-
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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
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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?
What word describes this?
-
/7r
a
E
.7 7z
8.
VeAre these figures alike in any way? YES NO
What word describes this?
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
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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
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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
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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
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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
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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
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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.
What have we proved?
238
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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°).
What have we proved?
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
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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
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PAPER II(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 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
2 Properties 21 4 3 of 43 Definition 40 1
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
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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
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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
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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
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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.
* Response Pattern Error
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
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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
ABCD is a square, BD is a diagonal.
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.
What have we proved?
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
In this figure AB and CB are the same length. AD and CD are the same length.
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
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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
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PAPER II
Concept Square Isosceles
triangle
Parallel lines Similarity
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
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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_
* Response Pattern Error
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
* Response Pattern Error
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
* Response Pattern Error
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
* Response Pattern Error
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
* Response Pattern Error
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
* Response Pattern Error
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.
What word describes this?
7. congruency, Level 1
B
What is true of A and B? What is true of C and D?
What word describes this?
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?
10. right triangle, Level 1
Which of these appear to be right triangles?
11. isosceles triangle, Level 1
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?
14. similarity, Level 1
Which figure appears to be similar to a?
15. congruency, Level 1
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.
20. parallel lines, Level 2
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
-
21. similarity, Level 2
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?
22. congruency, Level 2
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
ABCD is a square, BD is a diagonal.
(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 =
36. parallel lines, Level 3
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
Give reasons for your answers.
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
Give reasons for your answers.
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.
47. right triangle, Level 4
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.
49. isosceles triangle, Level 3
D
ABC is a triangle. A ADC === A BDC.
What kind of triangle is A ABC? Why?
274
-
50. isosceles triangle, Level 4
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.
What have we proved?
51. circle, Level 4
Figure 0 is a circle. 0 is the centre.AOB = ZCOD, so AB = CD.
What have we proved?
52. circle, Level 3isosceles triangle, Level 3
Figure C is a circle. 0 is the centre.
Prove that A AOB is isosceles.
275
-
53. parallel lines, Level 4
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°).
What have we proved?
54. parallel lines, Level 4
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.
55. congruency, Level 4
In this figure AB and CB are the same length. AD and CD are the same length.
Will LA and LC be the same size? Why or why not?
?76
-
56. congruency, Level 3
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)
Concept Level Question type Questionnumber
Score Questncriteria
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)
Concept Level Question type Questionnumber
Score Questncriteria
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
2 Properties 20 2 2 of 23 Definition 37 1
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
2 Properties 21 4 3 of 43 Definition 40 1
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
Parallel lines Similarity
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
Concept Item number Vector( s)
Weighting
Level
1
Level
2
Level
3
Level
4
Right triangle 3
10
17
26
27
32
44c
47
48
57
Arithmetic m erage
Degree of acquisition
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
Degree of acquisition
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
Degree of acquisition
-
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
Degree of acquisition
Concept Item number Vector( s)
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
Parallel lines Similarity
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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