This website is meant for general purpose Islamic WorldViews, while technical educational posts will be placed on a new blog “WEA Pakistan Chapter“, Re-launched and Re-activated, recently for this purpose. A previous post explained the most fundamental and basic concept of (Real) “Statistical Distributions”, a concept which is the fundamental building block of all more complex concepts in probability and statistics. Unfortunately, for complicated reasons, this concept is never actually explained or defined in conventional treatments of the subject. One of the reasons is that probability is treated in an abstract and theoretical way, using a mathematical and axiomatic approach. One of the keys to the “Real Statistics” approach is to discuss all concepts in relation to real world applications. Our treatment of this concept begins with a case where there are no complications created by probability and randomness. We provide a brief review of the previous post on “Understanding Statistical Distributions 1“. The main purpose of this post is to discuss some pedagogical problems in great detail. PP2 stands for Pedagogical Principle 2, which is building confidence of the students. For previous post in this sequence, see PP1 The First Principle of Pedagogy, which is all about making the right intentions.

**Review of Real Distributions**: Let P be a population consisting of N elements {h_{1}, h_{2}, … , h_{N}}. A function F(h) describes characteristics of each element of the population. For example, we could have the Age function, or the Gender function, or the Height Function, and many others. Each function creates categories within the original population – all members of the population which have common characteristics belong to the same category. For example, the Gender function divides the entire population into two categories. If Age is defined extremely accurately, then each person would have a different Age and every category would consist of exactly one person, since everyone would have a different Age. If we define Age in years, then all people with same Age in years would belong to the same category. Within this setup, the DISTRIBUTION of the function is just the proportion of the population which belongs to each category. Before going on to pedagogical principles, we provide some examples to illustrate this concept.

**Example to Illustrate the concept of Real Distributions**: Our population consists of 30 students enrolled in a Statistics Course at a University. The Age function of the students, rounded to the nearest year, creates five categories. There are 2 student of age 17, 5 students of age 18, 12 students of age 19, 6 students of age 20, and 5 students of age 21. The Age distribution of the students can be pictured as follows:

Alternatively, we can write the distribution by listing the percentage belonging to each category – (17:6.66%, 18:16.66%, 19:40%, 20:20%, 21:16.66%). The “laws of probability” say that each percentage is non-negative, and the sum of all percentages equals 100%.

**Pedagogical Principles**: The most important obstacle to learning is created by a lack of self-confidence. Every human being is born with infinite potential, as the best of the creations of God. Every human being can master any kind of knowledge that any other human being has acquired, if he or she receives the right training. *It is not true* that there are different types of human beings, some of whom are extra-ordinarily talented, and can master complex concepts, while other people are born without the required talent, and cannot ever understand more complicated ideas. Everyone has the capability of learning everything,

*if it is broken into steps which are small enough*.

**Learning Failures Caused by Big Jumps**: It is very important to understand the REASONS why we have failed to understand many things which were taught to us in the past. This is not because we are born with some learning defect, or some shortcomings in our background, culture, character. The fault lies in trying to take a learning step which is too large for our current capabilities. If someone can take a step of one foot, and we ask him to cover a distance of three feet in one step, he will fail to be able to do so. Different people have different capabilities, and other students might succeed in learning. Then the student will be convinced that the fault and defect is within him, that he cannot do what others can do easily. He will not understand that the step was too big for him, and he can get to the same goal just by taking smaller steps. It might take him three times the effort that it takes for other students who may have larger step size, but he CAN DO IT, if he puts in the time and effort, AND receives proper guidance.

**Learning Failures Caused by Lack of Connection to Reality: **Another major problem that makes learning difficult is that concepts are introduced which have no relation to anything ever experienced by the student. Almost all standard textbooks today introduce probability theory via the Kolmogorov Axioms on abstract probability spaces. Even though textbooks contain examples with coins, dice, and cards, which are concepts students can relate to, the mathematical formulation is so complex that no one understands how the random variables, defined as real-valued functions on sample spaces, are related to the facts of our daily experiences, such as flipping a coin, or drawing a card at random. Probability starts out on the basis of incomprehensible foundations, and builds superstructures which are even more impenetrable, making it impossible for students to understand what the subject is all about. The point of “Real Statistics” is that all concepts introduced should be related to experience of students. A “Real Distribution”, as defined above, is something easily understood, since classes of students, grouping into categories by age groups, and percentages belonging to each group, are concepts familiar to the students. Many educational experiments show that students will fail to solve problems which they know very well how to solve, if the problems are expressed in unfamiliar language, and in contexts which they do not recognize. At the next stage, we will introduce new concepts which are not familiar, but we will build them on the basis of known concepts so that all new and unknown materials are built out of known and familiar materials. It is failure to follow this Learning Principle that leads students to throw up their hands in despair, and give up on the idea of learning altogether.

**Learning Failures Caused by Inappropriate Challenge Level**: It is well known that learning requires mastering challenging tasks. If the task is too easy, then learning does not occur. If the task is too difficult, then again learning does not occur, and the student gets discouraged by failure, and learns to feel inadequate, creating lack of self-confidence which is a barrier to future learning. So, learning occurs when the student is given a task which is just suitable for his or her personal level of current skills. This means that lessons have to be tailored to students’ capabilities. This is very hard to do in our current educational systems which are designed for mass production of standardized human resources. When we try to move all students together at the same pace, then some of them will fail, and will be made to feel inadequate. These students need specialized attention, where the material that is too complex to grasp in one step, is broken down into smaller pieces so that they can arrive at the same place in a sequence of smaller steps. In our educational systems, the teacher is given the task of “COVERING” the book, regardless of whether students understand the material or not. The teacher proceeds at the speed required to cover all the chapters, even if this speed is too fast for the students to follow. As a result, only a very few students are able to learn anything. The rest of students acquire the feeling that they cannot learn anything, and they just try to survive by doing whatever is necessary to pass the course, without even TRYING to understand the material that is covered in class. Long and bitter experience has taught these students that they CANNOT learn, because whenever they tried to do so, they failed. They do not understand that the failing lies with the amount of material covered, the methodology of teaching, the mismatch between the size of the step required of them, and their own learning capabilities at that time.

**REMEDY**: The remedy involves avoiding the attempt to “teach fish to fly.” Courses have to be tailored to students. This requires re-designing the entire educational process. Models of such educational processes exist in the Islamic past. In fact, the concept of a university, where scholars would be shielded from worldly concerns so that they could devote their lives to learning, was first introduced in the Islamic civilization, because of the overwhelming importance placed on knowledge in Islamic teachings. Using modern technology, we should also be able to take advantage of methods of online learning, supplemented by groups working together. Creating radical changes in educational processes is a long-term project of great value. For immediate use, I would recommend the following steps to the students.

**Building Confidence**: Even though this is among the first steps, it is tricky and subtle. There is the trap of unjustified confidence: students are told to believe in their own abilities, without acquiring skills. This leads to overconfidence, not justified by performance. There is also the trap of under-confidence: learning skills without acquiring confidence. Confidence should be built in proportion to the amount justified by the competence acquired. The following steps are recommended for this purpose.

Give thanks for what you have acquired. Learn to appreciate small steps. For example, if you have understood the simple concept of the “real distribution”, thank Allah T’aala for the understanding you have been given.

الْحَمْدُ لِلَّهِ الَّذِي بِنِعْمَتِهِ تَتِمُّ الصَّالِحَاتُ

*Alhamdulillahil lathee bi ni’matihi tatimmus saalihaat *means: **All praise and thanks are only for Allah, the One who, by His blessing and favor, perfected goodness/good works are accomplished.**

Appreciate the little knowledge that you have been given, instead of worrying about what you have not been given. Giving thanks leads to the multiplication of the gifts of God. Have confidence in the words of God, that we have been created in the best of forms, the best of the creations of God. Trust in Allah for providing you with whatever knowledge you need, despite our own lack of competence and capability. Repair your relationship with God by seeking forgiveness for past carelessness and sins. Cleansing the heart by Istighfar is necessary preparation for the Noor of knowledge to enter the hearts. Regularly before going to sleep, make thanks for all the blessings you have been given that day, and make a promise and a commitment to Allah T’aala to give him the best performance that you can for the next day. Learn the prayers for acquisition of knowledge and use them in the mornings, and before starting studies. Ask Allah T’aala to develop your capabilities so that you provide your best services to the creation of Allah, for the sake of the love of Allah. Understand that knowledge is among the greatest treasures given to Man by Allah T’aala, and seek this knowledge with dedication, passion, and hard work. These spiritual practices will lead to steady progress in learning, so that a journey of a thousand miles can be completed, one step at a time.

In every course I teach, the first lecture is devoted to inspiring and motivating students to give their best performance. In this connection see my lecture for teachers on “How to Inspire and Motivate Students“. See also “ The Ways of the Eagles ” for a lecture to students on how to overcome training to think low, like crows, and instead, learn to soar the skies, like eagles.