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My Pedagogic Creed
Modeled after John Dewey’s Pedagogic Creed is a statement of my philosophy of education:
ARTICLE I--What Education Is
I believe that education is a birth-given right to all individuals under the founding principles of the United States of America. This requires equal opportunity for all, regardless of any external factors. In terms of a formal education, a free education should be available to all individuals throughout secondary schooling. In some ways, education also occurs naturally as humans are always learning innately. As a whole, education extends far beyond the traditional K-12 classroom hours, academic curriculum, and assessments. Thus, the school system only partially encompasses the learning process. In this way, schooling becomes a subsect of one’s education. As John Dewey outlines, education begins by birth as beings are constantly influenced, stimulated, and challenged by the world they grow up in. The situations one is faced with, the people one is surrounded by, and the environment one lives in all serve to teach the individual in the same way that content learning takes place in the classroom. Yet, this is not to say that the individual is shaped completely by their educational experiences.
I believe that not everything a person becomes has been learned or indoctrinated in them in some way. Dewey describes that "the child has his own instincts and tendencies, but we do not know what these mean until we can translate them into their social equivalents." Thus, I would say that the development of the individual is by part nature and by part nurture. Clearly, true education spans every knowledge and skill an individual accrues. As a result, a proper education is one that balances and blends the academics of the classroom with the life skills of the outside world. Education serves as a process of one's present life, whereas teaching serves to prepare for the future.
ARTICLE II--What the School Is
I believe that the school is a safe space for students to grow academically, socially, athletically, and individually. This environment demands a relationship founded on mutual respect for both the educator and the student. Misbehavior in the form of physical or verbal abuse to any other individual in the school should not be tolerated. While education is a free right in this country, the access to such is still a privilege on the global scale. Students should be encouraged and made to enjoy the school experience. If not, it is the utmost poor reflection on the administration.
I believe that the school serves to foster learning both inside and outside the classroom. While curriculum standards are a must, the learning process should not treat students like objects to feed knowledge or test-scores on a piece of paper. Thus, the American education system should have some measure of progress and feedback when it comes to learning, but first needs to de-emphasize standardized testing and the pressures of such. Today, this has transformed from learners 'demonstrating understanding' to students learning how to be test-takers. Tests certainly should not be the central goal of any classroom, and far too often now students are not learning curriculum, but instead how to take a test. In my master’s research, I have found that the New York State switch to Common Core curriculum has significantly hurt students with learning disabilities and English language learners. Increasing exam difficulty serves no purpose other than to discourage students and create test anxiety. Students should be made to feel capable in the classroom, and it is the job of the teacher and the school to support such.
I believe that educational standards need to be more unified across the country. As content and learning standards are currently left up to the decision of each state, this does not seem to encompass equal opportunity in education. In fact, while a few states have adopted the Common Core standards, New York is one of the only states to actually implement them. The Ravitch text states that "the law they wrote said that every state should write its own standards, pick its own tests, and be accountable for achievement." There is no doubt that states vary in terms of resources and demographics, which ultimately reflects in the quality of high school education. With accountability at such a high value, the school needs to become a more centralized body.
ARTICLE III--The Subject-Matter of Education
I believe that when it comes to formal schooling, the subject-matter of education should be well-rounded and neutral in stance. That is, curriculum must encompass a wide variety of material, not purely the basic academic subjects. The role of high school need not only be to prepare for college readiness, as this is not the only successful path in life. Thus, students should be provided with a diverse education that allows them to pursue a multitude of careers or trades. All these avenues should be valued equally, as teaching is meant to prepare students for the future, which may or may not include college. It is evident that not every student learns the same, and that every student has different talents and passions. As a result, the school should teach students Home Economics, Finances, Auto-Mechanics, Cosmetology, etc., in addition to the traditional STEM, Humanities, Art, Music, Health, and Foreign Language.
I believe that teaching subject-matter extends far beyond delivering factual information to students. What students should be learning and who gets to make that decision is clearly a controversial topic, but at the forefront the issue creates an overemphasis on the accruement of academic knowledge in the school system. Again, it is quite naïve to define teaching by simply conveying information. Chomsky on Miseducation exposes a key issue in this topic, that in many ways teachers are restricted to impose "an official truth predetermined by a small group of people..." that leaves students treated simply as beings needed to be filled up with this knowledge. In my personal experience, I always think of the mathematics teachers I had growing up. The ones that took the class 'off the page' and demonstrated that they truly cared were the ones that always left students feeling encouraged by the fact that they enjoyed fostering learning. This is how subject matter should be conveyed, without rigid content pressures and an overload of assessments. In college, my experience has granted me courses where the professor has tailored the material to the students. Whether it be incorporating topics that I find interesting or cutting out a lesson that appears too challenging, academic interest is peaked when learning is not restricted to specific standards and ideas. This flexibility should define the education system at all levels.
I believe, however, that is crucial for primary and secondary education to have guidelines that serve to unify the academic experience for students at this level. In this case, some degree of uniformity is a must to ensure an even playing field for students with content exposure. Thus, I think it is quite important that New York State has content and learning standards for all of its regents courses, and that creating the common core curriculum was meant to shift students away from memorization and towards demonstrating understanding. However, simply creating standards is less than half of the battle-- without implementation, standards mean nothing.
ARTICLE IV--The Nature of Method
I believe that the act of teaching is one that serves to support and prepare students for the future. At the base, this requires the inclusion and instruction of all students. Differentiation then becomes the most crucial part of the education system. As is always reiterated, not all students learn the same, and it is the responsibility of the teacher to teach all students. Thus, the goal of the teacher should be to make every student feel capable and supported in their academic journey. No matter the means, no matter how much planning and learning strategies it takes, every student should be learning in the classroom. If a student walks away from a class saying things such as “I just am not math-brained” or “I am not good at writing,” this is a reflection that the teacher has not done their job. The Kirby paper states that "when questioned about inclusion, some teachers saw it as a privilege for students with disabilities to be included with their peers in the general education classroom," which is absolutely absurd. All kids living in the United States have the right to public education, and thus equal educational opportunity for all, so this seems to be the utmost violation of the foundations of teaching.
I believe that true learning is sparked from emphasizing the "why" along with the "what" and understanding that this is not fed by a simple "right" or "wrong." In Palmer’s The Courage to Teach, it reads that "teaching always takes place at the crossroads of the personal and the public, and if I want to teach well, I must learn to stand where these opposites intersect.” Thus, a proper education is one that does not take on bias or opinion but provides students with a well-rounded view of topics and solutions. Again, students are not to be indoctrinated, but rather given the tools to think freely. Eisner says, "the first thing a student learns is to provide the teacher with what the teacher expects." This extends further, as I would say that many teachers learn to simply provide students with what the state expects. An utmost example of this occurred for me in high school, when the common core was first introduced in a manner that forced my school year to end up with two regents examinations every year in mathematics. As if double the standardized testing was not enough, the teachers ended up teaching to the common core test during the school year, and then cramming the traditional curriculum in the week leading up to the regular regents. Once teachers learn to move past this curriculum crunch, students can actually begin learning.
ARTICLE V-The School and Social Progress
I believe that diversity in the classroom is a crucial asset to one's education. Growing up near the city and in the public school system, I had the privilege of receiving a diverse high school education. Although this may not be attainable in all areas, students should still be exposed to the cultural, financial, etc. diversity the world has to offer as much as possible. This, to me, ensures that education is not simply indoctrination and that students are not ignorant. Rather, students should be exposed to diversity and allowed to formulate their own thoughts and opinions.
I believe, as teachers, that it is our duty to advocate for all of our students. It is clear that this starts with educating ourselves, as I was not previously aware of how lacking and flawed special education policy is. The Greene article writes that "in 1970, before the enactment of the federal protections in IDEA, schools in America educated only one in five students with disabilities. More than 1 million students were excluded from public schools and another 3.5 million did not receive appropriate services." Thus, the education system needs to put inclusion at the forefront of its reform. There is no right or wrong way to learn, so all students must not be measured for success in the same manner. At the very start, the school needs to ensure literacy above all else. Without it, content cannot come into play. All students must be taught to effectively read, write, think, and speak in some form. A true educator understands that this is not a uniform process, but an individually unique expression amongst students.
I believe that the educator should strive to be a role model for students. When we think of who we want our children to learn from, there is no surprise that educators should have good character and morals. As kids are growing up, arguably in the most influential time of their lives, an effective teacher is one that instills their students with both academic knowledge and good values. All around, the students are the responsibility of the teacher. Students are to be protected, kept healthy, and academically fulfilled. The teacher, the administration, and the school should make every effort to create a safe, encouraging environment for individuals to grow in all aspects.
My mathematics research as a part of Iona College’s Scholars Day.
More info: https://www.iona.edu/students/student-experience-and-activities/campus-events/annual-campus-events/iona-scholars-day-3-2
A Mini-lesson Model
What a great example of how to teach content off of a student's background knowledge. Brief, to the point, and well-grasped!
Framework for Teaching
Here's something new I learned in my literacy and learning course. Meeting the guidelines of domain three helps create the perfect mini-lesson, and employ learning strategies to extend to a full lesson. The biggest take away? Student engagement!
Algebra 1 - Linear Equations
Check out these word problems that I worked on with my Algebra 1 student after we learned how to solve a system of linear equations by graphing, substitution, and elimination! My particular favorite is the questions about animal legs 🤣
Dysgraphia: Within the Geometry Classroom
By: Frances Hoey
In the mathematics classroom, the importance of writing may not appear as clear as it would in subjects like english or social studies. When students reach high school, they are introduced to their first mathematics course that is not purely computation based. Much of the geometry curriculum involves writing proofs, which centers students’ grades on writing skills rather than analytical abilities. As a result, this can create challenges for students with writing struggles such as dysgraphia. These students have trouble with transcription, from forming letters to holding a writing utensil. Thus, it is the responsibility of the educator to identify and implement the proper strategies in the geometry classroom to ensure success for their students with dysgraphia.
When it comes to the signs of dysgraphia, it is important to understand that this disorder is purely based on struggles with transcription. These students do not have challenges with written expression, but having to place so much focus on handwriting, typing, and spelling causes written performance to suffer. With some extra help, students with dysgraphia are fully capable of comprehending and implementing geometry curriculum. This may even already be apparent by a comparison between students’ written and oral skills.
The challenges of dysgraphia impact students’ academic performance in a variety of ways. As this is a transcription disorder, the most prevalent struggles involve handwriting functions. In fact, “dysgraphia literally translates to difficult (dys-, English) writing (-graphia, Greek). Viewed largely as a handwriting impairment, dysgraphia also affects a person’s ability to draw lines and shapes” (Kelly & Kelly, 2019). Thus, students face barriers with note-taking that are often blindly met with cumbersome accommodations. A research study based on an engineering student with dysgraphia outlines his learning experiences without the proper strategies for success. He did not take the SAT as “the accommodations he would need would have made it a three-day endeavor. He cringed as he recalled the embarrassment of having someone read him the driver’s permit test because there wasn’t a computer reader available… In middle school, he was two math levels ahead of grade level, but he was in remedial reading classes” (Kelly & Kelly, 2019). Thus, it is evident that a lack of accommodations creates major setbacks and unnecessary challenges for students with dysgraphia.
These students face additional STEM-specific challenges when learning with dysgraphia. In the geometry classroom, signs of dysgraphia appear as difficulty in drawing shapes and completing geometric constructions such as sketching triangles, parallelograms, etc. A large component of the geometry curriculum is using the compass, protractor, and straightedge to construct bisected angles and shapes. As students with dysgraphia already struggle with holding their writing tools, the addition of a construction tool will only create more of a challenge. Another component of the geometry class is writing proofs on the congruence or similarity of shapes. The difficulty and time-consuming nature of writing in a straight line for students with dysgraphia creates a serious impediment to the proof-writing process.
As a result, there are many effective strategies to implement in the geometry classroom to ensure the success of students with dysgraphia. In order to improve the fine-motor skills that writing relies on, students with dysgraphia may benefit from occupational therapy. Therapists will work with these students on motion and posture, but there are also effective tools for inside the school system. For example, students can practice “picturing the letter they are writing and making the shape with the hand in the air. This can also be done by practicing letters and numbers with hand or finger motions” (Deh, 2019). Technology also plays a crucial role, as it can account for many of the struggles associated with dysgraphia. Since students struggle with handwriting, tablets can be implemented in the classroom as there exists a plethora of note-taking apps and dictation software. As a whole, typing suits the needs of dysgraphia students as it eliminates the struggle of forming letters and shapes, while also allowing for easy erasing. This creates an alternate style of learning, rather than a challenged one. As a bonus, there little difference between these digital dysgraphia tools and that of any other student in a technology-heavy society. It is the school’s responsibility to serve these students, as “the school system should assess and provide the necessary supports for the student’s needs in the educational setting” (Chung et al., 2020). With the resources available through an IEP or 504 plan, dysgraphia students must have access to writing accommodations, from scribes to assistive technology. All of these tools serve to alleviate the challenges and stresses that arise from writing-based classroom.
Yet, when writing becomes a necessity in the mathematics classroom, there remain effective strategies to aid students with dysgraphia. “Specific devices may be utilized, such as larger pencils with special grips and paper with raised lines to provide tactile feedback. Extra time can be permitted for homework, class assignments, and quizzes/tests. Depending on the student’s comfort level, alternative ways of demonstrating knowledge (e.g., oral or recorded responses) can be considered” (Chung et al., 2020). Educators can implement fill-in the blank notes and graphic organizers that will serve to benefit all students, as less focus is placed on blind note-taking and more time is allotted for comprehension.
Overall, while students with dysgraphia may struggle with the writing skills in the geometry classroom, traditional pencil and paper is clearly not the only path for success. With the proper accommodations, from technology to assistive writing tools, students with dysgraphia are given the foundations to achieve and succeed with confidence in the geometry classroom!
Works Cited
Chung, P. J., Patel, D. R., & Nizami, I. (2020). Disorder of written expression and dysgraphia: definition, diagnosis, and management. Translational Pediatrics, 9 (Suppl 1), S46.
Deh, T. H. (2019). THINKING IN PICTURES: A WAY FORWARD FOR DYSLEXIA, DYSGRAPHIA AND DYSCALCULIA IN GHANA. European Journal of Public Health Studies.
Kelly, D. P., & Kelly, D. L. (2019). Drawing the Line: The Challenges of Dysgraphia in Introductory Graphics Communication Courses. Journal of Technology Studies, 45(2), 60-66.
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Abstract Algebra Presentation on Geometric Constructions: Looking at constructible numbers, roots of polynomials, and fields. This information can be extended to complete impossibility proofs.
My presentation on geometric constructions for my abstract algebra class! Using fields, I’ll show you why you can construct certain numbers, why you can’t construct certain numbers, how this relates to roots of polynomials.
Theory of Numbers Project on Analyzing Polynomials in Modular Arithmetic
A switch-up from the typical education post, to explore the math side of things, here is an interesting presentation by my fellow math education classmate Willow and I on Analyzing Polynomials in Modular Arithmetic!
Spotlight on writing scaffold
As seen in the previous post, writing is a big component of any proof-based math classroom. When it comes to preparing students to become effective writers, we have seen that first and foremost it is the teacher’s responsibility to model concise writing to students. We also know that students vary in writing abilities when it comes to the classroom. Some may be very fluent writers, while others may struggle to make connections. Some students may also require differentiated instruction for various learning needs.
No matter the situation, joint construction is a strategy aimed to use teacher modeling to create a classroom collaboration with students that practices effective writing skills. Students will engage in academic discussion as the teacher scribes their input. The step-by-step process encourages analytical thinking and applies speech to text.
For example, in the geometry class room, I may ask students to prove that two triangles are congruent. By drawing the diagram on the board, students have one collaborative template for their answers. I will ask students to walk me through their solution process, and ask critical thinking questions to ensure their understanding. I may start with “what is the first thing I should write down?” and continue with “what should I do next?” as students work to problem-solve. I will write down the “givens” and “reasons” for each step of the proof in a chart for students to model, making sure to use complete sentences each time. All in all, we work together to create a clear, concise geometrical proof that serves to improve both students’ understanding of writing and content.
Writing in the Math Classroom
With subjects like english and history, the function of writing in the classroom is rather obvious. In the math classroom, it is not quite as obvious. Yet, writing serves a huge purpose in mathematics. This is especially true for the geometry classroom, where much of the curriculum involves writing proofs. Unlike other math courses, this one centers a student’s grade around writing skills more heavily than just analytical abilities.
Here is the rubric for a free response question on the 2015 NYS Geometry Common Core Regents. A perfect score of [4] is achieved by a complete and correctly written proof!
Likewise, here is the sample response for that same question, displaying what the criteria “a complete and correct proof is written that includes a concluding statement...” looks like.
I believe high school geometry is crucial for establishing a foundation of mathematics writing skills for students. Students that will go on to pursue mathematics beyond high school will need such a background to eventually be writing intense proofs and research papers!
Here’s an example to compare with the proof above. The following is a pretty basic proof that I wrote for a 400-level college math course:
As Pauline Gibbons mentions, journal writing and reflection (“what have you learned?”) is a big tool when it comes to effective writing. She observes that “the teacher-guided reporting sessions influenced the way the students wrote: their writing reflected wordings that they had used in interaction with the teacher or that had been part of the teacher’s recasting” (Gibbons, 91).
Thus, it is important for the teacher to model proper academic language and formality through both writing and conversation. While students must be able to write in their own words, they need to know how to properly format their proofs, and what proper language is needed to convey specific ideas.
Also, it’s worth noting that while high school is mostly traditional hand-writing of proofs, the college-level shifts emphasis to digital typing and text for formality. Thus, I think that it is important to prepare students early by introducing them to typed proofs in the geometry classroom.
Overall, it is evident that writing skills are a MUST for any geometry classroom!
Works Cited
Gibbons, P. (2015). Scaffolding Language, Scaffolding Learning: Teaching English language learners in the mainstream classroom. Portsmouth, NH: Heinemann.
Introducing New Vocabulary
When it comes to language, reading, and writing, none of these tools would be useful without a range of academic vocabulary to accompany them. As students climb the ranks of schooling, the task of comprehending and implementing new vocabulary becomes increasingly demanding as academic difficulty increases.
Thus, it is the educator’s responsibility to understand the what, why, when, how's to introducing students to new academic vocabulary. For the most part, repetition is key. The more students interact with new words, the more comfortable they will become with applying them in the correct settings.
Rather than just handing students a word box, teachers must get students actively engaged with new words in a variety of contexts. As a result, there are multiple instructional activities to consider:
Vocabulary Notebooks
Perhaps the most straightforward, students can create vocabulary notebooks with word logs, definitions, and examples. In a geometry classroom a student might log the word supplementary with the example “two angles are called supplementary when their measures add up to 180 degrees.”
Examples/non-examples
This strategy involves giving students a correct and incorrect example of a word and having them determine which one is the proper usage. E.g. Congruent
1. lines AB and CD are in a ratio of 1:3 (incorrect)
2. lines AB and CD have the same length (correct)
Altered Definitions
Likewise, this strategy serves to introduce students to various contexts of a word. E.g. Perpendicular
1. two lines intersecting to form a 90deg angle
2. the right angle of a 30-60-90 triangle
3. the sides of a square
Quiz Games
One version of this may have students competing against a timer to answer true/false questions about the definition of a new vocabulary word. E.g.
1. Parallel lines never intersect (true)
2. Similar triangles cannot also be congruent (false)
Mix and Match
This strategy, perhaps the most common, provides students with a column of new vocabulary words next to a column of definitions. It is the students’ job to match the correct word to the correct definition.
It is not just enough, however, to provide students with these materials. Teachers are equally responsible for ensuring that students retain their new vocabulary. Therefore, it is important to get learners involved in follow-up activities to reinforce the knowledge they have accrued. By consistently placing emphasis on vocabulary in the classroom, teachers are preparing students to achieve all-around academic success.
Works Cited
Beck, I. L., McKeown, M. G., & Kucan, L. (2013). Bringing words to life: Robust vocabulary instruction. Guilford Press.
Developing Reading Skills
Creating bridges to reading for ELLs in the secondary classroom
When it comes to mathematics, reading is an essential skill that may not appear quite as obvious as writing and language. What comes to mind may be english and history; the likes of reading textbooks and stories. Although it may look different, math challenges reading abilities in just the same ways.
We have seen that from word problems to complex vocabulary, reading skills are a must for mathematical success. As I discussed last week,
a sizable amount of the high school geometry curriculum is based on proofs. As this is something students have never encountered before, they must understand what it means to prove something before they are able to apply it. Along with that, they have to accrue a whole new set of vocabulary that corresponds to concepts needed in order to create effective proofs.
This can be tough for students of all language levels, so for ELLs, this provides an extra challenge. Here are some of the geometry vocabulary terms, with students being introduced to majority for the first time:
Before students can effectively solve problems, they must be able to remember the meaning of some of these more complex words. Rather than just having students read the text and focus on memorization, there are much more successful tools to encourage the development reading skills in the classroom. While some range from as basic as having students summarize and question the text, others provide more of a creative, visual approach to reading and vocabulary.
One that particularly interests me is the concept of Key-ring Words. This idea is provided by Pauline Gibbons as a strategy to help students remember vocabulary through physical representation. Since math is a highly visual subject, this is a great tool to explore. Here is how this strategy may be implemented into the geometry classroom:
Students will be given index cards
From the list of new vocabulary words above, students will write each word at the top of a card
They will then draw a visual representation of the word that helps them to remember and understand its meaning
Once all cards are complete, they will get hole-punched and placed in a key-ring for each student
For example, here is a key-ring that I made from some of the geometry terms. For the term supplementary, I drew a vector intersecting with a straight line. I then labeled the two resulting angles and wrote the fact of A+B=180°.
This activity provides ELLs an opportunity to process new words in a way different than just language. By taking words “off the page” in this manner, it furthers student understanding in a creative way, while also providing feedback for the teacher on where students may be struggling.
Instead of wrapping this post up, I will leave it open-ended as I have a teaching strategy simulation video of implementing key-ring words in the classroom up and coming- stay tuned!
Works Cited
Gibbons, P. (2015). Scaffolding Language, Scaffolding Learning: Teaching English language learners in the mainstream classroom. Portsmouth, NH: Heinemann.
Ahh! 👻😂😂😂
Reading Skills for Reluctant Readers
When it comes to reading, mathematics is likely not the first subject to come to mind. However, reading and language play an integral role in learning and understanding all types of math. From word problems to math vocabulary, reading skills are a must for success.
Yet, many high school students struggle with reading and comprehension for a variety of reasons:
“The two types of struggling readers most often encountered in secondary ary schools are resistive readers and word callers. Resistive readers can read but choose not to. Word callers can decode the words but don't understand stand or remember what they've read” (Tovani).
Just as many result to what Cris Tovani terms as ‘fake reading’ to get by in the classroom. In terms of a high school geometry classroom, these students may:
- skip straight to reading the textbook solution
- look at the word problem without paying attention
- read without understanding what the question is asking
- get distracted and forget the information in the problem
- avoid reading the problem all together
It is worth noting that, for the teacher, this is not something that is easy to spot as students with their heads buried in the textbook or worksheet appear to be actively engaged in the material.
As a result, it is up to educators to provide their students with the exercises necessary to produce good reading skills. Thus, there are comprehension strategies to consider in helping students to become effective readers. Here are three that I believe are particularly valuable to the math classroom:
The following question is taken from Pearson Common Core Geometry.
- Use Background Knowledge
A = S x S = S^2
Square can be broken into two right triangles
Right triangles can use the Pythagorean theorem
a^2+b^2 = c^2
Creates a 45-45-90 triangle
Rules for special right triangles
- Asking Questions
What do I already know about the area of a square?
Have I used all important givens of the question in my answer?
- Determine What’s Important
I don’t need to know that it is a garden, yet the shape is crucial
I should pay attention to units
Question is asking for length
I need to somehow find the diagonal
Most importantly, teachers should model this behavior for students. By being passionate about the subject material, it will in turn pull students’ focus and motivation to read. By using real-life applicable questions such as the example above, it gives students a purpose to read and makes class more interesting. Teaching by example allows students to see these strategies in action.
Giving students plenty of opportunities to read is the best way to improve their skills, but rather than just assignments and quizzes, teachers should have students reflect and discuss what they read. Analysis is a key component to any math curriculum, both verbal and written. This can help further student understanding as well as allow the teacher to check in and provide instant feedback on such.
Overall, it is necessary to redefine reading and reflection in the classroom, rather than eliminating it all together. By preparing students with reading comprehension strategies, it opens the door for academic understanding, confidence, and success in even the toughest of math courses.
Works Cited
Tovani, Cris. I Read It, but I Don't Get It: Comprehension Strategies for Adolescent Readers. Kindle Edition.