Elvia Nidia González

Professor Abrahamson

Education 195C

8 May 2009

Finding Five Inches

I. Abstract

Educational activities that promote intellectual understanding, engage students of all

levels, and help an educator measure success can be problematic for the novice educator to

implement. Assessing the understanding of students through educational activities is an essential

skill that experienced educators are constantly perfecting. I was given the opportunity to attempt

an educational activity with a classroom of seventh graders. The activity I chose involved

challenging these students to find exact measurements on an eight and a half by eleven inch

letter-sized paper. Although this activity varied from the normal seventh grade curriculum, it was

a rich learning experience for all of those involved. Students were able to come up with several

possible solutions with a bit of guidance and help from their peers. The activity proved to be

challenging and engaging for the students involved.

II. Introduction

Longfellow Middle School provides an excellent learning environment for all students

enrolled. The school is made up of just a small student body of 445 students; participates in a

nutrition and gardening program; insists that all eighth graders be enrolled in algebra; and offers

tutoring through their Extended Day Program (B.U.S.D.). I was placed in Ian Bleakney’s math

and science classroom. Mr. Bleakney’s students are also enrolled in another academic program, a

foreign language immersion program. All of their math and science work was taught solely in

Spanish. These students are all expected to read, write, and speak Spanish, which will give them

an advantage when they are adults entering into the workforce. The instructor and students I

worked with were all very pleasant and willing to try new educational avenues.

During the planning phases of my presentation, I found it challenging to plan an activity

that would promote intellectual understanding, engage students of all levels, and help me, as a

future educator, measure success from the activity. We were offered several options during the

early weeks of our Education 195c class at UC Berkeley. They varied from puzzles that required

little work to solve to problems that could be solved using the techniques learned in higher high

school math.

Many of them seemed inappropriate for the students I was working with. Ian Bleakney’s

seventh grade classroom at Longfellow middle school had barely covered the Pythagorean

Theorem, linear equations, and graphing. Students in his class were expected to have mastered

basic arithmetic, have an understanding of fractions, and geometric shapes, so I chose an activity

that made use of all three concepts. The lesson asked students to find five inches, then six inches,

and finally, one inch on an eight and a half by eleven letter paper, without a ruler.

On the surface this activity may seem like it is nothing more than kids playing with paper.

It may be hard to grasp how getting students to manipulate these sheets could help them learn

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anything at all besides how to make paper airplanes, but I was certain this activity would be

challenging and engaging for all students.

Students are used to solving normal arithmetic problems. Many classrooms drill

elementary aged students on arithmetic often and while timing them. I remember rushing through

sheets of fifty plus problems during weekly timed tests. I could tell anyone the solution to eleven

minus five or six plus five but I did not understand how to express those ideas concretely. Those

tests meant nothing more than a mix of numbers, “plus signs,” and “minus signs,” simple

memorization. These habits can persist as students get older while they continue their studies in

mathematics and other mathematics-related fields (like physics, computer science, etc).

JD Bransford makes an interesting point in Chapter 4 of How People Learn regarding

memorization and learning. He goes through his teaching experiences as a physics educator to

mention, “…far too many students have the expectation that physics is a set of simple ‘facts’ and

a list of equations, which, if memorized, should permit them to get an A in my class.” It is no

surprise that students rely heavily on memorization rather than understanding to get through their

academic careers. They are being groomed to use simple memorization techniques, rather than

actual understanding, from an early age.

I noticed this same trait in the students I worked with. The first part of the activity

required finding exactly five inches. There are several ways to find five inches. Once found, the

five inches will lie along the eleven inch side of the paper (see Diagram 1). The second part of

the activity required students to find six inches. Now, simple arithmetic tells us that as soon as

you find five inches on the eleven inch side, you have your six inches as well. We all understand

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at a very young age that six plus five is eleven or that eleven minus five is six. There is however,

a gap when it comes to applying this concept to a real-life situation. Students did not

immediately see that they had inadvertently found six inches along with their five inches from

step one.

Diagram 1

Furthermore, the final portion of the activity required that students find one inch. Again,

this is a simple arithmetic problem. Six minus five equals one. Students learn this in

kindergarten. My students hesitated though. Few seemed comfortable enough to use their

existing measurements to find a single inch. “The huge majority of students are not able to apply

their mathematical classroom experiences, neither in the physics or chemistry school laboratory

nor in the most trivial situations of daily life” (Freudenthal). The solution is simple enough. Just

fold the paper along your five / six inch mark as in Diagram 2.

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Diagram 2

My second goal was to keep all students engaged in the activity at hand. Since this

youthful classroom seemed to bore easy when crunching numbers or listening to lectures, I knew

I needed a hands-on project, something that would keep their hands, along with their minds,

actively participating in the learning process.

Most students were on task for the entire activity; even the students that were not

contributing to the solution were working with their paper at their desks. There only a few

students that volunteered their solution to the class. Most were too shy and did not want to have

the classes attention focused on them, but despite their reluctance to speak in front of the class,

they were participating.

My final goal was to understand how to measure learning success in varied situations.

Though supporters of easily graded Scantron® tests may protest, having students fold paper to

explore dimension, arithmetic, and problem solving, can give an educator a much broader

perspective of their students’ abilities. It is not possible to simply create a multiple choice quiz

after to see what students learned. Several observations must be made about individual students,

groups, and the classroom as a whole.

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Students had to think creatively, collaborate, abstract mathematical concepts, perform

arithmetic, and think critically to achieve their goal of finding five inches. All of these things,

though not simple to measure, are important to measure. As future educators we must remember

to, “measure what is important, not just what is easy to measure” (Burkhardt). After studying

mathematics for the majority of my life, I have seen what many others before me figured out.

“Nobody who knows mathematics thinks that short multiple-choice items really represent

mathematical performance” (Burkhardt). Folding paper with Mr. Bleakney’s seventh grade class

served my own studies into pedagogy as much as it helped the students understand mathematical

concepts.

III. Design

The day of my actual presentation I was focused on maintaining my goals of challenging

the class intellectually, engaging students of all levels, and understanding how to measure

success. The paper folding activity started off well. The instructor made sure the class knew I

would be working with them by adding my activity to the agenda at the beginning of class. He

also made an announcement was made immediately before I began. When I first began to speak,

“If I ask you to measure five inches, what would you do?” was met with silence in the class. The

students did not know how to respond to someone other than their normal instructor.

After a brief pause, I continued, “Can anybody tell me? What would you do to measure

five inches?”

Student: “Use a ruler.”

Nidia: “What is your name?”

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Student: “Alfonso.”

Nidia: “Alfonso, you would get a ruler?”

Alfonso: “Yeah.”

Nidia: “And if you don’t have a ruler, what can you do to find five inches?”

Silence, yet again, in the classroom. I began to explain the details of the activity to the

students, “This is what we will do today, without rulers and without estimating, you will try to

find me five inches. I am going to give each of you a sheet of paper. You can fold the paper in

any way. If you need a second sheet of paper, just raise your hand and I'll give you another.”

Although I had been less than five minutes into my activity, I was certain that my

students were being challenged intellectually. This became very clear when a second student,

Juan, asked a very clever question, “Can we know how many inches the paper is?” This student

beat me to the punch line. The dimensions of the paper are vital to this activity. Without the

dimensions, there would be no way to find exact measurements unless a ruler is used. My initial

plan had been to give the students the dimensions of the paper, then help them understand how to

use them. I had not given out any information on dimension nor had I hinted at why it was

important to this activity; Juan just considered this without guidance.

Nidia: “The small part, on top, is eight point five inches and here [pointing to the longer

side of a standard letter sized paper] we have eleven inches. So now take two, three minutes to

work alone or with your friends around you and try to find me five inches without using a ruler.”

The students all began to talk amongst themselves and fold paper. I knew another one of

my goals was being fulfilled. The students were all working to solve this problem together.

Within another minute, the first student, Alfonso, asked, “What is half of eleven?” I had to make

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sure the students were working out the simple arithmetic so I did not tell him that half of eleven

is five and a half. Instead I told him to compute it. This class had been allowed the use of

calculators for lessons and in-class work so I continued with this precedent and allowed them the

use of calculators in my activity.

After a few more moments of working in a group, the very talkative Alfonso again made

his voice heard. He asked, “Can it be five point five?”

Nidia: “What is five point five?”

Alfonso: “If you fold it in half.”

I acknowledged him for what he announced, told him we would discuss his findings

shortly, and asked him to keep working. I needed the class to work alone for another minute. If I

had Alfonso share his findings to the class too soon, some students may be left behind by having

been given a lead towards the solution without time to consider creative folding options on their

own.

While I waited and gave the class more time to fold, I walked around the classroom so I

could hear the side conversations students were having. There were several going on

demonstrating that the classroom was engaged in the activity (my second goal) as a whole. One

side conversation between the instructor and two students revolved around fractions of eleven.

Instructor: “Okay, you got to five point five?”

Student A: “One quarter of eleven… three?”

Instructor: “One quarter of eleven…. One quarter of eleven is three?”

Student A: “Oh wait, no…”

Although the student did not finish computing one quarter of eleven, learning was taking

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place. It was obvious that the student simply found one quarter of twelve. Oftentimes in our own

mathematical careers, we estimate to make the mental arithmetic easier. This student was doing

just that. Eleven is very close to twelve on a number line, and it is much easier to find one

quarter of twelve than it is to find one quarter of eleven. The student learned that in this situation,

an estimate was not appropriate. Only the actual quarter of eleven (two and three quarters) would

be an acceptable response.

At that point it was time to bring the class together. I had given the students three minutes

to work collectively and now it was time to share ideas as a class.

Nidia: “Okay everyone. Attention up front please. Now that you have had the opportunity

to fold your paper … Who has some idea, who wants to come up front and show us what you

have done with your paper? … Even if you haven't found five inches, but just ideas about how

you folded your paper? … Who thinks they have found five inches?”

I called on several students by name and they all responded the same. They had not found

five inches. This is exactly where I had expected them to be after such a short period of time. The

point of the first three minutes was not to race to a solution; the point was to explore ideas about

how folding paper along any edge changes its dimensions. I asked a second question, “Who has

found five point five inches?” Several hands went up and the teacher was even heard to exclaim

on video, “Oh Wow!” Most students were able to quickly compute half of eleven by folding the

paper with respect to its longest edge (see Diagram 3).

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Diagram 3

No student was being pushed to a quick solution. I knew these students understood the

arithmetic but I needed to gauge if they could use it, both to ensure they have a beneficial

learning experience and to help me learn to measure academic success in a classroom. This is

why I consistently urged them to explore their ideas, even the ideas that did not seem to be

leading to a solution.

Exploration in a math classroom is vital. Many facets of a student’s understanding cannot

be assessed by simple multiple choice exams. In one reading by Burkhardt there is an excellent

example of an exploratory math problem. "Consecutive Addends," quoted below from page 79 of

the article, "Some numbers equal the sum of consecutive natural numbers

5 = 2 + 3

9 = 4+ 5 = 2 + 3 + 4

Find out all you can about consecutive addends. "

This is an excellent example of an open investigation, “where students have to formulate

questions as well as answer them” (Burkhardt). This is the same type of learning that the students

had just been experiencing. They were told to share their ideas, even if they had not found five

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inches, with the class. Any idea was acceptable and no student would be told they were wrong

for finding five and a half inches rather than just five inches during the first three minutes of the

activity.

My final objective, learning how to measure success in the classroom, was meant

completely for me and my educational benefit. This activity was promoting several small

successes with my students. They were learning to apply their abstract ideas to concrete

situations, exploring other possible questions (like quarters of eleven) that could be solved while

participating in this activity and they were connecting older knowledge with newer knowledge.

This helped make it evident to me that there was some academic success happening in the

classroom.

Furthermore, students were interacting with each other and collaborating on their ideas.

They were asking insightful questions and thinking creatively as they progressed towards

solutions. The majority of the class, although not talkative, was working with their paper to come

up with their own solutions. The small side conversations also made it evident that learning,

beyond the arithmetic of the paper folding problem, was taking place. Asking the students to find

five inches without a ruler, met my lesson objectives entirely.

IV. Methods

As the activity continued, conversations continued to materialize throughout the

classroom. Students were asking questions to each other, the instructor, and to me. Alfonso, the

student that had found five and a half inches before most other students during the initial

brainstorm, volunteered to share how he found five and a half inches.

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Nidia: “Okay Alfonso, where did you fold the paper?”

Alfonso: “I folded it in half.”

Since his response was a bit vague (there is more than one way to fold a paper in half) I

decided to have him clarify. I asked, “This way?” (see Diagram 4) as a gestured with my arm

towards the middle portion running parallel to the eleven inch side.

Diagram 4

Alfonso: “No, the other half.”

Nidia: “This way?” (I gestured with my arm towards the middle of the paper parallel to the eight

and a half inch side).

Alfonso: “Yes.”

Nidia: “The half…”

Alfonso: “Yeah?”

Nidia: “And where are the five point five inches?”

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What follows is very interesting. The student relates the paper folding to actual fraction

arithmetic. He instructs me on finding half of eleven, using division, then, after I write a fraction

on the white board, he finds and corrects a mistake.

Alfonso: “And then I divided. Eleven divided by five, because, umm…”

Nidia: Writing 11/5 = on the dry erase board, “And what is it?”

Alfonso: “No! Eleven divided by two because…”

Nidia: Erasing the 5 and replacing it with a 2, “Eleven divided by two not five.”

Alfonso: “And then it equals five point five.”

Alfonso managed to see his own mistake seeing his work written up on the board.

Oftentimes students leave out their arithmetic on homework assignments or exams. They do not

show their work and when their answer is not correct, fail to see their error because of the

omitted arithmetic. Alfonso saw his arithmetic on the whiteboard and understood that you do not

divide by five but by two in order to find a half of some number because of it. He also

understood the application of his paper folding to division and fractions, another success.

To assist the class in exploring their folds beyond halves of the paper, I decided to give

them some guidance without giving away the solution. I posed another question to the students,

using another diagram on the white board (see Diagram 5), “What would happen if you would

take your paper from this part here [top right corner] and you fold it [along the diagonal

pictured]? What measurements would you get?”

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Diagram 5

There were no responses so I continued by asking students what the top edge (prior to

fold shown in Diagram 5) measured. The usual suspect, Alfonso, answered with, “Eight, eight

point five.” I continued and asked, “And if we take this top part [8.5 inch top side] and we move

it to this side [left 11 inch side], how much will this [the same top side just in a new position]

measure?”

Alfonso: “Eight point five.”

Nidia: “Thank you Alfonso. And this [the remaining portion of the 11 inch left side] piece?”

The next moment was triumphant. The class muttered, in unison, “Ooooooh!!!!” They

were seeing the connection with the diagonal fold and subtraction from eleven.

Nidia: “Who can tell me what this piece measures?”

Alfonso: “THREE!”

Juan: “Two?”

Luis: “Eleven minus eight point five.”

Nidia: “Okay… what is it? If you want to use a calculator it is okay. Anyone else want…”

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Lizette: Interrupting “TWO POINT FIVE!”

Eureka! The students had found a measurement, through folding, that was exact, and not

immediately obvious from simple half folds. The first question was almost answered. Two point

five is half of five. We were very close to a collective solution.

Nidia: “Okay, we have two point five. Now how are we going to find five if we have two point

five?”

Alfonso: “You can multiply it by two.”

Nidia: “Let me see. Show me on your paper.”

He did not continue at this point. It seemed as if Alfonso was not certain on how to apply

“multiplication by two,” as he had just told me, to the creased paper sitting on his desk, even

though he had previously understood division by two on his paper. It was evident that

multiplication, with respect to their folded paper, was an abstract notion to him. Multiplying by

two is simply doubling. Folding the two and a half inch portion (see Diagrams 6 and 7) in half is

nothing more than double (multiplying by two). Alfonso had not made that connection yet.

Another student decided to chime in.

Rian: “Me!”

Nidia: “Okay Rian. What did you do?”

Rian pointed at his paper without speaking. I responded with, “Okay, and what do you

have here?”

Rian: “Two point five.”

Nidia: “And where is your five?”

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Again, Rian is silent. He simply folds the two and a half inch portion of the paper, marks

a line, unfolds his paper, and then points to that mark. I then asked him to come up in front of the

class so we could all learn through his solution. He was too shy but another student, Juan,

volunteered to come up front. Juan was too shy to do much more than mumble at first. He said

something along the lines of “two point five,” and that was all I could make out in class and

while reviewing the video later. The only way I could think to help him was by asking him if he

wanted to show us on a sheet of paper. This did the trick. He went to his desk and grabbed his

paper. He showed it only to me at first until I asked him to hold it up so the class could see.

Without words he folded the top diagonally towards one side (see Diagram 5). He continued

with, “I folded it one more time…” (see Diagram 6).

Diagram 6

He simply folded the bottom two and a half inch portion upward, then he demonstrated

his final step after I asked, “And where are your five inches?”

He replied with, “Um… here.” He was pointing to the part of his paper where the top of the

folded two and a half inch side touched the rest of his paper (see Diagram 7).

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Diagram 7

I asked him to mark it with his pencil so I could verify, with a ruler, that he had indeed

found five inches. The students in class were astonished to see that he had found five inches and

I was equally astonished that several students had found the same solution even though they

were working independently. I made sure to walk around the class with his solution and the ruler

so they could all see his work.

V. Results and Discussion

Since I was running out of time (I was given the last twenty minutes of class to work with

students before they headed off to lunch), I decided to move on to the next question, otherwise I

would have liked to explore different solutions with students, longer. I asked the class, “How can

you find six inches? … If you have five inches, how can you find six?”

The very talkative Alfonso jumped right in again with another idea, one that I was saving

for my final question.

Alfonso: “Add one more inch?”

Nidia: “Okay, but how can you find one inch?”

Alfonso’s comment pushed the lesson ahead a bit quicker than I had anticipated. I was not

expecting any student to connect the idea of finding any measurement to addition by an inch so

quickly. It was becoming clear to me that I had underestimated this group of students. I

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remember working on this activity in class with other Cal students and not making the

connection Alfonso had just made. The remainder of the activity proceeded as if the last two

questions were combined into a single question. While asking students to find six inches, I also

asked them to find one inch.

Another student, Kate, which had been very quite throughout the entire activity, spoke up

at this point. She had taken the same initial steps as Juan to find five inches then found a unique

path to six inches. She explained, “If you fold the paper down the middle [of the eleven inch

side] and you have five inches, you know the space that is left until the middle is one half of one

inch.” (see Diagram 8)

Diagram 8

“And then you can fold it again to have six inches.” (see Diagram 9)

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Diagram 9

I verified her technique with a ruler and she had found six inches. Furthermore, she had

found the solution to every question of the day. This solution left her with two very important

creases. The first was her five inch mark which the activity had been leading up to. It is the same

five inch mark that Juan found. Her second crease was at five and a half inches, which was the

initial measurement most students found prior to finding five inches. She quickly recognized that

five and a half inches minus five inches results in half an inch. Doubling her half inch gave her

one inch, six inches, and five inches simultaneously.

Her solution was ingenuous. Kate managed to find one inch in the very middle of eleven

inches. This split her paper up symmetrically giving her five inches on either side of one inch. At

the same time, she could combine the center inch with either block of five inches to make her six

inches. Her clever solution did not mean we could end the activity. There were still other

solutions, and about a few more minutes of class time left to explore them. Alfonso helped me

transition to additional solutions beautifully. As soon as Kate sate down he shouted, “I HAVE

IT!”

He made his paper and his hands visible to the class with the use of the class projector (it

actually has a mounted camera that projects images on to the class white board.

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Alfonso: “Here you have five, so since this whole side is eleven, here you have six, because five

plus six is eleven.” (see Diagram 1)

Nidia: “Using your method, can you find one inch?”

Alfonso: “No.”

Alfonso had made a great leap from just a few minutes ago. When I first asked the class

to find six inches, although he suggested adding one inch, he could not find one inch. At that

time he had suggested estimating one inch with his fingernail. In that time period he did manage

to make a lot of connections between arithmetic and our real situation of folding paper. He

already knew that the sum of six and five gives eleven but it took more thinking for him to

connect the sum of six and five, to the eleven inch side of his paper. Right now he needed a little

more time to think about the difference of five and six before he could move on to one inch.

I decided to give my second hint of the day. It would not give the solution away it would

only help students focus using arithmetic to find a solution. I asked, “Think about the numbers

five and six. If you take five from six, what do you get?”

Alfonso: “One.”

Nidia: “Okay. So how will you find one inch?”

Several students caught on at that moment. I heard a few voices on the video say, “Yes,”

and, “Fold it like this.” Two students wanted to share their solution at that moment. Both Juan

and Alfonso, who had been working together off and on throughout the class activity, had a

solution, the same solution to share. I asked them if they wanted to explain together. Neither

student wanted to, but considering I had lost track of time, it was for the best. Juan handed me

his paper, folded along the five inch/six inch line (see Diagram 10).

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Diagram 10

Since class was almost over, I got a few more words in and one more thought in.

Nidia: “…you just fold your paper one more time. Here you have five, here you have six, and

here is one inch. This is another way to have an inch.” I was hoping to give the students

something to reflect upon at home so I added, “On your own time, think to yourselves, if you

have one inch, can you find any measurement you’d like?” I doubt any student thought about it

much because Alfonso excitedly shouted out, “YES!” At the same time, he was folding his inch

over itself on his paper.

VI. Reflection

Though I was not able to ask my questions in the manner I had originally intended all

three measurement questions were discussed. Several students got to solutions and even a unique

solution was found. This activity did meet my original goals. It promoted intellectual

understanding through the connections between arithmetic and real life situations (paper

folding), all students were participating, even if they were not speaking in front of the class, and I

did learn more about how to measure learning within a classroom through the reactions of

students to different portions of the activity.

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Next time I work through this lesson with students, I would like to tie it in other ideas

that are taught in the class. These students had already learned the Pythagorean Theorem prior to

this activity, and triangles came up in a side conversation. As I walked around after the students

had been given a hint on folding (see Diagram 5) I heard the instructor talking with a few of the

students.

Instructor: “What type of triangle is that?”

Miah: Isosceles?

Instructor: “No come on!”

Miah: “Equilateral?”

Instructor: “…don’t give it a name. What is the rule you know?”

Alfonso: “Hypotenuse, no… no… leg times leg equals….”

Juan: “RIGHT TRIANGLE!”

The students continued discussing this rule and triangles until they actually remembered

the Pythagorean Theorem is leg squared plus leg squared equals hypotenuse squared.

I think this could be incorporated into the paper folding activity. Once students have a

solution for five, six and one inch, the second part of the project could use the leftover creases.

These creases make very interesting shapes. Students could figure out the area of each individual

piece and verify they did their work correctly by computing the area of the paper.

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The creases also have their own measurements that the students could fill in. Most likely,

only one length, the hypotenuse created by the fold in Diagram 5, would need to be computed.

So long as students were able to see that a crease parallel to the eight and a half inch side

measured eight and a half inches (and for the eleven inch side), this extension of the activity

would be very simple. If a student did not see that, this extension would still not be much more

difficult. It may just take some group discussion to see the connection between the lengths of the

sides and the lengths of the creases.

For a classroom with students studying geometry, the angles made by the intersections of

the creases with each other and with the edges of the paper could also be explored. Students

could write in which angles are complementary, supplementary, and use alternate interior angles

to figure out which angles are the same.

Lastly, my field placement helped me reflect on my previous teaching and tutoring

experience. I enjoy learning math and I realized long ago that I learn it best when I teach it to

others. Because of this, I made sure to start tutoring wherever I could. I worked with

homeschooled children, as a private tutor, and even worked with students in the community

college system. All of these different settings helped me gain experience and insight into helping

others learn math. These experiences, however enriching, had not prepared me to enter a middle

school classroom.

Before this semester, I had never dealt with classroom management and I never had to

assist large groups, of sometimes unwilling students, learn. The environment at Longfellow

Middle School is very different than an adult classroom. At the community college there were no

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disciplinary issues. While I stood in front of adults explaining algebra (I was paid through the

school for a full year to assist professors in the math department), there was no need to threaten

with detention. In the middle school, the instructor had to. There were many moments when his

class would begin to get out of hand and he would, with a sort of calm that can only come with

years of experience, manage to gain control of his classroom once again. Seeing how classroom

management fits in with actual learning was the gap missing in my training as an educator.

Moreover, while working with small groups of students I noticed that many were behind

in key concepts needed to continue their mathematical journey. This was similar to what I saw in

the community college, however, I did not expect students to be so unwilling to fill in the

missing gaps of information. With adults, especially if they are funding their own education, I

saw more willingness to work to maintain good grades. Those that were not willing to put in the

effort were rarely seen or heard from. Middle school students do not have the option to ignore

school; they have to attend. These created several situations where students that were behind

would just socialize or disrupt the class rather than put in the effort, or as adults do, leave.

Working with these young learners throughout this semester (not just the day of the paper

folding activity) has been a positive step forward in my own education. Coordinating and

carrying out this activity has added priceless experience for me. I learned how to being with a

concept, set goals, plan a presentation, and work with a large (sometimes unwilling) group to

meet my goals. I will use this experience at Longfellow Middle School, along with my previous

educational experiences, to move forward in my own academic career.

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Works Cited

Bransford, J.D., Brown, A.L., & Cocking, R.R. (1999). How People Learn: Brain, Mind, Experience, and School. Washington D.C.: National Academy Press.

Burkhardt, H (2007). Mathematical Proficiency: What is important? How can it be measured? In A.H. Schoenfeld (Ed.), Assessing Mathematical Proficiency.

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