Posts Tagged ‘math’
Pairing math and music in integrated teaching method? And Most of us will love doing math?
Posted by: adonis49 on: May 1, 2020
Pairing math and music in integrated teaching method?
And Most of us will love doing math?
Like “If a student can clap about a beat based on a time signature, well aren’t they adding and subtracting fractions based on music notation? We have to think differently.”
Jazz composer Herbie Hancock later studied electrical engineering at Grinnell College before starting his jazz career full-time.
He says there is an intrinsic link between playing music and building things, one that he thinks should be exploited in classrooms across the country, where there has been a renewed emphasis on science, technology, engineering and math (STEM) education.
Hancock joined a group of educators and researchers Tuesday at the U.S. Education Department’s headquarters to discuss how music can be better integrated into lessons on math, engineering and even computer science, ahead of International Jazz Day this weekend.
Education Secretary John B. King Jr. said that an emphasis on math and reading — along with standardized testing — has had the unfortunate side effect of squeezing arts education out of the nation’s classrooms, a trend he thinks is misguided.
“English and math are necessary but not sufficient for students’ long-term success,” King said, noting that under the Every Student Succeeds Act, the new federal education law, schools have new flexibility to use federal funding for arts education.
Hancock is the chairman of the Thelonious Monk Institute of Jazz, which has developed MathScienceMusic.org, a website that offers teachers resources and apps to use music as a vehicle to teach other academic lessons.
One app, Groove Pizza, allows users to draw lines and shapes onto a circle. The circle then rotates and each shape and line generates its own distinct sound.
It’s a discreet way for children to learn about rhythm and proportions. With enough shapes and lines, children can create elaborate beats on the app, all in the context of a “pizza” — another way to make learning math and music palatable to kids.
Another app — Scratch Jazz — allows children to use the basic coding platform Scratch to create their own music.
“A lot of what we focus on is lowering the barriers to creative expression,” said Alex Ruthmann, a professor of music education at New York University who helped develop the Groove Pizza app.
Other researchers discussed their experiments with music and rhythm to teach fractions and proportionality, a challenging concept for young students to grasp when it is taught in the abstract.
Susan Courey, a professor of special education at San Francisco State University, developed a fractions lesson that has students tap out a beat.
“It goes across language barriers, cultures and achievement barriers and offers the opportunity to engage a very diverse set of students,” Courey said.
In a small study, students who received the music lesson scored 50 percent higher on a fraction test than those who learned with the standard curriculum. “They should be taught together.”
Hancock thinks that the arts may offer a better vehicle to teach math and science to some students. But he also sees value in touching students’ hearts through music — teaching them empathy, creative expression and the value of working together and keeping an open mind.
“Learning about and adopting the ethics inherent in jazz can make positive changes in our world, a world that now more than ever needs more creativity and innovation and less anger and hostility to help solve the challenges that we have to help deal with every single day,” Hancock said.
[Top business leaders, 27 governors, urge Congress to boost computer science education]
The Shortest-Known Paper Published in a Serious Math Journal:
Two Succinct Sentences
In Math| April 13th, 2015
Euler’s conjecture, a theory proposed by Leonhard Euler in 1769, hung in there for 200 years.
Then L.J. Lander and T.R. Parkin came along in 1966, and debunked the conjecture in two swift sentences.
Their article — which is now open access and can be downloaded here — appeared in the Bulletin of the American Mathematical Society.
If you’re wondering what the conjecture and its refutation are all about, you might want to ask Cliff Pickover, the author of 45 books on math and science. He brought this curious document to the web last week.
The math behind basketball’s wildest moves
Posted by: adonis49 on: October 7, 2016
The math behind basketball’s wildest moves
My colleagues and I are fascinated by the science of moving dots. So what are these dots?
it’s all of us. we’re moving in our homes, in our offices, as we shop and travel throughout our cities and around the world.
And wouldn’t it be great if we could understand all this movement? If we could find patterns and meaning and insight in it.
luckily for us, we live in a time where we’re incredibly good at capturing information about ourselves. So whether it’s through sensors or videos, or apps, we can track our movement with incredibly fine detail.
0:47 it turns out one of the places where we have the best data about movement is sports. So whether it’s basketball or baseball, or football or the other football, we’re instrumenting our stadiums and our players to track their movements every fraction of a second.
what we’re doing is turning our athletes into moving dots.
we’ve got mountains of moving dots and like most raw data, it’s hard to deal with and not that interesting. But there are things that, for example, basketball coaches want to know. And the problem is they can’t know them because they’d have to watch every second of every game, remember it and process it.
And a person can’t do that, but a machine can. The problem is a machine can’t see the game with the eye of a coach. At least they couldn’t until now. So what have we taught the machine to see?
we started simply. We taught the machine things like passes, shots and rebounds. Things that most casual fans would know. And then we moved on to things slightly more complicated. Events like post-ups, and pick-and-rolls, and isolations.
if you don’t know them, that’s okay. Most casual players probably do. Now, we’ve gotten to a point where today, the machine understands complex events like down screens and wide pins. Basically things only professionals know. we have taught a machine to see with the eyes of a coach.
how have we been able to do this? If I asked a coach to describe something like a pick-and-roll, they would give me a description, and if I encoded that as an algorithm, it would be terrible. The pick-and-roll happens to be this dance in basketball between four players, two on offense and two on defense. And here’s kind of how it goes. So there’s the guy on offense without the ball and he goes next to the guy guarding the guy with the ball, and he kind of stays there and they both move and stuff happens, and ta-da, it’s a pick-and-roll.
that is also an example of a terrible algorithm. So, if the player who’s the interferer — he’s called the screener — goes close by, but he doesn’t stop, it’s probably not a pick-and-roll. Or if he does stop, but he doesn’t stop close enough, it’s probably not a pick-and-roll.
Or, if he does go close by and he does stop but they do it under the basket, it’s probably not a pick-and-roll.
Or I could be wrong, they could all be pick-and-rolls. It really depends on the exact timing, the distances, the locations, and that’s what makes it hard.
So, luckily, with machine learning, we can go beyond our own ability to describe the things we know.
how does this work? Well, it’s by example. So we go to the machine and say, “Good morning, machine. Here are some pick-and-rolls, and here are some things that are not. Please find a way to tell the difference.”
And the key to all of this is to find features that enable it to separate. if I was going to teach it the difference between an apple and orange, I might say, “Why don’t you use color or shape?”
And the problem that we’re solving is, what are those things? What are the key features that let a computer navigate the world of moving dots?
figuring out all these relationships with relative and absolute location, distance, timing, velocities — that’s really the key to the science of moving dots, or as we like to call it, spatiotemporal pattern recognition, in academic vernacular. Because the first thing is, you have to make it sound hard — because it is.
The key thing is, for NBA coaches, it’s not that they want to know whether a pick-and-roll happened or not. It’s that they want to know how it happened.
And why is it so important to them? So here’s a little insight. It turns out in modern basketball, this pick-and-roll is perhaps the most important play. And knowing how to run it, and knowing how to defend it, is basically a key to winning and losing most games.
it turns out that this dance has a great many variations and identifying the variations is really the thing that matters, and that’s why we need this to be really, really good.
here’s an example. There are two offensive and two defensive players, getting ready to do the pick-and-roll dance. So the guy with ball can either take, or he can reject. His teammate can either roll or pop. The guy guarding the ball can either go over or under. His teammate can either show or play up to touch, or play soft and together they can either switch or blitz and I didn’t know most of these things when I started and it would be lovely if everybody moved according to those arrows.
It would make our lives a lot easier, but it turns out movement is very messy. People wiggle a lot and getting these variations identified with very high accuracy, both in precision and recall, is tough because that’s what it takes to get a professional coach to believe in you. And despite all the difficulties with the right spatiotemporal features we have been able to do that.
Coaches trust our ability of our machine to identify these variations. We’re at the point where almost every single contender for an NBA championship this year is using our software, which is built on a machine that understands the moving dots of basketball.
not only that, we have given advice that has changed strategies that have helped teams win very important games, and it’s very exciting because you have coaches who’ve been in the league for 30 years that are willing to take advice from a machine.
it’s very exciting, it’s much more than the pick-and-roll. Our computer started out with simple things and learned more and more complex things and now it knows so many things. Frankly, I don’t understand much of what it does, and while it’s not that special to be smarter than me, we were wondering, can a machine know more than a coach? Can it know more than person could know? And it turns out the answer is yes.
The coaches want players to take good shots. So if I’m standing near the basket and there’s nobody near me, it’s a good shot. If I’m standing far away surrounded by defenders, that’s generally a bad shot. But we never knew how good “good” was, or how bad “bad” was quantitatively. Until now.
what we can do, again, using spatiotemporal features, we looked at every shot. We can see: Where is the shot? What’s the angle to the basket? Where are the defenders standing? What are their distances? What are their angles?
For multiple defenders, we can look at how the player’s moving and predict the shot type. We can look at all their velocities and we can build a model that predicts what is the likelihood that this shot would go in under these circumstances?
why is this important? We can take something that was shooting, which was one thing before, and turn it into two things: the quality of the shot and the quality of the shooter. So here’s a bubble chart, because what’s TED without a bubble chart?
Those are NBA players. The size is the size of the player and the color is the position. On the x-axis, we have the shot probability. People on the left take difficult shots, on the right, they take easy shots. On the [y-axis] is their shooting ability.
People who are good are at the top, bad at the bottom. So for example, if there was a player who generally made 47 percent of their shots, that’s all you knew before. But today, I can tell you that player takes shots that an average NBA player would make 49 percent of the time, and they are two percent worse.
And the reason that’s important is that there are lots of 47s out there. And so it’s really important to know if the 47 that you’re considering giving 100 million dollars to is a good shooter who takes bad shots or a bad shooter who takes good shots. Machine understanding doesn’t just change how we look at players, it changes how we look at the game.
there was this very exciting game a couple of years ago, in the NBA finals. Miami was down by three, there was 20 seconds left. They were about to lose the championship. A gentleman named LeBron James came up and he took a three to tie. He missed. His teammate Chris Bosh got a rebound, passed it to another teammate named Ray Allen. He sank a three. It went into overtime.
They won the game. They won the championship. It was one of the most exciting games in basketball. And our ability to know the shot probability for every player at every second, and the likelihood of them getting a rebound at every second can illuminate this moment in a way that we never could before. Now unfortunately, I can’t show you that video. But for you, we recreated that moment at our weekly basketball game about 3 weeks ago.
And we recreated the tracking that led to the insights. So, here is us. This is Chinatown in Los Angeles, a park we play at every week, and that’s us recreating the Ray Allen moment and all the tracking that’s associated with it. So, here’s the shot.
I’m going to show you that moment and all the insights of that moment. The only difference is, instead of the professional players, it’s us, and instead of a professional announcer, it’s me. So, bear with me.
9:52 Miami. Down three. Twenty seconds left. Jeff brings up the ball. Josh catches, puts up a three!
10:03 [Calculating shot probability]
10:06 [Shot quality]
10:08 [Rebound probability]
10:11 Won’t go!
10:12 [Rebound probability]
10:14 Rebound, Noel. Back to Daria.
10:17 [Shot quality]
10:21 Her three-pointer — bang! Tie game with five seconds left. The crowd goes wild.
10:27 (Laughter)
10:29 That’s roughly how it happened.
10:30 (Applause)
10:31 Roughly.
10:33 (Applause) That moment had about a nine percent chance of happening in the NBA and we know that and a great many other things. I’m not going to tell you how many times it took us to make that happen.
10:46 (Laughter)
10:48 Okay, I will! It was four.
10:50 (Laughter)
10:51 Way to go, Daria.
But the important thing about that video and the insights we have for every second of every NBA game — it’s not that. It’s the fact you don’t have to be a professional team to track movement. You do not have to be a professional player to get insights about movement.
In fact, it doesn’t even have to be about sports because we’re moving everywhere. We’re moving in our homes, in our offices, as we shop and we travel throughout our cities and around our world. What will we know? What will we learn?
Perhaps, instead of identifying pick-and-rolls, a machine can identify the moment and let me know when my daughter takes her first steps. Which could literally be happening any second now.
11:47 Perhaps we can learn to better use our buildings, better plan our cities. I believe that with the development of the science of moving dots, we will move better, we will move smarter, we will move forward.
I hate to talk, read, and write. Oh, and I hate math: Different teaching resolutions
Posted by: adonis49 on: February 7, 2009
I hate to talk, read, and write. Oh, and I hate math: Different teaching resolutions…
I got this revelation.
Schools use different methods for comprehending languages and natural sciences. Kids are taught the alphabet, words, syntax, grammars, spelling and then much later, they are asked to compose essays. Why this process is not applied in learning natural sciences?
I have strong disagreement on the pedagogy of learning languages.
First, we know that children learn to talk years before they can read. Why kids are not encourage to tell verbal stories before they can read? Why kids’ stories are not recorded and then translated into the written words to encourage the kids into realizing that what they read is indeed another story telling medium?
Second, we know that kids have excellent capabilities to memorize verbally and visually whole short sentences before they understand the fundamentals. Why don’t we develop their cognitive abilities before we force upon them the traditional malignant methodology? The proven outcomes are that kids are devoid of verbal intelligence, hate to read, and would not attempt to write even after they graduate from universities.
Arithmetic and math are used as the foundations for learning natural sciences. We learn to manipulate equations; then solving examples and problems by finding the proper equation that correspond to the natural problem (actually, we are trained to memorize the appropriate equations that apply to the problem given!). Why we are not trained to compose a story that corresponds to an equation, or set of equations (model)?
If kids are asked to compose essays as the final outcome of learning languages, then why students are not trained to compose the natural phenomena from given set of equations?
Would not that be the proper meaning for comprehending the physical world or even the world connected with human behavior?
Would not the skill of modeling a system be more meaningful and straightforward after we learn to compose a world from a model or set of equations? Consequently, scientists and engineers, by researching natural phenomena and man-made systems that correspond to the mathematical models, would be challenged to learn about natural phenomena. Thus, their modeling abilities would be enhanced, more valid, and more instructive!
If mathematicians are trained to compose or view the appropriate natural phenomenon and human behavior from equations and mathematical models then the scientific communities in natural and human sciences would be far richer in quality and quantity.
Mathematics: a unifying abstraction?
Posted by: adonis49 on: October 26, 2008
Article #52, (September 12, 2006)
“Mathematics: a unifying abstraction for Engineering and Physics Phenomena”
A few examples of mechanical and electrical problems will demonstrate that mathematical equations play a unifying abstraction to various physical phenomena of entirely different physical nature.
Many linear homogeneous differential equations with constant coefficients can be solved by algebraic methods and their solutions are elementary functions known from calculus such as the examples in article 51. For the differential equations with variable coefficients, the functions are non elementary and they fall within two classes and play an important role in engineering mathematics.
The first class consists of linear differential equations such as Bessel, Legendre, and the hyper geometric equations; these equations can be solved by the power series method.
The second class consists of functions defined by integrals which cannot be evaluated in terms of finite many elementary functions such as the Gamma, Beta, and error functions (used in statistics for the normal distribution) and the sine, cosine, and Fresnel integrals (used in optics and antenna theory); these functions have asymptotic expansions in the sense that their series may not converge but numerical values could be computed for large values of the independent variable.
Entirely different physical systems may correspond to the same differential equations, not only qualitatively, but even quantitatively in the sense that, to a given mechanical system, we can construct an electric circuit whose current will give the exact values of the displacement in the mechanical system when suitable scale factors are introduced.
The practical importance of such an analogy between mechanical and electrical systems may be used for constructing an electrical model of a given mechanical system. In many cases the electrical model provides essential simplification because it is much easier to assemble and the values easily measured with accuracy while the construction of a mechanical model may be complicated, expensive, and time-consuming.
An RLC-circuit offers the following correspondence with a mechanical system such as: Inductance (L) to mass (m), resistance (R) to damping constant (c), reciprocal of capacitance (1/C) to spring modulus (k), derivative of electromotive force to the driving force or input force, and the current I(t) to the displacement y(t) or output.
Here are a few elementary examples:
5) Ohm’s law: Experiments show that the voltage drop (E) in a close circuit when an electric current flows across a resistor (R) is proportional to the instantaneous current (I), or E = R* I.
Also, that the voltage drop across an inductor (L) is proportional to the instantaneous time rate of change of the current, or E = L*dI/dt.
Also, the voltage drop across a capacitor (C) is proportional to the instantaneous electric charge (Q) on the capacitor, or E = Q*1/C. Note that I(t) = dQ/dt.
6) Kirchhoff’s second law: The algebraic sum of all the instantaneous voltage drops around any closed loop is zero, or the voltage impressed on a closed loop is equal to the sum of the voltage drops in the rest of the loop. Thus,
E(t) = R*I + L*dI/dt + Q*1/C.
For example, a capacitor (C = 0.1 farad) in series with a resistor (R = 200 ohms) is charged from a source (E = 12 volts). Find the voltage V(t) on the capacitor, assuming that at t = 0 the capacitor is completely uncharged.
7) Hooke’s Law: Experiments show that when a string is stretched then the force generated from the string is proportional to the displacement of the stretch,
or F = k*s. If a mass (M) is attached to a string, then when the string is stretched further more (y) after the system is in a static equilibrium, then: F = -k*s(0) – k*y.
Newton’s second law for the resultant of all forces acting on a body says that:
Mass * Acceleration = Force, or My” = -k*y.
Furthermore, if we connect the mass to a dashpot, then an additional force come into play, which is proportional to the rate of change of the displacement due to the viscous substance with constant (c). The equation is then a homogeneous second order differential equation: M*y” + c*y’ + k*y = 0. Depending on the magnitude of (c) we have 3 different solutions: either 2 distinct real rots, 2 complex conjugate roots, or a real double root [c(2) = 4*M*k)} corresponding respectively to the conditions of over damping, under dumping, or critical damping.
For example, determine the motions of the mechanical system described in the last equation, starting from y = 1, initial velocity equal zero, M = 1 kg, k =1 for the various damping constant: c = 0, c = 0.5, c = 1, c =1.5, and c = 2.
8) Laplace’s equation is one of the most important partial differential equations because it occurs in connection with gravitational fields, electrostatic fields, steady-state heat conduction, and incompressible fluid flow. The solutions of the Laplace equation fall within the potential theory.
For example, find the potential of the field between two parallel conducting plates extending to infinity which are kept at constant potentials; or the potential between two coaxial conducting cylinders; or the complex potential of a pair of opposite charged sources lines of the same strength at two points.
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