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PAEMST 2012-2013 - Dimensions of Outstanding Teaching Written Response Page 1Name: Alvar Garcia Teacher ID: 32327Dimensions of Outstanding Teaching Written ResponseDimension One: Mastery of mathematics or science content appropriate for the grade leveltaught.The narrative for Dimension One should be about three pages.1a. Discuss the mathematical or scientific ideas that are fundamental to understanding the chosentopic or concept.Early this school year in my AP Calculus BC class, we made sure to review suchconcepts as functions and relations, inverse functions and one-to-one functions as well as explicitversus implicit definitions of said functions or relations. In the lesson prior to this one, we usedalgebraic techniques to solve for the standard form of a conic section. Standard form conics areeasily graphed, as they have horizontal and vertical symmetry, without a Graphing Calculator aswell as with a Graphing Calculator in Function Mode.Today’s lesson focused on what I call Twisted Conics. These conics are not easily writtenin standard form as these graphs do not have horizontal or vertical symmetry. We discussedrewriting these conics in Polar Form using Trigonometric Identities. Further, these conics aredifficult to graph by hand due to a rotational transformation. However, Twisted Conics arereadily graphed on a Graphing Calculator in Polar Mode:We have that cos(Ɵ) =xrand that sin(Ɵ) =yr, 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 generalform conic expression, Ax^2+Bxy+Cy^2+Dx+Ey+F=0, and solve for r as an explicit function ofƟ. In this manner, we can graph conic sections in Polar Mode on our Graphing Calculator. Ideveloped this method for graphing Twisted Conics when we cannot complete the square andgraph in Function Mode. I don’t recall ever seeing this method in a textbook. The method used inour text is very convoluted. I think I may have to publish this method one day.
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PAEMST 2012-2013 - Dimensions of Outstanding Teaching Written Response Page 2Name: Alvar Garcia Teacher ID: 32327There are many topics in pre-Calculus that overlap with our Junior year NYS Regentscourse, Algebra II & Trigonometry at the High School. When I teach pre-Calculus, I make sureto emphasize those topics needed in AP Calculus AB and BC that are not covered as thoroughlyas 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 ConicSections 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 morecomplex 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 firstlesson of the year [Supplemental Page 3]. We have now come full circle and managed to reviewmost 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 APReview. The last class before review was about calculating derivatives,dydx=dydtdxdt=dydt dtdx,and arc length,dldt=dxdt dxdt+dydt dydt, 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) anddydx=f(t ) cos(t)+f ’(t) sin(t)- f(t) sin(t)+f ’(t) cos(t).Applying the Parametric Arc Length formuladldt=dxdt dxdt+dydt dydtwe have:
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PAEMST 2012-2013 - Dimensions of Outstanding Teaching Written Response Page 3Name: Alvar Garcia Teacher ID: 32327(–f(Ɵ) sin(Ɵ) + f (Ɵ) cos(Ɵ))2= f(Ɵ)2sin2(Ɵ) – 2 f(Ɵ) f (Ɵ) sin(Ɵ) cos(Ɵ) + f (Ɵ)2cos2(Ɵ)(f(Ɵ) cos(Ɵ) + f (Ɵ) sin(Ɵ))2= f(Ɵ)2cos2(Ɵ) + 2 f(Ɵ) f (Ɵ) sin(Ɵ) cos(Ɵ) + f (Ɵ)2sin2(Ɵ)dldƟ= f(Ɵ)2+ f (Ɵ)2) and, therefore, l= Ɵ1Ɵ2f(Ɵ)2+ f (Ɵ)2dƟ.Further, using the definition of a radian, the arc length of a circle s=r Ɵ and the area of asector of a circle we have:dAdƟ=12f(Ɵ)2and, therefore, A =Ɵ1Ɵ2f(Ɵ)22dƟ.1c. Discuss the misconceptions or misunderstandings that students typically have with regard tothis topic or concept.Many students find Polar Notation very daunting at first as they are typically veryconfused by even the simplest of Trigonometric concepts. I like to have fun with my studentstelling 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 totrigonometry 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 thatsin(Ɵ) =opposite leghypotenuse, cos(Ɵ) =adjacent leghypotenuseand tan(Ɵ) =opposite legadjacent leg. The teacher waswracking his mind over and over as to how to help his students remember these ratios. Finally, itcame to him! ‘SOHCAHTOA East of Java!’ will be easy for them to remember.” This mnemonicwas indeed easy to remember as, at the time in 1969, there was a new hit disaster movie playingin 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 overcomethese obstacles. It is helpful to know some of their varied interests and hobbies [Video 23:36].
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PAEMST 2012-2013 - Dimensions of Outstanding Teaching Written Response Page 4Name: Alvar Garcia Teacher ID: 32327Dimension Two: Use of instructional methods and strategies that are appropriate for thestudents 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 thetopic or concept chosen in Dimension One.Over the past 25 years, I have been fortunate to have had a very supportive administrationat my High School. I have always been encouraged to bring new teaching techniques andtechnologies into the Mathematics classroom to benefit my students. I have also beenencouraged to share these techniques and technologies with the rest of the faculty. As a result, Ihave been ScreenCasting many of my lessons on YouTube [Supplemental Page 7] for at least 5years. Now I can say that I have “Flipped the Classroom” [Video 4:41]. The Flipped, orInverted, Classroom refers to a teaching method becoming more and more popular in Scienceclasses all across the nation whereby students are assigned classwork for homework andhomework for classwork. Some of the homework consists of watching a screencast of newmaterial before it is covered in class. This approach works well in Science classes as the teacherfrees up extra class time for lab work. For Mathematics class, I have modified the flippedclassroom 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 newmaterial after the material has been taught in class. These ScreenCasts are also useful for thestudent 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 onlineforums. These colleagues and I form a Professional Learning Network. From this learningnetwork, not only did I learn about the Flipped Classroom, but I also learned aboutWhiteBoarding. I have been WhiteBoarding in my classes for over 2 years and I am really
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PAEMST 2012-2013 - Dimensions of Outstanding Teaching Written Response Page 5Name: Alvar Garcia Teacher ID: 32327impressed with the result [Video 18:00]. My students use small, portable WhiteBoards to sharetheir 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 aChemistry Lab. Then all the groups come together to share their results. They are all working onthe same problem. However, their data may be different making the presentations of group workvaried even though the students arrive at the same conclusions. My students typically work ingroups of 4 for about 10 minutes on a new problem after I present a mini lesson. Each group willwork 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 isaddressed in your video and in your general teaching strategies.Students’ prior knowledge for this topic is a very basic understanding of Right TriangleTrigonometry and Trigonometric Identities. I tease these topics out of my students at the verybeginning of the class, before developing my method of graphing Twisted Conics. To aid in thisprocess, 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 forclasswork and homework. We covered exercises 1 and 2 in class the day before. We coveredquestions 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 eachlesson 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 notto use names so as to have all the families file the Video Permission slip. I always try to makemy students feel welcome [Supplemental Page 2] by greeting them before class [Video 1:25],
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PAEMST 2012-2013 - Dimensions of Outstanding Teaching Written Response Page 6Name: Alvar Garcia Teacher ID: 32327making an inviting environment [Supplemental Page 6] in my classroom [Video 22:00] andcalling on them by name during class. I did slip a few times because, despite the fact that thiswas the first week of school, I knew many of these students from prior classes (pre-CalculusHonors, AP Compter Science, Scientific Computing Lab) and clubs (Chess Club, ContinentalMathematics League, Computing Independent Study). I can readily assess what my studentsalready 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 allstudents, challenging those with stronger knowledge while ensuring learning for lessaccomplished students.When my students are WhiteBoarding, they work in groups. I break up the class intogroups of 3 or 4 students. These groups are usually comprised of a Scribe, an Advocate, a Criticand an Orator. The Scribe is in charge of writing the solution [Video 20:38] on the WhiteBoardwith input from all members of the group. The Orator gives an oral presentation explaining thesteps given on the WhiteBoard [Video 24:50], [Video 27:12]. The Advocate is the student thatcomes 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 theRule Of Four which I refer to as GNAWing at each new question [Video 29:48]. Any newproblem 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 toknow how to solve traditional Calculus problems Analytically and Verbally. However, wheneveranalytic or algebraic techniques fail, we can approach the problems Graphically and Numericallyusing a Graphing Calculator. I installed Linux and several computing environments on each PC.Our PC Desktop has a Virtual TI-83 and TI-89, the Unit Circle and a local SAGE Server. I find itbeneficial for my students to be well versed in all four calculating environments [Video 30:50].
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PAEMST 2012-2013 - Dimensions of Outstanding Teaching Written Response Page 7Name: Alvar Garcia Teacher ID: 32327Dimension Three: Effective use of student assessments to evaluate, monitor, and improvestudent learning.The narrative for Dimension Three should be about three pages.3a. Describe how you assessed student learning and achievement for the topic discussed inDimension One and shown on the video, and how you use what you learned from the assessmentto improve your teaching.This topic was addressed over a period of 3 days [Video 10:03] during the first week ofschool [Supplemental Page 1]. Day 1 was about an algebraic approach using Completing TheSquare to solve for Standard Form Conics. On Day 2, the subject of this video, we usedtrigonometric identities to graph Conic Sections in Polar Form. Finally, Day 3 was a day forsharing our summaries as seen in this student sample [Supplemental Page 5]. We even usedSAGE to confirm some of our results [Supplemental Page 8].My teaching style was the same all three days. I alternated between 5-10 minute minilessons and 5-10 minute WhiteBoard sessions. In each case, I used the WhiteBaording session towalk around the room as a facilitator, not the “sage on stage.” In so doing, I could assess studentunderstanding on an individual basis. I accomplished this by walking from group to groupquizzing one student at a time about their understanding of the topic at hand. In this fashion, Ican easily see when students are struggling with a topic either individually or as a group. Ifseveral 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 fewstudents are struggling with the subject, then I ask them to come to extra help where we canwork 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 isorganized like FaceBook. However, Edmodo pages are private for each class a teacher creates. I
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PAEMST 2012-2013 - Dimensions of Outstanding Teaching Written Response Page 8Name: Alvar Garcia Teacher ID: 32327use Edmodo extensively to share assignments, ScreenCasts and SmartNotes with my students ona daily basis. Very often I will post a discussion question on Edmodo to see how my students aredoing with their homework or preparation for a test.3b. Discuss other specific ways that you routinely assess and guide student learning. You mayinclude examples of formative or summative techniques, including student presentations,projects, quizzes, unit exams, or other methods.Every week has at least one Multiple Chioce (MCQ) Monday, Take Home (TH) Tuesday,Think-a-lot Thursday or Free Response (FRQ) Friday. MCQ Mondays consist of a mock APExam, taken individually, one week. The following Monday becomes an AP Review sessiongoing over those questions. Problems are then solved in groups at the WhiteBoards and solutionsare presented at the SmartBoard. I grade each mock AP as a test to assess student progress. Igauge 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 ResponseQuestions, one week, to be completed at home in groups and graded as a test. The questions aredue 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. Icritique 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 IgniteThursday where students present and discuss short videos they find on the internet that arerelated to any topic we have covered so far. These videos could be Math and Science specificdocumentaries. Some videos are funny song parodies reviewing topics such as the Chain Rule orRiemann 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.
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PAEMST 2012-2013 - Dimensions of Outstanding Teaching Written Response Page 9Name: Alvar Garcia Teacher ID: 32327Most of my lessons are a combination of I Do, We Do, You Do. “I Do” refers to my minilessons. I assess student understanding during a mini lesson by asking pointed questions toindividual students throughout the lesson. “We Do” refers to group work such as WhiteBoardsessions or Take Home Exams. I gauge student understanding while students are WhiteBoardingby mingling with each group. I assess Take Home exams as a group grade. “You Do” refers towatching ScreenCasts at home or completing in-class tests. I sometimes give pop quizzes the dayafter a ScreenCast is assigned. I grade in-class tests individually.3c. Provide evidence of your teaching effectiveness as measured by student achievement onschool, district or state assessments, or other external indicators of student learning orachievement.I am constantly surprised at my students’ accomplishments as evidenced by this collegerecommendation [Supplemental Page 10]. I use Edmodo [Video 4:41] and student visitations tokeep in contact with graduates and have received great feedback as to their success in Math andScience 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 andteaching them how best to use it. I make sure they have everything they need at home tocomplete 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 hasdifficulty, I make sure they know I’m available for extra help after school. If a student fails atest, he or she may take a new test after reviewing the old test in extra help. If the studentimproves the grade, I use the average of the two assessments for my electronic grade book entry.
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PAEMST 2012-2013 - Dimensions of Outstanding Teaching Written Response Page 10Name: Alvar Garcia Teacher ID: 32327Dimension Four: Reflective practice and life-long learning to improve teaching and studentlearning.The narrative for Dimension Four should be about two pages.4a. Discuss the more successful and less successful aspects of the instructional activities shownin 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]. Mystudents were not yet familiar with my teaching style. This was only the second class where theyworked in groups showing their work on WhiteBoards. Neither were these students familiar withthe Inverted Classroom Paradigm. In retrospect, I would have liked to have had smaller groupsso more students could scribe and orate. I prefer several groups of two students. I tried to usesmaller groups in later classes. With smaller groups, more students will also be able toparticipate. 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 itstarted to respond very slowly. I was forced to think on my feet and redo my lesson the oldfashioned way on a more traditional WhiteBoard. Thank goodness, the SmartBoard startedresponding 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 classroominstruction. You may provide examples of lessons or activities you revised based on thisreflection.It is very important for me to reflect on my teaching practice every single day. In fact, Ibegan 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 aboutteaching strategies I tried in class or new technology we used. I would blog about successfulstrategies as well as those that did not work. I also write at length about successful uses oftechnology as well as solutions that had to be reevaluated. After a while, my blog became quite
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PAEMST 2012-2013 - Dimensions of Outstanding Teaching Written Response Page 11Name: Alvar Garcia Teacher ID: 32327popular as colleagues across the country appreciated my honesty as well as all the materials Ishared. 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 EdmodoVideo 2:40]. I post SmartNotes to my blog every week linked to Edmodo as well. In so doing, Iam constantly reviewing, rethinking and reediting everything I do in class. My using technologyin 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, describehow your participation in these activities has improved your teaching and enhanced studentlearning.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, andPrincipal, Susan Knors, who have always encouraged me to try new and innovative techniques. Iam particularly humbled by the recommendation from my mentor, Judy Broadwin. Were it notfor 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 havelearned from several past PAEMST winners at such professional development opportunities(Steven Conrad, Joseph Quartararo, Ann Davidain). However, I have learned the most fromJudy’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, Ilearned everything I ever knew about Graphing Calculators from Judy. I remember takingseveral Summer Institutes with Judy at SUNY Old Westbury in the early 1990s. Even then, Judywas teaching with Graphing Calculators well before most teachers had ever heard of thistechnology. I have since extended the use of calculating environments to Computer AlgebraSystems such as SAGE [Supplemental Page 8] and Programming Languages such as Python.
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PAEMST 2012-2013 - Dimensions of Outstanding Teaching Written Response Page 12Name: Alvar Garcia Teacher ID: 32327Dimension 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 throughactivities 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 yearsincluding such topics as Word Processors, Spreadsheets, Data Bases and Programming inMathematics 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 oftechnology in the pre-Calculus and Calculus curricula. Some of the topics I’ve presented are:TI83 BASIC Programming, TI89 BASIC Programming, Solving Differential EquationsGraphically with Slope Fields, Solving Differential Equations Numerically with Euler’s Method,“Look Ma, No Calculator,” SAGE Advice: pre-Calculus, SAGE Advice: Calculus. I have alsomentored 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, ornational level.I speak regularly at LIMACON organized by SUNY Old Westbury and at T^3 hosted byMolloy 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 willbe 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 DiscussionGroups sponsored by The College Board. I have also learned a great deal from my participationin SAGE-EDU sponsored by SageMath.org and EDU-SIG sponsored by Python.org. Theseforums are a great resource for me and constitute my Professional Learning Network.
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