Tid32327 dimensions alvar_garcia_tid32327_dimensions_essay

PAEMST 2012-2013 - Dimensions of Outstanding Teaching Written Response Page 1

Name: Alvar Garcia Teacher ID: 32327

Dimensions of Outstanding Teaching Written Response

Dimension One: Mastery of mathematics or science content appropriate for the grade level

taught. The narrative for Dimension One should be about three pages.

1a. Discuss the mathematical or scientific ideas that are fundamental to understanding the chosen

topic or concept.

Early this school year in my AP Calculus BC class, we made sure to review such

concepts as functions and relations, inverse functions and one-to-one functions as well as explicit

versus implicit definitions of said functions or relations. In the lesson prior to this one, we used

algebraic techniques to solve for the standard form of a conic section. Standard form conics are

easily graphed, as they have horizontal and vertical symmetry, without a Graphing Calculator as

well as with a Graphing Calculator in Function Mode.

Today’s lesson focused on what I call Twisted Conics. These conics are not easily written

in standard form as these graphs do not have horizontal or vertical symmetry. We discussed

rewriting these conics in Polar Form using Trigonometric Identities. Further, these conics are

difficult to graph by hand due to a rotational transformation. However, Twisted Conics are

readily graphed on a Graphing Calculator in Polar Mode.

We have that cos(Ɵ) = x

r and that sin(Ɵ) =

y

r , Therefore, x = r(cos(Ɵ)) and y=r(sin(Ɵ))

[Video 14:45]. It is now a very simple matter to replace all the x and y variables in a general

form conic expression, Ax^2+Bxy+Cy^2+Dx+Ey+F=0, and solve for r as an explicit function of

Ɵ. In this manner, we can graph conic sections in Polar Mode on our Graphing Calculator. I

developed this method for graphing Twisted Conics when we cannot Complete The Square and

graph in Function Mode. I don’t recall ever seeing this method in a textbook. The method im our

text is very convoluted. I think I may have to publish this method one day.



There are many topics in pre-Calculus that overlap with our Junior year NYS Regents

course, Algebra II & Trigonometry at the High School. When I teach pre-Calculus, I make sure

to emphasize those topics needed in AP Calculus AB and BC that are not covered as thoroughly

as I would like in Algebra II. These topics include Functional Analysis, Coordinate Geometry,

Vectors, Conic Sections, Matrices, Sequences and Series. I selected the topic of my video,

“Conic Sections and Polar Coordinates,” as an example of this sort of material. I address Conic

Sections early in the school year in AP Calculus BC because this is a rich topic full of Algebra,

Trigonometry and Graphing Calculator techniques [Video 4:36] needed for the rest of the year.

1b. Explain why this topic or concept is important for students to learn and how it relates to more

complex concepts that students will encounter in subsequent lessons, grades, or courses.

I find it providential that the most complex topic of the year is based on the very first

lesson of the year [Supplemental Page 3]. We have now come full circle and managed to review

most every topic of the year in this one unit right before the AP Exam.

We recently finished the AP Calculus BC curriculum for this year and started AP

Review. The last class before review was about calculating derivatives, dy

dx =

dy

dt

dx

dt

=

dy

dt

dt

dx ,

and arc length, dl

dt =

dx

dt

dx

dt +

dy

dt

dy

dt , of Vector Valued Functions and Polar Relations.

We attacked this challenging topic by converting from Polar Notation to Vector notation.

First, we converted from polar r=f(Ɵ) to parametric form r(t) =

x(t)

y(t) =

f(t) cos(t)

f(t) sin(t).

Now we have that r’(t) =

x '(t)

y '(t) =

–f(t) sin(t) + f '(t) cos(t)

f(t) cos(t) + f '(t) sin(t) and

dy

dx =

f(t ) cos(t)+f ’(t) sin(t)

- f(t) sin(t)+f ’(t) cos(t).

Applying the Parametric Arc Length formula dl

dt =

dx

dt

dx

dt +

dy

dt

dy

dt we have:



(–f(Ɵ) sin(Ɵ) + f '(Ɵ) cos(Ɵ))2 = f(Ɵ)

2 sin

2(Ɵ) – 2 f(Ɵ) f '(Ɵ) sin(Ɵ) cos(Ɵ) + f '(Ɵ)

2 cos

2(Ɵ)

(f(Ɵ) cos(Ɵ) + f '(Ɵ) sin(Ɵ))2

= f(Ɵ)2 cos

2(Ɵ) + 2 f(Ɵ) f '(Ɵ) sin(Ɵ) cos(Ɵ) + f '(Ɵ)

2 sin

2(Ɵ)

dl

dƟ = f(Ɵ)

2 + f '(Ɵ)

2) and, therefore, l=

Ɵ1

Ɵ2

f(Ɵ)2 + f '(Ɵ)

2 dƟ.

Further, using the definition of a radian, the arc length of a circle s=r Ɵ and the area of a

sector of a circle we have: dA

dƟ =

1

2f(Ɵ)

2 and, therefore, A =

Ɵ1

Ɵ2

f(Ɵ)2

2 dƟ.

1c. Discuss the misconceptions or misunderstandings that students typically have with regard to

this topic or concept.

Many students find Polar Notation very daunting at first as they are typically very

confused by even the simplest of Trigonometric concepts. I like to have fun with my students

telling them little stories at times like these when they feel overwhelmed with a particular topic.

Said stories are usually interesting historical anecdotes or funny in some way. When it comes to

trigonometry review, I tell my charges about SOHCAHTOA. I mentioned SOHCAHTOA

[Video 12:45] in this lesson as a reference to the story I told the day before: “Once upon a time,

a teacher was explaining how to solve Right Triangles. The students could not remember that

sin(Ɵ) = opposite leg

hypotenuse , cos(Ɵ) =

adjacent leg

hypotenuse and tan(Ɵ) =

opposite leg

adjacent leg . The teacher was

wracking his mind over and over as to how to help his students remember these ratios. Finally, it

came to him! ‘SOHCAHTOA East of Java!’ will be easy for them to remember.” This mnemonic

was indeed easy to remember as, at the time, there was a new hit disaster movie playing in all the

movie theaters in the country called “KRAKATOA East of Java!”

Keeping a good rapport with students goes a long way in motivating them to overcome

these obstacles. It is helpful to know some of their varied interests and hobbies [Video 23:36].



Dimension Two: Use of instructional methods and strategies that are appropriate for the

students in the class and that support student learning.

The narrative for Dimension Two should be about three pages.

2a. Describe the instructional approaches you used in the video to help students understand the

topic or concept chosen in Dimension One.

Over the past 25 years, I have been fortunate to have had a very supportive administration

at my High School. I have always been encouraged to bring new teaching techniques and

technologies into the Mathematics classroom to benefit my students. I have also been

encouraged to share these techniques and technologies with the rest of the faculty. As a result, I

have been ScreenCasting many of my lessons on YouTube [Supplemental Page 7] for over 5

years. Now I can say that I have “Flipped the Classroom” [Video 4:41]. The Flipped, or

Inverted, Classroom refers to a teaching method becoming more and more popular in Science

classes all across the nation whereby students are assigned classwork for homework and

homework for classwork. Some of the homework consists of watching a screencast of new

material before it is covered in class. This approach works well in Science classes as the teacher

frees up extra class time for lab work. For Mathematics class, I have modified the flipped

classroom concept in such a way that I ScreenCast my lesson whenever I cover new material.

Then the students can watch the ScreenCast before doing their homework to reinforce new

material after the material has been taught in class. These ScreenCasts are also useful for the

student who is absent or a student that needs to review old or confusing topics.

In addition, I have learned a great deal from other teachers at conferences and in online

forums. These colleagues and I form a Professional Learning Network. From this learning

network, not only did I learn about the Flipped Classroom, but I also learned about

WhiteBoarding. I have been WhiteBoarding in my classes for over 2 years and I am really



impressed with the result [Video 18:00]. My students use small, portable WhiteBoards to share

their answers when working in groups. Actually, my WhiteBoarding strategy is a variant of

“Think, Pair, Share.” In this strategy students are paired together to complete a task, such as a

Chemistry Lab. Then all the groups come together to share their results. They are all working on

the same problem. However, their data may be different making the presentations of group work

varied even though the students arrive at the same conclusions. My students typically work in

groups of 4 for about 10 minutes on a new problem after I present a mini lesson. Each group will

work on a different problem of a given type. Then each group presents their solution to the class.

2b. Explain how you identify and build on students’ prior knowledge, and how this knowledge is

addressed in your video and in your general teaching strategies.

Student’s prior knowledge for this topic is a very basic understanding of Right Triangle

Trigonometry and Trigonometric Identities. I tease these topics out of my students at the very

beginning of the class, before developing my method of graphing Twisted Conics. To aid in this

process, I often find it necessary to supplement the textbook with my own handouts

[Supplemental Page 4]. This handout includes all the theory and examples we need for the lesson

[Video 29:25]. I also supply my students with another packet including practice problems for

classwork and homework. We covered exercises 1 and 2 in class the day before. We covered

questions 6 and 7 in class today. I left questions 3 and 5 [Video 39:26] for practice. Also,

questions 4 and 8 were summary questions [Supplemental Page 5] extending students’

knowledge based on guided practice [Video 41:26], [Video 42:45]. I try to summarize each

lesson before the end of class [Video 41:51].

I found it difficult to teach this lesson without using each student’s name. I was told not

to use names so as to have all the families file the Video Permission slip. I always try to make

my students feel welcome [Supplemental Page 2] by greeting them before class [Video 1:25],



making an inviting environment [Supplemental Page 6] in my classroom [Video 22:00] and

calling on them by name during class. I did slip a few times because, despite the fact that this

was the first week of school, I knew many of these students from prior classes (pre-Calculus

Honors, AP Compter Science, Scientific Computing Lab) and clubs (Chess Club, Continental

Mathematics League, Computing Independent Study). I can readily assess what my students

already know before a given lesson, what their weaknesses are and what may be new to them.

2c. Discuss the instructional strategies and techniques you use to meet the learning needs of all

students, challenging those with stronger knowledge while ensuring learning for less

accomplished students.

When my students are WhiteBoarding, they work in groups. I break up the class into

groups of 3 or 4 students. These groups are usually comprised of a Scribe, an Advocte, a Critic

and an Orator. The Scribe is in charge of writing the solution [Video 20:38] on the WhiteBoard

with input from all members of the group. The Orator gives an oral presentation explaining the

steps given on the WhiteBoard [Video 24:50], [Video 27:12]. The Advocate is the student that

comes up with the initial solution and the Critic typically makes corrections [Video 21:25].

Whenever I present a new problem, or my students work on their own, we emphasize the

Rule Of Four which I refer to as GNAWing at each new question [Video 29:48]. Any new

problem we are presented with can be attacked using one or more of the following modalities:

Graphical, Numerical, Analytic and Wordy (aka Verbal). In this course, the students need to

know how to solve traditional Calculus problems Analytically and Verbally. However, whenever

analytic or algebraic techniques fail, we can approach the problems Graphically and Numerically

using a Graphing Calculator. I installed Linux and a virtual calculator on each PC. Our PC

Desktop has a Virtual TI-83 and TI-89, the Unit Circle and a local SAGE Server. I find it

beneficial for my students to be well versed in all 4 calculating environments [Video 30:50].



Dimension Three: Effective use of student assessments to evaluate, monitor, and improve

student learning.

The narrative for Dimension Three should be about three pages.

3a. Describe how you assessed student learning and achievement for the topic discussed in

Dimension One and shown on the video, and how you use what you learned from the assessment

to improve your teaching.

This topic was addressed over a period of 3 days [Video 10:03] during the first week of

school [Supplemental Page 1]. Day 1 was about an algebraic approach using Completing The

Square to solve for Standard Form Conics. On Day 2, the subject of this video, we used

trigonometric identities to graph Conic Sections in Polar Form. Finally, Day 3 was a day for

sharing our summaries as seen in this student sample [Supplemental Page 5]. We even used

SAGE to confirm some of our results [Supplemental Page 8].

My teaching style was the same all three days. I alternated between 5-10 minute mini

lessons and 5-10 minute WhiteBoard sessions. In each case, I used the WhiteBaording session to

walk around the room as a facilitator, not the “sage on stage.” In so doing, I could assess student

understanding on an individual basis. I accomplished this by walking from group to group

quizzing one student at a time about their understanding of the topic at hand. In this fashion, I

can easily see when students are struggling with a topic either individually or as a group. If

several groups are confused, I can alter the next mini lesson to accommodate their needs.

Whenever a topic proves particularly difficult, I readdress said topic the next day. If only a few

students are struggling with the subject, then I ask them to come to extra help where we can

work one on one at a pace more suited to each individual.

Edmodo [Video 2:40] is a website that is very easy for my students to use as it is

organized like FaceBook. However, Edmodo pages are private for each class a teacher creates. I



use Edmodo extensively to share assignments, ScreenCasts and SmartNotes with my students on

a daily basis. Very often I will post a discussion question on Edmodo to see how my students are

doing with their homework or preparation for a test.

3b. Discuss other specific ways that you routinely assess and guide student learning. You may

include examples of formative or summative techniques, including student presentations,

projects, quizzes, unit exams, or other methods.

Every week has at least one MCQ Monday, TH Tuesday, Think-a-lot Thursday or FRQ

Friday. MCQ Mondays consist of a mock AP Exam, taken individually, with only Multiple

Choice Questions one week. The following Monday becomes an AP Review session going over

those questions. Problems are then solved in groups at the WhiteBoards and solutions are

presented at the SmartBoard. I grade each mock AP as a test to assess student progress. I gauge

improvement in student understanding as they work in groups and on subsequent tests.

TH Tuesdays are comprised of a mock Take Home AP Exam with only Free Response

Questions (FRQ) one week to be completed at home in groups and graded as a test. The

questions are due the following Tuesday when we review solutions at the WhiteBoards.

Think-a-lot Thursdays consist of review questions given the day before a unit test.

Groups compete for bonus points applied to the test. Points are assigned for the best solutions. I

critique solutions much as an AP Reader would. The unit test, given the next day, is FRQ Friday.

In addition, every week has at least one TED Tuesday, YouTube Wednesday or Ignite

Thursday where students present and discuss short videos they find on the internet that are

related to any topic we have covered so far. These videos could be Math and Science specific

documentaries. Some videos are funny song parodies reviewing topics such as the Chain Rule or

Riemann Sums. Many videos are produced by the students themselves as extra credit projects.

Students can also write extra credit reports on a STEM related current event news article.



Most of my lessons are a combination of I Do, We Do, You Do. “I Do” refers to my mini

lessons. I assess student understanding during a mini lesson by asking pointed questions to

individual students throughout the lesson. “We Do” refers to group work such as WhiteBoard

sessions or Take Home Exams. I gauge student understanding while students are WhiteBoarding

by mingling with each group. I assess Take Home exams as a group grade. “You Do” refers to

watching ScreenCasts at home or completing in-class tests. I sometimes give pop quizzes the day

after a ScreenCast is assigned. I grade in-class tests individually.

3c. Provide evidence of your teaching effectiveness as measured by student achievement on

school, district or state assessments, or other external indicators of student learning or

achievement.

I am constantly surprised at my students’ accomplishments as evidenced by this college

recommendation [Supplemental Page 10]. I use Edmodo [Video 4:41] and student visitations to

keep in contact with graduates and have received great feedback as to their success in Math and

Science majors due to their experiences in my class. My students do well on the SAT Math II,

AP Physics and AP Calculus exams and in competitive clubs such as Mathletes, and Academons.

I pride myself in empowering my students by giving them the technology they need and

teaching them how best to use it. I make sure they have everything they need at home to

complete homework and study for tests by posting all class materials online. However,

sometimes the volume of material can be overwhelming. I make sure to give my students choices

[Video 4:05]. I tell my students to limit their homework to 30 minutes viewing the ScreenCast,

30 minutes reading the textbook and 30 minutes completing exercises. Whenever the student has

difficulty, I make sure they know I’m available for extra help after school. If a student fails a

test, he or she may take a new test after reviewing the old test in extra help. If the student

improves the grade, I use the average of the two assessments for my electronic grade book entry.



Dimension Four: Reflective practice and life-long learning to improve teaching and student

learning.

The narrative for Dimension Four should be about two pages.

4a. Discuss the more successful and less successful aspects of the instructional activities shown

in the video and describe what you might do differently to improve student learning.

This lesson was recorded during the first week of class [Supplemental Page 1]. My

students were not yet familiar with my teaching style. This was only the second class where they

worked in groups showing their work on WhiteBoards. Neither were these students familiar with

the Inverted Classroom Paradigm. In retrospect, I would have liked to have had smaller groups

so more students could scribe and orate. I prefer several groups of two students. I tried to use

smaller groups in later classes. With smaller groups, more students will also be able to

participate. However, it is very difficult to assess student learning if we have too many groups.

I also had a bit of technical difficulty with the SmartBoard [Video 7:38] whereby it

started to respond very slowly. I was forced to think on my feet and redo my lesson the old

fashioned way on a more traditional WhiteBoard. Thank goodness, the SmartBoard started

responding later in the class [Video 28:36] so I could use Virtual TI to graph Twisted Conics.

4b. Describe how reflection on your teaching practices helps you improve your classroom

instruction. You may provide examples of lessons or activities you revised based on this

reflection.

It is very important for me to reflect on my teaching practice every single day. In fact, I

began my BlogSpot [Supplemental Page 6] as a diary to keep track of said reflections.

Originally, my blog was just a “note to self.” On the blog I would write observations about

teaching strategies I tried in class or new technology we used. I would blog about successful

strategies as well as those that did not work. I also write at length about successful uses of

technology as well as solutions that had to be reevaluated. After a while, my blog became quite



popular as colleagues across the country appreciated my honesty as well as all the materials I

shared. I recieve many positive comments on my BlogSpot and my Youtube channel.

I post ScreenCasts to YouTube [Supplemental Page 7] every day with a link to Edmodo

Video 2:40]. I post SmartNotes to my blog every week linked to Edmodo as well. In so doing, I

am constantly reviewing, rethinking and reediting everything I do in class. My using technology

in this way has made it easy for me to reflect upon and improve my teaching practice.

4c. Using one or two of the professional development experiences cited in your résumé, describe

how your participation in these activities has improved your teaching and enhanced student

learning.

I was glad that Peter Hoffmeister, a 30 year veteran Mathematics teacher, nominated me.

I was honored by the recommendations I received from my Chairperson, Geetha Murthy, and

Principal, Susan Knors, who have always encouraged me to try new and innovative techniques. I

am particularly humbled by the recommendation from my mentor, Judy Broadwin. Were it not

for Judy, I would never have considered teaching Calculus all these many years.

I have attended many conferences and in-service courses over the years. In fact, I have

learned from several past PAEMST winners at such professional development opportunities

(Steven Conrad, Joseph Quartararo, Ann Davidain). However, I have learned the most from

Judy’s AP Calculus Summer Institutes and LIMACON [Supplemental Page 9] presentations.

Judy inspired me to use technology extensively inside and outside of class. For example, I

learned everything I ever knew about Graphing Calculators from Judy. I remember taking

several Summer Institutes with Judy at SUNY Old Westbury in the early 1990s. Even then, Judy

was teaching with Graphing Calculators well before most teachers had ever heard of this

technology. I have since extended the use of calculating environments to Computer Algebra

Systems such as SAGE [Supplemental Page 8] and Programming Languages such as Python.



Dimension Five: Leadership in education outside the classroom.

The narrative for Dimension Five should be about one page.

5a. Describe how you have supported other teachers, student teachers or interns through

activities such as induction, mentoring, leading professional development activities, or co-

teaching.

I have taught many in-service courses [Résumé] at Baldwin High School over the years

including such topics as Word Processors, Spreadsheets, Data Bases and Programming in

Mathematics class. In addition, I have presented at several conferences such as LIMACON

(Long Island Math Conference), T^3 (Teachers Teaching with Technology) and BSHS SDD

(Baldwin Senior High School Staff Development Day). I have spoken about the use of

technology in the pre-Calculus and Calculus curricula. Some of the topics I’ve presented are:

TI83 BASIC Programming, TI89 BASIC Programming, Solving Differential Equations

Graphically with Slope Fields, Solving Differential Equations Numerically with Euler’s Method,

“Look Ma, No Calculator,” SAGE Advice: pre-Calculus, SAGE Advice: Calculus. I have also

mentored several teachers in my department helping them to use technology in the classroom.

5b. Describe how you contribute to educational excellence at the school, district, state, or

national level.

I speak regularly at LIMACON organized by SUNY Old Westbury and at T^3 hosted by

Molloy College. For example, I was speaker #43 at LIMACON 2013 [Supplemental Page 9]

where I ran a workshop on the use of SAGE in pre-Calculus class replacing the TI-83/84. I will

be speaking at Molloy this November about teaching Calculus with SAGE instead of a TI-89/92.

I contribute regularly to the AP-Calculus, AP-Physics and AP-CompSci Educational Discussion

Groups sponsored by The College Board. I have also learned a great deal from my participation

in SAGE-EDU sponsored by SageMath.org and EDU-SIG sponsored by Python.org. These

forums are a great resource for me and constitute my Professional Learning Network.