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When Is Seeing Not Believing: A Look at Diagrams in Mathematics Education Aaron Brakoniecki Michigan State University Leslie Dietiker Michigan State University 1
When Is Seeing Not Believing: A Look at
Diagrams in Mathematics Education
Aaron BrakonieckiMichigan State University
Leslie DietikerMichigan State University
1
Problems
1
Is Seeing Believing? The Role of Diagrams in Geometric Reasoning
Abstract: Geometry students are expected to make deductive claims based on information that
may include a diagram. However, sometimes students make inappropriate claims when diagrams
are misleading. This paper looks at the role of diagrams in the high school geometry classroom
and attempts to illustrate how texts expect students to make claims based on diagrams. Several
strategies are offered to help students understand how diagrams may be misleading.
If you are a high school geometry teacher, you probably have had a student make a claim
based on a relationship he or she derived from a diagram. For example, when a student is
identifying a shape like the one found in a common high school geometry textbook (see fig. 1), it
is not uncommon for that student to assume that the angles of the quadrilateral are right angles
and therefore conclude that the shape is a rectangle. While the shape in the diagram visually
appears to have right angles, why does that student feel justified to make the claim that it has
right angles in a deductive argument?
Tell whether the shape at right is a parallelogram, rectangle, rhombus, or square. Give all the names that apply.
Figure 1. Textbook example adapted from a high school geometry text (E. B. Burger, et al., 2007, p. 423)
The fact that most students have previously been expected to make claims about shapes
specifically from visual diagrams may be surprising. While learning about shapes in elementary
school, students are often asked to identify shapes using visual evidence, in tasks similar to the
one shown in figure 2. However, a shift occurs by the time students are in high school, where a
From a High School Text
What is the name of this shape?
From an Elementary School Text
Expected Answer: Parallelogram
Expected Answer: Rectangle
2
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
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
Theoretical Framework• The van Hiele levels have often been used to try to describe
students’ development of geometric understanding
The van Hiele framework is a hierarchal framework that describes different levels of students’ reasoning in geometry. The five levels used in the framework describe different ways that students could reason about geometric topics, from visualization and analysis at lower levels to abstraction, deduction and rigor at higher levels (Burger & Shaughnessy, 1986). The van Hiele framework explains that different individuals (such as the teacher and a student) can use the same words (such as “rectangle”) with different intension, and thus essentially talk past each other. For example, a teacher might want to talk about a quadrilateral with four right angles, but the student might instead think of the image of a rectangle and not even recognize that it has four right angles. Thus, for classroom 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
So what are the different ways that geometric diagrams are used in mathematics textbooks to convey information? In this section, we examine three different tasks adapted from actual prompts found in a variety of United States texts across different grade levels. For each, we discuss the diagram(s) provided in the task, as well as how it might be perceived by students of different van Hiele levels. van Hiele Level 0 Reasoning Task
The task in Figure 2 was adapted from an activity in a text intended for a kindergarten classroom. As part of the task, students are presented with pictures of real world objects. We note here that the pictures do not include any additional markings added to indicate
The van Hiele framework is a hierarchal framework that describes different levels of students’ reasoning in geometry. The five levels used in the framework describe different ways that students could reason about geometric topics, from visualization and analysis at lower levels to abstraction, deduction and rigor at higher levels (Burger & Shaughnessy, 1986). The van Hiele framework explains that different individuals (such as the teacher and a student) can use the same words (such as “rectangle”) with different intension, and thus essentially talk past each other. For example, a teacher might want to talk about a quadrilateral with four right angles, but the student might instead think of the image of a rectangle and not even recognize that it has four right angles. Thus, for classroom 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
So what are the different ways that geometric diagrams are used in mathematics textbooks to convey information? In this section, we examine three different tasks adapted from actual prompts found in a variety of United States texts across different grade levels. For each, we discuss the diagram(s) provided in the task, as well as how it might be perceived by students of different van Hiele levels. van Hiele Level 0 Reasoning Task
The task in Figure 2 was adapted from an activity in a text intended for a kindergarten classroom. As part of the task, students are presented with pictures of real world objects. We note here that the pictures do not include any additional markings added to indicate
The van Hiele framework is a hierarchal framework that describes different levels of students’ reasoning in geometry. The five levels used in the framework describe different ways that students could reason about geometric topics, from visualization and analysis at lower levels to abstraction, deduction and rigor at higher levels (Burger & Shaughnessy, 1986). The van Hiele framework explains that different individuals (such as the teacher and a student) can use the same words (such as “rectangle”) with different intension, and thus essentially talk past each other. For example, a teacher might want to talk about a quadrilateral with four right angles, but the student might instead think of the image of a rectangle and not even recognize that it has four right angles. Thus, for classroom 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
So what are the different ways that geometric diagrams are used in mathematics textbooks to convey information? In this section, we examine three different tasks adapted from actual prompts found in a variety of United States texts across different grade levels. For each, we discuss the diagram(s) provided in the task, as well as how it might be perceived by students of different van Hiele levels. van Hiele Level 0 Reasoning Task
The task in Figure 2 was adapted from an activity in a text intended for a kindergarten classroom. As part of the task, students are presented with pictures of real world objects. We note here that the pictures do not include any additional markings added to indicate
5
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
Results
• We found that texts used geometric diagrams differently across and within grade levels.
Diagram as Objectvs.
Diagram as Representation
7
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 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
8
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
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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 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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9
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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9
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
is, undoubtedly, preferred. However, several interesting things can occur when there is a non-alignment in the ways that students are reasoning about geometry and the reasoning required to use a diagram in its intended way. If a student is at a lower van Hiele reasoning level than what the diagram is requiring, students may end up using empirical evidence from the diagram in their justifications of naming the shape. The diagram may not be intended to be used in this manner, especially if it is not drawn to scale. As stated previously, this may occur when students try to claim a figure is a rectangle, while a problem may be set up only to indicate that the figure is a quadrilateral.
The mismatch of visual rhetoric for different van Hiele levels can also occur in the other direction as well. Students may be at a higher van Hiele reasoning level than the
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11
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
is, undoubtedly, preferred. However, several interesting things can occur when there is a non-alignment in the ways that students are reasoning about geometry and the reasoning required to use a diagram in its intended way. If a student is at a lower van Hiele reasoning level than what the diagram is requiring, students may end up using empirical evidence from the diagram in their justifications of naming the shape. The diagram may not be intended to be used in this manner, especially if it is not drawn to scale. As stated previously, this may occur when students try to claim a figure is a rectangle, while a problem may be set up only to indicate that the figure is a quadrilateral.
The mismatch of visual rhetoric for different van Hiele levels can also occur in the other direction as well. Students may be at a higher van Hiele reasoning level than the
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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
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Object or Representation?
A High School Task Adapted from a US Textbook
6
However, this transition may be even more challenging due to the fact that high school texts
sometimes expect students to make claims based on geometric diagrams as well. For example,
consider a task that asks students to confirm a relationship via measurement or visual
comparison, such as the one in figure 4. Tasks such as these expect students to make geometric
claims based on the information in the diagram, and not to view the diagram as a representation
(and possible misrepresentation) of imaginary geometric objects. If these objects were only
representations of imaginary shapes, and could therefore be misrepresentations, then how would
comparison be possible? What if the right-hand circular diagram was actually drawn to be
0.01% larger in diameter than the left-hand circle? Or the right-hand angle is actually 1° smaller
than the other? If the student is expected to treat the diagram as only a representation of a
geometric object, then the proper claim would be “not possible to tell.” However, such splitting
of hairs is not the point of this task. Instead, this task offers students an opportunity to
demonstrate their understanding of the meaning of congruence, as well as to revisit the meaning
of angle measure (that it does not depend on the visual length of the rays, but instead on the
rotation from one ray to the other).
For each pair, decide whether the two figures are congruent. Explain your reasoning.
Figure 4. Problem where the diagram represents the geometric object, adapted from Geometry. (The CME Project, 2009, p. 164)
Other textual examples from high school which expect students to treat diagrams as
geometric objects are regularly found in tasks involving measurement or transformations. For
circles angles
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Object or Representation?
A High School Task Adapted from a US Textbook
5
very close to 90° might appear to be right angles but actually are not, and which could help
others recognize that the assumption that the angles are right angles is unjustified.
However, other diagrams found in geometry tasks are instead used to represent a general
class of shapes rather than the particular shape. For example, for the task in figure 3, students
are supposed to use measures of lengths (albeit that do not indicate units) from the diagram to
make a claim about the third length. However, if a student were to treat the diagram as the
geometric object itself, clearly the unknown side could not possibly be shorter than 10 units
because visually, it is longer than the side labeled 10 units. Therefore, students are instead
supposed to view the diagram as one possible triangle from an infinite set of triangles with two
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 Geometry. (E. B. Burger, et al., 2007, p. 371)
Making Claims from Diagrams As Geometric Objects
Given that elementary texts regularly ask students to make geometric claims from
diagrams (such as whether or not a shape is a rectangle), it is not difficult to understand why it
may be challenging for students to transition to avoiding claims based on geometric diagrams.
10 8
13
Concluding Comments
14
Possible Strategies
Use mis-drawn figures to identify students who are using visual cues when reasoning shapes or
properties
8
Helping Students Recognize the Relative Validity of Claims Based on Diagrams
As shown previously, there are times in high school when diagrams are the object of
study, and other times when diagrams are only a possible representation. Helping students learn
when it is appropriate to make claims from diagrams is clearly important for teachers. Below, we
introduce several different strategies that teachers might use to address this challenge.
One way that teachers can help students recognize how diagrams can misrepresent
relationships is to provide clearly misleading diagrams. For example, consider a task that asks
students to prove that a shape is a parallelogram based on a given set of statements (see fig. 6).
Since the diagram accompanying the prompt is intentionally drawn to not look like a
parallelogram, students are less inclined to make claims from visual cues (such as parallel sides)
in their reasoning. At the end of this task, the teacher can then revisit the prompt and ask students
about the diagram provided. This offers an opportunity for students to recognize how the
diagram misrepresented relationships and explore what a more accurate representation might
look like.
Given: AB = CD and AD = BC
Prove: ABCD is a parallelogram.
Figure 6. Task with misrepresented geometric diagram.
Another strategy to help students recognize the weakness of claims made based on visual
appearance is to prepare several diagrams intentionally designed to misrepresent special
relationships using a dynamic geometry software program and ask students to name each shape
(see fig. 7). If students claim that the angle in figure 7 is a right angle, have the dynamic tool
measure the angle and display that its measure is something other than 90°, such as 89.87° as
shown in the example. Continue this strategy with other shapes (such as non-square rectangles
A
CD
B
15
Possible Strategies
5
very close to 90° might appear to be right angles but actually are not, and which could help
others recognize that the assumption that the angles are right angles is unjustified.
However, other diagrams found in geometry tasks are instead used to represent a general
class of shapes rather than the particular shape. For example, for the task in figure 3, students
are supposed to use measures of lengths (albeit that do not indicate units) from the diagram to
make a claim about the third length. However, if a student were to treat the diagram as the
geometric object itself, clearly the unknown side could not possibly be shorter than 10 units
because visually, it is longer than the side labeled 10 units. Therefore, students are instead
supposed to view the diagram as one possible triangle from an infinite set of triangles with two
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 Geometry. (E. B. Burger, et al., 2007, p. 371)
Making Claims from Diagrams As Geometric Objects
Given that elementary texts regularly ask students to make geometric claims from
diagrams (such as whether or not a shape is a rectangle), it is not difficult to understand why it
may be challenging for students to transition to avoiding claims based on geometric diagrams.
10 8
In a triangle with side lengths 8 units and 10 units, which ofthe following CANNOT be the length of the unknown side?
Use multiple student-generated diagrams to emphasize the generic nature of some diagrams
16
Possible Strategies
Investigating shapes that aren’t quite what they appear to be
9
that appear square) until students start to recognize ambiguous relationships that might not be as
they visually seem.
What is thename of thisshape? m!ABC = 89.87°
B
A
C
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
AC
DE
17
Possible Strategies
Use of language in questions
9
that appear square) until students start to recognize ambiguous relationships that might not be as
they visually seem.
What is thename of thisshape? m!ABC = 89.87°
B
A
C
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
AC
DE
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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?
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