Educational Views of Michel Montaigne

By the 16th century, the Renaissance was in full swing, the Protestant Reformation had already been around for over a generation and people had serious doubts about the intellectual and spiritual grip the Church had on society. Since the Church also controlled education people began to question these methods. As this wave of humanism swept Europe.

It was during this time of doubt and skepticism that Michel Montaigne (1533-1592) arrived on the scene. Montaigne was not so much an educator as he was a person who had a strong opinion of how education should be. He also knew how to write witty insightful essays on the subject of education along with other subjects of his interest. This post will take a brief look at his life and educational philosophy

Montaigne’s Life

Montaigne was born into a well to do family in France in 1533. He was natural brilliant and was able to speak Latin, in addition to his mother tongue of French by the age of six. Yes, Montaigne was brilliant but he also had a German tutor who did not know French and used Latin to communicate with the child.

By 13, Montaigne had finished college. He turned his attention to politics and was a member of parliament by the age of 20. Soon after, he became mayor of his 20. Despite what looked to be a brilliant political future Montaigne left politics after becoming Mayor to live a life of quietness. Since he was already well off he did not need to endure the rigors of financial gain and power to maintain his livelihood.

It was in this semi-retirement that Montaigne began writing his famous humanistic “Essays” on various subjects. In fact, Montaigne was one of the first people to popularize the idea of an essay, which is now standard practice in school today. Our attention will be on his views on education.

Views on Education

Montaigne views on education were almost a reaction against Church views on education. Montaigne believed in a wholistic education of the man and not to divide him into pieces. He also criticized the study of Latin and Greek because he supported the development of the mother tongue first. This debate over mother tongue use is a recurring theme in early language education.

Montaigne also criticized the study of the classics as it did not prepare students for practical life but rather bade them conceited. Another target of criticism was the teaching methods of the day, which were often lecture-style. Montaigne views this pouring knowledge into the mind and not useful for the student.

Montaigne supported a more interactional approach to teaching in which the students and teacher take turns talking and listening. THrough action came understanding in his opinion.

Finally, Montaigne was a critic of corporal punishment. He viewed almost as if one was training an animal rather than a person. Again most of these criticisms were of common practices in education at that time period and the education was mostly controlled by the church.

Conclusion

Montaigne was a theoretician on education but not much of a practitioner. His experience as a student led him to write strong reactionary criticisms against education. In spite of his lack of practical experience Montaigne’s thoughts are highly insightful and somewhat applicable to this day.

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Common Data Types in Python

All programming languages have a way of storing certain types of information in variables. Certain data types or needed for one situation and different data types for another. It is important to know the differences in the data types otherwise serious problems could arise when developing an application. In this post, we will look at some of the more commonly used data types in Python.

Making Variables

It is first important to understand how to make a variable in Python. It is not that complicated. The format is the following

variable name =  data inside the variable

You simply type a name, use the equal sign, and then include the data to be saved in the variable. Below is an example where I save the number 3 inside a variable called “example”

example=3
print(example)
3

The “print” function was used to display the contents of the “example” variable.

Numeric Types

There are two commonly used numeric data types in Python and they are integers and floating point values.

Integers

Integers are simply whole positive or negative numbers. To specifically save a number as an integer you place the number inside the “int” before saving as a variable as in the example below.

example=int(3)

print(example)
3

You can check the data type by using the “type” function on your variable. This is shown below.

type(example)
Out[17]: int

The results are “int” which stands for integer.

Floating-Point Types

Floating-point numbers are numbers with decimals. If your number includes a decimal it will automatically be stored as a floating type. If your number is a whole number and you want to save it as a floating type you need to use the “float” function when storing the data. Below are examples of both

#This is an example of a float number

example=3.23

print(example)
3.23

#This is an example of converting a whole number to a floating point

example=float(3)

print(example)
3.0

Floating points can store exponent numbers using scientific notation. Floating point numbers are used because decimals are part of the real world. The downside is they use a lot of memory compared to integers.

Other Types

We will look at two additional data types and they are boolean and string.

Boolean

A boolean variable only has two possible values which are True or False. This seems useless but it is powerful when it is time to have your application do things based on conditions. You are not really limited to True or False you can also type in mathematical expressions that Python evaluates. Below are some examples.

#Variable set to True

example=True

print(example)
True

#Variable set to True after evaluting an expression

example=1<2

print(example)
True

String

A string is a variable that contains text. The text is always enclosed in quotations and can be numbers, text, or a combination of both.

example="ERT is an awesome blog"

print(example)
ERT is an awesome blog

Conclusion

Programming is essentially about rearranging data for various purposes. Therefore, it only makes sense that there would be different ways to store data. This post provides some common forms in which data can manifest itself while using Python.

Luther and Educational Reform

Martin Luther (1483-1546) is best known for his religious work as one of the main catalysts for the Protestant Reformation. However, Luther was also a powerful influence on education during his lifetime. This post will take a look at Luther’s early life and his contributions to education

Early Life

Luther was born during the late 15th century. His father was a tough miner with a severe disciplinarian streak. You would think that this would be a disaster but rather the harsh discipline gave Luther a toughness that would come in handy when standing alone for his beliefs.

Upon reaching adulthood Luther studied law as his father diseased for him to become a lawyer. However, Luther decided instead to become a monk much to the consternation of his father.

As a monk, Luther was a diligent student and studied for several additional degrees. Eventually, he was given an opportunity to visit Rome which was the headquarters of his church. However, Luther saw things there that troubled him and in many laid the foundation for his doubt in the direction of his church.

Eventually, Luther had a serious issue with several church doctrines. This motivated him to nail his 95 theses onto the door of a church in 1517. This act was a challenge to defend the statements in the theses and was actually a common behavior among the scholarly community at the time.

For the next several years it was a back forth intellectual battle with the church. A common pattern was the church would use some sort of psychological torture such as the eternal damnation of his soul and Luther would ask for biblical evidence which was normally not given. Finally, in 1521 at the Diet of Worms, Luther was forced to flee for his life and the Protestant Reformation had in many was begun.

Views on Education

Luther’s views on education would not be considered radical or innovative today but they were during his lifetime. For our purposes, we will look at three tenets of Luther’s position on education

  • People should be educated so they can read the scriptures
  • Men and women should receive an education
  • Education  should benefit the church and state

People Should be Educated so they Can Read the Scriptures

The thought that everyone should be educated was rather radical. By education, we mean developing literacy skills and not some form of vocational training. Education was primarily for those who needed it which was normally the clergy, merchants, and some of the nobility.

If everyone was able to read it would significantly weaken the churches position to control spiritual ideas and the state’s ability to maintain secular control, which is one reason why widespread literacy was uncommon. Luther’s call for universal education would not truly be repeated until Horace Mann and the common school. movement.

The idea of universal literacy also held with it a sense of personal responsibility. No one could rely on another to understand scripture. Everyone needs to know how to read and interpret scripture for themselves.

Men and Women Should be Educated

The second point is related to the first. Luther said that everyone should be educated he truly meant everyone. This means men and women should learn literacy. The women could not hide behind the man for her spiritual development but needed to read for herself.

Again the idea of women education was controversial at the time. The Greeks believed that educating women was embarrassing although this view was not shared by all in any manner.

WOmen were not only educated for spiritual reasons but also so they could manage the household as well. Therefore, there was a spiritual and a practical purpose to the education of women for Luther

Education Benefits the Church and the State

Although it was mentioned that education had been neglected to maintain the power of the church and state. For Luther, educated citizens would be of a greater benefit to the church and state.

The rationale is that the church would receive ministers, teachers, pastors, etc. and the state would receive future civil servants. Therefore, education would not tear down society but would rather build it up.

Conclusion

Luther was primarily a reformer but also was a powerful force in education. His plea for the development of education in Germany led to the construction of schools all over the Protestant controlled parts of Germany. His work was of such importance that he has been viewed as one of the leading educational reformers of the 16th century.

Logarithmic Models

A logarithm is the inverse of exponentiation. Depending on the situation one form is better than the other. This post will explore logarithms in greater detail.

Converting Between Exponential and Logarithmic Form

There are times when it is necessary to convert an expression from exponential to logarithmic and vice versa. Below is an example of who the expression is rearranged form logarithm to exponential.

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The simplest way to explain I think is as follows

  • for the logarithm, the exponent (y) and the base (a) are on opposite sides of the equal sign
  • For the exponent form, the exponent (y) and base (a) are on the same side of the equal sign.

Here is an example using actual numbers1.png

As you can see the exponent 3 and the base 2 are on opposite sides of the equal sign for the logarithmic form but er together for the exponential form.

When the base is e (Euler’s Number) it is known as a natural logarithmic function. e is the base rate growth of a continual process. The application of this is limitless. When the base is ten it is called a common logarithmic function.

Logarithmic Model Example

Below is an example of the application of logarithmic models

Exposure to noise above 120 dB can cause immediate pain and damage long-term exposure can lead to hearing loss. What aris the decimal level of a tv with an intensity of 10^1 watts per square inch. 

First, we need the equation for calculating the decibel level.

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Now we plug in the information into the word problem for I and solve

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Our tv is dangerously loud and should include a warning message.  We dropped the negative sign because you cannot have negative decibel level.

Conclusion

Logarithms are another way to express exponential information and vice versa. It is the situation that determines which to use and the process of concert an expression from one to another is rather simple. In terms of solving actual problems, it is a matter of plugging numbers into an equation and allowing the calculator to work that allows you to find the answer.

Education During the Reformation

By the 16th century, Europe was facing some major challenges to the established order of doing things. Some of the causes of the upheaval are less obvious than others.

For example, the invention of gunpowder made knights useless. This was significant because now any common soldier could be more efficient and useful in battle than a knight that took over ten years to train. This weakened the prestige of the nobility at least temporarily while adjustments were made within the second estate and led to a growth in the prestige of the third estate who were adept at using guns.

The church was also facing majors issues. After holding power for almost 1000 years people began to chaff at the religious power of Europe. There was a revival in learning that what aggressively attacked by monks, who attacked the study of biblical languages accusing this as the source of all heresies.

The scholars of the day mock religion as a superstition. Furthermore, the church was accused of corruption and for abusing power. The scholars or humanists called for a return to the Greek and Romans classics, which was the prevailing worldview before the ascension of Catholicism.

Out of the chaos sprang the protestant reformation which rejects the teachings of the medieval church. The Protestants did not only have a different view on religion but also on how to educate as we shall see.

Protestant Views of Education

A major tenet of Protestantism that influenced their view on education was the idea of personal responsibility. What this meant was that people needed to study for themselves and not just listen to the teacher. In a spiritual sense that meant reading the Bible for one’s self. In an educational sense, it meant confirming authority with personal observation and study.

Out of this first principal springs two other principles which are education that matches an individual’s interest and the study of nature. Protestants believed that education should support the natural interest and ablities of a person rather than the interest of the church.

This was and still is a radical idea. Most education today is about the student adjusting themselves to various standards and benchmarks developed by the government. Protestants challenged this view and said education should match the talents of the child. If a child shows interest in woodworking teach this to him. If he shows interest in agriculture teach that to him.

To be fair, attempts have been made in education to “meet the needs” of the child and to differentiate instruction. However, these goals are made in order to take a previously determined curriculum and make it palpable to the student rather than designing something specifically for the individual student. The point is that a child is more than a cog in a machine to be trained as a screwdriver or hammer but rather an individual whose value is priceless.

Protestants also support the study of nature. Be actually observing nature it reduced a great deal of the superstition of the time. At one point, the religious power of Europe forbade the study of human anatomy through the performing autopsies. In addition, Galileo was in serious trouble for denying the geocentric model of the solar system. Such restrictions stalled science for years and were removed through Protestantism.

Conclusion

The destabilization that marks the reformation marks a major break in history. With the decline of the church came the rise of the common man to a position of independent thought and action. These ideas of personal responsibility came from the growing influence of Protestants in the world.

Secular Education During the Middle Ages

The Middle Ages (500-1500 CE) is often viewed as a low point in the world of education. This was a time a strong superstition among people and a lack of scientific progress.

The European world was divided into three classes or estates which were the Priest, Nobility, and lastly, everyone else. These were the three estates. The Priestly estate held significant power over the other two estates. The priests would use the psychological terror of removal from having access to the sacraments of the church to maintain power.

When an individual was denied the sacraments it was called excommunication, when a region loss access to the sacraments it was called an interdict, final if an entire province or kingdom was denied the sacraments war was then declared and this was called a crusade.

There were two common forms of education below the university and these were the Knightly schools and the Burgher schools.

Knightly Schools

Knightly schools trained boys to become knights. The training was divided into 3 segments of seven years each. The first segment was from 0-7 years of age under the care of the mother. From ages 7-14, the boy would live with another knight perhaps as a page. The third stage from 14-21 had the boy serving as a squire. At the age of 21, a young man was declared a knight.

The subjects taught in the KNightly practicum focused on the physical, artistic, and strategic. Music, chess, manners, poetry, and military training were all part of the curriculum. There was almost no intellectual training but an obsession with practical learning.

Burgher Schools

Burgher Schools were for tradesmen and artisans and provided a basic education. The subject taught included reading, writing, and arithmetic as well as geography, history, natural science, and Latin.

THere was a constant power struggle between commoners and the priest for control of these schools. Locals wanted to control these schools themselves. However, technically only the church had permission to teach. This resulted in alternating back and forth in terms of control.

Teachers in these schools were paid almost nothing and traveled from school to school as vagrants. Teaching was not seen as a noble profession at this time thus having a powerful effect on the quantity and quality of education.

Conclusion

Education in the Middle Ages was designed to meet the needs of the three estates. People would often attend school corresponding to their rank in society. This system had an air of stability until rapid social changes brought about the decline of this system.

Expontial Models

There are times when we want to understand growth that is not constant. An example of this would be the growth of a virus. As time goes by the virus growth rate increases more and more. Another example would be in the world of finance when we are dealing with interest.

In situations like the ones mentioned above, it is critical to understand the use and application of exponential models. This post will go through examples of the use of exponential models.

Finance Example

One common exponential model in finance is for compounded interest. The equation is as follows…

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Below is a simple word problem that calls for this equation

You invest $10,000 in a mutual fund to prepare for retirement. The interest rate is 5% compounded monthly, how much will be in the account when you plan to retire in 25 years. 

Below is what we now

  • balance = ?
  • principal = $10,000
  • rate = 0.05
  • years=  25
  • times in year = 12 * 25 = 300

Now, we simply plug this information into the equatiom to get the answer.

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The answer is shown above. The initial investment would grow to almost $35,000 dollars over 25 years.

Continuous Growth

In some fields, such as the life sciences, you want to now the growth of a virus or bacteria. Unlike in finance where the balance grows several times a year,  a bacteria is growing continuously. This leads to a slightly different exponential model as shown below.

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e is an irrational number that serves as the base. With this information, we can address the problem below

A student starts their experiment with 10 bacteria. He knows the bacteria grow 100% every hour. He will come back and check in 12 hours. How many bacteria will he find?

Here is what we know

  • final size =?
  • initial size = 10
  • rate = 1/hour
  • time = 12

We plug this into the equation to get the answer

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As you can see, the growth of the bacteria is almost incomprehensible in such a short time. This is the power of exponential growth.

Conclusion

Exponential models provide another way to find answers to questions people have. Whether the growth is over a certain number of times or continuously the model can be adjusted to deal with either of this situations.

Monastic Schools

During the early Middle Ages (500-1000 CE) monastic schools began to take shape and heavily influence education. Their influence was felt for over a millennia providing education directly or indirectly to a countless number of people.

Background

The monastic schools grew out of the philosophy of Asceticism. Asceticism is the belief in a life of severe self-denial from the viewpoint that the body was evil. Practitioners of asceticism would forego marriage, financial gain, and most earthly pursuits, in order to focus on spiritual development usually in isolation. This a strong reaction to the non-Christian world’s focus on eating a drinking

There were two common ways to follow Asceticism. Hermits would often live in nearly complete isolation to pursue spiritual development. Monks, on the other hand, practice asceticism as well but would stay near communities of people in order to provide spiritual care for others. In addition, monks would live together in monasteries to support and encourage each other. Of course, at least in the past, monks were only men. Women could become nuns if they desired to live in similar conditions among women.

The largest order of monks was the Bendectin Order. The monasteries served as an asylum for the oppressed, as a missionary station, and most importantly as a preserver of knowledge.

Curriculum

The curriculum of the monastic schools consisted of the 7 liberal arts. These seven subjects can be broken into two categories, which are the trivium and the quadrivium.

The Trivium consisted of three subjects which were Latin, logic, and rhetoric. Latin was the lingua Franca of the Chuch at the time so its grammar was taught extensively. Logic was derived from the ideas of Aristotle and included deductive and inductive reasoning. Rhetoric is another term for public speaking and this was studied for the purpose of developing communication skills.

The quadrivium consisted of four subjects which were arithmetic, geometry, astronomy, and music. Many of the subjects are not studied as they are today. Arithematic study the mysterious or gnostic properties of numbers. Geometry was studied superficial and of little use. Astronomy was treated almost the same as astrology.

However, music was studied for the purpose of worship. The chants that the monks sang came to be called Gregorian chants named after Pope Gregory who had the chants codified. This is some of the earliest written version of western music. The notational system was different from modern notation using four lines instead of five and use squares instead of ovals to indicate notes.

The significance of Gregorian chants cannot be overestimated as they laid the foundation for modern music. Chants in the halls of monasteries provided the beginnings of most music found today.

Conclusion

Radical views in terms of the body led to the idea of asceticism. From this focus on self-denial comes the idea of living among like-minded people in monasteries. While in monasteries the monks would pursue education for personal development. This led to the liberal arts curriculum that is still used in part to this day.

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Education in the Early Church

The early church provides a unique look at the development of a system of religious education fairly recently in history. With the death of the apostles believers who were still alive had to face the reality of two major problems.

  1. What do we do with our children in terms of their education?
  2. How do we educate people who want to join the church?

The answers to these two questions intersect in many ways. This post will examine education in the early church.

Education of Children

The education of children was a problem for the early church. Children needed an education but state-run schools were not really an option. The reason has to do with the difference in philosophy of Christian education and state education.

Christian education is focused on character development and being prepared for eternity. In contrast, state education is focused on skill development and the here and now as eternity is often not a concern. As a result of this, Christians did not consider state-controlled schools as an option for their children.

In addition, it was common for state-led schools to mix Roman worship with education and for the Christians this was unacceptable. It is also important to realize that Christians were frequently persecuted as atheists during this time so it was impossible to go to school when one’s life was in danger

The solution to this was the one that the Jews used which was homeschooling. The focus of the child’s training was to develop a trust in the Christian God. By keeping the child at home he or she was protected from the influence of the world for a time. This led to a simplicity of taste that non-Christians found bewildering.

The Bible was the sole book for most children. The stories within it served as nursery tales. Scripture was memorized and the Bible was even used for learning to read.

With the focus on character development and a sense of morals, Christian education was vastly different from the education of other societies. Even without the focus on the classics and even technical training Christians were a spectacle to the world at this time. In terms of the results of this education among women one heathen author exclaimed “What wives these Christians have” indicating his awe in how these people conducted themselves.

Catechetical Schools

As the church grew, it became difficult to address new members. In particular, there were concerns over how to prepare prospective members for church membership. One answer to this problem was the development of Catechetical schools which were a place for prospective and current members to receive training in Christian beliefs.

For people considering baptism, the training could last anywhere from a few months to as long as three years. The curriculum consisted of learning the Ten Commandments, Lord’ Prayer, other parts of scripture, and as well as a confession of faith.

For people who were already Christian, they could receive advanced training that would prepare them for ministerial work as a teacher or leader. Some of the subjects covered for believers included philology, rhetoric, math, and philosophy.

The most prominent of these schools was found in Alexandria, Egypt. For several centuries after this,  Alexandria has a powerful influence on the Christian church.

Conclusion

The purpose of education is to me the needs of the people in the context in which it is needed. The Early Christian Church had the dilemma of having to be separate from the world while still developing skills needed to survive in it. This led to the development of the homeschool for children and the Catechetical School for new converts.

Intro to Python

Python is a highly popular programming language. It is so popular that it is now the most commonly used programming language for machine learning/data science purposes having surpassed R.

However, Python is not limited to just statistical tools. Python is also used by many companies for a host of reasons including Yahoo, Dropbox,  Google, NASA, IBM, and Mozilla.

One secret to Pythons popular is its flexibility. When using Python it is possible to employ several different coding styles. Below is just some of them.

  • Procedural: This is the simplest form of coding and involves executing each line of code sequential.
  • Functional: Functions are used to transform data as found in mathematics.
  • Imperative: Employs statements to achieve a goal
  • Object-oriented: The use of objects (aka data structures) to model the real world. Not fully implemented in Python.

You can mix these styles together to make powerful applications.

Using Python

You can download Python by searching for “Anaconda Python” in Google. The Anaconda version of Python downloads several additional features to besides Python including the Spyder IDE which is what we will use here.

Once you download and install Anaconda, on your computer you need to search for the program called “SPyder”. When you open it you will see the following.

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Here is what each pane represents.

  • To the left is the text editor, you can type code that you want to save here.
  • In the top right is the variable explorer. Here you can find a list of the objects you have made.
  • In the bottom right is the Interactive Python console or “IPython” console for short. Here you can type code quickly without the need of storing it for long-term use. In addition, the results of any code execution is normally displayed here as well

When writing code remember that you can save it long term in the text editor or just execute it quickly in  the console,

First Line of Code

We will now run our first line of code. Followed by the output in the IPython Console. Below is what we typed into the Console

print("Hello to Python")

Here is what it looks like in the console

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Here is what we did.

  1. We typed “print(“Hello to Python”)” into the console. This is an example of the use of a function.
  2. The output provides several pieces of information.
    • The blue shows what line this is in the console. In other words, this is the third line of code I had typed in the console. You may have a different number.
    • The purple is the function being used which for us is the “print” function which simply displays the input
    • The green is the argument that the function is changing. Our argument is a string of text that is put in quotes deliberately
    • The text in black is the actual output

Of course, there is much more to Python then this. However, this serves as an introduction for a future post.

Conclusion

Python is a popular programming language used in a variety of application. The source of its popularity has to do with it general-purpose philosophy. There’s a little bit of something for everybody in this language which encourages its use. Using the Spyder IDE will allow you to experience Python for the purpose of acquiring new skills.

Education in Ancient Rome

The Roman Empire was around in one form or another for over 1,000 years. To attempt to try and cover the educational approach of an empire over such a long period is not practical in a blog post. Instead, certain key ideas will be highlighted to provide a brief picture.

Background

The Romans had a war-like spirit due in part to the context in which they found themselves. They were surrounded by enemies on all sides and had no choice but to fight for their survival. This war context influence education in that the Romans were focused on a practical utilitarian education for their children. This is in stark contrast to the aesthetic education of the Greeks who loved beauty for beauty sake.

Another unique characteristic of Roman society was the status given to women. Women in Roman culture were often viewed as Queens of the Household and wielded tremendous power behind their husbands.

What they Taught

The Romans taught the same basic subjects of many other ancient cultures. Some of the subjects included reading, writing, math. grammar, poetry. However, due to their practical nature, the early Roman empire did not have a strong aesthetic culture. This came later as Rome began to absorb and imitate Greek life.

How Learning was Organized

Education was divided into three main stages of life. The first stage lasted from birth until about the age of 7 and was under the mother. Basic life skills were taught and not too much in terms of academics. Later, the mothers would reject this responsibility and leave their children in the care of a pedagogue but this did not happen until Rome began to decline.

From ages 7-12 a child went to elementary school and studied under a literature. Being a literator was often viewed negatively as someone who had failed in life. Therefore, primary education was full of washed up men. Corporal punishment was common as well and stern discipline was instilled.

From age 12-16 a boy would receive advanced training under a literatus. Unlike the primary teacher, the literator, the literatus was highly respected and could earn a great deal of money from his occupation.

At the age of 16, a boy was considered an adult and would pursue his life work which could be anything such as agriculture, law, politics, military, etc. were some of the many options available.

Conclusion

Roman education was focused on what was necessary to improve the practical life of the people. There quest for conquered lands help them to spread their influence over the entire planet.  Therefore, Rome is remembered for their sense of independence that is still remembered until this day.

Completing the Square

One method for solving quadratic equations is called completing the square. This approach is a little confusing but we will try to work through it together in this post.

What is Completing a Square

Completing the square is used when your quadratic equation is not a perfect square. Below is an example of a perfect square quartic formula =. The first is in the standard quadratic form and the second is after it has been simplified.

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However, not all equations are this easy, consider the example below.

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There is no quick way to factor this as there is no perfect square. We have to use something called the binomial square pattern.

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This is where it gets confusing but essential what the binomial square pattern is saying is that if you want to find the third term (b squared) you must take the second term and multiple it by1/2. We multiplied by 1/2 because this is the reciprocal of multiplying by 2 as shown in the equation. Lastly, we square this value. Below is the application of what we just learned from our problem equation.

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By taking the second term, multiplying by 1/2 and squaring it we were able to create the trinomial we needed to create the perfect square. By doing this we also solved for x if this was a full equation.

Examples

When using the completing the square approach with a quadratic formula there are some additional steps. We will work through an example below

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We are missing the third term and we need to find this first. Our second term is 8 so we will plug this in to find the third term

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We take this number 16 and add it to both sides which is a rule whenever manipulating an equation. Therefore, we get the following.

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We can now factor the left side as shown below.

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To remove the square we need to square root both sides. In other words, we are employing the use of the square root property.

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This leads to our two answers as shown below

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There are variations of this but they all involve just moving some numbers around before the steps shown here. As such, there is not much need to discuss them.

Conclusion

Completing the square provides a strategy for dealing with quadratic formulas that do not have a perfect square. Success with this technique requires identifying the terms you know and do not know and taking the appropriate steps to calculate the third term for the trinomial.

The Quadratic Formula

The quadratic formula is used for solving quadratic equations. The actual creation of this formula is somewhat complex. Creating it requires the use of completing the square as well as square root property. Below is what the equation looks like.

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For our purposes, we will go through an example that solves a quadratic equation using the quadratic formula. In addition, we will also explore the idea of the discriminant as it relates to quadratic formulas.

Example

The mechanics of solving a quadratic formula using this approach is similar to most other methods. You simply plug in the substitutes in the equation to get your actual answer. Below is an example,

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We will now plug in the values and determine x.

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Discriminant 

The discriminant of a quadratic equation is used to determine the type if the answer you would get if you solve the equation. THere are three types of answers that you can get when solving a quadratic equation.

  • Two real solutions-This happens when the discriminant results are positive.
  • One real solution-Happens when the discriminant results are zero
  • Two complex-Happens when the discriminant is negative

A complex solution involves the use of an imaginary number. This happens when the square root number is negative, which is technically impossible. To deal with this in math the letter i is used instead of the negative sign below is an example.

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The actual formula for calculating the discriminant is already in the quadratic formula. You simply calculate only the information under the square root. This is shown below.

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IN our first example, we got two real solutions. We will now confirm this by calculating the discriminant.

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Are answer is positive, which means that we can expect to calculate to real solutions for this particular problem.

Conclusion

The quadratic formula provides another way to solve a quadratic equation. This is probably the easiest method to learn as it is simply a matter of plugging numbers into the formula. This may explain why the quadratic formula is frequently the first method algebra students learn for solving quadratic equations.

The discriminant is a shortcut calculation that allows you to determine the quality of the solutions you would get if you solve the equation.

Education in Ancient Athens

In many ways, Athens is the home of Western thinking. Countless philosophers were either from Athens or at least spent time there. In this post, we will take a look at education in Ancient Athens.

Background

Athens is located in Central Greece and during antiquity had a population of about 500,000 with about 80% of this population being slaves. This huge disparity between freemen and slaves makes it more amazing that a population of only 100,000 could contribute so much to history.

Generally, slaves and women were not educated. It was considered embarrassing for women to obtain an education. It was the father’s responsibility to educate his son for usefulness.  Failure to do so meant the father forfeited whatever support his son would give him in old age.

The government was shaped largely by Solon. As a democracy, Greece was revolutionary for its time. Solon also established other laws such as outlawing the selling of children and requiring fathers to train their children.

What they Taught

The Athenian education was focused on aesthetics. The idea of beauty influenced everything that was taught.  Subjects taught in Ancient Athens included reading, writing, rhetoric, math, philosophy, music, and poetry. Music and poetry often worked together as poems were set to music. Music was viewed positively as a hobby but professional musicians were looked down on as common laborers.

Physical education was also rigorously taught as beauty was so important. Subjects include swimming, wrestling, running, jumping. One field of study that was often neglected was moral training. The Greek gods were not the best role models.

In place of morals, Greek boys were taught to be patriotic, respect religious rights, and generally to always strive to maintain a good appearance in public.

The teaching methods involved primarily transmission approaches. The teacher would read or say something and the student wrote it down. This was how most subjects were taught.

How they Organized Education

There were essential four levels of education in Ancient Greece. From 0-6 years of age, a boy was under domestic training under his mother or a nanny. Nannys were for the rich.

From 7-14 years of age, the boy was placed under a guardian called a pedagogue and sent to school. There he studied with private teachers the basics of education.

From 14-18 there was a split, the rich continue their education while the poor would branch off and focus on learning a trade from their fathers. For the rich, they would study more complex subjects such as philosophy or higher match. At 18 years of age, a boy would enter military service.

Conclusion

The education found in Ancient Athens was unique in its focus on aesthetics. However, there was at times an indifference to substance and there was almost no interest in moral development. However, educational systems have their flaws and even Ancient Athens is without exception in this regard.

Quadratic Equations and the Square Root Property

A quadratic equation is an equation that includes a variable raised to the second power. Below is a common format for a quadratic equation.

1

This characteristic makes it difficult to rely on linear equation tricks of addition, subtraction, and multiplying to isolate the variable. One trick that we often have to use now is factoring as shown below.

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An alternative way to solve quadratic functions is through having knowledge of the square root property which is shown below.

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Below is the same example as our first example but this time we use the square root property.

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This trick works for numbers that cannot be factored.

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This leads us to the point that the square root property is used for speed or when factoring is not an option.

With this knowledge, all the other possible ways to solve a linear equation can be used to solve a quadratic equation

Division

In the example below is a quadratic formula in which you have to divide to isolate the variable. From there you solve like always1.pngFraction

To remove a fraction you must multiply both sides by the reciprocal as shown below.

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When we got the square root of 18 we had to further simplify the radical by finding the factors of 18. In the second to last line if you multiply these numbers together you will get 18 because  9 * 2 = 18. Furthermore, if you square root 9 you get 3 but you cannot square root 2 and get a whole number. This is why the final answer is 3 * the square root of 2.

Conclusion

 

Quadratic formulas are common in algebra and as such there are many different ways to solve them. In this post, we looked at an alternative to factoring called the square root property. Understanding this approach is valuable as you can often solve quadratic equations faster and or they can be used when factoring is not possible.

The Life of Pythagoras

 

Pythagoras was a highly influential educator during the time of ancient Greece. In this post, we will take a brief look at his life and impact on education.

Early Life

Pythagoras was born around 570BC on the island of Samos. His early life was spent in private study. However, as a young man, Pythagoras traveled to Egypt to acquire additional education.

Down To Egypt

While in Egypt, Pythagoras studied with the Egyptian priest. The Egyptian priest were the masters of education in Egypt and was the only class in Egypt that received an advanced education. Under their tutelage, Pythagoras was exposed to various math and science subjects as well as some of the religious practices of Egypt. He was particularly touched by their way of life and it led him to develop his own style of living that would eventually be called Pythagoreanism.

After completing additional studies in Egpyt, Pythagoras moved to Italy and founded his own school. The school had essentially two levels which were the exoteric and esoteric. Students began in the exoteric studies and stayed there for at least 3 years. After completing exoteric studies a student would begin esoteric studies with Pythagoras himself.

In Italy

The subjects taught at Pythagoras’ school includes physics, geography, medicine, math and even metaphysics. In terms of math, it was Pythagoras who gave algebra students the Pythagorean theorem which states that the square of a  hypotenuse of a right triangle is equal to the sum of the square of the base and the square of the height as shown in the expression below

 base2 + height2 = hypotenuse2

Pythagoras also had distinct metaphysical views. He believed in one true called as a form of monotheism. This was in stark contrast to the commonly held beliefs of Greece at the time. Christianity was a rather strong religion at this time and it is possible that Pythagoras may have come into contact with this religion.

Pythagoras also believed in the transmigration of the soul. This essentially means that when an animal died they would come back as a lower animal. This is in many ways a form of reincarnation. It was simply another way of saying “You shall not really die” which was an idea shared in a garden by a snake to a woman.

Pythagoras’ school was known for being authoritative, strict, and even have a habit of being aristocratic. This along with other ideas made Pythagoras school unpopular. So unpopular that a mob would eventually burn his school down to the ground.

Conclusion

It is not clear if Pythagoras died in the flames or lived on as scholars are still debating this. What can be seen is that Pythagoras view of education has continued to live on to this day. His way of life had an influence on many people and his contribution to mathematics has touched the life of practical every algebra student on the planet.

Education in Ancient Sparta

With Ancient Greece there was a small city-state called Sparta. MAny today know of Sparta because of the movies that have been made of this war-like people. Spartan education was primarily a one about military training.

The reason for this emphasis on developing soldiers was due in part to the context in which the Spartans lived. In their own country, they were a minority with a large population of neighboring freeman and an even larger population of slaves. The only way in the Spartans minds to maintain power was through the use of strength. As such this was the focus of their education.

Background

The founder of the government of what makes up classical Sparta was Lycurgus. After spending time in Egypt Lycurgus came to Sparta and developed their constitution. Some of the practices he made lawe included the making all money out of iron to discourage greed and to require men to live in barracks together to encourage unity towards the military and state over the family.

By discouraging greed and familial affections Spartan men were focused on developing strength and military prowess almost to the exclusion of anything else. What else is there for a man to do when he cannot acquire wealth or enjoy his family?

One last point to mention is that children were seen as the property of the state. In a rather cruel way, weak children were eliminated at birth and only the strong were allowed to live. This further strengthens the idea of the state over family.

What They Learned

The training was primarily physical in nature. Young boys were taken from their homes at the age of 7 to live in the state barracks. Once there, they were given a minimum amount of clothes and food. The cold and hunger often compelled the boys to steal. Stealing was actually encourage as it taught stealth. However, being caught was punished severely because it indicated carelessness, which could prove deadly on the battlefield.

Gymnastics, wrestling, and the use of weapons were also emphasized. Despite the contradiction in encouraging stealing the Spartan education also strongly inculcated moral training as well. Boys were to control their appetites, respect the aged as well as their parents, and to be indifferent to suffering. It was considered shameful to lose control of one’s behavior in any way. This naturally discourages such behaviors as drunkness.

Unlike other ancient cultures, the Spartans loved music and spent a large of amount of free time developing this skill. Songs were frequently about war and brave acts.

Women also received an education and the focus was on the development of the physical nature.

How Were They Taught

Spartan boys were taught primarily by the senior citizens or the aged of the society. The old would spend time with the young boys. The common forms of instruction involved a question and answer format. This instilled a great deal of practical wisdom in the youth.

Another primary method of learning was imitation. Young people would learn simply through copying the actions and behaviors of the aged. This imitation of the aged rather than of other young people help Spartans to mature and develop a seriousness to them that would be hard to find in young people today.

COnclusion

The Spartans were a military culture with a strong state apparatus. Their educational system was developed to suppress the people around them in an attempt to maintain their own safety. This desire to survive contributed to a highly oppressive system from the viewpoint of an outsider but perhaps a saving grace for the Spartan.

Solving Single Radical Equations

This post will look at how to solve radical equations. The concepts are mostly similar to solving any other equation in terms of isolating terms etc. However, for people who are new to this, it may still be confusing. Therefore, we will go through several examples.

Example 1

Our first example is a basic radical equation that includes a constant outside the radical. Below is the equation.

1

Solving this problem requires to main steps.

  1. Isolate the radical
  2. Remove the radical by squaring it

Doing these two steps will lead to our answer. We will have two answers but the reason for this will become clear as we solve the equation.

First, we will isolate the radical by subtracting 1 from both sides

1

Now, to remove the radical we will square both sides. This new equation will need to be simplified and will become a quadratic equation.

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With our new quadratic equation we will factor this and as expected get two answers.

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Index Other than 2

For a radical that has an index other than  2, the process involves raising the radical to whatever power will cancel out the radical. Below is an example that has an index of 3. We will first subtract the constant from both sides.

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In order to remove the index of 3, we need to raise each side of the equation to the power of 3. After doing this, we solve a simple equation.

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Radicals as Fractions

One a number is a raised to a power that is a fraction it is the same as a radical. Below is an example.

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This means that the steps we took to solve equations with radicals can be mostly used to deal with equations with powers that are a fraction.

below is an equation. To solve this equation you must raise each side of the equation to the power of the denominator in the fraction.

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As you can see both sides were raised to the 4th power because that is the number in the denominator of the fraction. On the left side of the equation, the 4th power cancels out the fraction. Now you can simply solve the equation like any other equation.

1

Hopefully, this is clear.

Conclusion

Solving radical equatons in not that diffcult. Usually, the ultimate goal is to remove the radical. The difference between this and solving for other equations is that with radical equations you want to first isolate the radical, remove the radical, and then solve for the unknown variable.

Education in Ancient Egypt

Ancient Egypt is perhaps one of the oldest if not the oldest civilization on the planet. With a rich history dating several thousand years Egypt also had a reputation for education as well. This post will discuss education in Egypt with a focus on training by caste.

Background

Egypt was famous for their wisdom and architectural work. In terms of architecture, we are probably all familiar with the pyramids that are still standing after several thousand years. In terms of wisdom, Egypt was so highly regarded in the past that the Greeks sent several future philosophers and leaders to Egypt to study. Among those who went include Plato perhaps the greatest philosopher of all time, Lycurgus, the founder of Sparta, and Solon, the famous Athenian Statesman.

Egypt also had a strong caste system similar to India’s. There were essentially three classes. At the top were the priests, second, was the military, and the lowest classes was everybody else. The lowest class was also sub-divided into three subclasses of farmers/boatmen, then mechanics and tradesman, and lastly the herdsman, fishermen, and laborers. A person was born into their class and it was almost impossible to move from one to the other.

The priestly class was also exempt from taxes and owned as much as 1/3 of the land in Egypt. Their skills and training also commanded high salaries. Egypt was essentially a priests’ country in terms of status and privileges.

What They Study

The education an Egyptian received was heavily influenced by the caste they came from. The priest received the most extensive training. They studied philosophy, natural history, medicine, math, history, law, etc. With this training, a person from the priestly caste could be a physician, historian, surveyor, customs inspector, judge, counselor, etc.

Everyone else received a basic education depending on their occupation. Merchants learned how to read, write, and perform simple math. Tradesman only learned their trade from their parents.

The writing was also divided along class lines. THere were two types of writing systems. The Demotic style was for the masses while the Hieratic style was for the priestly class. The main difference between these two styles is the proportion of hieroglyphics used.

Two subjects commonly ignored in Egyptian education was gymnastics and music. Gymnastics was considered dangerous due to the risk of bodily harm. Music was considered to have an effminate influence on a man if studied to excess.

Conclusion

Ancient Egyptian was unique in terms of the dominance of the priestly class. The priest was allowed to study extensively while everyone else did not seem to enjoy the same access to education. This allowed the priest to wield tremendous informal power within Egypt and to quietly work behind the scenes to achieve goals

Roots, Radicands, and Radicals

Roots, radicands, and radicals are yet another way to express numbers in algebra. In this post, we will go over some basic terms to know.

Roots

A square root is a number that is multiplied by itself to get a new number. Below is an example

1

In the example above 5 is the square root of 25. This means that if you multiply 5 by its self you would get 25.

Another term to know is the square. The square is the result of multiplying a number by its self. In the expression above 25 is the square of 5 because you get 25 by multiplying 5 by its self.

Square Roots

Square roots, in particular, have a lot of other ways to be expressed. To understand square roots you need to know what roots, radicands, and radical sign are. Below is a picture of these three parts.

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The radical sign is simply a sign like multiplication and division are. The radicand is the number you want to simplify by finding a number that when multiplied by itself would equal the value in the radicand. We also call this new number the square root. For example,

1

What the example above means is that the number you can multiply by itself to get 100 is 10.

The index is trickier to understand. It tells you how many times to multiply the number by its self to get the radicand. If no number is there you assume the index is 2. Below is an expression with an index that is not 2.

1

What this expression is saying is that you can multiply 2 by its self 3 times to get eight as you can see below.

2 * 2  = 4 * 2 =  8

Additional Terms

There are some basic terms that are needed to understand using radicals. Generally, when every we are speaking of multiplying two times we call it square. Multiplying three times is referred to as cub or cubic. Anything beo=yond 3 is called to the nth powered. For example, multiplying a number by its self 4 times would be called to the 4th power, 5 times to the 5th power etc. However, some people referred to the square as the 2nd  power and the cube as the 3rd power if this is not already confusing. Below is a table that clarifies things

Number Power Example
2 square n2
3 cube n3
4 4th power n4
5 5th power n5

Conclusion

There are many more complex ideas and operations that can be performed with radicands and radicals. One of the primary benefits is that you can avoid dealing with decimals for many calculations when you understand how to manipulates these terms. As such, there actual are some benefits in understanding radicands and radicals use.

Education in Ancient China

As one of the oldest civilizations in the world, China has a rich past when it comes to education. This post will explore education in Ancient China by providing a brief overview of it. The following topics

  1. Background
  2. What was Taught
  3. How was it Taught
  4. The Organization of what was Taught
  5. The Evidence Students Provided of their Learning

Background

Ancient Chinese education is an interesting contrast. On the one hand, they were major innovators of some of the greatest invention of mankind which includes paper, printing, gunpowder, and the compass. On the other hand, Chinese education in the past was strongly collective in nature with heavy governmental control. There was extreme pressure to conform to ancient customs and independent deviate behavior was looked down upon.  Despite this, there as still innovation.

Most communities had a primary school and most major cities had a college. Completing university study was a great way to achieve a government position in ancient China.

What Did they Teach

Ancient Chinese education focused almost exclusively on Chinese Classics. By classics, it is meant the writings of mainly Confucius. Confucius emphasized strict obedience in a hierarchical setting. The order was loosely King, Father, Mother, then the child. Deference to authority was the ultimate duty of everyone. There is little surprise that the government support such an education that demanded obedience to them.

Another aspect of Confucius writings that was stressed was the Five Cardinal Virtues which were charity, justice, righteousness, sincerity, and conformity to tradition. This was the heart of the moral training that young people received. Even leaders needed to demonstrate these traits which limited abuses of power at times.

What China is also famous for in their ancient curriculum is what they did not teach.  Supposedly, they did not cover in great detail geography, history, math, science, or language. The focus was on Confucius apparently almost exclusively.

How Did they Teach

Ancient Chinese education was taught almost exclusively by rote memory. Students were expected to memorized large amounts of information.  This contributed to a focus on the conservation of knowledge rather than the expansion of it. If something new or unusual happened it was difficult to deal with since there was no prior way already developed to address it.

How was Learning Organized

School began at around 6-7 years of age in the local school. After completing studies at the local school. Some students went to the academy for additional studies.  From Academy, some students would go to university with the hopes of completing their studies to obtain a government position.

Generally,  the education was for male students as it was considered shameful to not educate a boy. Girls often did not go to school and often handle traditional roles in the home.

Evidence of Learning

Evidence of learning in the Chinese system was almost strictly through examinations. The examinations were exceedingly demanding and stressful. If a student was able to pass the gauntlet of rot memory exams he would achieve his dream of completing college and joining the prestigious Imperial Academy as a Mandarin.

Conclusion

Education in Ancient China was focused on memorization, tradition,  and examination. Even with this focus, Ancient China developed several inventions that have had a significant influence on the world. Explaining this will only lead to speculation but what can be said is that progress happens whether it is encouraged or not.

Polynomials

Polynomial is an expression that has more than one algebraic term. Below is an example,

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Of course, there is much more to polynomials then this simple definition. This post will explain how to deal with polynomials in various situations.

Add & Subtract

To add and subtract polynomials you must combine like terms. Below is an example1

All we did was combine the terms that had the y^2 in them. This, of course, applies to subtraction as well.

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In the example above, the terms with “a” in them are combined and the terms with “n” in then are combined.

Exponents

When dealing with exponents when multiplication is involved you add the exponents together.

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Notice how like terms were dealt with separately.

Since we add exponents during multiplication we subtract them during division.

1

The 2 in the numerator and denominator cancel out and  7 – 5 = 2.

When parentheses are involved it is a little more complicated. For example, when the exponent is negative you multiply the exponents below

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For negative exponents, you fill the numerator and denominator around and make the negative exponent positive.

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There are other concepts involving polynomials not covered here. Examples include long division with polynomials and synthetic division. These are fascinating concepts however in the books I have consulted, once these concepts are taught they are never used again in future chapters. Therefore, perhaps they are simply interesting but not commonly used in practice.

Conclusion

Understanding polynomials is critical to future success in algebra. As concepts become more advanced, it will seem as if you are always trying to simplify terms using concepts learned in relation to polynomials.

Education in Ancient Israel

The Nation of Israel as described in the Bible has a rich and long history of several thousand years. This particular group of people believed that they are the keepers of the knowledge of the true God. Their influence in religion is remarkable in that a large part of the theology of Christianity is derived from Hebrew writings.

In this post, we will only look at a cross-section of Hebrew education around the time of the time of the monarchy period of David and Solomon.

What Did they Teach

The goal of Hebrew education was to produce people who obeyed God. This is in stark contrast to other educational systems that emphasized obeying earthly rulers. The Hebrew system stress first allegiance to God and then allegiance to man when this did not conflict with the will of God. When there was a disagreement in terms of what man and God commanded the Hebrew was taught to obey  God. This thinking can be traced even in Christianity with the death of martyrs throughout Church history.

The educational system was heavily inspired by their sacred writings. At the time we are looking at, the majority of the writings were by Moses. The writings of Moses provide a detailed education of health principles, morality, and precise explanation of performing the rites of the sacrificial system.

The sacrificial system in the Hebrew economy is particularly impressive in that the ceremonies performed were all meant to help the Israelites remember what God had done for them and to be shadows of the life, death, and resurrection of Christ as understood by some Christian theologians.

In order for children to learn all of the laws and sacred writings of their nation, it required that almost everyone learn to read. People were held personally responsible to understand their role in society as well as in how to treat others and God’s will for them. Again this is in contrast to other religions in which people simply obeyed the religious leaders. The Jew was expected to know for themselves what their religion was about.

How Did They Teach

Despite the theocratic nature of the government and the details of the religious system, the educational system in Ancient Israel was highly decentralized. The school was the home and the teachers were the parents. Most nations that reached the strength and level of the Monarchy of Israel had a state ran educational system. However, the Hebrews never had this.

The decentralized nature of education is unusual because secular leaders normally want to mold the people to follow and obey them. In Israel, this never happens because of the focus on serving God. The personalized education allowed children to grow as individuals rather than as cogs in a nation-state machine. The idea of allowing parents to all educate their children as they decide would seem chaotic in today’s standardized world. Yet the Israel monarchy lasted as long as any other kingdom in the world.

How Did They Organize 

Once a child completed their studies they would learn a trade and begin working. Higher education was not focused on secular matters and was often reserved for the priestly class to learn skills related to their office. Example include law, sacred writings weights and measures, and astronomy to determine when the various feast days would be.

Another form of additional education was the Schools of the Prophets. Apparently, these were independent institutions that provided training in the scriptures, medicine, and law. At least one author claims that the Schools of the Prophets were established because Hebrew parents were neglecting the education of their children.  In other words, when the parents began to neglect the education of their children is when the nation begin to decline as well.

Conclusion

The Israelite educational system during the early monarchy period was an interesting example of contrasts. Highly detailed yet decentralized in execution, focused on obeying God yet having a monarchy that probably wanted to keep power, and little regard for higher education while producing some of the most profound theological works of all-time.  The strength of this system would be considered a weakness in many others.

Cramer’s Rule

Cramer’s rule is a method for solving a system of equations using the determinants. In order to do this, you must be familiar with matrices and row operations. Generally, it is really difficult to explain that is a simple matter but there are two main parts to completing this

Part 1:

  • Evaluate the determinants using the coefficients aka D
  • Evaluate the determinants using the constants in place of x aka Dx
  • Evaluate the determinant using the constants in place of y aka Dy

Part 2:

  • Find x by  Dx / D
  • Find y by Dy / D

This is modified if the system is 3 variables. Below we will go through an example with 2 variables.

Example

Here is our problem

2x + y = -4
3x – 2y = -6

Below is the matrix of the system of equations

1

We will first evaluate the Determinant D using the coefficients. In other words, we are going to calculate the determinant for the first two columns of the matrix. Below is the answer.

1
1

What we have just done we will do two more times. Once two find the determinant of x and once to find the determinant of y. When we say determinant x or y we are excluding that column from the 2×2 matrix. In other words, if I want to find the determinant of x I would exclude the x values from the 2×2 matrix when calculating. Below is the determinant of x.

1

1

Lastly, here is the determinant of y

1

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We no have all the information we need to solve for x and y. To find the answers we do the following

  • Dx / D = x
  • Dy / D = y

We know these value already so we plug them in as shown below.

1

You can plug in these values into the original equation for verification.

The steps we took here can also be applied to a 3 variable system of equation. In such a situation you would solve one additional determinant for z.

Conclusion

Solving a system of equations using Cramer’s rule is much faster and efficient than other methods. It also requires some additional knowledge of rows and matrices but the benefits far outweigh the challenge of learning some basic rules of row operations.

Classroom Discussion

Classroom discussion is a common yet critical aspect of the educational experience. For many, learning happenings not necessarily when students listen but also when students express their thoughts and opinions regarding a matter. This post will look at reasons for discussion, challenges, and ways to foster more discussion in the classroom.

Reasons for Discussion

Discussion is simply the flow of ideas between individuals and or groups. It is a two-way street in that both sides are actively expressing their ideas. This is how discussion varies from a lecture which is one-sided and most question and answers learning. In a discussion, people are sharing their thoughts almost in a democratic-like style.

Classroom discussion, of course, is focused specifically on helping students learn through interacting with each other and the teacher using this two-way form of communication.

Discussion can aid in the development of both thinking and affective skills. In terms, of thinking, classroom discussion helps students to use thinking skills from the various levels of Bloom’s taxonomy. Recalling, comparing, contrasting, evaluating, etc are all needed when sharing and defending ideas.

The affective domain relates to an individual’s attitude and morals. Discussion supports affective development through strengthing or changing a students attitude towards something. For example, it is common for students to hold strong opinions with little evidence. Through discussion, the matter and actually thinking it through critical students can realize that even if their position is not wrong it is not sufficiently supported.

Barriers and Solutions to Classroom Discussion

There are many common problems with leading discussions such as not understanding or failing to explain how a discussion should be conducted, focus on lower level questions, using the textbook for the content of a discussion, and the experience and attitude of the teacher.

Discussion is something everybody has done but may not exactly know how to do well. Teachers often do not understand exactly how to conduct a classroom discussion or, if they do understand it, they sometimes fail to explain it to their students. How to discuss should be at a minimum demonstrated before attempting to do it

Another problem is poor discussion questions.  The goal of a discussion is to have questions in which there are several potential responses. If the question has one answer, there is not much to discuss. Many teachers mistakenly believe that single answer questions constitute a discussion.

The expertise of the teacher and the textbook can also be problems. Students often believe that the teacher and the textbook are always right. This can stifle discussion in which the students need to share contrasting opinions. Students may be worried about looking silly if disagreeing, One way to deal with this is to encourage openness and trying to make content relevant to something it the students lives rather than abstract and objecive.

In addition, students need to know there are no right or wrong answers just answers that are carefully thought out or not thought out. This means that the teacher must restrain themselves from correcting ridiculous ideas if they are supported adequately and show careful thought.

Conclusion

Discussion happens first through example. As the teacher show how this can be done the students develop an understanding of the norms for this activity. The ultimate goal should always be for students to lead discussion independent of the teacher. This is consistent with autonomous learning which is the end goal of education for many teachers.

Evaluating the Determinant of a Matrix

There are several different ways to solve a system of equations. Another common method is Cranmer’s Rule. However, we cannot learn about Cranmer’s rule until we understand determinants.

Determinants are calculated from a square matrix, such as 2×2 or 3×3. In a 2×2 matrix, the determinant is calculating by taking the product of the diagonal and finding the difference.

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Here is how this look with real numbers

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Determinant for 3×3 Matrix

To find the determinants of a 3×3 matrix it takes more work.  By address, it is meant the row number and column letter. To calculate the determinant you must remove the row and column of that contains the variable you want to know the determinant too. Doing this creates what is called the minor. Below is an example with variables.

1

As you can see, to find the determinant of a1 we remove the row and the column that contains a1. From there, you do the same math as in a 2×2 matrix. When using real numbers you may need to add the row letter and column number to figure out what you are solving for. Below is an example with real numbers.

Find the determinant of c2

1

The number at c2 is -3. Therefore, we remove the row and the column that contains -3 and we are left with the minor of c2 shown below.

1

IF we follow the steps for a 2×2 matrix we can calculate the determinant of c2 as follows.

4(4) – (-2)(-2) =
16 – 12 =
4

The answer is 4.

Expand by Minors

Knowing the minor is not useful alone, The minor of different columns can be added together to find the determinant for a 3×3 matrix. Below is the expression for finding the determinant of 3×3 matrix.

1

What is happening here is that you find the determinant of a1 and multiply it by the value in a1. You do this again for b1 and c1. Lastly, you find the sum of this process to evaluate the determinant of the 3×3 matrix. Below is another matrix this time with actual numbers. We are going to expand from the first row and first column

1

All we do not is obtain the determinant of each 2×2 matrix and multiply it by the outside value before adding it all together. Below is the math.

 2(-4 – 0) + 3(-6 – 0) – 1(-3 – (2-2)) = 
-18 – 18 + 1= 
-25

Conclusion

This information is not as useful on its own as it is as a precursor to something else. The knowledge acquired here for finding determinants provides us with another way to approach a system of equations using matrices.

Making Tables with LaTeX

Tables are used to display information visually for a reader. They provide structure for the text in order to guide the comprehension of the reader. In this post, we will learn how to make basic tables.

Basic Table

For a beginner, the coding for developing a table is somewhat complex. Below is the code followed by the actual table. We will examine the code after you see it.

\documentclass{article}
\begin{document}
   \begin{tabular}{ccc}
      \hline
      Vegetables & Fruits & Nuts\\
      \hline
      lettace & mango & almond\\
      spinach & apple & cashews\\
      \hline
   \end{tabular}
\end{document}

1

We will now go through the code.

  • Line 1 is the preamble and tells LaTeX that we are making an article document class.
  • Line 2 is the declaration  to begin the document environment
  • Line 3 is where the table begins.  We create a tabular environment. IN the second set of curly braces we used the argument “ccc” this tells LaTeX to create 3 columns and each column should center the text. IF you wan left justification to use “l” and “r” for right justification
  • Line 4 uses the “\hline” declaration this draws the top horizontal line
  • Line 5 includes information for the first row of the columns. The information in the columns is separated by an ampersand ( & ) at the end of this information you use a double forward slash ( \\ ) to make the next row
  • Line 6 is a second “\hline” to close the header of the table
  • Line 7 & 8 are additional rows of information
  • Line 9 is the final “\hline” this is for the bottom of the table
  • Lines 10 & 11 close the tabular environment and the document

This is an absolute basic table. We made three columns with centered text with three rows as well.

Table with Caption

A table almost always has a caption in academics. The caption describes the contents of the table. We will use the example above but we need to add several lines of code. This is described below

  • We need to create a “table” environment. We will wrap this around the “tabular” environment
  • We need to use the “\caption” declaration with the name of the table inside curly braces after we end the “tabular” environment but before we end the table environment.
  • We will also add the “\centering” declaration near the top of the code so the caption is directly under the table

Below is the code followed by the example.

\documentclass{article}
\begin{document}
   \begin{table}
      \centering
      \begin{tabular}{ccc}
         \hline
            Vegetables & Fruits & Nuts\\
         \hline
            lettace & mango & almond\\
            spinach & apple & cashews\\
         \hline
      \end{tabular}
         \caption{Example Table}
   \end{table}
\end{document}

1.png

Conclusion

We explored how to develop a basic table in LaTeX. There are many more features and variations to how to do this. This post just provides some basic ideas on how to approach this.

Solving a System of Equations with Matrices: 2 Variables

This post will provide examples of solving a system of equations with 2 variables. The primary objective of using a matrix is to perform enough row operations until you achieve what is called row-echelon form. Row-echelon form is simply having ones all across the diagonal from the top left to the bottom right with zeros underneath the dia. Below is a picture of what this looks like

1.png

It is not necessary to have ones in the diagonal it simply preferred when possible. However, you must have the zeros underneath the diagonal in order to solve the system. Every zero represents a variable that was eliminated which helps in solving for the other variables.

Two-Variable System of Equations

Our first system is as follows

3x + 4y = 5
x + 2y = 1

Here is our system

1

Generally, for a 2X3 matrix, you start in the top left corner with the goal of converting this number into a 1.Then move to the second row of the first column and try to make this number a 0. Next, you move to the second column second row and try to make this a 1.

With this knowledge, the first-row operation we will do is flip the 2nd and 1st row. Doing this will give us a 1 in the upper left spot.

1.png

Now we want in the bottom left column where the 3 is currently at. To do this we need to multiply row 1 by -3 and then add row 1 to row 2. This will give us a 0.

1

We now need to deal with the middle row, bottom number, which is -2. To change this into a 1 we need to multiple rows to by the reciprocal of this which is -1/2.

1

If you look closely you will see that we have achieved row-echelon form. We have all 1s in the diagonal and only 0s under the diagonal.

Our new system of equations looks like the following

1x + 2y = 1
0x +1y = -1 or y = -1

If we substitute -1 for y in our top equation we can solove for x.

1

We now know that x = 3 and y = -1. This indicates that we have solved our system of equations using matrices and row operations.

Conclusion

Using matrices to solve a system of equations can be cumbersome. However, once this is mastered it can often be faster than other means. In addition, understanding matrices is critical to being able to appreciate complex machine learning algorithms that almost exclusively use matrices.

Education in Ancient Persia

The Persian Empire was one of the great empires of ancient civilization. It was this Empire that defeated the Babylonians. This post will provide a brief examination of the educational system of Persia.

Background

The religion of Persia was Zoroastrianism. The priestly class of Persia were called Magi. They responsible for sacred duties as well as the education of princes.

These are the same Magi that are found in the Bible in reference to the birth of Jesus. Due to their priestly responsibilities and knowledge of astronomy, this information merged to compel the Magi to head to Jerusalem to see Christ as a small child.

Teachers for the commoners were normally retired soliders. Exemption from the military began at the age of 50. At this age, if a male was able to live this long, he would turn his attention the education of the next generation.

What was Taught

The emphasis in Persian education was gymnastics, moral, and military training. The physical training was arduous, to say the least. Boys were pushed well nigh to their physical limits.

The moral training was also vigourously instilled. Boys were taught to have a strong understanding of right and wrong as well as a sense of justice. Cyrus the Great shared a story about how, as a boy, he was called to judge a case about coats. Apparently, a large student had a small coat and a small student had a large coat. The large student forced the small student to switch coats with him.

When Cyrus heard this story he decided that the large boy was right because both boys now had a coat that fitted him. The large boy had a large coat and the small boy had a small coat. However, Cyrus’ teacher was disappointed and beat him. Apparently, the question was not which coat fit which boy but rather which coat belonged to which boy.

Something that was neglected in ancient Persian education was basic literacy. The reading, writing, and arithmetic were taught at a minimal level. These skills were left for the Magi to learn almost exclusively.

How Was the Curriculum Organized

From the age of 0-7 education was in the home with the mother. From 7-15 boys were educated by the state and were even considered state property. After the age of 15, students spent time learning about justice in the marketplace.

Girls did not receive much of an education. Rather, they focused primarily on life in the home. This included raising small children and other domestic duties.

Conclusion

Persia education was one strongly dominated by the state. The purpose was primarily to mold boys into just, moral soldiers who could serve to defend and expand the empire. This system is not without merit as it held an empire together for several centuries. The saddest part may be the loss of individual freedom and expression at the expense of government will.

Making Diagram Trees with LaTeX

There are times when we want to depict hierarchical relationships in a diagram. Examples include workflow chart, organizational chart, or even a family tree. In such situations, the tree diagram feature in LaTeX can meet the need.

This post will provide an example of the development of a tree diagram used the tikz package. Specifically, we will make a vertical and a horizontal tree.

Vertical Tree

First I want you to see what our final product looks like before we go through each step to make it.

Screenshot 2018-04-17 09:02:18

As you can see it is a simple tree.

To develop the tree you need to setup the preamble with the following.

\documentclass{article}
\usepackage{tikz}
\begin{document}
\end{document}

There is nothing to see yet. All we did was set the documentclass to an article and load the tikz package which is the package we will use to make the tree.

The next step will be to make a tikzpicture environment. We also need to set some options for what we want our nodes to look like. A node is a created unit in a picture. In our completed example above there are 5 nodes or rectangles there. We have to set up how we want these nodes to look. You can set them individually or apply the same look for all nodes. We will apply the same look for all of our nodes using the every node/.style feature. Below is the initial setup for the nodes. Remeber this code goes after \begin{document} and before \end{document}

   \begin{tikzpicture}
      [sibling distance=10em,level distance=6em,
      every node/.style={shape=rectangle,draw,align=center}]
   \end{tikpicture}

The options we set are as follows

  • sibling distance = how far apart nodes on the same level are
  • level distance = how far apart nodes on different adjacent levels are
  • every node/.style = sets the shape and text alignment of all nodes

We are now ready to draw our tree. The first step is to draw the root branch below is the code. This code goes after the tikzpicture options but before \end{tikzpicture}.

\node{root branch};

Screenshot 2018-04-17 09:17:18

We will now draw our 1st child and grandchild. This can be somewhat complicated. You have to do the following

  • Remove the semicolon after {root branch}
  • Press enter and type child
  • make a curly brace and type node
  • make another curly brace and type 1st child and close this with a second curly brace
  • press enter and type child
  • type node and then a curly brace
  • type grandchild and close the curly braces three times
  • end with a semicolon

Below is the code followed by a picture

\node{root branch}
   child{node{1st child}
      child{node{grandchild}}};

Screenshot 2018-04-17 09:24:14

We now repeat this process for the second child and grandson. The key to success is keeping track of the curly braces and the semicolon. A Child node is always within another node with the exception of the root. The semicolon is always at the end of the code. Below is the code for the final vertical tree.

\node{root branch}
   child{node{1st child}
      child{node{grandchild}}}
   child{node{2nd child }
      child{node{grandchild}}};

Screenshot 2018-04-17 09:02:18.png

Horizontal tree

Horizontal trees follow all the same steps. To make a horizontal tree you need to add the argument “grow=right” to the options inside the brackets. Doing so and you will see the following.

Screenshot 2018-04-17 09:29:52

Conclusion

As you can see, make diagram trees is not overly complicated in LaTeX. The flexibility of the tikz package is truly amazing and it seems there are no limits to what you can develop with it for visual representations.