Friday, December 2, 2016

Notions of limits (written lesson)

In the previous lesson on limits, I introduced the lesson by assigning some videos that you have to watch. Today we get to the written part of the lesson.

I am going to describe this lesson and give you the assignments that you should do.

Before I continue I have to tell you that your learning should not be limited to what your teacher gives you. There are plenty of resources that you can use to learn. You can learn from a teacher, from someone who knows a subject well and can teach it to others, from books, from electronic resources like CDs, from electronic communications like radio and televisions. More importantly today there is plenty of resources in the internet that you can use if you know how to access them. You should make yourself comfortable to all types of resources that you can use for learning. Videos are great to learn something but you can't limit yourself to this only. If you want to learn something deeply you have to get the written materials. The written materials allow you to get an overview of what you  are going to learn and give you also the content. You can choose which parts to learn first or which parts to drop depending on your interests. The most important thing also is you can review the materials as much as you can. If your reading skills are good you can learn a lot from written materials but for math there isn't a lot to read. You have to do the reading and memorize certain things. You have to practice a lot.

Here is the link for the lesson but before you start read the following;

Description of the lesson

This lesson starts by a definition of limits and shows you the three methods of limits using examples. The lesson ends by giving you some problems to do. Below I give you the readings that  you have to do under each sub-title and the tasks you have to do for each lesson.


1. Objectives

You should start by reading the objectives again then read the definition of limits. The first video that you watched on the previous lesson with videos gave you verbally an idea of what a limit is. Now you are going to have a written idea of limit and three methods that allow you to calculate a limit.

2. The Idea.

You read the paragraph giving you an idea of what a limit is. You already have a video demonstration giving you the idea of a limit. Now you have a conceptual definition of a limit. You should try to state this definition either in your own words without compromising the concept or verbatim for more accuracy. Now that you have a definition of limit you are going to learn in written words how to find the limit of a function using three methods: graphtable, algebra.

3. Methods for determining limits

a) The graph method

Under this title you should see a problem named "Example 1". This problem asks you to find three limits using the graph on the right. This problem is already solved for you. You are going to do two things with this problem.

Task I 

Read the example and its solution. Read the explanations provided for the solution of the problem in case you don't understand it. Below is a guide to the explanations.

Explanation of the solution a) 

You should be able to understand the solution easily. I provide the explanations of the solution in case you don't understand it. I give you a method to understand the solution. It's graphic. You should read and do it.

Explanation of the solution of b) and c)

The same method is used for the solution of a) and b)

Explanation of the  solution of d)

You can use the same method for the solution of d) but this time notice that the function doesn't have a limit.

Task 2 

Do a pencil and paper to do the example yourself without referring to the solution. After you finish verify that your answers are correct. 

Task 3

Do Practice I.

b) Table method

In this method you are going to use two tables to find a limit. You start by giving x some values to the left of the given value of x and you group the values of x and f(x) in a table. You do the same thing to the right of the given value of x to have a second table. Even though I don't mention the tasks in the lesson you are going to do them in the same way you do for the previous problem.

Task I

Start by reading the problem they ask you to find the solution. Then read the solution. I didn't provided any explanation of the solution because the solution is explanatory by itself. Below is a guided explanation


You start by giving x a value less than 0 and closer to 0. This value has to be to the left of 0. Then you calculate the value of f(x). You give to x a second value and closer to zero than the previous one You calculate the second value of f(x). You continue to give some values to x closer and closer to x and calculate the corresponding values of x. You do a table grouping all the values of x and f(x) in a table. When you look at the table you notice that f(x) gets close to 1 to the left as x gets closer and closer to 0 to the left.

Now you give x values to the right of 0 but closer to 0 and you calculate the corresponding values of f(x). You do a table grouping the values of x and f(x). When you look at the table you notice that as x gets closer and closer to to 0 to the right f(x) gets close to 1 to the right.

Since f(x) gets close to 1 as x gets closer and closer to 0 both to the left and right to 0 we conclude the limit of f(x) is 1 when x gets close to 0.

Task 2

Take a pencil and a piece of paper to do the problem by yourself.

Task 3

Do the practice problem

Algebra method

This method is very simple but it involves some calculations to do. In this method you substitute x in the function

Task 1

Read the problem first. Then write its solution. Below is a guided explanation.


You substitute x in the function and you do the calculations to find f(x). The value of f(x) is the limit of the function

Task 2

Do the problem by yourself

Task 3

Do the practice problems

Review problems

Do the review problems involving the three methods.

Interested in learning more about limits get this free course Introduction to Calculus
You can also be enrolled in the complete Calculus course

Friday, November 4, 2016

How to learn a subject deeply

Learning a subject deeply means you know its theories and are able to apply it. Very often people learn a subject because they are required to without knowing its applications or if they would ever apply it. People learn practical subjects and are not able to apply them. These subjects require practice. But when you learn a subject deeply its practice becomes easy.

To learn a subject deeply requires to know "how to learn". You start by learning the concepts or key words in the subject. Sometimes there are words that are not known or are not well understood. Having a clear definition of these words helps to learn the subject deeply. Besides knowing key vocabulary it is necessary to master the theories. It is also important to have a clear understanding of the concepts of the subject. This can be done by having a clear mental picture of these concepts in one's mind. If it's not possible to imagine the concepts one can try to represent them by a visual representation. In order to learn a subject deeply it has to be absorbed gradually. so that the previous concepts can be applied to the following ones.

Deeper learning is the ability to apply knowledge to new situations. Deeper learning is associated  with better life and work outcomes according to a 2012 report.

Superficial  learning is associated with poor performance. On the 2012 Program for International Student Assessment (PISA), a test that measures students' abilities to apply their knowledge to real-world problems U.S fifteen years old scored 26th of the 34 industrialized nations in mathematics.

Schools that practice deeper learning have their students graduated and attended college at higher rates than schools that don't use deeper learning.

Students who practice "deeper learning" take responsibility for their learning. In "deeper learning" students master their subjects deeply. They know the concepts, can apply them and reflect deeply on the subject.

Deeper learning is defined by 6 competencies: mastering content, critical thinking, effective written and oral communication, collaboration, learning how to learn and developing academic mindsets.

Deeper learning is associated with practice and reflection. In practical subjects learners can build things. Imagination, intuition and inspiration are some of the characteristics of deeper learning.

Deeper learners cultivate academic mindsets. They make the most out of their learning experiences. They hold the following key beliefs:
"I can change my intelligence and abilities through effort"
"I can succeed"
"This work has value and purpose for me"

Beliefs and learning skills bring success for learners.

If you are interested in getting some help in learning Math, French, ESOL (English to Speakers of Other Languages), Spanish, visit New Direction Education Services at to get the contact information . If you need help in AP Calculus take this Introductory Course for free. You can access the complete course here (click the word "here")

Friday, October 28, 2016

Notions of limits

Lesson: Introduction to the notion of limits


At the end of this lesson the learner should be able to:

1) Have an idea of what a limit is
2) Be able to calculate a limit using the graph, table and algebra method

This lesson is part of a series of lessons on the AP Calculus course I will be teaching throughout this blog. This lesson is the first lesson on Chapter I of the course. It is made of two parts. The first part is made of video lectures. The second part consists of the written lesson and the activities.

Video lectures

1) Introduction to the notion of limit

Here you'll have to watch this video that will introduce you the notion of limits. Here is the link: Introduction to the notion of limit

2) Methods for determining limits

The three methods for determining limits are: Graph, Table and Algebra method. You will have to watch three videos on the Graph method, two on the Table method and two on the Algebra method.

a) Graph method. 

Here are the links to watch the videos for this method:
Two-sided limits from graph
Limits examples Part I
Limits examples Part II

b) Table method

Here are the links to watch the links  for the Table method
Finding limits numerically with tables
Determine a limit numerically

c) Algebra method

In the Algebra method you are going to watch two videos.

1. In this video you are going to learn how to evaluate a limit using the substitution method and verifying the result using a graph. The notion of continuous functions is introduced to help to determine the limit. A continuous function is one that goes without interruption. The notion of continuity is introduced later in the Calculus course. Notice that the first function is a constant. As such the limit is a constant. This is one of the limit properties that will be introduced later. Since you don't know this property the limit is determined using a graph. The limits of the other 3 functions are calculated using the notion that if a function is continuous for a value x = c its limit is f(c). These 2 functions are continuous for any value of x. Therefore f(x) exists for any value of x and the direct substitution method is applied.

 Here is the link of the video to watch; Determining limit using direct substitution

2.. In this video you are going to use the three methods to evaluate a limit
2.1 Direct substitution
2.2 Factoring
2.3 Conjugation
Here is the link of the video to watch: How do you evaluate limits

In the next post I will introduce you to the written lesson that includes the assignments that you will have to do. You can also subscribe to the free Introductory Calculus course. If you want to get the complete Calculus course you can get it here (click on the highlighted word here)

Friday, April 1, 2016

Learning Calculus by following a simple model of learning

Many learners find it difficult to learn a subject or anything that they want to learn. The difficulties come from the fact that people have always thought that in order to learn something somebody has to teach it in the first place. Learning doesn't always come from someone else. One can learn by oneself. In fact learning happens throughout life mostly in the informal way. Life would be impossible without learning. Learning happens explicitly after birth. Babies learn to cry to get fed. This is a natural process of a simple stimulus-response conditioning. A natural stimulus is used in order to get a response. The baby cry is a natural stimulus to get a response which is food. Learning viewed this way is a change of behavior. Later comes complex stimulus-response conditioning. The complex stimulus-response conditioning is known as classical conditioning of Pavlov. In complex stimulus-response conditioning a second stimulus is introduced, which stimulus is neutral. Dog naturally salivate when they see meat but Pavlov was able to teach a dog to salivate at the sound of a bell  by associating the sound of a bell to the presentation of the meat to the dog. By repeating several times the association meat with the sound of a bell the dog learns to salivate when the bell rings. This process of conditioned learning has been used by humans to live and to create different structures in society.

Learning happens whether we want it or not. In order to learn more complex things ways of learning are necessary. One cannot depend exclusively one someone else to learn as if this person isn't present learning cannot take place. A teacher doesn't force learning to take place. He facilitates and creates conditions for learning. This starts by believing that you can learn. Then you learn the study skills and habits. You need to know the theories, rules and processes in order to learn math.

 Mathematics play an important role in human activities. They are used from simple everyday activities such as personal budgeting, checkbook balancing, groceries shopping to more complicated disciplines such as Economy, Science, Computers, Engineering, etc. The buildings we live in the roads we use, the computers, cellphones, tablets, televisions, etc are designed by people who know math. Calculus is an important branch of mathematics used in various disciplines taught at the college level. The notions of limits are fundamental in understanding some very important notions in Calculus such as Continuity, Derivation and Integrals. I have designed two Calculus courses for learners taking AP Calculus or who will take it. If a student plans to take Calculus as their next math course it's good to start taking them now so that it doesn't look strange to them. They are also designed for students at the high school or college level who need a remediation course. The first one is a free Introductory Calculus course. The second is a complete Calculus course at an affordable price.

The instruction process for this course is designed in the following manner:
1. Students will watch an introductory video. The videos introduce the lessons to the learner
2. There will be some readings to do. The readings expose the learners to the theories of different topics.
3. There will be some problems completely solved. Students should master the solution process of these problems.
3. They will have to solve practice problems demonstrating an understanding of the topics.

Courses in Basic Algebra, Algebra I & II, Geometry, Trigonometry, Pre-Calculus and math for adults are also available. Other face-to-face and online courses in French and  English to Speakers of other Languages  are available on demand. Online and face-to-face tutoring are also available in these subjects.For more information visit New Direction Education Services at If you are interested in the 2 Calculus courses, click on the link at the end of this post. If this is not for you please share the link to people who might be interested. Here is the link: Free Introductory Calculus Course. Complete Calculus course.

Tuesday, March 1, 2016

Mindsets impact mathematics achievement

Study done by the educator Carol Dweck and her colleagues shows that everyone has a learning mindset, a core belief about how they learn. People can have a growth mindset or a fixed mindset. In the Psychology of Learning a growth mindset is the attitude of people who believe that their intelligence can increase with hard work. The learning ability of people with a growth mindset tends therefore to increase. People with a fixed mindset believe that their intelligence is fixed and cannot go beyond their fixed levels. They think that their learning ability is limited. Because of this mindset they think that they can't learn a subject fully and realize great performances at it. These two types of mindsets lead to different kinds of learning behaviors and consequently to different learning outcomes. Learners with a fixed mindset give up easily while those with a growth mindset persist even though their work is hard.

Mindsets impact math achievement. A survey was given to students in a 7th grade class to measure their mindset. The researchers monitored their math achievement over a two years period. The study yields to important results according to the type of students' mindsets. The math achievement of students with a growth mindset tends to progress increasingly while the math achievement of students with a fixed mindset stays constant.

A study about the relationships between beliefs and brain activity shows that the brain  of people  with a growth mindset  reacts differently than that of people with a fixed mindset when they make a mistake. Those with a growth mindset are more aware of their mistakes and willing to fix them. This attitude is different for those with a fixed mindset. Another study supports that students with a growth mindset experience  heightened brain activity and are able to pay more attention to their mistakes.

The brains of all participants to the latter study show some type of activity but  the brain of those with a growth mindset  is likely to show subsequent activities allowing them to be aware of their mistakes.

What are the implications of these studies in learning math or any other subject? These studies show that it's not natural that some individuals are more intelligent than others. Beliefs and mindsets play a great role in people's level of intelligence and their ability to learn. People with a growth mindset or who believe that they can learn if they put some effort have have higher levels of intelligence and increased learning abilities. Those who have a fixed mindset think that their intelligence and leaning abilities are limited. Because of these beliefs they aren't making any effort to learn a subject.

I presently teach and tutor face-to-face and online Math, French, ESL and Spanish. If you believe that you can't learn Math and any of the other subjects above I can work with you to help you to develop a growth mindset. I give away two freebies: a few tutoring sessions in any of the subjects mentioned above and a free Calculus course. For the free Calculus course click this link Introduction to Calculus: notions of limits. For free tutoring by email fill out this form Free Tutoring by Email
For paid tutoring and courses face-to-face and online visit New Direction Education Services at  and click on the contact information button. You can also reach me by email at

Saturday, January 9, 2016

Visualization in mathematics helps students in math learning

There are different ways by which we acquire information. We mainly acquire information from our senses. The two senses mostly used in learning are the eyesight and the hearing. The multiple intelligences theory by Gardner. presently debated, show also other senses besides the traditional senses involved in learning. Teacher's lectures, videos, written materials, manipulatives are the primary ways by which we learn. Written information is widely used in learning and day-to-day activities. Reading and writing play an important part in learning and life. The command of these two techniques can help us tremendously in learning and life. Writiting comes as visual information in symbols. The comprehension of written information involves the mastery of different structures of a language. Visual information comes also in pictures and shapes that aid in the understanding of written information. The word "visualization" is a common word used in computer, psychotherapy, etc. Pictures that can come in different forms and shapes are easier to decode than symbols because they are more related to our personal experiences. Therefore they bring more clarity to coded information. In this article is highlighted the importance of visualization techniques to facilitate the learning of mathematics. We can approximately define "visual mathematics" as the represention of mathematics that are symbolic or not through shapes that correspond more to our actual experiences. Three main highlights are discussed in this article

Visual mathematics are used in basic and high levels of mathematics

Educators in beginning classes of mathematics use manipulatives, games, shapes and pictures to help learners to understand mathematics. Visualization techniques are also used in higher levels of mathematics. Mathematics don't deal only with numbers. Visual representation is a part of the structure of mathematics. Consider algebra that is mainly symbolic. Different shapes are used to represent abstract relations. Diagrams, tables, graphs are used to represent relations and functions . Visualization techniques can be used even in abstract theories and problems. One can invent pictures, graph or any sort of visualization technique to represent abstract situations. The visual representation can especially be useful when it facilitates understanding, higher order of thinking and develops ideas.

Brain research shows that visual mathematics improve student's math performance

Researchers found that when students used visual mathematics they activated another area of their brain besides the one used when using numbers and symbols. The communication and working of these two areas of the brain facilitate math learning. They even state that visualization techniques are more beneficial than numerical techniques of learning math even when students are essentially learning numerical mathematics. It's obvious that when the concrete is used to explain the abstract the understanding of the latter becomes clearer.

Visual mathematics help students to solve problems in different ways.

Visual mathematics are nothing but a visual representation of abstract mathematics. Visual mahematics facilitate individualized learning since students can have different views on visual representation. Not only visual representation facilitates understanding it develops imagination and allows communication to take place between students. They can compare their individual work between each other. They can also discuss problems together. Educators can favor this type of learning by asking students to come up with different ways of solving problems.


There is no doubt that visualization represents an important tool that can facilitate learning. However it can be used for some specific purposes but not as an obsession. Sometimes it might not be needed. When understanding is clear and  there is no need for clarification and depth one can move further.

It is also important to note that a math educator can use different learning techniques to facilitate student's learning comprehension. One is the use of sequential learning. Math is sequential meaning each concept is based on the previous one.It is important that students master previous concepts in order to understand the concept that is actually learned. The sequential nature of mathematics is obvious in the learning of the four basic arithmetic operations. The learning of subtraction is based on that of addition. Without knowing addition one can't do multiplication. Division implies the learning of multiplication and subtraction. As an educator I have found that students who have math learning difficulties don't master the basic calculations. They also don't love mathematics, which is linked to the learning deficiencies in the fundamental notions of mathematics. Study skills are also important in the study of mathematics. I have written about different study techniques in this blog. As a math educator and tutor. my primary task is to instill the love and usefulness of math in students. If you or your child is interested in math tutoring don't hesitate to contact me. You can also refer other learners to me.

Interested in learning to use effective study skills? For free tutoring by email fill out this form:Free Tutoring by Email . For paid tutoring and courses visit New Directions Education Services at