October 8, 2023
Measuring Up
The History of Geometry as a Branch of Mathematics in Classical Antiquity
Math has been around longer than you think. Here’s a little inside scoop to the early stages of one of our most common math practices.
As we all know, many disciplines within Mathematics cater to particular scenarios, theories, and areas of quantitative knowledge. One of the more common and foundational disciplines aside from algebra would be geometry!
Like algebra, geometry is one of the oldest branches of mathematics still being utilized today! Merriam Webster’s formal definition of Geometry is articulated as “a branch of mathematics that deals with the measurement, properties, and relationships of points, lines, angles, surfaces, and solids”. Many of us initially learn the concept of geometry through shapes and graphs, but it seeps into so many more configurations! The purpose of geometry is to determine spatial relationships in our real-life scenarios and environments.
Where did geometry come from, you ask? Well, the term geometry originates from the two Greek phrases “gēo” (“Earth”) and “metron” (measurement). The Greek historian Herodotus (484-425 BC) was the one to credit Egypt with the subject’s origination. However, the Babylonian, Chinese and Hindu civilizations were the first to put it into practice; it was passed to the Greeks and Egyptians who popularized it through their obvious technological and cultural advancements.
If you’re familiar with the Babylonians, they were the ones to utilize clay tablets as an early form of documentation. Some of the tables found on said tablets indicate the need for square roots, the area of various polygons, even reciprocals. Not only that but it is believed the Babylonians were also the first to calculate measurements of a circle, specifically the circumference. This instigated the long process of discovering the infamous, infinite number of pi (π).
Unlike the Babylonians, the Egyptians kept records on papyrus scrolls. The Egyptians utilized geometry primarily for land surveying and construction; that’s how the Pyramids came into fruition! These groups of people were cognitively able to create and maintain streamlined systems of production! Geometry is such an elemental factor in the construction of our lives, it’s no wonder many ancient civilizations made great use of it as well.
The early Greeks were the ones to adopt geometry as a more rigid process. Greek philosopher Thales of Miletus (620-546 BC) was initially credited with bringing math from Egypt to Greece. After inhabiting the subject from their predecessors and neighboring communities, the early Greeks began prioritizing reasoning over results. Their emphasis on logic is incredibly representative in many influential works by early Greeks mathematicians.
One of the most infamous Greek documentations of math is known as Euclid’s Elements. It is a collection of 13 books the mathematician Euclid (appx. 300 B.C.) contrived. This series was intended to exemplify the functionality of many geometrical strategies. Plane geometry, geometric algebra, the geometry of a circle, elementary number theory, and proportions are just a few of the many subjects covered.
Euclidian math clenched the reins on math education for many centuries, yet many other Greek mathematicians were eager to contribute newfound knowledge. Infamous figures such as Pythagoras and Archimedes were able to provide foundational parts of geometry we still use today!
Because geometry was a perfect strategy to determine the location and environmental factors, the Greeks incorporated it into other practices such as astronomy. They philosophized so much about the universe they orbited, therefore the Greeks felt compelled to calculate what we now recognize as our solar system.
The most famous mind in early astronomy and classical antiquity is Ptolemy (circa 2nd century B.C.). Ptolemy was an Egyptian astronomer well known for his advancements in the model of our universe. He argued that our Earth was the center of the universe, thus formulated the geocentric solar system; this is what’s currently known as the Ptolemaic system.
The application of geometry in astronomy consumed much of Greek thought. Their passion and prioritization of logic over result fueled much of their fire for many of their geocentric theories. Much later on, Greek cosmologists were the ones to apply the practicality of geometry to the Earth’s measurements and its orbital cycle in terms of time, location, even season.
Many centuries passed before other civilizations throughout Asia and Europe were able to get their grip on geometry. Many geocentric Greek theories were debunked, and other communities took it upon themselves to study astronomy for results more than reason. Geometry has even gone on to evolve into different kinds, such as analytical geometry and progressive geometry.
Universally applying these dimensional techniques was to not only grow humankind’s knowledge but the many empires. Without the study of these dimensional elements in our ever-growing reality, much of our structural and technological integrity would be lacking today.
October 1, 2023
Out of This World – How Math Helped Formulate Astronomy
Through the early mathematical and philosophical learnings made by classical and ancient civilizations, we’ve come to better acquaint ourselves with the machinations of our Universe.
It’s no secret that much of our knowledge of the Universe was closely related to the fundamental discoveries both in Math and Science. Astronomy has proven especially resourceful in how we’ve been able to comprehend our place in the never-ending cosmos.
Despite the few early misconceptions, there have been many astronomers, mathematicians, and philosophers who’ve produced profound learnings we still apply today. Some of the most influential minds in math and astronomy derived from classical and ancient civilizations such as Greece, Rome, Babylonia, Egypt, even China, and India.
There’s distinctive material evidence to prove Babylonia was among the first to adopt mathematical practices when studying celestial occurrences. Back when they utilized cuneiform tablets (a pictographic system) for writing and recording, the Babylonians left behind various charts to indicate they were geometrically calculating astronomical placements!
The Babylonians, around 300 to 400 BC, started to use math as a way to calculate what is now known as the Zodiac chart! They divided the path of the Sun, Moon, and planets equally into 12 “phases”, then named them based upon nearby constellations. This was one of their many ways to compute where the planets, Sun, and Moon were which helped them eventually decipher what time of year it was!
It was one of the first and more advanced exemplifications of math being applied to astronomical affairs, but certainly not the last. Their early strategies centered around mathematical astronomy helped them calculate predictions of the position and track of Jupiter’s orbit!
The Babylonians, as well as the Egyptians, went on to adopt early astronomical calculations to formulate advanced calendars. They were the civilizations to primarily adopt astrology to better understand the machinations of our Universe. Unlike the earlier Mesopotamian communities, the Romans and Greeks used math and astronomy for other purposes.
Early Romans were very familiar with celestial beings! Ptolemy, a prominent Egyptian philosopher (when Egypt was under Rome’s rule) was most known for formulating an Earth-centric depiction of our solar system. Ptolemy’s findings were so heavily inspirational to the Romans, they even created mythological figures to represent their understanding of the planets!
The ancient Romans interpreted the role of the seven planets (they knew at the time): the Moon, the Sun, Mercury, Venus, Jupiter, Mars, and Saturn. It was the Greeks who ended up naming the planets, however, the Romans applied their own gods’ names to fit their narrative. We typically refer to these planets as their Roman names:
- Mercury (aka Mercurius): named after the god of commerce, eloquence, travelers,
- Venus: named after the goddess of love for its bright light and softer appearance
- Mars: for the god of war, rightfully named due to the planet’s bold red color
- Jupiter: the biggest planet in the system named for the head Roman deity, the god of sky and thunder, and King of the Roman gods in their mythology
- Saturn: the father of Jupiter, god of agriculture. Fun fact: based on Ptolemy’s model, Saturn is also named after Saturday (Saturn’s day)
Once the other planets in our solar system were discovered, they also received names in conjunction with Roman mythology. Uranus was named for the Roman personification of the sky, while Neptune was named after the god of the sea for its beautiful blue-green hue.
The Romans weren’t just accredited with naming the solar system; they created our current and most used yearly calendar! Before they adopted a prominent amount of Greek astronomy, the Romans paid meticulous attention to the placements of stars and planets in the sky to determine the lunar cycle. This aided them in growing a ten-month cycle to the twelve-month cycle.
Much of our current understanding of astronomy would be nowhere without the Greeks. They were able to provide many other civilizations with specific findings of more minute planetary aspects at the time- for instance, our own Earth. Because the helio-centric depiction of our solar system didn’t arise until the discoveries of Copernicus (1473-1543), the Greeks took observational astronomy to try and detail more about Earth.
One of Greece’s innovative philosophers and mathematicians, Pythagoras, pushed many discoveries forward with his work. However, he based many of his revelations on mathematical perfectionism rather than genuine quantitative reasoning. For instance, Pythagoras was the first to propose that the Earth was spherical not because it made “sense”, but because the sphere is considered a “perfect” 3D shape.
Pythagoras was not the only Greek to hypothesize major astronomical functions. Along came Aristarchus of Samos (310-230 BC) with his much more realistic contributions revolving around the Earth’s role in our Universe. He was the inspiration for Copernicus’s work.
Aristarchus focused mostly on the movement of the Earth and its size concerning both the Sun and Moon through eclipses! He procured three core premises that helped him articulate his findings.
While observing a lunar eclipse, Aristarchus confirmed through geometrical analysis that the size of Earth’s shadow on the moon further proves that the Sun is of greater size than the Earth. Although Aristarchus made true statements about the measurements of our planet & its surroundings, he still followed the inaccurate geocentric model of our solar system (rather than the heliocentric one).
Many many years passed before math and astronomy were applied to the RIGHT depiction of our solar system. Thanks to Copernicus and his proposal of an accurate heliocentric model, we were able to better detail why our Earth experiences the natural occurrences it does and how it affects other planets in our system.
It took quite a long time, and many other astronomers, to apply relevant math to confirm our place in the Universe. However, without the ancient and classical civilizations passionate about investigating our vast world, we would not have a strong basis to formulate our findings.
September 24, 2023
What’s The Big Deal About Math?
What really makes this educational discipline so integral to our learning as humans?
Math, as a whole, has become such an intrinsic part of comprehensive education.
Without the calculated use of numbers, shapes, equations, and so much more- we wouldn’t have made nearly as many advancements we use in our daily lives today! Isn’t that crazy?
Although math is a subject many people have a hard time grasping, it’s greatly applicable to assure our general wellbeing. Those who are more adept in the field have the power to alter the course of our evolution- FOREVER. And even those who don’t choose to make a career out of it can still possess applicable life skills to make their daily living that much easier.
Math was originally discovered and used to construct empires and their systems of living in ancient times. Nowadays, it’s applied to nearly everything from cooking to architecture to dance choreography. We measure and calculate and articulate so many things in our habitual lifestyle it’s easy to overlook how much math we exercise consistently.
What makes math SO important, you ask?
Well, here are six core reasons why Math is incredibly important to include in a diversified model of education:
- Math promotes critical thinking skills: Critical thinking is an important skill covering a breadth of other cognitive abilities such as analysis, interpretation, inference, explanation, self-regulation, open-mindedness, and problem-solving. Math utilizes its kind of critical thinking to get students (or anyone using it) to brainstorm reasoned decisions to approach particular problems. Rather than guessing or applying a rule without testing its validity, they have to be able to concoct educated judgments that best suit the necessary solution to the problem.
- Truth be told, STEM is the future: STEM includes varying math-oriented disciplines such as science, engineering, and technology. These are fields able to help our technological evolution progress, so it’s incredibly important to implement the fundamental learnings that provide the solid basis for pursuing and studying such prominent endeavors.
- Logical thinking is a communicable life skill: Unlike critical thinking, logical thinking is how to evaluate the processes behind the reasoning. Critical thinking uses logic and its core principles to separate truth from falsehood. However, it doesn’t focus on said reasoning specifically. Solidifying a strong math education, especially for those at a young age, can empower them to make creative yet careful decisions.
- Math productively teaches about failure: Look, we’ve all experienced an excruciatingly difficult math problem we SWORE we would be unable to solve. But sure enough, that’s one of the most crucial lessons math teaches us- failure is the opportunity to learn even more. When someone fails to solve a problem, they have the option to give up or attempt the problem again. If they choose to try again, this strengthens their learning process; if not, then they’re stuck in a limiting belief they can’t achieve a resolution. Math subtly emphasizes that failure is a part of reality, and we have the choice to turn it into something greater.
- Math ALSO exemplifies the importance of a growth mindset: Math has the power to teach us that competence is an ever-evolving process- NOT a limited set of skills. Combining a growth mindset with the reality of failure prepares young students to attack their decision-making with patience and an awareness that they have the power to reach their end goal! These are two very essential skills in optimal social, emotional, and cognitive development.
- It helps strengthen financial literacy skills for the future: There comes a day where many of us are given the choice to inhabit the responsibility of supporting ourselves (and in some cases, others). Math is so deeply ingrained into financial processes, it’s unavoidable when taking on those monetary tasks. If you assess and incorporate strong math education at an early age, financial literacy is much more likely to be retained at an easier and faster rate!
Mathematics is so deeply incorporated into our daily functions, and the mechanic behind our most used tools, that it’s nearly impossible to avoid. Our societal standard of success is based greatly upon our inherent ability to make informed, sound, logical decisions; we don’t even realize how much math helps us to condition this technique!
If you encounter a young student with a difficulty in learning math, now is the time to encourage these key principles. Math can be taught in environments outside of the traditional classroom, such as the kitchen or certain extracurriculars. Although that young child may not NEED math for their future professional endeavors, it’s still going to be a functional skill once they reach adulthood.
Even though math is a meticulous discipline, it has the power to better our means of thriving. And without it, who knows where we’d all be!
September 17, 2023
Let’s Go Over This Again, Shall We?
Why Review in Math Education is ESSENTIAL for Student Retention- and Good Grades
It’s very common among educational institutions that some students will present a much more difficult time retaining learning. Whether it be a learning divergency, inability to pay attention, or the content itself is too difficult to grasp- many students all over the world struggle with education retention.
It isn’t entirely their fault. Young minds are so malleable they all have various ways of learning new content. Memory is an especially intricate function of the human brain; it only makes sense that a young child’s ability to remember is slightly warped due to their lack of biological maturity as well as experience. They take in SO much information throughout a day, it’s only rational to understand the overwhelm they must feel! According to an article by Neuron, forgetting is “a critical component of a healthy mnemonic system”.
However, the instructors and teachers in place have a responsibility to best try and coordinate different modalities for learning to ensure every student is receiving what they need.
What To Do About Forgetfulness in Education
A very traditional yet consistently successful way to increase learning retention is constant, even meticulous review. This has proven to be especially beneficial when it comes to teaching young students Math. Math is one of the more objectively logical subjects in school, with a multitude of tiny moving parts and rules.
Because math is a particularly dense yet detailed subject, with LOTS of steps to completing certain problems, children have a hard time keeping it all fresh and at their disposal.
Extended breaks typically diminish the potency of learned knowledge, which makes review all that more beneficial. To review means to look back at all the lessons taught, read through, maybe even practice problems again and again for optimal comprehension.
Math has been proven a subject that strongly implicates the importance of a growth mindset. This is especially representative through the learnedness of the review!
In math, a student is given the option to attack a failed problem or equation two ways: try again with a new approach, or disregard it entirely. How does this tie into Math review?
If a student (given the proper resources) REVIEWS the mathematical fundamentals they’re currently learning, they’ll eventually discover the flaws in their problem solving, thus being more motivated to try again with a more developed solution. Converging review with a growth mindset encourages the students to practice patience, resilience, and a more optimistic outlook.
Additionally, the review is representative of a growth mindset because it encourages students to look outside themselves for resources (when available).
The vital aspect of the review is to be able to connect the content. If a student has an upcoming assessment with more than a few mathematical techniques being addressed, then they must be provided the insight into how they’re all related. The more the students are assisted in recognizing the interconnectedness of math, the more retention they constitute.
Fun Ways to Implement Math Review
Yes, there are FUN and exciting ways to integrate math review into a student’s coursework. There is a multitude of approaches, ranging from collaborative work to games even to additional class resources!
Here are just some (of the many) ways to enjoyable implement math review- in a classroom AND at home!
Games: Everybody loves a good game! Math is not everyone’s favorite subject to learn at school, so incorporating fun into the learning process makes it all that more digestible. These types of reviews can vary based on inspiration, like Math Jeopardy or Math Candyland where each student has to complete a problem to advance on the board. A Math trivia is also a great option- Kahoot has become a very fun option for classroom review!
Collaborative Work + Peer Review: It’s amazing what kids can learn from one another. Throughout a unit of lessons (that’ll be included on the same assessment), continuously include time for peer work on classwork problems so they can go through them TOGETHER. This is also a wonderful opportunity to strengthen their collaborative teamwork skills!
Extra Help Opportunities: Sometimes, the best resource is the teacher. If you are an instructor of any sort, try to allot time for students to have the option for extra help. Not only will this provide wonderful one-on-one focus to address the specific obstacles of each student, but it’ll strengthen trust in the class overall!
Study Guides: Depending on the age of the student, take-home study guides can work wonders! Teachers can subliminally add questions they plan to include on a test, especially the more difficult ones. Study guides are a great, direct framework for what you expect the students to know and complete!
Access to Classroom Resources: This can also vary depending upon the classroom. For some students, they learn best when they can look back at the notes and reread what they were taught. If the students are at an appropriate age to access work via computers, create a page entirely dedicated to your Powerpoints (or in-class exercises) as a review option!
Computer Games: This is a GREAT At-Home tool! Kids of many ages and levels of math difficulty can do this. There is an assortment of computer games online that a child can access to review what they’re learning. If you know your students have utilized Kahoot for review, this is something they can access at home (given computer access). Additionally, smaller and younger children who enjoy games will probably experience less resistance to math learning if there is fun included.
An Important Factor To Consider
As recently stated, there are going to be students who don’t receive Math well if they don’t particularly feel passionate about the subject. That’s why in any learning or review capacity it’s important to incorporate Math problems students will ENJOY. Regardless of what review is being implemented, students typically respond better when they can relate to what’s being given. This can range from (consensually) including students’ names in word problems, adding their favorite characters to problems, or finding acronyms and fun nicknames to describe mathematical concepts.
September 10, 2023
Put Your Mind To It
How The Use of Mindfulness Can Benefit Math Students
Mindfulness encapsulates an overall state of being. Generally, it is the human capacity to be conscientious of our existence, more specifically the emotional and mental sensations we experience. This cultivates a varying amount of different skills; for instance, someone who practices mindfulness is present in the moment, intensely sensing their feelings, being reflective, and many more intuitive behaviors.
This is a specific kind of meditation many people practice. However, mindfulness in itself is a quality whereas meditation is an action. You can acquire mindfulness THROUGH meditation; all in all, the two can exist independently or together.
Having a moment-by-moment awareness with a more nurturing approach is incrementally beneficial. Rather than focusing on the future or past to gain a sense of control over your reality, practicing mindfulness relieves you of stress that your focus is entirely invested in the PRESENT. Reducing that anticipatory rumination will alleviate feelings of frustration, stress, anger, and even helplessness. Mindfulness is also especially helpful to increase metacognitive awareness, or the awareness for how you think.
Why does mindfulness matter in the first place? Because it’s an avenue one can take to overall improve their mental and/or emotional health. Mindfulness bleeds into all sorts of life experiences, like interpersonal relationships. Aside from the external benefits, mindfulness can overall better one’s relationship or perception of themselves.
There have been studies and discussions about the physical benefits of mindfulness as well, but the primary influence is within the conscious self.
How does Mindfulness help with Math?
By learning to address mental or emotional frustrations, you’re establishing a safe space to grow your perspective. Actively practicing mindfulness helps you to create a growth mindset, which can be helpful in times of challenge.
Due to math’s more dense and logical functionality, it’s one of those disciplines that challenges a majority of people in MANY ways. Mindfulness, especially when fused with math, can strengthen divergent thinking skills, patience, and improve many life skills.
When being faced with an obstacle, for example a challenging equation, mindfulness can be used to navigate the challenge with a more hopeful approach. If you were to fail or get the math equation wrong, you can still utilize mindfulness by removing yourself from the stressful environment, gaining clarity, then returning with a newfound approach that may attack the problem correctly (maybe even more efficiently).
Noticing the reactions you experience when partaking in mathematics can guide you to find a new attitude better suited for the task. Identifying those feelings will also pave the way to cultivate a more thoughtful, intentional response!
Younger children with less emotional regulation may find it difficult to master mindfulness during math on their own. Don’t worry, familiarizing them with the process of mindfulness can be introduced in AND out of the classroom. Implementing mindfulness at an early age can prove to be expediently helpful.
If you are a teacher or educator looking to provide a supportive environment for optimal learning, there are many ways to incorporate mindfulness! First and foremost, encourage the student(s) to be fully present and focused on what’s currently occurring! To prepare students for an engaging experience, instruct and/or guide them to calm their bodies and minds for the upcoming educational lesson.
Can you fuse mindfulness with math? ABSOLUTELY. There are a few different methods that can be integrated into a young mind’s lifestyle, whether it be during math time at school or when they experience a stressful situation at home.
3 Practices That Blend Mindfulness With Math
Count Your Breathing:
There are LOTS of specialized kinds of breathing able to sound the physiological sensations of stress or anxiety. There’s deep breathing, belly breathing, box breathing… you get the idea. Regardless of what KIND of breathing is being used, counting is a great way to control respiration!
Instruct the student or class to get comfortable in their seat(s). Count an inhale for four counts, have them hold it for another four counts, then fully exhale for the final four counts. The purpose of the counting is to bring attention to this typically involuntary action. Becoming more aware of the flow of oxygen in and out of the body will instantaneously calm the nervous system, which means the students will feel more relaxed and present for their lesson.
Nourish The Mind AND Body:
Cooking has proven to be a fun yet engaging task that incorporates various math techniques! The human body needs to be nourished with yummy, wholesome food to function at its best, therefore this is an exciting yet immersive way to bring a child back into their body and enjoy caring for themselves. Try including the child in fun cooking exercises that get them excited to feed their bodies (and minds).
Some fun ways to mindfully practice math is to have a child count their pieces of snacks, measure ingredients for their favorite food, time the cooking process of their highly anticipated treat, and so much more!
Time to Shake It Out:
Kids feel so pent up and sometimes frustrated when they cannot exert their energy in the classroom, so physical movement may help them out!
Aside from recess at school, take a few minutes at different times during school to allow children to move and wiggle their bodies. Encourage them to (safely) shake out their frustration, attune to their bodily sensations, and wiggle away any negative feelings. This can be used as a good transition or break between subjects, lessons, and used before or after a long lesson!
September 3, 2023
What Makes A GREAT Math Tutor?
What are the qualities & skills to look for when searching for an effective math tutor for your student?
As many of us know, every student is different when it comes to learning. It’s no secret that not every student can retain or learn math easily and enjoyably. That’s quite alright! Unfortunately, it’s still something they must encounter throughout their schooling. What’s a kid to do?
To minimize the challenges that math presents, it’s a great idea to seek a potential resource outside of the child’s traditional classroom. There are various tutors available, both in-person and virtually, able to teach math in new or different ways than conventional school teachers. Not only are they outside their typical learning environment, but it’s a much more individualized and intimate setting that will give the child the attention they need to tackle their math struggles.
What should you exactly look for in a tutor? An effective and experienced tutor will encompass a wide array of useful skills and qualities to establish a worthwhile experience for the student.
Below are just a handful (of the ever-expanding) qualities of a GREAT math tutor.
Qualities + Skills Of A Great Math Tutor
Great, But Growing, Knowledge of Math: It’s important to give a young student a tutor that knows what they’re teaching. Having a prominent amount of mathematical knowledge is one of the most fundamental traits when looking for a good tutor. Not only should they have a foundational knowledge of the subject, but they should be open to expanding their perspective as math evolves with the times.
Humility: A humble, grounded tutor will provide an environment for your students that’s in THEIR best interest. The student is there to receive encouraging guidance on how to tackle math, not to boost the ego of the tutor. That’s why a great tutor should recognize their skills as worthwhile and useful, but utilize them for the good of others!
Ability to Individualize Strategies for ALL Students: Not every student is the same, especially when it comes to HOW they retain dense or difficult knowledge. Math, although very logical and quantitative, cannot be taught to everybody the same way. An experienced math tutor should take the time and care to assess the individual learning styles of every student they tutor to best plan their approach to teaching math at hand.
Accountability: There is a certain level of responsibility a great tutor must be able to take on. They have a wide assortment of students to educate, a schedule to maintain & update, and even families to interact with to communicate student progress. Accepting this role with an ethical and honest attitude ensures they will be able to address the unique circumstances of each tutoring relationship in a way it wholly deserves.
Enthusiasm for Learning + Growing Math Skills: How can you expect a young child to not get excited to learn about math if the tutor can’t be? Showing up to the tutoring session with a passion will motivate the student to TRY; the contagiousness of the enthusiasm may even get them excited to learn. An excellent math tutor will attack their job with fervor to help in any way they can, and even be excited to learn more about math themselves!
Flexibility: There will be times when a last-minute cancellation or rescheduling occurs, or maybe you’re a virtual tutor tackling technical obstacles. Regardless of the challenge, a great math tutor will maintain a flexible yet understanding mindset! The accessibility of a student will vary, so the tutor themselves should take as much responsibility as they can to do their part to ensure the students receive the resources they need.
Professionalism: In reality, the tutor isn’t just interacting with the student! There are probably guardians, parents, even fellow educators who want to be kept in the loop on the student’s progress; that can be overwhelming! An experienced tutor knows that providing an approachable, personable, and well-composed attitude so the dynamic remains beneficial and positive!
Patience AND Positivity: Learning is a marathon, not a sprint. Students will only learn as far as their motivation, focus, and mood can allow in various sessions. That’s why an exceptional math tutor will be patient yet challenge the student within certain boundaries. Not only that but maintaining positivity that the student will succeed in due time will rub off on them; children tend to energetically absorb what’s occurring around them. If you want the student to have a positive attitude, lead by example!
Able to TEACH, not just lecture: To fully retain or understand a mathematical concept, a student may need an in-depth explanation about a specific equation or concept. More likely than not, they won’t comprehend the math well if they’re just lectured to. It’s up to the math tutor to engage in discourse that fully encompasses the use, function, and steps of a math problem. Additionally, not many people learn effectively by just being talked to; they need various forms of education for their brain to absorb what’s being taught.
Motivation: Evidently, there will be sessions and/or weeks where the students will just not have the same determination to do the work they usually do. Despite the challenges, especially an energetic ebb and flow, a wonderful math tutor will persevere by brainstorming new yet engaging ways to teach; they will also know when to dial back on the studying to energetically match the student in the session. As long as the tutor has the drive to do the best they can to teach to the specific individual, it’ll be worthwhile.
Reliability + Preparation: Being a good math tutor doesn’t just mean showing up. It’s about being able to facilitate a productive session about the content the student needs to learn. That comes with showing up consistently, putting in your all, and most importantly coming PREPARED. A good math tutor will prepare for a session by outlining the lesson and its concepts whilst providing lots of exercises and reviews to effectively conceptualize the lesson.
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.