Examples of physics quiz with H5P content type

This post has two examples from the same problem using H5P multiple choice and fill in the blank content types.

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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

$F ( x ) = − \cos x + C$

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.

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.)

An example use of H5P item for creating hotspots

Just trying out different H5P items. This item has a flower  and hotspots indicating parts of the flower.

Click on the question marks to know more..

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.

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

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

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.

Experiments, Data and Analysis

There are many sad stories of students, burning to carry out an experimental project, who end up with a completely unanalysable mishmash of data. They wanted to get on with it and thought that they could leave thoughts of analysis until after the experiment. They were wrong. Statistical analysis and experimental design must be considered together…
Using statistics is no insurance against producing rubbish. Badly used, misapplied statistics simply allow one to produce quantitative rubbish rather than qualitative rubbish.
–  Colin Robson (Experiment, Design and Statistics in Psychology)

Why philosophy is so important in science education

This is a nice article whicH I have reposted from AEON…

Each semester, I teach courses on the philosophy of science to undergraduates at the University of New Hampshire. Most of the students take my courses to satisfy general education requirements, and most of them have never taken a philosophy class before.
On the first day of the semester, I try to give them an impression of what the philosophy of science is about. I begin by explaining to them that philosophy addresses issues that can’t be settled by facts alone, and that the philosophy of science is the application of this approach to the domain of science. After this, I explain some concepts that will be central to the course: induction, evidence, and method in scientific enquiry. I tell them that science proceeds by induction, the practices of drawing on past observations to make general claims about what has not yet been observed, but that philosophers see induction as inadequately justified, and therefore problematic for science. I then touch on the difficulty of deciding which evidence fits which hypothesis uniquely, and why getting this right is vital for any scientific research. I let them know that ‘the scientific method’ is not singular and straightforward, and that there are basic disputes about what scientific methodology should look like. Lastly, I stress that although these issues are ‘philosophical’, they nevertheless have real consequences for how science is done.
At this point, I’m often asked questions such as: ‘What are your qualifications?’ ‘Which school did you attend?’ and ‘Are you a scientist?’
Perhaps they ask these questions because, as a female philosopher of Jamaican extraction, I embody an unfamiliar cluster of identities, and they are curious about me. I’m sure that’s partly right, but I think that there’s more to it, because I’ve observed a similar pattern in a philosophy of science course taught by a more stereotypical professor. As a graduate student at Cornell University in New York, I served as a teaching assistant for a course on human nature and evolution. The professor who taught it made a very different physical impression than I do. He was white, male, bearded and in his 60s – the very image of academic authority. But students were skeptical of his views about science, because, as some said, disapprovingly: ‘He isn’t a scientist.’
I think that these responses have to do with concerns about the value of philosophy compared with that of science. It is no wonder that some of my students are doubtful that philosophers have anything useful to say about science. They are aware that prominent scientists have stated publicly that philosophy is irrelevant to science, if not utterly worthless and anachronistic. They know that STEM (science, technology, engineering and mathematics) education is accorded vastly greater importance than anything that the humanities have to offer.
Many of the young people who attend my classes think that philosophy is a fuzzy discipline that’s concerned only with matters of opinion, whereas science is in the business of discovering facts, delivering proofs, and disseminating objective truths. Furthermore, many of them believe that scientists can answer philosophical questions, but philosophers have no business weighing in on scientific ones.
Why do college students so often treat philosophy as wholly distinct from and subordinate to science? In my experience, four reasons stand out.
One has to do with a lack of historical awareness. College students tend to think that departmental divisions mirror sharp divisions in the world, and so they cannot appreciate that philosophy and science, as well as the purported divide between them, are dynamic human creations. Some of the subjects that are now labelled ‘science’ once fell under different headings. Physics, the most secure of the sciences, was once the purview of ‘natural philosophy’. And music was once at home in the faculty of mathematics. The scope of science has both narrowed and broadened, depending on the time and place and cultural contexts where it was practised.
Another reason has to do with concrete results. Science solves real-world problems. It gives us technology: things that we can touch, see and use. It gives us vaccines, GMO crops, and painkillers. Philosophy doesn’t seem, to the students, to have any tangibles to show. But, to the contrary, philosophical tangibles are many: Albert Einstein’s philosophical thought experiments made Cassini possible. Aristotle’s logic is the basis for computer science, which gave us laptops and smartphones. And philosophers’ work on the mind-body problem set the stage for the emergence of neuropsychology and therefore brain-imagining technology. Philosophy has always been quietly at work in the background of science.
A third reason has to do with concerns about truth, objectivity and bias. Science, students insist, is purely objective, and anyone who challenges that view must be misguided. A person is not deemed to be objective if she approaches her research with a set of background assumptions. Instead, she’s ‘ideological’. But all of us are ‘biased’ and our biases fuel the creative work of science. This issue can be difficult to address, because a naive conception of objectivity is so ingrained in the popular image of what science is. To approach it, I invite students to look at something nearby without any presuppositions. I then ask them to tell me what they see. They pause… and then recognise that they can’t interpret their experiences without drawing on prior ideas. Once they notice this, the idea that it can be appropriate to ask questions about objectivity in science ceases to be so strange.
The fourth source of students’ discomfort comes from what they take science education to be. One gets the impression that they think of science as mainly itemising the things that exist – ‘the facts’ – and of science education as teaching them what these facts are. I don’t conform to these expectations. But as a philosopher, I am mainly concerned with how these facts get selected and interpreted, why some are regarded as more significant than others, the ways in which facts are infused with presuppositions, and so on.
Students often respond to these concerns by stating impatiently that facts are facts. But to say that a thing is identical to itself is not to say anything interesting about it. What students mean to say by ‘facts are facts’ is that once we have ‘the facts’ there is no room for interpretation or disagreement.
Why do they think this way? It’s not because this is the way that science is practised but rather, because this is how science is normally taught. There are a daunting number of facts and procedures that students must master if they are to become scientifically literate, and they have only a limited amount of time in which to learn them. Scientists must design their courses to keep up with rapidly expanding empirical knowledge, and they do not have the leisure of devoting hours of class-time to questions that they probably are not trained to address. The unintended consequence is that students often come away from their classes without being aware that philosophical questions are relevant to scientific theory and practice.
But things don’t have to be this way. If the right educational platform is laid, philosophers like me will not have to work against the wind to convince our students that we have something important to say about science. For this we need assistance from our scientist colleagues, whom students see as the only legitimate purveyors of scientific knowledge. I propose an explicit division of labour. Our scientist colleagues should continue to teach the fundamentals of science, but they can help by making clear to their students that science brims with important conceptual, interpretative, methodological and ethical issues that philosophers are uniquely situated to address, and that far from being irrelevant to science, philosophical matters lie at its heart.

Subrena E Smith
This article was originally published at Aeon and has been republished under Creative Commons.

Science Education and Textbooks

What are the worst possible ways of approaching the textbooks for teaching science? In his book Science Teaching: The Role of History and Philosophy of Science pedagogue Michael Matthews quotes (p. 51) Kenealy in this matter. Many of the textbooks of science would fall in this categorisation. The emphasis lays squarely on the content part, and that too memorized testing of it.

Kenealy characterizes the worst science texts as ones which “attempt to spraypaint their readers with an enormous amount of ‘scientific facts,’ and then test the readers’ memory recall.” He goes on to observe that:
Reading such a book is much like confronting a psychology experiment which is testing recall of a random list of nonsense words. In fact, the experience is often worse than that, because the book is a presentation that purports to make sense, but is missing so many key elements needed to understand how human beings could ever reason to such bizarre things, that the reader often blames herself or himself and feels “stupid,” and that science is only for special people who can think “that way” … such books and courses have lost a sense of coherence, a sense of plot, a sense of building to a climax, a sense of resolution. (Kenealy 1989, p. 215)

What kind of pedagogical imagination and theories will lead to the textbooks which have a complete emphasis on the “facts of science”? This pedagogical imagination also intimately linked to the kind of assessments that we will be using to test the “learning”. Now if we are satisfied by assessing our children by their ability to recall definitions and facts and derivations and being able to reproduce them in writing (handwriting) in a limited time then this is the kind of syllabus that we will end up with. Is it a wonder if students are found to be full of misconceptions or don’t even have basic ideas about science, its nature and methods being correct? What is surprising, at least for me, that even in such a situation learning still happens! Students still get some ideas right if not all.
A curriculum which does not see a point in assessing concepts has no right to lament at students not being able to understand them or lacking conceptual understanding. As Position Paper on Teaching of Science in NCF 2005 remarks

‘What is not assessed at the Board examination is never taught’

So, if the assessment is not at a conceptual level why should the students ever spend their time on understanding concepts? What good will it bring them in a system where a single mark can decide your future?