Students’ Understanding of Volume and How it Affects Their Problem Solving A Classroom Research Project submitted to The Master of Arts in Teaching Program of Bard College by David Price Annandale-on-Hudson, New York May 24, 2012
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Students’ Understanding of Volume and
How it Affects Their Problem Solving
A Classroom Research Project submitted toThe Master of Arts in Teaching Program
of
Bard College
by
David Price
Annandale-on-Hudson, New York
May 24, 2012
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Contents
1 Introduction, Question, and Purpose 3
2 School and Classroom Context 5
3 Literature Summary 7
4 Description of Method 11
4.1 Outline of surface area and volume unit . . . . . . . . . . . . . . . . . . . . 114.2 Pre-assessment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.3 Clinical interviews . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
5 Analysis and Findings 15
5.1 Pre-assessment analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155.2 Case study: Travis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215.3 Case study: Nadia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245.4 Case study: Yolanda . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275.5 Case study: Ryan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305.6 Case study: Isabella . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335.7 Case study: Martha . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
6 Reflection and Implications 40
Bibliography 44
Appendix 45
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List of Figures
5.2.1 Travis’s response to Question 9 on the pre-assessment. . . . . . . . . . . . . 225.3.1 Nadia’s work on the last question in our interview. . . . . . . . . . . . . . . 265.5.1 Ryan’s work on the last question in our interview. . . . . . . . . . . . . . . 335.6.1 Isabella’s pre-assessment definitions of area, surface area, and volume. . . . 345.6.2 Isabella’s solution to Problem 8 on the pre-assessment. . . . . . . . . . . . 345.7.1 Martha’s solution to Problem 2 on the pre-assessment. . . . . . . . . . . . 375.7.2 Martha’s definition of surface area. . . . . . . . . . . . . . . . . . . . . . . 38
5.7.3 Martha’s answer to Question 9 on the pre-assessment. . . . . . . . . . . . . 39
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1
Introduction, Question, and Purpose
This project is centered on my students’ understanding of surface area and volume, specif-
ically in the context of problems similar to those on the New York State Regents Examina-
tion in Geometry. I wanted to see what my students’ conceptual understandings of surface
area and volume were and then explore how their conceptual understandings correlated
with their use of traditional formulas for finding those quantities for various solids such as
prisms and cylinders. This led to the following guiding question for my research: how are
my students’ understandings of volume and surface area tied to the methods they use to
grapple with Regents Geometry problems on those topics?
I was attracted to this line of questioning when my students had difficulty keeping
straight the formulas for area and circumference of a circle in a previous unit; this led to a
discussion of the notion of area and how the concepts of area and perimeter are reflected
in the formulas that students must memorize. The surface area and volume formulas
required of students on the Regents examination are only more numerous and difficult to
apply (even with a reference sheet), and thus I wanted to learn more about the thought
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1. INTRODUCTION, QUESTION, AND PURPOSE 4
processes of my students when they were deciding which formula to use and deciding
between finding surface area and finding volume.
One major sub-question in this project is how my students understand and reconcile
two notions of volume. The first is that of “packing”; this notion is highlighted in questions
on rectangular prisms and questions involving the number of unit cubes that can fit into
a shape. It is connected to an understanding of dimension; a segment has length, a square
has area, and a cube has volume. The second notion of volume is that of “filling”; this idea
is apparent when students are asked to verify empirically the volume of a cone or pyramid.
Of course these two understandings are of the same quantity, but I wanted to investigate
how my students’ grasp of these two different notions did or did not affect their ability to
solve traditional volume problems.
Another important factor I found myself investigating was how students grab context
clues from problems to help themselves decide whether to calculate surface area or volume.
For example, students with a grasp of the three-dimensional nature of volume were often
quick to figure out that it was necessary to find volume in a problem which mentioned
cubic feet. Perhaps more interesting were some of the incorrect ways in which students
brought their prior knowledge of surface area and volume (both abstract and applied) to
bear on problems they were solving for the first time.
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2
School and Classroom Context
This research project took place with students from the three sections of Geometry taught
at the Bronx Academy of Letters. Effective class size ranged from approximately twelve
students in the smallest section to approximately 25 students in the largest and best-
attended section. Most students were in the tenth grade, although a handful were in
ninth, eleventh, or twelfth. When this project was conducted, approximately 40 out of 65
students were registered to take the New York State Regents High School Examination in
Geometry in June 2012. Student desks were gathered in groups of three or four, although
some students made the choice or had been asked to sit and work independently during
class time.
The school-wide grading system at BAL is based on “learning targets” rather than
traditional grades. Students’ grades are calculated completely on whether or not they
have achieved “mastery” on particular targets. For the geometry students studied here,
mastery was assessed in the form of a weekly or biweekly period-long written assessment
which included up to ten learning targets from the current semester. On each target,
students received “mastery” (full credit), “approaching mastery” (no credit), or “remote
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2. SCHOOL AND CLASSROOM CONTEXT 6
mastery” (no credit). A student’s grade on a given target was his or her mastery level
as of the last time assessed; this was in the interest of reflecting long-term retention and
deemphasizing how long it took a student to master a given target.
Geometry students in my placement had one of four goals for the class: achieve a score
or 85 or more on the Regents exam, 75 or more, 65 or more, or simply pass the class. Each
student’s goal was determined by their prior performance and an individual conversation
between my mentor teacher and the student. Students taking the Regents took a more
difficult version of each Learning Target Assessment. The total number of targets out of
which a student was graded was also determined by their goal for the class. For example,
when there were 18 targets total at one point in the semester, the 85+ group was graded
out of 16 targets and the 75+ group was graded out of 11 targets.
The Bronx Academy of Letters consists of a high school and a middle school. The high
school has an enrollment of approximately 330 students, drawing the vast majority of
its student body from the surrounding neighborhoods in the South Bronx. Approximately
85% of the student population is eligible for free or reduced lunch, the four-year graduation
rate is 83% (with that figure rising to 94% for six years), and 19% of the student body
is deemed “college-ready” by the New York City Department of Education. 10% of the
school is classified as ELL and 21% as Special Education; these students are integrated into
the classes studied. The student body is 34% Black or African-American, 65% Hispanic or
Latino, and 1% Asian or Native Hawaiian/Other Pacific Islander. All demographic data
is from [9].
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3
Literature Summary
Childrens’ conceptions of volume and three-dimensional space have been studied exten-
sively. In [11], Piaget and Inhelder describe the difficulty of genuine two- and three-
dimensional thinking:
Hence the systematic difficulty found by children at earlier levels when tryingto relate areas and volumes with linear quantities. The child thinks of the areaas a space bounded by a line which is why he cannot understand how linesproduce areas. We know that the area of a square is given by the length of itssides, but such a statement is intelligible only if it is understood that the areaitself is reducible to lines, because a two-dimensional continuum amounts toan uninterrupted matrix of one-dimensional continua (350).
In other words, a deep understanding is difficult to achieve since it requires a huge leap
in the understanding of multiplication, from repeated addition of discrete quantities to
a continuous operation. This same phenomena is at work when children experience diffi-
culty understanding and working with volume. Indeed a rigorous treatment of area and
volume (i.e. measure theory) is significantly more difficult than the computation generally
required of high schoolers when solving volume problems. As put in [8], “developing mea-
surement sense is more complex than learning the skills or procedures for determining a
measure. . . however, classroom instruction is mainly focused on memorizing the formulas
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3. LITERATURE SUMMARY 8
to solve problems requiring low level of cognitive demand” (62). Volume and surface area
are two topics for which developing a rigorous and robust concept of the ideas in play
is significantly harder than any of the mathematical operations required to solve many
related problems.
Furthermore, different notions of volume and analogies used to describe volume serve
to color individuals’ understandings of volume, as well as their ability to parse and solve
different volume problems. The primary distinction made in the literature (e.g. [2,3,5]) is
that of “volume (packing)” versus “volume (filling),” terminology used in [3] and through-
out this paper. The notion of volume (packing) is related to the analogy of cubes; the
number of unit cubes which can fit inside a three-dimensional object is a measure of that
object’s volume. Volume (filling), on the other hand, is a notion of volume as measured by
a fluid; to find the volume of a shape, fill it with water and measure the volume of the wa-
ter in another container. The notion of volume (packing) is inherently three-dimensional,
while volume (filling) is more linear; indeed, in [3], it is claimed that an understanding of
area is a prerequisite for an understanding of volume (packing), while only an understand-
ing of length is required to acquire a notion of volume (filling). This distinction is further
supported in [5], in which it is shown that students arrive at a concept of volume (filling)
only with strong three-dimensional visualization skills, while such ability is less necessary
when volume is taught in terms of filling or capacity. In short, while both concepts of
volume are of course accurate, they are arrived at in very different ways, require different
sets of prerequisite skills, and thus stand to affect how students solve volume problems.
Another issue encountered when teaching and learning about volume and surface area
is the gap between concrete and abstract understandings of volume. In [7] it is noted that,
especially with volume and other physical quantities, “because students have experienced
and thought about the world, they do come to class with ideas, often ill-formed, hazy, and
inappropriate, but ideas nevertheless” (742). This means that a unit on volume must aim
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3. LITERATURE SUMMARY 9
to help clarify, extend, and abstract the prior knowledge of students, something that may
be particularly difficult when teaching a topic that can range from the very concrete to
the very abstract. In [1], Battista argues that the first level of geometric thinking consists
of interacting with concrete objects in isolated episodes, in which “students identify and
operate on shapes and other geometric configurations as visual wholes; they do not ex-
plicitly attend to geometric properties or to traits that are characteristic of the class of
figures represented” (88). For a student at this level of thinking, volume may indeed still
seem only a confusing list of unrelated formulas with no theoretical unifying idea tying
them together. This transition from the concrete and specific to the abstract and general is
always a difficult jump in mathematical thinking; according to Piaget in [10], the difficulty
of understanding spatial concepts lies in this jump, and “the problem is in fact none other
than that of the physical and experimental nature of mathematics as opposed to its being
of an a priori and purely intellectual character” (380).
This leaves last the issue of how to access students’ understandings of volume, since one
of the difficulties many students experience is precisely that their personal concept of vol-
ume is somehow flawed and is the driving force behind their incorrect answers to problems.
Clinical interviews are one type of tool that can be used to access these understandings,
since, according to [6], “the interviewer makes every effort to be child-centered–to see the
issues from the child’s point of view” (113). This sort of low-stakes and investigative con-
versation allows the interviewer to discover more about a child’s thinking than is possible
in a traditional classroom setting. At the same time, interviews can inform teaching, since
“good teaching.. . sometimes involves the same activities as those comprising formative
assessment: understanding the mathematics, the trajectories, the child’s mind, the obsta-
cles, and using general principles of instruction to inform the teaching of a child or group
of children” ([6], 126). By using interviews to get at student ideas (as opposed to mere
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3. LITERATURE SUMMARY 10
answers), it is possible to understand more deeply the source of their misunderstandings
about volume and other mathematical concepts.
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4
Description of Method
4.1 Outline of surface area and volume unit
The field component of this project took place from May 9 to May 18, 2012. Starting
with a pre-assessment task on area, surface area, and volume, the data collection consisted
primarily of work collected from the pre-assessment and clinical interviews conducted with
students, most of whom attended “office hours” mandatory for those signed up for the
Regents Examination (approximately three fourths of the students fall into this category).
Students were selected for interviews based on availability and output produced on the
pre-assessment. The following is a timeline of the unit and data collection for this project.
More detailed lesson plans and tasks for the days which pertain to this project can be
found in the appendix.
Day 1: Students were given 25 minutes to work on a pre-assessment task involving area, sur-
face area, and volume. Questions included those asking for students’ “own words”
definition of surface area and volume, as well as traditional volume problems, and
problems asking students to determine what quantity was being asked for in a par-
ticular word problem. See the appendix for a copy of the pre-assessment.
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4. DESCRIPTION OF METHOD 12
Day 2: Objective: To discover the properties of rectangular prisms and investigate their vol-
umes.
Students began to work through a task involving the volume of rectangular prisms.
In addition to more traditional volume and surface area questions, students also in-
vestigated the surface area to volume ratios of their prisms and collected data from
their classmates. A copy of the lesson plan and task for this day is included in the
appendix.
Day 3: Objective: To strengthen student knowledge of rectangular prisms’ volume and sur-
face area.
Students continued the task of the previous day and solved more traditional rectan-
gular prism problems.
Day 4: Objective: To extend student knowledge of rectangular prisms to non-rectangular
prisms.
Students investigated and solved problems involving the surface area and volume of
triangular prisms, framed by the following questions: “How can we use our knowledge
of rectangular prisms on other prisms? What still works and what doesn’t?”
Day 5: [Continuation of previous day’s work.]
Day 6: First Learning Target Assessment to include volume and surface area.
In addition to six targets from earlier units, three targets were included on volume
and surface area. A copy of these targets and test questions can be found in the
appendix.
20: “I can define volume and surface area and recognize them in context.”
21: “I can find the volume of basic prisms and use it to solve problems.”
22: “I can find the surface area of basic prisms and use it to solve problems.”
Two clinical interviews (Travis and Yolanda) were conducted on this day.
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4. DESCRIPTION OF METHOD 13
Day 7: Objective: To investigate the volume and surface area of cylinders; to solve problems
involving these quantities.
Students investigated the volume of cylinders by finding the volume of beakers and
graduated cylinders. They also investigated the lateral area of cylinders by measuring
canned goods and their labels. One interview (Nadia) was conducted. A copy of the
plan and task can be found in the appendix.
Day 8: Most students absent on field trip. With students who were present, continuation of
cylinders and comparison to cones. Three interviews (Ryan, Isabella, Martha) were
conducted.
Day 9: Continuation of cones (beyond the scope of this project).
Day 10: Volume of cones and pyramids (beyond the scope of this project, end of unit).
Day 11: Second Learning Target Assessment of volume and surface area unit. Two new learn-
ing targets:
23: “I can find and apply facts about the volume of cylinders, cones, and pyramids.”24: “I can find and apply facts about the lateral area of cylinders and cones.”
4.2 Pre-assessment
All of my geometry students who were present on May 9 were given a 25 minute pre-
assessment to help me gauge their prior knowledge concerning area and volume. The
pre-assessment consisted of a question each about the surface area and volume of a rect-
angular prism, questions asking for students’ notions of area, surface area, and volume,
two questions involving nets which folded into three-dimensional shapes, two Regents ge-
ometry questions on volume (with the formula provided for one and not the other), a
question asking how to find the volume of an irregular shape, and questions which asked
students whether finding surface area or volume was more appropriate in a certain context.
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4. DESCRIPTION OF METHOD 14
Students were allowed calculators but no Regents examination formula sheets. A copy of
the pre-assessment can be found in the appendix.
4.3 Clinical interviews
Over the course of the unit, I interviewed six students during various free times in the
school day. Due to the fact that this unit culminated on the last day of my student
teaching placement, I was unable to wait until after the unit to interview any students.
I attempted to order my interviews by the individual Regents/non-Regents goals of the
students selected. This gave me a rough idea of who I could interview earlier in the unit;
earlier interviews are with students from the “higher” brackets, and so on. Each student’s
goal is included in the analysis of their interview. This ordering of interviews was only
approximate for logistical reasons and thus the day of each student interview is included
to put their answers in the context of the unit. When selecting students, I considered
their work on the pre-assessment, the types of questions they asked in class, and their
willingness to help me with my project.
An outline of the interview questions and the sheets shown to the students interviewed
can be found in the appendix. While the format of every interview differed, they each fol-
lowed the same basic structure and all included the slightly modified Regents examination
questions. According to [4], it is crucial during such interviews to “not include questions
that are outside the realm of what the respondent can answer” (103) and thus interviews
with students with weaker computational skills were often shorter. In addition to this,
these students often had the most interesting conceptual ideas of surface area and volume,
and so a larger portion of their interviews is centered on their conceptual understandings.
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5
Analysis and Findings
5.1 Pre-assessment analysis
The first two pre-assessment questions which are analyzed in this project asked students
to find the volume (Q2) and surface area (Q3) a rectangular prism measuring 5 inches by
7 inches by 4 inches, phrased in terms of one inch cubes and wrapping paper, respectively.
Counting a numerically correct answer with incorrect units as correct, 26 out of 52 students
found the volume of the rectangular prism correctly and none found the surface area (most
papers being left blank on that question).
The next questions asked for students’ definitions of area, surface area, and volume;
their responses are included in Table 1, with a blank space denoting a blank answer and
brackets denoting descriptions of a drawing or a formula. Notable patterns included several
students whose concepts of area and volume were tied to that of rectangles and rectangular
prisms, respectively. A notable trend in the definitions of surface area was comparison to
the concept of perimeter. Several students also used the word “mass” in their definition
of volume.
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5. ANALYSIS AND FINDINGS 17
I found that my students did not do nearly as well on the pre-assessment as I had
predicted. For example, I had expected that at least some students would correctly com-
pute the surface area of a rectangular prism and the volume of a triangular prism, but
no students did. With this in mind, there were several ways in which I found myself re-
evaluating the needs of my students as a group as we went into the surface area and
volume unit. First, as I had been warned by my mentor teacher, my students struggled to
translate mentally between three-dimensional objects, two-dimensional pictures of solids,
two-dimensional nets which folded into three-dimensional objects, and verbal descriptions
of three-dimensional objects. This led me to build in more hands-on activities with solids
than I had previously planned and spend more time during debriefs and mini-lessons
discussing the visualization of three-dimensional objects. Second, many of my students
seemed to confuse volume and surface area, two definitions which I then tried to reinforce
over the course of the unit with multiple explanations and analogies. Third, my students’
grasp of dimension was often shaky. For example, many students would multiply only
two lengths to find the volume of an object, an operation which would be dissonant to
a student who understood in a deeper way the invariantly three-dimensional nature of
volume and how units in different dimensions relate to each other. This led me to focus
on units and dimension more than previously planned, albeit in small ways in the course
of an activity or mini-lesson.
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5. ANALYSIS AND FINDINGS 21
5.2 Case study: Travis
Travis is a tenth grade student in my afternoon section of Geometry. While often classified
as a “troublemaker” with low grades, he clearly has a competitive streak and strong
memory. His personal goal for the Geometry class is to take and pass the Regents exam
with a score of 85 or above (i.e. the highest bracket). He can be sloppy when working on
mathematics problems, but is often able to provide more precision when prompted.
From Travis’s pre-assessment, it is clear that he has some prior knowledge of area,
surface area, and volume, but that they are somewhat muddled. For his definition of area,
he provides “the amount inside a figure,” for surface area, “the distance around a figure,”
and for volume, “the mass of a figure.” When calculating the volume of both prisms on
the pre-assessment, he finds all three dimensions and squares their product, thus for the
rectangular prism giving the square of the correct answer and for the triangular prism
giving the square of twice the correct answer. Also notable is that when asked for the
surface area of a rectangular prism he provides the volume. When calculating the volume
of a pyramid on the pre-assessment, he uses one dimension of the base rather than the
area of the base when applying the formula for volume. All of this data suggests that
his understanding of the three-dimensional nature of volume is shaky in a computational
context. In contrast to this, his answer on Question 9 is simple, clear, and seemingly
correct (see Figure 5.2.1).
My interview with Travis took place on the fourth day of the unit following the pre-
assessment, at which point we had explored the surface area and volume of rectangularprisms, triangular prisms, and cylinders. By this point, he seemed to have more articulate
descriptions of surface area and volume. For example, he describes “surface area” as “all
the areas of faces added up of a 3D shape,” an understanding emphasized in class when
studying prisms (but rephrased in his own words).
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5. ANALYSIS AND FINDINGS 22
Figure 5.2.1. Travis’s response to Question 9 on the pre-assessment.
Near the beginning of the interview, I present Travis with two rectangular prisms of
equal volume and different dimensions (a red 2 cm by 5 cm by 12 cm prism and a green
3 cm by 5 cm by 8 cm prism). It is here that he still demonstrates some confusion on the
interaction between different measures of three dimensional shapes. For example, he (like
most of the interviewees) selects the green prism. When asked why he chose the green
prism, he replies “because the height is bigger than this [red] one. . . oh no, but the width
is the same. . . I think the height makes the difference.” When asked why the height made
the difference, he replies that multiplying by the height is the last step to find volume. It
seems then that part of his confusion stems from his procedural understanding of how to
find volume of a rectangular prism. However, he gives a correct process for comparing the
volume numerically and seems satisfied by his process for comparing the volume of the
two prisms.
When asked two “which would you compute” questions, it becomes clearer that Travis’s
understanding of volume is heavily dependent on the context of the problem. For example,
he correctly claims that he would want to find the volume of a rectangular prism in order
to answer the question “How much air is in this room?” For this simpler question, the
language allows him to answer “because. . . it’s inside of the room, and volume is inside.”
However, when asked about how to find the amount of stone to build one of the Great
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5. ANALYSIS AND FINDINGS 23
Pyramids of Giza (of which I show him a picture), he quickly chooses surface area. His
reason: “It’s a structure you build from the outside not inside. . . you build a pyramid from
the outside so I say surface area.” His confusion on the problem has nothing to do with
his abstract notions of volume or surface area; rather he seems to have some difficulty
recognizing which is appropriate to compute in situations that are more ambiguous.
Again his prior understandings outweigh clues in the problem when presented with
the concrete slab problem. Asked to find the cost of a concrete building foundation, he
claims that “this is surface area, because we’re looking for the foundation.” He correctly
computes the surface area of the shape, even remembering unprompted to include ft2 as
the appropriate units. Especially in light of the fact that the phrase “cubic foot” is in the
problem twice, I wonder what has caused Travis to be so confident that finding surface
area is appropriate, and so I ask him. He replies that the foundation of a building is a like
a base, which makes him think of surface area. Again, his ability to parse a problem is
inhibited by some aspect of a procedure, this time the vocabulary used consistently when
finding volume and surface area. While I did not get a chance to ask him, this exchange
also makes me wonder what his understandings are of units such as ft2 and ft3 and how
they relate to “square feet” and “cubic feet.”
When asked about sizes of his favorite drink, he again shows that his understanding
of volume (and especially how scaling affects it) is heavily context-dependent. He quickly
answers “twice as wide” when asked whether he would rather double the height or the
width of his favorite drink. However, he is unable to elaborate on his explanation until he
looks back to the red and green prisms. He asks the clarifying question “twice as wide,
what happens to the width?” and I motion with my hands. At this point, he is able to
articulate; “Oh the length gonna get bigger too, if you do it the wide way, it’s going to be
the width and the length.”
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5. ANALYSIS AND FINDINGS 24
This response convinces me that Travis’s understanding of the different measures of a
shape (dimensions, surface area, and volume) are clearest when working with rectangular
prisms and realistic questions like “How much air is in the room?” While he is shakier
dealing with other shapes, he is often able to draw analogies or break down a problem
into simpler cases in a way that suggests he has a reasonable conceptual understanding
of volume supported by healthy mathematical habits of mind. The questions I asked him
gave him more trouble the more parsing was necessary. Overall, our interview suggested
that while his abstract notions of surface area and volume are relatively strong, they are
not flexible or abstract enough to guide him through a problem with “mixed context clues”
like the concrete problem. After his interview, I used more time in further interviews trying
to get at what specific words in problems led to students’ decisions.
5.3 Case study: Nadia
Nadia is another strong student and like Travis is in the 85+ Regents group. At the time of
our interview, she had the highest grade of all the students in all sections. She is articulate
and often able to reflect on her own understandings and correct her own mistakes, both
computational and conceptual. She seems to be one of the few students who never confuses
volume and mass, answering on the pre-assessment that neither quantity is immediately
related to answering the question “How heavy is a box of books?” On the pre-assessment,
her ability to find the volume of objects still seems rooted in rectangular prisms; like
approximately a dozen other students, she gives twice the correct answer for the volume
of a triangular prism (no students answered the question correctly) and gives “the depth,
width, and height” as a definition of volume.
In our interview (which took place after we started looking at cylinders), it seems she
has taken care to refine her ideas. Her description of volume is not only correct but far
from any notion discussed in class: “how much can fit into something. . . like I imagine a
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bowl, and the cover of the bowl, and how much will fit into the bowl, that won’t break the
cover.” When asked about surface area, she gives the example of the Earth’s crust. When
asked which rectangular prism has a greater volume, she refuses even to pick one, instead
confidently stating “you would find their volume and compare; length, width, height.”
When asked about finding the amount of air in the classroom, she gives a precise ex-
planation: “Volume. . . because it’s how much is in a room. . . my main words are ‘in’ or
‘around,’ volume is ‘in’ and surface area is ‘around.”’ Like Travis, she is good at looking
for key words in the problem given and says that when she sees one of those two words, she
usually doesn’t think any harder about which to compute. Also like Travis, she answers
that one would find surface area to calculate the amount of stone necessary to build a
pyramid, and gives a similar answer, showing similarly that her abstract understanding
of surface area, volume, and their distinction is not what is throwing her off, but rather
the “real-world” context of the problem itself. This is further confirmed by the ease with
which she described the process for finding the volume and surface area of a complex solid
(a square pyramid on top of a rectangular prism).
DP: How would you find the volume of this weird shape?Nadia: Cut it up into shapes. [Draws the pyramid and prism separately andshows me.] I’d break it up into different shapes, you’d add them together.DP: What about finding the surface area of that shape? How would you dothat?Nadia: I’d break it up again? But that base is shared, I wouldn’t necessarilyadd that base twice. It’s one base, I wouldn’t add it twice. . . wait, wait, youwouldn’t add it all, it’s practically inside it!
When asked how she would solve the concrete foundation problem, Nadia starts by
drawing a rectangular prism. When asked why she did this, she gives an interesting ex-
planation:
When I think of buildings, because I live in New York, I think of them as tallrectangles, flat tops!. . . the foundation. . . that’s underneath it right? [DP: Yes.]It’s for one of the bases, probably for the bottom base, so I’d find the areaof the bottom base. . . wait! Cubic foot, so that means. . . when I think of cubicfoot I think of volume. . . oh yeah, you probably just find the volume of that. . .
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Like Travis, she absorbs much of the context of the problem when figuring out where
to start. However, it seems that she also searches for mathematical clues which cause
her to revise her first ideas. It is this response that leads me to believe her conceptual
understanding of volume is quite deep in that she can weather misleading words and
information in such problems. While able to isolate context clues, she does not lean on
them to the exclusion of her abstract notion of volume.
Figure 5.3.1. Nadia’s work on the last question in our interview.
Her notion of volume is strong enough that she tentatively makes statements in our in-
terview like “sometimes I think about a pyramid and cone as the same, maybe I shouldn’t.”
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Without having formally been taught about these shapes, she recognizes that pyramids
and cones are related in that their volumes are found in similar ways, yet when asked why
she is concerned about conflating the two, she says that it is because they probably have
different formulas. It seems that even she still sees volume formulas as less related than
they are.
She is also the only student interviewed to prove to me that doubling the width of a
cylinder increases the volume more than doubling the height, and even recognizes that
she need not compute the volume of the “original” cylinder (see Figure 5.3.1). She is
the only student interviewed who quickly switches over into the language of cylinders
and volume unprompted. This suggests that while other students answered this question
correctly and were able to compute the volume of a cylinder when called such, much of
Nadia’s mathematical strength is in knowing when to switch in and out of the context of
a problem. She is often able to attain hints from the context of a problem, but does not
rely on it so heavily as to let context clues mislead her.
5.4 Case study: Yolanda
Yolanda is absent from class more often than Travis and Nadia, and has voiced her frustra-
tion to me over her “gaps.” However, from previous units it was already clear to me that
she is strong when it comes to reading and parsing geometry problems, but that she has
difficulty with computation. Our interview was conducted approximately four days into
the unit, at which point we had explored prisms. She is part of the 75+ Regents group
and our interview was the first conducted for this project.
On the pre-assessment, she attempts to calculate the volume of a rectangular prism
by adding two copies of each dimension, getting “32 inch cubes,” suggesting that she is
confused about the three-dimensional nature of volume, instead trying to find some sort of
three-dimensional version of perimeter. She gives no definition of surface area or volume
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on the pre-assessment. However, her struggles to calculate volume do not seem to stem
from any lack of spatial ability; she is one of only four students who seems to correctly
identify both nets on question 5 of the pre-assessment, a question with so few correct
answers it is not even included in Table 1 above.
During our interview, her description of volume is a bit more fleshed out, if still shaky.
She describes volume as “what could fit into an object. . . you have an object and you put
something else in it, it has to be a certain size. . . you know how you measure it in like,
cubes, cubic and stuff? So I think about it in cubes.” This is one of the notions of volume
that had been discussed in class.
While the cubes notion seems to convey a solid idea of volume, it seems that Yolanda has
not abstracted this into a more universal concept of volume, for when asked to compare
the volume of the green and red rectangular prisms, she says “this one is wider. . . it could
be bigger cubes, but then again this one could be like tiny ones, you could fit a lot of tiny
little ones and its going to be more than this [green] one.” When asked to compare their
surface area, she refers back to a (correct) formula from memory, stating that she would
calculate and compare. However, with her discussion of smaller cubes and larger cubes in
mind, it seems that she might not have a strong concept of what that formula is helping
her to compare. Her notion of volume is related to that of “volume (packing)” described
in [3], but a superficial component of that notion (the size of a single cube) has led to a
misconception.
Her reliance on particular formulas for particular shapes to inform her idea of volume
shows up again when discussing how to calculate the amount of air in the classroom:
DP: Would you rather calculate volume, surface area, or neither to find outhow much air is in this room?Yolanda: Volume.DP: Why do you say volume?Yolanda:The air is in the room, and the volume is what’s inside the thing.DP: How would you do that?Yolanda: Go by the formula, with the base and the height, so you get the
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5. ANALYSIS AND FINDINGS 29
bottom, that would be the base, and the height, right? No,no, no, no, it’s asquare, so the length, the width, and height!
Even though her first method would have worked, we have only used that method in class
to find the volume of non-rectangular prisms, and thus it seems that Yolanda thinks that a
formula that works for all prisms only works for non-rectangular ones. This suggests that
perhaps that she does not always have a unified concept of volume when solving problems,
where by “unified,” I mean an idea that the same type of answer is desired regardless of
the shape at hand.
Yolanda is the only student interviewed to claim confidently that one needs to find the
volume of a pyramid to find the amount of stone required to build it; in fact, she makes
a case for needing to know both volume (to find the total amount of stone) and surface
area (to find the stone on the outside). When we look at the formula for the volume of a
pyramid, which was provided in the interview on a formula sheet but not yet discussed in
class, she quickly provides a reasonable rationale for the coefficient 1
3.
DP: Let’s look at the formula. . . why do you think we need the 1
3, the base and
the height?Yolanda: The height is included in the thing, the base, you need to calculatewhat’s inside. . . and the third because its not a whole, like, rectangle. . . youknow how for a triangle it’s half, so I’m guessing for that it’s a third.
Even though her definition of volume is shaky, she is skilled at reasoning by analogy.
Indeed in this case her analogy is especially apt in light of the fact that the cofficients in the
formulas for area of a triangle and volume of a pyramid can be viewed as the coefficients
that arise when taking the antiderivate of a linear and quadratic function, respectively.
This is another piece of evidence that Yolanda relies heavily on a combination of her
knowledge of formulas and her ability to reason out a problem through pattern-matching
and analogy, perhaps without ever having a clear concept of volume.
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This shows up in particular when she struggles with the concrete foundation problem.
First, she asks what a slab is (note: the word slab is used in the original Regents exam
problem and there is no accompanying figure), suggesting that she is heavily reliant on
non-mathematical context clues, especially in light of the fact that she answers in the
negative when I ask if the phrase “cubic foot” helps her to solve the problem. Second, she
begins by saying “you go by the formula,” first using a method for finding the volume of
a triangular prism before correcting herself.
From our interview it seems that Yolanda uses the context of problems and her strong
memory of vocabulary and formulas to help her solve problems, all the while still struggling
to develop an abstract notion of volume.
5.5 Case study: Ryan
Ryan is one of the few ninth graders in my Geometry classes, having passed the Algebra
Regents exam in middle school. He is a member of the 85+ Regents group, but his perfor-
mance on tests in the class ranges from failing grades to one of the highest grades across
all the sections. He is a very quiet student but is willing to participate in class discussion
when directly asked to do so.
From Ryan’s pre-assessment (and interview), it seems that he has conflated the notions
of mass and volume and that he has difficulty translating between the idea of volume and
questions such as asking how many unit cubes would fit in a given rectangular prism. For
example, on the pre-assessment, he gives an answer of “34 boxes” to a question asking
how many 1 inch cubes would fit inside a box measuring 5 inches by 7 inches by 4 inches.
Since I forgot to ask him about this question during our interview, the only interpretation
of his answer I can make is that perhaps he found twice the sum of each dimension and
made some minor calculation error. It also seems that he has difficulty with visualizing (or
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5. ANALYSIS AND FINDINGS 31
at least describing) three-dimensional shapes; on the pre-assessment, he gives “triangle”
as the shape formed by a net which actually folds into a pyramid.
In our interview, he articulates his definition of volume, which seems to be a mix of
correct and incorrect ideas: “volume. . . is basically, like, the inside of an object, like how
much the mass is. . . any type of object, could be a chair, a desk, could be a paper, a box.”
Notably he gives concrete examples as opposed to many of the other interviewees who
stick to prisms, cylinders, and so on unless prompted by the context of a problem.
Like most of the interviewees, he claims that, of the red and green prism, the green
has a greater volume. Like many of the others, his evidence is that “the height is taller,”
but follows up quickly with “you would find the volume, so you would do length times
width times height for both and see what your answer is and compare.” He seems at ease
describing the method for finding volume as he refers to the physical objects in front of
us, showing me the length, width, and height.
When asked how to find the amount of air in the room, Ryan replies “volume. . . because
the air is in the room, you’re not trying to find like how big the room is.” To him, it seems
that the notion of “how big” is more closely tied to surface area, while volume is a property
of the material of which an object is made. His notions of surface area and volume are
rooted in concrete examples, which makes me wonder if there is a connection between his
conceptual understandings and the fact that he seems to (unusually for the class) have a
harder time with abstract computational volume problems than problems in context. This
shows up again when I ask him how to find the volume of a shape I’ve drawn, a cylinder
with a cone attached at one end and a hemisphere at the other. He asks first “is that a
cone, with ice cream on top of it?” before proceeding to answer that one would “find the
volume of the half circle [i.e. hemisphere, again with visualization problems] first and then
the cone type and then the cylinder and add them” to find the total volume.
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5. ANALYSIS AND FINDINGS 32
Like many of my geometry students, Ryan often uses names of polygons and other
two-dimensional shapes when referring to three-dimensional shapes that resemble them
(e.g. “triangle” for pyramid). However, in our interview, he states that he has a hard
time seeing and drawing three-dimensional shapes, which suggests that, at least for him,
confusing the terminology is not just a vocabulary-based slip-up. When looking at the
concrete problem, he says “you start by drawing a rectangle, of course it would be 3D,
but I don’t know how to draw it like that,” and struggles to do so for a short while. In
this problem he leans on the context of the problem, and like several other interviewees
concludes that surface area is necessary, explaining that the concrete is the foundation of
a building, perhaps associating the foundation with a face of a three-dimensional object,
not unlike when he answers “surface area” for the Pyramid of Giza question.
It seems then to me that Ryan’s difficulties in solving surface area and volume problems
come from his difficulty visualizing abstract three-dimensional objects. He is quick to solve
problems when physical shapes (such as the red and green prisms) are in front of him,
and seems less likely in these situations to confuse surface area, volume, and mass. While
he is confident when working with manipulatives, he has a hard time taking a problem in
context and abstracting it, thus shying away from visualizations.
He was the only student I interviewed who showed me work on paper during our inter-
view and never drew a shape. On the “favorite drink” question, he tried to show to me
that both volumes were equivalent by starting with values for the area of the base and
the height of a cylinder, doubling each in turn, and substituting into the volume formula
(see Figure 5.5.1). It seems then that Ryan’s problem solving strategies involving volume
and surface area avoid visualization perhaps because he finds it difficult. His reliance on
concrete objects when giving examples and solving problems further suggest that his ab-
stract and concrete understanding of volume and surface area are both somewhat correct
but relatively disconnected.
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Figure 5.5.1. Ryan’s work on the last question in our interview.
5.6 Case study: Isabella
Isabella was one of two students I interviewed not signed up to take the Regents exam,
instead taking the class only for mathematics credit. While “low-skilled” in some ways,
she can be a hard worker, willing to re-examine her own ideas and check her work, and
often takes the time and effort to clear up her own misunderstandings and ask for help.
It seems from her pre-assessment that, in addition to associating these quantities with
rectangular prisms (“the box”), she has two misconceptions about area, surface area, and
volume. The first is that, like Ryan, she confuses volume with another quality of three-
dimensional objects; her definition is “the weight of the box.” Second, she seems to have
a misconception about area (see Figure 5.6.1).
This shows up in two questions on the pre-assessment; when finding the volumes of a
triangular prism and pyramid, she correctly multiplies her answer for the area of the base
by the height (and by 1
3for the pyramid), but in each case finds the area of the base by
adding the dimensions given (see Figure 5.6.2).
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5. ANALYSIS AND FINDINGS 34
Figure 5.6.1. Isabella’s pre-assessment definitions of area, surface area, and volume.
Figure 5.6.2. Isabella’s solution to Problem 8 on the pre-assessment.
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5. ANALYSIS AND FINDINGS 35
Isabella is the only interviewee who states explicitly that her concept of volume has
changed over the course of the unit: “When I think about volume, when I first thought
about volume, I thought it was like something like, how much the object weighs or how
much it holds, but after we learned about what volume is or whatever, I learned that
it’s about what can fit inside, how big the shape is.” However, her working knowledge of
volume still seems shaky, since when asked about the green and red prisms, she answers
“green. . . because when we did the project in class, we learned that the smaller the object
is, the more volume that it had.” (The project to which she refers included a discussion of
surface area to volume ratio, the major observation being that smaller boxes usually had
a higher ratio.) As a student, she claims that she often has to rely on formulas, and the
green/red prisms question is one for which she does, saying “I think you have to do length
times width times height to figure out the volume. . . you could do the same equation on
both.”
Isabella seems to lack confidence in her own computational skills and ability. For exam-
ple, she interprets my questions in the following way:
DP:Would you rather figure out volume, surface area, or neither in order tofind out how much air is in this room?Isabella: For the room, I would rather figure out how much volume is in it,because I figure volume is easier because you just have to figure out the lengthwidth and the height of the room and you could figure out all those measure-ments with a ruler and then figure out the volume.DP [later]: Would you rather figure out volume, surface area, or neither to findout how much stone to build this (very famous) pyramid?Isabella: Surface area than the volume, because finding the length, the width,and the height of the pyramid, it would take a lot to figure that out. . . becauseyou could easily measure the surface area of the pyramid.
For each of these questions, Isabella does not seem to consider which quantity would be
more appropriate, but instead takes my question as asked and provides which quantity she
would prefer to calculate. Previously, I thought that part of the wording of the question
was unimportant, but Isabella’s responses showed me how a student, especially one not
confident in their problem-solving ability, might perceive what I am looking for when
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5. ANALYSIS AND FINDINGS 36
asking such a question. In short, her responses made me wonder later what was her concept
of where a “right answer” comes from in mathematics.
Her responses when working with the concrete question supported this interpretation
of how she solves problems, as well as why she (and many other students) have a hard
time figuring out whether to find volume or surface area:
Isabella: I want to say volume because I feel like it’s more to work with thanit is for surface area. . . I don’t know. . . I could use the 15 feet by 15 feet by 2as length, width, height, yeah!DP: How do you know that finding volume would help you answer the ques-tion?Isabella: Because. . . I don’t know. . . the fact that it gives us the measure-
ments. . . it makes me think that I could use volume to answer this question,but once it comes up to the cost. . . and then I would try to divide it by two ormultiply by two to get a possible answer.
She does not seem to notice that the measurements could also be used to find surface area,
suggesting that one of the reasons my students have difficulty deciding whether to find
surface area or volume is that problems involving either often provide the same numerical
information. It seems that, somewhat like Yolanda, she is able to “get by” solving many
problems by pattern matching and using the fact that most of the problems given her
provide precisely the amount and type of information necessary to solve the problem.
Similarly, she explains the coefficient of 13
in the volume formula for a pyramid by analogy
to the relationship between a rectangle and a triangle. She does not seem to be as strong
as Yolanda when it comes to memorizing formulas and the names of three-dimensional
shapes, and thus she has a harder time picking what procedure to use.
Isabella strikes me as representative of many of the sharp students in my classes who
are struggling. She uses her critical thinking skills to search for patterns, draw analogies,
and identify important information, all good mathematical habits of mind. However, she
has a hard time distinguishing between mathematical patterns and patterns found in high
school math problems (e.g. only appropriate information is ever given). Furthermore, some
of her work is internally consistent, but incorrect due to some older misunderstanding (e.g.
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5. ANALYSIS AND FINDINGS 37
her misconception of area). My interview with Isabella left me primarily with the question
of how I can better harness the reflective ability of students like her to prevent and repair
such misconceptions.
5.7 Case study: Martha
Martha is one of the students in my classes with the weakest skills. She is not signed up
for the Regents exam and is classified as a special education student. However, she often
participates actively in discussion in class, asks questions, and provides a mix of correct
and incorrect answers during mini-lessons. Unlike most of the students in my classes, her
grasp of surface area seems to be stronger than that of volume. For example, to find the
volume of a prism on the pre-assessment, she adds the dimensions, but seems to give a
more accurate definition for surface area, drawing an arrow pointing to the lateral area of
the cylinder (see Figures 5.7.1 and 5.7.2).
Figure 5.7.1. Martha’s solution to Problem 2 on the pre-assessment.
By the time of our interview (following prisms and cylinders), she seems to be more
confident in her understanding of volume, even reaching an understanding that allows her
to circumvent her own computational difficulties. Her definition of volume is “how much
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5. ANALYSIS AND FINDINGS 38
Figure 5.7.2. Martha’s definition of surface area.
something can fit inside, like a box, or like in a cylinder, or something like that” and
surface area is “the whole perimeter of a shape, like all the faces.” Like all interviewees,
she selects the green prism as having more volume, but is unique among the interviewees
in her method for verifying her answer:
DP: Why the green prism?Martha: It may be a little bit small, but it has much more space than the redone.DP: Is there a way you could prove to me that the green prism has a largervolume than the red one?Martha: We could put stuff inside the green one and put stuff inside the redbox.
DP:. . . how would that tell us if we were right?Martha: We could just put the same amount, how much it could fit in thegreen one, we could put, say, beans or something, we could put it here [green]and the same amount here [red] and see if like, which one holds more.
She returns to the concept of volume (filling) mentioned in [3] when asked how she would
find the volume of a cylinder attached to a cone. In light of her method for finding volume,
her answer to Problem 9 on the pre-assessment makes much more sense (see Figure 5.7.3).
Like Ryan, she seems to see volume as a quantity less intrinsic to an object than surface
area. For example, when asked about the amount of air in the room, she decides on surface
area, “because youre gonna need how big, how large is the room in order to see how much
air you’re gonna need.” Although she and Ryan have different answers, their answers to
this question reveal how they might associate surface area more closely with an object
than volume.
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5. ANALYSIS AND FINDINGS 39
Figure 5.7.3. Martha’s answer to Question 9 on the pre-assessment.
Martha definitely has the hardest time with the last three problems we look at in our
interview, and her answers suggest her notions of surface area as the “size” of a shape and
volume as the “inside” of a hollow shape are at the heart of her misconceptions.
Martha: [in reference to the concrete problem] Wouldn’t you have to figure outboth, the surface area and the volume?DP: Why would you have to figure out both?Martha: Because first the surface area, because you’ve gotta see the amountof space you’re gonna take, and the volume because of how many things couldfit inside. . .DP: [later, on favorite drink question] Why would you choose twice as wide?
Martha: Wide is means its more, tall is just like a little bit, it’s just in abigger. . . because twice is wide is like, it’s bigger and it has a lot of fruit punchinsdie, and twice as tall, it’s tall but it has the same amount.
Martha’s notion of volume seems tied to whether or not a three-dimensional object is
hollow. It seems to me that, to her, the volume of an object is related to, but physically
disjoint from, the object itself. Like many of the other interviewees, her conceptual under-
standing was shaky and sometimes overwritten by mathematically unimportant aspects
of the contexts of the problems.
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6
Reflection and Implications
First, this project has led me to reflect on the way that my students have developed a
conceptual understanding of volume. The students who had the most accurate definitions
and concepts of volume did not always solve the most problems in our interviews, leading
me to wonder what other factors were in play and how their concept of volume did or did
not help them through a problem. Many of the students interviewed in this project had a
concept of volume rooted in their strongest areas. For example, Ryan defined volume in
terms of concrete objects (since he has difficulty drawing and visualizing abstract three-
dimensional shapes) and Martha’s notion of volume is close to that of volume (filling)
described in [3], perhaps to account for her relatively low computational and spatial skills.
Many students’ concepts of volume also led me to believe that some of their misconceptions
were actually rooted in their misunderstanding of length and area, leading me to wonder
how I would remediate such misunderstanding while simultaneously moving forward with
new material.
Second, I came to understood over the course of this project that each mathematical
question I ask of a student is often much more than just a mathematical question to
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6. REFLECTION AND IMPLICATIONS 41
them. For example, several students interviewed answered “surface area” to the concrete
problem and in most cases this answer stemmed from confusion over what the foundation
of a building was. Especially given that the concrete problem was a real Regents exam
problem with only the numbers changed, I am concerned that too many math problems my
students encounter are confusing instead of hard. While I or some other mathematically
well-trained person is skilled at accurately abstracting away the context of a problem
precisely when necessary, even many of my strongest students are not. This has led me to
believe that this ability is a separate mathematical habit of mind that must be focused on
and taught explicitly in mathematics classrooms, since so often I found that the question
I intended to ask was not the question my students heard.
My students’ conceptions of volume and their interpretation and parsing of problems
interacted in the following way. Those who were most comfortable thinking about volume
abstractly and articulating their ideas about volume were often most able to solve the
problems presented in our interviews and provided stronger justifications for their answers.
In constrast, many of my students have a moderate grasp of volume, but their concept
of volume is still rooted in some artificial aspect of the analogy they use. For example,
Yolanda was unable to answer the prism question completely since here cube-based idea of
volume was flawed in that she considered the number of cubes in a shape without regard
to their volume. The students with these types of understandings seemed more likely to be
misled by the context of the problems we looked at, and this has led me to conclude that a
strong conceptual understanding of volume is not just nice to have, but crucial for solving
problems that go beyond basic formulas and computation. This further implies that an
analogy used in class may not be received by students in the same spirit in which it was
given, and that we as teachers must always be careful to (1) identify where analogies break
down and (2) provide as many of them as possible in order to provide as many different
entry points to understanding difficult abstract concepts.
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6. REFLECTION AND IMPLICATIONS 42
The benefits of this deeper level of understanding show up in the problem-solving habits
of students like Nadia (and Travis to a certain extent). With her strong, flexible, and
unified notion of volume, she was able not only to extract useful clues from the context of
problems, but also to sort out and throw away misleading or mathematically superficial
components of problems. It is this sort of confidence in problem solving that I hope to
build in my own students. Before this project, I feel that I saw knowledge of mathematical
concepts and problem solving habits as critical, but more separable than they actually
are. By watching a student like Travis break down a problem in front of him into simpler
problems, I came closer to understanding that this sort of leap was only possible due to
a robust notion of volume. While it may seem obvious that these are related, I feel that
it is too easy for me and others teaching mathematics to keep mathematical habits and
mathematical concepts too far apart from each other even when focusing on both.
As I tried to include hands-on activities in my unit and manipulatives-based questions
in my interviews, I was surprised by what understandings my students did or did not carry
over with them to more abstract problems or problems without accompanying physical
objects. I assumed that some time playing with real rectangular prisms made out of
cardstock would make it easy to move on to more standard forms of assessment on prisms.
Instead, I found that, especially for students with a weaker conceptual notion of volume to
begin with, leaving manipulatives behind was not just difficult, but so difficult that it made
me question how many of my students had actually deepened their knowledge of volume
and surface area. They may have been better at computational volume problems, but how
much did their ability to memorize formulas mask a continued lack of understanding that
working with manipulatives was supposed to clear up? This has left me with one major
resolution for my future classroom. If I am going to use manipulatives, then I must include
in my plans for how they are used how my students will grow out of using them as well.
Furthermore, I should not just use them because they temporarily make solving particular
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6. REFLECTION AND IMPLICATIONS 43
problems easier; instead, it is all the more important that I focus on what deeper notions
I want my students to gain.
In terms of the data collection of this project, I encountered several difficulties that have
certain implications for my first year of teaching. First, it is difficult to collect meaningful
data. With dozens of students turning in several assignments a week, it is hard to sort out
what reveals what about my students’ thinking processes (as opposed to their ability to
provide answers). Students are absent, they have bad days, they have crazy days, they have
days where they answer every question correctly but refuse to show their work. Second,
as intended, my interviews with my students provided me with much more insight into
their thought processes than their classwork or my shorter interactions with them in class.
Many of them were much more articulate and deliberate than in class, making me wish
that such a conversation was a requisite part of the class for each student.
This has made me wonder how to create this atmosphere in my future classroom. One
aspect of doing this would be to hold myself back more from correcting students as they
solve problems, something which I sometimes struggled to do during interviews. Further-
more, while I don’t think I’m quite yet ready to implement one-on-one conferences in
my classroom, I would like somehow to build space for these sorts of low-stakes extended
mathematical conversations, especially with my struggling students. This is crucial since
every interviewee (even Nadia) surprised me at least once not just with a misconception,
but with the cause of that misconception. For example, Travis thought the Great Pyra-
mids at Giza were hollow and Yolanda thought that unit cubes could have different sizes.
It seems then that one of the greatest difficulties I or any math teacher faces is how to
find these misconceptions and help students recognize and correct them; I am sure that
many of my efforts at self-improvement in my teaching will be towards this goal.
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Bibliography
[1] M.T. Battista, On Greeno’s Environmental/Model View of Conceptual Domains: ASpatial/Geometric Perspective , Journal for Research in Mathematics Education 25
(1994), 86-94.
[2] M.T. Battista and D.H. Clements, Students’ Understanding of Three-Dimensional Rectangular Arrays of Cubes , Journal for Research in Mathematics Education 27
(1996), 258-292.
[3] M. Curry and L. Outhred, Conceptual Understanding of Spatial Measurement , PME
Conference (2006), 265-272.[4] Beverly Falk and Megan Blumenreich, The Power of Questions: A Guide to Teacher
and Student Research , Heinemann, Portsmouth, NH, 2005.
[5] D.L. Gabel and L.G. Enochs, Different Approaches for Teaching Volume and Students’ Visualization Ability , Science Education 71 (1987), 591-597.
[6] Herbert P. Ginsburg, The Challenge of Formative Assessment in Mathematics Edu-cation: Children’s Minds, Teachers’ Minds , Human Development 52 (2009), 109-128.
[7] M.G. and P.W. Hewson, Effect of Instruction Using Students’ Prior Knowledge and Conceptual Change Strategies on Science Learning , Journal of Research in ScienceTeaching 20 (1983), 731-743.
[8] M. Isiksal et. al., A Study on Investigating 8th Grade Students’ Reasoning Skills on Measurement: The Case of Cylinder , Education and Science 35 (2010), 61-70.
[9] The New York State School Report Card Accountability and Overview Report 2010-2011: Bronx Academy of Letters , New York State Education Department, 2011.
[10] J. Piaget and B. Inhelder, The Child’s Conception of Space , Routledge and KeganPaul, London, 1956.
[11] J. Piaget et. al., The Child’s Conception of Geometry , Basic Books, New York, 1960.
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Appendix
Following is a selection of materials used in this project. In order, they are:
1. A copy of the pre-assessment given on May 9.
2. An outline of clinical interview questions asked of students.
3. A copy of the questions that were shown to students on paper during interviews.
4. The lesson plan and task from the day during which students investigated rectangularprisms.
5. A copy of the pertinent problems on the Regents version of the Learning TargetAssessment of May 16.
6. The lesson plan and task from the day during which students investigated cylinders.
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