Common Algebraic and Calculus Errors


The topics covered here are errors that students often make in doing algebra, and not just errors typically made in an algebra class. Typically, these mistakes are made by students in all level of classes, from algebra classes up to senior level math classes. Some of the mistakes in this article actually come from calculus courses.

Many of these are caused by student’s getting lazy or getting in a hurry and not paying attention to what they’re doing. By slowing down and paying attention to what you’re doing and paying attention to proper notation you can avoid the vast majority of these mistakes.

Many times students may understand the calculus concepts, but the algebra gets me every time. Having an efficient command of algebra is critical in mastering not only an algebra course but also in the study of calculus.

The article begins with common algebraic errors including trigonometry, logarithmic and functions. The article concludes with some common calculus errors including derivatives and integrals.

We begin with some common algebraic errors.

Algebraic Errors

\sf \frac{a}{x+b} \neq \frac{a}{x} + \frac{a}{b}

To see this error, a = b = x = 1.

\sf a - b(x - 1) \neq a - bx - b

Remember to distribute negative signs. The equation should be:

\sf a - b(x - 1) = a - bx + b

To divide fractions, invert and multiply.

\sf \frac{\left(\frac{x}{a}\right)}{b} \neq \frac{bx}{a}

The equation should be:

\sf \frac{\left(\frac{x}{a}\right)}{b} = \frac{\left(\frac{x}{a}\right)}{\left(\frac{b}{1}\right)} = \left(\frac{x}{a} \right) \left(\frac{1}{b} \right) = \frac{x}{ab}

\sf \frac{a + bx}{a} \neq 1 + bx

This is one of many examples of incorrect cancellations. The equation should be:

\sf \frac{a + bx}{a} = \frac{a}{a} + \frac{b}{x} = 1 + \frac{bx}{a}

\sf \frac{1}{x^{1/3} - x^{1/3}} \neq x^{-1/2} - x^{-1/3}

This error is a sophisticated version of the first error.

\sf (x^2)^3 \neq x^5

The equation should be:

\sf (x^2)^3 = x^2 x^2 x^2 = x^6

\sf (a + b)^2 \neq a^2 + b^2

The equation should be:

\sf (a + b)^2 = a^2 + 2ab + b^2

\sf |-2(x + 3)| \neq -2|x + 3|

The equation should be:

\sf |-2(x + 3)| = 2|x + 3|

\sf \sqrt{x + y} \neq \sqrt{x} + \sqrt{y}

There is no simplified way to write this expression.

\sf \sqrt{x^2+a^2} \neq x + a

To see this error, let x = 3 and a = 4

\sf \sqrt{-x^2 + a^2} = -\sqrt{x^2 - a^2}

We can’t factor a negative sign outside of the square root.

Solve for x:

\sf x(x - 2) = 1

\sf x = 1 or \sf x - 2 = 1

This is incorrect because the right hand side of the equation is not zero and the factors x and (x – 2) have been set equal to that non-zero number, that is, 1.

The zero product rule states that if ab = 0 then either a = 0 or b = 0. This rule is applicable only if the right hand side of the equation is zero.

\sf x(x - 2) = x^2 - 2x = 1

\sf x^2 - 2x - 1 = 0

Then try to factor, but since this will not factor nicely, use the quadratic formula to solve for x.

Trigonometric Errors

\sf \cos(10) = -0.839071529076 \text { radians}

\sf \cos(10) = 0.984807753013 \text { degrees}

Always use radians when dealing with trig functions.

\sf \cos(x + y) \neq \cos(x) + \cos(y)

For example, let \sf x = \pi and \sf y = 2\pi

\sf \cos(\pi + 2\pi) \neq \cos(\pi) + \cos( \pi)

\sf \cos(3\pi) \neq -1 + 1

\sf -1 \neq 0

\sf \cos(3x) \neq 3\cos(x)

For example, let \sf x = \pi

\sf \cos(3\pi) \neq 3\cos(\pi)

\sf -1 \neq 3(-1)

\sf -1 \neq -3

Remember: \sf \cos(x) is not multiplication; it represents the cosine function.

\sf \frac{\sin2x}{x} \neq\sin2

You can only cancel terms in the numerator and denominator of a fraction if they are not inside anything else and are just multiplying the rest of the numerator and denominator. The function \sf \sin2x is not \sf \sin2 multiplied by x. If the fraction had been written as:

\sf \frac{\sin(2x)}{x}

it would be harder to make such an error.

If \sf n is a positive integer then:

\sf \sin^2(x) = (\sin x)^n

However, the following two trig functions have different meanings:

\sf \tan^2x and \tan x^2

In the first case we are taking the tangent then squaring result and in the second we are squaring the \sf x then taking the tangent.

\sf \cos^{-1}(x) \neq \frac{1}{\cos(x)}

The -1 in the trig function \sf \cos^{-1}(x) is not an exponent, but to denote an inverse trig function.

Another notation for inverse trig functions that avoids this problem:

\sf \cos^{-1}(x) = \arccos(x)

However, this notation is seldom used.

Logarithmic Errors

Logarithmic term is expressed as \sf \log_aY , where the symbol a is known as the “base”.

If the base is 10, normally we will leave the logarithmic term as \sf \log Y , without writing the base 10. The examples shown use base 10 for simplicity.

The study of logarithms involves 3 basic laws. With these laws, any logarithmic expressions can be easily simplified. Here are the 3 laws:

1) Product Law:

\sf \log (XY) = \log X + \log Y the log terms are added.

2) Quotient Law:

\sf \log(\frac{X}{Y}) = \log X - \log Y the log terms are subtracted.

3) Power Law:

\sf \log X^n = n \log X the power \sf n is brought in front of the term.

Common mistakes:

\sf \log X + \log Y \neq \log (X + Y)

They are not equal.

Writing \sf \log (X/Y) as \log X / \log Y
It is “X divided by Y” before being “log”.

Thinking that \sf \log and X are separated; they are together as in: \sf \log X

Functional Errors

The exponent –1 should be treated carefully. When applied to a number, or an algebraic expression, it means the reciprocal. When applied to the notation of a function, it means the inverse.

For example, \sf 3^{-1} = \frac{1}{3} and \sf (x^2 + 2x + 1)^{-1}=\frac{1}{x^2 + 2x + 1}. But \sf \sin^{-1}(x) \neq \frac{1}{\sin(x)} and in general \sf f^{-1}(x) \neq \frac{1}{f(x)} therefore \sf \sin^{-1}x stands for the inverse sine function and \sf f^{-1}(x) stands for the inverse function of \sf f(x).

When a function \sf f(x) is raised to an exponent \sf k, \sf k \neq -1, it is written as \sf f^k(x). In other words, \sf f^k(x) = [f(x)]^k. This notation is especially common in trigonometry, e.g., \sf \sin^2x = [\sin x]^2 , \sf \tan^{1/2}x = \sqrt{\tan x}, etc.

Calculus Errors

\sf \frac{d}{dx} [2^x] \neq x2^{x-1}

The correct answer is \sf 2x\ln2 . The power rule only applies if the base is a variable and the exponent is a constant, as in \sf x^3 .

\sf \frac{d}{dx} [\sin(x^2+1)] \neq \cos(2x)

This is a typical example of the kind of mistakes made when applying the chain rule. The correct answer is:

\sf \frac{d}{dx} [\sin(x^2+1)] = \cos(x^2 + 1)\times 2x

\sf \frac{d}{dx} [\sin(x^2+1)] \neq \cos(x^2 + 1) + \sin(2x)

Another common way in which the chain rule is misapplied. This time the product rule has been used where the chain rule was the way to go.

\sf \frac{d}{dx} [\cos x] \neq \sin x

The answer should be \sf - \sin x . This is an extremely common error.

\sf \frac{d}{dx} \left[\frac{f}{g} \right]\neq \frac{fg^\prime - gf^\prime}{g^2}

This is backwards. It should be:

\sf \frac{d}{dx} \left[\frac{f}{g} \right]= \frac{gf^\prime - fg^\prime}{g^2}

\sf \frac{d}{dx}[\ln 3]\neq \frac13

The quantity \ln 3 is a constant, so \sf \frac{d}{dx}[\ln 3]= 0 . The same is true for all constants. So \sf \frac{d}{dx}[e]=0 and \sf \frac{d}{dx}[\sin \frac\pi2]= 0 as well.

\sf \int{x dx}\neq \frac{x^2}{2}

The correct answer is:

\sf \int{x dx} = \frac{x^2}{2} + C

Picky profs penalize points pedantically.

\sf \int{\frac{1}{x} dx} \neq \frac{x^0}{0} + C

The power rule for integration does not apply to \sf x^{-1}. Instead,

\sf \int{\frac{1}{x} dx} = \ln|x| + C

\sf \int{\tan x dx} \neq \sec^2 x + C

It is the other way around. \sf \frac{d}{dx}[\tan x] = \sec^2 x . The correct answer is:

\sf \int{\tan x dx} = \ln|\sec x| + C

as can be found by u-substitution with u = \cos x.

Forgetting to simplify mistake, for example:

\sf \int{x\sqrt x dx}

is easy if you notice that \sf x\sqrt x = x^\frac23 and then apply the power rule for integration. If you try to do this problem by using integration by parts or substitution, you will find this approach much more difficult, if not impossible to solve.

Not substituting back to the original variable mistake

\sf \int{2xe^{x^2} dx}

does not equal \sf e^u + C . It equals \sf e^{x^2} + C .


The worst mistake many students make is to think they know the material better than they really do. It is easy to fool yourself into thinking you can solve a problem when you’re looking at the answer book or at a worked out solution. Test your knowledge by trying problems under exam conditions. If you can do them under that restriction, the exam should be a breeze.

Do not fear mistakes. We can learn from these mistakes, but, after looking and understanding these mistakes, we should correct and not make them again.

We hope you found this article informative and wish the best to your child’s success.

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Why Is Math Hard


Have you ever wondered why math is easier for some students and not for others? According to a Gallup Poll in 2005, students were asked to name the school subject that they considered to be the most difficult, mathematics came out on top. This is probably not a surprise to teachers and parents. The question this raises is what is it about math that makes it so difficult?

In this article we will discuss why some students have a difficulty with math and how to improve their chances. Most students have the capability to succeed in math, so let’s being our discussion with patience and persistence.

Patience and Persistence

The key word here is difficult implying something that is not easy, hard or complicated. It is important to keep in mind when learning math that it takes patience and persistence. Math is not a subject that is intuitively obvious requiring little effort on the part of the student; it requires effort. For most students this means devoting lots of time and energy in understanding the material presented in class and most importantly, doing the homework.

We live in a world of instant gratification. We can quickly find answers to our questions and solutions to our most difficult problems via the Internet. With this technology available to students, is it no wonder that patience and persistence is lacking amongst students?

Importance of Homework

Teachers assign homework to give students the extra practice needed to be successful in math. Students with the highest grades in the class almost always have perfect or near-perfect homework scores. The worst grades in the class inevitably go to students with poor homework scores. For many students the problem has little to do with dominate brain types; it is mostly a matter of will power.

Dominate Brain Types

Many researchers and scientists put forth the notion of left brain vs. right brain dominate students in being able to understand math quickly. Basically, many scholars believe that some students are wired with better math skills than other students. According to some researchers, students that are left-brain dominant tend to have stronger math abilities than right brain dominate students. They argue that left-brain dominant students may grasp concepts quickly while right-brain dominant students do not.

There will always be opposing views on any topic and the process of human learning is subject to debate just like any other topic.

Cumulative Discipline

Math is a cumulative subject. This means students must understand the material before they move on to the next. Without a solid understanding of basic math skills, some students will fall behind their peers as they advanced through the mathematical curriculum.

Students begin their mathematical understanding in elementary school where they learn rules for addition, multiplication, fractions, etc. Understanding these basic rules, along with others taught in a typical basic arithmetic class is important in future math courses.

Next is middle school where students first learn about formulas, operations and word problems, etc. Typically, these new topics also include material taught in previous math courses and introduce abstract thinking. Again, students must master this material before they move on to more advanced topics to build on their repertoire of mathematical knowledge.

For some students, a problem starts to appear sometime between middle school and high school. This may be in part that students move on to a new grade or subject before they are ready. Students who earn a “C” in middle school have absorbed and understood about half of what they should, but they move on anyway. Some reasons they move on or are moved on are:

  • They think a C is good enough
  • Parents do not realize that moving on without a full understanding poses a big problem for high school and college
  • Teachers don’t have time and energy enough to ensure that every single student understands every single concept

So these students move to the next level with a weak mathematical foundation. This weakness in their mathematical understanding can only lead to a serious limitation when it comes to future courses and real potential for failure.

Making Math Less Difficult

We have discussed a few of the more important topics on why some students have a difficulty in understanding math. They are summarized here:

  • Are not patient enough to give math time and energy it deserves
  • Are satisfied by just getting by
  • Move on to study more complex concepts with a weak foundation
  • A weak mathematical foundation can only lead to collapse at some point

Although this may sound like bad news, the solution may not be that difficult. No matter where you are in your math studies, you can excel if you backtrack far enough to reinforce your foundation. You must fill in the holes with a deep understanding of the basic concepts you encountered in middle school math. The trick is to be patient enough. Here are some guidelines that may help in improving your mathematical abilities:

  • If you’re in middle school do not attempt to move on until you understand pre-algebra concepts fully.
  • If you’re in high school and struggling with math, download a middle school math syllabus. Make sure you understand every single concept and activity that is covered in middle grades.
  • If you’re in college, backtrack all the way to basic math and work forward. This won’t take as long as it sounds. You can work forward through years of math in a couple weeks.
  • Most importantly, do your homework.
  • Get a tutor if necessary.


Some people are better at math than others, just like some people run faster than others, and some people need glasses to see while others don’t. The truth is that most people are capable at being competent at math. Just like most people without serious health issues or disabilities can learn to run 3 or 10 miles. For some students, the hardest part is to get started and believe that you can do it. Next identify your weaknesses and take the appropriate steps to overcome your weaknesses.

No matter where you start and where you struggle, you must make sure you acknowledge any weak spots in your foundation and fill the holes with practice and understanding.

We hope you found this article informative and wish the best to your child’s success.

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Why Is Mathematics Important


It is sometimes difficult for students to appreciate the importance of mathematics. They often find the subject boring and hard to understand. It is not uncommon for many students to ask their math teacher why learning math is important.

Parents attitudes, whether it be positive or negative can also influence their children’s attitudes towards math and ultimately their performance. It is important to encourage your child to excel in math, as they will function better in society and have better opportunities in securing a rewarding career.

In this article we will discuss the importance of math in everyday life and what it teaches. We will also explore why companies like to hire students with good math skills and increasing the likelihood of securing scholarships.

Let’s begin our discussion on how understanding math fits into everyday life.

Everyday Life

It is difficult to live and function in society without a good set of basic math skills. In early elementary grades students learn about counting money and progress to calculating percentages and fractions. These skills are necessary in order to follow a recipe, evaluate whether or not an item on clearance is a good deal and to manage a budget to name a few. Students proficient at math will be able to buy a car without getting taken for a ride.

Logic and Critical Thinking

Math teaches logical reasoning, order and quantitative calculations. A mathematical equation has a predictable outcome and precise steps must be followed in order to reach that result. The disciplines that math teaches can carry over into everyday life. Here are just some of the important skills students learn in their math courses:

  • Problem solving skills
  • Ability to see relationships
  • Logic and critical thinking skills
  • The ability to identify and analyze patterns

Companies realize the importance of math and will hire math majors based on the presumption that students who are good at math have learned how to think. Math can also provide a means through which critical thinking skills are put into practice and refined. For example, when they are required to explain how they arrived at a solution to a complex problem or to describe the ideas behind a formula or procedure.

Education and Career

Entry-level jobs in fields seemingly unrelated to mathematics require math skills. Cashiers must be able to count money accurately or a customer service representative may need to discuss a discrepancy in a customer’s bill. Students skilled in math and want a college degree will find that high paying careers such as engineering, medicine and research now become available to them.

Even students not interested in these careers must have highly developed math skills. This is typically a requirement to graduate with a bachelor’s degree in most colleges or universities. Students who have better math skills than their peers are more likely to obtain scholarships based on their performance on assessment tests such as the SAT and ACT.


A good math teacher should incorporate techniques that are designed to demonstrate the relevancy of math and how it may apply to their future careers in their lessons. Math teachers should provide realistic examples explaining the rationale behind the concept they are teaching. This will avoid students from repeatedly asking why math is important and may make math more interesting to learn.

We have covered just a few reasons why math is important from a practical view. Obviously, there are other reasons, but the focus of this article is to get students at an early age interested in math and the opportunities available to them by learning math.

We hope you found this article informative and wish the best to your child’s success.

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When to Get a Math Tutor for Your Child


We all want to see our children to do well in school, especially in math. Some children are able to motivate themselves, while others need a push to catch up or a little help in their math skills. When it comes to building math skills, there is no reason to postpone giving your child that extra push.

In this article we will examine some of the indicative signs your child may need a math tutor and the potential risk by postponing such help. We will then examine hiring a tutor and the role of the tutor as well as setting goals for a successful tutoring session and how you can help at home. Lastly, we will summarize some of the key concepts discussed in this article.

Let’s begin with identifying if your child needs a math tutor.

Signs Your Child May Need a Math Tutor

The easiest sign is to look at your child’s grades. This is a quick way to determine whether or not your child might need help. Falling grades can indicate anything from a straight-A student getting their first B to a child not being able to keep up in class. These may be signs that extra help may be necessary.

Another sign is to look for a lack of enthusiasm for math. Typically, elementary school children are interested in learning about math. They like to learn about counting, money, telling time and other math-related topics. When you see enthusiasm slip in math that could be a sign your child may need a math tutor.

Loss of interest in math may be a sign that your child needs help, but it can also mean your child is bored. That is when a tutor can help. Tutoring is good for children who are highly talented and is not limited to children in need of academic help. If the math course is not challenging enough that might mean that your child is probably smart in math and in need of extra challenges.

Talk with your child’s math teacher to get more insight on how your child is doing in class. It is important for the teacher to know your child’s relationship with math has changed. Let the teacher know if your child used to love math in one grade but suddenly dislikes it in the next grade. Since you cannot be in the classroom, having a dialogue with your child’s the teacher will help you identify how best to help.

Falling grades, being unable to keep up in class and lack of interest are not the only signs your child may need a tutor. Disruptive and/or similar behavior, stemming from a plethora of reasons can also be a sign and is beyond the scope of this article. However, the ones discussed in this section are probably the most indicative your child may be in need of tutoring.

Postponing Help

Whether you choose to hire a tutor or provide more learning opportunities at home, it is important to identify your child’s signs of needing extra help early on. Mathematics is typically a linear structured curriculum. Each new lesson or course builds on the previous one. Should your child miss a lesson or does not master a particular skill; it becomes increasingly difficult for your child to succeed in future lessons or courses.

Delaying getting your child the help they need and you may run the risk of letting your child slip further behind. This may lead them to lose the confidence they need which is essential to continued learning, especially in mathematics.

Hiring a Tutor

Once you have determined that your child may need help in math, take your search seriously. You may want to read Tips For Finding A Good Math Tutor to help select a tutor for your child.

Obviously, you want someone who is properly educated in the field of math that will assess your child’s abilities and provides lessons that are age appropriate. Depending on the age of your child, on-line technology such as a visual Internet communication (e.g., Skype) or virtual classrooms may be an alternate and helpful resource as a learning resource. However, the most effective tutoring is where the tutor and your child work together. You may want to read Tips On A Successful Tutoring Session for additional information.

Depending on the particular needs of your child and age, it may be necessary for the tutor to have contact with their teacher. This way the tutor can get a better understanding of curriculum and classroom goals, teaching styles and practices and gaps the teacher is noticing with your child.

Tutoring can also be used as enrichment and not necessarily for remedial support. A good math tutor can stimulate students to prepare them for advanced math courses. Parents may feel their child is not being challenged enough and a tutor may provide the additional challenges lacking in the classroom.

The tutor should be supportive of the learning process by emphasizing or accelerating the material covered in the classroom. It is important that the tutor be an advocate for your child’s learning and success as well as a support for yourself. Also, the tutor should make learning math interesting and fun.

Setting Realistic Goals

When selecting a tutor make sure you explain what you and your child expect. Discuss with your child the goals you want the tutor to focus on. For example, does your child need help to catch up, keep up or get ahead in class? Does your child need help studying for better test grades? Does your child need help organizing and assimilating the classroom material? A good tutor should ask you some of these questions to help set realistic goals.

Also ask how the tutor likes to work to provide the best tutoring session. Most tutors should provide a quiet workspace. Some tutors do not prefer the parent to be hovering during the tutoring session, but be in earshot to listen on how the session is progressing. It is important that the parents explain to their children that tutoring is not a punishment, but is intended to help them succeed in the classroom.

Helping at Home

Although math may not have been your best subject in school you can help your child by refreshing your math skills and knowing the vernacular. If your child asks you to look at their homework assignment, you want to be ready to relate as best you can.

When you suspect that your child is having a little trouble in math that may be the time to start boning up on stuff that you have forgotten since you were in a math class. You do not have to be in expert in math, but should at least have the vocabulary to know what your child is talking about. You can also ask your child’s teacher or tutor for ways to provide support.

It should be noted that you are not required to be the teacher. If your child is struggling, let the teacher know that they need more help and is having a hard time with certain assignments. You can encourage your child by giving them time and place to do their homework.


In this article we explored the signs that your child may need a tutor. Here we focused on three easy signs: falling grades, lack of enthusiasm and difficulty to do the assignments. We discussed that tutoring is not restricted to those in need of help, but even the talented child can benefit from tutoring.

We discussed the importance of determining if your child needs a tutor and the risk of postponing such action. We discussed the importance of mastering each lesson prior to the next in order for your child to succeed in math and the potential risk.

We discussed hiring a tutor and the role they should play. Setting up realistic goals can make the tutoring session more effective. Lastly, we discussed how you can help at home and to provide the setting for your child to succeed in math.

Some items to keep in mind regarding teachers and tutoring:

  • Teachers typically provide a general teaching level that fits most kids in the class.
  • Children with special challenges may need more individual attention and help from someone that understands their personal situation.
  • Children with exceptional abilities may need the individual stimulation that may be lacking in their class.
  • Encourages your child to learn new concepts and provides motivation.
  • Saves valuable time by providing hints and guidance.
  • Provides the necessary skills in general logical thinking and problem solving.

It is important that parents tell their child that tutoring is not a punishment nor should they feel embarrassed but rather as a alternate means in learning.

Obviously, each child is different in their abilities as well as their age and academic background. What may apply for elementary students may not necessarily apply for high school or college level students. However, the intention of this article is to give parents a general guideline to determine if their child may need a math tutor and how they can help.

We hope you found this article informative and welcome your comments. Whether you seek the services from a good math tutor or provide the necessary environment yourself, we wish the best for your child’s success.

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Tips On A Successful Tutoring Session


A successful tutoring session is one where a lot is accomplished. However, a wide range of descriptions can characterize the result of a tutoring session. The student and tutor can have different views on how the tutoring went. In the worst case, the student may feel it was a waste of time and still feels lost and unsure on what to do. In the best case, the student may feel they learned more in that one session than they ever learned in all the math classes ever taken. Obviously, most student descriptions would be somewhere in the middle of these two extremes.

In the worst case, the tutor may think this student is lost and needs to retake one or more previous courses to be prepared for this material. In the best case, the tutor may think that this student is completely in tune with the material and doesn’t need a tutor. Again, most tutor thoughts are somewhere in the middle of these two extremes.

In this article, we will focus on a successful tutoring session. The key to a successful tutoring session is being prepared. The tutor and the student should to be prepared prior to the tutoring session. Preparedness between the tutor and student is discussed separately.

Let’s begin with The Tutor.

The Tutor

A prepared tutor should be skilled in a variety of ways. Following is a list of skills a prepared tutor should possess. Although not an exhaustive list, it should serve as a guide to a successful tutoring session.

Mastery of the Material

This is necessary in order to understand where a students’ understanding is breaking down and to be able to provide instruction to guide the student to understanding.

Good Listener and Communicator

The tutor must be able to listen to the student in order to understand where the students’ understanding is breaking down. The tutor must be insightful and be able to communicate well in order to deliver instruction that is catered to the students’ level of understanding. The goal is always to have the student attain understanding and to become proficient. If the tutor loses sight of this goal and not stay on task, both student growth and session effectiveness are at risk.

Service Oriented and Responsible

A responsible tutor realizes they are being paid to provide a service. The service they provide is concentrated and focused teaching. The tutor is not being paid for their own personal past performance in the subject matter they are providing guidance in.

Respectful and Motivational

The tutor should be respectful and personable to the student. If not, the student is most likely to neither respect the tutor nor respond well to the tutors’ instruction. The tutor should be patient and view at the students’ mistakes as an opportunity to gain insight into the students’ thought process. This gives the tutor a better understanding on how to guide the student. The tutor should praise the student for a job well done since this will encourage and motivate the student to keep up the good work.

The tutor should be approachable and to make the tutoring session pleasant. The tutor should never force the student onto new material before the student is ready; that simply does not work. This can only make the student dislike the tutor and dread the next tutoring session.

The Student

The student knows what they understand and therefore is able to help guide the tutor into the areas they are having difficulty. This way tutoring session is spent giving the student the guidance they need most and not going over problems or material they already understand.


The student should be well prepared by making a concentrated effort to do the work on their own. Ideally, the student should bring any class notes, texts and the results of the material they are having problems. This way the tutor can get a better understanding where the student is having difficulties and to begin to guide the student.

Tutor and Student

Both tutor and student should work together with the goal of having the student gain an understanding of the material they are studying. This will encourage communication between tutor and student, giving them a sense of responsibility and accomplishment.


When the above is true, the amount accomplished during individual tutoring sessions is maximized. In reality though, both tutors and students are human and are not perfect. We should always keep our expectations in check and hope for the best. Both the tutor and student should always be striving to excel. The tutor should strive to deliver better instruction and presentation. The student should strive to better understand the material and to be well prepared prior to the next tutoring session.

Hope this article helped and good luck on your next tutoring session.

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Professional Tutoring Services – Is It The Best Option


It seems that professional tutoring services are all over the place, whether it be T.V. commercials or on the Internet. Even though these services were probably started with good intentions, make no mistake, these services are run as businesses; and businesses have challenges. They have overhead such as: rent, advertising and employees. Professional tutoring services need customers which they must acquire customers and retain. In addition, they want their customers to use their service repeatedly.

In this article we will take a look at a common sense view on how professional tutoring services operate to see if they are right for your tutoring needs. First, let’s take a look at marketing strategies and their role in professional tutoring services.

The Role of Marketing Strategies

So how do professional tutoring services get customers? They employ professional marketers and/or professional marketing techniques to attract customers. It is not the marketer’s job to ensure that the quality of the service provided is set to any standard. Good marketers do their job well. They entice and manipulate potential customers to obligate themselves to the tutoring service. They do not have your best interest in mind; that is not their job. Their job is to get people in the door. That is just the way businesses operate.

When competitors employ these same techniques, they take market share away from other tutoring services. This causes the business owner to employ the same techniques or go out of business. After all they have a business with expenses that need to be met. The business owner has no choice but to employ the same technique of doing whatever it takes to get the new customers.

Typical business Model

Typically professional services employ a business model of 3 students to 1 tutor, which they attempt to sell as a feature. They say this works well because after a tutor explains concept(s) to a student, the student has time to apply the concept on their own until the tutor comes back to see how they are doing. This is a recipe for failure. The tutor needs to watch the student apply the concept(s) to problems. Students’ typically have difficulty understanding how to apply concepts and the tutor needs to guide the student through a few problems. The student will have plenty of time to try and apply the concept(s) on their own when the tutoring session is over and they are at home.

The tutor should dedicate the time in making sure the student understands the concept(s) and how to apply them. However, professional tutoring services cannot afford to provide this level of service. They have expenses and therefore need to have multiple paying customers for each tutor they have on the clock. This is not the best way to help a student with math. Ideally is to have a tutor sit with the student and watch the student attempt to apply the concept(s). This way the tutor can better understand where the student is having difficulties in understanding mathematical concepts. The tutor can teach to the student instead of simply explaining concept(s) the way the tutor understands them. Obviously, tutoring is an art.

Relationship Between Tutor and Student

Professional tutoring services cannot afford to have exceptional tutors on staff. They would have to pay them too much. Probably most tutoring services do have “some” good tutors working on staff. These tutors probably do not want to run a business and would rather rely on the tutoring service to find students. They could certainly make a better hourly wage if they found their own students. Also, the business model of 3 students to each tutor prohibits quality tutoring you would expect. Most competent tutors will tell say it takes insight to determine where the students’ understanding is breaking down. This way, the tutor can cater their explanations to the students’ level of understanding. As a result, what you get from professional tutoring services is less than ideal. Add to this the fact that sometimes employees don’t show up for work. Do you think tutoring services will turn away students when the tutor to student ratio is less than 1:3? Obviously not they need to maintain the 3:1 ratio, so the student winds up with another tutor to maintain that ratio with another tutor.

Students are not guaranteed to get the same tutor each time they go to the tutoring service for tutoring. This is a negative because the tutor and the student develop a rapport over time. This rapport is very important to the students’ understanding of the subject matter they are studying. This rapport allows for the tutor to know how verbose they need to be when making their presentation. In addition, this allows the tutor to know where they need to slow down and where they can speed up. It also allows the student to have confidence in the tutor. The student becomes aware that the tutor is an expert in the subject matter and is willing to listen when the tutor wants them to think about something in a different way or wants to guide them in a specific direction.

Unnecessary Costs

Tutoring services are experts in charging fees, some of which are unnecessary. They charge registration fees and want students to take costly assessment tests. Registration fees can cause the customer to feel locked in to a tutoring service. After all it will cost time and money to switch from one tutoring service to another. Tutoring services cite that assessment tests will pinpoint where and what a student does not understand.

Students typically want help with their homework. They are already taking a class. The school they attend has already assessed the student and has put them in a class that is appropriate. Most likely, the tutor never makes any use of the assessment test. The tutor simply knows how to do the work and shows the student how they would do the problem. This is less than ideal and the assessment test is a waste of customer money.

Additional Charges

Another thing tutoring services do is charge more for tutoring in subjects that are considered more difficult like Calculus. Does this make sense? Typically, High School teachers want to teach Calculus. This is because when you teach Calculus, the students are more experienced in math. In a Calculus class most students do not have difficulties with concepts like “order of operations” or fractions, etc. The students have a sophisticated understanding of computational techniques. Students who understand computation are always much easier to teach than students who do not understand how to apply factoring, distribution, common denominators and a myriad of items that are taught at the lower levels of math. Students at this higher-level can do the computation and understand the theory; they are ready to expand their understanding. This type of student makes it all the easier to tutor. Probably most tutors would charge more for Pre-Algebra, Algebra I and Algebra II. Unfortunately they can’t; the typical customer probably would not understand such cost differences.

Trigonometry, Pre-Calculus and Calculus subjects are all much more enjoyable and probably easier to teach than lower levels of math. That is probably why most High School teachers would rather teach Calculus than Algebra I. Obviously, there are exceptions to this such as Statistics. Statistics is a subject that is inherently difficult to teach. The concepts are involved and it takes a lot of focused effort to explain. The types of problems encountered in a Statistics class are usually word problems where you need to be especially focused and detailed when reading the problems. To a lesser extent, probably Geometry is another such subject. Geometry is usually a student’s first real encounter with applying concepts and their first exposure doing proofs. This can be difficult for some students who do not possess strong computational skills and/or good study habits.


We hope you found this article helpful in understanding how professional tutoring services may fit in to your pursuit of education. Probably, the only real benefit to professional tutoring services is that they are easy to find. Good luck.

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Tips For Finding a Good Math Tutor


Many students and parents hire math tutors unaware of what they should get out of the tutoring process. After finding a math tutor, there are key things you can do to evaluate whether the tutoring experience will be successful and whether you have found a good tutor.

So what sort of qualifications should you look for when choosing a math tutor?

  • Experience as a schoolteacher
  • High standardized test scores and other impressive academic achievements
  • Years of tutoring experience

Seems important, right? Well, not as important as you might think. Let’s take a look at what not to look for in a math tutor and a tutor in general and then follow it up with some helpful guidelines in determining a good math tutor.

Assumptions About Teachers

Many people feel that being a teacher is an important qualification to look for in a tutor after all, a math teacher has years of experience teaching kids and is state licensed.

Let’s take a look at licensing. A state license is just that; a legal document that allows someone to teach in a public school, private schools have no such requirements. We all have had both good and bad experiences with licensed professionals such as: doctors, lawyers, mechanics, contractors and teachers are no different. There are many good teachers, but having a license does not guarantee that someone will be a good teacher, let alone a good tutor.

Teaching and tutoring use different skill sets. A teacher focuses on a group, but tutoring is personal. A good tutor is less of an authority figure and more of guide, helping the student discover math and figuring things out for themselves. Don’t assume that a trained public speaker would be an equally good therapist; so you can’t assume that a teacher would be a good tutor either.

Academic Achievements

Academic achievements typically get our attention. Attending an Ivy League college, having a high GPA, amazing test scores, accolades and advanced degrees all make a potential tutor stand out from the pack.

Unfortunately, being great at something doesn’t mean that you can teach it.


There are two circumstances when it makes sense to seek out an exceptionally academically talented tutor.

The first is when you have an overachieving, type-A kid. For example, if you child wants to get perfect SAT’s scores or go to an Ivy League college.

The second is when your child is exceptionally smart but completely unmotivated.


It is good to select a tutor with a good amount of experience. You don’t want to work with someone that is completely inexperienced. But what is a good amount of experience? A year, 2 years, 5 years, 10 years 20 years?

The most important and effective quality that a tutor can have is not so much experience, proficiency, professionalism or something that can be taught; it’s enthusiasm.

Most new tutors are full of enthusiasm and energy, even though they don’t have much experience. On the other hand, a tutor that has been working with people for 10 or 15 years may be quite seasoned, but are they still excited by helping people learn or has it become just a way to pay the bills?

Guidelines For a Good Math Tutor

Now that we discussed what not to look for in a tutor, let’s focus on some key things you can do to evaluate whether the tutoring experience will be successful and whether you have found a good math tutor.

  1. The math tutor should be asking questions involving the student during the session. If the student is falling behind in school due to lack of engagement on the part of his or her teacher or text, a tutor who falls in the same category will not help. If your child is the one being tutored, don’t be too obtrusive. Listen from another room to hear if the math tutor asks questions and encourages involvement from the student. Some students will initially be shy but if you have hired a good tutor, you should begin to hear them speak up more in response to questions after a couple of tutoring sessions.
  2. A good math tutor should be able to give the student, parent and teacher appropriate progress reports. The student should get detailed and accurate feedback on his or her work. A parent should know how well prepared their child is for a test and a teacher should be informed of what mathematical concepts the student is having difficulty.
  3. After a few sessions, a good math tutor should be adapting to you or your child’s learning style. The tutor should be able to explain the way your child learns so the parent and teacher can better reach the student. If you are the student, ask the tutor about your learning style and how you can apply it in all your subjects. If you are the parent, ask the same about your child. If the math tutor cannot give you an answer, this may be a bad sign.
  4. A good math tutor will be able to come up with extra examples, metaphors and resources to help explain the material. Ask yourself or your child if the tutoring sessions consist solely of watching the student do math homework problems and making comments. If so, your tutor is not doing his or her best to help you or your child understand the material.
  5. It is important that the math tutor encourage the student to think independently. The tutor should show the student how to recognize and where to apply mathematical concepts so that the homework and tests become easier. Depending on the academic situation, this might not happen right away. However, a good tutor should be able to explain how this will eventually help the student be able to work more independently. For example, the tutor might explain how to “translate” key words in word problems.
  6. A good math should be able to help you or your child come up with general study skills that correspond with personal learning styles and are applicable to other school subjects.
  7. Tutor and student personalities should be compatible. If you are the one being tutored, you can make this call yourself. If you are dealing with a child, remember that he or she probably doesn’t like “tutoring” in general. Ask your child this question: “Is ________ a good tutor if you have to have a tutor?”
  8. If your child has a hard time paying attention, listen to see if the math tutor uses creative ways to keep your child focused and incorporates visual information. The same holds true if you are the student. Math tutoring sessions should be more visual, engaging and helpful rather than doing schoolwork on your own. Sometimes short conversational tangents can help students stay interested in the material and place trust in the tutor. A good math tutor will quickly work these back into the work at hand and avoid a child’s attempts to spend the session talking.
  9. A good math tutor will be truthful and realistic, but supportive and optimistic when questioned about a student’s immediate grade potential. The tutor should not promise you or your child all A’s, but will do the best to help the student understand the material to get the best grade possible with your child’s individual circumstances. Often math tutors are faced with having to bring a student up to date on past concepts that are lacking. If this is the case, the tutor should be able to explain to the student or parent how he or she will divide up the time between review and current material.


In short, you can find a Yale graduate with years of experience as a teacher and a tutor, perfect S.A.T. scores, piles of awards and accolades. However, none of these qualifications will guarantee a successful tutoring experience. They will, however, guarantee a hefty hourly rate.

Ultimately, what we really want is someone that just works well with us and our kids. A good math tutor that possesses those difficult to describe personal qualities that makes things click.

Hope you found this article helpful and good luck in finding a good math tutor.

Brought to you by:

Personal Tutoring

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Your Personal Tutor in Mathematics and Computer Science

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