Mathematical Sciences, Secondary Education
MY STUDENTS
Inclusion Geometry
Math IV / QCC MAT-099
ELL Algebra I
Inclusion Geometry
The students at Worcester Tech alternate between "academic" and "shop" weeks. As a result, these students take a single class period of mathematics each week during their former two years
(although the specific period of the day may change per a weekly basis), yet they only take a double class period of mathematics biweekly during their latter two years.
As such, the individual class periods at WTHS are labeled as 1–8 during an "A-week" sequence
(academic week for some, shop week for others), and as 9–16 during a "Z-week" sequence (where academic and shop weeks switch for each student, based on their specific shop, grade level, and personal schedule).
SOPHOMORE
5–9
My fifth-ninth period sophomores were the very first class that I took over during Week 3, following full-time observation of Mr. Fitzpatrick's teaching practices and lesson lay-outs. As such, my mentor teacher had already had a chance to establish a repertoire with the students and had covered his specific classroom expectations with these students just a few weeks prior, making it rather odd to jump in and take over the classroom with little credibility, in the eyes of the students, on my end (other than from the one time on the third day of school when I had substitute taught for a morning using my mentor teacher's lesson plans, rather inorganically).
​
These students were the first Inclusion students that I had particularly encountered as well, as I had always grown up in rather advanced classrooms and had never seemed to cross paths with students on Special Ed or 504 plans. Additionally, I never actually received a list of all of the students who were on special plans in my class, nor a specific idea of their accommodations. Although this made me rather nervous going into an environment where I could be held personally accountable for students' learning had their parents decided to complain, I adapted as best as I could to the various needs and adjustments that my students presented to me.
​
I had one student, for example, whom I'll call A, who presented particular challenges, even as I got to know her as both a student and a person. A gets easily distracted when working on individual work, and it is difficult for her to complete more than a single problem or two within a timeframe that is allotted for an entire worksheet or quiz. A had a lot of anxiety in the beginning of the year when it came to asking questions during class, as I suppose that she expected judgement from myself and from her peers. However, I observed A at the Worcester Tech Fall Formal, where I chaperoned, and I noticed that she is actually quite the social butterfly, with many friends and connections. This is when I realized exactly how important peer influence and reverence is to her. From that point,
I became extra conscious of my efforts to prevent singling students out, respecting the beauty of error and misunderstanding in enhancing learning that much more. I began to catch myself much sooner when responding to students' inquiries that may have been frustrating given the content that we had just covered (such as an incident where we had just finished covering right triangles, and when I drew two right triangles on the board as mirror images of one another, A herself pointed to the one with the right angle on the left side of the triangle and asked which theorem to use if we encountered a "Left Triangle." I am rather proud of how patiently I handled her question and used it as an opportunity to point out some facts about congruence and orientation, preventing the class from laughing at A or making her feel embarrassed about her question, especially since she had asked it while trusting that I would respond honestly and carefully). I made sure to keep A's peers occupied as she worked in class and to talk to her frequently about the lesson to keep her mind on track. This realization also aided me in owning up to my own mistakes, such that when a student pointed out an acute angle that I had labelled as 110º, or a polygon that I had lettered incorrectly,
I was nothing but proud of my students for their reasoning in picking out my mistakes.
SOPHOMORE
1–13
My first-thirteenth sophomores were the second class that I picked up, and I did so about two weeks after tackling my first full-takeover. I was expected to teach both of my Inclusion Geometry classes the same topics and lessons each day, and the three most difficult aspects of this were the ever-distinct class sizes, especially as some students switched to honors classes (SO.5–9 only had 12 students by the end of the semester, as opposed to SO.1–13's 18 kids!), the different dynamics of each class period (SO.5–9 was a much rowdier period at the beginning of the year and slowly dwindled into falling extremely quiet, whereas SO.1–13 remained widely participatory and even had a larger range of students asking questions by the end of my practicum), and the flipping periods each week (my two geometry classes were first and fifth each day, with the specific class demographic switching periods each week for their shops). Thus, the earlier periods in the day were always quieter and rarely had energy (although when they felt compelled, it was rather fun to teach at 7am), and it definitely did not help when I had an observation of the other geometry period early in the morning when this class period, both being larger and in the afternoon on those days, consistently ended up receiving better, altered, more responsive (and effective) lessons.
​
Nonetheless, the students in this class challenged my views towards pacing and expectations behind learning. They also opened my eyes to the different ways in which students can show respect (as with one student, D, whom I thought greatly disregarded me since she was rather guarded, before she became sad that I was leaving and starting asking me a lot more questions, even calling me "amazing"). I had a student, H, who is a brilliant mathematician and who loves a challenge. As the weeks progressed, he started greeting me each day with a smile and updating me on how he felt about yesterday's lesson, even commenting on something that I was wearing or on some detail on the board that I thought negligible. He became a milemarker for me when I evaluated how the class was grasping material, and when H did not get something, I had to re-evaluate a bit. What surprised me about H, however, was that I could not always expect him to grasp everything immediately. Although sharp, he was still on a SpEd plan, and he occasionally had moments where he completely misconstrued a question or needed an alternate explanation. Through H, I learned to focus strongly on presenting multiple examples and strategies for each problem or lesson, such that I could reach every student equitably (especially with my ELLs, too).
​
I had another student, F, who was not only the "newcomer" in the class (as in she had not had Mr. Fitzpatrick for Algebra I the prior year), but she had a very specific accommodation (she was legally blind and needed for me to use only blue or black marker on the board) and was very bright once she understood a topic. However, it often took much explaining and practice for her to grasp material, and I could never predict when that lightbulb moment would hit. This taught me patience and made me feel helpful whenever she asked for Chemistry or life advice (she is passionate about BioTech) or simply lit up when she realized that she knew an answer all along. I also knew that my Problem of the Day strategies worked when she completed them, as she often rebelled against this strategy and did not complete problems that she knew that I would cover on the board shortly.
FRESHMAN
2–10
My freshmen were the final class that I took over for most of my practicum, although I did take over all 5 of my mentor teacher's class periods eventually for the final few weeks. These students were my English Language Learners (ELLs), ranging from Levels 1–6 in English Proficiency. Most of these students were from Spanish-dialect-speaking countries, although we had representation from about 13 countries in the 30-student room alone. As this was a large class size for Worcester Tech, when I first observed and aided individual instruction before fully taking over, I especially spent time with my Levels 1 and 2 students in the back of the room, who struggled with the content due to not only a language barrier, but from a content gap. I realized that content gaps are a genuine issue with capable ELLs, especially in a diverse city such as Worcester. One student, E, demonstrated this with her pure character and drive to learn, even though she very often required arithmetic reinforcement, such as with times tables and negatives, even in the ninth grade. She had such courage and focus during each lesson, and she would not rest until she understood a topic, learning barrier and all.
Many of my ELL students, in fact, often had issues in class because they either needed extra help or reiteration with different vocabulary (due to the language barrier -- our Teaching Assistant, Mrs. McKnight, was extraordinarily helpful for individual students in this sense, as were Miss Bigelow and Mrs. Bonofilio in SO.5–9 and SO.1–13, respectively, since Inclusion and ELL students think and process material in very similar ways from their constant barriers), or because they were simply ahead of the pack from outside preparatory instruction upon entering the USA. The disparity was rather large in this sense, and students' varying confidence levels made it rather difficult to isolate those who needed extra help or who just needed more practice, which made parent input all the more useful. During our Parent-Teacher night during Week 6, I got to interact with the mother of one of my students, C, who is a generally soft-spoken student who sits in the corner of the room and who generally goes (I must admit) unnoticed during a regular lesson. His mother was so passionate about his education, however, and was such a champion for his character and intelligence that I felt somewhat guilty for not interacting with C more often! I learned that C even had a twin brother who was at a different mathematical level from him, and I began to see the brothers all over the school and also at the dance that I chaperoned. It gave me a clearer sense of the lives that I was impacting with my work and of the value of individual instruction; I performed a type of experiment during the following weeks and particularly attempted to gather a sense of how C was grasping the material as we progressed. There were times where I glanced at his work and he totally got it, and there were rare occasions where I checked in with him specifically and he asked for help. I feel like I needed that interaction with his mother to open my eyes to how well students can hide difficulty, or to how students may either feel excluded based upon their seating location, or perhaps even enjoy the isolation or lack of obligation for class participation. This may, however, affect the quality of education for that student.
Upon meeting with my mentor teacher following the parent night, I mentioned C and suggested that we rearrange seats more often or make an effort to check in with students who may seem the most assured or even neglected, due to the mere nature of teaching a large class of ELLs (especially for me, as a new teacher).
I believe that seating arrangement has a large impact on how students process material or feel connected with their class, and whether they enjoy the isolation of sitting in a back corner or not, they definitely will get more overall attention from the teacher and reassurance in their learning if they experience sitting closer to the action.
SENIOR
6–7, 14–15
This class was designated as a "cover-all" class for seniors who did not wish to retake a prior math course or to step into an Advanced Placement class. It was called "Math IV" by WPS standards, however, Mr. Fitzpatrick not only ensured that it covered all content that would normally follow MAT-099 standards at Quinsigamond Community College, where he used to instruct intermittently, but he encouraged students to take the MAT-099 final exam at the end of our course so that they could sign up for bonafide college credit, free of charge, after simply passing our class! Of course, it was up to the students to seize this opportunity if they so chose.
​
Even though I did not officially take over these classes until Week 13 of my practicum, I did have the privilege of helping out throughout the prior twelve weeks each day during the "work period" of the classes. Each senior block contained a double-period to make up for shop weeks, so each class was generally split up into a single "teaching" period followed by a single "work" period. We used the teaching period to go through the lessons that we outlined in the packets that we gave to students each week (for organizational purposes and to aid those who had more background in the topic prior to our instruction and who wished to work ahead), and we allowed them to spend the work period completing exercises, group-work sheets, and book problems, all the while answering individual questions and clarifying common issues on the board as they arose. We occasionally switched this format up to allow for a larger group activity during the earlier period, where groups copied their work onto the board and presented it to their peers for solution review, or by assigning online modules on EdCite or as "shop week" work (due as homework).
​
The seniors were an interesting bunch. Throughout even the first half of the semester, almost half of each class period switched out of the class and was replenished with a new group of students. Some of the students whom I had somewhat bonded with by that point were replaced by students who had no idea who I was to be wandering around their classroom and offering math help and why I looked so young, even during Week 8! However, I strangely formed an even closer bond with some of these students than I had with the previous bunch. One student, Y, I happened to see at an outing for my sorority somewhat early in the practicum, and she got a bit angry that I could not recall her name! She eventually warmed up to me, though, after seeing me in a social setting, especially at one as laid-back as the roller-rink that she worked at. I also enjoyed helping out two students, K and P, who sat right next to each other at the back of the room during Y's same period, 6–7. Both students were far too bright to be in Math IV, in my opinion, but they always respected me, despite my baby face, and very often asked questions. K liked to ask questions about real life (outside of high school) and how my education has applied every day; P and I bonded over the sport of bowling, which we have both loved since a very young age, and I felt so proud the day that I learned that he had bowled a 300 game (a perfect score) and had still finished his homework! My 14–15 class also had two students, G and S, who were rather apprehensive toward me to begin with, yet who were very bright, so as I demonstrated my expertise to them and encouraged questions no matter how small or "dumb," they began to take me more seriously.
During Week 10 of my practicum, I administered a paper student feedback survey to my SO.5–9, SO.1–13, and FR.2–10 classes to gain some perspective on my performance thus far and on things upon which I could improve within my remaining six weeks. I recorded each response in a spreadsheet, weighing "Strongly Agree" with a 4, "Strongly Disagree" with a 1, and so on, and taking a weighted average in the final column to get a quick overview of my perceived performance in each area, per the students' perspectives. I had only had my FR.2–10 students full-time for two weeks at this point, so it was particularly interesting to see my freshmen's viewpoints on how I handled their class as the teacher of record and not just as an assistant.
​
Some of the responses contained comments on how I could improve or continue my teaching. From these, I gathered that the students appreciated my help and guidance and that I brought some new ideas and passion for learning into the classroom. However, a few responses told me that I needed to work on my strategies for re-explaining specific topics or for answering individual questions, as I tended to repeat my original lecture thinking that it contained all of the information that I needed to convey, and I did not branch out as much in the ways that I could get a message across. Due to this, I made sure to incorporate more diverse and culturally responsive examples into my lessons and extra help (such as equating a proof example to an every-day application or using physical objects to demonstrate congruence), increasing the number of channels through which students could receive information and process it in their own way. In fact, I actually observed an uptick in student questions and participation, specifically in class, when I adopted this perspective and shifted my teaching methods a bit.
SO. 5–9
My 5–9 Sophomores seemed to near-unanimously agree that I often used a positive tone and that I demonstrated mistakes as an integral part of learning, but they wished to see particular improvement in my tendencies to assess their understanding in-class to summarize what they had learned in a lesson. They also wanted for me to encourage them to work with their peers and to foster increased peer respect in class, which would help with exchanging ideas in the prior category. Although I did already have a habit of asking the students if they had understood something or if they were ready to move on (and thus I suspected that they may have misconstrued the question about "rating their understanding," Question #14, since it is worded in a rather strange manner), I did seek to improve in peer collaboration and evaluation. After all, peer opinion is a large factor in whether students will engage or participate confidently in class and feel comfortable with their personal pace of learning. I began to ask students to correct each other's work during the lesson, and I set them up with a group project in their Proofs unit in an attempt to change things up from my usual "elbow buddy" pairings, which apparently did not successfully allow for peer feedback or reassurance.
SO. 1–13
My 1–13 Sophomores gave very similar responses to their fellow Inclusion Geometry students in almost all of the same categories, even though this class is comparably more sizable. These students, however, provided me with more written, specific feedback, and so I was able to use these survey responses to round out my teaching methods for all of my classes, especially with the comments about repeating material too frequently.
