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Maci Eisenhower

February 11, 2013

EDU 4250

Curriculum Concept Paper Draft

The Diversity of the Pythagorean Theorem

The Pythagorean Theorem is well known among all who have completed a geometry

course.  Even those who have no interest in math can usually recite the Pythagorean Theorem

and state that it applies to triangles.  While this theorem is very well known and essential to math

far beyond geometry, it is not always taught in a way that accommodates all learning styles and

social groups.  Not only are there hundreds of proofs for the Pythagorean Theorem that could

cater to every learning style, but there is also a rich mathematical history of this theorem that is

often glazed over or many times not taught.  Examples for where the Pythagorean Theorem can

be useful are many times not applicable to students’ lives and many times they cannot relate at

all to these examples.  Throughout this paper, I will give specific examples from an 8th grade

curriculum where the teaching of the Pythagorean Theorem contributes to the oppression of a

social group.  Then I will present ideas to modify or supplement the curriculum such that no

social group would be oppressed and all students could broaden their global perspective along

with their mathematical knowledge.  Finally, I will address opportunities for differentiation on

the topic of the Pythagorean Theorem.

The following problem is from an eighth grade curriculum textbook. "On a standard

baseball diamond, the bases are 90 feet apart.  How far must a catcher at home plate throw the

ball to get a runner out at second base?" (1 p. 2)

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In many mathematics classrooms, teachers use

real world examples often so students can see

the relationship between mathematics and

their everyday lives.  Yet, often times these

examples are not applicable to students’ lives

and many times assume students know

something about a subject that they know

nothing about.  While a baseball example might be a great example for a handful of students, it

could confuse others even more.  Many more examples and applications need to be given in a

large diverse classroom, because many times the students do not understand examples that are

only related to the teachers interests.  Throughout my time as a student I have seen other

examples given such as finding the long side of a sailboat's sail, finding the height of a ladder

against a house, finding the height of a tree or a building, or finding the size of a television

screen.  While using a wide variety of examples is necessary in a diverse classroom, many of

these examples assume students are from middle class backgrounds.  There could be a handful of

students who have never seen a sailboat or owned a television.  There could be students who

have no interest in finding the height of a tree in the Redwood Forest.  Because of the wide

variety of backgrounds and student learning needs in a classroom, differentiation and non-

oppressive examples are needed for all students to flourish.

"Because they [the Pythagoreans] held Pythagoras in such him regard, the Pythagoreans

gave him credit for all of their discoveries.  Much of what we now attribute to Pythagoras,

including the Pythagorean Theorem, may actually be the work of one or several of his

[1 p. 2]

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followers." (1 p. 33)  A large part of the Pythagorean Theorem that is often taught incorrectly is

whom the theorem is attributed to.  While this particular book acknowledges the fact that not all

of Pythagoras' ideas were his own, it excludes the fact that hundreds of years earlier the

Babylonians had already discovered the Pythagorean Theorem.  While this may not seem like

important information, many history books are written from the view of the victor and thus leave

out perspectives of others who were involved in a historical event.  The history of the

Pythagorean Theorem many times forgets to mention the contributions of the Babylonians.

Students learn in many different ways. Some need to see concepts visually, some

algebraically, and others learn best kinesthetically. In some classrooms, students are just given

the formula a2+b2=c2 and told to apply it to triangles with missing side lengths. Because of this,

students have misconceptions about the Pythagorean Theorem, along with a very shallow

understanding of its meaning and significance. While all concepts can be taught to different

learning styles, the Pythagorean Theorem is one mathematical concept that literally has hundreds

of ways to be taught, and the teacher should utilize a handful of these ways.

An easy, non-oppressive way to begin teaching the Pythagorean Theorem is to use

example problems that are applicable to the students’ lives. At the school I am currently student

teaching at, the large majority of students walk to school. Calculating the shortest distance to

walk to school using right triangles could be a project that is applicable to the students’ lives;

they could even implement this new walking route into their morning commute if geographically

possible. Depending on the culture and diversity of the classroom, there are many different

examples that could be chosen that the students could relate to, rather than “classic baseball

field” textbook examples.

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Along with using current real world examples, solving problems the ancient world used

the theorem for could give students a context for when and why the theorem was so useful.

Through this historical perspective, credit could be given to the ancient civilizations, like the

Babylonians, for their development of the theorem years before the Pythagoreans. Not only

would students be seeing multiple ways to apply the theorem, but they would also see a broad

perspective of mathematical history. Students could also learn about the significance of the

credibility of historical sources. This can be related to and extended beyond much more than

mathematics into all other subjects and areas of studies like language arts, social studies, science,

or history.

Along with giving a diverse type of example problems, teaching to the diverse learning

styles in the classroom is necessary. There are students who need to see the algebraic

computations to fully understand the concept. There are hundreds of algebraic proofs for the

Pythagorean Theorem that can be shown to students with understanding from middle school to

high school. There are also geometric proofs; shown below is a picture of a common geometric

proof.

Seeing the visual of the Pythagorean Theorem is

necessary for other students to understand it. This

diagram can be changed to a hands-on learning activity

where students are moving square tiles from the square

with the side length of a and the square with the side

length of b, to the area with the side length of c, to see

they are equal.

[1 p. 46]

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All three of the teaching methods could be used in one classroom, as each one could be further

clarifying of the meaning of the theorem for students.

Finally, using differentiated instruction to teach the Pythagorean Theorem can increase

students’ understanding even more. Using stations in the classroom is a great way to reach each

student’s need, along with enabling students to work with others. When using stations, the

teacher has the opportunity to work with small groups of students with similar readiness levels,

interests, or learning styles. It also gives students a chance to see different aspects of the

theorem as they travel around the classroom. If done well, using stations could be a discovery

learning activity in which students are using their own knowledge and tools to “discover” the

Pythagorean Theorem. Another differentiation strategy is giving students a larger, individual

project to work on that is related to a certain aspect of the theorem. Students could choose to

write a song, rap, or poem about the theorem; they could even go further and make a music video

about the theorem. Some students could develop a plan for building a road system, house, or city

and show how the Pythagorean Theorem could be incorporated into it. Then others could write a

paper about the history of the theorem along with historical places where its use was significant.

The Pythagorean Theorem is incredibly important to much math beyond geometry. It

also has a complex and culturally rich history. Through this topic other subjects can be

incorporated into the mathematics classroom. This topic is the perfect opportunity for teachers

to teach to the diversity of their classrooms, along with differentiating instruction for different

learning styles and interests.

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Works Cited

[1] Lappan, Fey, Fitzgerald, Friel, and Phillips. Looking for Pythagoras. 2nd ed. Boston: Prentice

Hall, 2006. Print.

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