Wolfram STEM Software

 

This is what Wolfram advertises for on their website:

“Educators and schools are under pressure to prepare students for entry into the workforce, while maintaining high test scores. Thousands of schools have adopted Mathematica to help them engage students, increase understanding of STEM concepts, and improve performance. […]

Mathematica helps educators make their classrooms dynamic—where students can interactively explore concepts in math, physics, economics, biology, engineering, or virtually any subject.

  • Easily solve equations and analyze mathematical and scientific data
  • Visualize functions or data in 2D or 3D
  • Access real-world data or use your own for analysis
  • Create and present lessons, lectures, and class materials featuring interactive content
  • Get started quickly with point-and-click palettes and free-form linguistic input

[…]

Wolfram Demonstration Project

Instantaneous Rate of Change: Exploring More Functions with the First and Second Derivatives  Snowboarding over Derivatives   The Greatest Fenced Area along a Barn   A Generalization of the Mean Value Theorem …..

 

Conceived by Mathematica creator and scientist Stephen Wolfram as a way to bring computational exploration to the widest possible audience, the Wolfram Demonstrations Project is an open-code resource that uses dynamic computation to illuminate concepts in science, technology, mathematics, art, finance, and a remarkable range of other fields.

Its daily growing collection of interactive illustrations is created by Mathematica users from around the world who participate by contributing innovative Demonstrations.

Interactive computational resources have typically been scattered across the web. Moreover, their creation requires specialized programming knowledge, making them difficult and expensive to develop. As a result, their breadth and reach are limited.

With its debut in 2007, the Wolfram Demonstrations Project introduced a new paradigm for exploring ideas, providing a universal platform for interactive electronic publishing. The power to easily create interactive visualizations, once the province of computing experts alone, is now in the hands of every Mathematica user. More importantly, anyone around the world can freely use these thousands of fully functional Demonstrations.

From elementary education to front-line research, topics span an ever-growing array of categories. Some Demonstrations can be used to enliven a classroom or visualize complex concepts, while others shed new light on cutting-edge ideas from academic and industrial workgroups. Each is reviewed and edited by experts for content, clarity, presentation, quality, and reliability.

Demonstrations run freely on any standard Windows, Mac, or Linux computer. In fact, you do not even need Mathematica. You can interact with any demonstration using the free Wolfram CDF Player—for most platforms this happens right in your web browser. If you have Mathematica you can also experiment and modify the code yourself.

Demonstrations can be created with just a few short lines of code. This opens the door for researchers, educators, students, and professionals at any level to create their own sophisticated mini-applications, then publish and share them with the world using Wolfram’s Computable Document Format (CDF).

You can embed Demonstrations on your website or blog. Just copy and paste the snippet of JavaScript code from the Share section of the Demonstration page into your page or post. The code will also add the author’s name and links to the Demonstrations page and the CDF Player Download page. Anyone with CDF Player installed will be able to interact with the Demonstration right on your website.

The Wolfram Demonstrations Project is part of the family of free online services from Wolfram Research—these include Wolfram|Alpha, the world’s first computational knowledge engine; MathWorld, the number one mathematics website; as well as The Wolfram Functions Site,WolframTones, and more.

Watch this video about the project: http://demonstrations.wolfram.com/about.html

Source for exploration of projects: http://demonstrations.wolfram.com/

Topics for math: http://demonstrations.wolfram.com/topics.html?Mathematics#2

Uses for the CDF-Player: http://www.wolfram.com/cdf/uses-examples/textbooks.html

Escaping the Education Matrix – The Human Factor

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“We tell a story about the power of learning that is very different from what we practice in traditional models of school,” says Steve Hargadon, education technology entrepreneur, event organizer, and host of the long-running Future of Education podcast series. If we really want children to grow up to become self-reliant and reach their full potential, “we would be doing something very different in schools. We live in a state of cognitive dissonance.”

His comments are informed by a recent cross-country tour facilitating community discussions on education, as well as more than 400 interviews he’s logged with a broad spectrum of education practitioners, analysts, and innovators.

“What are most kids getting out of 12 years of school?” he asks. “The honest answer is they’re learning how to follow, and that was the original intent. Public schools were based on the belief that what was needed was a small group of elites who would make the decisions for the country, and many more who would simply follow their directions” — hence a system that produces “tremendous intellectual and commercial dependency.”

And the notion that the smartest students rise to the top, regardless of family and social circumstances, “sends a message to the majority of students that they are losers,” Hargadon notes, which doesn’t square with a professed belief in the inherent value and capacity of every child.

“How do you tell a story that opens the door to rethinking what people have believed for decades?”

The system’s fundamental design also leads to a host of unintended consequences, including bullying. “We’re placing kids in an artificial environment,” he says, “telling most of them they’re not good at things, and then expecting them not to explode at each other? Of course they will. The ‘mean girls’ thing is not a natural part of childhood—it’s more a reflection of how kids are being treated than a reflection of kids. It’s shocking that we put up with it.”

The reason so many adults find the situation tolerable, he says, may stem from the fact that they experience little control over their own lives. Additionally, they themselves are products of the system and, as such, find it difficult to envision an alternative. “People are almost in this Matrix-like existence,” Hargadon says. “They don’t question schooling. How do you tell a story that opens the door to rethinking what people have believed for decades? So much in their lives depends on that story being what they think it is. How do you tell a new story that involves people reclaiming their destinies, children not being defective, and learning not being owned by one organization?”

There are also vested interests in the status quo. “The people who benefit from us not being active citizens, from all buying the same things, and being willing to take jobs that demand we leave our personal values at the door—they all benefit from the current schooling system, because it produces a populace that does not feel confident in being critical,” he notes. “At an institutional or personal level, those who benefit don’t have much incentive to promote changes in education that would lead people to question their motives or challenge their practices.”

To Drive Real Change, Focus on the Human Factors

An economic crisis (perhaps the one we’re already experiencing) may provide the financial imperative to overhaul the system, Hargadon says. But something even more powerful may take precedence: He’s noticed “more and more resonance with the idea of having a moral imperative for education,” pointing to the growing backlash against high-stakes testing as one indication of a shift in thinking.

He sees a need for more people to “stand up and say: ‘This is not the right thing for children—it’s not a healthy childhood.’” But families must also reclaim ownership of learning, rather than viewing it as the responsibility of schools and government, and also resist the tendency to make decisions for others. “In some ways, traditional schools have co-opted a lot of traditional parental responsibilities,” he says. “That’s really unhealthy, and it becomes self-fulfilling. And when society says it knows better than the family, it’s a recipe for disaster. Some family circumstances are not ideal, but it’s a slippery slope. It’s about trusting and respecting the capacity of individuals to make choices.”

Technology can support a transformation, but it’s not a silver bullet. The Internet has ushered in an era of “digital democracy” and increased people’s capacity to question the status quo. Widespread access to unlimited information has also opened many doors. But “the process of becoming a self-directed, independent learner is a very human process,” Hargadon says. “Recognizing the different needs of every student, and the desire to help each one become personally competent as a learner and find productive things to do in life—that won’t happen online.”

The temptation to “solve all these problems with data” must also be tempered, he says. “Data does not define the core things in education, such as someone opening your eyes to something.” There’s a lesson to be learned from the world of business, he adds, where “the true value of the ‘total quality’ movement came not from tracking, but from involving workers themselves in using the data for self improvement.”

A Future Marked by Greater Freedom and Collaboration

For models of healthier ways to frame education, Hargadon suggests looking to food and libraries. “No one says that from age six to 17, we will give you all the same food, at the same time, regardless of your individual circumstances or needs,” he says. He envisions a world where families can similarly choose where, how, and what they learn.

What might that world look like? He considers libraries good examples of places that already facilitate such mandate-free learning. “The reason we have a hard time conceiving [an alternate reality],” he says, “is because we so strongly associate education with control. If I ask you how you choose your own food, you’d probably say that it’s just what you do: Depending on your circumstances at the time, you may go to a farmer’s market or grocery store or restaurant or grow your own food. The difficulty is dismantling something that’s taken away our conception of having that kind of agency. But when I imagine that world, it includes things like community college classes, apprenticeships at businesses, educational certification programs. You have a range of choices, depending on the child’s interests.”

Hargadon sees connecting people to each other as the most effective way to get from here to there, hence his recent tour. “The tour convinced me that policy changes are not the answer, and that change needs to come from us,” he says. “As individuals, families and communities, we need to reclaim the conversation around learning, and to do so in such a way as to recognize the inherent worth and value of every student, with the ultimate goal of helping them become self-directed and agents of their own learning.”

Hargadon thinks one way change agents get tripped up is by promoting a particular model, rather than a process by which people can develop (or adopt) models that best fit their needs. He considers deep, meaningful conversations a useful starting point for people to use to shape the future, and to that end, he’s planning to host a series of national conversations in 2014 that probe the deeper questions around education and can serve as models for conversations people initiative in their own communities.

“Living in a democracy means involving people in decision making,” Hargadon says. “You can’t just create a new system to implement top down; you have to provide the opportunity to talk about it and build it constructively.”

By Luba Vangelova / MindShift / 8 January 2014

 

(ε, δ)-definition of limit


In calculus, the (ε, δ)-definition of limit (“epsilon-delta definition of limit”) is a formalization of the notion of limit. It was first given by Bernard Bolzano in 1817, followed by a less precise form by Augustin-Louis Cauchy. The definitive modern statement was ultimately provided by Karl Weierstrass.

Let f be a function. To say that

 \lim_{x \to c}f(x) = L \,

means that f(x) can be made as close as desired to L by making the independent variable x close enough, but not equal, to the value c.

How close is “close enough to c” depends on how close one wants to make f(x) to L. It also of course depends on which function f is and on which number c is. Therefore let the positive number ε(epsilon) be how close one wishes to make f(x) to L; strictly one wants the distance to be less than ε. Further, if the positive number δ is how close one will make x to c, and if the distance from to c is less than δ (but not zero), then the distance from f(x) to L will be less than ε. Therefore δ depends on ε. The limit statement means that no matter how small ε is made, δ can be made small enough.

The letters ε and δ can be understood as “error” and “distance”, and in fact Cauchy used ε as an abbreviation for “error” in some of his work. In these terms, the error (ε) in the measurement of the value at the limit can be made as small as desired by reducing the distance (δ) to the limit point.

The (\varepsilon, \delta) definition of the limit of a function is as follows:

Let f(x) be a function defined on an open interval containing c (except possibly at c) and let L be a real number. Then we may make the statement

 \lim_{x \to c} f(x) = L \,

if and only if:

If the value of x is within a specified \delta units from c, this implies that f(x) is within a specified \varepsilon units from L.

or, symbolically,

 \forall \varepsilon > 0\ \exists \ \delta > 0 : \forall x\ (0 < |x - c | < \delta \ \Rightarrow \ |f(x) - L| < \varepsilon).

All that the statement

 0 < | x - c | < \delta

means is that x is within \delta units of c, since all it really states is that the magnitude of the difference between x and c is greater than 0 and no more than \delta. In this sense the condition term of the requirements for a limit to exist first asserts that an arbitrary \delta should be picked, and then the range of surrounding x values calculated. In exactly the same manner, the conclusion, that

 |f(x) - L| < \varepsilon,

in plain English reduces to that f(x) must remain within \varepsilon units of L. In other words, in order for the limit to exist, one must be able to pick a small x window around c, and deduce that the value of the function f(x) must remain bounded within a certain calculable range.

Let us prove the statement that

\lim_{x \to 5} (3x - 3) = 12.

This is easily shown through graphical understandings of the limit, and as such serves as a strong basis for introduction to proof. According to the formal definition above, a limit statement is correct if and only if confining x to \delta units of c will inevitably confine f(x) to \varepsilon units of L. In this specific case, this means that the statement is true if and only if confining x to \delta units of 5 will inevitably confine

3x - 3

to \varepsilon units of 12. The overall key to showing this implication is to demonstrate how \delta and \varepsilon must be related to each other such that the implication holds. Mathematically, we want to show that

 0 < | x - 5 | < \delta \ \Rightarrow \ | (3x - 3) - 12 | < \varepsilon .

Simplifying, factoring, and dividing 3 on the right hand side of the implication yields

 | x - 5 | < \varepsilon / 3 ,

which looks strikingly similar in form to the left hand side now. To complete the proof, we are granted the mathematical freedom to choose a delta \delta such that the implication holds. A quick look at the expression

 0 < | x - 5 | < \delta \ \Rightarrow \ | x - 5 | < \varepsilon / 3

encourages that one choose that

 \delta = \varepsilon / 3 .

Substituting back in to the above yields

 0 < | x - 5 | < \varepsilon / 3 \ \Rightarrow \ | x - 5 | < \varepsilon / 3 ,

which is clearly true since the two sides are equivalent. And thus the proof is completed. Though it may seem unnecessary, the key to the proof lies in the ability of one to choose boundaries in x, and then conclude corresponding boundaries in f(x), which in this case were related by a factor of 3, which in retrospect is entirely due to the slope of 3 in the line

 y = 3x - 3 .

 

 

I’ve never been very satisfied with the epsilon-delta applets I’ve seen. This applet is almost precisely what I’ve always wanted, since it highlights the values of f(x) that make your delta not work and makes it clear when it does work:

Delta/Epsilon Definition of Limit (Created with GeoGebra – Shared by ddrake)

The applet enables students to visually identify the definition of limit.