###### August 18, 2021

### Is zero a rational number?

If you are currently studying integers in your math class, you may find that rational numbers are the next topic addressed by your teacher. They may ask the class “is zero a rational number?” Although you know the definition of a rational number, this question can stop you in your tracks.

Knowing basic properties regarding integers and rational numbers may seem like a one-time lesson in your mathematics career, but being able to fully comprehend the purpose and the rules of rational numbers can directly help you with other complicated math concepts. Instead of just thinking the old narrative that learning a math lesson will only pertain to your specific lesson, quizzes, and tests, truly understanding rational numbers can help you learn how to solve multi-step problems in the future.

In addition, learning rational numbers and mastering integers helps you to understand how these math concepts all relate to one another as you progress in your education. Learning the relationship between rational numbers makes mathematics easier to understand as a whole.

By using critical thinking skills to find answers to questions such as “is zero a rational number?”, you can help simplify complicated equations and find the solution.

**First – what is a rational number?**

Before you can determine if zero is a rational number, we need to brush up on your definition of rational numbers. A **rational number** is any number that can be presented as ** a** over

**, where**

*b**b*is not equal to zero. Like this:

Therefore, practically every fraction that you have worked with in the past can be defined as a rational number. The only condition is that the “bottom” number in the fraction – also known as the denominator – cannot be equal to zero. Therefore, all whole numbers are rational, since any rational whole number can be expressed as a fraction with 1 in the denominator slot.

But how can you identify rational numbers from irrational numbers? Irrational numbers are not able to follow the rule that it can be presented as **a** over **b**, where *b* is not equal to zero.

In this case, you might come across irrational numbers that look like the following:

When looking at a fraction, sometimes the numerator can be zero. This means that **a** can equal zero, but *b* is not equal to zero. For new learners, just seeing a zero in the equation can sometimes throw off their thought process. However, keep in mind that a rational number with zero as a numerator can easily be simplified as such:

Since the greatest common denominator between 0 and 12 is 0, you can simplify a rational number fraction with a zero in the numerator by dividing both integers by zero. If you see a fraction with zero in the numerator, the answer will **ALWAYS** be zero.

**Examples of rational numbers **

Some examples of rational numbers include ¼, ⅔, 0/1, 8/6, and so on. As you can see, one of the numbers included here was 0 over another number – this is considered a rational number since the denominator is either a positive or negative number and is not equal to zero.

**Operations on rational numbers**

Just because a number is a fraction or contains a zero within the fraction, this does not mean arithmetic operations are impossible. Arithmetic operations are the basic processes of functions you can use with integers, including addition, subtraction, multiplication, and division.

- With addition, you can add two fractional rational numbers by making the denominator the same value.
- When carrying out subtraction, make the denominator the same number before subtracting one rational number from the other one.
- In the case of multiplying two rational numbers, make sure you multiply both the numerator and the denominator of the rational numbers.
- When it comes to dividing a fraction, you must calculate the inverse of the rational number. In this process, switch the numerator and denominator.
- For example, 2/4 will then switch to 4/2. You can double-check your work by multiplying the two values together to see if they equal 1, with the equation being (2/4)x(4/2)=1.

**What are irrational numbers?**

There are clear differences between rational and irrational numbers. Now that you know rational numbers are any type of integer that can be expressed as a/b where b does not equal zero, you can narrow down all integers that do not follow this formula to be considered irrational numbers.

Rational numbers are either positive numbers, negative numbers, or equivalent to zero. However, irrational numbers cannot be written in the form of a/b, but must be written as a decimal.

One of the most common irrational numbers is Pi (), which has endless digits after the decimal point. If you have studied geometry in the past, you know that Pi is the ratio of the circumference of a circle to its diameter – a number that comes out to approximately 3.14. However, this solution is just an approximation, but not accurate enough to be written as a fraction or classified as a rational number. Since the decimal digits of Pi are nearly infinite, you **cannot** use a simple fraction that represents Pi.

Furthermore, Euler’s number () is another irrational number that cannot be simplified or written as a fraction. This number is the base of natural logarithms and is an important concept used in the exponential function. Euler’s number is typically written as 2.718, but contains over 1 trillion digits in the decimal – making it impossible to compose in fractional form.

Another common irrational number is the Golden Ratio (), a ratio called “divine proportion”. This unique ratio is a common number that is found in various aspects of our life and culture, not just in mathematics. Appearing in paintings, artwork, sculptures, and buildings, the Golden Ratio is known for being aesthetically pleasing. This number is typically approximated as 1.618, but cannot be expressed in simple fraction form.

**So – is zero a rational number?**

Now that you know the difference between rational and irrational numbers, you may still be asking yourself – is zero a rational number? You know that zero cannot be the denominator of a rational number, since this would classify the rational function as undefined. However, zero can be the numerator in a fraction, which qualifies zero as an eligible rational number.

Since zero is an integer, it can be written as A in the formula A/B (ex: 0/1, 0/3, 0/6, etc.). Writing zero as the numerator over a positive or negative denominator confirms that zero can be considered a rational number (ex: 0/2, 0/-2, etc.).

Keep in mind that 0 cannot be used as a denominator in a rational number equation. If this was the case, any number divided by 0 would result in infinity – as we learned earlier, irrational numbers cannot be written as a fraction with integers as both the numerator and denominator. Any integer divided by 0 is an irrational number.

**Why is it important to know if zero is a rational number?**

If you are struggling in your math class and you have **finally cracked the code** to the rational vs. irrational number debate you might still be wondering how this will help you understand difficult math concepts.

Knowing that zero is a rational number can give you deeper insight as to how integers can be calculated and how they function within different equations. Being able to distinguish between rational and irrational numbers can help you understand more difficult concepts that you previously had a difficult time understanding.

Learning rational numbers, understanding how to interpret rational numbers, and identifying rational vs. irrational numbers can help students understand how to represent rational numbers when composing equations, how to determine which fractions are greater than others, and how to simplify rational numbers in fractional form.

###### July 23, 2021

### How To Find A Statistics Tutor

“As with almost anything, you benefit most by being taught by someone who has a solid knowledge of the fundamentals, has real world experience in the area, and has the ability to communicate effectively.”

― Ron Glaser, P.H.D., retired US Government statistician and retired UC Davis Statistics Professor, on finding the right statistics tutor.

If you are looking for a statistics tutor, there is a good probability you are looking at a math syllabus full of unusual words: biostatistics, linear models, regression analysis, data mining, survey sampling—that kind of thing.

Statistics isn’t like algebra or geometry, it’s a whole other animal. Even calculus professors can be lousy when it comes to statistics. And if you need help, you may find your math tutor doesn’t make the best statistics teacher either. So, where do you find the elusive statistics pro? Craigslist? The local college career office? Is there a young, fun descendant of the Father of Modern Statistics Sir Ronald Aylmer Fisher living in Portland?

If you actively seek out a statistics specialist, there’s a ninety-nine percent chance you will find what you’re looking for. But if you want a 100% guarantee, here’s what you need to do…

**Where to Find a Statistics Tutor**

If you sense you need help in statistics, you are already ahead of the curve. Many struggling students can be too stubborn to get help and end up tanking their grade point average by attempting to take on statistics on their own. If they pursue a career that uses statistics, such as sales or computer programming, they might be at a loss, frantically searching the internet for a review course. But it doesn’t have to be that way!

It’s important for anyone seeking a career that involves statistics to find a statistics tutor with a well-rounded understanding of both statistical computations and how it relates to a career in data science or statistics. The best place to find private statistics tutors is online. Don’t be overwhelmed with all the tutoring websites out there because there are ways to trim the fat and find the perfect fit.

Look for websites that have been around for a while, have helpful contact information, and use official email addresses or phone lines (as opposed to tutors’ personal contact information, which can be a sign of inexperience as a business). Another trait of a trustworthy tutoring site is a review section where users can provide feedback on their tutoring experience. This way you can validate a good reputation within your local tutoring community.

**What Qualifications Should Your Tutor Have?**

According to Ron Glaser, a retired Lawrence Livermore Lab statistician and university statistics professor, it’s ideal to find a tutor who has a degree in math or statistics, or a college student with marked success in statistics coursework. Glaser would not recommend engineers or scientists who have not had formal statistics training as tutors, because they tend to bluff on knowing lessons students need, but don’t actually know themselves.

You will need someone who can prepare you for college-level coursework in statistics, which translates to someone who has at least a B.S. in statistics or a closely-related field, such as biostatistics, applied mathematics, or computer science. If you are searching on the Internet, think *grad student*. If a tutor is actively working toward a Masters or PhD in statistics, they will have the necessary experience to teach course material and apply it to everyday life.

However, having academic experience isn’t always enough. Consider finding someone who has experience applying statistics in the workforce or has experience teaching or tutoring. The combination of education and experience in the field will be your best bet for a statistics tutor.

To take the search one step further, here is a checklist for just some of the key elements we look for when hiring statistics tutors at Tutor Portland:

- Can this tutor explain tough concepts in five different ways?
- Can this tutor use metaphors that relate to the student’s life?
- Can this tutor adopt an active approach with in-depth discussions about statistics?
- Does this tutor embody integrity and virtue?
- Can this tutor effectively teach us, the Tutor Portland team, before we hire them?

**How much does a good statistics tutor cost and how to ask?**

Okay, so you found yourself a qualified tutor named Sir Ronald Aylmer Fisher Jr. and he’s fabulous, but you need to know how much he charges per hour. Here’s a statistic for you: The cost of living in Portland is 29 percent higher than the national average. On a college student’s salary of $21,000 per year, with a meal card from the folks, can you afford a tutor?

**Here’s the rundown on the costs: **

At Tutor Portland, for instance, we offer online and in-home tutoring backed by a Better Grades Guarantee for our Portland peeps with different budgets. Let’s say you need a statistics tutor for your coursework at Portland State University. We charge $96 per hour for our Silver Plan academic coaching or $384 monthly for one hour per week. Your hours never expire, rolling over to future months, and the more hours you book, the more discounts you receive.

If you decide to hire an independent tutor without such payment models, you should consider asking their rates upfront. The more education, work experience, and overall skills the tutor has, the higher the rates may be. And don’t be shy— after all, statistics tutors are used to talking numbers.

**How to Get the Most out of a Tutoring Session**

It’s all about your personal needs. So the key to maximize your time with a tutor is to study between sessions. By being observant (and hopefully excited) about how statistics play out in the news, in school, in work-life, and at home, you will be mastering the subject in a meaningful way. Then when you meet with your tutor to review the lessons, you can have a deeper understanding of the course material in less time.

**Now that You’ve Aced Statistics … **

If statistics clicks with your brain after a tutoring course, you may even choose it as a career. Among the industries that hire statisticians, the median wages range from $70,000 to more than $100,000 per year. Statisticians are in high demand, and according to Northeastern University, employment for mathematicians and statisticians is expected to grow 30 percent from 2018 to 2028.

Now that you found your tutor who is qualified, affordable, and can get you ahead in your statistics class, what’s next? Maybe another class! TutorPortland offers tutoring in test prep, science, Spanish, and English, so contact us if you’re looking for some extra help.

###### March 24, 2021

### How Tutoring Can Help Students Master Math in the Era of Virtual Learning

If your kids are taking math online right now, I can see why you might be worried. It’s hard enough to hope for As in math classes when kids are attending school in person, but 2020 made everything more complicated.

Now that class is online, it’s even easier for kids to find themselves falling behind. Students are more distracted and less engaged. Class discussions are virtually nonexistent. Getting individual attention from a teacher seems harder than ever. How in the world are students supposed to master complicated math concepts in this environment?

While the online format isn’t ideal, tutoring can help! For kids who are missing that one-on-one interaction, a math tutor can bring them the connection they crave, now from the comfort of their home. Classes are now almost always recorded, giving tutors a way to glimpse course material and work through it with students. Virtual pod-based tutoring can help kids supplement the social aspect they might be yearning for. So if you worry about your kid grasping the math curriculum, don’t give up just yet. Here’s how tutors can make a difference…

**Engaging Students**

Many parents worry that Zoom is less engaging than in person math class. There’s definitely some truth to that–it’s hard for kids to get invested in the material when they’re just another black square in a sea full of turned-off cameras. It’s tempting for kids to start playing Minecraft or scroll through Instagram during a lecture as there’s no teacher in the room. Discussions aren’t as fluid or active and it’s hard for kids to make themselves known. In this virtual environment, students are not as involved in the subject at hand.

However, getting a great tutor can help solve this problem. No longer will your kid be an anonymous member of a huge Zoom class. Tutoring gives kids a personal, individualized approach to learning, even through the internet.

At Tutor Portland, we give one-on-one tutoring that’s totally focused on engaging the student. Our tutors build a strong rapport with students, ensuring that kids feel safe to express their frustrations and explain their own way of thinking about problems. Tutors check back with students each week and give them the positive encouragement they need to strive for growth. Even though kids have to learn through a screen, with this approach, we can keep kids excited and interested in the material.

That being said, even if students remain involved, they might not be able to be able to keep up. Math is cumulative, meaning if students miss one concept, they can easily get behind. Working through recorded lectures with a tutor might be able to change that…

**Keeping Students From Falling Behind**

Fast-paced math lectures are so common, it’s a wonder how kids can keep up at all. This has been a problem long before online learning. It’s a byproduct of our tendency to teach math with little to no discussion among students, let alone individual questions and feedback. Now that many lectures are virtual, students might have even less opportunity to slow down the class and inquire about a tricky concept.

Luckily, in this new online educational environment, lectures are almost always recorded. These recorded lectures are made available to students at any time. This means students can go back and rewatch whenever they need to in case they get stuck on homework. They can pause and rewind or go to a specific section when they may have overlooked a detail. Plus, if they want to review for a midterm or a final, they have every lecture laid out for them online. Instead of having one chance to get the teacher’s lecture, they can access it any time.

This is a great tool that tutors and kids can use together. Previously, if a kid struggled to comprehend an idea, tutors would have to guess how the teacher taught it to them. Now, tutors can see exactly how the concept was explained to the student, so the tutor can more deeply understand their student’s learning process. Students and tutors can go through recorded videos and take a new look at any material the student doesn’t quite understand.

Beyond having relationships with teachers and tutors, students also need to relate to each other and work together. But, how can tutoring help kids get that experience without in-person classes?

**Incorporating the Social Aspect**

Although socializing with others may seem extraneous to learning equations, having discussions with others about material really helps struggling math students. Studies show that students understand concepts better when they’re able to verbalize their thinking and have back-and-forth exchanges with other individuals.

However, online school makes it impossible for kids to talk about material as they walk the halls, and makes forming study groups a lot more challenging. This is especially true if your student is new to the class or doesn’t have a group of classmates to collaborate with.

Tutoring can help by not only connecting students with a great tutor, but a network of their peers as well. My brother and I had this social component in mind when we created **Zoom Tutor**, a website made specifically to help kids through the online educational landscape. On Zoom Tutor, not only is one-on-one tutoring offered, but pod-based tutoring with a group of students as well.

This means students are organized into small groups that meet each week to go over material. Although they may not be able to gather in person (yet), these regular sessions give kids the ability to meet and work with an entirely new group of peers. Instead of working through problem after problem alone, they can have a community to share their frustrations and victories. This can fundamentally transform students’ attitudes towards math. Instead of stale repetition, these peer groups help make math fun and social.

**Get Your Kid Tutoring Today!**

Even though learning math online may seem like an impossible challenge, tutoring can help students make the most of it. When Zoom leaves kids uninvested, good tutoring can help them engage with material again. Working through recorded lectures with tutors can guide kids towards success if they find themselves falling behind. Incorporating group work into the online tutoring format can help students have meaningful discussions that allow them to get a grasp on complicated math subjects..

So if you want to help your kid make the most of learning math online, you can sign up for your first free Tutor Portland session **here**, free of charge. You can also check out Zoom Tutor **here**! If we can give students access to a meaningful learning experience despite the tribulations of the online format, we can help kids come out of this challenging time on top, ready to achieve their goals.

###### December 15, 2020

### The Ultimate Guide to Math Placement Tests

So your child has completed their college apps and they’ve started to narrow down their higher education choices. Whether they want to attend a junior college, attend school part-time, or are taking the plunge into a four-year university, their big decision will help shape their future. You’ll want to be sure that they’re as prepared as they can be.

Before they even step foot in a classroom, their skills will need to be tested. Math placement tests are a required examination prior to enrolling in specific classes. If they’re worried about what lies ahead for them when it comes to math placement tests and scheduling math classes, there’s no need to worry.

We have the ultimate guide right here for parents and students like you to better understand what math placement tests are all about.

**What is a Math Placement Test?**

A math placement test is designed to measure a student’s math skills and gauge the most appropriate math classes they should take for the upcoming semester. Before starting college or university, students must complete a math placement test at home. This happens after a student has been admitted to a school and is a normal part of the enrollment process.

There is no passing or failing a math placement test. The point of these exams is to see how competent the student is in the subject. They are more of an assessment of personal skills rather than an analysis of mastery. A math placement test is not a measure of intelligence, but a measurement of personal experience and how well a student demonstrates that experience.

After the test, the school will tailor a choice of math class to the student’s best strengths. If a student is a top scorer, they’ll be rightly placed in advanced classes that will properly challenge them. If they score lower on the math placement test, then they will be placed in less intense math courses.

**What to Expect on a Math Placement Test**

Although all schools require students to take a math placement test, there is no universal standard they adhere to. Each university or college will create their own tests that best measure math skills according to their own set of standards.

However, there will be some similarities across the board. Questions will be pulled from a wide variety of math topics such as algebra, geometry, trigonometry, and some precalculus and calculus, but the questions will not dive deeply into each subject. The range of questions will be wide, not deep.

The wide-range of questions about basic concepts is followed by longer word problems. These word problems focus on the application of concepts. Students will apply the previously mentioned concepts to help solve longer story problems. Further, the test will ask students to provide written analysis to test if they can fully understand and demonstrate mathematical concepts.

These tests usually do not have a time limit and are mostly multiple choice. The point of the exam is to measure skill, not speed. They’re usually administered online too, so there is no need to travel to a testing location. The tests are often completed from the comfort of a student’s own home.

**What’s The Deal with High Placement Scores?**

Preparing for math placement tests can save students and parents time and money down the road. If your university requires payment for classes by the unit, students can save money by testing out of classes that would otherwise be required for their degree.

By scoring well, students can bypass entry-level courses and qualify for more challenging (and more interesting!) math classes. By scoring poorly, the school may place students in lower intensity remedial courses and take time away from more enriching classes.

If a student plans to enter a field that is less math-intensive, they can bypass math classes altogether if they score high enough. By testing out of basic math classes, they free their schedule to take other classes that are more relevant to their field of study. This will save time and money as students will skip a few steps on their way to complete their degree.

**How to Prepare for a Math Placement Test**

Before taking a placement test, I encourage students to brush up on basic mathematical concepts. A math placement test will have questions about basic arithmetic such as addition, subtraction, multiplication, division, fractions, proportions, averages, decimals, and integers. It sounds like a lot, but it’s nothing students haven’t mastered before!

Here are our tips to **maximize your studying habits!**

However, it’s important not to put too much weight on the outcome of the test. There is a downside of over-preparing. If students prepare for the test in anticipation to score “well” or “ace” the exam, then they could be setting themselves up for a difficult situation. By studying for hours to ace the exam, students may earn placement in an advanced level math course they are not ready for. They might become overwhelmed by the higher level material because of their inflated placement score.

Remember, there is no passing or failing these placement tests, so over-preparing won’t always pay off. Ending up in a less intense math course might be the perfect situation for a student who isn’t as mathematically gifted as their peers.

Conversely, if a student enters the math placement test completely unprepared, then they could be placed in a math course that is far below their skill set. These students might get stuck paying for a class that is repetitive, boring, and ultimately a waste of their time because they did not demonstrate higher level math skills on the placement test.

**Final Thoughts on Math Placement Tests **

If a higher education institution accepts you or your child into their program and requires you to take math placement tests, look into hiring a tutor. When you hire a tutor to work one-on-one, you can focus on reviewing the areas that are the best use of your time. Meeting with a tutor to hone math skills could make a positive difference for your future in higher education.

Experts from **Tutor Portland** or **Zoom Tutor** can assist in tailoring a plan for you or your child. Having a tutor that understands how the math placement test works will make a huge difference in your educational experience. By fortifying math experience, you will be ready to take a math placement test and best serve your educational career.

###### September 12, 2020

### The Mystery of Who Invented Calculus

Let’s say you were the creator of one of the world’s most useful and life-changing inventions, but someone else discovered that invention *at the same time* and wanted to claim the credit. Wouldn’t you be … I don’t know … furious?! Wouldn’t you fight tooth and nail to be recognized for coming up with the essential mathematics behind engineering, computer science, and economics? That’s exactly what happened between Sir Isaac Newton and Gottfried Leibniz over the claim of who invented calculus. The problem is, there’s still some major controversy over who really created it despite there being hard evidence of both their work. Maybe you can decide for yourself…

While today’s scandals circulate around notorious twitter feeds, Newton and Leibniz’s battle for recognition over who invented calculus rocked the entirety of Europe in the 17^{th} century. England was so invested in the mathematician from their country that they stuck with Newton’s theory setting them back 100 years in mathematical progress. So does that mean Leibniz was the first inventor? Well it’s not actually that simple.

And up close, the debate gets ugly and personal. There are intimidation tactics, receipts, slander, cliques (looking at you British Royal Society), and lots of shade. This isn’t the stuff you’ll find in your textbooks, but it was enough to make two scientists go to war to prove their accomplishments. They knew that being named the inventor of calculus was too significant to be left in the dust, forgotten. If they fought hard enough for the title, they knew we’d be talking about them today.

**Background**

When you think about who invented calculus, you might imagine some pompous aristocrat in a powdered wig. But early traces of calculus-related math were actually initiated in the time of the toga, long before the 1600s.

Around 250 BC, Archimedes of Syracuse was the first person to conceptualize the tangent, which is a straight line that grazes a curve and an instrumental concept in today’s calculus. This concept was described in the *Archimedes Palimpsest*, a document that uses a variety of rudimentary concepts to solve problems now treated by modern calculus. This kind of math would later be used by both Newton and Leibniz.

Other Greek philosophers built off Archimedes’ work in calculus, and in 500 AD India, mathematicians wrestled with the same early concepts. The astronomer and mathematical theorist, Aryabhata, began using infinitesimals to study the rate of change. He was later succeeded by astronomer Vatasseri Parameshvara Nambudiri, who invented calculus fragments, like an early version of the mean value theorem^{ [1]} from the 15^{th} century.

So, if calculus has been around for a while, why are Newton and Leibniz fighting over who gets all the royalties? Weren’t they just two links in a long chain of people who invented calculus before? Because it didn’t have the legs to stand on as an independent type of study before they put all the pieces together. In order to understand their contributions and why it’s important to know who invented calculus, you first need to know what calculus is in the first place.

Essentially, calculus is a branch of mathematics that quantifies how a thing changes. It’s kind of like how geometry examines the properties of shapes and how algebra is the study of arithmetic (numbers). More specifically, calculus uses infinitesimals, or an endlessly smaller measurement than real numbers, to describe the size of a change. So, the person who invented calculus would be recognized as a genius! This is what Newton and Leibniz were at war over.

Now, our mystery of who invented calculus takes place during The Scientific Revolution in Europe between 1543 – 1687. This was a time when developments in math, physics, astronomy, biology, anatomy, and chemistry dramatically changed how people thought about nature. Seriously, people began challenging archaic religious beliefs with scientific discovery about the center of the universe^{[2]} and fundamental human rights. ^{[3]}

You can imagine how heartbreaking it must have felt for both Newton and Leibniz to get their names among the ranks of Galileo, Kepler, and Copernicus. They created a feasible means of doing math with numbers that didn’t totally exist yet! Not only were they fighting to protect their honor and pride in their work, but the chance to become an A-list celebrity. These guys were fighting over Oprah vs. Robert Downey Jr. levels of fame. Indeed, being the person who invented calculus is the kind of stuff that puts you in history books and is precisely what was at stake for both theorists.

**Who are the Defendants?**

Let’s start with the more well-known contender for who invented calculus. Sir Isaac Newton was born on December 25, 1642 (a Christmas baby!) in the United Kingdom and became a mathematician, physicist, astronomer, theologian, refusing to stay sequestered on his mother’s low-income farm.

Isaac is recognized for discovering gravity when he allegedly witnessed an apple falling while contemplating the literal forces of nature. He later realized that gravitational forces constantly exist between all objects and used this theory to explain that the Earth moves in relation to the Sun.

Newton also has many other accomplishments. He discovered ways to describe how thing’s move with force [4], how colors are actually made up of light, and he initiated groundbreaking studies on the speed of sound! Newton was also mentored by Isaac Barrow, one of the foremost thinkers of calculus-ish mathematics^{[5]} during the European Enlightenment. It wouldn’t be that surprising if such an accomplished mathematician as Newton came up with this branch of the subject. Why not say he was the one who invented calculus as well and call it a day?

Because…

Gottfried Leibniz was also an ambitious scholar and a passionate thinker in his own right. Born on July 1^{st}, 1646 to wealthy German parents, Leibniz inherited an impressive collection of advanced philosophy at the age of six when his Father passed away. Many of the works he read weren’t available to general school audiences, so he took an interest in the exclusive material because, well, it’s not like they had Youtube back then.

By the age of 18, Leibniz graduated with a Master of Philosophy and was awarded a Bachelor of Law after just one year of legal studies. He later developed the binary system,^{[6]} invented modern formal logic, predicted problems later addressed by Albert Einstein, and daydreamt about computing systems that could do algebra. Leibniz’s accomplishments made him a valid response to the question of “who invented calculus.”

Leibniz even created his own theory of truth. Seriously, he thought he could explain the entire universe through simple logic statements. He was so famous, even before being considered as a potential mathematician who invented calculus, that the internationally renowned German chocolate covered biscuits were named Leibniz-Keks. Apparently that’s what happens when you’re the most notable person to reside in Hanover, where the manufacturer is located.

Now that you’ve got an idea of who Isaac Newton and Gottfried Leibniz are and what they’re capable of, it’s time to figure out who invented calculus and review the hard evidence.

**The Evidence**

Despite the controversy over who invented calculus, there is actually a well-documented record of each mathematician’s findings over the years. We’ll use this here to paint a picture of how things went down so you can really be the judge of who deserves the sole title of inventor.

**1666**

This is Newton’s annus mirabilis, the “miracle year” when he discovered gravity and started to theorize about colors. In 1666, Newton was sent home from Cambridge University after graduating due to the bubonic plague. With all this time on his hands, he initiated work on calculus, calling it then, “the method of fluxions.” Newton focused on geometry and the physical world in his work rather than theoretical concepts.

However, all of this thinking was off the books and in the early stages of development. Nothing had been published yet and his work largely remained in his personal collection or resided in letters to correspondents…

**1674**

Leibniz started working on his theory of calculus a few years before making a breakthrough in 1675. While he was in Paris on November 11^{th}, 1675, he made a breakthrough, inventing a new system of notation. This breakthrough is important because it was unique from Newton’s methodology in that it approached calculus from a completely different side of math. Ultimately, it was more robust and practical.

**1686**

Leibniz published the *Discourse on Metaphysics*, his formal explanation of calculus in 1686. This book is significant because it means that Leibniz published his calculus work before Newton.

**1687**

A year after Leibniz’s publication, Newton published his findings in the *Philosophiæ Naturalis Principia Mathematica. *This is the big one, the publication that is praised as one of, if not, the greatest science book of all time. Despite some early problems in the work, Newton used this new calculus in his book to support the theories in his book.

The *Principia Mathematica* is where Newton presents his findings about gravity, motion, and planets, basically all the stuff that he’s famous for. It swiftly won him presidency of The Royal Society, the highest independent scientific academy from the United Kingdom that still operates today. Newton’s esteemed position as the head of the most influential scientific group garnered him a steep advantage over Leibniz in terms of cultural recognition.

**The Rivalry**

At first, Newton and Leibniz didn’t have an ugly conflict. They were both highly publicized members of the scientific community, so it was likely that they had seen each other’s work. But since Leibniz had published first, it was Leibniz not Newton who had sole credit for inventing this amazing new field of mathematics. The recognition Leibniz won for *Discourse on Metaphysics*, may have left Newton with the need to center his own accomplishments.

- Leibniz published work on calculus in
*Discourse on Metaphysics*: 1686 - Newton’s published work on calculus in
*Principia Mathematica*: 1687

Newton notably led The Royal Society through an aggressive agenda to publish the *Commercium Epistolicum, *a ledger containing letters from both Leibniz and Newton to correspondents about their work on calculus. The point of publishing this ledger was to publicize dated documents proving that Newton started working on calculus before Leibniz.

The two key pieces of evidence from the *Commercium Epistolicum* that Newton used to prove he was truly the one who invented calculus include: 1) Letters submitted to The Royal Society from Isaac Barrow, Newton’s mentor. These letters showed that Newton’s work in calculus began in 1666, and 2) Letters submitted to The Royal Society from Leibniz, explicitly expressing an early interest in calculus dated in 1673, clearly after Newton’s work began on the subject.

- Newton’s letters to Isaac Barrow about calculus: 1666
- Leibniz’s letters to The Royal Society expressing interest in calculus: 1673

But Newton wasn’t satisfied. Proving that he started working on calculus first wasn’t enough; he wanted to completely annihilate Leibniz’s credibility. After all, Leibniz’s method is superior to Newton’s, and Newton knew it. Newton also knew that this branch of mathematics was too important to become irrelevant in its history, so he embarked on a complete smear campaign to prove that Leibniz plagiarized.

It devolved into an all-out war to prove who invented calculus. Both Newton and Leibniz tried publishing several statements expanding on how each one’s math was superior to the other’s. But Newton had a huge advantage: he was the president of The Royal Society, and no one dared challenge the establishment.

The one notable scientist that stood up on Leibniz’s behalf was a mathematician named Johann Bernoulli. When Bernoulli publicly announced support for Leibniz’s credibility as the one who invented calculus, Newton swooped in with a silent brigade of intimidation tactics. Bernoulli eventually retracted his statements, and Leibniz was left alone, again, to fend for himself.

The finishing blow of Newton’s attack came in the form of a devious claim: Newton argued that because his letters were circulating around the scientific community at the time, Leibniz must have seen them and started copying off Newton’s invention. However, Leibniz’s methodology is dramatically different from Newton’s. More than anything, Newton’s evidence proves that both Newton and Leibniz independently discovered calculus using their own approaches to the work.

**Conclusions**

In the end, The Royal Society deemed their president the sole discoverer of calculus in 1715, using Newton’s name to proudly answer “who invented calculus” while Leibniz’s credibility dwindled until his death a year later. As a result of the UK’s nationalistic pride in Newton, they refused to use Leibniz’s superior method and suffered for it as the rest of Europe progressed without them.

Today, the scientific community recognizes both mathematicians for their work and respects that they individually made discoveries about calculus independently. Knowing who invented calculus is the kind of history that isn’t taught in textbooks, but it can make math more interesting when you learn where it came from.

Clearly, it’s important to engage in ideas beyond the text with peers, mentors, tutors, and teachers. You might even be able to bring up who invented calculus at a party as a secretly dramatic tale of scandal and sabotage.

**Sources & Citations **

^{[1]} The mean value theorem is a type of formula that calculates the sine, or a specific angle’s dimensions.

^{[2]} In 1543, Nicholaus Copernicus argued that the Sun was near the center of the Universe and that the Earth along with all the other planets orbited around it in circular paths.

^{[3]} John Locke argued for the fundamental human rights of “life, liberty, and property” in *An Essay Concerning Human Understanding* in 1689.

^{[4]} Newton’s Three Laws of Motion: 1. An object is equal to its mass times its acceleration, 2. The force of an object is equal to its mass times its acceleration, and 3. When two objects collide, they apply force to each other of equal magnitude and opposite direction

^{[5]} Barrow pursued the concept of the derivative and linked calculus concepts like differentiation and integration.

^{[6]} The binary number system serves as an alternative to the decimal system that is used in our day to day actions. Rather than using a 10-digit-based system, the binary number system uses 1’s and 0’s to simplify the design of technology. Essentially, it makes up the basis of all contemporary electronics and computers.

###### March 28, 2020

### Math Tutor Portland Oregon

**Math is a hard subject.** It’s *really, really* hard. I know. I’ve been there. My name is Eric M Earle and I’m the founder of Tutor Portland. As a middle school and high school student, I fell behind in mathematics. [I mean, I fell *way* behind]. And that made math and science really challenging for me because math is a cumulative subject. Math is inherently based on what came before it.

Courses and teachers don’t wait. They move on. This teacher had 25 other students to focus on. I simply wasn’t their top priority. I got lost as their attention was consumed by others. And my ability and confidence in mathematics suffered.

I wish I knew back then the things that I know now. At the time, I didn’t realize that math was important! I thought I would never need to know it. But I didn’t understand that math and science are everywhere in society. And additionally, there are a number of “mental-models” or frameworks for viewing the world—big concept ideas—that you can learn by studying math and science. You can apply these “mental models” in any career: finance, law, entrepreneurship, health-care, teaching—whatever the case, you can always use the big lessons from mathematics and apply those to everyday life and situations.

But when I was little I didn’t think about these things. I wish someone told me! My parents spent time looking for a private math tutor, but they didn’t find one because hiring and finding the right tutor can be challenging. As a result, I fell behind and was largely unsuccessful at math and science in high school. I fell behind and never caught back up. And *that’s* why I didn’t like math. I wasn’t good at it and because I didn’t focus on it. My lack of skill in mathematics caused me a lot of stress and anxiety. I remember once in grade school when I sat at the kitchen counter pulling my hair out and crying 😢 because I simply couldn’t understand math. I wanted to work on it but I was too far behind to even begin.

It would have been possible for me to catch up, but I wasn’t **resourceful enough.** I had all the resources 📚 in the world. My parents could have helped me. I could have asked my teacher. I could have found someone to help me. But I didn’t use those resources. I wasn’t resourceful. I was too young and didn’t have the drive and motivation to succeed. I look back now and realize that the right mentor or tutor could have really helped me. I wish that 27-year-old Eric could have had a conversation with my younger self.

*What would you have told him? *

I would have told him how important math is. I would have encouraged him to work harder, be persistent, and just sit down with tough ideas and eventually he would start to figure them out. I would have shown him that he could learn anything if he set his mind to it. And I would have sat down with him and helped him.

When you’re trying to learn a hard subject, it is helpful to have someone sitting down next to you. They can be the one reminding you: “together, we can learn anything” and “we are going to figure this stuff out.” I often make comments such as these with my clients. I want to reassure them that we are going to get through it. We can solve this. We can figure this out.

I ask my students to get the syllabus of the course. That way we can look at and analyze their course rationally—see what every assignment is worth—and craft a winning game plan for their course.

A lot of learning is about having the right mental attitude and frame of mind. Actually, so much of life is about that!

Sometimes learning challenging concepts and ideas takes time. You have to commit to them. **Sit with them.** You have to sit down and start researching and learning. But when you’re a beginner and lack confidence in yourself, this can feel so overwhelming that many students fail to even start. I know this—because I used to be one of those students. *And that is such a hard place to be in.*

When thinking about tutoring and some of the big ideas & themes in education and education research—I often sit back and think to myself—what type of qualities would a tutor have needed in order to help me? There are a few core ideas that always come to mind:

- A great understanding of math & and the ability to explain difficult concepts in simple ways 👍
- A pleasing personality and pleasant demeanor 👍
- Somewhat that I looked up to 👍
- Strong understanding of communication and psychology 👍
- The drive to teach and improve others 👍

Wow! What a great list of attributes. Those are the top qualities I’d want in any great tutor. Tutors are able to help in multiple ways. They provide resources and assistance to students. They can also teach students how to be more resourceful and self-directed learners themselves. The best tutors *empower* their students to achieve their own learning goals. The best tutors also have a sense of persistence. They sit down and say—in a reassuring way, “we are going to figure this stuff out, okay?” That alone is incredibly helpful to a student. Normally, when I say this, I see the student visibly relax. I can almost feel my sense of calm and confidence being transferred to the student. The best tutors come from a place of knowing confidence. The best tutors are people who themselves had to struggle with the subject. And because they were able to overcome their own learning struggles time and time again, they are now in a blessed place to help others.

## My Story (continued…)

I got decent grades in math and science, but I never understood the subjects. It wasn’t until my early to mid-twenties that I developed the desire to start relearning mathematics. I had a life changing volunteer trip to India. I was volunteering at a neurology clinic up in northern India and I witnessed first-hand the incredible amount of poverty [and absolute lack of access to healthcare]. **And it moved me.** I saw people

*suffering*and I realized that I had to help. That’s when I became a pre-medical post-grad student. At that point, my mathematics was so terrible that I couldn’t get into a college math class. I took the placement test at Portland State University, but it was so bad I

**couldn’t even get into math 70 [high school pre-algebra].**

*So I started working with a tutor.*We met at the Multnomah Athletic Club twice a week. And slowly I started to improve. By fall term I was ready and enrolled myself into math 95. I remember once telling a woman this and she laughed at me. She laughed because I was starting back in such a low level of math. This didn’t bother me because I knew deep inside of me that in due time I would be taking advanced mathematics courses. And that is exactly what happened.

**I earned straight A’s in math 95, 111, and 112.**

**I fell in love ❤️ with mathematics **because of my trigonometry professor, who taught our class with **active learning** strategies. [This is something we now employ at Tutor Portland]. These **active learning techniques** helped me learn math easily and effortlessly. **I went on to earn over 100% in both Calculus I and Calculus II. **

Learning math has continued to pay benefits. It has helped me in every area of my life. Learning math teaches you mathematical reasoning and “number sense.” It gives you the ability to look at numbers and data and just *make sense* of them without extensive analysis. *You just start to *get* math and numbers.* This has taught me that I can truly learn anything. I took my most challenging subject and mastered it. *That* is empowering. *That* is what we strive to do for students at **Tutor Portland**.

## Eric’s Perspective

👍 All of this has given me perspective. It has given me a unique point of view on 1) how hard it can be to learn math and 2) how rewarding it can be to finally understand it! As a 20-something, I started tutoring students in the liberal arts. I taught public speaking, communication, writing, and Spanish. Soon after I started, parents began asking if I knew any good Portland, OR math tutors. I began interviewing math tutors. Eventually, I found a mechanical engineering student at PSU who was a good fit. He had a great ability to explain difficult concepts in simple ways. Soon I became a math tutor as well.

I realized that my unique view on math and learning was something that had to be available to more Portland families. Together, my math tutors and I have developed and honed our **active tutoring approach** that works to engage students in their course material and get them thinking deeply about math. We have also developed our own **private mathematics curriculum**. All of the research today shows that** students learn math when they hear themselves verbalize their mathematical thinking. **So that’s what we focus on. We ask questions which encourage students to engage with and think critically about the course material. 👍

## If I can learn it…

As you have read already, math used to be a huge challenge for me. I avoided it at all costs. And my attitude was that “math doesn’t matter.” This is a defensive and protective attitude. And I was holding it because I was afraid. I was engaging with math from a place of fear and lack.

Now, that has all changed. I relate to math from a place of confidence, abundance, and curiosity. My mathematical reasoning skills are highly developed. I can **use and apply** the mathematics that I know to real-world situations. And math has provided me with different **mental models, **or ways to think about and analyze the world. Mental models are things all disciplines have. They are frameworks—or guiding visions—that help us see the world in certain ways. That’s why an **interdisciplinary approach** to learning is so favored because it allows you to think about complex problems through various lenses.

## Tutor Portland’s Mission

That’s why we’ve made it our mission to help all students—from middle or high school to adult learners—understand mathematics and apply their learning to other subjects. We’ve made it our mission to help children and families overcome the stress and anxiety around math.

[And – hey! – this is totally normal. It’s normal to have anxiety about math]. But it’s not acceptable to never overcome your fear. The world needs young people who know mathematics. More and more the jobs and positions are shifting towards students who understand STEM (science, technology, engineering, and mathematics). All of the research shows this. America is being an economy built on math, technology, and science. The best employees in the future will need to know these ideas and be able to think critically about math, science, and more.

High school students often wonder what the point of studying math is. “When will I ever use this?” They ask.

I used to ask those exact same questions!!!

We get it. It’s natural to ask that. 👍

It’s our mission to help them learn math! (Leave that stuff up to us!!!!!). Our goal is to help students understand how critically important math is to their future. Do they want to be a doctor, an engineer, a scientist, a businessman? All of these occupations require math! Math will make you better at whatever it is that you decide to do. At Tutor Portland, we have an element of mentoring in all the tutoring that we do. We believe it’s important to inspire students and **make them want to learn math, **instead of forcing them. We’ve found that by connecting a student’s future goals with what they are learning today, they become ten times more motivated to study and put the effort it. 👍

But it’s important not to push subjects on students. We often hear about teacher and parents who, in a very well-meaning way, tell their students the reasons they need to learn math. This doesn’t work 🚫. Countless studies have shown that people become **more motivated** to do things when they **hear themselves give their reasons** for **why** they want to do something. Therefore, we like to ask questions. We ask students questions like: “So, we know you probably aren’t interested in learning all this mathematics stuff—especially not on Sunday!—but if you were interested, why might you be?” **Questions like this do a lot. **First, they provide students with autonomy. Autonomy is something that is important for *anyone,* but it is especially, especially important to teenagers. Second, this question gets the student thinking about why they might want to learn math. There are several other follow up questions to ask.

It’s important to note that questions such as these need to be asked **in the setting of a relationship with strong rapport. **This is where many teachers and tutors go wrong. They try to “motivate” their students without first developing a relationship with them. This erodes trust. **Students don’t need to be “motivated.”** Students have all the motivation they could ever need already inside of them. We just need to coax it out. **But it’s elusive.** That’s why tutoring is an art as well as a skill.

## Math Tutors in Portland, Oregon

😃 At Tutor Portland, we’ve learned a few things about how to find a great math tutor. The first, most important thing that we look for is integrity and character. In order to gain the trust and respect of a student, it is necessary that a tutor must first show and demonstrate these deeply held qualities. In addition, it’s really important that a prospective tutor knows how to teach math. I once hired a physics tutor who was very brilliant. He had a Ph.D. in physics and was clearly smart. But he couldn’t get through to kids. He couldn’t explain challenging ideas in simple ways. This is the most important thing that is required of a tutor. I spoke with our clients regarding the Ph.D. tutor. These families told me that he was explaining concepts in a way that was over the students’ ability to grasp them. Eventually, we had to let him go. It’s not enough to be good at a subject, tutors have to know how to teach ideas and concepts in multiple ways. In fact, **the ability to teach tough concepts in easy effortless ways is the number one ability of a good tutor. **Understanding the material should be a given. That is baseline stuff. A lot of people understand mathematics. **But very few people know how to teach math in a way that anyone can understand. **

Over time we have developed a comprehensive checklist regarding what we look for in math tutors. Here are *just some* of the key elements that we look for:

- 👍 Can this tutor explain tough concepts in 5 different ways?
- 👍 Can this tutor use metaphors [which are relevant to the student’s life]?
- 👍 Is the tutor able to adopt an
**active learning approach**where he engages the client in in-depth discussions about mathematics? - 👍 Does this tutor embody virtues and integrity?
- 👍 How adept is this tutor at teaching mathematics?

😃 These are some of the key questions that we can ask ourselves before hiring any math tutor.

If you want to learn more about us, please feel free to email us or call us at (503) 347 7102.

*Post updated March 2020.*

Tutor Portland | Math Tutoring

950 SW 21st Ave #1117

Portland, OR 97205

eric@tutorportland.com

(503) 347-7102

We offer **Math Tutor** services in:

- Lake Oswego
- Milwaukie
- Vancouver WA
- West Linn
- NW Portland

We offer tutors in the following specific Math subjects:

- Tutoring algebra
- Geometry tutors
- Calculus tutors
- Statistics tutors
- Basic mathematics tutoring

###### August 24, 2019

### In-Home Math Tutoring

**What is in-home math tutoring?**

In-home math tutoring is one of the most common and popular types of tutoring for several reasons. In-home tutoring is incredibly popular because families love the convenience and lack of stress that comes with having a tutor come out directly to your home at a time of your choosing. Families have busy schedules these days! We get it. Everyone is packed full and overloaded. You simply don’t have time to drive all over town. That’s why in-home math tutors come to you! Another reason this type of tutoring is so popular is that math is one of the most difficult subjects for students to learn. Time and time again we see students struggling in courses such as geometry and algebra and trigonometry.

**How does in-home math tutoring work?**

In-home math tutoring is really pretty simple. The first step is scheduling your **free session**. That’s right, every first time Tutor Portland member gets access to a free tutoring session. You will have an initial phone call with **our founder, Eric**. In this quick free introductory call, we will cover all the details prior to the session. We will ask about your child and your learning goals. We will ask some initial discovery questions to assess where exactly your student is at, and what skills and tools will benefit them the most. Then we will pick a tutor who we believe to be the best match. And we will set a time for your initial free session. **99% of the time this first session is a good fit. **And nearly everyone who receives the session signs up to become a paying member.

**What is an in-home math tutoring membership?**

An in-home math **tutoring membership** is something that we offer at Tutor Portland. Our memberships are designed to be incredibly customer-focused. Our mission is to be the most customer-focused company in the world. **We believe in the customer. **Our founder was tired of brands not treating customers right. He wanted his brand, Tutor Portland, to stand for something. **And we chose to stand for the customer. **That’s why our tutoring membership is flexible. **You can cancel anytime.** The sessions are yours to keep, **they never expire. **And **the first session is always free. **We also have a **better grades guarantee. **This means that for our top two membership tiers we guarantee that your child’s grades will improve.

We set up our membership this way because we realized that customers are the most important thing in the world. **Your satisfaction means everything to us. **The only reason that we are here in business is **to serve you. **Please consider letting us do just that.

**Why learn mathematics?**

There are **countless reasons why it’s important to learn mathematics**. Many students complain about learning math. They say: “when will I ever need to learn this?” I get it! I used to be one of those students. But there are so many **jobs that require math**. Here are just a few positions which are generally filled by mathematicians or which math majors are normally hired for:

- Accountant
- Biostatistician
- Computer Programmer
- Computer Systems Analyst
- Doctor
- Financial Analyst
- Financial Manager
- Information Systems Analyst
- Insurance Underwriter
- Inventory Control Specialist
- Investment Manager
- Market Research Analyst
- Professor
- Risk Manager
- Securities Analyst
- Statistician

These are some of the most common jobs you can get from learning math.

Two of the most common responses for why it’s important to learn math is because A) it can prepare you for STEM-related jobs and positions and B) because math is inherently beautiful. Some mathematicians have called these the **classical reasons**. These classical reasons are indeed very strong reasons to learn mathematics. But there are also other reasons why someone might want to learn mathematics. Learning mathematics can add to the diversity of your knowledge. Mathematics—especially ideas such as ratios, rate of change, and simple algebra concepts—can be used to think about our world in unique and compelling ways. Mathematics can make you a better person by giving you a mental framework to think about the philosophical problems of fairness and equity. In addition, learning mathematics will make you more rational and reasonable. It will help you see that most of life is statistics, it comes down to odds and chances.

**What are Tutor Portland’s beliefs about math education?**

Tutor Portland believes that mathematics tutors and teachers should take an active approach. Most students don’t do well at math because they aren’t interested and they are not engaged in the course. This is usually because the course is lecture-based and they fall behind in the course. However, small & active group work allows students to engage with others. Working with others allows students to attempt to put their mathematical thinking into coherent sentences. Normally, when doing this, students will begin to see the holes and gaps in their own learning and reasoning. They can then ask questions and try to dive deeper.

Therefore, at Tutor Portland, we take an **active learning approach**. We don’t believe in bland memorization of facts or rules. We want students to understand the rules. We want them to feel what a derivative is. To experience it. To know what it is before they are taught the formula. This type of learning is **sticky. **But it also takes more time to plan and more skill to execute. That is why most teachers stick to lectures and most tutors simply do low-level homework help. **Creating a good active learning environment is really hard.**

We also follow what is known as the **peer-assisted learning** approach. This approach is built upon research showing that students make the largest learning gains when they actively teach another individual how to complete or solve a problem. We, therefore, implement that into our tutoring. Once we have sufficiently taught a skill to one of our students through active learning strategies. We then employ peer-assisted strategies and **have our students teach *us* the material. **This does a number of things. First, it challenges the student because they have to become the teacher for a few minutes. Second, it shows us how much the student truly understands and where they might be falling short. And lastly, this strategy leads to the biggest learning gains.

**What is active learning?**

Most teachers and tutors believe that their learning environment is inherently an “active” one. But that is far from always being true. Active learning environments do not generally arise spontaneously or from chance. They need to be built, crafted, composed. Essentially, courses or tutoring sessions in which students only listen are the worst teaching methods possible. These lead to the lowest learning gains. Research shows that students need to think, discuss, write, and engage. They need to perform higher-level thinking: synthesis and analysis. Interestingly, **students actually prefer to be more** **engaged**—and they learn more when they are. That’s why we take an active learning approach with our in-home **math tutoring**.

###### August 22, 2019

### Portland Math Tutors

**What are Portland math tutors?**

When your child starts to struggle in mathematics, complain, and bring home poor report card grades, the last thing in the world you want to have to focus on is re-learning mathematics and trying to teach them. Having the best Portland Math Tutors available at your door within hours can be a massive relief. Whether your child scored and B+ on a recent exam, or whether they have been struggling with and failing math all year long, Tutor Portland is there to cover for you and help your child start earning A’s again.

In a world that is increasingly being dominated by science and mathematics, the best thing your child can do is master those subjects, and we can help with that.

**How do Portland math tutors work?**

Math tutoring is simple with Tutor Portland. You pay a monthly membership fee in order to have access to a top hand-picked math tutor who comes out to your home for sessions at a time that is most convenient to you.

If your child is wanting help with homework, preparing for a quiz, or learning conceptual math problems, just give us a call and we can set up tutoring. With our monthly memberships, you can rest assured knowing that a math tutor is always available for you. Most families to choose to pick one tutor and stick with them. Together, they find a weekly time and meet every week to work on homework, higher-level thinking, and general study strategies.

You are only responsible for making sure your child is fed, ready, and prepared for the tutoring session. We will cover everything else! We know how challenging it is for parents and family members to try and tutor each other. We have heard countless stories about this not working out. That’s why many families choose to leave their tutoring up to us!

**What topics do Portland math tutors cover?**

Portland math tutors cover a variety of topics. This includes algebra, geometry, calculus, and other concepts and courses your child might be taking. We have different membership tiers, which allow you to focus on any of these subjects for a certain number of hours per week. We are proud to offer an unlimited membership, which means that you can do as many tutoring hours as you’d like or as many as you can schedule. With this, **there are no limits to the amount of tutoring that we can do**. There is also no limit to the concepts or topics we can cover.

What’s Covered:

- Arithmetic
- Pre-Algebra
- Algebra
- Functions
- Modeling
- Geometry
- Statistics
- Probability
- Pre-Calculus
- Calculus
- Trigonometry

**How much do math tutors in Portland cost?**

Hiring math tutors in Portland can range in price depending on your location, zip code, travel distance, subject, and more. Usually, tutors who teach more challenging subjects charge more than those who teach foundational material. Some organizations offer group tutoring. At Tutor Portland, we focus on individualized personal tutoring. We charge a monthly **membership** fee based on the number of sessions (hours) that you want to use per month. We believe this makes tutoring simply and more accessible because the rate is standardized regardless of where you live or what subject you need to learn. This allows you to efficiently budget for your children over the course of their academic career. To look up more information about our pricing, you can visit **our membership pag**e to see what each different membership tier costs.

**When should I get math tutoring for my child?**

This really depends on you, your situation, and your goals. We recommend math tutoring first and foremost if your child is struggling with math or if they don’t enjoy math. These two things often go hand-in-hand. People don’t like math because they are bad at it! Normally, once students start tutoring they get better at math and start to really enjoy it.

We also recommend tutoring for students who want to stay ahead of the curve. Many families today realize that college is getting more and more competitive. They are also discovering and reading about how the best careers often involve science, technology, and mathematics. Therefore, we’ve worked with many families that want to simply keep their kids ahead of the curve—and ensure that they are really mastering the material.

Some families even ask for tutoring for their kids who are very gifted at math. Often times they will ask for help preparing for an AP or IB examination. Other families have asked us to help prepare their kids for the **SAT**, **SSAT**, or **ACT**.

**Why are math tutors important?**

Math tutors are becoming more important for several reasons.

- With the increase in technology, students today are having trouble focusing. This is hindering their ability to learn mathematics. Some research has found that
**calculators and other technology are reducing children’s mathematical reasoning**ability and conceptual understanding. Research has found that high-tech classrooms equipped with “interactive whiteboards” actually decrease a student’s math performance. In addition, mathematics learning software has been shown to have no benefit on student’s standardized test scores. One thing is clear: learning math the old fashioned way is what works. Learning math**should be hard.**It requires hours of sitting in front of a math textbook and working on it. Tutors can help explain some of these difficult ideas and get students working in the right direction. **Colleges are getting more competitive**. It takes higher math, science, and SAT scores to get into the top colleges these days. We can feel the rising stress among families even just since 2015 when we were started our local tutoring company. Acceptance rates are falling across the nation. Standardized test scores are inching higher and higher. Colleges such as Harvard reject students with perfect SAT scores year after year. Many students are now using the**common application**, which lets them hedge their bets and apply to more and more schools. In order to boost their children’s grades and mathematical understanding, many families are hiring tutors.- All of the
**to****p students are working with tutors**. This is causing the best students to pull further and further ahead from the pack. That’s one reason why it’s becoming more and more difficult for average good and great students to make it into great colleges. Top students are highly adept at mathematics. In addition to this, they are highly skilled in writing and English. This shows that they are well-rounded. This is why many parents are hiring subject tutors for various subjects.

**What math courses do colleges expect you to have taken?**

The majority of colleges expect students to have taken at least 3 years of math in high school. Many expect four years. **At Tutor Portland, we highly highly highly recommend that students take four years of high school math. **Colleges officially say that they don’t require it. But four years of math is largely accepted—and three years are often frowned upon by the admissions team.

Top schools also expect advanced courses such as Honors, IB, or AP mathematics. For students who want to become doctors or pursue other STEM degrees or programs in college, it is **highly **recommended that these students complete 4 years of **advanced **high school math **with excellent marks **in those courses. This will help set your child apart.

There are three different AP math courses offered to high school students: **AP Calculus AB**, AP Calculus BC, and **AP Statistics**. You can learn more about **AP Exams** and how they are graded **here**.

However, across the board, geometry and algebra are the bare minimum courses that need to be completed in order to graduate from high school. The vast majority of states have adopted the common core curriculum and standards, which were discussed earlier in this article.

Most high schools follow this progression of courses:

- Algebra 1
- Geometry
- Algebra 2/Trigonometry
- Pre-Calculus
- Calculus

Many students will commonly stop at pre-calculus. And this order is not the same at every school. There are some schools in Portland that teach geometry first and then algebra 1 and 2 following that.

Many students believe they will never need to use mathematics, so why bother studying it? And, at first glance, that appears to be true for people studying the humanities. Why do English majors need to know algebra? But these days colleges are realizing the value of having strong interdisciplinary students on their campuses.

**Why should I choose Tutor Portland?**

Tutor Portland is a local tutoring company that time and time again has gotten results for students. We take our clients very seriously and want to ensure that they have an incredible experience with us. And we will do whatever it takes to make that happen!

**What makes Tutor Portland unique is our focus on active tutoring.** We believe in having engaging tutoring sessions. Learning should be dynamic. That’s why we focus on **active learning strategies** which are based on research and have been proven to drive learning results. There is a lot of recent research on math and science education. The data clearly shows that **students learn mathematics when they attempt to verbalize their mathematical reasoning and thinking. **This helps them realize what they know, think through challenging concepts, and fill in gaps in their own mental models. That’s why we take an active approach. We ask questions which get students thinking and which nudge them to explain their mathematical reasoning.

###### July 25, 2019

### Portland Math Tutor

# The Benefits Of Hiring A Portland Math Tutor

Only around 25% of children in America are meeting the standard proficiency levels for math. Parents with children who are struggling to keep up in their math classes know firsthand just how frustrating it can be for both themselves and their children. At first, most parents take the approach of helping teach their children alone. Unfortunately, this is not always the best approach and most parents can only help so much because of their limited knowledge. It takes more than know-how to effectively teach a student properly.

The solution to a struggling student is to hire a **math tutor in Portland**. There are many benefits of offering the gift of a personal tutor to your child when they are struggling in their classes. If you are on the fence, below are some of the advantages to be gained from using a Portland math tutor.

**More Confidence in Math And Learning**

One of the best benefits of hiring a tutor for your child is it will help boost their confidence levels. As a parent who loves their children, a few minor mistakes on tests or a couple of low scores might not seem so bad. However, there is a lot of pressure for students these days to get great scores on all tests in order to get into the classes and schools they want.

All kids make mistakes, but when you have a child making numerous mistakes and they don’t understand how they keep making them, it can become very frustrating and their confidence levels will take a hit. They may become hesitant to ask questions or speak up for help in fear they will be made fun of or made to feel less capable than other students.

Tutors can help make the subject of math more fun for your child and less intimidating. Time spent with a professional tutor will give your child the confidence they need to keep trying. It can give your child a great foundation for the future.

**One-On-One Attention**

You have the option of choosing an online tutor or an in-person tutor. No matter which choice you go with, your child will be getting one-on-one attention. **Teachers in school** do not have the time to devote attention to each child in their classes. Because of this, some children end up suffering the consequences. Even if your child’s math teacher offers extra help after school, it still may not be enough to make a difference. A tutor will be able to give them full attention and effectively identify where they are having issues understanding their coursework.

This type of help rarely happens inside a classroom. Teachers simply don’t have the time or motivation to help their students individually. They are focused on other things. They miss the individual trees and instead see the forest. They are too focused on deadlines, exams, and ensuring they are on track with their common core standards. In addition, more teachers are burned out and underpaid. They lack real and true motivation to help your child learn.

In addition, we have found that it’s the students who struggle with math who most need and most benefit from having one-on-one individually personalized tutoring.

**Improve Grades and Test Scores in Math**

Low test scores are often an indication of a lack of motivation or a lack of understanding. When you hire a tutor, they will be able to help your child with both of these issues. Everyone learns a little differently. Tutors are able to learn about the student and find ways to help them according to how they learn for the best results. Over time, test scores and grades will improve. This is essential for preparing your children for important standardized tests in the future. Math is critical to do well on the ACT and the SAT. And it’s also important for other exams. Does your child want to attend business school? The GRE and GMAT both require extensive mathematics. In addition, famously, the MCAT does not allow students to use a calculator.

**Develop A Growth Mindset**

Many students believe that they “suck at math” or that they simply can’t learn it. Often times, these students have a **fixed mindset **when it comes to learning subjects like math and science. Having a fixed mindset is a type of attitude. People who are fixed in their mindset believe that their skills are locked in place. They falsely believe that no matter the amount of effort they put in, they are doomed to fail. Therefore, they wrongly conclude, they might as well not even try. When they earn a D or F, they think “I’m stupid.” This internalizes the failure. They think it is part of their being or their essence.

On the other hand, students who develop a growth mindset believe that they can learn anything they set their mind to. If they happen to get a D, they would think something like: “Wow—that really stinks. I must not have studied hard enough for that exam. **Next time I will step it up.” **They realize that often time learning “failures” are simply a result of having the wrong strategy.

Working closely with a trusted tutor is important because it can help develop a growth mindset within the student. This can radiate out to every area of their life.

**Help Them Learn The Most Important Subject**

A **Gallup poll** showed that Americans rank mathematics as being the most valuable and beneficial subject in their lives. There has been a big push these days to teach more communication, arts, and English skills in schools. These skills are really important. And they should be taught. However, the data here show that students value mathematics. This is likely becoming more and more prevalent as the world moves towards an increasing focus on STEM-related fields and businesses. Hiring a tutor can help your child learn this valuable subject!

**Ease Of Transition**

When children have to change schools due to a move, it can have a significant impact on their studies. The new school may not be following the same math learning schedule your child was getting used to. This can cause confusion and difficulty for your child to acclimate. The trouble with their classwork can be made exasperated by the pressures of being the new kid in class. A tutor can help them behind the scenes to transition easier into their new math class. They can quickly be caught up if they are behind and helped to understand concepts that were missed out on while moving from one school to another. Your child will be able to start off on the right foot at their new school.

**Improve Mathematical Reasoning**

Hiring a math tutor can help your child develop their mathematical reasoning skills. These skills are thought to be the glue that binds all math and science subjects together. Mathematical reasoning is the ability to think about, understand, and explain reasoning about math problems. These skills are vital because often in math there is not one single solution. There are often multiple ways to solve complex problems. Students who have strong mathematical reasoning skills are able to see this and analyze problems before they attempt to solve them. Becoming a strong mathematical reasoner is also crucial to developing independence as a mathematical thinker. It allows students to see patterns in the problems that they encounter.

**Encourage Strong Interdisciplinary Skill & Thinking**

Mathematics is a special discipline because the core concepts in math are interdisciplinary, they are very useful in other fields. To the majority of students, math is an abstract concept that they will “never use in real life.” One benefit of working with a tutor is that the student can learn interdisciplinary thinking. At Tutor Portland, our tutors are all well-versed in mathematics, science, and other subjects as well. This means they are able to relate key concepts and ideas between subjects and incorporate those into their lesson plans.

Tutors can teach students that math isn’t only important during 50 minute periods throughout the week. Math is important all the time. You can use math to think about concepts in other fields. And other fields are starting to realize this. Many professional degrees and programs are now requiring applicants to have a strong interdisciplinary background.

**Prepare Your Student for a Great Career**

Many of the best—the most meaningful, well-paid, and highest status—careers require mathematics. Becoming a doctor requires high levels of math in order to succeed in subjects like general chemistry and general physics, and in order to do well on the MCAT. Engineers also need to be highly skilled in math. Any science field requires math. People high in the realms of computer science require mathematics. Statistics is built on math. For a long time, these careers have all required math. And the world is shifting even more so in this direction.

Many articles discuss how **the future of work is STEM** (Science, Technology, Engineering, Mathematics). This article discusses research from the Department of Education showing that students who study mathematics in college have higher rates of employment and better wages and careers post-college. This is likely because **STEM positions are growing at a faster rate **than non-STEM positions are.

**Improve Your Child’s Financial Skills**

Having a strong sense of mathematics helps students to better manage their resources or their monetary “capital.” Understanding the concepts of fractions, percents, ratios, proportions, and rate of change allow people to mentally think about their finances. It also improves their ability to strategically allocate their capital. Much of life comes down to the decisions that we make. And the decisions we make surrounding money are really important. Where are we going to choose to live, to go to school, and what investments might we make?

Warren Buffett has famously said that all the mathematics you need to be a good investor is a strong understanding of algebra. He says that he uses high school algebra every day as he analyzes and values companies. Now, your son or daughter may never become Warren Buffett. But, knowing algebra could help prevent them from making poor investment decisions.

**Our Tailored Math Tutoring & Mentoring Process Creates Inner Drive**

There is a lot of research these days showing that **inner drive and study strategies** are *the* most important factor determining which kids excel at mathematics. IQ is not the most important factor. Research shows that kids learn math when they are driven to learn it by internal motivation and when they have the most refined study strategies. The study strategies are easy to implement, but you have to know the right ones to apply. We can teach those to your student quickly. The more challenging thing it to gently kindle their inner drive to learn mathematics. Our math tutors know how to do this. We have a developed a specialized simultaneous tutoring and mentoring process that allows us to help with this.

At that the same time that we are tutoring your child, we are also mentoring them. I like to say that the mentoring aspect is the most important thing we do. We help students develop **their own reasons **for why *they* want to learn mathematics. We never force reasons upon them. Their reasons have to come from within! This is multiple orders of magnitude more powerful.

**Develop Math Skills During School Breaks**

All kids love their spring breaks and winter breaks. The downside to these long breaks is that it can be very easy for a child to forget the skills they have learned during the year. Thankfully, a tutor can help your child remember the skills they have already learned and then teach them new skills before going back to school. They will be better prepared for the new lessons their teacher will be giving. Your child will already have a leg up on the curriculum before they head back to class. With reinforcement, a tutor can help students recall their skills much easier. Their memories and skills of complicated math formulas will greatly improve as well.

As you can see, there are many reasons to consider hiring a tutor for your child. The right tutoring professionals can help children become more receptive to the struggles they are having in their math class and learn the skills to overcome them. Their comprehension, confidence and learned skills will offer them a better future both in mathematics and beyond.

#### Finding A Great Math Tutor in Portland

If you live in the Portland, Oregon area and are looking for a great math tutor for your child, consider us. No matter what area your child is struggling in or wants to improve in, our tutors will give them the necessary skills to improve their grades and be better prepared for testing.

Tutor Portland | Math Tutoring

950 SW 21st Ave #1117

Portland, OR 97205

(503) 347-7120

#### Math tutoring Services Include:

- Mathematics tutoring
- Calculus tutors
- Algebra tutors
- Elementary math tutors
- Math test prep

## We also offer math tutoring services for:

- Spanish tutors
- Chemistry tutors

# Math

###### March 19, 2019

### 10 Ways Portland-Based Math Tutors Can Help Your Teen

High school can be a challenging time for many young students. They have new social groups, sports teams, and classes. We’ve been around in Portland since 2015 and have worked with students from nearly every high school. We’ve found that there are 10 ways our **Portland-based math tutors** can help your student improve their grades and their confidence.

### Our Approach To Math Tutoring

We focus on a personalized approach to tutoring. It is one thing for math tutors to provide stock answers to questions and challenges, but we seek to move beyond that by customizing our tutoring and providing what we call: tutoring in the moment. Our tutors are brilliant at adapting to the needs and demand of students.

Every student learns differently, and therefore we must teach through new examples and methods until one works. We believe it is our responsibility to teach an idea or topic, not the student’s problem to learn it.

That’s why during our interview process we look for Portland-based math tutors who have not only an understanding of the subject matter, but who can consistently demonstrate an ability to teach topics and ideas in new and exciting ways.

The end result of all of this is that your middle school or high school student will start to excel at math.

### The Benefits of Our Portland-Based Math Tutors

- Better Grades
- Lower pressure and stress
- More confidence
- Improved understanding
- Greater speed and accuracy
- Stronger mathematical reasoning (which is very important for the SAT, ACT, and GRE tests)
- Higher performance
- A new-found interest in the subject
- A reliable support system
- Better grades in other courses, too! (Math is the foundation for many sciences. And the skills and tools they learn prepare them for chemistry, physics, and even biology.)

### Important Factors to Consider

The most important part is to choose the right kind of tutor. Just because someone has strong understanding and knowledge of the subject, does not make that person a good tutor. Dealing with teens requires immense patience, care, and an ability to teach ideas in different ways. At Tutor Portland, we developed a team of professional tutors who know and understand the best ways and practices that can help your teen achieve success in mathematics. Math is not as complicated as most people believe it to be, once the fear is driven out, interest and capability take hold. Contact us about a free tutoring session and information phone call, to see if Tutor Portland might be a good fit for you.