 of Steamboat, HBCSE's popular science series. Today, we have with us Dr. Shweta Naik, who will uncover the history, context, and complexity of accurate measurement. A little about our speaker, Dr. Shweta Naik is a researcher in mathematics education at the Homi Baba Center for Science Education and believes in teaching mathematics with justice. Her research intends to develop materials for teaching mathematics that take into account the social context, history, and interdisciplinary connections of math ideas. So, warm welcome to you, Shweta, and let's get the session started. Over to you. Thank you. Thank you so much, Aditi. So, hello, everyone, and let me share my screen and take it. So, as Aditi said, I am going to talk about measurements, a topic that is very familiar to all of us. It's also a part of our everyday life. So, basically, what's measurement? Measurement, people will call it as it's counting. They'll say it's something of a continuous quantity, then we compare it with some unit. So, measurement is roughly, if you have to say it's some, it's kind of giving a number to some length or area. We want to give some number to these quantities so that we can do manipulation over those. So, in Hindi, we might say it's just like, I'm, may kuch maap rahi. But measurement also means the action of coming up with some numbers, some units, for the quantities that we want to use. We want to manipulate. And each of us must have measured something. So, people who do cooking, they do a lot of measuring. People who do carpentry, they do very accurate measurements. So, each of us has measured one or the other thing. And you also might have some interesting experiences of measurement. So, when we go through this session, you can also keep those in mind that what were your interesting memories of measurement. So, when we measure, we sometimes measure by comparison. We do use some strategies. But the most common way of measurement right now in today's time is that we have a standardized unit which we repeat and we find the measure. In case of area, we find some measures and then do some manipulations over it and find area in terms of volume also we do that. So, what I'm going to do today because the field is so vast and there is no way that I can cover that in 45 minutes. So, I have created three stories for you around measurement. And I'll tell you these three stories and we'll reflect on it at the end and we'll talk about what sort of things we learned from this. So, the first story is about the standardization of length measurement. So, I've taken an example of length. So, many, many years before, all of you might know that how we were measuring. You must have some ideas. So, I know you must be thinking that I can measure, people must have measured with their feet, people must have measured with their handspan. So, things like that. So, yes. So, these were sort of measurements like one handspan, one forearm. We could use this for measurement and we have used this for measurement or we could also use one feet, one stride. Even today sometimes for some purposes we do make use of this. In the schools though, in elementary grades, to give an idea of the process of measurement that repeating the unit, students do these kinds of measurement where they'll take some object and then repeat it again and again and measure the other objects using that. But here, the way you say that the pencil is as long as 12 cubes that we used. So, the unit name is used in the, giving the measurement of the object that we measured. So, this is what we, some of you must have done in the school. But why don't we continue measuring like this? Why do we have these units? We could always use one or the other objects, measure any other object. So, the question is about standardization. So, we know that if I measure using my handspan, my handspan and other people's handspan is not going to say. There is something we call as universal, that scientists and many people who work in the industry of construction, manufacturing, have realized their importance that we have to reach to some sort of a universalization where when I say five units of those, it remains same whether I'm sitting in India or I'm sitting in Europe. So, it doesn't depend on my body or any other measures which will differ when I change the place. So, how did we come to this universal idea of unit and how there were attempts in the history to reach some sort of a calibration or standardization that we've seen and this is our part of the first story. So, this is a selected story again. So, there are lots of, lots of scales that you will find in different cultures. So, I have chosen two cultures to talk about which are one of the oldest sort of cultures. So, one is Egyptian, which is almost 4,500 years old culture that has been there. And this scale that you see, it's called as Egyptian Royal Tribute. You can see the year of it. So, I hope everyone knows what BCE means. I have written BC, but the general usage is BCE. It says before Common Era. In your textbook, it might be BC which also means before Christ. This means that if today is 2022, which means we started somewhere, there was some zero year. So, before those years. So, roughly when I said 1,300 years before, so then we are already in 2022 and 1,300 years before, that will be 3,500 years ago. So, this scale that you are seeing is almost 3,500 years old. And you can... So, this is called the Royal Tribute. So, we'll a little bit learn more about this. So, this is a 52.3 centimeter to 52.9 centimeter long scale. It has seven divisions on it and the each division is the make one palm. So, basically this palm is of their king. So, this is again hence some bodily aspect of it. So, this for the palm size that you can see the number three here in this picture is their king's palm size. And the seven of those are repeated to make this scale. And then the each palm is divided into smaller units. So, this is made. It also has some pictures on it. So, this is the Egyptian language in the ancient times that is called logographic. So, basically in that language, what happens is each picture has a meaning as well as it has a place. So, depending on a place and the picture, there is meaning to it. Right now, it is describing some units for measurement that they will use. So, this is called as the scale is still called as the anthropic unit. It's still like a... Yeah, it is still human because there is a connection of human element here because it's the palm of a king. It is an anthropic unit. And the story around it is that because it's a royal unit, it's created only one. But there were other qubits like who were made using wood or other materials. And they're given to the workers who are doing construction work or for example, measuring that high tide or measuring the water level. So, they are also given these qubits but the replicas of it. And every full moon night, they all have to assemble and they have to measure, they have to compare their qubits with this royal qubit. So, royal qubit is kept in the king's palace. It's not accessible to everyone but everyone can come and compare their qubits every full moon day and check whether it has some distortions, it has reduced in length and they have to adjust it to maintain the accuracy. So, this process itself is like, we know it's called calibration. So, this is almost 4,000 years back, they are worried about calibration because they understood the importance of having an exactly same skill when you're doing construction, when you're making things. And they're doing this local kind of calibration here. You can see this Digio step pyramid and there is a historical record that this particular royal qubit is used in this construction of this pyramid. Yeah. So, let's go ahead a little bit. We'll talk about another historical story, another civilization. This is connected to us, it's induced value civilization. This also happened 3,300 BCE, which again means from now on, it's almost 5,000 years, 5,500, 5,000 years, yeah. So, there are a lot of studies that has happened around induced value civilization, like people have studied water in it, people have studied geometry in it, but mostly art and culture and social interactions, different meanings that these construction gives, it's all under study. This site is also spread across different countries. So, the studies has been a bit challenged, but still people are studying many things in it. So, what I'm going to tell you is another sort of a qubit kind of thing here, which is they call ivory scale. So, it's made from ivory and a similar logic. So, there is one scale which is made in ivory and it has, so if you can see that the scale also has one inch, that they call it as induced inch. And this was found in Lothal site, which is in Gujarat, 2004. So, it's dated 2,400, it was found in 1900 recently, but it's dated 2,400 back years. So, this scale is basically has the one inch that they call it as induced inch. It's today's 1.32 inch. So, it is very, very closer to what we have right now. So, their inch is equal to 1.32 inch of today's inch. And the another contribution, which this scale did is the smallest one. Of course, there is some contestation about what is the smallest unit length because different scholars are studying it. And there are different sets have found or evidence have found. So, some say it's 1.6 mm, some say 1.7 mm, some say 0.6 mm. So, this is with the history that people do have different ways of figuring out from the pieces of what they found, but it's valued for, it found the smallest unit. And you can think about it, why having a smallest unit is considered as a good value. Because more, if you can measure the smallest, smallest value, it makes the measurement accurate. And that is why it's valued when it has a smallest sort of a measurement possibility. There are also other scales or counters that they found in, and I'm just showing one picture, but I think if you are interested, you should pursue this interest and find out what sort of other counters of scales in this value civilization had. So, this is one of it where there is no definite understanding of what was this used for measuring. Whether it was a counter, whether it was measurement, whether it was used as a log representing some number. So, there are several theories. Some people say that this is some base three understanding and it was used to represent the number of whatever is there inside that room. So, there will be markers like this to maintain the count. Some say that it is used for telling forecast. And some others have also figured out some ways based on the number of dots that are there. So, maybe you can also do some exercise and think about what this could be used for. So, yeah. So, there is a map in the civilization, which is still mysterious and one can actually spend some time in that. So, for longer time in the history, there were always scales like qubit, like ivory scale. And some cultures developed something for measurement because they all constructed, they all did some structure. And they also figured out ways to calibrate locally. Like, so there is always the element of royal or leader when you go and calibrate your scales. So, this practice was there. But of course, there was no way to have something as a universal. And the recent, even recently, this is a 19-4 story that I'm going to tell you. Even at that time, countries lacked nationalized standardized units. So, this is in pictures from Baltimore fire. It's called as the Big Fire. This is from 1904, 1904, where basically the fire was very big. So, there was the support for fire brigades was called from nearby states as well. And nearby states actually reached with their hoses and everything. But the hydrant that water pipes that were used in Baltimore, they had different hoses units and the hoses that other states brought in, they were on different sides. So, basically there was no standardization, like a national standardized, this is from the USA, story from America. So, different states used different manufacturers, different manufacturers used different units, and it led to something that they can't use each other's even water hose. So, because of this, the fire took much, much bigger form than what it could have been and the losses were very, very high. So, actually this time, there was the definition of meter, one definition of meter, but it was still not popularized and we see this instance. So, let's understand how what, where is like a meter standardization as of today? So, the first definition of meter came in 1793. It was by Science Academy of France. So, you know, when France revolution happened, when the whole royal, so there was a revolution which took over all the royal families and there was also scientists who came forward and push made push for the making use of units, standardized units. So, and the science academies were formed in France. So, they came up with the first definition of meter. So, what they decided was a definition of meter as a one upon one per hour times of the half meter. So, let me just explain what half meter is. Many of you might know. So, this is a picture of like a tracing of Earth. And you can see that these lines, of course these lines does not exist there physically. This is something human thought that for the measurement for angles and for multiple various things. So, Meridian is the shortest distance between the North Pole and the equator. And so, basically what they did is the half distance. This is the half distance, Meridian is the full distance. So, what they basically did is they took the one upon one per hour times of this half Meridian distance and decided it as a meter. But how do you do that? Because you basically do not have any other measurement to even measure the Meridian. And there is no Meridian as in real line which is there. So, what you basically do is you travel and in different places wherever it is possible. Of course, not possible in water, you take estimation. You use some unit to make estimation of these numbers. So, John Baptis and Michen, Peré Michen were assigned to do this work. One was astronomer, one was mathematician. And they took almost seven years to do this work. To figure out key, whether what standardized unit they have thought, whether it fits with the scale of this scale. And after the seven years, they made some entrance. But trans was very proud of their achievement because they defined something as one meter and they have a concrete piece which is placed in the museum, which people can now replicate and use. But of course, this happened in Europe. But this is the first definition of meter. People were still not happy with this definition because there was something called Earthly, something still specific. Because this Meridian distance was taken from Meridian which passes through Paris because this was done by the science academy. So, scientists were still not happy because they wanted some measure as something which is very universal. It's not about any country, it's not about this earth. So that time, there was a research, there were findings about how can we find a light emission from elements, so Maxwell's work. So there was a constant attempt to figure out whether we can use this transmission, transmitted light to define meter as well as second. And then finally, when Mitchell Sen, who is also a Nobel Laureate who figured out that one can use Krypton's wavelength to define, he basically came up with Interferometry, which is a device where you can interfere into the wavelength. So this was a big thing in science and measurement. And people could now measure very, very small distances. It actually opened up a completely different branch in physics. And scientists have spent times in figuring out accurate measurements of very small magnitude. So they use this Krypton's wavelength and one meter is now equal to these many wavelengths of the orange-red emission line in the electromagnetic spectrum of Krypton 86. Krypton 86 is one of the isotope of Krypton. Yeah, I might not be talking here about isotopes, but there are some stable forms of Krypton. So Krypton 86 is one of the most stable forms. So this definition was there and even with this definition, scientists were not happy. And the reason for that was because it's connected to Krypton, it's one specific element, why Krypton, all those kinds of questions. And right now, the meter that we use uses this definition, which was decided in 1980. So it basically depends on the speed of light. So we know speed of light, which is this number 2997924550. So what they decided that in vacuum, in one second, sorry, in this part of the second, so like a partial of second, which is the speed light. So speed of light is defined as in one second, how long the light travels, which is that distance. So basically, they use that number to say that in that much of second, how much light travels is our speed. And again, calculation of this is again a different, entirely different history and very interesting to figure out from this definition, how did they actually came up with this speed and bar? That's something that you should look for every. But the point of this entire story is that we use this scale in our hand, we use it so often, but so much has gone behind it. So for this definition of one meter, so much of math, so much of science has gone behind it. It has led to many other branches of science and math. And all this is for what? It is for achieving universal accuracy. Achieving accuracy and achieving in a way where it is not dependent on any Raja or any particular person or any particular country, but it's universal, it's for all. So this is also one aspect of mathematics and science where we go from individual to like particular to universal. So, but it takes a lot of effort and energy to reach to that stage. So this is one part of a measurement story that I wanted to tell you. Let's go to the second story. So the second story is about accuracy. So I'm going to tell you something about volume of sphere and mechanics of how Archimedes did it. So all of you must have seen a balance. So before we go to volume of sphere, let's understand another thing that Archimedes did. So we need to understand balance, but then you all must have seen when we buy something at the store, there is a balance. Nowadays there is an electric balance which turns to exact weight, but the traditional balance is where to keep the weight in one pan and then you put things here which balances and then we know that both weights are exactly the same. Yeah, these are some examples of the balance that you'll see either in the market or in your labs. Yeah. So what if I ask you that you have a 3 kg weight and you have a 5 kg weight? Now balance it. In the sense, I'm asking you that put it in pans, such fact they are all, they both are at the same level, they're balanced. Is this really possible? Yeah, so most of you must be thinking that it's not possible because both are different weights. So all of you know that 5 kg will go little down and 3 kg will be up. So the 5 kg is heavier, so it will go down. When you play seesaw, you experience this. So let's try to understand how can we do this? So there is this beam, this I'm going to call as a beam and there are two weights on it, 3 kg and 5 kg. So this will not be balanced like this, but for timing, I'm just using them as a picture. So let's say there are 3 kg weight and 5 kg weight and this is the fulcrum. So fulcrum is something that we are going to... So when you see the balance, what you see in the middle which holds, it's called fulcrum. It's on the top but you can also have fulcrum in the bottom. So instead of asking question that whether we can balance 3 kg and 5 kg, I'm asking you another question. Where do you place this fulcrum, such that 3 kg and 5 kg, both will be at the same level. They're both at the same level, okay? So to decide that, let's do something else. So I distributed 3 kg as 1, 1 kg. I just made 3 kg into 3, 1 kgs. And similarly, I distributed 5 kg into 1, 1 kgs of 5 kgs. Now our question is, where do I place this fulcrum, such that the beam will remain balanced, okay? So I still can't answer. So what do I do? I still don't know where to put it. So what I basically do is, now this 8 kg is equally distributed on this beam, okay? There is a... They're placed equidistant. Each is 1 kg and exactly 1 kg. So now where do you put fulcrum? Yes, so now we must have got a sense. Where do I place the fulcrum? So that will be exactly in the middle of this. So such that there is 4 kg here and then there is 4 kg here. Now we have no doubt that it will be balanced at that place, okay? But our initial question was 3 kg and 5 kg. So let's see how can we do that. So we know that this is 3 kg and we know that this is 5 kg. So I still know that because in the middle weight we'll have the center of mass. So I can put this here and the beam will not move. It will be still the same because this is... If I put this to here, there will be no movement because the center of mass is here. So all this 3 kg weight, the center of mass will be in the middle of it. So I put that and I get 3 kg there. Similarly for 5 kg, I'll do the same. Where will be the center of mass here in the middle? That's right. So basically I put it here and I get 5 kg. Now 3 kg and 5 kg will be balanced. Will be exactly at the same level. But what is different here? So like your balance, the fulcrum is not at the center. So in the balance what happened? It is exactly in the center. It's not. It's basically a little bit near to 5 kg and far from 3 kg. So this is what... This is how Archimedes thought about it. So Archimedes thought that it's... This is called law of liver. He understood that it's not only balancing is about weight and the distance from the fulcrum, where you are balancing. So it's weight 1 times the length is equal to weight 2 times the length. So this is going to remain... This equality is going to be maintained. So more the weight, the fulcrum will be towards that weight. So Archimedes... There are a lot of fun cartoons you'll find around this where Archimedes is saying that you can basically balance anything. If you can find the right liver and if you can find the right distance. And there is like a cartoon picture where he's trying to balance the earth also. So do look for it and try to make sense of that. So this is law of liver. And Archimedes is again... Archimedes is here, 287 BC. Now you know how to make sense of those years. How many years before that has happened. So this is what we learned is called law of liver. Now we'll try to use this in how he figured out the volume of this liver. So what's volume? Okay, so all of you have studied it in the school. You might know some formulas. But what is volume of the concept? So if there is like your milk vessel at home. So how do you find out the volume? So volume is basically how much things you can fill into that vessel. So it's a space occupied which is inside a 3D object. Whereas the area is basically, again region occupied but it's a 2D place. So there is no layer in area. Whereas in volume there is layer. So that's sort of a difference. Let's see how Archimedes found the volume of this liver. So before we begin, I'm just going to tell you what he knew. So what he knew was he knew what is the volume of cylinder that he figured out first. Because there is always help there because you know that with cylinder, the easier part is the surface. The bottom surface is surface. And then the other two things are exactly straight. So you can imagine that it is discrepancy those many times. And then he could find out the volume. So basically if you know this area, pi r square, and you know the height of the disc, then you can repeat it. It's right now it sounds very easy to say that, but during that time and the way the mathematics of it is, which is called integral calculus, is the notation is very challenging. But meaning-wise it's now accessible for us to see that cylinder can be seen as many discs. And then you can basically know that the pi r square is the surface area. And if you just have d1, d2 as different heights, you multiply and you sum, and you know that the basically pi r square multiplied by the height, which is d. That is going to be the volume of cylinder. Then there was also realization that if you take the cone of the same diameter like this to be, then if you fill this cone with water or sand and try to fill in the cylinder, the three of the cones will fit into cylinder. So which also tells us that the volume of cone is going to be the 1 third of volume of cylinder. So this information was there. So now what's the volume of sphere? So there is another way, yeah. There is another way to look at this problem and maybe we'll come to it later, but let's see how I'll come to this part. Okay. So I'm going to show you first what he proved and then how he proved. So he proved that on a beam, on a beam, on one side, he put this cylinder, which is a diameter 2d, radius d, is exactly equal. You have to assume this as a solid field with same density material. Was balanced with cone and the sphere where cone has the same radius as cylinder, but sphere has the half radius as the cylinder and cone. So this is what he proved. So what does this mean? What does this prove mean? That volume of this equal to volume of this cone plus volume of this sphere. So now, if you know this relationship, this is what he proved. The volume of sphere is now, how can you find? Volume of cylinder minus volume of the cone. Okay, this is what you can do. So let's see the mechanism in which he did. So I'll show you some, I'll show you a video. So, yeah, let's see. Can you see the screen now? Yeah, let me play from the beginning. So just see this. So there is our cylinder, there is our cone and there is our sphere. And you can see the diameters are, so radius is D here for cylinder and cone and D by two for sphere. So let's see what he did. Okay. See, you have to understand that these are all solid objects. So it's not physically possible to place them on top of each other. So this is a sort of a thought experiment and it's a very intelligent thought experiment. So see how he imagines that he basically puts sphere inside the cone and cone inside the cylinder. So this is the first thing that he's imagined. What if I did this? Okay. So this is the first thing. And now you understand why we have taken sphere also the D by two radius. So he imagined that these objects are kept together. He already knows the relationship of sphere, cone and cylinder. So he knows the possibility. Now he takes the disk surface. Okay. And he thinks that how each disk will be balanced and he knows law of limit. So the question for him is where to put the fulcrum. Because we can now balance any weight. You know that we can balance any weight with any weight if we know the right place for the fulcrum and if we have a right limit. So he basically puts first disk and puts the first disk of sphere and first disk of cone. And similarly he places each and every disk like that and what is the end product is sorry. So let's see this again. I'm not able to show you the end product. So we'll see in the picture. I'll show you the static picture of picture. Okay. So let's see the picture. So this is what he does first. He has imagined it this way. And then he's going to take the slides of it. So this figure is... I'm not going into the mathematics of it but just briefly when you take a slide. So this is the cylinder. Then this is sphere and this is cone. So when you take a slide basically what every time what you can figure out is what is the volume of that slide. Each disk because when I cut it here I get one disk from cylinder. I can find out its volume. I get one disk from cone. I can find out its volume. And I get one disk from sphere. So every time, so there is some mathematics, some geometry you'll have to prove about the size but you can find it. And then this is some equations about how can one find out using the formula of area and then multiplying it by the thickness and density which is same for all three of them. Those things are assumed. And when I cut the disk the thickness is assumed same and its densities are also the same. So basically what you do is you start placing it and you realize you have to place the cylinders disks like this which comes up to fulcrum. So the first disk you have put it as a height D. So this is the mechanics of what you used. It's very interesting mathematics. And whenever you get time try to read more about this. But the point of this story is that it's not, there is no assumption, there is no rough measure for sphere. Because you know cylinder, you know cone. And now you can just try some other things that will tell you what would be the volume of sphere. But this is an accurate measure of sphere because what you are showing is using mathematics you're showing it's exactly equal to cylinder minus cone. So what you find is accurate measure. So there have been this very fascinating stories about finding accurate formulas, accurate measurements. This is another aspect of measurement. Set this, of course this story, this particular story is very interesting. It was not found much earlier. It was found recently. This Archimedes-Heathard of finding volume of sphere was written on some papers and in ancient times paper was not very easily available. So they were often reused. So I think in 1960s this paper was found as reused in some writing, some poems in one of the church. And it came, one mathematician encountered it and then historian, the mathematician encountered it and then he removed the writing and traced what was the original writing and he found this method of finding the volume of sphere. So that this story is also fascinating from the point of view of accessing history. So yes, so the thing is, so said this, we said there is a push for standardization. We said that there is a push for accuracy and said this and the third story is about how sometimes we use the units. So I'm going to tell you the story from Shabnam. So Shabnam is a school student and she helps her parents after her school hours. They do a lot of different works in leather and textile industry. So they make the wallets and purse and she helps them in that. So this is basically, what you're seeing is the cuttings of the purse for sticking leather on it. She also does some collar work, so making standard collars. This is bag making, she also participates in that. She also does this very work. The rework is where you make some patterns and things done a lot. And these patterns are very symmetric. So there is a lot of other mathematics is also going on. So there is a math of using material on it, like you can see these binbies on it. So they have to figure out how many binbies to use, how to place them, but they also have to constantly figure out other geometry like symmetry and things like that. And they use measurements and other sense of measurements while doing that, like which is called spatial geometry. So what I found was two strategies that Shabham used in her work. One is comparison and another is estimation. So she does this, this is called Lutton, which is used in the Zari work to stick to the clothes or saree or even for some purse. So what she does is she makes a standard Lutton for her, which she makes from the jewellery, so Tali tropes it. She uses the jewellery, she applies wax to it. So it's kind of making it safe unit. And then she uses that standard as a way to make other patterns. It's very common sensible, you will think. It saves time for her and it also maintains the accuracy because she has made it once, careful measurement. She has waxed it. It's not going to change it, not going to move it. So it's something that she's trying to achieve accuracy, accuracy at the same time when she's trying to save time. This is another example where she uses something called pharma in leather bags. So it includes several measurements in the first place, but she makes one pharma, maybe it came from the word form. And then she uses that to make leather bags. There's also the use of sequin. This is Dindi's sequin, it's called sequin and she has to use them in the Zari work. But there is a specific amount that she has to use for per object. She can't just be luxurious because she's using it to make profit. So she can't luxurious means it. She has to be standardized about it. So what she did was she took her piece full of sequin and she measured it once in grams. How many grams of that? And then every time she doesn't have to make it. It's roughly, there is some roughness, there is some estimation about it. But it's also that she has measured it. So she's also keeping track of some accuracy in it. So it's like the in-between of accuracy and estimation. Then other thing she did is that you know that people use with their fingers and hands all the time. But what she does is that she measures her fingers and hands and she has written some measures on it. Using the standardized scales, like a centimeter scale or meter scale say, she has measured her hand and some distances that she can often use to maintain the symmetry. So every time she doesn't have to go to sleep. And she, so she's using body units, but this is different than what we saw in the Egyptian Royal period. It was in the unit was defined using King's hand where the here Shabnam's hand is defined using the universal unit. So the direction is reversed now. Okay. So basically Shabnam learned some standardized units in her school and she uses that along with her work unit and makes something called composite unit. And this composite unit is an efficient unit because it maintains accuracy and it also saves her time and increases efficacy of material using efficiently, like without wastage. Because every, if you measure more time using a balance, you're going, it's a more process. Whereas you know the measurement of your fistful or whatever comes in your book, then it's a little bit saving material. Yeah. So the point I tried to make today was in measurement, I, we saw three things. We saw standardization, we saw accuracy and we saw how it is used. So this is sort of a three perspectives of measurement, which is there. There is like immense efforts and science that is going on to make measurements using ourselves. These immense efforts are going on in mechanics, figuring out things to maintain accuracy. But then when we use it, then there is this social context that comes into play. And the social context is not only that there is, this is humanly used, but the context of profit and loss, context of sustainability and how do I make the efficient use of the material and that also impacts the base of measurement. This is like a three story part that I told you about measurement. And I want you to think more on these lines, find out other histories around things and see how people have historically come to units and what is the practice of those. Lastly, just to say a few words about educators in the group is that NCF does call for making connections between real life and the school curriculum. And measurement is one of the prime topic that you can actually do that. So asking students to create some hybrid units, composite units, asking students to ways to improve accuracy, asking students to way to be comfortable with social aspects, social context and what should be. So for example, if you want to measure something, you can't remove. So for example, I want to measure an exact measure of some mountain. You don't remove everything from the mountain. So you have to coexist with the world and the complexity and still find the accurate measurements. And the measurement, for example, even the measurement of population, for example, you have to coexist. So there are a lot of complexities, pandemic, illness, birth rate. And we try to find measurements making use of, making consideration of all these complexity. So this is what I think in the school curriculum, this complexities of real life could be brought in. Then a very, very dry and standardized way of thinking about it. This is something is five meter and we measure. So learning to measure is one thing. But learning to measure in reality with all the complexities is something would connect real life and school life. So yeah, I'll stop here. These are some of the links in which I'm connected. I also do work. And if you can take screenshot of this, you can find more resources at each of this link. I want to thank each and every one of you for watching this video. I hope it has given you some thoughts to think about. Of course, this session is part of the Steamboat series, as Aditi told you in the beginning. And I thank the entire team for giving me guidance on the teachers and students who always work with me. The drawings were made by my colleague, Malosh Nair. And the animation was made by my colleague, Subramaniam and Malosh Nair. So the last pictures that you saw were also part of one of my colleague and now his work is in another place. From his work, PhD work, I think number six. So thank you, everyone. And over to you, Aditi. Hi, Shweta. That was a very interesting session. And we just want to inform all our viewers that we have already had this session in Marathi. So please do go check out our YouTube channel. Shweta, I think we have time for just a question or two which came up during the conversations with students in the Marathi session. So I thought we could just discuss that briefly here. The first thing is, we talked about these hybrid units. If you could just elaborate on that a bit, that why did we say that Shabnam used hybrid units? Thank you so much, Aditi. Yes. So right now, the hybrid world is not unfamiliar to all of us because of the online and offline media. You have seen the hybrid media is like that you use, some of them are online, some of them are offline. So hybrid in Shabnam's case is very interesting because she has, because she goes to school, she has access to the standardized units. But for, we saw this history of almost 4,500, since 4,500 years that humans were motivated towards using this body units. You all also said that we will use this hand span or my feet. So it was always there in us as human beings to measure using our body parts. There are even cultures who count using body parts. So coming up with body parts, it was like a tendency that humans had and they measured the world for many, many years using body parts and creating some units that are dependent on. What Novel Shabnam did is, she used this standardized units, this speed of light definition of meter-valent scale. She uses that definition and she imposes that on her body units. So now it's a hybrid unit which uses the body part which is convenient for me to use like this. But I know the exact measurement of this in centimeters or millimeters because she also uses the, so she sort of is a, according to me, she's a scientist who sort of created this fusion of human tendency to the, another human tendency of reaching the universal unit and combine the two for reaching the hybrid unit. Yeah. Does that make sense, Aditi? Yeah, definitely. And I think it also sort of brings to life the last point that you were making in the talk about the relevance of mathematics in real life, like doing mathematics in real life. So that was really interesting. And I hope all viewers have enjoyed this session like I have. And we'd also like to let all of you know that we have our steamboat sessions on every second Sunday of the month. So do, you know, do join us for our future sessions and we'll call this an end now and we wish you all a good day and we look forward to interacting with all of you for our next steamboat session which is due on 13th of March. So till then, bye-bye. Bye. Thank you, Aditi and thanks everyone for watching. For teachers, please do send this video to your students and get their opinion about the points that are made for you. Thank you.