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283 Pages·2012·6.92 MB·English

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WPI Teaching Practicum

An Interdisciplinary Qualifying Project
Submitted to the faculty of
Worcester Polytechnic Institute
in partial fulfillment of the requirements for the
Degree of Bachelor of Science

Submitted By:
Warren Anderson

Submitted To:
Professor John Goulet

Date: April 19, 2011
[email protected]
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The simplest schoolboy is now familiar with truths for which
Archimedes would have sacrificed his life.
- Ernest Renan
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Table of Contents
Chapter 1: Demographics ............................................................................................................................. 7
Introduction .............................................................................................................................................. 7
Worcester Demographics: ........................................................................................................................ 7
Enrollment ................................................................................................................................................ 9
Selected Populations ............................................................................................................................... 12
Technology .............................................................................................................................................. 12
Plans of High School Graduates .............................................................................................................. 13
Graduates Attending Colleges and Universities ..................................................................................... 14
Mobility Rates ......................................................................................................................................... 15
Graduation Rates and other Indicators .................................................................................................. 16
Recent MCAS Results .............................................................................................................................. 20
MCAS Annual Comparisons..................................................................................................................... 21
Item by Item MCAS Results ..................................................................................................................... 23
Grade 10 Mathematics MCAS Results of all Worcester Public High Schools ......................................... 27
Grade 10 Mathematics MCAS Results by Subgroup ............................................................................... 28
MCAS Participation ................................................................................................................................. 29
Advanced Placement Results .................................................................................................................. 29
SAT Results .............................................................................................................................................. 30
Teacher Data ........................................................................................................................................... 31
Staffing by Ethnicity and Gender ............................................................................................................ 32
Staffing Age ............................................................................................................................................. 32
Teacher Specialties ................................................................................................................................. 33
Conclusion ............................................................................................................................................... 33
Chapter 2: The Curriculum .......................................................................................................................... 34
Introduction ............................................................................................................................................ 34
Course Levels .......................................................................................................................................... 34
Course Sequence..................................................................................................................................... 35
Worcester Public Schools Mission Statement ........................................................................................ 36
The Massachusetts Curriculum Frameworks and the Common Core Standards ................................... 37
New 2011 Mathematics Standards in the Massachusetts Curriculum Frameworks .............................. 37
The Reauthorization of the Elementary and Secondary Initiatives ........................................................ 37
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National Council of Teachers of Mathematics Curriculum and Evaluation Standards ........................... 38
The Effect the MCAS has on Honors Advanced Algebra and Pre-Calculus ............................................. 39
Syllabus for Advanced Algebra (H) .......................................................................................................... 40
Textbook for Advanced Algebra (H) ........................................................................................................ 43
Course Outline for Advanced Algebra (H) (Short Form) ......................................................................... 45
Course Outline for Advanced algebra (H) (Long Form) ........................................................................... 47
Prerequisite Skills for Advanced Algebra (H) .......................................................................................... 52
Syllabus for Pre-Calculus ......................................................................................................................... 52
Textbook for Pre-Calculus ....................................................................................................................... 55
Course Outline for Pre-Calculus (Short Form) ........................................................................................ 56
Course Outline for Pre-Calculus (Long Form) ......................................................................................... 58
Prerequisite Skills for Pre-Calculus ......................................................................................................... 69
Revisiting MCAS Question #20 ................................................................................................................ 69
Conclusion ............................................................................................................................................... 70
Chapter 3: Course Materials ....................................................................................................................... 71
Introduction ............................................................................................................................................ 71
Do Now ................................................................................................................................................... 71
Example Do Now ..................................................................................................................................... 72
Homework............................................................................................................................................... 72
Example Homework Assignments .......................................................................................................... 73
Instruction and Lesson Plans................................................................................................................... 74
Example Lesson Plan ............................................................................................................................... 76
Using Color with the White Board .......................................................................................................... 81
The ELMO ................................................................................................................................................ 81
Graphing Calculators ............................................................................................................................... 82
Paper ....................................................................................................................................................... 82
Writing Utensils ....................................................................................................................................... 83
Email ........................................................................................................................................................ 83
Tests ........................................................................................................................................................ 83
Example of a Test .................................................................................................................................... 83
Conclusion ............................................................................................................................................... 88
Chapter 4: The Students ............................................................................................................................. 89
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Introduction ............................................................................................................................................ 89
Survey...................................................................................................................................................... 89
Learning Styles ........................................................................................................................................ 91
Behavioral Issues ..................................................................................................................................... 92
Absenteeism and Tardiness .................................................................................................................... 93
New Students .......................................................................................................................................... 95
Handicaps ................................................................................................................................................ 95
Parental Issues ........................................................................................................................................ 95
Teaching Method .................................................................................................................................... 96
Conclusion ............................................................................................................................................... 96
Chapter 5: Assessment ............................................................................................................................... 97
Introduction: ........................................................................................................................................... 97
Do Now ................................................................................................................................................... 97
Class work ............................................................................................................................................... 97
Homework............................................................................................................................................... 98
Quizzes .................................................................................................................................................... 99
Pre-Calculus Quizzes ............................................................................................................................. 100
Tests ...................................................................................................................................................... 102
2.4-2.6 Pre-Calculus Test (Period 1) ...................................................................................................... 105
2.4-2.6 Pre-Calculus Test (Period 3) ...................................................................................................... 106
2.4-2.6 Pre-Calculus Test (Period 6) ...................................................................................................... 109
2.4-2.6 Pre-Calculus Test Results .......................................................................................................... 111
3.1-3.4 Pre-Calculus Test (Period 1) ...................................................................................................... 112
3.1-3.4 Pre-Calculus Test (Period 3) ...................................................................................................... 113
3.1-3.4 Pre-Calculus Test Results .......................................................................................................... 114
3.6-3.8 Pre-Calculus Test (Period 1) ...................................................................................................... 114
3.6-3.8 Pre-Calculus Test Results .......................................................................................................... 115
Advanced Algebra Chapter 3 Test ......................................................................................................... 116
Advanced Algebra Chapter 3 Test Results ............................................................................................ 117
Extra Credit ........................................................................................................................................... 118
Extra Credit Assignment 1 ..................................................................................................................... 118
Extra Credit Assignment 2 ..................................................................................................................... 119
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Conclusion ............................................................................................................................................. 119
Closing Remarks ........................................................................................................................................ 120
Appendix A: 2011 Massachusetts Curriculum Frameworks for Algebra 2…………………………………..………..121
Appendix B: 2011 Massachusetts Curriculum Frameworks for Pre-Calculus…………………………………………126
Appendix C: 2000 Massachusetts Curriculum Frameworks for Algebra 2…………………………………………….132
Appendix D: 2000 Massachusetts Curriculum Frameworks for Precalculus………………………………………….134
Appendix E: NCTM Curriculum Standards for Grades 9-12…………………………………………………………………..136
Appendix F: Course Outline for Algebra 1…………………………………………………………………………………………….140
Appendix G: Course Outline for Geometry…………………………………………………………………………………………..144
Appendix H: Pre-Calculus Lesson Plans………………………………………………………………………………………………..147
Appendix I: Advanced Algebra Lesson Plans…………………………………………………………………………………………231
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Chapter 1: Demographics
Introduction:
In this chapter, several pieces of statistical data pertaining to the specific population Doherty Memorial High School
serves is presented and discussed. In education, it is absolutely critical to understand the type of students that
teachers must instruct. For example, if the large majority of the student population expresses an interest in
graduating college, it is essential that the school provides an abundant foundation to prepare students toward that
goal. Similarly, if a large portion of students want to find a job immediately after school, which is especially
common among the several technical high schools around the state, the school ought to prepare the students for the
workforce. It is also critical that faculty do not forget about the minority, because everyone is a minority in one
sense or another. A particular student may have a special need or a handicap, or may be a member of an unusual
race or religion. That is, a school should provide a comprehensive education, and without knowing the needs of
the students and how diverse they are, that is nearly an impossible goal. Without this knowledge, it is like shooting
an arrow in a random direction, and hoping that it meets the desired target.
Worcester Demographics:
Throughout this chapter, numerous comparisons will be made between the demographics between the school and the
city, and between the city and the state. Performing comparisons such as these enable an educator to identify how
the city is unique compared to the state, and how the school is unique compared to the city. Similarities can also be
identified between the three. By making these comparisons, it is possible to identify characteristics that the school
may have, but the city lacks, and vice-versa.
From this point forward, when the word school is mentioned, it is understood as Doherty Memorial High School,
located in Worcester, MA. The words state is understood as Massachusetts, and the words city and district will be
both understood as Worcester, MA. Also, all the data in this chapter are the most recent data, primarily provided by
the United States Census Bureau and the Massachusetts Department of Education.
Table 1: State and District Demographics1
2010 Data (* indicates 2009 data) Worcester State
Population 181,045 6,547,629
People under 5 years 6.60% 6.50%
People under 18 years 22.10% 24.00%
People 65 years and over 11.70% 13.00%
Median Household Income $45,944* $64,057*
Male: Female Ratio 974:1000 968:1000
Residence Living in Poverty 17.9%* 10.3%*

What is the second largest city in New England? If you ask most people in the region this question, they will say
that it is Providence, RI. In actuality, however, Worcester, MA has become the second largest city in New England.
(Providence has a declining population of 171,909.2) This common misconception presents an excellent reason why
statistical data is necessary. What we believe is not always what is true, and choosing actions based on belief and
not on factual data can have great consequences.

1 Source: http://quickfacts.census.gov/qfd/states/25/2582000.html
2 Source: http://www.city-data.com/city/Providence-Rhode-Island.html
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From the data above, it is evident that the age distribution of Worcester is not considerably different from the state
as a whole. In particular, the percentage of school age children can be calculated by subtracting the percentage of
the population that is under 18 by the percentage of the population that is under 5. The results are 15.5% for the city
and 17.5% for the state. Combining this information with the populations of the city and state, the number of school
age children can be calculated. There are approximately 28,061 school aged children in Worcester and
1,571,430 school aged children in Massachusetts.
Another important statistic is how the median household income of the city compares to the state. The median
household income in Worcester is $45,944, compared to the state average of $64,057. In other words, the
household income of an average family in Worcester is approximately 71.7% of that of an average
Massachusetts family. Additionally, 17.9% of the people living in Worcester live in poverty (as defined by the
national poverty level) compared to only 10.3% of the people living in Massachusetts. The monetary wealth of a
city can have great consequences on the education sector. The less income people earn the less money a city can
spend on public schools. This is of course alleviated somewhat by federal and state aid, but not entirely. Even if
external aid could completely satisfy the needs of such schools, individual families can still struggle paying for
school supplies and even basic needs such as food and clothing. A family in poverty is far less likely to have
educationally enriching devices, such as the computers, graphing calculators and the internet than wealthier families.
It is all too obvious that educational resources cost money, and the less money that is available, the greater the
challenge it is to educate without reducing the quality of education. In short, Worcester is a community that requires
wise financial distribution in order to deliver a comparable education to that of wealthier communities. This is
indeed possible, but achieving this may require students to share more resources, such as calculators, and to use
resources more economically, such as using both sides of a piece of paper.
Table 2: Ethnic Composition of Worcester3 Table 3: Ethnic Composition of MA4

White 59.6% White 76.1%

Hispanic 20.9% Hispanic 9.6%

Black 11.6% Black 6.6%

Asian 6.1% Asian 5.3%

Multi-Race 4.0% Multi-Race 2.6%

Native Hawaiian/Pacific Islander 0.0% Native Hawaiian/Pacific Islander 0.0%

Native American 0.4% Native American 0.3%

Ethnicity is another major demographic that must be considered. When evaluating a community based on its
ethnicity, it is vital to avoid stereotypes, (such as Asians are good at math, or African Americans love watermelon)
because stereotypes are very rarely accurate, and are plainly offensive. Rather, evaluate ethnicity in terms of
cultural diversity. A population with a strong ethnic diversity is far more likely to be culturally diverse than a
community that is not ethnically diverse. For example, the cuisine and religions of Caucasians are significantly
different than the cuisine and religions of Asians. A culturally diverse community can be more enriching than one
that is not, because individuals encounter an abundance of ideas, beliefs, and traditions and will have a greater
understanding of humanity as a whole. Although culturally diverse communities can be more simulating, they are
arguably more difficult to educate because their ideas and traditions differ. For example, in a Catholic school, there
is little question on what religion will be encouraged and taught. However, at a public school in which people have
significantly different religions, it would be outrageous to favor one religion over the other. In public schools,
religious matters are much trickier to teach than at patriarchic schools.

3 Source: http://quickfacts.census.gov/qfd/states/25/2582000.html
4 Source: http://quickfacts.census.gov/qfd/states/25000.html
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Segregation and prejudice are also issues regarding ethnicity. Two different things can happen in an ethnically
diverse community. Individuals may either favor their own “kind” and develop hatred for those that are unlike
them, or they may learn to embrace the richness of various cultures and develop love for one another. It is almost
too obvious that the latter is more desirable. Therefore, schools that educate students from several ethnicities (and
even schools that are not ethnically diverse) should strive for an appreciative and accepting environment for every
race, culture, religion, etc. A school should be a community in which every culture is celebrated.
Form Tables 2 and 3, it is apparent that Worcester is more ethnically diverse than Massachusetts. In particular, the
White population in Worcester is significantly less represented (although still a majority) while every other race on
the list is more represented. This fact should be interpreted optimistically, because a greater ethnic diversity means a
greater opportunity to learn about each other.
It should be noted that the data from the two tables above are directly from the Census Bureau. The Census Bureau
allows certain people to be included in more than one category, so the sum of all the different percentiles is above
100%.
Enrollment:
The tables in this section are data pertaining to the enrollment of Doherty Memorial High School.
Table 4: Student Enrollment by Grade (2010-11)5
Grade # of Students
9 330
10 348
11 323
12 326
Tab le 5: Enrollment by Gender (201 0 - 2011)6
Gender School District State
Male 661 12,471 490,363
Female 666 11,721 465,200

Table 6: Enrollment by Race (2010-2011)7
Ethnicity School (%) District (%) State (%)
African American 13.6 13.6 8.2
Asian 10.9 8.1 5.5
Hispanic 25.9 38.3 15.4
Native American 0.7 0.3 0.2
White 47.2 36.5 68.0
Multi-Race (Non-Hispanic) 1.7 3.1 2.4
Native Hawaiian, Pacific Islander 0.0 0.0 0.1

5 Source: http://profiles.doe.mass.edu/profiles/student.aspx?orgcode=03480512&orgtypecode=6&
6 Source: http://profiles.doe.mass.edu/profiles/student.aspx?orgcode=03480512&orgtypecode=6&
7 Source: http://profiles.doe.mass.edu/profiles/student.aspx?orgcode=03480512&orgtypecode=6&
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School Ethnic Composition (2010 - 2011)
Multi-Race (Non-
Hispanic)
1.7%
African American
13.6%
Asian
10.9%
White
47.2%
Hispanic
25.9%
Native American
0.7%

District Ethnic Composition (2010 - 2011)
Multi-Race (Non-
Hispanic)
3.1%
African American
13.6%
Asian
White
8.1%
36.5%
Hispanic
38.3%
Native American
0.3%