education

The Mitr

Learning science progresses funeral by funeral

So said one of the founders quantum mechanics Max Planck. But I think this quote applies to other areas of human endeavour as well. I have been working in the area of learning for major part of my adult life. During my own learning, when computers were just getting mainstream (late 1990s and early 2000s) I experienced first-hand how learning experience can be enhanced by proper use of computers. Another aspect of proliferation of computer which are connected to the internet is that you have access to sum of almost all human knowledge available to you. Even 20 years back this was not the case. I remember when I discovered that there are accessible resources about physics on the web, it was almost a revelation. And the resources grow day-by-day, becoming more and more accessible to everyone. Even with a smartphone you can access all the information on the web. Most modern web designers are adopting a mobile first policy.

I have been musing about these impacts on learning experience ever since. But there is a strong opposition to use technology in the classroom. This mostly comes from people in two categories. One is a old lot who grew and learned in a world without accessible technology and other is a younger lot who have weird (read extremist) ideas about teaching and learning. The younger lot is a lost tribe who live on the Eastern pole. Both these two categories of people opposed to use of technology in the classroom think that they are “progressive” and are fighting against “oppressive” technology.

A note on the term “technology”: Here I am using the term “technology” in a narrow sense of computer technology. A more inclusive sense would include blackboards, printed textbooks and the classroom itself as forms of technology.

I will try to present this perspective of opposition to technology in classroom and dismantle them giving a rebuttal. In some cases, the holders of these ideas are beyond redemption, and quote of the Max Planck which is the title of the post applies to them. They will die off and their technophobia will die with them. A newer generation of pedagogues conversant and comfortable with technology will emerge in the next generation and will be in tune with the need of the time.

Let us start with the older lot. Many of the progressive pedagogues grew in India that was deprived of any computer technology. This was the era of many socialist inspired people’s movement which aspired for egalitarian approach to education, particularly the sections of society which are low in socio-economic order. The approach was to enlighten the masses inspired from the socialist ideas. Now till 90s, the computer technology was expensive and its use even in the developed countries was rather limited. And most of the people in the older lot I am talking about did spend their formative and working years in this era.

 

Now it is not to say that all of the people did not have any contact with computers at all. Some of these inspired people were highly qualified individuals who did their research work in some of the best institutions in the world. Some of them had some experience of using the computers. But computers were never a second nature to them, as they are not to many people even now. And a lot of them never used computer, because in their era it was an expensive technology and hence they didn’t have access to it. Hence it made computer technology an alien artefact for them in that era.

And when computers finally became accessible, their own years of learning new things had long gone by. Some of them did adopt newer computer technology, able to see the potential to transform both learning and dissemination of knowledge, but most didn’t. Apart from the ideological commitment to a “non-computer” approach to learning, I think their own fears and phobia of being unable to learn and use the new technology also played a role in their opposition. This was the situation in early 2000s, which was still acceptable as computer and internet penetration was not good. These pedagogues threw anything to do with computers as too Western (hence sitting on Eastern pole from where every direction is West).

But by 2010, smart phones were becoming more and more common as were the desktop computers and laptops. By 2015, the access to cheap smart phones with fast internet exploded. Now, here were are in the mid 2020s when proliferation of computer devices in the form of smartphones, tablets and laptops is increasing by the day. The dreams of last mile connectivity are not far off.

The Covid-19 pandemic forced us to shift to online classes. Of course, it did have its issues particularly for students who lacked infrastructure in terms of devices and connectivity. But it did show that even with present conditions something is still possible. Yet, people had their doubts. Now its been

Yet, the resistance from the older pedagogues continues. They cannot get away from ideas about computers that were formed 4 decades back, when computers were still primitive and expensive. And they continue to the same arguments even today. Questions like “Have computers reached everyone?” and since they have not we cannot use them.  Or they give  overarching statements like “The most downtrodden will be neglected in this”. They are like classical physicists who could not accept ideas of modern physics at the turn of the last century.

To objections like these, I have two rebuttals, one is historical-pedagogical and other is on the nature of computer technology in particular. Let us look at the first objection: “Have computers reached everyone?”, of course, they have not! But what about other technologies like the classroom and blackboard? Yes they are technologies! Have they reached everyone? Of course not! But then you don’t give the same arguments, let school reach every child (or every child reach the school) and only then we will allow/accept school as a viable mechanism for learning. And that is something they will never accept, just because they are comfortable/conversant with technology school-classroom-black-board-textbooks. That is a given for them. But even that “technology” has access issues, and comes loaded with challenges of its own for learning. I mean these are the very challenges that many of these pedagogically oriented movements addressed.

So this argument about last-mile connectivity applies to the existing technologies to teaching and learning as well. Why should it be singled out for “computer” technology? This is only because the older lot is not familiar (rather don’t want to accept) with the potential of the computer technology as it would destroy their anachronistic cherished notions of teaching and learning.

Other major assumption in this notion is that the teacher and textbook are the (sometimes the only) source of knowledge, almost an axiom in the Euclidean sense.

Is this why there was so much focus on developing text-based teaching learning materials. But this is no longer true. We now have almost entire sum of human knowledge accessible literally at fingertips to anyone with a connected device. But now with Open Education and internet this is being challenged in a serious way. Added to this is the absolutely disruptive technology of AI bots like chatGPT. Why should learning be limited to a centralised textbook which usually does not take into account the context of learners written by folks sitting in ivory towers, which is not updated for years?

Now we have the technology and appropriate legal licenses to change this by really empowering learners to bypass the filters of textbooks and teachers. But still we are hung on cherished notion of teacher in the constructivist classroom.

Now, I come to aspects of the nature of technology and young learners. The nature of “computer” technology is such that younger learners adapt to it very quickly. They are still in the phase of learning about the world. A very young child given a smartphone will try out everything and figure out how it works (or doesn’t) and start playing with it as if its any other toy. Parents often ask help from their very young children to solve technological challenges they face.

Same is true for teachers. I have seen enough examples during my field work in very rural areas. Learners when exposed to computer technology even for a very short time several were first-generation learners who were using computer for the first time, could out-pace the teacher in using the computer for the task at hand. Now in the traditional approach (even the progressive ones) the knowledge of teacher is almost never surpassed in a teaching-learning setting. The teacher is always the “more-able-peer” in the Vygotskian sense and is considered as an a priori truth. Now, I am not denying that in many senses this is correct, but if you give access to technology to young learners in many cases the need for teacher is bypassed. This is in the true sense that a child constructs knowledge with the only difference being that it is not mediated by the teacher (or even if it is teacher is exactly a mediator). Constructionist microworlds provide excellent examples of such learning by the learners on their own. By denying access to computer technology, this is what is being missed.

Of course there are examples and examples of bad use of technology in the form of PPT/click books etc which is often rightfully criticised. But that is missing forest of the trees. Another point in this regard is that educational technology abhors vacuum, if any technology is not opted by good pedagogy, it will be co-opted by a bad one. So we need to stake claim, otherwise poor pedagogical approaches which just replicate what is done without a computer to be done on a computer. To give Papert’s analogy it would be like attaching a jet engine to a horse-wagon!

Now we come to the younger lot. They typically have grown with technology in their formative years. And as reasearchers and activists they use computers and internet and are familiar with the technologies. Yet they give the same arguments of “oppression” as the older lot. I mean it doesn’t occur to them they are using the same computers for their own work because it suits them. But when it comes to use in the classroom it is not to be used. Double standards much. If they think the computers are oppressive as much, they should stop using it themselves. But then how will they post social media updates on Facebook and Twitter?

The younger generation of people who oppose technology in the classroom fits very well in the category of people without skin in the game (after Taleb). For things they think computer is useful for themselves, they will make full use of it be it data analysis, report writing or other work. But they are not ready to give same concession to children (especially in resource deprived areas) who need such scaffolding more. Instead they want to deprive the children of a learning companion because it does not work with their ideological world view centred at Eastern pole. And some of these same researchers, when it comes to their own children will provide them with computers and tablets for learning. But when it comes to the children who need it more…

I could go on and on.. But you get the point, the opposition is not based on factual aspects but ideological and there too they are on thin ice. But the opposition is waning funeral by funeral and computers are the new normal…

 

 

 

 

 

 

The Mitr

Remarkable Curves 1: The Straight Line and the Circle

 

 

 

 

The Straight Line and the circle

(Note: All the images are interactive, move the points around to see. Dynamic mathematics interactive web page with Cinderella )

A moving point describes a straight line when it passes from one
position to another along the shortest possible path. A straight line
can be drawn with the help of a ruler; when a pencil runs along the
edge of a ruler it leaves a trace on the paper in the form of a
straight line.

When a point moves on a surface at a constant distance from another
fixed point on the same surface it describes a circle. Because of this
property of the circle we are able to draw a circle with the help of
compasses.

The straight line and the circle are the simplest and at the same time
the most remarkable curves as far as their properties are concerned.

You are no doubt more familiar with these two curves than with
others. But you should not imagine that you know all of the most
important properties of straight lines and curves. For example, you
may not know that if the vertices of the triangles $ABC$ and $AB’C’$
lie on three straight lines intersecting at the point $S$ (Fig. 1),
the three points of intersection $M$, $K$, $L$ of the corresponding
sides of the triangles, the sides $AB$ and $A’B’$, $BC$ and $B’C’$,
and $AC$ and $A’C’$, must be collinear, that is, they lie on a single
straight line.

(Note: this image below is interactive, move the points to see the dynamic change!)

You are sure to know that a point $M$ moving in a plane equidistantly
from two fixed points, say $F_1$, and $F_2$, of the same plane, that
is, so that $MF_{1}= MF_{2}$, describes a straight line (Fig. 2).

But you might find it difficult to answer the question:

What type of curve will point $M$ describe if the distance of $M$ from
$F_1$, is a certain number of times greater than that from $F_2$ (for
instance, in Fig. 3 it is twice as great)?

The curve turns out to be a circle. Hence if the point $M$ moves in a
plane so that the distance of $M$ from one of the fixed
points. $F_{1}$ or $F_{2}$, in the same plane is always proportional
to the distance from the other fixed point, that is

$$
MF_{1} = k \times MF_{2}
$$

then $M$ describes either a straight line (when the factor of
proportionality is unity) or a circle (when the factor of
proportionality is other than unity).

 

This is a post to create interactive mathematics elements using Cinderella a Free Software alternative to GeoGebra which is no longer a Free Software. The files have been exported from Cinderella at html interactives)

The Mitr

What is the concept of basic education?

Introduction

The Government of India are keenly interested in promoting the pattern of Basic education in the country and h ave adopted a number of measures for this purpose. These include programmes of expansion as well . as steps to improve the quality of work ·in Basic schools and their methods and techniques. If this
objective is to be achieved, it is essential that all those who work in the field should have a clear concept of what Basic education means and should be able to distinguish between its essential features and what are mainly matters of detail to b e adjusted in the light of local needs and developing experience.

This Statement about the Concept of Basic Education , which has been prepared by the Basic Education Standing Committee of the Central Advisory Board of Education, is meant to highlight its significant features and. to remove possible misunderstandings. I hope our educationists will ponder over it and try to work out the scheme in the spirit that it envisages, so that Basic education may play its proper role in inculcating the right attitudes and ideals of character and efficiency in our children.
A.K. Azad

 

The Concept Of Basic Education

The term ‘Basic Education’ has been interpreted — and sometimes misinterpreted — in a. variety of ways. This is, to some extent understandable because it is a comparatively recent development and its concept and technique are still in the making. It seems necessary therefore, to state clearly what is meant by
Basic education.

Broadly speaking, it may be stated that the concept of Basic education is the same as defined in the Report of the Basic National Education Committee (the Zakir Husain Committee) and elucidated by the Central Advisory Board of Education. It is clear that the basic principles and techniques, as made out in that Report, should guide and shape educational reconstruction in India.So far as the provision of eight years of compulsory universal schooling and the use of the mother tongue as the medium of instruction are concerned, there is now no difference of opinion about them. They have come to be universally accepted and need no further elucidation, except in so far as it may be necessary to stress the intrinsic wholeness of the entire period of Basic education, covering the Junior as well as Senior Basic grades. The other implications and features of Basic education that need to be clarified and stressed are the following:

  1. Basic education, as conceived and explained by Mahatma Gandhi, is essentially an education for life, and what· is more, an education through life. It aims at creating eventually a social order free from exploitation and violence. That is why productive, creative and socially useful work in which all boys and girls may. participate, irrespective of any distinction of caste or creed or class, is placed at the very centre of Basic education.
  2. The effective teaching of a basic craft, thus, becomes an essential part of education at this stage, as productive work, done under proper conditions, not only makes the acquisition of much related knowledge more concrete and realistic but also adds a powerful contribution to the development of personality and character and instills respect and love for all socially useful work. It is also to be clearly understood that the sale of products of craft work may be expected to contribute towards part of the expenditure on running the school or that the product s will be used by the school children for getting a midday meal or a school uniform or help to provide some of the school furniture and equipment.
  3. As there has been controversy and difference of opinion regarding the position of craft work in Basic schools, it is necessary to state clearly that the fundamental objective of Basic education is nothing less than the development of the child’s total personality which will include productive efficiency as well. In order to ensure that the teaching of the basic craft is efficient and its educative possibilities are fully realised we must insist that the articles made should be of good quality, as good as children at that stage of their development can make them, socially useful and, if necessary, saleable. The acquisition of skills and the love for good craftsmanship have deeper educative significance than merely playing with the tools and raw materials which is usually encouraged in all good activity schools. This productive aspect should in no case be relegated t o the background as has been usually the case so far, because directly as well as indirectly, efficiency in the craft practised undoubtedly contributes to the all-round development of the child; but on the other hand, never should the productive aspect be allowed to take precedence over the educational aspect. It sets up before children high standards of achievement and gives them the right kind of training in useful habits and attitudes like purposeful application, concentration, persistence and thoughtful planning. While it may not be possible to lay down specific targets for productivity at this stage, it should be the teacher’s endeavour to explore its economic possibilities fully with the emphatic stipulation that this does not in any way conflict with the educational aims and objectives already defined. However, it has to be stated that, in the upper classes of Junior Basic schools and in the Senior Basic schools, it should not be difficult for States to lay down certain minimum targets of production in the light of carefully assessed experiences.
  4. In the choice of basic crafts which are to be integrated into school work, we should adopt a liberal approach and make use of such crafts as have significance from the point of view of intellectual content, provide scope for progressive development of knowledge and practical efficiency. The basic craft must be such as will fit into the natural and social environment of the school and hold within it the maximum of educational possibilities. The idea that has been wrongly created in the minds of
    some people that the mere introduction of a craft in a school, e.g., spinning, can make it a Basic school does grave injustice to the concept of Basic education.
  5. In Basic education as, indeed, in any good scheme of education, knowledge must be related to activity, practical experience and observation. To ensure this·, Basic education rightly postulates that the study of the curricular content should be intelligently related to three main centres of correlation viz., craft work, the natural environment and the social environment. The well trained and understanding teacher should be able to integrate most of the knowledge that he wishes to impart to one or the other of these centres of correlation, which form the important and natural foci of interest for the growing child. If, therefore, in-the Junior Basic stage he is not able to do so, it either means that he lacks the necessary ability or that the curriculum has been burdened with items of knowledge which are not really important and significant at that particular stage. It should also be realised, however, that there may be certain items in the syllabus which cannot be easily correlated directly with any of the three above centres. In such cases, which should occur only infrequently, there should be no objection to these being taught according to the methods of teaching adopted in any good school. This means that even in the case of .such lessons, the principle of interest and motivation and the value of expression-work will be utilised. In any case, forced and mechanical ‘associations’ which pass for correlation in many schools should be carefully avoided.
  6. The emphasis on productive work and crafts in Basic schools should not be taken to mean that the study of books can be ignored. The Basic scheme does postulate that the book is not the only or the main avenue to knowledge and culture and that, at this age, properly organised productive work can in many ways contribute more richly both to the acquisition of knowledge and the development of personality. But the value of the book, both as a source of additional systematised knowledge and of pleasure cannot be denied and a good library is as essential in a Basic school as in any other·type of good school.
  7. The Basic scheme envisages a close integration between the schools and the community so as to make education as well as the children more social-minded and cooperative. It endeavours to achieve this, firstly, by organising the school itself as a living and functioning community — with its social and cultural programmes and other activities — secondly, by encouraging students to participate in the life around the school and in organising various types of social service to the local community. Student self-government is another important feature in Basic education which should be envisaged as a continuous programme of training in responsibility and in the democratic way of living. In this way, the Basic school not only helps in cultivating qualities of self-reliance, cooperation and respect for dignity of labour, but also becomes a vital factor in the creation of a dynamic social order.
  8. Basic education should no longer be regarded as meant exclusively for the rural areas. It should be introduced in urban areas as well, both because of its intrinsic suitability and also to remove the impression that it is some inferior kind of education designed only for the village children. For this purpose, necessary modifications may have to be made in the choice of basic crafts for urban schools and even in the syllabus but the general ideals and methods of Basic education should remain the same.

 

from

The Concept of Basic Education, Ministry of Education and Scientific Research Government of India (1957).

The Mitr

What is the mathematical significance of the constant C in an indefinite integral?

As we had seen in an earlier post, calculus bottleneck, calculus presents one of the most difficult topics for the students in higher mathematics. But the problem is not just limited to the students. Teachers feel it too. Too often the emphasis is given on how to solve integration and differentiation problems using “rules” and “methods” while the essence of what is happening is lost. Recently, I asked this question in an interview to a mathematics teacher who was teaching indefinite integration. This teacher had almost a decade of experience in teaching mathematics at +2 level. The teacher tried to answer this question by using an example of the function \(x^{2} + 5 \). Now when we take the derivative of this function, we get

\[ \dv{  (x^{2} + 5)}{x} = \dv{x^{2}}{x} + \dv{5}{x} = 2x \]

as derivative of a constant (5 on our case) is 0). Now the teacher tried to argue, that integration is the reverse of the derivative), so

\[ \int 2x \, \dd x = \frac{2x^{2}}{2} + C = 2x + C \]

After this the teacher tried to argue this \( C \) represents the constant term (5) in our function \(x^{2} + 5 \). He tried to generalise the result, but he was thinking concretely in terms of the constant in the form of the numbers in the function. The teacher could understand the mechanism of solving the problem, but was not able to explain in clear mathematical terms, why the constant \( C \) was required in the output of the indefinite integral. This difficulty, I think, partly arose because the teacher only thought in terms solving integrals and derivatives in a particular way, and knew about the connection between the two, but not in a deep way. He did in a way understood the essence of the constant \( C \), but was not able to understand my question as a general question and hence replied only in terms of concrete functions. Even after repeated probing, the teacher could not get the essence of the question:

why do we add a constant term to the result of the indefinite integral?

To put it in another words, he was not able to generalise an abstract level of understanding from the examples that were discussed. The teacher was thinking only in terms of symbol manipulation rules which are sufficient for problem solving of these types. For example, look at the corresponding rules for differentiation and integration of the function \(x^{n} \).

\[ \dv{x^{n}}{x} = n x^{n-1}  \iff  \int x^{n} \dd x = \frac{x^{n+1}}{n} + C \]

Thus, we see according to above correspondence that adding any extra constant \( C \) to the derivative formula will not affect it. So the teacher claimed it is this constant which appears in the integration rule as well. In a way this is a sort of correct explanation, but it does not get to the mathematical gist of why it is so. What is the core mathematical idea that this constant \( C \) represents.

Another issue, I think, was the lack of any geometrical interpretation during the discussion. If you ask, what is the geometrical interpretation of the derivative you will get a generic answer along the lines: “It is the tangent to the curve” and for integration the generic answer is along the lines “It is the area under the curve”. Both these answers are correct, but how do these connect to the equivalence above? What is the relationship between the tangent to the curve and area under the curve which allows us to call the integral as the anti-derivative (or is the derivative an anti-integral?). I think to understand these concepts better we have to use the geometrical interpretation of the derivative and the integral from their first definitions.

The basic idea behind the derivative is that it represents the rate of change of a function \( f \) at a given point. This idea, for an arbitrary function, can be defined (or interpreted) geometrically as:

The derivative of a function \( f \) at a point \( x_{0} \) is defined by the slope of the tangent to the graph of the function \( f \) at the point  \(x =  x_{0} \).

The animation below shows how the slope of the tangent to the sine curve changes. Point \( B \) in the animation below plots the \( (x, m) \), where \( m \) is the slope of the tangent for the given value of \( x \). Can you mentally trace the locus of point \( B \)? What curve is it tracing?
Now, the tangent to any point on a curve is unique. (Why is it so?) That means if one evaluates a derivative of a function at a point, it will be a unique result for that point.
This being cleared, now let us turn to the indefinite integral. One approach to understanding integration is to consider it as an inverse operation to the derivative, i.e. an anti-derivative.

An anti-derivative is defined as a function \( F(x) \) whose derivative equals an initial function \(f (x) \):

\[ f(x)= \dv{ F(x)}{x} \]

Let us take an example of the function \( f(x) = 2x^{2} – 3x \). The differentiation of this function gives us its derivative \(f'(x) = 4x – 3 \), and its integration gives us anti-derivative.
\[ F(x) = \frac{2}{3} x^{3} – \frac{3}{2} x^2 \]

This anti-derivative can be obtained by applying the known rules of differentiation in the reverse order. We can verify that the differentiation of the anti-derivative leads us to the original function.

\[ F'(x) = \frac{2}{3} 3 x^{2} – \frac{3}{2} 2 x = 2x^{2} – 3x \]

Now if add a constant to the function \( F(x) \), lets say number 4, we get a function \( G(x) = \frac{2}{3} x^{3} – \frac{3}{2} x^2 + 4 \). If we take the derivative of this function \( G(x) \), we still get our original function back. This is due to the fact that the derivative of a constant is zero. Thus, there can be any arbitrary constant added to the function \( F(x) \) and it will still be the anti-derivative of the original function \( f(x) \).

An anti-derivative found for a given function is not unique. If \( F (x) \) is an anti-derivative (for a function \( f \) ), then any function \( F(x)+C \), where \( C \) is an arbitrary constant, is also an anti-derivative for the initial function because
\[
\dv{[F(x)+C]}{x} = \dv{ F(x)}{x} + \dv{ C}{x}= \dv{ F(x) }
\]

But what is the meaning of this constant \( C \)? This means, that each given function \( f (x) \) corresponds to a family of anti-derivatives, \( F (x) + C \). The result of adding a constant \( C \) to any function is that it shifts along the \( Y \)-axis.

Thus what it means for our case of result of the anti-derivative, the resultant would be a family of functions which are separated by \( C\). For example, let us look at the anti-derivative of \( f (x) = \sin x \). The curves of anti-derivatives for this function are plotted in will be of the form

\[
F ( x ) = − \cos x + C
\]

A family of curves of the anti-derivatives of the function \( f (x) = \sin x = –  \cos x \)

And this is the reason for adding the arbitrary constant \( C\) to our result of the anti-derivative: we get a family of curves and the solution is not unique.

Now can we ever know the value of \( C\)? Of course we can, but for this we need to know the some other information about the problem at hand. These can be initial conditions (values) of the variables or the boundary condition. Once we know these we can determine a particular curve (particular solution) from the family of curves for that given problem.

Further Reading

Lev Tarasov – Calculus – Basic Concepts For High Schools (Starts with and explains  the basic mathematical concepts required to understand calculus. The book is in the form of a dialogue between the author and the student, where doubts, misconceptions and aha moments are discussed.)

Morris Kline – Calculus – A physical and intuitive approach (Builds the concepts in the context of the physical problems that calculus was invented to solve.  A book every physics student should read to get an understanding of how mathematics helps solve physical problems.)

Richard  Courant and Fritz John – Introduction to Calculus Analysis (In 2 Volumes) (Standard college level text with in-depth discussions. First volume is rigorous with basic concepts required to conceptually understand the topics and their applications/implications.)

The Mitr

Interesting LaTeX Packages – Drawing the Solar System

Many times we need to quickly illustrate the solar system or the planets. Usually we use photos to illustrate the planets. But sometimes the photos can be an over kill. Also to draw the entire solar system is a typical and can be used for illustrative purposes. Though most of the illustrations of the solar system in the school and other text books are horribly out of scale, (with no indication that the figure is not to scale!). For example,  look at the illustration in the Class 6 Science Textbook from NCERT.

A good visualisation will always present a scale, and/or indicate whether the visualisation is to the scale or not. Though in the visualisation above the distances are given, they are not to scale.

 

Coming back to the topic of our post, a simple way to draw solar system diagram in latex is to use the PStricks package solarsystem. The package can create “Position of the visible planets, projected on the plane of the ecliptic” at a given time and date. This feature might be useful sometimes.

From the package manual:

As we can not represent all the planets in the real proportions, only Mercury, Venus, Earth and Mars are the proportions of the orbits and their relative sizes observed. Saturn and Jupiter are in the right direction, but obviously not at the right distance.
The orbits are shown in solid lines for the portion above the ecliptic and dashed for the portion located below.

The use of the command is very simple, just specify the date of observation with the following parameters, for example:
\SolarSystem[Day=31,Month=12,Year=2020,Hour=23,Minute=59,Second=59]
By default, if no parameter is specified, \SolarSystem gives the configuration day 0 hours to compile.

The resulting output for the above code:

The output also provides the longitude and latitude of the planets at the time given.

Another package that is useful to create free standing planets is the tikz-planets package which we will see next.

The Mitr

John Tukey on data based pictures and graphs

John Tukey‘s wisdom on importance and value of graphics and pictures in making sense of exploring data.

Consistent with this view, we believe, is a clear demand that pictures based on exploration of data should force their messages upon us. Pictures that emphasize what we already know — “security blankets” to reassure us — are frequently not worth the space they take. Pictures that have to be gone over with a reading glass to see the main point are wasteful of time and inadequate of effect. The greatest value of a picture is when it forces us to notice what we never expected to see. (p. vi emphasis in original)

John Tukey – Exploratory Data Analysis

The Mitr

Galileo’s Experiments on Accelerated Motion

A short account of Galileo’s description of his own experiment on accelerated motion — a short account of it, the apparatus he used and the results he got.
The first argument that Salviati proves is that in accelerated motion the change in velocity is in proportion to the time (𝑣 ∝ 𝑡) since the motion began, and not in proportion to the distance covered (𝑣 ∝ 𝑠) as is believed by Sargedo.

“But for one and the same body to fall eight feet and four feet in the same time is possible only in the case of instantaneous (discontinuous) motion; but observation shows us that the motion of a falling body occupies time, and less of it in covering a distance of four feet than of eight feet; therefore it is not true that its velocity increases in proportion to the space. (Salviati)

Also, he proves that the increase in proportion is not of simple doubling but larger. They agree upon a definition of uniformly accelerated motion,

“A motion is said to be equally or uniformly accelerated when, starting from rest, its momentum receives equal increments in equal times. (Sargedo)

To this definition Salviati adds an assumption about inclined planes, this assumption is that for a given body, the increase in speed while moving down the planes of difference inclinations is equal to the height of the plane. This also includes the case if the body is dropped vertically down, it will still gain the same speed at end of the fall as it would gain from rolling on the incline This assumption makes the final speed independent on the profile of the incline. For example, in the figure below, the body falling along𝐶 → 𝐵, 𝐶 → 𝐷 and 𝐶 → 𝐴 will attain the same final speed.

This result is also proved via a thought experiment (though it might be feasible to do this experiment) for a pendulum. The pendulum rises to the height it was released from and not more.
After stating this theorem, Galileo then suggests the experimental verification of the theorem. of The actual apparatus that Galileo uses is an wooden inclined slope of following dimensions: length 12 cubits (≈ 5.5 m, 1 cubit ≈ 45.7 cm), width half-cubit and three-finger breadths thick . In this plank of wood, he creates a very smooth groove which is about a finger thick. (What was the thickness of Galileo’s fingers?) The incline of this plank are changed by lifting one end. A bronze ball is rolled in this groove and time taken for descent is noted.

“We repeated this experiment more than once in order to measure the time with an accuracy such that the deviation between two observations never exceeded one- tenth of a pulse-beat.

Then Galileo performed variations in the experiment by letting the ball go different lengths (not full) of the incline and “found that the spaces traversed were to each other as the squares of the times, and this was true for all inclinations of the plane”. Each variation was repeated hundreds of times so as to rule out any errors. Also, the fact that for different inclines the times of descent were in noted and were in agreement with the predictions.
Since there were no second resolution clocks to measure time, Galileo devised a method to measure time using water. This was not new, water clocks were used earlier also.

The basic idea was to the measure the amount of water that was collected from the start of the motion to its end. The water thus collected was weighed on a good balance.This weight of water was used as a measure of the time. A sort of calibration without actually measuring the quantity itself: “the differences and ratios of these weights gave us the differences and ratios of the times”

Galileo used a long incline, so that he could measure the time of descent with device he had. If a shorted incline was used, it would have been difficult to measure the shorter interval of time with the resolution he had. Measuring the free fall directly was next to impossible with the technology he had. Thus the extrapolation to the free fall was made continuing the pattern that was observed for the “diluted” gravity.

“You present these recondite matters with too much evidence and ease; this great facility makes them less appreciated than they would be had they been presented in a more abstruse manner. For, in my opinion, people esteem more lightly that knowledge which they acquire with so little labor than that acquired through long and obscure discussion. (Sargedo)

 

Reference

Dialogues Concerning Two New Sciences

William Francis Magie A Source Book in Physics, New York, Harvard University Press 1935
The Mitr

Bertrand Russel’s proof of naïve realism being false

What is naïve realism you may ask? To put simply naïve realism is a belief that whatever you see with your senses is the reality. There is nothing more to reality than what your sense perceptions bring to you. It is a direct unmediated access to reality. There is no “interpretation” involved.

In philosophy of perception and philosophy of mind, naïve realism (also known as direct realism, perceptual realism, or common sense realism) is the idea that the senses provide us with direct awareness of objects as they really are. When referred to as direct realism, naïve realism is often contrasted with indirect realism.

Naïve Realism

To put this in other words, naïve realism fails to distinguish between the phenomenal and the physical object. That is to say, all there is to the world is how we perceive it, nothing more.

Bertrand Russel gave a one line proof of why naïve realism is false. And this is the topic of this post. Also, the proof has some implications for science education, hence the interest.

Naive realism leads to physics, and physics, if true, shows that naive realism is false. Therefore naive realism, if true, is false; therefore it is false.

As quoted in Mary Henle – On the Distinction Between the Phenomenal and the Physical Object, John M. Nicholas (ed.), Images, Perception, and Knowledge, 187-193. (1977)

Henle in her rather short essay (quoted above) on this makes various philosophically oriented arguments to show that it is an easier position to defend when we make a distinction between the two.

But considering the “proof” of Russel, I would like to bring in evidence from science education which makes it even more compelling. There is a very rich body of literature on the theme of misconceptions or alternative conceptions among students and even teachers. Many of these arise simply because of a direct interpretation of events and objects around us.

Consider a simple example of Newton’s first law of motion.

In an inertial frame of reference, an object either remains at rest or continues to move at a constant velocity, unless acted upon by a force.

Now for the naïve realists this will never be possible, as they will never see an object going by itself without application of any force. In real world, friction will bring to halt bodies which are moving. Similar other examples from the misconceptions also do fit in this pattern. This is perhaps so because most of the science is counter-intuitive in nature. With our simple perception we can only do a limited science (perhaps create empirical laws). So one can perhaps say that learners with alternative conceptions hold naïve realist world-view (to some degree) and the role of science education is to change this.