What's New for Physics Education: Formulas and Quantities

The major, one of the first major things that we've been working on this past year is expanding the step by step functionality that you find in our formulas on WolframAlpha. So prior to this year you could find our step by step functionality if you went through to a formula like Ohm's law and press the step by step solution button. But that was really only available for the simplest of formulas. So simple algebraic equations that only have 111 equations, something simple like this, no complicated empirical factors or anything like that. So one area that we have recently expanded into is covering multiple equations. So this is situations where you have a formula that's solving for multiple unknowns. For example, the horizon's distance formula, which is sort of simple geometric formula where you're looking at the distance from a point at a given altitude out to the horizon and as well as the distance along the curve of the earth. And so we've reworked our step by step functionality so we can handle multiple equations both in terms of formatting them, inserting the values appropriately, and doing the various unit conversions. We've also sort of improved our unit conversions to have sort of factors here that you would see in education. And this is all should be on WolframAlpha as of next month. This also includes handling for cases where you need to do a recursive solution. So for example, here, taking an example from photography, the depth of field depends on both the far and near limits of the depth of field. And so if we look at the equations here, once we substitute in these input values, we will have those limits, but we won't have a solution yet for our depth of field values. So there's code in place to recursively insert the solutions that we already have into the remaining equations so that we can get the final solution for all of the unknowns. And then another major area that we've been expanding is just there are places where we could have step by step solutions, but we didn't for various reasons. One major one was a case where we needed to add a third button, essentially. So when you call a formula without naming it, without giving it the proper name, by default we will suppress the showing the formula as part of our style. So here we're calling for, we're giving a mass and a volume and we're saying density, but we're not saying that we're looking for the mass density formula. And so we would get before just this more units button and the show formula button. But we have reworked our buttons so that we can now include the step by step solution. And again this will be coming live on the website next month. The major area where this helps is cases where you trigger multiple formulas at once. So like if we go back to chemistry for a moment here, the dilution equation, that's, that's a name that's used both by obviously the dilution equation formula, but also the ideal dilution equation because you might be meaning that version. And so in that case, because we have multiple results, we would suppress showing the formula, which then changes how our result pods are constructed. And so there needed to be additional work so that we could get those step by step buttons to appear. Another area that's currently under development and should be, should be working in the new year is support for quantity factors. So these are cases where you have some quantity, usually empirically derived, that then we need to address in our step by step solutions. So for example, if we look at earth science here, the surface wave magnitude formula, which is right here, includes two factors, a nanometers per second and at 111.2 kilometers. And this is just experimentally derived as part of this formula. So we want those to make, to solve this step by step, we would want to have those units be compatible with whatever units that we're using for the inputs. And so we added the introduct. We introduced another step to convert these factors into units that are compatible with what we're using elsewhere. Another example of this is from medicine, a concentration of oxygen in the blood. That formula uses a interesting, let's call it factor of milliliters over liters per millimeter of mercury. It's helpful just to eliminate that milliliters per liter to do the full calculation. Then another feature that we've added moving beyond extension is just improving our unit conversions. So we already saw some of that in that we've sort of clarified how we're simplifying our units when we're summing units as well as how we're multiplying them, applying them to our calculations like I showed before with the horizon distance, as well as improving handling in more complicated cases where say in this example here we are summing PA and then this complicated expression here. And so being able to handle the unit conversions on a per term and a summation basis is a significant improvement in clarifying how these steps are conducted. And then lastly, here is the addition of unit choice. So now you always had on the website the results in several different units and you could always click on more units to get a larger set to look at. But the step by step Solutions were always limited to whatever that first result was. But now you'll be able to choose whether you want to solve for it in centimeters per second or feet per second or miles per hour, for instance. And so it'll change this first step here to acknowledge that we are solving for this in miles per hour and then it'll change the final case where now we will convert the result to miles per hour. Okay, and with that I'm going to turn this over to Daniele who's going to talk about some of the step by step improvements in more detail.

Okay, so as Jason mentioned, we have been working improving the way we visualize formulas, we handle step by step formula and we are working on also multiple fronts on adding better explanation, more detailed ones. And so one feature that we were lacking was a step by step guide on how to some quantities which appear. This is my first block for the future to have a full step by step simplification of quantities. So I'll give you example here of things we have worked on. So for example, for a simple case, here we have steps for summation of quantities. So simple one are pretty fine like 3kg plus 2kg. It's an easy calculation. But we also are able to handle more complicated ones. So when we sum quantities with different units, so we handle unit conversion in step and have them summed up inside the step by step. So that's fine. And also I want to point out that this is quite flexible. You can choose time here, you can summon some trapped different quantities and here I choose parts cover day. So as you can see, you can play with it basically and make your own unit and see what happens summing them. So one thing that's important is that we can select the final unit and have the step by step for that. So yeah, here is in parsec per day which is not a common unit. But just to show off what are our capabilities. And moving on, another front that we have been worked on, we have been working on is to provide better, at least more correct, a more correct approach to step by step formula. Instead of saying this is a relationship between this quantity, this is the formula we want a way that is more accessible to students and teachers and hopefully it's more explanatory. So whenever we're using, I don't know, a definition, we changed our words to match our phrasing just to say, look, this is a definition. We define the average angle velocity as displacement, angular displacement divided by 10. So this is an improvement. This is done in, this has been done in A multiple formulas should be available in the website soon. And also we provide this is not only for definition, but we also provide some sort of inline explanation. If those are available in some ways, we can explain the formula in basic in a few steps. We can show that to the student. That's for simple formulas. Also still not yet new website that they're going to be soon adding graphs there so we can show off what are the relevant quantities we are talking about when we show in the step SF format. So here, this is a graph for the centripetal acceleration and this is for the momentary inertia. So those are going to be available soon and hopefully improve our step by step formula coverage. But our long term plan is to move away from just giving the formula and instead provide a detailed description, physical description on how we know things. So for example, how do we know that the Here I provided a few example in mechanics. So example, how do we know that the final velocity of a projectile launched from the from a certain height is going to touch ground at a certain time? So based on the website we just provide the formula. But what we are, what we have been working on is having a full physical description of that which starts from a few basic assumptions. So I took out the assumptions from kinematics, for example, where you start with the equation of motion and then you do physical physical consideration to get to the final result. So as you can see here, for example, we start with the equation of motion on the vertical direction. We choose the height to be zero and we derive the formula. We do some consideration, both physical but geometrical, to get to the endpoint, which is to get to the result like you will do in a classroom. So in the end you get a formula and you plug in the number using the step by step formula that Jen just shown off and you get the result. So this way we are trying to move from a more just calculate things, don't understand them and go to a more detailed detail and hopefully trying to clarify what are the physical principles at. And this is quite flexible also because the explanation changes according to the initial values of the parameter that you're choosing. So is not just a step by step for a single problem, but you get different consideration, different steps for those. And also there are. This is more clear in dynamics. Also we are providing different explanation according to the different levels of the students that are looking at those. So for example in dynamics the simple pendulum problem and we solve it, we find for example here the period both using an explanation for the high school level students and which uses a more simple approach, having done a force diagram like that and also a calculus based one which is oriented to students who already know calculus and can use that to full force. And one nice thing that I think is worth mentioning is that when doing this calculation we are also able to use some of the stuff, the calculus, step by step that has already been worked through. So for example, if you are given this and the student wants to know how you get from this differential equation, which is the Newton equation, with a few modifications that we work through using physical, physical, physical assumptions to this solution. So what we do is actually integrate in a show details button the solution, the analytical solution. So you use whatever we build for calculus to, to get to the final point. So those are the equation for the position of pendulum at a certain time. And yeah, also one of the main things that is going to come is also we can use different ways to get the same result. So the student can choose if they want to use conservation laws instead of the kinematics. And I saw here, for example, for the projectile motion problem, we can show we can use the conservation energy to get the maximum here in this example, the maximum height. And so like we said, we get the, and then we use the, all the machinery we have used, we have in step as a formula for wrapping the numbers and get the result. And yeah, with a more an approach more focused on the physical explanation, geometrical meaning of the things, we want to see how to get the formula and show the student how to reason around the problem. So I think this is also. I'll hand the talk back to Jason for the final part.

So now to take us to Wolfram Alpha Notebook Edition and the Wolfram language in general. So all of this step by step functionality will ultimately be available through the Wolfram language, through the normal channels, double equals and so forth. But we've also been making improvements to other aspects of our physics education. So for example, the language, the interpretations for these formulas. So you could always say, ask WolframAlpha what is the equivalent Wolfram language form for a given formula. And for many formulas this will be formula data, you know, the simple, easy to understand ones. But there are many cases where we have formulas that don't line up, that aren't algebraic or could be better described using a more complicated function. And so over the past few years we've been adding alternative interpretations for these so that we can actually have a useful result for people who are looking to explore this data in more, more detail. So for example, if we have a built in Function, for example, with like say human growth data, which is purely empirical and we're not going to have a algebraic expression for that, so it can't be in formula data. But we do have a human growth data function. We have similar thing for Planck's radiation law and psychometric properties and those sorts of things. Expanding that even further, make use of the Wolfram function repository. So if we don't want to have a built in function for that, we can create our own function repository item which can address that formula in all the same detail that the formula itself does on on the website or beyond that. So we have some for heat index. You should be seeing some of these for some of our mirror and lens formulas coming up soon, where it will return a function repository item that was developed by one of our interns that creates a manipulative object where you can shift around the object and see how the lines of light trace as they pass through the lens or off of the mirror. And we've also been trying to help users in certain cases where it might not be clear what input values they should be putting into for the formula. You can always use formula data to find out what the physical quantities involved are for for the various parameters. But there are cases, for example, where we have a dropdown menu, I might be looking at a monatomic gas or diatomic gas, and if you haven't been looking at the website, it might not be clear what that number is. Here it's fairly clear because we have a three and a five. But other cases that parameter might be less clear, and if you didn't specify it in your initial query, you might be at a bit of a loss. So in those cases we're now supplying a default value just to give people a helpful way to get started with using the formula. Now, in terms of formula data itself, we've recently just made somewhat substantive improvement where we introduced classes of formulas. So this categorizes all of our formulas within formula data into different classes and subcategories. So here we have a table showing all of the different subcategories of physics formulas with a count of the number of formulas and formula data that fall within them. And so you can use formula lookup to find these formulas, find both all the classes, or if you query for a specific case, all the different formulas that fall within that category. Some of these are a list, and those are referring to a category and a subcategory. So we can get all of the relativistic formulas that we have available through this method and you can use the same tools you might use in formula lookup to require certain physical quantities to be part of the formula or to exclude others. You can get a subset, for instance, and then of course you can use formula data on a particular formula to find out what classes ITLAT is within. So that brings us to the end of our talk. So our future plans are to continue to expand the expanded definitions that Daniele mentioned. He showed you some of the mechanics ones. We also have work underway for electromagnetism and thermodynamics and we hope to have a whole host of those available come this spring. We're also incorporating the code that showed for solving these problems from physical principles into the formulas. So you should expect to see that in several months. And I'm continuing to expand our coverage of step by step functionality to more formulas by handling formulas that have otherwise hidden conversion factors or that include some piecewise functionality. And lastly, we are working on incorporating significant digits into the formula step by step and in particular supporting around and intervals in formula data. And you should expect to see that in a few months, if not sooner. So thank you for your attention and hope you enjoyed our talk. Let me know if you have any questions or.

I think you mentioned black body radiation. Can you show us the formula or the graph?

Sure. You mean Planck's radiation law?

Yes.

Yeah, we can. Let's use the natural language processing. There we go. So that's the built in formula. This is the various usages. Let me go see if we can bring up the. There we go. So here's our. This is. This. This. This function has been part of the Wolfram language for a while now. I forget when I introduced it looks like version 10, but it lets you calculate the. The spectral radiance for different wavelengths and frequencies as well as a few other properties such as what the peak wavelength is and spectral plots and so forth.

It looked like for a second you were showing us a pull down menu that showed the options you just mentioned. Can you show us that menu again? Is there a pull down that goes with this equation?

Oh, you mean. I think you mean just the, the tools for. It's like here, this little button that pulls in pulls up all the usage messages.

Yeah, it's a little hard to read. I, I see what you're saying. It is complete but it just the, the font size is so small.

Yeah, the, the pop ups on here, I don't think.

Okay, okay, I'll check it at home. Thank you.

Yeah, yeah. If you hover over a function, it should. Should eventually pop up. That. Okay. Oh, thank you, everybody.