Alvar Garcia TID42859_Dimensions_Essay_New

PAEMST 2014-2015 - Dimensions of Outstanding Teaching Written Response Page 1
Name: Alvar Garcia Teacher ID: 42859
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 mathematics or science concept that is fundamental to understanding the chosen
lesson.
We approached the topic of Anti-Differentiation both numerically and algebraically over
a period of 3 days during the last full week of November [Supplemental Page 1]. Day 1 was
about a numerical approach using The Fundamental Theorem of Calculus and Riemann Sums to
evaluate Definite Integrals as areas. On Day 2, the subject of this video, we used an algebraic
approach to find Anti-Derivatives as seen in this student sample [Supplemental Page 5]. Finally,
Day 3 was a day for combining both approaches and solving Variable Separable Differential
Equation Word Problems. We even used SAGE to confirm our results [Supplemental Page 8].
Today’s lesson focused on the difference between Definite Integrals and Indefinite
Integrals. The day prior to this lesson, we were working with numerical methods for estimating
Definite Integrals as Riemann Sums using many rectangles by programming our TI nSpire CX
CAS Graphing Calculators. By this point in the school year, we’ve also written programs for
solving Quadratics and for Newton’s Method. We will also write a program in TI-BASIC for
Euler’s Method. Now, we move on to a more algebraic approach finding Anti-Derivatives
analytically whenever possible. We made the case that the Anti-Derivative is not unique [Video
33:45] and that any two Anti-Derivatives may be equivalent if they only differ by a constant.
Graphically speaking, two such functions are equivalent Anti-Derivatives if their graphs are
simply vertical shifts, one from the other. For example, x dx = x2
+C, so x2
+1 and x2
+2 [Video
34:29] are equivalent Anti-Derivatives as the slopes of the tangent lines along each curve will be
the same at any given value of x in the domain of the Anti-Derivative. All the while, we are

emphasizing the connection between the Derivative and the associated Anti-Derivative as
described in the Fundamental Theorem of Calculus (FTC):

a
b
f(x) dx = F(b)-F(a) if and only if F’(x)=f(x) and f(x) is continuous ˅xϵ[a,b]
My Calculus students took Honors preCalculus last year. This is a standard preCalculus
class I teach with SAGE, a Computer Algebra System (CAS), ending with an introduction to the
basic concepts of Calculus. One of the main points of our preCalculus class was what makes a
relation a function. We also conduct a survey of several different functions throughout the year.
As such, the idea of geometric transformations of functions such as dilation, reflection and
translation is very familiar to them. I think it’s great to show that several equivalent Anti-
Derivatives only differ by a constant because the derivative function, namely the integrand, is the
same [Video 33:15]. Appealing to the student’s graphical intuition that these Anti-Derivative
functions are vertical shifts of each other drives the point home even better [Video 34:29]!
Finally, I am constantly talking to my students about Derivative Functions as Rates of
Change and Definite Integrals as Accumulator Functions for those changes based on the FTC.
1b. Explain why this 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 amazing how central this topic is for the rest of the year. If the students don’t
understand the difference between Definite and Indefinite Integrals as well as the relationship
between Derivatives and Anti-Derivatives as stated in the FTC, my students will be at a loss to
understand all that comes next in the curriculum. The rest of the year builds on these concepts
every day as we deal with more and more complex integration techniques and differential
equations that we can solve analytically with Indefinite Integrals using, for instance, Integration
By Parts or Partial Fractions to find exact answers. Further, if we cannot deal with a Definite

Integral analytically, we do so numerically using functions built into the Graphing Calculator
(GC) and programs we write. In this vein, we go on to develop Euler’s Method, Slope Fields and
Power Series for numerical estimates. We have a lot of fun periodically taking a break from
theory and applying what we have learned to real world applications. Said applications usually
take the form of modeling problems we can solve by writing our own Differential Equations. We
mostly encounter exponential growth and decay, exponential approach and logistic models.
1c. Discuss the misconceptions or misunderstandings that students typically have with regard to
this concept and explain how you address them in your chosen lesson.
My students will often forget to use a “+C” in their work when finding an Anti-
Derivative analytically and arrive at an incomplete answer. I like to use “Filks” in class to help
my students remember to watch out for these kinds of errors. What we call a Filk is a
recognizable tune or folk song, such as “My Darling Clementine,” where we change the words to
make a funny, sometimes satirical, song about Math or Science to make a point [Video 23:10].
We found just such a Filk by Dan Kennedy, the author of our text book, about the very problem
of mixing up Definite and Indefinite Integrals and when to use “+C.” As per my lesson plan
[Supplemental Page 3], I encouraged my students to solve the Free Response Question from
1988 (AB6) related to the new lyrics of Dan Kennedy's version [Supplemental Page 4]. This is
an entertaining and educational cautionary tale about a poor girl named Betty Sue who managed
to get 0 out of 9 points on this question even though she arrived at the correct final answer! To
avoid having this happen to my students when it comes to their AP Calculus exam in May, I
make a point of practicing AP Free Response Questions in class all year long as unit tests and
pre-tests. We also practice Multiple Choice questions from old AP exams throughout the year!
My students had some AB Calculus experience in Honors preCalculus last year, so we started
reviewing Multiple Choice Questions from old AB Calculus exams early in the year!

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
concept you have chosen to address in Dimension One.
Over the past 30 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. To this end, I
have been ScreenCasting many of my lessons on YouTube [Supplemental Page 7] for at least 6
years now. As a result, I think I can say that I have “Flipped the Classroom!” 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 my Mathematics class, however, I have modified the
flipped classroom concept in such a way that I ScreenCast [Video 2:10] 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 who 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 4 years and I am really
impressed with the result [Video 7:30]. 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 [Video 20:20] to complete a
task, such as a Chemistry Lab. Students work on said task individually at first, then in pairs.
Finally, 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. However, my students typically work in
small groups for about 10 minutes on a new problem after I present a mini lesson. Each group
will work in parallel on a different problem. Then each group presents their solution to the class.
2b. Explain how you identify and build on students’ prior knowledge, when and how this
knowledge is addressed in your video, and how you address this knowledge in your general
teaching strategies.
Students’ prior knowledge for this topic is a basic understanding of functions,
transformations, derivatives and the FTC. I tease these topics out of my students during the
course of the lesson. To aid in this process, I often find it necessary to supplement the textbook
with my own handouts. One of today’s handouts included a copy of a Multiple Choice Question
from the 1998 BC exam (Part IA #15) the students worked on individually [Video 1:08]. Another
handout was a copy of the Free Response Question discussed above [Supplemental Page 7]. Yet
a third handout included the lyrics to the Betty Sue song [Supplemental Page 4]. I usually
summarize each lesson before the end of class [Video 34:52].
I did find it difficult, however, to teach this lesson without using each student’s name. I
was told to avoid using names so as to have all the families file the Video Permission slip.
However, I do try to make my students feel welcome [Supplemental Page 2] by greeting them
before class, making an inviting environment [Supplemental Page 6] in my classroom and calling
on students by name during class. I did slip a few times as I know many of these students from

prior classes (pre-Calculus, Computer Math, AP Computer Science, Scientific Computing Lab)
and clubs (Chess Club, Continental Mathematics League, Computing Independent Study,
National Computer Science Honor Society). So, I think 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 usually break up the class
into groups of 3 or 4 students. These groups are comprised of a Scribe, an Advocate, a Critic and
an Orator. The Scribe is in charge of writing the solution [Video 12:20] on the WhiteBoard with
input from all members of the group. The Orator gives an oral presentation [Video 13:10]
explaining the steps given on the WhiteBoard when the group work is done. The Advocate is the
student that initially comes up with a solution [Video 17:36] and the Critic typically makes
corrections [Video 16:45] on the WhiteBoard. Each group is also made up of students of varied
ability. I find students in such a setting work well together and learn from each other.
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. Any new problem can be
attacked using one or more of the following modalities: Graphical, Numerical, Analytical or
Wordy (aka Verbal). In this course, we need to know how to solve traditional Calculus problems
Analytically and Verbally. Whenever analytic or algebraic techniques fail, we solve problems
Graphically and Numerically using a GC or a CAS. I installed Linux and several computing
environments on each student PC. Our PC Desktop has a Virtual TI-84C, a Virtual TI nSpire CX
CAS, the Unit Circle [Video 20:42] as a PC DeskTop BackGround image and a local SAGE
Server. I find it beneficial for my students to be well versed in all four calculating environments.

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 concept discussed in
Dimension One and shown on the video, and how you use what you learned from the assessment
to improve your teaching.
I typically alternate between 5-10 minute mini lessons and 5-10 minute WhiteBoard
sessions. In each case, I use the WhiteBoarding session to walk around the room as a facilitator,
not the “sage on stage.” In so doing, I can assess student understanding on an individual basis. I
accomplish 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. If a topic proves particularly difficult, I readdress said
topic the next day. If only a few students are struggling with the subject, I ask those students to
come to extra help where we can work one on one at a pace more suited to each individual.
Edmodo [Video 0:16] 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 to share assignments, ScreenCasts, SmartNotes [Video 22:00] and SageCells
[Supplemental Page 8] with my students on a daily basis. I will often 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 Multiple Choice (MCQ) Monday, Take Home (TH) Tuesday,
Think-a-lot Thursday or Free Response (FRQ) Friday. MCQ Mondays consist of a mock AP
Exam, taken individually, 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, 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 Filks 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 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 to keep in contact with graduates and
have received great feedback as to their success in STEM majors. My class also competes in the
Continental Mathematics League (CML). This competition is like a Calculus based Mathletes
where my Calculus students have excelled placing in the top 3 schools in our region (NY, NJ)
I pride myself in empowering my students by giving them the technology they need and
teaching them how best to use it. For example, last Summer I ran a successful Donorschoose,
http://www.donorschoose.org/calcpage2010, campaign to fund a class set of TI nSpire CX CAS
Graphing Calculators. This year, I gave each student one of these calculators just like a textbook.
The volume of material can be overwhelming. I try to give my students choices. When
we have new material, I tell my students to limit their homework to 20 minutes viewing the
ScreenCast, 20 minutes reading the textbook and 20 minutes completing exercises. If we are
reviewing a topic, I expect my students to devote up to 60 minutes on assigned exercises only.
I try to make my students feel welcome in my classroom. I often meet with them after
school in a club setting either to practice for the CML or for our Computing Independent Study
(CIS). Many of my Calculus students are also in my AP Computer Science class. Students that
took AP Computer Science last year take part in CIS. This year we studied Android Application
Development. Last year we set up and programmed a Linux Cluster (see fractals on the walls in
the video). CIS students also join our chapter of the National Computer Science Honor Society
which I started last year. All these extra-curricular activities make for a good rapport between my
students and myself. I am privileged to be their teacher and honored to get to know them. So,
whenever a student has difficulty, they know I am always available for extra help after school.
Also, if a student fails a test, they may take a new test after reviewing the old test in extra help.

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.
I prefer several groups of two or three students at the most. Later in the year, I tried to use
smaller groups after purchasing more WhiteBoards. With smaller groups, more students were
able to participate. However, it is difficult to assess student learning if we have too many groups.
I had some technical difficulty with the SmartBoard [Video 32:27] whereby I could not
enter input on our TI nSpire CX CAS emulator. I was forced to sit at a keyboard at the front of
the room to enter input for the emulator when we were checking our work [Video 31:00]. I also
use this classroom for my AP Computer Science students. In that class, I will sometimes sit at a
keyboard in the back of the room to demonstrate a programming technique using Java on the
Linux Desktop I installed and maintain. The room is set up so I can see all the monitors from the
back of the room and keep everyone on task. The keyboard in the front of the room is sometimes
convenient when ScreenCasting as my mic is up front attached to the SmartBoard [Video 1:08].
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 blog about successful strategies
as well as those that do not work. I also write at length about successful uses of technology as
well as solutions that need 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

receive many positive comments on my BlogSpot, http://shadowfaxrant.blogspot.com, and my
associated YouTube channel, http://www.youtube.com/calcpage2009. I post ScreenCasts to
YouTube [Supplemental Page 7] every day with a link to Edmodo. I post SmartNotes and
SageCells [Supplemental Page 8] for my students to my blog 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.
For example, when I started WhiteBoarding, I assigned the same problem to every group. Upon
reflection, I found a more efficient use of time was to assign many problems to work in parallel!
4c. Using a professional development experience cited in your résumé, describe how your
participation in this activity has improved your teaching and enhanced your students’ learning.
I was honored to have received recommendations from Principal Susan Knors [Baldwin
Rec] and Director of Continuing Education Lou Cino [Molloy Rec] who have always
encouraged me to try innovative techniques, technologies and curricula. I am very proud of the
recommendation from Stuart Kaplan [NCC Rec]. Stu was one of my very first students at
Baldwin Senior High. Now, Stu is my Chairman when I teach NCC Summer Sessions! I am also
particularly humbled by the recommendation from my long-time mentor, 1990 NYS PAEMST
Judy Broadwin [Baruch Rec].
I have attended many conferences over the years to help improve my craft. I have learned
from several past PAEMST winners at such professional development opportunities (Steven
Conrad 1985, Joseph Quartararo 1987, Ann Davidian 2001). However, I learned the most from
Judy’s AP Calculus Summer Institutes. I remember taking several Summer Institutes with Judy
at SUNY Old Westbury in the early 1990s. Even then, Judy was teaching with GCs before most
teachers had heard of this technology. I have since extended the use of calculating environments
to CASs 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 and how their teaching practice has since improved.
I have taught many in-service courses [Résumé] over the years including such topics as
using Word Processors, Spreadsheets, Data Bases and Programming in Mathematics class. In
addition, I have presented at several conferences such as the Long Island Math Conference and
T^3. I have spoken about the use of technology in Calculus and Computer Science curricula.
Some of the topics I’ve presented are: TI83 BASIC Programming, Solving Differential
Equations Graphically with Slope Fields, Look Ma: No Calculator, SAGE Advice: pre-Calculus.
I have also mentored several teachers in my department helping them to use technology in the
classroom. Teachers report positive experiences using these techniques. One of my colleagues
was observing this very lesson sitting in the back of the room [Video 2:40]!
5b. Describe how you contribute to educational excellence at the school, district, state or national
level.
I am regularly invited to speak at LIMACON hosted by SUNY Old Westbury and at T^3
hosted by Molloy College [Molloy Recommendation]. Both conferences attract educators from
around the country who want to share ideas for improving the teaching of Mathematics with
technology. For example, I was speaker #43 at LIMACON two years ago [Supplemental Page 9]
where I ran a workshop on the use of SAGE in pre-Calculus class replacing the TI-83/84. 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, and the conferences listed above, greatly enhance my Professional Learning Network.

Alvar Garcia TID42859_Dimensions_Essay_New

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