=Paper=
{{Paper
|id=Vol-2128/scitech1
|storemode=property
|title=Constructing and Defining 3D Polyhedra: A Design Study Fostering Early Mathematical Practice and Visualization
|pdfUrl=https://ceur-ws.org/Vol-2128/scitech1.pdf
|volume=Vol-2128
|authors=Megan Wongkamalasai,Rich Lehrer
}}
==Constructing and Defining 3D Polyhedra: A Design Study Fostering Early Mathematical Practice and Visualization==
https://ceur-ws.org/Vol-2128/scitech1.pdf
Constructing and Defining 3D Polyhedra: A Design Study Fostering
Early Mathematical Practice and Visualization
Megan Wongkamalasai, Rich Lehrer, Vanderbilt University
megan.j.wongkamalasai@vanderbilt.edu , rich.lehrer@vanderbilt.edu
Abstract: Developments within STEM fields increasingly rely upon new tools and methods of
visualization; yet, current early elementary mathematics curriculum largely neglect cultivating
children’s mathematical visualization skills. During a six day classroom design experiment, we
studied how young children co-developed concepts of space and 3D shape and practices of
defining and visualization. Our design is shaped by our commitments to theoretical developments
in the learning science that conceptualize learning as based in participation within local
mathematical practices that emerge from everyday activity.
Supporting early mathematical practice through space and visualization
Geometry and spatial mathematics are often overlooked in early mathematics education. When forms of spatial
mathematics are taught, instruction is often constrained to the naming and the decomposition and composition of
shapes (NRC, 2009). Yet, as developments in STEM fields increasingly rely upon visualization tools and spatial
reasoning, there is an increasing attention to what forms of early geometry experience might be foundational for
students’ long term STEM learning and engagement (NRC, 2006). Recent work in early geometry has helped to
identify forms of activities and tools that might support children’s geometry learning (Hawes et al., 2017; Ng &
Sinclair, 2015; Clements & Sarama, 2007). However, less is known about how geometry activities and tools operate
within the learning ecology to support young students in co-developing mathematical concepts of space and
mathematical practices. This is critical, as studies from the learning science have demonstrated that visualization is
intricately tied to disciplinary epistemic practices or ways of knowing (Stevens & Hall, 1998; Goodwin, 1994).
Thus, if the goal is to systematically develop students’ visualization skills, these skills must develop in relation to
the practices and goals of a discipline. In our instructional design experiment, we aimed to study how the learning
ecology and forms of visualization support young children’s (grade 1) co-development of the mathematical practice
of defining and concepts relating to 3D polyhedra and shape (primarily pyramids and prisms).
Context
The context of this work is a six day instructional unit on defining properties and classes of 3D polyhedra. This is
the first implementation of this instructional unit developed by the research team. We shared a written form of the
unit with the focus teacher, Mrs. B; however, this document acted as a boundary object between the research team
and teacher to help negotiate and guide classroom instruction and activities rather than as a set curriculum. Thus, the
instructional design was emergent over the six days to account for developments in student thinking, history of
activity, teacher input, and classroom sociomathematical norms. At the end of the unit, each student engaged in a
one-on-one post-interview with the first author, M.W.
The study took place in first grade classroom in a school located in a Southeastern, rural town in the U.S.
The classroom has 21 students with a range of cultural backgrounds (48% of students are emergent bilinguals). The
classroom teacher, Mrs. B, has been teaching first grade for 10 years; however, this is the first time that she has
engaged with this particular content area. Additionally, Mrs. B has a practice of eliciting student thinking and often
asks for students to share their solutions, but she has not engaged students in defining. Thus, not only is the practice
of defining new for students, but it is also new for Mrs. B in the context of mathematics. Despite these new defining
goals, Mrs. B acted as the lead teacher throughout the instructional unit, and M.W. served as a co-instructor. In the
capacity of co-instructor, M.W. talked with students during small group activities, posed questions to the class
during whole group to redirect discussions, and helped Mrs. B make instructional decisions around what next steps
would be most fruitful.
Instructional design and implementation
In this section, we first describe how our initial conjectures about what instructional supports and activities might
help meet the goal of co-developing students’ defining practice and concepts of shape. These conjectures informed
the initial design of instruction, and they convey our theoretical commitments and assumptions. Next, we provide a
summary of the enacted instructional design that highlights salient moments of student learning. We conclude with
results from the post-interviews to provide a more individual look at students’ development of defining and concepts
of shape.
Design conjectures and theoretical perspectives
We build on sociocultural views of mathematics that position mathematics as situated in everyday activity and
defined by the evolving practices and goals of the local community (Saxe & Esmonde, 2012; Lehrer & Lesh, 2003;
Gravemeijer, 1999; Lave et al.,1984;). Thus, in designing instruction to support students’ development of
mathematical concepts and practices, we not only attend to the development of students’ concepts, but also to how
the activities and instructional support help foster a mathematical community defined by their evolving collective
goals and practices. Because we are working with young children, we take particular care in considering how
children’s existing forms of activities and goal can serve as the foundations for continued mathematical learning and
engagement. This consideration helped inform the activity context of the unit—3D construction and design.
We aimed to look at how 3D construction and design activities can be leveraged to develop children’s
concepts of 3D shape (e.g., properties and measures of 3D polyhedra) and defining and visualization practices. Not
only is 3D construction an activity familiar to many children, but we conjectured that the goals children bring to this
task might align with mathematical goals of defining. For example, children often build structures for self-
expression, and they take great pride in describing their structures to others. The practice of defining in mathematics
plays a similar communicative function in mathematics; however, unlike in construction play, defining in
mathematics goes beyond self-description and instead requires negotiation around the words used to describe in
order to establish shared meaning (Kobiela & Lehrer, 2015). We hoped that having children build and describe 3D
structures to others would create a need for a shared language of description. In order to develop a language of
description focusing on mathematical properties of shape, we chose to use construction materials that would likely
elicit children’s existing mathematical resources, Magformers or 2D magnetic polygons (see Figure 2).
Definition in mathematics also plays a critical epistemic role by helping establish relations and distinctions
between concepts that can lead to new mathematical objects and questions for inquiry (Kobiela & Lehrer, 2015). In
alignment of our goals to establish a practice of defining and concepts of 3D shape, we conjectured that students
would need to do more than describe their 3D structures to each other in order to support defining. Thus, we
conjectured that by having students compare their structures to teacher identified sets of structures, students would
be more likely to attend to salient properties and relations between properties (i.e., number of faces in relation to the
shape of the base of a pyramid).
Finally, mathematical conceptions of space require visualizing and attending to patterns of invariances and
change through dynamic transformations (NCTM, 2014). A second goal of our study was to support children’s
mathematical visualization; thus, rather than considering 3D shapes as static, we emphasized defining properties of
space dynamically. By considering properties of space in relation to motions and transformations (i.e., rotations) we
hoped that this would support students in using visualization to further refine relations and properties of shape. In
the unit reported here, students primarily used rotations to consider how their sense of different properties was or
was not consistent with a new position in space. In our ongoing project, students develop further conceptions of
transformations to consider properties in relation to rotation and reflection symmetries; however, we thought it
important that students first develop a shared language to describe what properties of space are preserved by
different transformations.
Implementation
Day 1-2: Student constructions and descriptions of their structures
Mrs. B began the unit by communicating to students the goal of the unit: to construct and describe 3D structures to
others. To elicit some initial description words, Mrs. B presented the class with two different structures—a closed
tetrahedron and a tetrahedron missing one face. She then asked students how they would describe these structures to
others. Students generated a list of descriptive words that included both properties and name of 3D polyhedra and
2D polygons (Figure 2A). When students described a property of 3D shapes using everyday language and gesturing
to the corresponding part on the tetrahedrons, Mrs. B would record the word used by the students and relate this to
the mathematical word (i.e., sides are faces). Additionally, not all of the words produced were accurate. For
example, students described the tetrahedrons as cones, pyramids, and triangular prisms. Mrs. B recorded all words
with the hopes that clarification of these words would emerge over the course of the unit.
After developing a list of words that could be used to describe 3D structures, students used Magformers to build
whatever 3D structures they liked, and as students built their structures, Mrs. B and M.W. circled the classroom
asking students to describe their structures using the words previously produced. When students introduced a new
descriptive word, such as corners, Mrs. B would pause the class and introduce the new word before adding it to the
chart of descriptive words. Figure 1 shows a representative sample of the types of structures produced by the class.
After students had time to explore what the different types of structures they could make with the Magformers, Mrs.
B asked students to write down a description of one of their structures to share with the class, pointing to the list of
descriptive words as a guide. While students wrote down their descriptions, Mrs. B and M.W. looked for student
descriptions that ranged from broad (it is a closed 3D structure) to more specific (it is a pyramid with 4 triangle
faces).
Figure 1: Examples of students' intial constructions with Magformers
Next, students brought their written descriptions and structures to the floor for a whole group discussion.
Mrs. B selected students to read their descriptions of their structures, beginning with students with very broad
descriptions. Our intention in starting with broad descriptions and moving towards more specific descriptions, was
that students would begin to see the importance of precise descriptions in distinguishing between structures. As
students read their descriptions, Mrs. B asked students to identify what descriptive words the student used from their
list. She then wrote these words on a strip of paper and laid the paper in the middle of the circle. Students then had
to decide if their structures fit the description. The first student description was “a closed 3D structure.” Students
sorted their structures based on the property of closed vs. open. Most students had built a closed 3D structure; thus,
the result of this first sorting activity was a large pile of almost all of students’ structures. Mrs. B pointed out that the
initial description applied to lots of their structures. She then invited a student with a more precise definition to read
their description to the class. This students included pyramid in their description. Students then sorted their
structures into a “pyramid” group and a “non-pyramid” group (Figure 2B). This sorting activity revealed that
students had varied senses of what constitutes a pyramid. Some students included any structure with a pointy top or
triangles; other students appealed to image prototypes, stating that their structure “looks like pyramid.” Mrs. B and
M.W. attempted to get children to share whether they agreed with how students had sorted their structure; however,
the variability among students’ structures and their perceptions of what constitutes a pyramid was so great that there
was too little common ground to move forward.
(A) (B)
Figure 2: A) chart of student’ descriptive words for 3D structures,
B) students sorting of their structures based on their sense of
pyramids
Day 3-4: Comparing groups of structures and defining properties of shape
Because the variability of structures was overwhelming when including all students’ structures the previous day,
Mrs. B and M.W. chose examples of students’ structures to represent two sets of structures. The first set of
structures included examples of pyramids (hexagonal pyramid and square pyramid). The second set of structures
included examples of prisms (a rectangular prism and a pentagonal prism). In both sets we elected to include an
example of each class of shape that would likely be more familiar to students (a square pyramid and a rectangular
prism) and a less familiar example (hexagonal pyramid and pentagonal prism) in order to expand students’ image
sets of each class of shapes. In the remainder of implementation summary we trace how the example sets of
pyramids led to a classroom generated definition for pyramids; however, the development for prisms followed a
similar pathway.
Mrs. B began the third day of instruction by showing students the two example pyramids and a chart with
two columns—one for similarities and one for difference. Rather than telling students that both structures were types
of pyramids, Mrs. B simply asked students to help describe what were similarities and differences between the
structures. We decided based on students’ sorting of their own structures the previous day that students did not yet
understand names as classes of polyhedra; thus, we did not think it would be meaningful for students to compare the
structures knowing they were types of pyramids. During this discussion, a number of opportunities for defining
emerged as new properties were introduced and as disagreements about what properties the two examples shared.
For example, some students said that that the square pyramid was closed and that the hexagonal pyramid was open.
Previously, students described shapes as being open when you could “go inside the structure.” Because the hexagon
base was a frame with no cross bars and the square base had diagonal cross bars, students argued that you could fit
your hand inside the base of the hexagonal pyramid but not the square pyramid. Other students disagreed, and
argued that open meant that a structure was missing a face. To help resolve this disagreement Mrs. B brought back
to open and closed tetrahedron from day 1, and asked students whether the open tetrahedron was “open” in the same
way that the hexagonal prism. By comparing examples of two senses of open, students decided that a better way to
define open was when a face was missing. Thus, this led to a revision in their chart. Instead of one structure being
open and one structure being closed, they decided a new similarity was that both structures were closed.
A second defining opportunity arose when M.W. asked students what they meant by both structures being
“pointy.” While students pointed to the apex of each pyramids, others noted that the hexagonal pyramid was more
“pointy.” When asked what made one more “pointy,” one students noted that the hexagonal pyramid only had one
vertex. Other students quickly protested and began counting the number of vertices on each pyramid out loud. Mrs.
B asked for a student to come up and point to the vertices they were counting to help others see what they thought
was a vertex. By counting the number of vertices on each pyramid and ensuring that all students agreed on the
number of vertices, students clarified what was meant by vertex beyond just corners with small angles to include all
vertices regardless of angle measure. After establishing a new sense of vertices, M.W. turned back to the student that
said that the hexagonal pyramid only had one vertex. She asked the student to point out what vertex she was
referring to. When explaining why that vertex looks “different,” the student explained that all of the triangles meet
together and that’s what makes it “pointy.” M.W. took this opportunity to introduce the word “apex” to identify this
as a special vertex where all the triangles meet. She then asked if the other structure also had an apex. Students all
pointed out that the square pyramid also had an apex on the top vertex. To help further establish a definition for
apex, Mrs. B rotated the square pyramid so that it was resting on a triangle face and the position of the apex was
now longer appeared on the “top.” Mrs. B then asked students which vertex was now the apex. Some students said
that the pyramid now had two apexes—the two vertices closest to the ceiling, some students pointed to the top edge
of the pyramid pointing towards the ceilings, and some students argued that the apex had not changed, “it turns with
the structure.” M.W. directed students’ attention to the apex on the hexagonal pyramid, and asked students what
made this the apex. Students recalled that they had noticed that all the triangles “met together.” This helped students
reach a consensus that a pyramid always has one apex where all triangles meet rather than just the top of the
pyramid. These example of defining episodes show important development in students’ understanding of both
individual properties and relations between properties (i.e. number of faces and closure; vertices, apexes, and faces).
Day 4-6: Defining pyramids
Based on students’ list of similarities and differences between the examples of pyramids, we hoped that students
would next be able to generate a definition based on properties that can be generalized across all types of pyramids.
We first thought it would be important to expand the example set of pyramids; thus, we asked students to construct
examples of pyramids that we had not yet seen. From students’ constructions we selected a number of examples and
non-examples to present to the class. Mrs. B first showed students an octagonal pyramid (a pyramid with an octagon
base), and she asked to students whether this new structure was also a pyramid like the previous two examples. We
were surprised that some students argued that the structure was not a pyramid because it wasn’t “pointy” or it
couldn’t have an octagon; however, other students argued that it was a pyramid because it still had an apex. Mrs. B
told the class that their goal was to come up with a list of rules that would help them decide if the examples students
had built were pyramids. She placed a new chart title, “rules of pyramids” next to their list of similarities and
differences, and asked students what was a rule for pyramids. Students all agreed that the first rule was that all
pyramids have an apex, where all the triangles come together. Mrs. B turned students’ attention back to the
octagonal pyramid and asked whether this could be a pyramid based on their first rule. Once students agreed that
they could call this example a pyramid, Mrs. B pointed out that some people said that it couldn’t have an octagon as
a base. This led to the second rule, that a pyramid must have a base, but this base can be different shapes.
Next, Mrs. B introduced a non-example constructed by a student, a triangular prism. She asked students
whether this new structure was also a pyramid. Students objected, appealing to their rules by stating that the
structure does not a have an apex. M.W. suggested that maybe it could be a pyramid because it does have a base,
however, students objected and said that a pyramid has to have an apex where all the triangles meet and the
triangular prism didn’t have an apex because it didn’t just have triangles meeting together because it also had
rectangle faces. This exchange led to the final rule, that all the faces (besides the base) need to be triangles.
After adding this rule, a student suggested that another rule is that pyramids cannot be “skinny.” M.W.
presented a “skinny” triangular pyramid made up of a small equilateral base and three-long isosceles faces to help
demonstrate the students’ claim. The student then explained that the example built by M.W. was not a pyramid
because it was too skinny and there are no skinny pyramids in the real world, like in the desert. The rest of the class
quickly objected, and argued that it was a pyramid because it fit all of their rules. Seeing that the student was upset
by this interaction, M.W. took the opportunity to clarify a goal for defining in mathematics for students. She
expressed that mathematicians often have very precise ways of talking about shapes that they help them build and
imagine examples of shapes that maybe we don’t see every day. This exchange helped solidify the position of
students’ definition as a tool within their classroom mathematics community.
Figure 3: Classroom generated definition for pyramids
Post-interview results
At the conclusion of the instructional unit, M.W. conducted one-on-one post interviews with all 21 students. The
interview tasks consisted of six 3D structure student needed to sort as pyramids, prisms, or neither (the structure is
not a pyramid or a prism). Students also had to explain why they assigned a structure to particular group. M.W.
made available to students a copy of the rules for prisms and pyramids that the class had generated as a tool to
support students’ reasoning and to analyze to what extent students understand the definitions as a tool.
Each student was shown six 3D structures (2 prisms, 2 pyramids, and 2 neither) listed in Table 1. The items
were scored as correct or incorrect. Students reasoning were scored based on three categories: 1) systematic defining
(student used definitions as a tool for explaining), 2) listing of properties (student identified properties but did appeal
to definition), and 3) informal judgements (appealed to prototype reasoning).
90.5% of students were correct on greater than 80% of items (5 or more items), and 52.4% of student used
definitions systematically on at least 80% of items. It is also notable that students that did not use definitions
systematically on all items often improved in their use of definition throughout the interview. Given that this is a
new practice for all students, these findings suggest that defining is accessible to young children, and we expect that
in future iterations the number of students using definitions systematically will increase as we adjust the
instructional unit to account for important forms of instructional support
Table 1: Overall Student Performance on Post Interviews
3D Structure Correct Use of Definition
Triangular pyramid 100% 86% (systematic) ;14% (lists properties)
Rhombic prism 62% 38% (systematic); 62% (lists properties)
Hexagonal anti-prism 86% 62% (systematic); 33% (lists properties);
(neither) 5% (informal)
Cube (prism) 100% 71% (systematic); 29% (lists properties)
Octagonal pyramid 100% 86% (systematic); 14% (lists properties)
Rhombus base w/ triangle faces (neither) 86% 71% (systematic); 29% (lists properties)
Challenges
Despite greater attention to geometry in elementary mathematic standards, such as the Common Core, early
elementary classrooms still devote little instructional time to this area of mathematics. This is in part due to issues of
accountability testing that privilege number and arithmetic. Teachers and administrators feel greater pressure to
devote the majority of instruction to these topics. However, while the goal is to develop a more integrated and
cohesive mathematics instruction that privileges geometry, there is little research to help inform this transition
towards a more spatialized mathematics curriculum.
One consequence of this tension between accountability and weak research base is a lack of school settings to
conduct ongoing research. We have encountered numerous obstacles within schools to find partnering teachers who
are interested in developing their geometry instruction. We also have faced obstacles at the district level from
administrators who do not endorse and ultimately provide permission for research in geometry instruction because
they see it as taking away from instructional time. Finally, as geometry continues to be positioned at the periphery of
early elementary mathematics, preservice and teachers spend little professional development time improving their
knowledge of geometry and understanding of children’s thinking in relation to geometry. When we work with
teachers, they are often learning the geometry content at the same time as their instructing. This creates challenges
as it is difficult to support teachers in anticipating student thinking in order to provide instructional support when
they have little sense of where instruction is leading. Thus, our design study supports understandings of way to
support both children’s and teachers’ geometry learning.
Connection
Development within STEM fields increasingly rely upon new tools and methods of visualization; yet, current early
elementary mathematics curriculum largely neglect cultivating children’s mathematical visualization skills.
Additionally, while the learning sciences have helped establish new ways of conceptualizing learning that emphasize
students’ access to disciplinary practice, we still understand little about how young children can learn to participate
in these practices. Thus, development in both STEM fields and the learning sciences present new instructional goals
and challenges regarding how to design new form of mathematical experiences for children. In our instructional
design aimed at supporting young children’s co-development of mathematical practices and concepts of shape and
space, we attempted to address these new goals and challenges. Our design is greatly influenced by work in the
learning science regarding what it means to engage in mathematics and the situated nature of visualization, but it is
also focused on the horizon of where STEM fields are heading and the types of experiences that will prepare
students for long term STEM learning.
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