Friday, December 2, 2016

Notions of limits

 Before I introduce this lesson I'd like to make some considerations about math learning:

Nowadays many people turn to videos for learning. While this medium is great, written materials shouldn't put away. The written materials allow you to get an overview of what you  are going to learn and give you also the content in advance. 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 properties, formulas, etc. You have to practice a lot if you want to be proficient in math.

Here is the link of a detailed lesson on limits. Below is a guide  of the lesson  

Description of the lesson

This lesson starts by a definition of limits and shows you the three methods of limits using examples. Assignments consist of example assignment, practice assignment and review problems.  

 Outline

1. Objectives

The objectives of the lesson are given here. Read them to know the outcomes of the lesson. 

2. The Idea.

Here is given a conceptual definition of a limit. You can 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  how to find the limit of a function using three methods: graphtable, algebra.

3. Methods for determining limits

a) The graph method

The graph method allows to find the limit of a function using its graph. An example is given that allows to know how to find the limit of a function using its graph. 

After reading the example and mastering the process of finding the limit the function from its graph, you should the two assignments: example assignment and practice assignment. Example assignment consists in doing the example. The practice example consists in doing an exercise that is not solved.

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. If you see from the table that f(x) gets closer and closer to a certain value when x gets closer and closer to the given value, that value represents the limit to the left. You do the same thing to the right of the given value of x to have a second table. If the limit to the right equals to the limit to the left, this limit represents the limit of the function.

After mastering the process of finding a limit of finding a limit using the table method, you should do the example assignment.
  
 Algebra method

This method is very simple but it involves some calculations to do. In this method you substitute x in the function to determine the limit. Here you find the limit directly. There is no limit to the left or the limit to the right to determine..

After mastering the solutions of the exercises that are examples, you do the assignment examples.

 Review problems

The review problems have 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 www.ndes.biz 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

Objectives:

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 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. if you are interested in learning about Calculus, check out this link Center for Integral Development 


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 5 Calculus courses for learners taking AP Calculus who need remediation or who need supplemental resources. These courses can also be taken by independent learners for specific purposes or for their personal enrichment. Three courses are available to subscribe on the site of the courses. The first course covers all the main topics: limits, continuity, derivatives, integrals. The second is titled: "Introduction to Calculus: Functions, Limits, Continuity. The third is on Differential Calculus. The fourth is on Integral Calculus.  

Tutoring is also available on these courses. Tutoring based on other Calculus courses students are taken is also available, Tutoring in other math courses taken in class is also available.

Tutoring in French, ESL and Spanish is also available. All tutoring can be done face-to-face and online. 
 
To subscribe for Calculus courses and tutoring visit Center for Integral Development

For tutoring in other courses visit New Direction Education Services


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  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. For tutoring and private lessons visit in the subjects above visit New Direction Education Services. For learning Calculus online visit Center For Integral Development 

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.

Conclusion

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 www.ndes.biz