Sage Interact Essentials: Dynamic Table

Following up on an earlier post about tables in HTML, I put together a simple example of a Sage Interact manipulative where the table expands depending on the size of the data. I don't see much educational benefit to the manipulative; it simulates picking the order of a list of  n items and determines the experimental probability of getting a derangement. The table lists the probability of getting a derangement for lists of size 1 through n:

I've posted the code for this simple example on the Sage Interact Essentials page. If nothing else, it shows how an HTML table can make your manipulative look just a little bit nicer.

Graphics: Circles Inscribed in (Regular) Polygons

In the Graphics: Inscribed Polygons post of February 16, 2012 I posted a collection of inscribed polygons to the Graphics page to illustrate that the area of a regular polygon inscribed in a unit circle is a lower bound on the estimate of $latex \pi$. Today I've added a collection of unit circles inscribed in regular polygons (posted on the Graphics page) which can illustrate  the area of the regular polygons is an upper bound on the estimate of $latex \pi$. Before calculators were invented, mathematicians could figure out the length of each side of the regular polygon (if the angle was "nice") and use it to calculate the area of the regular polygon which would be greater than $latex \pi$.

In the case above, $latex x=\sqrt{3}$ so the length of each side of the triangle is $latex 2\sqrt{3}$. The area formula for the regular triangle with each side of length $latex 2\sqrt{3}$ is calculated using the formula $latex A=\frac{3(2\sqrt{3})a}{2}$ where a is the apothem. Since this value is 1, the area of the triangle is just $latex A=3\sqrt{3}$, hence $latex \pi <3\sqrt{3}$. That's not a good estimate of $latex \pi$ but, of course, that estimate gets better as the number of sides increases.

Sage Interact Essentials: Tables

Chances are, at some point, you'll find you want to create tables for your Sage Interact manipulative. Tables are a great way to organize information and can make your manipulative look a little more professional.

Having some examples to refer to as you create your own tables is useful. I've added this information on tables (along with some code) to the Sage Interact Essentials page. The output of the code looks like this:

Web Equation

If you're new to $latex \LaTeX$, learning all the commands to typeset math takes time. Chances are you'll make some mistakes as well and end up with code that doesn't compile. That leaves you with the task of debugging the code, slowing you down even more. If that's deterring you from using $latex \LaTeX$, you might be interested in a tool that can help: Web Equation. I stumbled upon it recently and thought it might be helpful to those not yet proficient with $latex \LaTeX$. Click on the garbage can to clear the screen, write the equation you want and it will show you what it thinks you want in the bottom left hand corner. If that's correct, all you need to do is copy and paste the $latex \LaTeX$ code it gives you from the bottom right hand corner. It even gives MathML output. If you make a mistake along the way, the arrows in the upper left hand corner of the yellow region allow you to undo and redo your strokes.

I've found the site does a pretty good job with a lot of the examples I tried: complex fractions, sequences, series, integrals, and more. It couldn't handle equations with matrices, though, and I felt at times it would be nice if they'd add an eraser to let you remove part of a stroke. I've added a link to the Web Equation site on the sidebar. Note that there are more tools on the Web Equation home page.



Common Core: math

Currently there are no national standards for teaching math in a public school. Each state has its own guidelines and the quality can vary dramatically from state to state. The CCSO helped design Common Core curricula standards in English, language arts and mathematics and these standards are are required if a state wants to a piece of the $4 billion in the Race to the Top competition of the Obama administration. As a result, more and more states are moving to the Common Core standard. By 2015 almost all the states will be teaching according to these standards.

The mission statement from the Common Core website says, in part, that the standards "provide a consistent, clear understanding of what students are expected to learn, so teachers and parents know what they need to do to help them. The standards are designed to be robust and relevant to the real world, reflecting the knowledge and skills that our young people need for success in college and careers.".

On one level, the idea of common core standards is a great idea. The Common Core website indicates that the standards:

  • Are aligned with college and work expectations;
  • Are clear, understandable and consistent;
  • Include rigorous content and application of knowledge through high-order skills;
  • Build upon strengths and lessons of current state standards;
  • Are informed by other top performing countries, so that all students are prepared to succeed in our global economy and society; and
  • Are evidence-based.

Having a common set of standards makes it easier for publishers of textbooks to create books acceptable to all states. Moreover, a common set of standards should make state curricula more uniform, which is good for those families with children that move from one state to another.

All this is background for a very interesting article I saw on Common Core math standards. Ze’ev Wurman and W. Stephen Wilson provide an excellent critique:

  • Common Core is vastly the standards in more than 30 states.
  • "Common Core standards are not on par with those of the highest-performing nations."
  • examples of how common core standards "..still tend to be wordy and hard to read."
  • "it seems that the majority of people in power think the three pages of Mathematical Practices in Common Core...are more important than the 75 pages of content standards, which they sometimes refer to as the “rote” mathematics. They are wrong."
  • "...the main authors of the Common Core mathematics standards had minimal prior experience with writing standards...How, otherwise, can one explain their selecting an experimental approach to geometry, teaching it on the basis of rigid motions, that has not been successfully tried anywhere in the world? Simple prudence and an ounce of experience would tell them either to stick to what is known to work or to recommend a trial phase before foisting it sight-unseen on a nation of 300 million."
  • "What should we make, then, of a recent study purporting to “validate” that Common Core standards indeed reflect college readiness? The study, led by David Conley, was published more than a year after Common Core standards were already certified as college-ready by…David Conley as a member of the Common Core Validation Committee. Paraphrasing Shakespeare, he doth attest too much."
  • "...their promise of college readiness rings hollow. Its college-readiness standards are below the admission requirement of most four-year state colleges."
  • "In other countries, if you say “learn to multiply whole numbers,” no one questions how this should be done; students should learn and understand the standard algorithm. In the U.S., even if you say “learn to multiply whole numbers with the standard algorithm,” some people will declare wiggle room and try to avoid the standard algorithm." (Observation: if teachers were hired on the basis of having a good math background there wouldn't be people declaring wiggle room and trying to avoid the standard algorithm.)
  • "No state has any reason left to aspire for first-rate standards, as all states will be judged by the same mediocre national benchmark enforced by the federal government. Moreover, there are organizations that have reasons to work for lower and less-demanding standards, specifically teachers unions and professional teacher organizations. While they may not admit it, they have a vested interest in lowering the accountability bar for their members."

That last point, which actually contains several points, seemed particularly on the mark to me. From what I've seen, public schools focus on the lowest level to meet because that's the standard by which school administrators are judged. It also leads to the minimum number of problems with parents and students; more failing students from tougher standards means more headaches from parents. Likewise, the difficulty in getting rid of bad teachers and the fact that many decisions are made on the basis of seniority make it clear that the best interests of the children take a back seat to other issues.

There are a lot of merits to the common core idea but it seems that, if everything goes well and according to plan, the proficiency in mathematics still won't be very high on an international level. A good start, I guess, but not much more than that.

Online LaTeX compilers

I haven't worked as much with LaTeX now that I'm teaching at a public school because getting LaTeX installed on my school computer is problematic. Without administrator rights to install it and with no administrator interested in sparing the time to install it, I've had to do my work at home and bring it in on a thumb drive. Since the school has Apple computers, MikTeX Portable isn't an option.

I'm not aware of any portable version of LaTeX for Macs, so the inconvenience and frustrations that come from working on LaTeX and bringing it in to school pushed me away from LaTeX. For example, if you forget to bring the thumb drive with the PDF output of your files then you can't get your work done. Or you notice a mistake (or need to increase the font size) in the PDF you brought in but you can't change it until you go home. Problems like that force you to plan further ahead, which isn't always possible. I just can't afford to forget the thumb drive because I know I'll be left scrambling to redo the work on a school computer.

So I finally decided to look for an online LaTeX compiler: Tex Stack Exchange to the rescue! This post was helpful in getting me started. I wanted the online LaTeX compiler to be free and I didn't want to be bothered by having to remember yet another password. I've settled on these 2 sites:


LaTeX-Online-Compiler V0.2

Looking at the log files from the output, neither compiler is up to date. The LaTeX-Online-Compiler is just a little bit more current, and it uses:

pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian)

Since the LaTeX files that I create for school are more basic, however, these websites are all I need. I've added both links to the sidebar. Now I can finally work with LaTeX at school.

Graphics: Inscribed Polygons

I've added a collection of inscribed polygons to the Graphics page. The intended use is to illustrate that the area of a regular polygon inscribed in a unit circle is a lower bound on the estimate of $latex \pi$. The area of a regular polygon can be calculated by breaking it up into isosceles triangles and using the formula $latex \frac{1}{2}ab\sin(C)$ (which is $latex \frac{1}{2}\sin(C)$ if inscribed in a circle of radius 1) on each of the pieces. When the isosceles triangles have a central angle for which the sine is known (e.g. 30 degrees) then the area of the polygon can be calculated exactly.

Sage Interact: Experimental Probability

Experimental probability comes up in a typical probability/statistics class but it's also a great topic to (re)introduce when your class is learning about the idea of limits (typically calculus or precalculus). That's because coin flipping is a relatively concrete and an easy to explain, natural example of a limit. Calculating the ratio of the number of heads flipped to the total number of coin flips is easy. Start by leading your class through the calculations for the ratio. If you do that for the first 10 or 20 flips your class should understand. They'll also get a chance to appreciate how quickly the ratio changes when there haven't been many coin flips. But as the number of flips increases, your class will definitely get bored and/or restless. That's where the Sage Interact manipulative comes in: they'll be able to see how the ratio becomes more stable when the coin has been flipped "many" times. This "stable" value, the limit, becomes our estimate of the probability of flipping heads on that particular coin.

If you're teaching prob/stats then the slider for setting the probability of getting heads on a coin flip might be useful in talking about a fair coin. If not then you can, of course, remove that section of the code.

The output is below. It's also posted on the Sage Output page:

The code is posted on the Python/Sage page.