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Theory and research is essential to professional development. Our world, and thus, our students are constantly changing with the times. Because of this, our methods of teaching must also change to reflect this. Keeping current on research allows teachers to try and experiment with new methods. Even more, teachers can share their learning with other teachers, which again, benefits students. Understanding educational theory and how to implement it in the classroom can bridge the gap between theory and practice. One hallmark of a great teacher is a teacher who is still learning.

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I believe that my final exam from EDSE 550 demonstrates the theory and research component of the USC Conceptual Framework because I reflect on a lesson that I enacted and support my conclusions and reasoning with research. The lesson was a high school probability and statistics lesson. The resources I cited justify various teaching strategies such as questioning, group work, discussions, and asking students to make connections among mathematical concepts.

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I believe that my final exam from EDSE 770 demonstrates the theory and research component of the USC Conceptual Framework because I reflect on a lesson that I enacted and support my conclusions and reasoning with research. This particular lesson was another probability and statistics lesson. This lesson was made to demonstrate how to use technology effectively in the classroom. The resources are technology resources as well as resources from academic literature that justify the ways I used or could have used the resources more effectively.

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I was recently asked in a teaching interview what types of technology I have used in the classroom. Thankfully, I had this technology padlet I created as a resource. This resource started as just an assignment for a class, however, I have continued to update it throughout my internship, and will continue to update it throughout my career. I believe that this continuous updating of the resource indicates me taking initiative to stay current on the various ways that technology can be implemented in the classroom. This is a clear indication of the research component.

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Diversity is so critical in education and teaching. Students come to us with a myriad of languages, races, cultures, exceptionalities, and viewpoints. This has to be reflected in the methods of teaching. I think that recognizing and acknowledging diversity in the classroom can only effectively be done by taking strides to increase equity and remedy the implicit inequities in a system that is part of a larger context. This can be done primarily by promoting tolerance and encouraging diversity in opinions, perspectives, language, cultures, and exceptionalities within the classroom. This widens all students' horizons, especially those students that are very used to mainly seeing others who look like themselves in positions of power or in media. When those students who have power and privilege can appreciate diversity, I believe this sets up our future generations well.

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Diversity is such a broad term that cannot be applied only in terms of race, language, or culture. In education, I think recognizing diversity in types of learners is crucial. This artifact is a text set about functions. During a unit on functions, these resources would be provided to students, with the purpose of providing access to the material and the learning at various levels and for different types of learners. This text set includes various types of media including Tik-Toks, Youtube videos, TED Talks, and online articles. This is also attributed to the fact that modern definitions of literacy must include technology.

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In this paper, I critically reflect on how teachers can help remedy the inequity of the hidden curriculum. The hidden curriculum refers to the difference in the quality of education according to socio-economic status. I support the strategies I provided with my learning and observations from observing an AVID class at Blythewood High School. In this paper I outline three strategies for teachers to use: providing resources for overcoming obstacles, cultivating a deep understanding of the community, and incorporating inquiry and creative thinking. This supports the principle of equity that teachers should hold high expectations and strong support for all students. More specifically, high expectations and strong support is addressed when providing resources for students when they come to you with obstacles to learning and incorporating that inquiry and creative thinking discussed.

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EDSE 764 Lesson Plan w/ Historical Component

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EDSE 764 Lesson Plan w/ Historical Component

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This lesson incorporates a brief history lesson to satisfy the historical component. I believe that it can be very beneficial to make math human so that students understand that humans developed these theories and processes. I believe that when students see mathematics as something that is man-made vs. something that computers do, it could make it appear more accessible for students. This lesson highlights a white man, which adds nothing new to the diversity landscape, however I have made it a point since this lesson to amplify the voices and work of mathematicians from different cultures such as Liu Hui and Zu Gengzhi. This will be addressed more in the historical development & perspectives section.

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September 28 Geometry Lesson w: Historical Component

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According to the USC website, decision-making is defined as follows: "through which one leads educational progress by planning appropriate goals, environments, actions, and assessment strategies based on in-depth professional knowledge, collaboration with other stakeholders, and the systematic collection and analysis of appropriate data (e.g., characteristics, needs, and abilities of learners; educational, environmental, and community contexts; perspectives of multiple stakeholders)". Any decision a teacher makes to facilitate learning based on appropriate data and knowledge can fall under this category. This includes but is not limited to the way a teacher arranges the classroom, the way they differentiate lessons, the classroom management plan they establish, community connections established with the classroom, the way they present information, etc. I believe the most important aspect of this component is that decisions are influenced by the teacher's knowledge/analysis of their students.

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This artifact exemplifies the environment portion of the decision-making component. This demonstrates the leading of educational progress by planning appropriate environments. In this classroom management plan, I establish some rules and procedures for my classroom. This allows decision-making to be more fair and consistent in the classroom when rules and consequences are set ahead of time. Beyond that, having clear expectations and procedures allows learning to take place more effectively and smoothly.

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This artifact required me to create a mid-range plan for a numbers and number systems unit in middle school. For this assignment, I had to order the standards and objectives in a logical and coherent way. Further, I had to determine how I would assess student learning through various formative and summative assessments. Then I had to determine how to differentiate instruction to accommodate all learners in my classroom. So many decisions were required to develop a coherent mid-range plan. I believe this was necessary practice in planning units and assessments and differentiating instruction. This is a crucial skill that will serve me well in practice.

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Significant Figure 1: Pythagoras and his disciples made various contributions to number theory. For example, they were the first to classify integers in groups such as prime or composite and even or odd. They also introduced triangular numbers, square numbers, pentagonal numbers, and more. They also provided a method of determining infinitely many Pythagorean triples.

Significant Figure 2: In the early 9th century, the Persian mathematician Muhammad ibn Musa al-Khwarizmi discovered the Hindu number system, which utilized nine symbols for numbers and a small circle to represent zero. Intrigued, Al-Khwarizmi recognized the potential of the place value system, where a number's position determined its value. This breakthrough led to the adoption of the Hindu-Arabic numeral system, which revolutionized mathematical calculations and paved the way for various technological advancements. Al-Khwarizmi's work laid the foundation for modern arithmetic and algebra, enabling developments ranging from the Industrial Revolution to contemporary technologies like cars, computers, space travel, and robotics.

Diverse Culture 1: The ancient Egyptians expressed numbers with different symbols for each place like the ones, tens, hundreds and so on within hieroglyphic writing. However, to complete arithmetic operations, this type of writing was not efficient, so they used hieratic writing instead. For multiplication, scribes used successive doubling then used addition to add the corresponding multiples. Division was the reverse process. Fraction operations are also evident in the hieratic writings.

Diverse Culture 2: The Babylonians made significant contributions to number theory through their development of a base 60 number system. They were the first to use place value notation, allowing for the representation of large numbers with more accuracy. Further, Babylonian mathematicians created mathematical tables, including tables for multiplication, division, and square roots, which aided with complex calculations. Their mathematical achievements extended to astronomy, where they used numerical methods to track celestial movements and predict astronomical events. A Babylonian tablet dating back to approximately 1700 BC also displays an extensive list of Pythagorean triples.

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Significant Figure 1: Diophantus, a Greek mathematician, also known as the "father of algebra" made various contributions to the field, most notably his work "Arithmetica". This work had a very significant influence on the development of number theory. Diophantus explores various types of Algebra in this work such as quadratics, making linear expressions into squares, powers between given limits, and more. During his work with quadratics he grouped them into three different categories because he lacked a symbol for zero, which we now have today and have one type of quadratic equation.

Significant Figure 2: Brahmagupta, an Indian mathematician and astronomer, is an immensely influential mathematician who aided in developing the rules for computing with zero.In the Brahmasphutasiddhanta , his 25 chapter work, he defined zero as the result of subtracting a number from itself. He also provides arithmetic rules in terms of fortunes (positive numbers) and debts (negative numbers). He goes on to provide formulas for cyclic quadrilaterals, rules for summing various series, solving quadratic indeterminate equations, and more, in addition to his work in astronomy.

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Significant Figure 1: Liu Hui, a Chinese mathematician, (approximately 220-280) made various contributions to the field of geometry. Liu Hui proposed a calculation for pi. Liu Hui proved an existing formula for the volume of a sphere incorrect, but was unable to find the correct formula for the volume of a sphere. Liu Hui also started Cavalieri's principle (before Cavalieri), with Zu Gengzhi finishing it. His assumption was as follows: "If an object with circular cross-section is inscribed in an object with square cross-section, and every circular cross-section is inscribed in the corresponding square cross-section, then the ratio of the volumes of the objects is equal to the ratio of the areas of a circle and a circumscribed square, i.e. π:4." He also advanced the field of cartography.

Significant Figure 2: Zu Gengzhi, a Chinese mathematician, (approximately 450-520) is most famous for working with Liu Hui to finish what is now inappropriately named Cavalieri's principle to provide the formula for the volume of a sphere. Zu Gengzhi's statement of Cavalieri's principle, before Cavalieri, is as follows: "If blocks are piled up to form volumes,
And corresponding areas are equal,
Then the volumes cannot be unequal." This follows from Liu Hui's assumption. With their combined statements, Zu Gengzhi developed the formula for the volume of a sphere: πd ³/6

Diverse Culture 2: Another culture to contribute to geometry was Egyptian culture. They are credited with using practical geometry in the surveying the land after the flooding of the Nile every year, building the pyramids, and storing resources such as food and water. This is pre-Euclidean geometry.

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Significant Figure 1: Blaise Pascal, a French mathematician, physicist, inventor, philosopher, and Catholic writer, made significant contributions to the field of probability theory during the mid-17th century. His correspondence with Pierre de Fermat led to the development of the probability theory, particularly in the context of gambling. Because of this, both men are credited with being the fathers of probability. Pascal's most notable work in this area is "Traité du Triangle Arithmétique," where he introduced Pascal's Triangle, a tool for calculating probabilities and combinations. Additionally, Pascal formulated what is known as Pascal's Wager, a philosophical argument regarding the belief in God based on probability considerations. His contributions laid the groundwork for the formalization of probability theory, influencing fields ranging from mathematics to economics and philosophy.

Significant Figure 2: John Graunt, an English demographer, made a significant contribution to the field of statistics. Graunt wanted to predict mortality rates. He wrote the paper, "Natural and Political Observations Made upon the Bills of Mortality" . He was the first to coin the term "excess deaths" in epidemiology. He also provided the first accurate male:female ratios based on his data. He also noticed the "urban penalty", that being that those living in urban areas die faster than those living in rural areas. He is regarded as the father of demography, epidemiology, and vital statistics.

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Significant Figure 1: Gottfried Wilhelm von Leibniz, a German mathematician, developed Calculus independently of Sir Isaac Newton. Leibniz thought of the variables as ranging over sequences of infinitely close values. He knew that dy/dx provided the tangent. Leibniz used very good notation in his journaling of his development of calculus and most of our modern notation can be credited to him.

Significant Figure 2: Sir Isaac Newton, an English mathematician, developed Calculus independently of Gottfried Wilhelm von Leibniz. When developing calculus, Newton considered the variables to be changing with the time. Newton also used x' and y' as quantities to represent finite velocities to compute tangent. Newton thought of calculus in more geometrical terms. While Leibniz was very conscious of notation, Newton was not. Instead, Newton used different notations by the day, which is why we instead use and accredit today's calculus notation to Leibniz. Both Leibniz and Newton used infinitesimals, which bothered later mathematicians when criticizing and tweaking the foundations of calculus.

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Significant Figure 1: Dr. Doron Zeilberger is an Israeli mathematician who is, in fact, still alive. He graduated from the University of London in 1972 and focuses his current work on teaching computers how to do math. His mathematical focus is combinatorics. In an interview, he states "Combinatorics is everything. All our worlds, the physical, mathematical, and even spiritual, are inherently finite and discrete, and so-called infinities, be their actual or potential, as well as the ‘continuum’, are ‘optical illusions’." (ECA, 2021). His biography states: "Professor Zeilberger has important contributions to the fields of hypergeometric summation and q-Series and was the first to prove the alternating sign matrix conjecture. He is considered a champion of using computers and algorithms to do mathematics quickly and efficiently, and his results have been used extensively in modern computer algebra software." (ECA, 2021). From this interview, one can gather that Dr. Zeilberger believes computer software math to be the way of the future. I believe that this has extreme implications on the way we teach math and use technology inside of the classroom.

Significant Figure 2: One cannot speak about discrete mathematics without discussing the Hungarian mathematician, Paul Erdos. Paul Erdos was a legendary mathematician who made significant contributions to discrete mathematics. He pioneered the study of combinatorics, with a focus on problems that involved discrete structures and finite objects. Erdos introduced the concept of Ramsey theory, which explores the order in random structures. I hated this in my undergraduate years. He also played an important role in developing the probabilistic method, a powerful tool for solving combinatorial problems. He was known for his collaboration with other mathematicians. This lead to the creation of the Erdos number, which measured collaborative distance. His work continues to inspire mathematicians working in the realm of discrete mathematics.

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This artifact exemplifies the decision-making component because I had to make various decisions regarding the actions and goals of my chosen reader based on the "collection and analysis of appropriate data (e.g., characteristics, needs, and abilities of learners; educational, environmental, and community contexts; perspectives of multiple stakeholders)", per the definition provided on the USC webpage. In this assignment, I had to interview my reader about their experiences with reading and reflect on this interview to guide my decision-making. Further, I used various reading strategies. Completion of the goals set forth in these reading strategies influenced my decisions about the goals and the actions that I expected during our next session.