The purpose of this blog is to teach Math subjects : Calculus, Pre-calculus, Algebra and Basic Math. Now it focuses on Higher Math. You can also find posts on learning theories, study skills and web resources for learning.
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")
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
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.
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
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
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.
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
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 atwww.ndes.biz
I am adding a few comments about the article: Not a math person: "how to remove obstacles to learning math" written by Katrina Schwartz and published in the online magazine Mind/Shift concerning the pedagogical techniques to remove obstacles in learning math. The author of the article wrote: "Neuroscience research is now showing a strong connection between the attitudes and beliefs students hold about themselves and their academic performance". Our brain is the command center of our physical, emotional, mental and spiritual life. The brain stores also a lot of information recorded consciously or unconsciously. It interprets our surrounding reality and draws its conclusions. One type of conclusions is "the beliefs and attitudes" mentioned in the quote. Our beliefs and attitudes strongly influence our actions and personality. If students believe they can't do math it's obvious that they are not going to make any efforts in order to learn the subject. Their actions will reflect negative views or attitudes about learning math. The beliefs and attitudes originate from the student himself and his social and educational environment. If the social environment including the family cannot do much about developing positive attitudes about learning math this is the role of the school system to favor the development of attitudes and beliefs necessary for learning math. Not only the math teacher develops techniques to help students to learn the subject but he encourages students to "like" the subject. I use the word "like" instead of any other complicated word because I found when a student likes a subject he tends to make efforts in order to perform strongly at it. "Neuroscientists now know that the brain has the abilities to grow and shrink". The fact that our brain grows simply means that we are able to use the brain to think, reflect and solve our problems. We know that every individual uses his brain to live. When we think for a certain period of time about something we solicit the brain's resources in order to help us in order to solve problems. We use a great portion of the "working memory" of our brain to the solution of these problems. However we need to use the brain resources and the "working memory" effectively. Background knowledge helps us to free some parts of the working memory in order to move quickly in the solution of problems. In fact a study done about the strong performance of chess players shows that the memorization of different chess positions helps players to think quicker and gain advantages on their competitor. The background knowledge is important in performing math. Background knowledge is the knowledge of facts and theories so that we don't have to demonstrate them each time we encouter them. The fact that we know the multiplication tables allow us to do the multiplication of different large numbers instead of figuring out each time the multiplication of single numbers. There are math problems that are so complicated that we have to know the method of solutions and the formulas instead of figuring out each time how to solve the same type of problems. Background knowledge of math theories help us to solve complicated math problems. A complicated and abstract subject like math cannot be mastered without knowing its theories. The other technique mentioned in the article is about visualization. Visualization allows to visualize a fact, theory, etc. It helps to figure out something more clearly. It is a tool. It doesn't substitute the knowledge of the subject. Here is an excerpt of the article:
Stanford math education professor Jo Boaler spends a lot of time worrying about how math education in the United States traumatizes kids. Recently, a colleague’s 7-year-old came home from school and announced he didn’t like math anymore. His mom asked why and he said, “math is too much answering and not enough learning.”
This story demonstrates how clearly kids understand that unlike their other courses, math is a performative subject, where their job is to come up with answers quickly. Boaler says that if this approach doesn’t change, the U.S. will always have weak math education.
“There’s a widespread myth that some people are math people and some people are not,” Boaler told a group of parents and educators gathered at the 2015 Innovative Learning Conference. “But it turns out there’s no such thing as a math brain.” Unfortunately, many parents, teachers and students believe this myth and it holds them up every day in their math learning.
There’s no such thing as a math brain.’Jo Boaler, Stanford professor of math education
“We live in a society with lots of kids who don’t believe they are good at math,” Boaler said at an Education Writers Association conference. “They’re put into low groups; they’re given low-level work and their pathway has been set.” But math education doesn’t have to look like this.
Neuroscience research is now showing a strong connection between the attitudes and beliefs students hold about themselves and their academic performance. That’s a departure from the long-held traditional view that academic success is based only on the quality of the teacher and curriculum. But researchers like Carol Dweck, Camille Farrington and David Yeager have shown repeatedly that small interventions to change attitudes about learning can have an outsized effect on performance.
Neuroscientists now know that the brain has the ability to grow and shrink. This was demonstrated in astudy of taxi drivers in London who must memorize all the streets and landmarks in downtown London to earn a license. On average it takes people 12 tries to pass the test. Researchers found that the hippocampus of drivers studying for the test grew tremendously. But when those drivers retired, the brain shrank. Before this, no one knew the brain could grow and shrink like that.
“We now know that when you make a mistake in math, your brain grows,” Boaler said. Neuroscientists did MRI scans of students taking math tests and saw that when a student made a mistake a synapse fired, even if the student wasn’t aware of the mistake. “Your brain grows when you make a mistake, even if you’re not aware of it, because it’s a time when your brain is struggling,” Boaler said. “It’s the most important time for our brains.”
A second synapse fires if the student recognizes his mistake. If that thought is revisited, the initial synapse firing can become a brain pathway, which is good for learning. If the thought isn’t revisited, that synapse will wash away.
A recent study of students with math learning disabilities found in a scan that their brains did behave differently from kids without the disability. “What they saw was the brain lighting up in lots of different areas while working on math,” Boaler said. The children were recruiting parts of the brain not normally involved in math reasoning.
The researchers tutored the group of students with math disabilities for eight weeks using the methods Boaler recommends like visualizing math, discussing problems and writing about math. At the end of the eight weeks, they scanned their brains again and found that the brains of the test group looked just like the kids who did not have math disabilities. This study shows that all kids can learn math when taught effectively. Boaler estimates that only 2 to 3 percent of people have such significant learning disabilities that they can’t learn math at the highest levels.
People who learned math the traditional way often push back against visual representations of math. That kind of thinking represents a deep misunderstanding of how the brain works. “When you think visually about anything, different brain pathways light up than when we think numerically,” Boaler said. The more brain pathways a student engages on the same problem, the stronger the learning.
An example of many ways to visually represent 18*5 (Jo Boaler/YouCubed) (to be continued)
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.wikidot.com
Success skills in school and learning fall in different categories such as: study skills, organization, environment, etc. In this post I will develop four important skills for success in learning and one to avoid.
1. Study skills
Success in study require the ability to use some skills to master the subject. These skills are related to some reading and study techniques. Reading skills are important for success in study. Reading begins before study. In the reading phase some techniques are used such as previewing, skimming, scanning, etc, In the previewing phase one seeks to get a panoramic view of the material without reading the details. One looks at the title, headings, subheadings, pictures of a single piece of the study material. Then one reads the introduction, the first sentence of each paragraph and the conclusion. These techniques are not easy to use especially when one is accustomed in reading line by line. In acquiring the general view one can sketch an outline with the main points of the subject. Once one has finished with the general view one starts by reading the full text. This reading is active since it involves different activities such as outlining important details, thinking, reflecting, etc. The last phase consists in memorizing the important details of the subject.
2. Organization
A calendar is important in order to find time to study. Everyone in school or not has different daily activities. It is important to schedule these activities in order to find time for study, If you are a student in grade school or at the university your time is divided in different blocks of activities, Classes take the majority of the time in school. Extra curricular activities and social events are also included in the school time. There are also personal activities outside the school. For adults who are in school they are very busy and share their time between work, personal activities and family responsibilities. One can use online calendars such as Google calendar to manage time for different activities. There are also some apps for reminders such as Google keep to remind about different activities. Students can use these reminders or hard and computer sticky notes to remind about textbook pages, assignments, etc. Google keep can also be used to take some notes.
3. Disconnection
Disconnect from the internet is the most difficult thing to do since the internet is also used for classwork. In addition one is addicted to social medias, emails and text messages. One has to set the rule that when it's time to study one has to avoid logging in these things. One can set a time for study and another time for the internet to avoid doing both at the same time.
4. Environment
Finding one's convenient environment is important. There are two elements in the environments: the place to study itself and the absence or not of complete quietness. A quite place such as the library or home place reserved especially for study is important. One should avoid to be bothered by other people, telephone, television, etc. If other people are doing other activities that prevent you from concentrating while you are studying this can distract you. If they ask you to do some things you lose time in your study. It doesn't mean that you can't interrupt your study to do some important things for someone else. But these things have to be really important and you are the sole person who can help in time and place. Some people like to learn while they listen to music, Others prefer not, The choice depends on your preferences. You can't study while watching the television, answering phones, sending text messages and logging in social medias. You decrease considerably the time dedicated to study and your concentration when you study and do these things at the same time. Observe the Ecclesiastes principle that says there is a time for each thing.
5. Cramming
Cramming is the process of studying material that one hasn't studied before an examination. Study should be a daily activity. It extends over a certain period of time if one wants to master the material. It involves also constant reviewing. Mastering a certain amount of materials that one hasn't had the opportunity to digest during times where the study of materials is allotted can be hard to do. This requires the display of a considerable amount of energy and can lead to exhaustion.
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