2017 PAEMST NARRATIVE ESSAY

Dimensions of Outstanding Teaching Written
Response
Dimension One: Mastery of content appropriate
for the grade level taught.
1a. Describe the concept that is important for students to learn and
understand within 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]. SAGE is
a Computer Algebra System available free online,
http://sagecell.sagemath.org
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, thin
PAEMST 2017-2018 - Dimensions of Outstanding Teaching Written Response
Name: Alvar Garcia-Fernandez
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Teacher ID: 42859

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, the Anti-Derivative of
x with respect to x is (1/2)x^2 + C. In other words, (1/2)x^2 +1
and (1/2)x^2 +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:
If F’(x)=f(x) and f(x) is continuous for all xε[a,b], then
Integral(f(x),x,a,b) = F(b)-F(a).
My Calculus students took Honors preCalculus last year. This
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Teacher ID: 42859

is a standard preCalculus class I teach with SAGE 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 Fundamental Theorem of Calculus.
1b. Discuss why this concept is fundamental for them 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
Fundamental Theorem of Calculus, my students will be at a
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Teacher ID: 42859

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 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.
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Teacher ID: 42859

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 preCalculus last year, so we started
reviewing Multiple Choice Questions from AB Calculus exams
early in the year!
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Teacher ID: 42859

Dimension Two: Use of instructional methods
and strategies that are appropriate for the
students in the classroom and that support
student learning.
2a. Describe the instructional approaches you used in the video to
help students understand the concept addressed in Dimension
One and explain how you identified and built on your students'
prior knowledge.
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
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Teacher ID: 42859

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.
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Teacher ID: 42859

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. Describe, in general, your decision-making process when and
how you utilize various instructional methods, strategies, or tools.
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:
Scribe is in charge of writing the solution [Video 12:20] on the
WhiteBoard with input from all members of the group.
Advocate is the student who initially comes up with a solution
[Video 17:36].
Critic typically makes corrections [Video 16:45] on the
WhiteBoard.
Orator gives an oral presentation [Video 13:10] explaining the
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Teacher ID: 42859

steps given on the WhiteBoard when the group work is done.
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 Graphing Calculator or a Computer
Algebra System.
I installed Linux and several computing environments on each
student PC. Our PC Desktop has a Virtual TI-84, 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. So I provide each student with their
own PC workstation! Please note that we have many handheld
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Teacher ID: 42859

TI84s as well so the use of our PC emulators does not violate
copyright.
2c. Discuss the instructional methods and strategies you use to
meet the learning needs of all students, challenging those with
stronger knowledge while ensuring learning for those with less
understanding.
Students’ prior knowledge for this topic is a basic
understanding of functions, transformations, derivatives and
the Fundamental Theorem of Calculus. 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
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Teacher ID: 42859

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 almost slipped a
few times as I know many of these students from prior classes
(preCalculus, Computer Science Honors (preAP), AP
Computer Science, Calculus Research 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.
Dimension Three: Effective use of student
assessments to evaluate, monitor, and improve
student learning.
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 through out the course of the
class. 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
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Teacher ID: 42859

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 very much 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 Question
Monday, Take Home Tuesday, Think-A-Lot Thursday or Free
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Teacher ID: 42859

Response Friday.
Multiple Choice Question Mondays consist of a mock AP
Exam (adjusted for a class period), 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 quiz to assess
student progress. I gauge improvement in student
understanding as they work in groups and on subsequent
tests.
Take Home 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 or Think-Pair-Share 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 the
same way an AP Reader would. The unit test, given the next
day is Free Response Question Friday.
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Teacher ID: 42859

In addition, every week has at least one TED Tuesday
(https://www.ted.com), YouTube Wednesday
(https://www.youtube.com) or Ignite Thursday
(http://www.ignitetalks.io) 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 (Science, Technology, Engineering and
Mathematics) or STEAM (Science, Technology, Engineering,
Art and Mathematics) 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.
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Teacher ID: 42859

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 or
STEAM majors.
My class also competes in the Continental Mathematics
League (http://cmleague.org). This competition is like a
Calculus based Mathletes where my Calculus students have
excelled usually placing in the top 3 schools in our region (NY,
NJ) every year.
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.
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Teacher ID: 42859

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 Continental Mathematics League meets or for
our Computing Independent Study class. Many of my Calculus
students are also in my AP Computer Science class. Students
that took AP Computer Science last year take part in
Computing Independent Study. 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). Computing Independent Study 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
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Teacher ID: 42859

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.
I have tried to meet the needs of my students by writing new
and innovative curricula:
CSH: Computer Science Honors (using python programming to
introduce AP Computer Science basics)
CSI: Computing Science Independent Study (students
research advanced topics after completing the AP Computer
Science)
CSL: Computing Science Lab aka Scientific Computing Lab
aka Calculus Research Lab (learning Calculus using SAGE)
Dimension Four: Reflective practice and life-long
learning to improve teaching and student
learning.
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
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Teacher ID: 42859

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 near the
end of this lesson [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. In that
class, I 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) on the SmartBoard.
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
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Teacher ID: 42859

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,
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Teacher ID: 42859

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 Caterina Lafergola [Baldwin Recommendation] and
Director of Continuing Education Lou Cino [Molloy
Recommendation] who have always encouraged me to try
innovative techniques, technologies and curricula.
I am very proud of the recommendation from Stuart Kaplan
[Nassau Community College Recommendation]. Stuart was
one of my very first students at Baldwin Senior High. Now,
Stuart is my Chairman when I teach Nassau Community
College Summer Sessions!
I am also particularly humbled by having Judy Broadwin 1990
NYS Secondary Mathematics PAEMST as my long-time
mentor. Judy also wrote a recommendation letter for me in the
past but I am limited to only 3 recommendation letters this
year, one being from my Principal.
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Teacher ID: 42859

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 to SAGE [Supplemental Page 8]
and the TI nSpire CX CAS as well as Programming Languages
such as Python.
Dimension Five: Leadership in education outside
the classroom.
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 (LIMACON at SUNY
Old Westbury) and Teachers Teaching with Technology (T^3
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at Molloy College). I have spoken about the use of technology
in preCalculus and Calculus curricula. Some of the topics I’ve
presented include: "TI83 BASIC Programming," "Solving
Differential Equations Graphically with Slope Fields," "TI89
BASIC Programming," "Look Ma: No Calculator, SAGE Advice:
pre-Calculus" and "Look Ma: No Calculator, SAGE Advice:
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 and/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 preCalculus
class replacing the TI-83/84.
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LIMACON (Long Island MAth CONference) is a conference
that is national in scope and is sponsored by the National
Council of Teachers of Mathematics, Nassau County
Association of Math Supervisors as well as SUNY Old
Westbury. I have attended this conference every year for over
25 years. I have been a presenter at LIMACON 5 times in the
last 10 years. LIMACON focuses on how best to teach
mathematics at the High School and College levels.
T^3 (Teachers Teaching with Technology) is regional in scope
and sponsored by Texas Instruments Incorporated and Nassau
County Math Teachers' Association as well as Molloy College.
I have been a speaker at this conference 4 times in the last 12
years. T^3 focuses on the use of technology in the
Mathematics classroom.
I care for a two people with significant disabilities in my family.
As a result, I am unable to travel for many other conferences
that may interest me. I am restricted to presenting at these two
conferences.
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
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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.
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2017 PAEMST NARRATIVE ESSAY

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2017 PAEMST NARRATIVE ESSAY