1 1,1 1 0 0 1,0,0 0 0 0 0 0 0 1,1,0,0

S36 1 1 1,1 1 0 0 1,0,0 0 0 0 0 0 0 1,1,0,0

S37 1 0 1,1 1 0 0 1,0,0 0 0 0 0 0 0 1,1,0,0

S38 1 1 1,1 1 0 0 1,0,0 0 0 0 0 0 0 1„1,0,0

S39 1 1 1,1 1 1 0 1,1,1 0 0 0 0 0 0 1„1,0,0

S40 1 1 1,1 1 0 0 1,0,0 0 0 0 0 0 0 1,1,0,0

S41 1 1 1,1 1 0 0 1,1,1 1 0 1 0 1 0 1,1,0,1*

S42 1 1 1,1 1 0 0 1,0,0 0 0 0 1 0 0 1,1,0,0

S43 1 1 1,1 1 0 0 1,0,0 1 1 0 0 0 0 1,1,0,0

S44 1 1 1,1 1 0 0 1,0,0 1 1 0 1 0 0 1,1,0,0

S45 1 1 1,1 1 0 0 1,1,1 0 0 0 0 0 0 1,1,0,0

S46 1 1 1,1 0 0 0 1,0,0 1 0 0 0 0 0 1,1,0,0

S47 1 1 1,1 1 0 0 1,0,0 0 0 0 0 0 0 1,1,0,0

S48 1 1 1,1 1 0 0 1,0,0 0 0 0 0 0 0 1,1,0,0

S49 1 1 1,1 1 0 0 1,0,0 0 0 0 0 0 0 1,1,0,0

S50 1 1 1,1 1 0 0 1,0,0 0 0 0 0 0 0 1,1,0,0

S51 1 1 1,1 0 0 0 1,0,0 0 1 0 1 0 0 1,1,0,0

S52 1 1 1,1 1 0 0 1,0,0 1 1 0 0 0 0 1,1,0,0

S53 1 1 1,1 1 0 0 1,1,1 0 0 0 0 0 0 1,1,0,0

S54 1 0 1,1 0 0 0 1,0,0 0 0 0 0 0 0 1,1,0,0

S55 1 1 1,0 0 0 0 1,0,0 0 0 0 0 0 0 1,0,0,0

S56 1 1 1,1 0 0 0 1,0,0 0 0 0 0 0 0 1,1,0,0

S57 1 1 1,1 0 0 1 1,1,0 1 1 1 0 0 0 1,1,1,0

S58 1 1 1,1 1 0 0 1,0,0 1 0 0 1 0 1 1,1,0,1*

S59 1 0 1,1 1 0 0 1,0,0 1 0 0 0 0 0 1,1,0,0

S60 1 1 1,1 1 0 1 1,1,1 0 0 0 1 0 0 1,1,0,0

S61 1 1 1,1 1 0 0 1,0,0 0 0 0 0 0 0 1,1,0,0_


257

Scoring for Right Triangle

Level 1

1 of 2.

Level 2

3 of 4

Level 3

5 of 7

Level 4

2 of 3

Results

Q 3 10 17 26 27 32 44c 47 48 57

SO1 0 1 1,0,0,0 1 0,0,0 0,0 0 0 0 0 1,0,0,0

S02 1 1 1,1,1,1 0 0,0,0 0,1 0 0 0 0 1,1,0,0

S03 0 1 0,0,1,1 0 1,0,0 0,0 0 0 0 0 1,0,0,0

SO4 1 1 1,1,1,1 0 1,1,1 0,0 0 0 0 0 1,1,0,0

SO5 1 1,0,1,1 0 0,1,1 0,0 0 0 0 0 1,1,0,0

S06 1 1,1,1,1 1 1,1,1 1,1 0 0 0 0 1,1,1,0

S07 1 0 1,1,1,1 0 1,1,0 0,0 0 0 0 0 1,1,0,0

S08 1 0 1,0,1,1 0 0,0,0 0,1 0 0 0 0 1,1,0,0

S09 0 1 1,1,1,1 1 0,1,1 0,0 0 0 0 0 1,1,0,0

S10 1 1 1,1,1,1 1 1,1,0 0,0 0 0 0 0 1,1,0,0

S 11 1 0 1,1,1,1 1 1,1,1 0,0 0 0 0 0 1,1,0,0

S12 1 1 1,1,1,1 0 1,0,0 0,0 0 0 0 0 1,1,0,0

S13 0 0 1,0,1,0 0 0,0,0 0,0 0 0 0 0 0,0,0,0

S14 1 1 1,1,0,0 0 1,1,1 1,1 0 0 1 0 1,0,1,0*

S15 0,0,1,1 0 0,0,0 0,0 0 0 0 0 1,0,0,0

S16 1,1,1,1 1 1,1,1 1,1 0 0 0 0 1,1,1,0

S17 1,1,1,1 0 0,1,1 0,0 0 0 0 0 1,1,0,0

S18 1,1,0,0 0 0,0,0 0,0 0 0 0 0 1,0,0,0

S19 1 1,1,1,1 0 1,1,1 1,1 0 0 0 0 1,1,1,0

S20 1 0 1,1,1,1 0 1,0,0 0,0 0 0 0 1 1,1,0,0

S21 1 0 1,1,1,1 1 1,0,1 0,1 0 0 0 0 1,1,0,0

S22 0 0,0,1,1 0 0,1,0 1,1 0 0 0 0 1,0,0,0

S23 1 1,0,0,0 0 1,1,0 0,0 0 0 0 0 1,0,0,0

S24 1 0 1,0,1,1 0 0,1,0 0,0 0 0 0 0 1,1,0,0

S25 1 1 1,1,1,1 1 1,1,1 1,1 0 0 1 0 1,1,1,0

S26 1 1 1,1,1,1 0 0,0,0 0,0 1 0 0 0 1,1,0,0

S27 1 0 1,1,1,1 0 1,1,1 1,1 0 0 0 0 1,1,1,0

S28 1 1 1,1,1,1 0 1,1,1 0,0 0 0 0 0 1,1,0,0

S29 1 1 1,1,1,1 1 1,1,1 1,1 0 1 1 1 1,1,1,1

S30 1 0 1,1,1,1 0 1,0,0 0,0 0 0 0 0 1,1,0,0

S31 1 1 1,1,1,0 0 1,0,1 0,0 0 0 0 0 1,1,0,0

258

Scoring for Isosceles Triangle

Level 1

1 of 2

Level 2

2 of 2

Level 3

8 of 12

Level 4

2 of 3

Results

Q 4 11 18 28 29 30 49a 42b 31 32 48 50 I 52

S32 1 0 1,1 0 1 1,1,1 1 0 0,1,0 1,1 1 0 0

1

1,1,1,0

1,1,1,0S33 1 1 1,1 0 1 1,1,0 1 1 0,1,1 1,0 0 0

S34 1 1 1,1 0 1 1,1,0 0 0 0,1,0 0,0 0 0 0 1,1,0,0

S35 1 1 1,0 0 1 1,0,0 1 0 0,1,0 1,1 0 0 0 1,0,0,0

S36 1 0 1,1 0 1 1,1,0 0 0 0,1,0 0,0 0 0 1 1,1,0,0

S37 1 1 1,0 0 1 0,1,0 0 0 0,1,0 0,1 0 0 0 1,0,0,0

S38 1 0 1,1 0 1 1,1,0 1 0 0,1,0 0,0 0 0 0 1,1,0,0

S39 1 0 1,1 1 1 1,1,1 1 0 1,1,1 1,1 1 0 0 1,1,1,0

S40 1 1 1,0 0 1 1,0,0 0 0 0,0,0 0,0 0 0 0 1,0,0,0

S41 1 1 1,1 0 1 1,1,1 1 0 1,1,1 1,1 1 1 1 1,1,1,1

S42 1

0

1,1 1 1 1,1,0 0 0 0,1,0 1,1 0 0 1 1,1,0,0

S43 1 1,1 0 1 1,1,1 1 0 0,1,0 1,1 0 0 0 1,1,1,0

S44 1 1 1,1 0 1 1,0,0 0 1 0,1,0 1,1 0 0 1 1,1,0,0

S45 1 1 1,1 1 1 1,1,1 0 0 0,1,0 0,1 0 0 0 1,1,0,0

S46 1 0 1,0 1 1 1,1,0 1 0 0,1,0 0,0 0 0 0 1,0,0,0

S47 1 1 1,1 0 0 1,1,0 0 0 0,1,0 0,0 0 0 1 1,1,0,0

S48 1 0 0,0 0 0 1,1,0 0 0,0,0 0,0 0 0 0 1,0,0,0

S49 1 0 1,0 0 1 1,1,1 0 0 0,1,0 0,0 0 0 0 1,0,0,0

S50 1 1 1,1 0 1 1,0,0 0 0 0,1,0 0,0 0 0 0 1,1,0,0

S51 1 0 1,1 0 1 1,1,0 1 0 0,1,0 1,1 0 0 0 1,1,0,0

S52 0 0 1,0 1 1 0,0,0 0 0 0,1,0 0,0 0 0 0 0,0,0,0

S53 1 0 1,1 0 1 1,1,1 1 0 0,1,1 1,1 0 0 0 1,1,1,0

S54 1 0 0,0 0 0 0,0,0 0 0 0,0,0 0,0 0 0 0 1,0,0,0

S55 1 1 0,0 0 0 1,1,0 1 0 1,0,1 0,0 0, 0 0 1,0,0,0

S56 1 1 1,1 0 1 1,1,0 1 0 0,0,0 0,0 0 0 0 1,1,0,0

S57 1 0 1,0 0 1 1,1,0 0 1 1,1,1 0,1 0 0 1 1,0,1,0*

S58 1 1 1,1 0 1 1,1,1 1 1 0,1,0 0,0 0 0 1 1,1,0,0

S59 0 0 1,0 0 0 0,0,0 C 0 0,1,0 0,0 0 0 0' 0,0,0,0

S60 1 1 1,1 0 1 1,1,1 0 0 0,1,0 0,1 0 0 1 1,1,0,0

S61 1 1 , 1,1 , 0 1 1,1,0 1 , 0 , 1,1,1 , 0,0 0 0 1 , 1,1,0,0


259

Scoring for Circle

Level 1

1 of 2

Level 2

5 of 6

Level 3

6 of 10

Level 4

1 of 2

Results

Q 5 12 19 33 33 35 44d 34 51 52

SO1 1 1 1 1,0,1,0,0 0 1,1,0,1 0 0,0,0,0 0 0 1,0,0,0

S02 1 1 1 1,0,1,0,1 0 1,1,1,1 0 1,1,1,1 0 0 1,0,1,0*

S03 1 1 1 1,1,1,1,1 1 1,1,1,1 0 0,0,0,0 0 0 1,1,0,0

SO4 1 1 0 0,0,1,0,1 0 0,0,0,0 0 1,0,1,1 0 0 1,0,0,0

SO5 1 1 1 1,0,1,1,1 1 1,1,1,1 0 1,1,1,1 0 0 1,1,1,0

S06 1 1 1 1,1,1,1,1 1 1,1,1,1 0 1,1,1,1 0 0 1,1,1,0

SO7 1 0 0 1,1,1,1,1 0 1,1,0,1 0 0,1,1,0 0 0 1,1,0,0

S08 1 1 1 1,1,1,1,1 1 0,1,1,1 1,0,1,1 0 0 1,1,1,0

S09 1 1 1 1,1,1,1,1 0 1,1,0,1 0 1,1,1,1 0 0 1,1,1,0

S10 1 1 1 1,1,1,1,1 1 1,1,1,1 0 1,1,1,1 0 0 1,1,1,0

Sll 1 1 0 1,1,1,1,1 0 1,1,1,1 0 1,1,1,1 0 0 1,1,1,0

S12 1 1 1 1,1,1,0,0 1,1,1,1 0 0,0,0,0 0 0 1,1,0,0

S13 1 1 0 1,0,0,1,1 0 0,0,0,0 0 0,1,1,0 0 0 1,0,0,0

S14 1 1 1 1,1,1,1,1 1 0,0,0,0 1 1,1,1,1 0 1 1,1,1,1

S15 1 1 1 1,1,1,1,1 1 1,1,1,1 0 1,1,1,1 0 O 1,1,1,0

S16 1 1 0 1,1,1,1,1 1 1,1,1,1 0 1,1,1,1 0 1 1,1,1,1

S17 1 1 1 1,1,1,1,1 0 1,1,1,1 0 1,1,1,1 0 0 1,1,1,0

S18 1 1 1 1,1,1,1,0 0 0,0,0,1 0 0,0,1,0 0 0 1,1,0,0

S19 1 1 1 0,1,1,1,1 1 1,1,1,1 1 1,1,1,1 1 CI 1,1,1,1

S20 1 0 1 1,1,1,1,1 1 1,1,1,1 0 1,1,1,1 0 CI 1,1,1,0

S21 1 1 1 1,1,1,1,1 1 1,1,1,1 0 1,1,0,0 0 0 1,1,1,0

S22 1 l 0 1,1,1,0,0 0 1,1,0,0 0 1,1,1,1 0 C) 1,0,1,0*S23 1 1 1 1,1,1,1,1 1 1,1,1,1 0 0,1,1,0 0 0 1,1,1,0

S24 1 I. 0 1,0,0,1,0 0 0,0,0,0 0 0,1,1,0 0 0 1,0,0,0

S25 1 1 1 1,1,1,1,1 1 1,1,1,1 0 1,1,1,1 0 1 1,1,1,1

S26 1 1 1 1,1,1,1 ,1 1 1,1,1,1 1 1,1,1,1 0 0 1,1,1,0

S27 1 11 1 1,1,1,1,0 0 1,1,1,1 0 1,1,1,1 0 () 1,1,1,0

S28 l 1 1 1,1,1,1,0 0 0,1,0,0 0 1,0,1,1 0 () 1,1,0,0S29 1 1 1 1,1,1,1,1 1 1,1,1,1 0 1,1,1,1 0 1 1,1,1,1

S30 1 1 0 1,0,1,1,0 0 1,1,0,1 0 1,1,1,1 0 0 1,0,1,0*

S31 1 0 1 1,0,1,1,1 0 0,0,0,0 0 1,0,1,1 0 0 1,1,0,0


260

Scoring for Parallel Lines

Level 1

1 of 2

Level 2

2 of 2

Level 3

6 of 9

Level 4

1 of 2

Results

Q 6 13 20 37 39 38 36 53 54

S32 1 1 1,1 0 0 0,0,1,1,1 0,0 0 0 1,1,0,0

S33 1 1 1,1 1 0 0,0,0,1,1 0,0 0 0 1,1,0,0

S34 1 1 1,1 0 0 0,0,0,1,0 0,0 0 0 1,1,0,0

S35 1 1 0,0 0 0 0,0,1,1,0 0,0 0 0 1,0,0,0

S36 1 1 1,1 0 0 0,0,0,0,0 0,0 0 0 1,1,0,0

S37 0 1 0,0 0 0 0,0,0,0,.) 0,0 0 0 1,0,0,0

S38 1 0 1,1 0 0 0,0,1,1,1 0,0 0 0 1,1,0,0

S39 1 1 1,1 1 0 0,0,0,0,0 1,1 0 0 1,1,0,0

S40 1 1,1 1 0 0,0,0,1,0 0,0 0 0 1,1,0,0

S41 1 1 1,1 0 0 0,0,1,1,1 0,0 0 0 1,1,0,0

S42 1 1,1 0 0 0,0,0,1,0 0,0 0 0 1,1,0,0

S43 1 1 1,1 0 0 0,0,0,0,0 0,0 0 0 1,1,0,0

S44 1 1 1,1 0 0 0,0,1,1,0 0,0 0 0 1,1,0,0

S45 1 1 1,1 0 0 0,0,0,0,0 0,0 0 0 1,1,0,0

S46 1 1 0,0 0 0 0,0,0,0.0 0,0 0 0 1,0,0,0

S47 0 1 1,0 1 0 0,0,0,0.0 0,0 0 0 1,0,0,0

S48 1 1,1 0 0 0,0,0,0.0 0,0 0 0 1,1,0,0

S49 l 1 1,1 0 0 0,0,0,0.0 0,0 0 0 1,1,0,0S50 1 1 1,1 0 0 0,0,0,0,0 1,1 0 0 1,1,0,0

S51 1 0 1,1 0 0 0,0,0,0,0 0,0 0 0 1,1,0,0

S52 1 1 1,1 0 0 0,0,0,0,0 0,0 0 0 1,1,0,0

S53 1 1 1,1 0 0 0,0,0,0,0 0,0 0 1 1,1,0,1*

S54 1 1 0,0 0 0 0,0,0,0,0 0,0 0 0 1,0,0,0

S55 1 1 1,1 0 0 0,0,0,0,0 0,0 0 0 1,1,0,0

S56 1 1 1,1 0 0 0,0,0,0,() 0,0 0 0 1,1,0,0

S57 1 1 1,1 0 0 0,0,0,0,0 0,0 0 0 1,1,0,0

S58 1 1 1,1 0 0 0,0,1,1,0 0,0 0 0 1,1,0,0

S59 1 1 1,1 0 0 0,0,0,0,0 0,0 0 0 1,1,0,0

S60 1 1 1,1 0 0 0,0,0,C,0 0,0 0 0 1,1,0,0

S61 1 1 1,1 0 0 0,0,1,1,0 0,0 0 0 1,1,0,0


261

Scoring for Congruency

Level 1

1 of 2

Level 2

3 of 4

Level 3

6 of 10

Level 4

1 of 2

Results

Q 7 15 22 43* 44 42c 55 56

SO1 0 1 0,0,0,0 0,0,0,0 0,0,0,0.0 0 0 0 1,0,0,0

S02 1 1 1,1,1,1 0,0,1,0 0,0,0,0.0 0 0 1 1,1,0,1*

S03 0 1 0,0,0,0 0,0,0,0 0,0,0,0.0 0 0 0 1,0,0,0

SO4 1 1 1,1,0,0 0,0,0,0 0,0,0,0.0 0 0 0 1,0,0,0

Z05 1 1 1,1,1,1 0,0,0,0 0,0,0,0 .0 1 0 0 1,1,0,0

S06 1 1 0,1,0,1 0,1,1,1 1,1,0,0,0 1 0 0 1,0,1,0*

S07 1 1 0,1,0,1 0,0,0,0 0,0,0,0,0 0 0 0 1,0,0,0

SO8 0 1 1,1,1,1 0,0,0,0 1,0,0,0,0 0 0 0 1,1,0,0

S09 1 0 1,1,0,1 1,0,0,0 1,1,0,0,0 1 0 0 1,1,0,0

S10 0 1 0,0,1,0 0,0,0,0 0,1,0,0,() 0 0 0 1,0,0,0

Sll 1 1 1,1,1,1 0,0,0,1 1,0,0,0,0 0 0 0 1,1,0,0

S12 0 1 1,1,0,1 0,0,0,0 0,0,0,0,0 0 0 0 1,1,0,0

S13 1 0 0,0,0,0 0,0,0,0 0,0,0,0,0 0 0 0 1,0,0,0

S14 1 1 1,1,1,1 1,1,1,1 1,1,0,1,1 1 0 1 1,1,1,1

S15 1 1 1,1,1,1 0,0,0,0 1,0,0,0,() 0 0 1 1,1,0,1*

S16 1 1 1,1,1,1 1,1,1,1 0,0,0,0,0 0 1 1 1,1,0,1*

S17 1 1 0,1,0,0 0,0,0,0 0,0,0,0,0 1,0,0,0

S18 1 0 1,1,1,1 0,0,0,0 0,0,0,0,0 1,1,0,0

S19 1 1 1,1,0,1 0,0,0,0 1,1,0,C,1 0 0 1 1,1,0,1*

S20 1 1 1,1,1,1 0,0,0,0 1,1,0,0,0 0 0 0 1, l ,0,0S21 1 1 1,1,0,1 0,0,0,0 1,0,0,0,0 0 0 0 1,1,0,0

S22 1 1 1,1,0,1 0,0,0,0 0,0,0,0,0 0 0 0 1,1,0,0

S23 1 1 1,1,1,1 0,0,0,0 0,0,0,0,0 0 0 0 1,1,0,0

S24 1 1 0,1,0,0 1,0,1,0 1,1,0,0,0 0 0 0 1,0,0,0

S25 1 1 1,1,1,1 1,1,0,0 0,0,0,0,0 0 1 1 1,1,0,1*

S26 1 1 1,1,1,1 1,1,1,1 1,1,1,1,0 1 0 1 1,1,1,1

S27 1 1 1,1,1,1 1,0,1,1 0,0,0,0,0 0 1 1 1,1,0,1*

S28 1 1 0,0,0,0 1,0,0,0 1,1,0,0,0 0 0 0 1,0,0,0

S29 1 1 1,1,1,1 0,0,0,0 0,0,0,0,0 0 1 1 1,1,0,1*

S30 1 1 1,1,0,0 0,0,0,0 0,0,0,0,0 0 0 0 1,0,0,0

S31 1 1 1,1,1,1 0,0,0,0 1,1,0,0,0L 0 0 0 1,1,0,0


262

Scoring for Similarity

Level 1

1 of 2

Level 2

3 of 4

Le` el 3

8 of 13

Level 4

1 of 1

Results

Q 8 l4 21 40 41 42 44e 58

S32 1 1 1,1,0,0 0 1,1,0,0,0,0 1,0,0,0,1 1 0 1,0,0,0

S33 1 1 1,1,0,0 0 0,1,1,1,1,0 0,1,1,1,0 0 0 1,0,0,0

S34 1 1,1,1,0 1 1,1,0,0,0,0 0,1,0,0,0 0 0 1,1,0,0

S35 1 1 1,1,0,0 0 0,0,0,0,0,0 0,0,0,0,0 0 0 1,0,0,()

S36 1 1 1,1,1,1 1 0,0,1,0,1,0 0,0,0,0,0 0 0 1,1,0,()

S37 1 1 1,1,0,0 0 0,1,0,0,0,0 0,0,0,0,0 0 0 1,0,0,0

S38 1 1 1,1,0,0 0 0,0,0,0,0,C' 0,0,0,0,0 0 0 1,0,0,0

S39 1 0 1,1,1,1 1 1,1,1,1,0,1 0,0,0,0,0 0 0 1,1,0,0

S40 1 0 1,0,1,1 1 0,0,0,1,0,0 0,0,0,0,0 0 0 1,1,0,0

S41 1 0 1,1,1,1 0 1,1,1,1,1,0 1,0,0,1,1 0 0 1,1,1,0

S42 1 1 1,1,1,1 0 0,1,0,0,0,0 0,0,0,0,0 0 ► 1,1,0,0

S43 1 1 1,1,1,1 0 0,0,0,0,0,0 1,0,0,0,1 0 CI 1,1,0,0

S44 1 1 1,1,1,1 0 1,1,1,1,0,1 1,1,1,0,1 0 0 1,1,1,0

S45 1 1 1,1,1,1 1 1,0,0,0,0,► 0,0,0,0,0 0 0 1,1,0,0S46 1 0 0,0,0,0 0 0,1,1,0,0,0 1,0,1,0,1 0 0 1,0,0,0

S47 1 1,1,1,1 1 1,0,0,0,0,► 0,0,0,0,0 0 0 1,1,0,0S48 1 1 1,0,1,0 0 0,0,0,0,0,0 0,0,0,0,0 0 0 1,0,0,0

S49 1 1 1,1,1,1 0 0,0,0,0,0,0 0,0,0,0,0 0 0 1,1,0,0

S50 1 1 1,1,0,0 0 0,0,0,0,0,0 0,0,0,0,0 0 0 1,0,0,0

S51 1 1 1,1,1,1 1 1,1,0,0,0,1 0,0,0,0,0 0 0 1,1,0,0

S52 1 1 0,0,0,0 0 0,0,0,0,0,0 1,0,0,0,1 0 0 1,0,0,0

S53 1 1 0,1,1,1 1 0,0,0,0,0,0 0,0,0,0,0 0 1 1,1,0,1*

S54 0 1 0,0,0,0 0 0,0,0,0,0,► 0,0,0,0,0 0 0 1,0,0,0S55 1 1 1,1,0,0 0 0,0,0,0,0,0 0,0,0,0,0 0 0 1,0,0,0

S56 1 0,0,0,0 0 0,0,0,0,0,0 0,0,0,0,0 0 0 1,0,0,0

S57 1 1 1,1,0,0 1 0,0,1,1,0,0 1,1,1,1,1 1 0 1,0,1,0*

S58 1 1 1,1,1,1 1 1,1,0,1,1, l 1,1,0,0,0 1 1 1,1,1,1

S59 1 1 1,1,0,0 0 0,0,0,0,0,0 1,0,0,0,1 0 0 1,0,0,0

S60 1 1 1,1,1,1 1 0,0,0,0,0,3 0,0,0,0,0 0 0 1,1,0 ,0

S61 1 1 1,1,1,1 1 0,0,0,0,0,1 0,0,0,0,0 0 0 1,1,0,0


263

Appendix G

AMENDED SET OF MAYBERRY TEST ITEMS

1. not assessed

2. square, Level 1This figure is which of the following?

a) triangle

b) quadrilateral

c) square

d) parallelogram

e) rectangle

3. right triangle, Level 1

/IAre all of these triangles? YES NOExplain:

Do they appear to be a special kind of triangle?If so what kind?

4. isosceles triangle, Level 1

A z\These appear to be what kind of triangles?

264

5. circle, Level I

Name this figure.

6. parallel lines, Level 1

Suppose these two lines will never meet no matter how far we draw them.


7. congruency, Level 1

B

What is true of A and B? What is true of C and D?


8. similarity, Level 1

z

Are these figures alike in any way? ES NOWhat word describes this?

265

a

d

c

a fd

9. square, Level 1

Which of these figures are squares?


Which of these appear to be right triangles?


d

Which of these figures appear to be isosceles triangles?

266

12 circle, Level 1.

aWhich of these are circles?

13.parallel lines, Level 1

a

b c d

Which pair(s) of lines appear to be parallel?


Which figure appears to be similar to a?


Which figure appears to be congruent to A?

267

d 1

I ►

d 2

16. square, Level 2

Draw a square.What must be true about the sides? What must be true about the angles?

17. right triangle, Level 2Does a right triangle always have a longest sick? If so, which one?

Does a right triangle always have a largest angle? If so, which one?

18. isosceles triangle, Level 2What can you tell me about the sides of an isosceles triangle?

What can you tell me about the angles of an isosceles triangle?

19. circle, Level 2

This figure is a circle. 0 is the centre.

Name the following line segments.OB is a of the circle.OC is a of the circle.AC is a of the circle.


If d 1 = d 2, what if true about lines 11 and 1 2?

If d 1 � d 2, what is true about lines 1 1 and 1 2?

268


Triangle ABC is similar to triangle DEF.

How long is ED? How do you know?

What is the size of ZEDF? How do you know?


A

These are congruent figures.

What is true about their sides?

What is true about their angles?

Al) =

B

23. square, (a) Level 2, (b) Level 3


(a) Name an angle congruent to Z ABD (b) How do you know?

269

24. square, Level 3Circle the smallest combination of the following which guarantees a figure to be a square.

a. It is a parallelogram.b. It is a rectangle.c. It has right angles.d. Opposite sides are parallel.e. Adjacent sides are equal in length.f. Opposite sides are equal in length.

25. square, (a) Level 2, (b) Level 3

(a). Name some ways in which squares and rectangles are alike? (b). Are all squares also rectangles? Why?

26. right triangle, Level 3Circle the smallest combination of the following which guarantees a triangle to be a righttriangle?

a. It has two acute angles.b. The measures of the angles add up to 130°.c. An altitude is also a side.d. The measures of two angles add up to 90°.

27. right triangle, Level 3QAB is a triangle.

a) Suppose angle Q is a right angle. Does that tell you anything about angles A andB? If so, what?

b) Suppose angle Q is less than 90°. Cou - d the triangle be a right triangle? Why?

c) Suppose angle Q is mote than 90°. Could the triangle be a right triangle? Why?

28. isosceles triangle, Level 3Circle the smallest combination of the following which guarantees a triangle to be isosceles?

a. It has two congruent angles.b. It has two congruent sides.c. An altitude bisects the opposite side.d. The measure of the angles add up to 1:i0°.

29. This question is not assessable

30. isosceles triangle, Level 3Suppose all we know about A MNP is that L M is the same size as Z_ N.(a) What do you know about sides MP and I\ P?

Suppose LM is larger than LN.b) What do you know about MP and NP" c) Could A MNP be isosceles?

270

31. isosceles triangle, Level 3(a) Triangle DEF has three congruent sides. It is an isosceles triangle?

Why or 'why not? (b) Is the following true or false?

All equilateral triangles are isosceles.

32. right triangle, Level 3, isosceles triangle, Level 3Which are true? Give reasons.a) All isosceles triangles are right triangle b) Some right triangles are isosceles triangles.

33. circle, (a) to (e) Level 2, (f) Level 3

(/) CN/CDa c

e

Tell why each of these figures is or is not a circle.a)b)c)d)e)

(f) Can you give a general rule to fit all the ab 3-ye answers?

34. circle, level 3

Figure A is a simple closed curve. Figure B a circle.

Is figure B a simple closed curve? How are these figures alike? How are they different?

(T—F) All simple closed curves are circles.

271

35. circle, Level 2

This figure is a circle with centre 0.Would the following be

a) certain

b) possible c) impossible.Give reasons for your answer.

1) OB = OA 2) OD = OA 3) 20B = AD 4) AD =


Suppose L 1 and L2 are congruent. What dc es that tell you about 1 1 and 1 2? Suppose L 1 is larger than L2. What does that tell you about 1 1 and 12?

37. parallel lines, Level 3How do you recognise lines that are parallel?

38. parallel lines, Level 3Are these lines or line segments parallel?

a) always b) sometimes c) neverGive reasons for your answers.

a) Two lines which do not intersect.b) Two lines which are perpendicular to he same line. c) Two line segments in a square.d) Two line segments in a triangle.e) Two line segments which do not intersect.

39. parallel lines, Level 3Circle the smallest combination of the following which guarantee that two lines are parallel?

a) They are everywhere the same distant e apart.b) They have no points in common.c) They are in the same plane.d) They never meet.

272

40. similarity. Level 3What does it mean to say that two figures are s milar?

41.Triangle ABC is similar to triangle DEF (in thz t order).Are the following a) certain b) possible, orGive reasons for your answers.

a) AB = DE b) AB > DE c) ZAd) LA >e) AB = EF f) ZA > zfD

similarity, Level 3

c) impossible?

42. similarity, (a) to (e) Level 3(a) and (d), square, Level 3(b), isosceles triangle, Level 3(c), congruency, Level 3

a)b)c)d)e)

Will figures A and B be similarI-always


Aa) a squareb) an isosceles trianglec) a A congruent to Bd) a rectanglee) a rectangle

imes or HI-never?

Ba square an isosceles triangle a A congruent to A a square a triangle

43.A ABC is congruent to A DEF (in that order).Are the following a) certain b) possible, orGive reasons for your answers.

a) AB = DE b) LA= ZE c) LA < ZD d) AB EF

congruency, Level 3

c) impossible?

44. congruency, (a) to (e) Level 3(b), square, Level 3(c), right triangle, Level 3(d), circle, Level 3(e), similarity, Level 3

Will figures A and B be congruentI-always


Aa) a squareb) a square with a 10cm sidec) a right triangle with a

10cm hypotenused) a circle with 10cm chorde) a A similar to B

II-some times III-never?

Bb)

a iriangle b) a square with a 10cm side c) a :-fight triangle with a

10cm hypotenuse d) a ,circle with 10cm chord e) a A similar to A

?73

45. square, Level 4ABCD is a four sided figure. Suppose we kno w that opposite sides are parallel. What are thefewest facts necessary to prove that ABCD is a square?

46. square, Level 4Figure ABCD is a parallelogram, AB BC and Z ABC is a right angle. Is ABCD a square?Prove your answer.


A D B

CD is perpendicular to AB. LC is a right anE le.If you measure LACD and LB, you find that they have the same measure.Would this equality be true for all right triangles? Why or why not?

48. right triangle, Level 4isosceles triangle, Level 4similarity, Level 4

Figures ABC and PQR are right isosceles triangles with angles B and. Q being right angles.Prove that ZA = LP and LC ZR.


D

ABC is a triangle. A ADC === A BDC.

What kind of triangle is A ABC? Why?

274


AB is the line segment with A and B the midpoints of the equal sides of the isosceles triangleXYZ.AY = BY and A AYB is similar to A XYZ so LA = LX and AB is parallel to XZ.


51. circle, Level 4

Figure 0 is a circle. 0 is the centre.AOB = ZCOD, so AB = CD.


52. circle, Level 3isosceles triangle, Level 3

Figure C is a circle. 0 is the centre.

Prove that A AOB is isosceles.

275


C

Line 1 is parallel to AB.Because of properties of parallel lines we can )rove that Ll = LA and Z3 = LB.Now, / is a straight angle (180°).



Prove: If 1 1 is parallel to 1 2 and 1 2 is parallel to 1 3, then 1 1 is parallel to 1 3.



Will LA and LC be the same size? Why or why not?

?76


These circles with centres 0 and P intersect at M and N.

Prove: A OMP A ONP.

57. right triangle, Level 4Prove that the perpendicular from a point not on the line to the line is the shortest linesegment that can be drawn from the point to theline.

58. similarity, Level 4(This item was not part of the original Mayberry items. It was created when it was found thatthe Mayberry items did not necessarily assess Level 4 for the concept similarity.)

What is the least additional information needei to ensure that a pair of right trianglesare similar?

:!-77

Appendix H

AMENDED MARKING FORMATS

PAPER I(amended assessment)



Mark Levelcriteria

Totalscore

Square 1 Name 2 11 of 2Discriminate *9 2 2/2=1

2 Properties 16 2

*3 of 423a 1*25a 1

3 Definition 24 1

*4 of 6

Class Inclusn *25b *1Relationships 42a 1

42d 144b 1

Implications 23b 14 Proof 45 1

1 of 246 1RightTriangle

Name 3 11 of 2Discriminate 10 4 3/4=1

2 Properties 17 4 3 of 43 Definition *26 1

5 of 7

Implications 27 3Class Inclusn 32 2Relationships 44c 1

4 Proof 47 1

2 of 348 157 1

Circle 1 Name 51 of 212 2 2/2=1

2 • 19 3 2/3=1

*7 of10

33 a-e 5*35 4

—33f

*4 of 6

s 44d 1*52 1*34 *3

4 51— 1 *1Con:ruenc 7 1

1 of 215 12 Properties 22 4 3 of 43 Relationships 43 4 Deduct

1 pointif missa and c

Implications 44 5*56 1

Class Inclusn 42c 1 *7 of11

4 Proof 55 1 *1

?78

PAPER :LI(amended assessment)



Mark Levelcriteria

Totalscore

Square 1 Name 2 1Discriminate *9 2 2/2=1 1 of 2

2 Properties 16 223a 1*25a 1 *3 of 4

3 Definition 24 1 1Class Inclusn *25b *1Relationships 42a 1

42d 144b 1

Implications 23b 1 *4 of 64 Proof 45 1

46 1 1 of 2IsoscelesTriangle

1 Nam , 4 11 :

18

1Discriminate 2

21/2=1 1 of 2

2 of 2,

Properties3 Definition *28 1

Implications 30 3_42 149 — 1*52 — 1

Class Inclusn ' *31 *2. 32 — 2 *7 of

—-

114 Proof 48 1

50 1 *1 of 2Paralle 1 Name 6 1Lines

_Discriminate 13 1 1 of 2


39 1Relationships 38 5Implications 36 2 6 of 9

4 Proof 53 1'54 1 1 of 2

Similarity 1 Name 8 1Discriminate 14 1 1 of 2


Relationships 41 642 5

Class Inclusn 44e 1 8 of 134 Proof *58 1

48 1 *onlyif

appropriate

*1 (of 2)

* indicates where changes have been made to the original Mayberry assessment.

279

Appendix J

AMENDED RESULTS

PAPER I

Square Right triangle Circle Con:ruenc

SOl 1,0,0,0 1,0,0,0 1,0,0,0 1,0,0,0

SO2 1,1,0,0 1,1,0,0 1,1,0,0# 1,1,0,0#

S03 1,1,0,0 1,0,0,0 1,1,0,0 1,0,0,0

SO4 1,1,0,0 1,1,0,0 1,0,0,0 1,0,0,0

S05 1,1,0,0 1,1,0,0 1,1,1,0 1,1,0,0

S06 1,1,1,0 1,1,1,0 1,1,1,0 1,0,0,0#

S07 1,1,0,0 1,1,0,0 1,1,0,0 1,0,0,0

S08+ 1,1,0,0 1,1,0,0 1,1,0,0# 1,1,0,0

S09 1,1,0,0 1,1,0,0 1,1,0,0# 1,1,0,0

S10 1,1,1,0 1,1,0,0 1,1,1,0 1,0,0,0

S 1 1 1,1,0,0 1,1,0,0 1,1,0,0# 1,1,0,0

S12 1,1,0,0 1,1,0,0 1,1,0,0 1,1,0,0

S13 1,1,0,0 0,0,0,0 1.0.0.0 1,0,0,0

S14+ 1,1,1,1 1,0,1,0- 1,1,1,0# 1,1,1,0#

S15 1,1,0,0 1,0,0,0 1,1,1,0 1,1,0,0#

S16 1,1,0,0 1,1,1,C 1,1,1,0# 1,1,0,1*

S17 1,1,0,0 1,1,0,C 1,1,1,0 1,0,0,0

S18 1,1,0,0 1,0,0,0 1,1,0,0 1,1,0,0

S19 1,1,0,0 1,1,1,0 1,1,1,1 1,1,0,0#

S20 1,1,0,0# 1,1,0,0 1,1,1,0 1,1,0,0

S21 1,1,0,0 1,1,0,0 1,1,0,0# 1,1,0,0

S22 1,1,0,0 1,0,0,0 1,0,0,0# 1,1,0,0

S23 1,1,0,0 1,0,0,0 1,1,0,0# 1,1,0,0

S24 1,1,0,0 1,1,0,0 1,0,0,0 1,0,0,0

S24 1,1,0,0 1,1,1,0 1,1,1,0# 1,1,0,1*

S26 1,1,0,0 1,1,0,0 1,1,1,0 1,1,1,0#

S27 1,1,0,0 1,1,1,0 1,1,1,0 1,1,0,1*

S28 1,1,0,0 1,1,0,0 1,1,0,0 1,0,0,0

S29 1,1,0,1* 1,1,1, L 1,1,1,0# 1,1,0,1*

S30 1,1,0,0 1,1,0,0 1,1,0,0# 1,0,0,0

S31+ 1,1,0,0 1,1,0,0 1,1,0,0 1,1,0,0#Change in results

*Response Pattern Error +Interview subject

280

PAPER II

Square Isosceles

trianEle


S32 1,1,1,0# 1,1,1,1# 1,1,0,0 1,0,0,0S33+ 1,1,0,0 1,1,1,0 1,1,0,0 1,0,0,0S34 1,1,0,0 1,1,0,0 1,1,0,0 1,1,0,0S35 1,1,0,0 1,0,0,0 1,0,0,0 1,0,0,0S36 1,1,0,0 1,1,0,0 1,1,0,0 1,1,0,0S37 1,1,0,0 1,0,0,0 1,0,0,0 1,0,0,0S38 1,1,0,0 1,1,0,0 1,1,0,0 1,0,0,0S39 1,1,0,0 1,1,1,1# 1,1,0,0 1,1,0,0S40 1,1,0,0 1,0,0,0 1,1,0,0 1,1,0,0S41+ 1,1,0,1* 1,1,1,1 1,1,0,0 1,1,1,0S42 1,1,0,0 1,1,0,0 1,1,0,0 1,1,0,0S43 1,1,0,0 1,1,1,0 1,1,0,0 1,1,0,0S44 1,1,0,0 1,1,0,0 1,1,0,0 1,1,1,-S45 1,1,0,0 1,1,0,0 1,1,0,0 1,1,0,0S46 1,1,0,0 1,0,0,0 1,0,0,0 1,0,0,0S47 1,1,0,0 1,1,0,0 1,0,0,0 1,1,0,0S48 1,1,0,0 1,0,0,0 1,1,0,0 1,0,0,0S49 1,1,0,0 1,0,0,0 1,1,0,0 1,1,0,0S50 1,1,0,0 1,1,0,0 1,1,0,0 1,0,0,0S51 1,1,0,0 1,1,0,0 1,1,0,0 1,1,0,0S52 1,1,0,0 0,0,0,0 1,1,0,0 1,0,0,0S53 1,1,0,0 1,1,1,0 1,1,0,1* 1,1,0,1*S54 1,1,0,0 1,0,0,0 1,0,0,0 1,0,0,0S55 1,0,0,0 1,0,0,0 1,1,0,0 1,0,0,0S56 1,1,0,0 1,1,0,0 1,1,0,0 1,0,0,0S57 1,1,1,0 1,0,1,0' 1,1,0,0 1,0,1,0*S58 1,1,0,1* 1,1,1,0# 1,1,0,0 1,1,1,1S59+ 1,1,0,0 0,0,0,C 1,1,0,0 1,0,0,0S60 1,1,0,0 1,1,0,e 1,1,0,0 1,1,0,0S61 1,1,0,0 1,1,0,0 1,1,0,0 1,1,0,0# Change in results *Response Pattern Error +Interview subject

281

Appendix K

GUTIERREZ et al MARKING FORMATS

PAPER I (Gutierrez)

Concept Item number Vector( s)

Weighting

Level

1

Level

2

Level

3

Level

4

Square 2

9

16

23

24

25

42a

42d

44b

45

46

Arithmetic aNerage

Degree of acquisition


Weighting

Level

1

Level

2

Level

3

Level

4

Right triangle 3

10

17

26

27

32

44c

47

48

57

Arithmetic m erage


282

Paper I (cnt)

Concept Item number Vector s s)

Weighting

Level

1

Level

2

Level

3

Level

4

Circle 5

12

19

33

34

35

44d

51

52

Arithmetic average


Concept Item number Vector(s)

Weighting

Level

1

Level

2

Level

3

Level

4

Congruency 7

15

22

42c

43

44

55

56

Arithmetic average


PAPER II (Gutierrez )

Concept Item number Vectors)

Weighting

Level

1

Level

2

Level

3

Level

4

Square 2

9

16

23

24

25

42a

42d

44b

45

46

Arithmetic av erage



Weighting

Level

1

Level

2

Level

3

Level

4

Isosceles

triangle

4

11.

18

28

29 no level assessable for a rote definition

30

31

32

42b

48

49

50

52

Arithmetic average

Degree of acquisition _

2f;4

Paper II(cnt)

Concept Item number Vector, s)

Weighting

Level

1

Level

2

Level

3

Level

4

Parallel lines 6

13

20

36

37

38

39

53

54

Arithmetic average

Degree of acqu: sition

Concept Item number Vector , ․ )Weighting

Level

1

Level

2

Level

3

Level

4

Similarity 8

14

21

40

41

42

44e

48

58

Arithmetic average

Degree of acqu[sition

Appendix L

GUTIERREZ et al RESULTS

PAPER I

Square Right Triangle Circle Congruency

HNNNSO1 CLNN CLN N CLNN

S02 CCIN CHIN CCLN CCLN

S03 CHNN CC *NN CCNN CLNN

SO4 CHNN CHL N CH*LN CC*LN

S05 CCNN CCLN CCL*N HHLN

S06 CCIN CCH N CCI*N CC*IN

S07 CINN CHI\ N CHNN CINN

S08 CHNN CHL N CHNN CHNN

S09 CCLN CHI\ N CHNN HHNN

S10 CCIN CCLN CCL*N HH*LN

S 11 CCNN CHL N CHNN CCLN

S12 CHNN CHI\ N CHNN HINN

S13 CINN LLN N CLNN HNNN

S14 CCHL* CCCN# CCCL CCHN

S15 CCNN CH*NN CCL*N CCLN

S16 CCLN CCH N CHL*N CCH*L*#

S17 CHNN CHI\ N CCN*N CLNN

S18 CHNN CH*NN CHNN CHNN

S19 CCLN CCL 'NI CHL*L* CCLN

S20 CCIN CCIN CCL*N CCIN

S21 CHNN CC*I4N CCNN HINN

S22 CINN HHN N CLNN CINN

S23 CHNN CINN CHNN CHNN

S24 CHNN CINN CLNN CINN

S25 CCIN CCH N CCIN CC IL*#

S26 CCLN CCLN CCHN CCHN

S27 CCLN CCL '`I\T CCIN CCIL*#

S28 CCNN CCLN CCNN CH*NN

S29 CCIN*# CCHL* CCHN CCIL*#

S30 CHNN CCI\1N CHNN CH*NN

S31 CCLN CHNN CCNN CCNN* Does not correlate with amended Mayberry results# Response Pattern Error removed

2 .36

Gutierrez Results (cnt)

PAPER II

Square Isosceles

triangle


S32 CCIN CCHN* CHNN CC*NN

S33 CCIN CC IN CHLN CC*IL

S34 CHNN CHNN CCNN CCNN

S35 CHNN CC*LN CC*NN CH*NN

S36 CHNN CCLN CCLN CCLN

S37 CINN CH*NN IINN CINN

S38 CCNN CHNN CCNN CH*NN

S39 CCIN CCHN* CCLN CCLN

S40 CHNN CH*NN CCNN CCNN

S41 CCHN*# CC1-EL* CCLN CCIN

S42 CCLN CCNN CCNN CCNN

S43 CHNN CC EN CCNN CCNN

S44 CCLN CCIN CCLN CCI-

S45 CCLN CC IN CCNN CCLN

S46 CHNN CC*NN CLNN LLNN

S47 CCNN CCLN HC*NN+ CCLN

S48 CHNN CH*NN CCNN CLNN

S49 CHNN CC*NN CHNN CHNN

S50 CINN CHNN CINN CINN

S51 CHNN CCNN CCNN CCNN

S52 CHNN IH*NN+ CHNN CINN

S53 CHIN CCHN CCIN*# CCIL*#

S54 CINN CLNN CLNN CLNN

S55 CLNN CLNN CH*NN CLNN

S56 CINN CHNN CHNN CLNN

S57 CHIN CC* [N# CCNN CC*IN#

S58 CCIL*# CC UN CHLN CCIL*

S59 CHNN IIININ CINN CINN

S60 CCIN CCLN CCNN CCLN

S61 CHNN CCLN CCNN CCLN* Does not correlate with amended Mayberry results# Response Pattern Error removed+ New Response Pattern Error created

287

Appendix M

EIGHT TYPES OF RESPONSES: GUTIERREZ et al

Type 0 No reply or answers that cannot be (;odified.

Type 1 Answers that indicate that the learner has not attained a given level but that give no

information about any other level.

Type 2

Type 3

Type 4

Type 5

Type 6

Type 7

Wrong and insufficiently worked ot t answers that give some indication of a given

level of reasoning; answers that con .ain incorrect and reduced explanations,

reasoning processes, or results.

Correct but insufficiently worked of t answers that give some indication of a given

level of reasoning; answers that con ain very few explanations, inchoate reasoning

processes, or very incomplete result s.

Correct or incorrect answers that clearly reflect characteristic features of two

consecutive van Hiele levels and that contain clear reasoning processes and

sufficient justifications.

Incorrect answers that clearly reflect a level of reasoning; answers that present

reasoning processes that are complel e but incorrect or answers that present correct

reasoning processes that do not lead to the solution of the stated problem.

Correct answers that clearly reflect a given level of reasoning but that are incomplete

or insufficiently justified.

Correct, complete, and sufficiently j astified answers that clearly reflect a given level

of reasoning.

Weights of different types of answers

Type 0 1 2 3 4 5 6 7

Weight 0 0 20 25 50 75 80 100

(Gutierrez, Jaime & Fortuny 1991, pp.240-241)

288

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references adams, r. j. & khoo, s. (1993). quest - the interactive test analysis ... · 2019. 3....

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