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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



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



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



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



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.



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



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].



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



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




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




answers), it is possible to understand more deeply the source of their misunderstandings

about volume and other mathematical concepts.



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.



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.




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.




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.



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.



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.




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).




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




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.”




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




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. . .




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.”




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




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




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.




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




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.




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.




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).




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.




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




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.




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




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.




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.



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



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.




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




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.



Bibliography

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[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.

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[11] J. Piaget et. al., The Child’s Conception of Geometry , Basic Books, New York, 1960.



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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