Scruffy's math program for kids

[ShahdagMountains]

Simplest case, is moving in a straight line, and not accelerating (so, constant velocity). This is called an "inertial" reference frame.

In such a reference frame, distance and angles confirm to a "local Cartesian coordinate system", meaning distance can be measured by a yardstick and time by a clock.

The fundamental principle of physics is that the laws of physics are the same from any reference frame. So if as you say the two asteroids are moving directly at each other, we should be able to exchange reference frames, in other words A looks to B exactly as B looks to A. This is what the Lorentz transformation does, it leaves the physical observables invariant when we swap reference frames.

The weirdo thing about our universe is, the geometry isn't flat, it's hyperbolic. In inertial reference frames this structure is called "Minkowski space". To understand it, you can define an "event" as occurring at a point in space and time. So, in a flat Euclidean geometry the distance between two events would be given by

D = t^2 + x^2 + y^2 + z^2

where t is the time difference and x, y, and z are the space differences.

But Minkowski space doesn't work like that. It works like this:

D = t^2 - x^2 - y^2 - z^2

It is an example of a "non-Euclidean" geometry.

So, using this, you can figure out the Lorentz transformation for any two inertial events.

First, in keeping with the fundamental principle of physics, the speed of light is the same (constant) from any reference frame. Therefore, you can calculate the Lorentz factor as follows:

L = 1 / sqrt (1 - v^2 / c^2)

where L is something like the "amount of compression" as the velocity increases. When v is near 0, the compression factor is 1, which means you see something like the actual velocity. But when v is near c, the compression factor grows without bound, meaning that what you see is always "a fraction of" the speed of light, and you'll never quite hit the actual speed of light (cause if you did you'd be dividing by 0, making the compression infinite).

You can derive this equation from the "Lorentz group", or alternatively using the Minkowski metric. Because of the reciprocity of reference frames, it only describes a situation where one of the reference frames can be translated (or rotated, or sheared) to the origin. And left fixed there, during the measurement of the spacetime events.

If you're accelerating, or equivalently if your spacetime is curved, you have a more complicated situation and you have to use general relativity instead of special relativity. In general relativity, the assumption of yardsticks and clocks breaks down (because of the curvature), and you have to use a "metric tensor" to measure distances and angles.

Scruffy, can you answer this one for me:

I’m a big fan of the Dulles Air and Space Museum.

And l’ve often wondered, in the context of planes taking off and landing, flying around the globe etc.

Given the Earth is a moving object, spinning on its own axis, at the same time circling around the sun, how does one moving object (the plane) interact with another moving object (the Earth)? Do the scientists have to figure all this stuff out?

 

[Sherlock Holmes]

The observer is ON one of the astroids.

OK.

The core of Einstein's approach to physics, is that there are laws of nature and those laws cannot depend on the situation of an observer, in other words all observers no matter their situation, will always observe the exact same laws.

Now consider, that the speed of light is itself a law, it pops out of Maxwell's electromagnetic theory. The speed emerges from Maxwell's equations as a constant speed, but relative to what? His equations do not define that, therefore they postulated the "aether" and assumed that the speed of light is relative to this aether.

Einstein recognized that we don't need an aether, if we assume that all observers see the same speed of light, then they all see the same law and instead must experience time differently, Newton assumed time was universal but it is in fact the speed of light that's universal.

As scruffy describes above, the equations describing the relationship between law and observer, become the same no matter who's moving toward whom.

 

[scruffy]

Scruffy, can you answer this one for me:

I’m a big fan of the Dulles Air and Space Museum.

And l’ve often wondered, in the context of planes taking off and landing, flying around the globe etc.

Given the Earth is a moving object, spinning on its own axis, at the same time circling around the sun, how does one moving object (the plane) interact with another moving object (the Earth)? Do the scientists have to figure all this stuff out?

I dunno, not a pilot, but it seems to me aeronautics is one of the better known sciences.

Up in NorCal they teach the novices how to avoid bald eagles.

I know a little about angular momentum, just a little though. I'm kind of a latecomer to the field, the rocket scientists had all that differential geometry stuff figured out a long time ago. The pattern seems to be they use the equations for long distance stuff, and let the computes take over when things get close to runways or targets.

 

[scruffy]

Scruffy, can you answer this one for me:

I’m a big fan of the Dulles Air and Space Museum.

And l’ve often wondered, in the context of planes taking off and landing, flying around the globe etc.

Given the Earth is a moving object, spinning on its own axis, at the same time circling around the sun, how does one moving object (the plane) interact with another moving object (the Earth)? Do the scientists have to figure all this stuff out?

Actually there is one interesting thing I discovered. It has to do with "numerical methods for integration".

Most of the flight computers use numerical integration from sensors. Some of it is for averaging, but some of it is also for control systems.

It turns out the standard integral we learn in calculus class (called the Riemann integral) is very picky and delicate, it depends on continuity and smoothness and other "good behavior".

Turns out a lot of control systems are moving to a less demanding integral called the Lebesgue integral. It comes from measure theory, and it's nowhere near as finicky.

For example for aircraft, let's say you encounter a sudden change in atmospheric (air) pressure. In the past, the control systems would "average" sudden jumps to prevent thrashing, but this isn't necessarily good for altimeters and other instruments that want accurate readings. With a Lebesgue integral you can minimize the averaging window and therefore respond much more quickly and accurately to discontinuities.

The numerical method isn't all that different, you're still looking for the area under a curve, but with a Lebesgue integral you can handle "poorly behaved functions", which is closer to the real world. Think for example, of a stock ticker, where every point jumps to/from the next and nothing is continuous. Such data might resemble the real world output from a pressure sensor, as you're flying through weather. In the stock market they do 30, 60, and 90 day averages - the smallest one I've heard of is 7 days, and very few people apparently use it. 7 days is a lifetime for an aircraft in flight, you want 7 seconds not 7 days. The idea is you have a much broader range of choices with a shorter window, in terms of how you're going to treat the data. In some of the single-engine stuff I worked on they even do "hardware averaging", which now becomes unnecessary if your software is able to process the fluctuations quickly enough

 

[Sherlock Holmes]

Scruffy, can you answer this one for me:

I’m a big fan of the Dulles Air and Space Museum.

And l’ve often wondered, in the context of planes taking off and landing, flying around the globe etc.

Given the Earth is a moving object, spinning on its own axis, at the same time circling around the sun, how does one moving object (the plane) interact with another moving object (the Earth)? Do the scientists have to figure all this stuff out?

May I respond?

For an airborne aircraft it is a question of vectors really, the aircraft at any point in time is being acted upon by several forces - vectors.

One is gravitation acting straight toward the earth's center, the others are life, the result of air pressure acting on the wings and the other is forward thrust as a result of the engine and finally drag, the pushing force of air resistance.

So when sitting in a plane these are the forces acting upon it and the fact that the earth is a ball or is rotating and so on, are not relevant.

The aircraft will follow a curved trajectory as it flies around the globe, that curve is due to the gravitation vector which is always perpendicular to the earth's surface, so in that sense I suppose the spherical shape does manifest itself.

 

[scruffy]

Here's a good one.

How can kids remember, when to multiply on the right and when to multiply on the left?

Matrix multiplication is non commutative, you get a different answer if you multiply in the wrong order. So how do you remember?

Well, here's a cute little trick a kid will never forget.

Every vector has a basis, and components. To get the dot product to work out correctly, the basis has to be a row and the components have to be a column.

And, the multiplication only works one way. Like this:

| ex ey | . | cx |
...............| cy |

Where E is the basis and C are the components.

So what you can do, is put an identity matrix BETWEEN the basis and the components, like this:

V = E . I . C

In the above example it would be a 2x2 identity matrix that looks like this:

| 1 0 |
| 0 1 |

Now you can SPLIT the identity matrix into any linear transformation and it's inverse, like this:

V = E . A . A(-1) . C

And now you can clearly see, that the basis gets multiplied on the right, and the components get multiplied on the left. Plus, you can see that the basis gets the forward transform because it's covariant, and the components get the backward (inverse) transform because they're contravariant.

This method works for vectors and tensors of any rank.

It's a little memory trick that lets kids derive the correct answer even on a bad day (Plus, it introduces them to cosets in group theory).

Back when I was 7 years old learning the resistor color code, a crusty old ham radio operator taught me "bad boys rape our young girls but violet gives willingly". Kids never forget stuff like that. It's a quick, easy, efficient way to make sure they pass their exams.

 

[scruffy]

 

[toobfreak]

Anyone who graduates from college should be conversational with mathematics.

What you describe is far more than conversational.

It doesn't mean you have to pull equations out of your butt, it just means you have to be familiar with the principles. You don't have to know what a Lorentz boost is unless you're a physicist, but you should know enough about coordinates systems so someone could explain it to you in 5 minutes. This business of graduating from college without knowing how to balance a checkbook has to stop. Scruffy says: if you want to graduate, you WILL learn math. If you can't do it, go to trade school instead.

I love math. That said, far better to master the basics than needing to be versed in everything you describe. Also, where most math teachers fail is in supplying the /application/ for which a given math solves. For example, a teacher will introduce you to a new type of math, then teach you how to use it without ever explaining what problem it solves, which is the exact opposite of what I'd prefer. Instead, show a situation to which no other math succeeds in explaining nor solving the problem, THEN teach the math which solves it! This is how it is done in real life--- first comes the problem, then comes the new math (like the calculus) aimed at being able to solve it. I find the math is much more relatable that way.

Two areas you missed including are fractions and Boolean algebra. There are many non-technical fields of learning where higher math would largely be a waste, but thinking/reasoning/logic is never wasted on anyone.

An area where I find many people in needing of a greater grasp of (other than general reasoning skills) is in the area of dealing with fractions, exponents and radicals. For instance, I often ask people to solve the following; can you give us the answer?

Faced with a simple problem like this, many people's face goes blank.

 

[ReinyDays]

What you describe is far more than conversational.


I love math. That said, far better to master the basics than needing to be versed in everything you describe. Also, where most math teachers fail is in supplying the /application/ for which a given math solves. For example, a teacher will introduce you to a new type of math, then teach you how to use it without ever explaining what problem it solves, which is the exact opposite of what I'd prefer. Instead, show a situation to which no other math succeeds in explaining nor solving the problem, THEN teach the math which solves it! This is how it is done in real life--- first comes the problem, then comes the new math (like the calculus) aimed at being able to solve it. I find the math is much more relatable that way.

Two areas you missed including are fractions and Boolean algebra. There are many non-technical fields of learning where higher math would largely be a waste, but thinking/reasoning/logic is never wasted on anyone.

An area where I find many people in needing of a greater grasp of (other than general reasoning skills) is in the area of dealing with fractions, exponents and radicals. For instance, I often ask people to solve the following; can you give us the answer?

View attachment 1090680

Faced with a simple problem like this, many people's face goes blank.

If x=1, then y=1 ... easy peasy ... ha ha ha ... you know I'll take that dare ... if x=64, y=1/4 ... or did I make myself into an idiot? ...

The math classes I took were more general ... yes, most of us use vector calculus, some of us will move on to tensor calculus, so everybody has to learn Analytical Geometry ... yeah, that'll weed out the fakers ... even though only a few of us move on to Differential Geometry and then the above mentioned Tensor Calculus ... in order to solve field values found in Modern Physics ... (the main reason I stick to Classical) ...

We may never see a field in our entire professional lives ... but if you finished calculus class, you'll know they exist ... part of that "Universal Education" promised by your diploma ...

I used calculus once in my life, as an electronic test/technician ... we were making modules that performed basic arithmetic on inputs using op-amps ... and then we got one that integrated, and I was the only one who even knew what that was ... and quickly solved for the various "set" values we needed to insure it functioned correctly ... [beams] ... made the engineers upstairs happy ...

 

[toobfreak]

If x=1, then y=1 ... easy peasy ... ha ha ha ... you know I'll take that dare ... if x=64, y=1/4 ... or did I make myself into an idiot? ...

I get that a lot, people always think they must solve for x. I originally created the problem for a Russian troll here called Selivan who used to troll us on the wonders and superiority of Russia.

But since it is an equality, it is OK to still have a virtual number in the solution since it is truly impossible to solve for x without more information, so, the real answer to x= can be two: either--

But you'd be surprised the number of people who do not know how to rework a fractional power.

 

[Sherlock Holmes]

I get that a lot, people always think they must solve for x. I originally created the problem for a Russian troll here called Selivan who used to troll us on the wonders and superiority of Russia.

But since it is an equality, it is OK to still have a virtual number in the solution since it is truly impossible to solve for x without more information, so, the real answer to x= can be two: either--

View attachment 1090898

But you'd be surprised the number of people who do not know how to rework a fractional power.

Well the first form is just a notation really, another more compact way of writing the other forms.

A word I never see these days is Surd:

nth root - Wikipedia

en.wikipedia.org

 

[ReinyDays]

I get that a lot, people always think they must solve for x. I originally created the problem for a Russian troll here called Selivan who used to troll us on the wonders and superiority of Russia.

But since it is an equality, it is OK to still have a virtual number in the solution since it is truly impossible to solve for x without more information, so, the real answer to x= can be two: either--

View attachment 1090898

But you'd be surprised the number of people who do not know how to rework a fractional power.

It's not used in accounting ... most folk have no use for this ... I use it for shorthand e.g. kg m s^-2 or W m^-2 K^-4 ... I thought you offer this as an equation, something to be solved ... it's a function ... needs her differential taken ... x^-1/4 (?) ...

 

[scruffy]

Conceptual.

I'll give you a great modern example.

Complex harmonic decomposition of critical states in the human brain creates sustainable low dimensional manifolds.

What does that mean?

It means: if you mix two chemicals and heat them up a little, you get this:

The patterns you see here are stable. That means, they're like standing waves.

Ordinarily, if you heat water and salt, you get bubbles. Bubbles are unstable, they swirl around and then they burst.

So how come we get this beautiful stable pattern when we mix the salt with bromic acid instead of water?

The answer is: long range interactions.

Ordinarily salt will only interact with the water that's immediately around it. This causes "chaos", in the sense of chaotic dynamics. Hence, bubbles. Lots of local swirling and bursting.

But sometimes, chaos in one place can couple with chaos in another - and when it does, the interaction can stabilize the local dynamics. In fact, if you add malonic acid and a little sulfuric to this reaction, the patterns change color. They start looking a lot like brain waves.

The math principle involved here is called "manifold reduction". Manifold is just a fancy name for surface, with the nuance that it's continuously differentiable (ie "smooth").

https://direct.mit.edu/neco/article/15/6/1373/6730/Laplacian-Eigenmaps-for-Dimensionality-Reduction

The "LaPlacian" is fundamental to all of physics. It's just a little geometry - it means the divergence of the gradient.

Students can learn this very easily. A gradient is a single picture. Divergence is a little harder, maybe two pictures and a few words. Eigenvalues are just matrix math. So if a student knows algebra all you have to do is show them a couple of pictures and explain a little bit of geometry. You don't have to tell them they're learning nonlinear nonequilibrium thermodynamics, cause that'll scare them. You just show them the pictures, and they go "wow, cool".

Then you surprise them by telling them the brain works exactly the same way. And that means, they don't have to do dangerous chemistry experiments with concentrated sulfuric acid, all they have to do is get on a computer and then they can see the exact same pictures there - in an "artificial brain".

This falls into the category of "machine learning", which is important in today's AI world. You can explain the difference between this and ChatGPT in one sentence: this is self organizing, whereas ChatGPT requires training.

If you're a teacher, all you have to do is bring the kids to the "wow, cool" moment. The rest will take care of itself. So you don't write down a pile of equations, you just show the pictures and explain the concepts. Divergence in the context of fluid flow is a natural, it takes less than 60 seconds to explain it.

You only start writing equations later, when you ask the kids to start solving real world problems. Eigenvalues from scratch takes maybe half a class - and again a picture is worth a thousand words. There are videos on YouTube that explains eigenvalues in terms of linear maps in less than 15 minutes, and they're full of cool graphics so the kids can visualize the concept without any hairy equations.

Then you can spend the actual class time explaining what Navier-Stokes is and why it matters, and what its relationship is to the reaction-diffusion equation they're witnessing in the above pic. In about 3 class days you can bring the kids from simple linear algebra to manifold reduction.

The simple idea in the pic is you have a reaction promoter and a reaction inhibitor, diffusing across the manifold at different rates. If you're a really good teacher you can take this all the way to stochastic resonance in 7 days.

Kids don't have to solve the Navier-Stokes equation, they just have to know what it is. Oh yeah I heard of that, it's fluid flow right? Yeah well, it's also climate and brain waves and a few other things... but yeah. If they want to see the actual math you can show them. Here it is:

Oregonator - Wikipedia

en.wikipedia.org

It's not hard, just some coupled equations. And now you can start talking about the coupling constants. Which in turn leads to a discussion of phase space, and eventually bifurcations.

Oregonator - Scholarpedia

www.scholarpedia.org

My take is, this should be about 12th grade math. A survey of dynamical systems. Kids should know about this stuff before they get to college. So when they take the multivariable calculus class, they're not intimidated by div, grad, and curl.

 

[Winco]

But you'd be surprised the number of people who do not know how to rework a fractional power.

And those might suggest you can't leave an answer with a RADICAL in the denominator.
You got a final solution?
It is called, Rationalizing the Denominator. Heard of it?

I get that a lot, people always think they must solve for x. I originally created the problem for a Russian troll here called Selivan who used to troll us on the wonders and superiority of Russia.

But since it is an equality, it is OK to still have a virtual number in the solution since it is truly impossible to solve for x without more information, so, the real answer to x= can be two: either--

View attachment 1090898

But you'd be surprised the number of people who do not know how to rework a fractional power.

How about this.

 

[ReinyDays]

And those might suggest you can't leave an answer with a RADICAL in the denominator.

There may be special situations where we can't divide by a radical ... I can't think of any right now ... but we divide by a fourth root in the Stefan-Boltzmann equation used in climatology ... T = ( S( 1 - a ))^(1/4) / (4oe)^(1/4) where T=temperature, S=solar constant, a=albedo, o=SB constant and e=emissivity

How about this. View attachment 1091094

... or 1/(2cos(π/6)) ...

 

[scruffy]

Numerical analysis is hard.

Kids should learn groups before they have to tackle that stuff.

Groups lead to rings, which then leads to numerical analysis (primes and such).

 

[scruffy]

Criticality and phase transitions are also intuitive, for kids.

You can demonstrate a first order transition by boiling water. The temperature related kinetics suddenly becomes more powerful than the ionic bonds, and then the bonds break and you get bubbles and gas escapes.

You can demonstrate a second order transition by heating a magnet with a torch. At some critical temperature the long range correlations will become more powerful than the local spin alignment, and the magnet will stop working. (That is, the local spins won't align anymore which means there will be no macroscopic magnetizatíon).

The difference is, the second order magnetic transition is continuous, you can differentiate it. And if you position the system exactly at the middle of the transition you get criticality. You can demonstrate criticality in two ways:

1. There is a huge spike in correlation at criticality, which disappears on either side

2. The ratios of the sizes of aligned and unaligned regions is fractal at criticality. No matter how far you zoom in or zoom out your microscope, you always see the same thing. The ratios can be plotted to derive the fractal index.

And then you explain what fractal means by discussing the size of the yardstick (or ruler), and show cool fractals like the Julia set.

So now, without solving any equations, the kids understand phase transitions, criticality, and fractals - and they're probably saying "wow, cool".

So now you give them a homework problem to calculate the correlation of two datasets (let's say, two sets of measurements of local spin from your magnet experiment). They're learning applied statistics. Then the next day you talk about it, show them Pearson correlation coefficients and all that stuff. Which leads to statistical distributions, which then leads to relative entropy, and now you're right back to gas and water and phase transitions, and of course the whole time you've been dropping Boltzmann's name so now you can show the class a Boltzmann machine for AI, operating on the edge of criticality so it can store 200x as much information as the ChatGPT they're using on their cell phones (which is also a Boltzmann machine, as is most generative AI in today's world).

 

[scruffy]

Here's a great resource for young kids.

GeoGebra - the world’s favorite, free math tools used by over 100 million students and teachers

Free digital tools for class activities, graphing, geometry, collaborative whiteboard and more

www.geogebra.org

Interactive, intuitive, accurate.

 

[scruffy]

Here's the real problem:

We have a bunch of old school math types teaching incorrect math.

It's gonna confuse the hell out of the kids when they get to college, because they're going to have to unlearn everything they learned, before learning the right way.

I'll give you a specific example. I'm up on Coursera doing a survey of their math classes (of which there are many). They have a class called "mathematics for machine learning", which is basically the linear algebra of matrices. It's taught by Imperial College London, which is a respected institution.

In this class, the first thing they cover is change of basis, which is probably the most important procedure in linear algebra. And they try to keep it simple, so they only show you 2d and 3d vectors, and small matrices.

However if you know machine learning, it's all about tensors, which are vectors on steroids. Vectors are an elementary form of tensors, so if you're studying vectors, it's important to learn them properly, so you won't get confused when you see the real thing.

The PROBLEM is, the course teaches old school linear algebra. It's not just old school, it's actually incorrect. They teach you to build a transformation matrix by writing the new basis as columns, and then "apply it to" the old basis vector by multiplying on the left. This is not correct, and it'll break the minute you start looking at a real tensor.

Basis vectors are ROWS, and they need to be multiplied on the RIGHT by the transformation matrix. If you do it the other way, you'll never be able to keep your forward and backward transforms straight. Not only that, but the Einstein summation convention won't work.

Kids need to know right up front, that a basis is something different from a set of coordinates. Basis is covariant, coordinates are contravariant. Covariant indices go on the bottom, contravariant indices go on top. In the old school, they didn't care. They just told you the most convenient way to get the calculations to work, but the problem is the convenient way is wrong.

So I'm looking at their educational video, they're showing their students a transformation matrix they call B, and a transformation that reads

new basis = B . old basis

and then they're showing you the new and old basis written as column vectors.

This is wrong! Completely incorrect. Mathematically, and especially from a machine learning point of view.

First of all, the matrix should be called F, not B. F is the forward transform, whereas B is actually the backwards transform. Confusion #1.

Then the conversion should read

new basis = old basis . F

with the basis elements written as row vectors. Because they are covariant, their indices go on the bottom, and that way the Einstein summation for the right hand side works out correctly. Confusion #2.

Then to make matters worse, they start talking about coordinate transformations by forcing you to calculate F(-1) even though they call it B(-1). Confusion #3.

And finally all their coordinates are also column vectors, so there's no distinction between columns and rows. Confusion #4.

I'm providing this as an example, but it's by no means the only example. And I'm not trying to single out Coursera or ICL, the problem is this is the way linear algebra was taught, even 20 years ago. Any teacher who needs a class syllabus will go to the internet, where they will find THIS outdated 20 year old material. To get anything different they would have needed a refresher class within the last 5 years, OR, they would have needed advanced math in tensor algebra or differential geometry.

Simple truth: the Chinese are teaching their kids the right way, and we're not. In our system, you have to get to differential geometry before you learn the right (consistent) way to do a simple change of basis.

And that is a bad thing. It does a disservice to our kids and turns them off from math. Not to mention creating a competitive disadvantage for American engineers.

We desperately need to revamp our math curriculum. We need a generation of geniuses, not a generation of dummies. Teach it right the first time. Give the kids a solid foundation they can build on for the rest of their lives.

 

[scruffy]

A further example of Scruffy's math program.

Introduce the concept of "algebraic characteristic" in first year geometry.

Show it to the kids by discussing "rotation groups" of the circle.

First year geometry has a lot to do with circles. It's a natural.

Characteristic is defined thusly:

The smallest number of multiplicative identities that sum to the additive identity.

In other words, 1 + 1 + ... + 1 = 0

If no such number exists, the field has characteristic zero.

Therefore, children have to learn what a "field" is, and they have to start thinking abstractly about the concepts of "addition" and "multiplication".

This is important BECAUSE:

1. When they become computer programmers in college (which everyone has to do these days, even sociologists lol) they'll intuitively understand what "overloading an operator" means, and they won't be intimidated by it

2. They are learning some basic truths about Lie groups and Galois theory, both of which are essential in every field of modern science and engineering. "Permutation groups" are essential in the study of computer algorithms, a class which every STEM student has to take.

So when the kids see an equation like "1 = 0" or "1 + 1 = 0", they'll know what it means. If you have a finite group defining the rotation of a circle by 360 degrees, then 1 = 0. If the rotation is only 180 degrees, then 1 + 1 = 0.

When the kids finally get to differential equations in college, they'll have to be experts in finding the roots of polynomials. The quadratic formula doesn't cut it anymore. If your roots are in the complex plane you can use Galois theory to find out which permutations will work, and you may need a Clifford algebra too if it turns out your operators are non-commutative (a situation which covers half of quantum mechanics - if you'd like an example Google on "Zassenhaus approximation" and read about the Campbell-Baker-Hausdorff formula).

This kind of knowledge is now FUNDAMENTAL for any kind of understanding of modern materials. A simple example being photonics, and a more complicated example being proteins with multiple stable shapes.

Scruffy gets around. He talks to lots and lots of engineers. The Chinese and Indians, they all understand this stuff. Even the 20 year olds. The American engineers? Blank stare. Glassy eyes.

I hate to say it, but we're way behind in education. We need to up our game. No more degrees in underwater basket weaving. You shouldn't be able to get a master's or PhD in any kind of STEM field without passing an advanced math test. Calculating reflections in a transmission line isn't good enough anymore. Navier-Stokes isn't good enough anymore. Differential geometry (using tensors) should be the MINIMUM for a bachelor's degree.

No more basket weaving. If you want to be a sociologist you better know more than statistics. If you want a master's or PhD in sociology and you've never heard of an Ornstein-Uhlenbeck process you FAIL and you need to go back to square one.