Part 1: Instructional strategies and theories

Academic vocabulary and language

This instructional strategy according to involve more precise or subtle forms of familiar words. In this exercise, worlds like trajectory, parabola, and apogee are used to illustrate logical concepts of projectiles that are much more related to the task at hand.

Adapting to learning styles/multiple intelligences

This strategy is based upon the concept that students learn, remember, perform, and understand in different ways. In this exercise, a variety of lessons were used that involved hands on activities, calculations, visual and mathematical methods – those being the use of computers to mechanical methods of research and learning.

Cooperative learning

These project based learning activities are at the heart of the group work concept.

Field experience, field trip, or field study

Getting out of the classroom is always a way to refresh the mind and take a different approach to learning.

Hands-on learning

These methods are truly center around this concept.

Modeling

The method of deliver, through a Youtube as I am the teacher who is actually doing the project in an example way, it indeed the essence of this instructional strategy.

Project-based learning

All of these exercises support an overall objective and are depended on open ended questions.

Part 2: Research/literature about algebraic reasoning

“Physics teachers are on a continuous quest to find or create physics laboratory experiences that have high interest, are realistic, and provide students with an opportunity to perform meaningful quantitative work,” according to Randy Jenkins (Jenkins, 1993). Jenkins’ article is older however it is difficult to find modern treatment of the model rocket concept used within the current standardized test curriculum drive. Jenkins used a pully, a photo gate, and an inferred beam to measure acceleration and velocity of Estes rockets. Quite complicated, however with modern video equipment, and Estes beginning to manufacture rocket cars, estimating velocity is much less cumbersome and can be done with younger children.

Jenkins ideas are more physics related than mathematics, however, Flath, Stoll, and Wagon clearly make the connection between angles of inclination and rocket height. “Our investigation led us to the classic height measurement problem where observers on the ground use a theodolite to measure both the elevation and the direction to the rocket (called an azimuth). For a stationary object such as the top of a tree or mountain, classic trigonometry turns these angles into the height. But a model rocket adds an extra dimension, since it does not stay in one place. All this adds challenge and mathematical interest to the problem of height determination. adds challenge and mathematical interest to the problem of height determination. A comprehensive approach involves interesting aspects of geometry, algebra, and statistics.”

Flath, Stoll, and Wagon deal with college level mathematics as they are concerned with a greater level of precision, using three observers and three non-colinear points, where we just used one.  This level of precision is not necessary  for in this middle school exercise we are exposing the students to the concept of angle of elevation. One observer is well within the concept of genuine exposure to the fact that an angle of elevation can determine the height of an object. The use of the theodolite is in sync with my exercise where the Estes Altitrack was used.

Herald, M. Svatek. and W. Svatek make great on the ideas that toys can allow students to “have a ball” while they are learning, including using cars to teach velocity and acceleration and projectiles.

Borlaug makes the clear connection to Algebra and extends it to Calculus by using a toy Tonka truck to demonstrate constant velocity, average velocity and instantaneous velocity. “It can be used to illustrate an application of slope in an algebra class or to introduce the derivative in a calculus class,” including position and time functions that I have illustrated in the lessons that I have presented. However, I am using the concept of video data to illustrate this concept at a much higher rate of speed and excitement with the rocket car and the wooden ball projectile.

Freidlander presents the usefulness of using spreadsheets in beginning-algebra courses taught in seventh grade Israeli students dealing with sequences and patterns. Through the applications of Excel, the author demonstrates the importance of using technology to teach algebra through modeling and justifying.  The author proposes the importance of spreadsheets as a learning tool as they release the momentarily burden of calculations so that the larger picture can be the focus of the mental processes. Freidlander claims that spreadsheets expand the realm of algebra concepts, allowing for more freer flow between numbers and algebra, where algebra can more easily be explored in a real-world context.

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For further demonstration, the following video is offered available at the links;

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https://www.youtube.com/watch?v=WwDsNGVtGD4

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https://www.youtube.com/watch?v=G0ZjfyBHOQM&t=134s

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https://www.youtube.com/watch?v=jUc21o7H2Mo&t=56s

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Abstract

            The true beauty of studying the history of mathematics is that through historical narratives and context, the “esoteric, dry way” (Maor, 1994). of traditionally teaching math has a cross curricular appeal that can expand learning and retention. While I may find math concepts fascinating, my interest in the topic needs little urging; however, as a secondary and elementary educator, I have found that the majority of math students show very little enthusiasm for the topic. Eli Maor, in his book, e:The Story of a Number, entertained this reality in math education. He writes, “As one who has taught mathematics at all levels of university instruction, I am well aware of the negative attitude of so many students toward the subject.”[1] He contends that math instructors and professors…

…tend to overwhelm students with formulas, definitions, theorems and proofs, but we seldom mention the historical evolution of these facts, leaving the impression that these facts were handed to us, like the Ten Commandments, by some divine authority.[2]

(Maor, 1994, p xii)

Maor proposes that math history be included in curriculums as a way to “correct these impressions” as a means to spur interest to the subject[3]. This study will focus on the use of historical narratives and multi-sensory context as an effective method for math instruction. This study will also demonstrate and justify the use of historical narratives and analogies, concepts of novelty and innovation, and other mechanisms (Reddy, 2016)[4] to diminish the student content familiarity deficit (Heath, 2006)[5] associated with understanding logarithms (Clark & Montell, 2011) [6]– combating what will be referred to as the “curse 

gap” [1]— a void of circumstantial knowledge differences between a mastery level learner and the novice[2], often referred to as the “Curse of Knowledge” (Heath, 2006)[3].

[1] See “Chapter 2:*chapter introduction”

[2] Chapter 2

[3] Introduction: (Heath, 2006). See also, Footnote[46], [120]

[1] See Footnote [68]

[2] Footnotes [84] , [111], [119], [129. The idea of not equating instruction to simply disseminating facts.

[3] See Footnotes [69], [107]

[4] Figure 3.1

[5] Introduction (Heath, 2016)

[6] Introduction: (Clark & Montelle, 2011)

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Introduction

In an edutopia.org article Christopher Reddy cites several articles and studies that identify a phenomenon known as the “Curse of Knowledge.” While Reddy does make the connection to how this phenomenon has negative consequences on content mastery (Reddy, 2016), a Harvard Business Review article provides a good explanation of the cognitive anomaly in more general terms. The article states that “The problem is that once we know something—say, the melody of a song[1]—we find it hard to imagine not knowing it. Our knowledge has ‘cursed’ us. We have difficulty sharing it with others, because we can’t readily re-create their state of mind” (Heath, 2006). Reddy concludes that because of this mental disconnect, teachers sometimes struggle to empathize[2] with their students’ “content familiarity deficit”[3] as they are novices to new concepts. Reddy proposes seven methods[4] to “lift the curse” focused on avoiding simple “fact teaching” and converging curriculum with “narratives,” “analogies and examples,” “emotion” and “novelty”[5] (Reddy, 2016).

Reddy’s seven methods can easily be put in sync with what Maor proposed when he advocated for the inclusion of math history in mathematics curriculums[6]. In addition, a Convergence article by Kathleen Clark and Clemency Montelle presents similar strategies directly related to logarithms. The Clark/.Montelle article is a self-proclaimed promotion of “the teaching of the logarithmic relation from its historical roots” (Clark & Montelle, 2011) such that math curriculums would better demonstrate the “crucial insights” and importance of the logarithm tool in progressing mathematics over time (Clark & Montelle, 2011). On this line of thinking, the following pages provide detailed examples[7] of how to break the curse of knowledge (Reddy 2016) in the secondary classroom, specifically in the case of teaching logarithmic and exponential concepts.

Chapter 1 deals with the methods and narratives of the Scottish inventor and mathematician John Napier who is generally given credit for the “invention” of a logarithmic system. This first chapter serves a dual purpose in that it provides an example of possible narrative content that can be incorporated into math instruction as well as an example of how the Reddy seven methods can be applied. Chapter 2 is dedicated to identifying and explaining the “curse gap” quantitatively as it exists in the current Texas secondary math curriculum – through the method of a keyword search examination of the TEKS[8]. Chapter 3 will discuss the modern academic framework of logarithms as functions[9] and will demonstrate how the Reddy “curse lifters”[10] and Maor’s historical context[11] -- using Chapter 1 as an example — are effective ways to generate interest to the topic of logarithms. This chapter will explain how multi-sensory[12], novel[13] examples that construct context through a historical references are a much more effective teaching method than simply presenting theory and fact with little explanation and background[14] (Maor, 1994). The third chapter also demonstrate the existence of the “curse gap”[15] in the Texas curriculum through categorically analysis of the TEKS paragraphs[16].

Chapter 4 is dedicated to Swedish clock maker Joost Burgi’s logarithm tables and the work on logarithms done by the Englishmen Henry Briggs, leading into a multi-sensory[17], novel[18], and historically connected (Maor, 1994) approach to understanding the current common logarithms. Chapter 5 will continue to apply Maor and Reddy’s methods with examples that demonstrate logarithmic properties transitioning into final considerations of the number e and natural logarithms.

[1] See Chapter 3:*quantifying the curse, Footnote[ 56]

[2] Usually applied to comprehending feelings of another, in this study is focused more on comprehending the cognitive struggle that accompanies learning new material.

[3] Abstract, Footnote [5]

[4] Figure 3.1

[5] See Figure 3.1:7.Teach Facts;4.Narratives;5.Analogies and Examples; 1.Emotion;6.Novelty respectively

[6] See “Abstract”, Footnote [3]

[7] Figure 3.1:5Analogies and Examples

[8] Texas Essential Knowledge and Skills

[9] As in Chapter 2: *chapter introduction, Footnotes [33],[43]

[10] Figure 3.1

[11] See “Abstract”, Footnote [3]

[12] Figure 3.1:2.Multi-Sensory Lessons

[13] Figure 3.1: 6.Novelty

[14] Figure 3.1:7.Teach Facts

[15] See “Chapter 2:*chapter introduction”, Footnote [53]

[16] Figures 3.5 and 3.6

[17] Figure 3.1:2.Multi-Sensory Lessons

[18] Figure 3.1: 6.Novelty

Due to health conditions, I was forced to retire from full time teaching. I taught high school math for five years but due to immense pain throughout my body due to the conditions of military service, I was forced to seek part time employment. During that time, I worked as a tutor at a tutoring company. We had all sorts of courses that just a few of us were tasked with helping students with. I was the math guy, but due to my disabilities and the institutional requirements that the employees maintain a part time status; some of the other tutors had to accommodate and ended up having a few sessions with advanced math students in AP courses and Calculus. On one occasion, a fellow tutor approached me and detailed the lengths he had gone through with a student involving imaginary numbers. His comfort level was about the second semester of the Texas curriculums first semester Algebra 2. He asked me, why do we study imaginary numbers? I pointed out that in engineering and other fields these theories on the imaginary number help to solve certain types of problems. He continued to inquire of their relevance. I explained the typical definitions of the imaginary number and how to work with them and the varying theorems that are associated with it. But he still seemed confused at the concept. Why he asked, if the number itself is undefined, why is it relevant. I have to admit that I failed to explain the concept. I accredit this to the way in which I was taught math – rigorous explanations of theorems and facts with little explanation of the history and concepts that led to a type of mathematical thinking, that is—the way we teach math in America.

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This brings this treatise to a concept proposed by Eli Maor, in his book, e:The Story of a Number, He stated an all too common dilemma faced by an American math teacher.  “As one who has taught mathematics at all levels of university instruction, I am well aware of the negative attitude of so many students toward the subject.” Maor points out that perhaps math instructors and professors…

…tend to overwhelm students with formulas, definitions, theorems and proofs, but we seldom mention the historical evolution of these facts, leaving the impression that these facts were handed to us, like the Ten Commandments, by some divine authority. The history of mathematics is a good way to correct these impressions. (Maor, 1994, p xii)

It is my continued contention that including math history in our curriculums is a valid premise for establishing a better math curriculum at all levels of the study – especially secondary education, and even in the elementary grades.

            This paper is an attempt to work through the processes in history concerning the imaginary number – its controversies and it value – in hopes that this treatise may serve to demonstrate the fascinating and intriguing study of math.

Abstract

            In lectures 1-10 we have focused on the development of mathematics during the end of Ancient Greece (approximately 100 B.C) to the end of the Renaissance(17th Century) and the virtual cusps of the coming Enlightenment (late 17th Century). In between this time frame their existed an age often referred to as the age of exploration (roughly 1450 to 1600). Christopher Columbus’ famous voyages to the Americas often come to the mind of most Americans. In fact, the title “Age of Exploration” is a very westernized viewpoint on exploration of the world. In fact, the great Chinese seaman and employer Zheng He had already ventured out from the Far East to east Africa and the Middle East almost a hundred years before Columbus crossed the Atlantic, kicking off the original spice trade (Hadingham, 2001). One could argue that the successes of the Arabian—African—China trading mechanism served as an impetus for the Europeans to follow suit. In fact, the Chinese and Arabians were trading with India a hundred years before Zheng He (Hadingham, 2001). In fact, Columbus was not even the first European to trek the Atlantic to the Americas. The Viking Leif Eriksson arrived in Newfoundland 500 years before (Klein, 2013).

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            This paper sets out to answer the questions surrounding the navigation devices and the mathematics that go along with them – those used by the Europeans in their exploration age as well as their predecessors in China, Arabia and the Scandinavia. This paper will describe the use and math behind the compass, the “kamal,” the quadrant, the cross staff, and the sunstone as well as providing commentary on their non-so-European origins.

In lectures 1 we have focused on a study of the dawn of mathematics, in particularly references to the Ishango Bone – which is said to be the oldest table of prime numbers. The bone was named after the people who lived in that region near Lake Edward on the boarder of the modern African countries of Zaire and Uganda. The bone, along with the mathematics inscribed on it, are dated to be about 20,000 years old (Williams, 2008). In addition to remarks concerning the Ishango Bone, Scott Williams also details the Lebombo bone, dated 37,000 years old, which appears to be a calendar and Williams remarked that this stone is one of “the oldest mathematical objects known” (Williams, 2008). Our discussion fasts forward to the earliest forms of writing – which according to record occurred somewhere around 3,000 B.C. with the Sumerian/Babylonian cuneiform written on clay tablets (British Museum, 2016), which includes counting inventories and even “what appear to school exercises in arithmetic and geometric problems” (Mastin, 2010A). In this paper, I wish to focus on the history of math that occurs between those first hash marks on a Ishango Bone(20,000 B.C) to the Babylonian clay tablets(3,000 B.C).

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            While the ancients of this time-period did not appear to have left any record of their mathematical study in written form, they have left us clues that hint to their thinking as related to math and astronomy. Granted, astrology is not math and vice versa; however, it is my contention that the mathematical theories of Ancient Greece would never had occurred were it not for the primitive “counting and tallying” behaviors of those that lived before them. I believe that humanity’s transcendence from natural numbers to mathematical thought and reasoning is a result of mankind’s observation of the natural world – of which is composed of the objects and occurrences of mathematics – in this case, astronomical observations. Albert Einstein once commented on the inverse of my contention, “How is it possible that mathematics, a product of human thought that is independent of experience, fits so excellently the objects of reality?” By observing and counting the patterns of the moon, sun and stars, I believe humankind began to see the universe as more predictable – a belief that is inherent within any man who endeavors to set out to proving something in the general case such as the deductive nature of mathematical proofs.

One of the true beauties of study the history of mathematics and university courses on the topic is that there is that the content goes beyond the “esoteric, dry way” of traditional methods of teaching the subject of math (Maoer, 1994). While I may find math concepts fascinating, as a secondary and elementary educator, my students as a majority did not. Eli Maor, in his book, e:The Story of a Number, entertained this reality in math education. “As one who has taught mathematics at all levels of university instruction, I am well aware of the negative attitude of so many students toward the subject.” He went on to point out that perhaps math instructors and professors…

         …tend to overwhelm students with formulas, definitions, theorems and proofs, but we seldom           mention the historical evolution of these facts, leaving the impression that these facts were             handed to us, like the Ten Commandments, by some divine authority. The history of                           mathematics is a good way to correct these impressions. (Maor, 1994, p xii)

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            Maor also discusses his early education before the days of the electronic calculator and explains the struggles he had using logarithm tables. “You got bored to death doing hundreds of drill exercises and hoping that you didn’t skip a row or look up the wrong column” (Maor, 1994, p xi). Maor’s book is entertaining, humorous at times, and above all just plain good writing. In that same spirit, this paper will be an exercise in combing history and math – including the quirky and the theoretical. In the following pages, this paper will discuss the father of logarithms, John Napier, the tables he published, their popularity among mathematicians and the version of logarithms we use today.