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News & Updates
August 27, 2023
Even in Ancient Times, Women Were Breaking Barriers
How the first recorded female mathematician Hypatia paved way for women in math and philosophy
The Great Alexandria
Many of us have been educated on humankind’s earliest civilizations. One of the more infamous yet mysterious communities we have collected knowledge on was Alexandria. It was a port city located in present-day northern Egypt, and one of the greatest Mediterranean cities to exist in our recorded history.
It was founded in approximately 331 BC by Alexander the Great, a fierce Macedonian ruler and military genius. Alexander left to pursue a takeover of nearby Persia, so Alexandria was left to be ruled by the Ptolemaic Dynasty for nearly three centuries; a famous ruler within this dynasty includes queen Cleopatra VII. Alexandria was known as the greatest city to ever exist and would later become a major hotspot for early Christianity, a center for religious turmoil from clashes between faiths.
A highly regarded figure that emerged from this time was true icon Hypatia.
Who was Hypatia?
Hypatia (355-415 AD) was one of the first well known feminine figures in philosophy, astronomy, and math.
“First” is a loosely applicable term in this discussion since the TRUE first female mathematician was Pandrosion, but Hypatia was the first to be well recorded and depict historical accuracy about one of the world’s most prolific civilizations.
She lived during the time of ancient Alexandria, but existed amidst an incredibly disorderly and violent era in the empire’s existence. Due to the turbulent chain of events that occurred during and after her time, it greatly accentuates Hypatia’s success to defy traditional odds inflicted upon Alexandrian women.
She was absolutely set up for greatness; her father was a well known mathematician, Theon of Alexandria (335-405 AD). It was most likely that she was taught and instructed by her father, which shows through the similarities in their separate works. Theon was most known for aiding in the preservation of Euclid’s Elements, a thirteen book collection encapsulating mathematical theories and proofs. Much of his work molded her own, as she took great effort in preserving the historical intellect of Greek heritage, especially in the fields of math and philosophy.
A well-loved Pagan, Hypatia was quite tolerant towards those of Christian faith.This is incredibly important being that she existed in a time where the Christians, Jews, and Pagans were all experiencing conflict with one another. Her well-composed demeanor helped establish a reputable relationship with the elite upper class; this dynamic even bled into responsibilities she held later in her life.
Hypatia’s realms of expertise were heavily male dominant fields of intellect, work, and discourse. Therefore, a woman entering that space with confident and productive contributions challenged the conventionality of Alexandria’s gender roles and dynamics.
What did she DO exactly?
Hypatia was credited with writing commentaries for various texts. She constructed a commentary focusing on Diophantus’ Arithmetica. This is a partially survived thirteen volume text harping on the development of number theory through equations. Her other prominent commentary revolved around Apollonius of Perga’s Conics. Conics is a dissertation about conic sections and their influence on modern and ancient analytical geometry.
Her commentaries greatly exemplified her high level of intelligence and understanding of math as well as philosophy. In fact, she was so well-versed that she became a very well-known
teacher at the Neoplatonic school of Alexandria; she taught philosophy and astronomy. This school grounded their particular education in the teachings of infamous figures Plato and Aristotle. It catered largely to both Christian and Pagan students, and many who were of the Pagan faith became loyal pupils and friends of Hypatia.
Aside from her educational endeavors, Hypatia also spent time constructing various tools for use in her fields, such as the hydrometer and astrolabes. Even though she didn’t invent them, she showed great familiarity with their functionality.
Hypatia’s great relationships with her pupils influenced the effects of historical events on her life. The religious divide in Alexandria was very tumultuous and chaotic, and possibly led to the burning of the Great Library of Alexandria. There were books from the Library believed to be kept in the Serapeum, a temple of the Greco-Roman god Serapis, led by Saint Theophilus of Alexandria. That temple was eventually destroyed as well due to its ties to the Great Library.
Coincidentally, Theophilus was associated with one of Hypatia’s devoted pupils, Synesius. These mutual ties temporarily permitted Hypatia to continue her work until her death. Once the theologian St. Cyril ascended into his power, he continued his uncle Theophilus’s work by enabling the violence against non-Christian civilians. Hypatia was gruesomely murdered by a group of fanatical Christians; Hypatia was a well-known Pagan and fell victim to their rioting.
Despite her demise, Hypatia remains a powerful feminist icon and influential mathematician. She represented great resilience, female empowerment and a perseverance to break the mold that’s been repeatedly inflicted upon feminine figures in the past and present.
August 20, 2023
Ancient Greece – Since The Beginning
How ancient Greece birthed an entire foundation of thought for modern day technology & intellect
Origins of Mathematical Knowledge
A multitude of classical and archaic communities paved the way for modern life and thought. Civilizations such as Rome, China, India, Egypt, Mesopotamia, Persia, and many more were able to start an array of practices- from SCRATCH. They were not privileged with many of the resources we have today, so they relied on futuristic thought and hard labor to create a life they were happy with (and proud of).
One of the fundamental empires is Greece, a civilization rich with philosophical thought, groundbreaking strategy, and a jubilant social nature. They provided a historical miscellany of concepts and contraptions, which lays ground for many of the practical modalities we modernized in order to build our own civilization. Humankind embodies many Greek contributions on an intellectual basis, and we don’t even know it!
There’s so much to dive into when exploring ancient Greek culture. Despite the ecosystem of knowledge they’ve nourished, there is a core contribution that initially planted the seed of their impact + influence on modern day thought.
Major Contributions from Ancient Greece
One of ancient Greece’s most influential contributions primarily involves the school of thought. Logic, philosophy, and academia was one of their strong suits, for many schools of thought were born and flourished. They all lay a foundation for other areas of thought to incubate. However, one of their greatest subjects of impact is mathematics.
Math is the backbone for many other areas of knowledge we’ve used such as Science, Astronomy, Architecture & Engineering, Warfare, even Agriculture. The Greeks needed to concoct a logical methodology to formulate tools, tricks, and processes that would build their empire as efficiently as possible.
LONG before modern technology, math was a slightly more laborious process. For instance, everything was written or through word of mouth; for instance, ancient civilizations kept charts and tables on clay tablets or papyrus scripts. And because the value of thought was so potent, the exchange was that much more impactful!
Between 685-525 BCE, (before the common era), Egypt’s ports along the Nile river opened up to Greek trade, breaking the barrier of interaction between them. With the migration of people & goods, both verbal and written communication acted as the vehicle to carry Egyptian ideas about math. That explains how and why much of Greek mathematics was adopted from the nearby civilization.
Egypt and their neighbors, like Mesopotamia and Ionia, had some of the finest math in the world. They utilized calculations for engineering purposes, to build structures for living and business such as the Great Pyramids or boats for trade & transportation. Unlike the others, the Greeks took these mathematical calculations to create practical applications for effective life skills.
Rigor was a major characteristic of Greek math. It was meticulous, exact, and at times super specific. They spent much of their effort contemplating deeper connotations behind the math they were working on. Even the word theorem evolved from the Greek word theoreo, which translates to “I contemplate”. Therefore, Greek math was intricately rooted in the association between mathematical review and analytical scrutiny.
Here are some of the most common contributions ancient Greek math gave to modern and even Western thought:
Ratios of a Triangle: Many of us have dabbled with the Pythagorean theorem, a tool proposed by one of ancient Greece’s most impactful mathematicians, Pythagoras. The 3:4:5 triangle was easily understood as a right triangle, but the Greeks were incredibly interested in the specificities of this abstract thought unlike their Egyptian benefactors.
Eventually, they expanded on it further by trying to calculate the longest side of the triangle (the hypotenuse) by calculating the similarity between the two smaller sides. This cracked open an intellectual revolution!
Numerical System: One of the most popular and widely-used tools the Greek created was their base system. By picking one core number, they formulated a number system for real-life usage that was easily divisible; this helped especially with fractions and proportions.
The ancient Greeks used the base number 60, which wasn’t as difficult to apply as we think. It’s a moderately divisible number with lots of other divisible factors, which made it a pretty flexible system to work with.
Square Root: This was an idea concocted a tad after the Pythagorean theorem swept civilization. With the new theorem begged a new question: if two sides of a right triangle are 1 unit, and the diagonal side equates to the square root of the two sides, what is it’s exact calculation?
After trying to find the square root of 2 and realizing it was irrational, this opened a world of questions regarding the square root of all numbers and what made them rational versus irrational.
Geometry: This is an entire discipline with math that works with the properties and relationships between lines, points, shapes, surfaces, and higher dimensional figures. With Greek architecture and engineering came the need for deep understanding of shapes and their dimensional properties.
Because it was a time of practicality, the Greeks were really using geometry as a logistical science to calculate land measurements. This was also a practice that originated with the Egyptian mathematical perspective; how do you think they built the Pyramids so beautifully?
Proofs: Known as one of the most tedious and difficult techniques to master, proofs are arguments based in inference and math logic to assure the answer to a problem is correct! Other theorems and math techniques can be applied to verify the validity of the proof, encouraging the practice of deductive reasoning with logic.
The first mathematical proof was credited to another Greek math icon, Thales of Miletus. He also proposed proofs that concerned ALL mathematical shapes and figures, not just the abstract ones! His contributions kickstarted the discussion of what the Universe was made of.
August 13, 2023
Math Fun
Have Your Fun, and Learn From It Too! — Why Games Can Make for a GREAT Option to Optimize Math Review
Let’s Have Some FUN
To sharpen learning retention in kids, it’s highly advantageous to have their full attention. And it’s even more advantageous to express the learning review in a way that’s digestible yet productive.
This is especially applicable and prevalent in math education, a discipline that requires a lot of practice and logistical awareness for the functionality of the techniques being taught. And hey, that can be unrealistic when you’re constantly lecturing them. Talking AT the students about math, or whatever they’re learning, doesn’t fully communicate the knowledge they need in order to understand what they’re being told. And even when you work one-on-one, instilling a two way path of communication about math, it may feel like pulling teeth with some kids. Admittedly, it isn’t the most enthralling of subjects for some young minds, and their resistance shows.
Another concern regarding communicable math review is the style in which you provide it. When addressing a class collectively, you’re catering to all kinds of learning styles. This can mean you have a cultivation of visual, auditory, kinesthetic, and even audio-visual learners. And although they each have individualized methods of retaining information, it is your job as an educator (or guardian) to mold a comprehensive yet constructive way to deliver the knowledge you need to.
Imagine being able to provide your students a tool, modality, or resource that not only TEACHES them the math they need, but grabs their attention in a positive and engaging way. Wouldn’t that be just AWESOME?
What’s the solution, you ask? Certainly NOT dumping a heavy review on them with no direction or energetic buffer. Instead, a better approach may be to keep it lighter in deliverance!
It’s become increasingly popular to include more lighthearted, enjoyable math reviews in the classroom through games. Yep, that’s right- games! Board games, computer games, CARD GAMES… that enjoyable pastime we highly encourage kids to participate in. And because it receives a moderately positive response from kids, it’s a prime candidate for utilization in school!
Why are they so beneficial you ask? Well, the primary appeal of games is that they’re designed to be lightly competitive but enjoyable. Not only that, they’re also easy to grasp once you play a few times (at least kid-friendly games are). If you were to find a way to incorporate math into a widely-known game structure, you create a refreshing review that actually gets the students excited to try it!
And what students love most about games is the childlike creativity or fun! If you bring a recognizable character, storyline, even movie into your classroom and use it to communicate your teaching, the students may grasp it more attentively! They naturally gravitate towards things they find enjoyable.
What games are best to use in a classroom? Well, that’s entirely situational.
If you have a smaller class size, you can split the class evenly and do multiple rounds of a one-on-one style game. For the larger-sized classes, team games are always a great way to instill community, teamwork, accountability, and enthusiasm! Students always receive teams well because they don’t want to feel isolated in their learning, and a few minds are always better than just one.
Ask yourself about these important factors when creating math review games:
Is this a game that is familiar to my students, or at least digestible in terms of difficulty?
If I give them the clear-cut directions and allow a few practice rounds, will they be able to effectively retain the knowledge while participating in the game?
Does this game have any offensive, discriminatory, or inappopriate content originally included that I can take out for my purposes?
Does this game ensure I can implement the exact math I’m teaching relatively easily?
What are some clear cut boundaries I need to set with my students?
After careful consideration, outline a game that you believe will deliver what you’re aiming to.
And if you’re looking for some guidance, look no further…
Games for Inspiration
Trivia/Jeopardy: Split the class into a few medium sized teams and create trivia style questions about math. You can give them whiteboards or paper to work out the problem on the screen, all the while collaborating with their classmates. Whoever has the correct answer first wins whatever reward was assigned to the question! This also broadens their knowledge of the discipline overall. A popular electronic trivia game is Kahoot!
What’s Missing: Memory is such an integral skill in math, and it should be exercised to the best of its ability! Present an equation, or visual representation of a problem (this is a great game for geometry-based questions) and ask the students what step or factor is missing! This helps them increase awareness for problem solving as well as formula-based questions.
True or False: Present a math equation or world problem (bonus if it’s on your assessments) and give the students an answer. It’s up to them to work on the problem on their own and decide if the answer you originally gave them was the correct one or a false one. This is also a great opportunity to work through the problem as a class AFTER you tell them if it’s true or false; some students may get it wrong, so if you show them the mistakes or misconceptions they’ll retain it. And this also helps them create a growth mindset!
Match the Equation: Matching card games are super quick, easy, and fun! You can create a deck that splits into: problems vs. answers, formulas vs. what they’re meant for, shapes vs. dimensions (size, shape, sides, etc). This is a flexible game style that’s suitable for all ages!
August 6, 2023
Watering Your Money Tree: The Money Mindset
How Implementing Basic Math Skills Can Greatly Improve Your Money Mindset
A Different Kind of Green Energy
Money is boundless energy proven quite influential to our current socioeconomic systems. Everything seems to come at a price, and drives much of how we contribute to the collective with work and life.
And yet, it’s able to be solved, deciphered, even managed with the use of common math. It’s no wonder why accounting, marketing, finance, and many more industries are booming- they utilize this core piece of our earliest education to structure it!
Let’s not be blind to the similarities; math and money are both numerical, logistical, and quantitative disciplines. If we didn’t have math to help us budget our personal finances or calculate economic relations, where would we be?
Which is one of the many MANY reasons why teaching math in a comprehensive way is super important. Money, and the mindset around money, has become a refreshingly popular topic of discussion these last few years. With a handful of economic crashes, loss of jobs, and the world in turmoil amidst a global pandemic, many people are coming to realize they’ve been perceiving their money all wrong.
So, What Exactly IS a Money Mindset?
And more importantly… WHERE does it come from?
A money mindset is your individualized collection of beliefs of money that fuel much of your attitude towards it. It forms long before you even begin to start handling your own finances, brewing inside the subconscious of your mind.
The sociologist Morris Massey theorized that there are three major periods of life in which life values, ideas, and perceptions are developed. There’s the Imprint period, Modeling period, and Socialization Period. Why is this important?
Well, between ages zero to seven years of age (the Imprint period), the human brain heavily resembles a sponge that absorbs any information it encounters. Much of your money mindset was developed there. You were exposed to others’ predisposed and existing money beliefs that were either expressed or inflicted upon you, especially by those who raised you.
While you may also acquire new or similar beliefs around money, the most profound influences on money occur during these years when your mind is most impressionable. And you may not even realize it, but you carry those impressions into your everyday monetary beliefs and behaviors. You subconsciously harbor an attachment style to your money, how you use it, and the emotions that arise; money is seen as a resource many use to take out their stress or success.
Essentially, the mindset you harbor and subsequently channel when making financial decisions like purchases and investments all have to do with your mindset around money itself. And you know what? It’s hard. Sometimes it can be brutal. It’s difficult to escape something that’s been conditioned in you since your earliest core memories- but you can alter them consciously to slowly assimilate or revitalize your entire view.
With the growth of knowledge through technology, it’s become significantly easier to be re-educated on how to manage your money. There are coaches, courses, media influencers, and so many more looking to enhance our financial consciousness. Because even though you live in a capitalist society with very clear-cut ideas about how money can be made and used, it may not always fit your values, needs, or background.
And even though how you use money is much more in your mind than in math, you can still incorporate mathematical skills and strategies in order to increase your awareness, and therefore your financial security.
Bring On the Math
On a fundamental level, using math can help you organize, plan, and allot for many different kinds of expenses! You can manage a business (if you’re an owner of one), save up for big life investments (car, house, college fund, etc.), put some money into stock, and even budget for a new item or experience you’ve been DYING to acquire.
Something critically important to remember about money is that it is an energy just as much as it is a resource. It is natural for it to come and go, and much of how you react or respond to these transactions will affect the ebb & flow of your financial wealth.
In terms of actual math skills, basic arithmetic such as addition, subtraction, multiplication, and division are practical starting methods. You can accumulate various answers for an assortment of situations; the use of arithmetic can help you calculate static or consistent numbers if you’re working with a steady income or payment.
Arithmetic is the more obvious and digestible math to use, but there are plenty of people who balance their finances with advanced strategies- and the help of technology. If you do not have access to such resources, then freshening up your mental math skills (and getting a calculator) is a great place to start.
If you’re beginning the journey to healing or bettering your money mindset, try keeping a money journal or spreadsheet (if you have access to a computer). This will be the place to record your daily and weekly expenses, which can help you craft your monthly budgets!
By keeping a detailed account of where your money goes and returns, it’ll increase your awareness. And when we broaden the scope of our awareness, we tend to feel more empowered and confident in taking aligned action.