1) The document examines how geometric diagrams are used differently in elementary and high school mathematics textbooks and how this relates to students' reasoning abilities. 2) Elementary textbooks often use diagrams as objects to identify, where students rely on visual appearance alone, fitting with van Hiele Level 0 reasoning. High school texts use diagrams as representations, requiring attention to precise definitions, as in van Hiele Level 2. 3) Tasks were analyzed through the lens of the van Hiele levels to understand how students at different levels might interpret the same diagram differently. Alignment between diagrams and students' reasoning levels is important for effective learning.

1. When Is Seeing Not
Believing: A Look at
Diagrams in Mathematics
Education
Aaron Brakoniecki
Michigan State University
Leslie Dietiker
Michigan State University
1

2. Problems
If you are a high school geometry teacher, you probably have had a student make a
ased on a relationship he or she derived from a diagram. For example, when a student is
entifying a shape like the one found in a common high school geometry textbook (see fi
not uncommon for that student to assume that the angles of the quadrilateral are right an
nd therefore conclude that the shape is a rectangle. While the shape in the diagram visua
ppears to have right angles, why does that student feel justified to make the claim that it h
ght angles in a deductive argument?
Tell whether the shape at right is a
parallelogram, rectangle, rhombus, or
square. Give all the names that apply.
gure 1. Textbook example adapted from a high school geometry text (E. B. Burger, et a
007, p. 423)
From a High School Text
What is the name of this shape?
From an Elementary School Text
Expected Answer: Parallelogram
Expected Answer: Rectangle
2

3. Thoughts on Diagrams
in Mathematics
• "A mathematician could always be fooled by his visual
apparatus. Geometry was untrustworthy. Mathematics
should be pure, formal, and austere.”
James Gleick (1987)
3

4. Purpose/Goals
• Explore how geometric diagrams are used in
different grade level textbooks as visual rhetoric to
convey meaning
• Describe possible curricular opportunities that may
assist students in adapting their reading of different
geometric diagrams
• Make recommendations for future studies
4

5. Theoretical Framework
• The van Hiele levels have often been used to try to describe
students’ development of geometric understanding
coincide with the van Hiele levels that students are reasoning at. For example,
discussions about the definitions and properties of shapes may not be effective if students
have only begun to reason about shape by comparing pictures of shapes to real world
objects that look similar.
This paper uses the van Hiele levels as a lens through which to compare the visual
rhetoric used in elementary textbooks compared to that found in high school textbooks. In
the same way that the effectiveness of words can be hampered by differing van Hiele
levels, we look at how the geometric diagrams used in mathematics at different grade
levels might be affected by differing van Hiele levels. To do this, we focus on the three
lowest of the van Hiele levels, denoted here as levels 0, 1, and 2. Burger and
Shaughnessy identified different indicators that suggest a student may be reasoning at a
particular. For the purposes of this paper, we included descriptions of some (but not all)
of the indicators at these three lowest levels.
Level 0
Students recognize shapes by their by appearance alone. The figure is perceived as a
whole, and the properties of the shape are not recognized.
Level 1
Students are able to sort shapes by single attributes and are able to recognize
properties of the shapes, but do not recognize relationships between the shapes.
Level 2
Students are able to define shapes and recognize that different criteria may be used to
define shapes. Students can sort shapes by a variety of mathematically precise
attributes and can recognize relationships between shapes (such as understanding that
a square is a special rectangle).
Task Descriptions and Student Reasoning
So what are the different ways that geometric diagrams are used in mathematics
discourse to be effective, the ways in which topics are discussed in the classroom should
coincide with the van Hiele levels that students are reasoning at. For example,
discussions about the definitions and properties of shapes may not be effective if students
have only begun to reason about shape by comparing pictures of shapes to real world
objects that look similar.
This paper uses the van Hiele levels as a lens through which to compare the visual
rhetoric used in elementary textbooks compared to that found in high school textbooks. In
the same way that the effectiveness of words can be hampered by differing van Hiele
levels, we look at how the geometric diagrams used in mathematics at different grade
levels might be affected by differing van Hiele levels. To do this, we focus on the three
lowest of the van Hiele levels, denoted here as levels 0, 1, and 2. Burger and
Shaughnessy identified different indicators that suggest a student may be reasoning at a
particular. For the purposes of this paper, we included descriptions of some (but not all)
of the indicators at these three lowest levels.
Level 0
Students recognize shapes by their by appearance alone. The figure is perceived as a
whole, and the properties of the shape are not recognized.
Level 1
Students are able to sort shapes by single attributes and are able to recognize
properties of the shapes, but do not recognize relationships between the shapes.
Level 2
Students are able to define shapes and recognize that different criteria may be used to
define shapes. Students can sort shapes by a variety of mathematically precise
attributes and can recognize relationships between shapes (such as understanding that
a square is a special rectangle).
Task Descriptions and Student Reasoning
discourse to be effective, the ways in which topics are discussed in the classroom should
coincide with the van Hiele levels that students are reasoning at. For example,
discussions about the definitions and properties of shapes may not be effective if students
have only begun to reason about shape by comparing pictures of shapes to real world
objects that look similar.
This paper uses the van Hiele levels as a lens through which to compare the visual
rhetoric used in elementary textbooks compared to that found in high school textbooks. In
the same way that the effectiveness of words can be hampered by differing van Hiele
levels, we look at how the geometric diagrams used in mathematics at different grade
levels might be affected by differing van Hiele levels. To do this, we focus on the three
lowest of the van Hiele levels, denoted here as levels 0, 1, and 2. Burger and
Shaughnessy identified different indicators that suggest a student may be reasoning at a
particular. For the purposes of this paper, we included descriptions of some (but not all)
of the indicators at these three lowest levels.
Level 0
Students recognize shapes by their by appearance alone. The figure is perceived as a
whole, and the properties of the shape are not recognized.
Level 1
Students are able to sort shapes by single attributes and are able to recognize
properties of the shapes, but do not recognize relationships between the shapes.
Level 2
Students are able to define shapes and recognize that different criteria may be used to
define shapes. Students can sort shapes by a variety of mathematically precise
attributes and can recognize relationships between shapes (such as understanding that
a square is a special rectangle).
Task Descriptions and Student Reasoning 5

6. Methods
• Surveyed texts and selected what appeared to be
typical tasks which asked students to use the
diagrams to reason about shape
• Used the van Hiele level framework to analyze how
students at different levels might interpret the
diagrams
6

7. Results
• We found that texts used geometric diagrams
differently across and within grade levels.
Diagram as Object
vs.
Diagram as Representation
7

8. Diagram as Objectadditional information about the objects. The only information students have to rely upon
are the visual images of the objects. Students are asked to identify the objects that are
circles as well as rectangles.
Figure 3. A kindergarten task adapted from a United States math textbook.
This task seems to fit very well with a type of reasoning about geometry that is
described in the Level 0 of the van Hiele framework. Here, students are expected to use
the visual qualities of the pictures of the objects to identify and sort the shapes. These
!
Directions: Have students mark an X on the objects that are circles. Then have
them put a circle around the objects that are rectangles.
A Kindergarden Task Adapted from a US Textbook
8

9. Diagram as Object
additional information about the objects. The only information students have to rely upon
are the visual images of the objects. Students are asked to identify the objects that are
circles as well as rectangles.
Figure 3. A kindergarten task adapted from a United States math textbook.
This task seems to fit very well with a type of reasoning about geometry that is
described in the Level 0 of the van Hiele framework. Here, students are expected to use
the visual qualities of the pictures of the objects to identify and sort the shapes. These
prototypes of the shapes are being used to help students pay attention to certain
characteristics of the objects. Thus, the student is expected to identify the dinner plate as
a circle and the clipboard as a rectangle based on appearance. However at this point,
students might not recognize the framed picture as a rectangle.
While this task seems fairly straightforward for someone who is reasoning at Level 0,
how might students at other levels of understanding of geometry interpret these shapes?
For students at Level 1, who begin to pay attention to the attributes of shapes, they might
be critical of the CD and the dinner plate, measuring the several different diameters to
check that it really is a circle and not an oval. These students might also argue that the
briefcase and the clipboard do not have straight sides, so therefore they are not rectangles.
However, students reasoning at Level 2, who begin to use mathematical definitions and
pay attention to mathematically precise attributes, may require additional information
about the shapes before they would classify the objects. They may want to know if the
sides of the clipboard are really parallel, or if the framed picture indeed has equal-length
sides.
van Hiele Level 1 Reasoning Task
The task in Figure 3 was adapted from a textbook intended for second grade
mathematics. Students are given a set of three triangles and asked to circle the one that
!
Directions: Have students mark an X on the objects that are circles. Then have
them put a circle around the objects that are rectangles.
A Kindergarden Task Adapted from a US Textbook
Level 0 Students might identify these
shapes based on visual qualities
Level 1 Students might recognize that the
properties of shapes may be
violated with the picture
Level 2 Students might not be able to
definitively identify the shapes
without additional information
8

10. Diagram as Object
A 2nd Grade Task Adapted from a US Textbook
“doesn’t belong.” Again we note that the images of the triangles are only the three sides
of the triangles, no additional information about the triangles are included.
Figure 3. A second grade task adapted from a United States math textbook.
This task seems to be aimed at those students who are reasoning in ways similar to
the ways described in van Hiele level 1. Here students may be sorting shapes based on a
single attribute, while ignoring other attributes. Indeed in each line of this task, two of the
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11. Diagram as Object
A 2nd Grade Task Adapted from a US Textbook
Level 0 Students might compare based on
the whole shape’s appearance
and not on attributes of the shape
Level 1 Students might sort shapes based
on a single attribute while
ignoring other attributes
Level 2 Students might not be able to
definitively separate these shapes
into categories without additional
information
“doesn’t belong.” Again we note that the images of the triangles are only the three sides
of the triangles, no additional information about the triangles are included.
Figure 3. A second grade task adapted from a United States math textbook.
This task seems to be aimed at those students who are reasoning in ways similar to
the ways described in van Hiele level 1. Here students may be sorting shapes based on a
single attribute, while ignoring other attributes. Indeed in each line of this task, two of the
triangles contain apparent right angles, while one triangle contains angles that appear to
measure 90°. This task seems to expect to have students identify the non-right triangle as
the figure that “doesn’t belong.”
Next, we consider how students who reason at a Level 0 might use these diagrams. In
the first line, students might identify the middle triangle as the one that does not belong
since its base is not horizontal, a common way that the image of triangles are portrayed.
A student reasoning at Level 2 might require additional information about the triangles,
and will not assume that angles that appear to measure 90° do.
van Hiele Level 2 Reasoning Task
The task in Figure 4 was adapted from a textbook intended for secondary school
geometry. In this task, students are given one diagram for several questions. In each
question, certain relationships between parts of the figure are stated. From these criteria,
students are expected to name the shape. It is noted that the diagram used does not appear
to satisfy any of the criteria of any of the questions. Also, while the questions all refer to
the same figure, some criteria are given for certain questions, while excluded from other
questions.
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12. Diagram as
Representation
A High School Task Adapted from a US Textbook
Figure 4. A high school geometry task adapted from a United States math textbook.
This question appears to require reasoning that is similar to the kind described in van
Hiele level 2. Students are expected to pay attention to the definitions of different shape
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13. Diagram as
Representation
A High School Task Adapted from a US Textbook
Figure 4. A high school geometry task adapted from a United States math textbook.
This question appears to require reasoning that is similar to the kind described in van
Hiele level 2. Students are expected to pay attention to the definitions of different shapes,
and explicitly reference them. They are also expected to sort shapes according to their
mathematically precise attributes. This prompt seems to expect students to pay attention
to the relationships described in each question, and not the visual shape of the diagram, in
determining the different shapes.
Consider how someone reasoning in a manner similar to Level 0 would engage with this
task. For this student, they have not yet begun to consider formal mathematical
definitions of shapes, or consider the implications of describing parts of the figure as
bisecting each other. Instead they might look at the diagram and conclude that the shape
does not look like any of the shapes they have special names for, regardless of the verbal
information printed in the text. Someone reasoning at a Level 1, might not be able to
consider a definition of figure in terms of its diagonals, instead of its sides.
Discussion
An alignment between student reasoning of geometry and reasoning about a diagram
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Level 0 Students might not be able to use
the characteristics described and
only use visual information from
the diagram
Level 1 Students might not be able to
define shapes based on diagonals,
only on sides
Level 2 Students might identify shapes
according to their mathematically
precise attributes
11

14. Object or
Representation?
A High School Task Adapted from a US Textbook
metric object, then the proper claim would be “not possible to tell.” However, such splitti
hairs is not the point of this task. Instead, this task offers students an opportunity to
monstrate their understanding of the meaning of congruence, as well as to revisit the meani
angle measure (that it does not depend on the visual length of the rays, but instead on the
ation from one ray to the other).
For each pair, decide
whether the two
figures are congruent.
Explain your
reasoning.
ure 4. Problem where the diagram represents the geometric object, adapted from Geometr
e CME Project, 2009, p. 164)
Other textual examples from high school which expect students to treat diagrams as
circles angles
12

15. Object or
Representation?
A High School Task Adapted from a US Textbook
because visually, it is longer than the side labeled 10 units. Therefore, students are inste
supposed to view the diagram as one possible triangle from an infinite set of triangles wi
sides of length 8 and 10 units.
In the triangle at right, which of the following CANNOT be the
length of the unknown side?
(a) 2.2
(b) 6
(c) 12.8
(d) 17.2
(e) 18.1
Figure 3. Problem where the diagram represents the geometric object, adapted from Geo
E. B. Burger, et al., 2007, p. 371)
10 8
13

16. Concluding
Comments
14

17. Possible Strategies
Use mis-drawn figures to identify students who are
using visual cues when reasoning shapes or
properties
arallelogram, students are less inclined to make claims from visual cues (such as parallel sides)
their reasoning. At the end of this task, the teacher can then revisit the prompt and ask student
bout the diagram provided. This offers an opportunity for students to recognize how the
agram misrepresented relationships and explore what a more accurate representation might
ok like.
Given: AB = CD and AD = BC
Prove: ABCD is a parallelogram.
igure 6. Task with misrepresented geometric diagram.
Another strategy to help students recognize the weakness of claims made based on visua
ppearance is to prepare several diagrams intentionally designed to misrepresent special
lationships using a dynamic geometry software program and ask students to name each shape
ee fig. 7). If students claim that the angle in figure 7 is a right angle, have the dynamic tool
A
C
D
B
15

18. Possible Strategies
supposed to view the diagram as one possible triangle from an infinite set of triangles wi
sides of length 8 and 10 units.
In the triangle at right, which of the following CANNOT be the
length of the unknown side?
(a) 2.2
(b) 6
(c) 12.8
(d) 17.2
(e) 18.1
Figure 3. Problem where the diagram represents the geometric object, adapted from Geo
E. B. Burger, et al., 2007, p. 371)
Making Claims from Diagrams As Geometric Objects
10 8
In a triangle with side lengths 8 units and 10 units, which of
the following CANNOT be the length of the unknown side?
Use multiple student-generated diagrams to
emphasize the generic nature of some diagrams
16

19. Possible Strategies
Investigating shapes that aren’t quite what they
appear to be
uare) until students start to recognize ambiguous relationships tha
eem.
What is the
name of this
shape? m!ABC = 89.87°
B
A
C
ple dynamic geometry diagrams that could be used to help demon
hat appear to exist are not necessarily true even when treating diag
ge in tasks can also help students recognize that perhaps diagram
17

20. Possible Strategies
Use of language in questions
Figure 7. Sample dynamic geometry diagrams that could be used to help demonstrate that
relationships that appear to exist are not necessarily true even when treating diagrams as objects.
Language in tasks can also help students recognize that perhaps diagrams are not what
they may seem. For example, in figure 8, by asking “Which of the figures below appear to be
parallelograms?” (italics added for emphasis), the text subtly indicates that even though shape B
may appear to be a parallelogram, it may not be.
Which of the figures below appear to be parallelograms? Explain.
Figure 8. Task which subtly indicates that the diagram may not represent the shape, adapted
from Geometry (The CME Project, 2009, p. 151)
Finally, we can help students develop a basic heuristic of recognizing assumptions when
looking at geometric diagrams by encouraging them to ask themselves, “What seems to be
true?”, “How do I know for sure that is true?”, and “What if that is not true?” This process can
help students start to use language like “this could be a right angle” (instead of “this is a right
angle”) or to make claims with conditional statements like “If this is a right triangle then…”.
B
A
C
D
E
18

21. Further Questions
• Where do diagrams as representations first appear in
textbooks?
• Do there exist diagrams in elementary texts that
use the diagrams as representations?
• What implications do the different roles of diagrams
have on assessments?
• How might different strategies assist students in
recognizing the roles of diagrams in texts?
19

22. Thank You!
Aaron Brakoniecki
brakoni1@msu.edu
Leslie Dietiker
dietike4@msu.edu
20