Good Graphs

School administrators, it seems to me, are crazy about technology in the classroom. Technology is the answer in search of a question. In my school, there's a steady drumbeat of pressure to incorporate technology into the classroom. The message is rarely "Use this technology tool because it will help you do x better"; the impression I'm left with is that using technology makes the school look cutting edge to each group that comes in to take a tour. The push for I-pads in the classroom is such an example, yet there are many students caught up in games to the exclusion of school.

I'll be the first to admit that technology should be an important part of education provided it is used judiciously and appropriately. Graphing calculators should be a requirement for higher level math classes such as precalculus and calculus; after students have mastered the basics. The problem is it's never very simple at the public high school level; students get passed on despite having a poor mathematical background. Combine that with schools which are eager use technology and you're potentially creating all sorts of problems. It's no surprise to me that Shanghai led the world in math scores for high school students even without an I-pad in the classroom.

One of the reasons for using technology, I hear, is to save students from the drudgery of math so they can focus on the concepts. Arguments like these are often short sighted and made by people who don't have the math background needed to make good decisions on math policy. Let's look at a typical example: graphing calculators, we're told, can graph functions quicker and easier than humans.  Obviously that's true but advocates seem to think that means a graphing calculator should be used to save students having to plot graphs by hand. But that's just wrong. Confused? Suppose, for example, you need to graph $latex x^4-4x^3-12x^2-6$. Plot that with your graphing calculator and maybe you get something like this:

The graph was created almost instantly; so are we done? Of course not, we're just getting started. The calculator hasn't solved the problem because the graph shown is lousy. Which brings us to today's post: good graphs. We need some criteria to decide if the graph created is a good portrayal of the function. I stressed the concept of a good graph by which I mean the graph should:

  • be centered
  • show all relative and absolute minimums and maximums
  • contain all horizontal and vertical asymptotes
  • display enough of the graph that the user can properly infer the information not on the screen (eg horizontal asymptotes or period of a sine function)

A good graph isn't always possible but for almost every function at the high school level it is. By the criteria listed, the graph above isn't very good. It takes some calculus for us to find the minimums and maximums and limits to find the behavior of the function "outside the screen". Knowing those basic skills allows us to plot the graph properly to highlight the essential features. I'll spare you the calculations but here's a good graph of the function $latex x^4-4x^3-12x^2-6$:

This graph is centered, shows 1 relative minimum, 1 relative maximum, and 1 global minimum. It has no asymptotes and the user can properly infer that $latex \lim_{x\to\infty}x^4-4x^3-12x^2-6=\infty$ and $latex \lim_{x\to-\infty}x^4-4x^3-12x^2-6=\infty$ from the picture. All the "action" is shown on the screen; that's a good graph.

To properly graph a function, I insisted that my students know the essential characteristics of a good graph are and be able to work (step by step) to establish:

  • $latex \lim_{x\to\infty}f(x)$
  • $latex \lim_{x\to-\infty}f(x)$
  • vertical asymptotes
  • minimums and maximums

Those who think that pushing some buttons on a calculator can replace having to master the basics shouldn't be setting math policy but (unfortunately) these just the people setting policy. If you teach calculus, the concept of a good graph can help you to push your students to master some basics.


How do I....? (align text [left, right, center])

I've added a page on aligning your text in $latex \LaTeX$ as part of the How Do I...? section. By default, your text is center aligned which means $latex \LaTeX$ expands or contracts the spacing between your words to make the line appear full. Many people are used to text which is left justified; this results in the lines of your text looking ragged on the right. Of course, $latex \LaTeX$ gives you that option as well as right justification.

You can access the page through the sidebar menu or press here.


Chess Basics: The Value of the Pieces

I've add a post to the Chess page on the value of the pieces. Although there is no agreement on the value of the pieces (because the value of the pieces depends on the particular position) relying on Fischer's value of the pieces will give the beginner a crude tool to help them evaluate the position and prevent them from trading a stronger pieces for weaker pieces. Should you really consider trading your queen for 2 bishops and a knight? Find out!

Graphics: PST-lens

In an earlier post, I added graphics for Riemann sums under the curve $latex y=x^2$ which was supposed to complement the later post on integrals and series. Today I've added graphics to polish that integrals and series post just a little bit more. The first graphics uses 25 rectangles and is just a trivial modification of the earlier file. For the student, showing 25 rectangles makes it clear that the estimate of the area under $latex y=x^2$ improves and, more importantly, is a better way to show off the PST lens package.

In both graphics your students will see the area of the rectangles is getting very close to the area under the curve but the addition of the magnifying glass is a little more elegant. If you're interested in more than the graphics file then you need to be aware that pstricks doesn't compile under PDFlatex, so I used XeLaTeX to compile it (If you use Gummi then all you need to do is change the engine through the Preferences setting).

Tikz lovers have the spy option to give magnification but the pstricks-add package makes Riemann sums so easy to implement, I went with pst-lens. The code is a little harder to understand; essentially you define the thing you want to magnify first and give it a name, in this case \Graph. After that  you include it in the pspicture environment.

\PstLens[LensSize=2, LensMagnification=2, LensRotation=90](.75,.5){\Graph}

The \PstLens command then sets the parameters of the magnifying glass and applies it to \Graph to produce the picture above.

As usual, the graphics can be found on the Graphics page.

Problem: natural logs

If you use a calculator you'll find that $latex \ln(3^{\ln(7)})$ is the same as $latex \ln(7^{\ln(3)})$. But are they approximately the same or exactly the same? For what values of $latex a$ and $latex b$ is $latex \ln(a^{\ln(b)})$ the same as $latex \ln(b^{\ln(a)})$. Prove your assertion.

Two articles

I ran across 2 articles this week and one of them is an outstanding piece that everyone should take the time to read fully.

The best article is related an earlier post where I commented that public schools which were able to follow the private school model of making students go through entrance exams coupled with other factors, such as faculty with advanced degrees and smaller class sizes, were able to do create top notch schools. This article refers to them as "exam schools" and the focus of the article is, in their words, "Could the selective public high school play a larger role in educating our country’s high-achieving pupils?". The article then tries to describe the features of these schools: who goes to them, what type of classes do they offer, how are they governed and financed. Most importantly, are they effective and would be better off as a country in having more of these schools? Here's a few of the numerous interesting facts that they discovered.

  • "Their overall student body is only slightly less poor than the universe of U.S. public school students. Some schools, we expected, would enroll many Asian American youngsters, but we were struck when they turned out to comprise 21 percent of the schools’ total enrollment, though they make up only 5 percent of students in all public high schools. More striking still: African Americans are also “overrepresented” in these schools, comprising 30 percent of enrollments versus 17 percent in the larger high-school population. Hispanic students are correspondingly underrepresented, but so are white youngsters. Individual exam schools often qualify as racially “imbalanced”: in nearly 70 percent of them, half or more of the students are of one race."
  • "The schools we visited were serious, purposeful places: competitive but supportive, energized yet calm. Behavior problems (save for cheating and plagiarism) were minimal and students attended regularly, often even when ill. The kids wanted to be there, and were motivated to succeed. (Bear in mind that many of the schools seek such qualities in their applicants.)"
  • "We also came upon other kinds of specialized and advanced courses, in addition to or in lieu of AP and IB. Schools with a STEM focus or university affiliations, for example, reported an array of upper-level science and math courses that few ordinary high schools—even very large ones—could offer. Among them were Human Infectious Diseases, Chemical Pharmacology, Logic and Game Theory, and Vector Calculus."
  • "One assumption about selective public schools is that they have more and “better” teachers. It turns out, however, that their pupil-teacher ratio is actually a bit higher (17:1) than in all public high schools (15:1). (One likely reason: not much “special ed.”) The percentage with doctoral degrees is higher, too (11 vs. 1.5 percent), as is the percentage with master’s degrees (66 vs. 46 percent.) Nontrivial numbers of teachers also have experience in industry, science, and universities."
  • "A handful of responding schools said either that they are not required to hire teachers with state certification, or that other credentials (e.g., PhD in relevant field) preempt certification, at least for several years. In general, however, routine regulations and contract provisions prevail. We were struck by how few schools reported explicit freedom from them. Principals did say, however, that they could usually “work things out” as needed."
  • "Leaders of these schools felt doubly vulnerable as attention—and resources—were concentrated on low-performing schools and students. (“Smart kids will do fine, regardless, and in any case are not today’s priority” was the undertone they picked up.)"
  • "Nearly every school on our list offers a host of AP courses and has a huge number of students enrolling in them (either by requirement or by choice) and racking up solid scores on the AP exams."
  • "Much like private schools, which are more apt to trade on their reputations and college-placement records than on hard evidence of what students learn in their classrooms, the schools on our list generally don’t know—in any rigorous, formal sense—how much their students learn or how much difference the school itself makes."
  • "The marketplace signals, however, are undeniable: far more youngsters want to attend these schools than they can accommodate. Many applicants go to exceptional lengths to prepare for the admissions gauntlet, which may well lead to more learning in earlier grades than the same youngsters might have absorbed without this incentive. And we also know that most of those who are admitted stick with it through graduation; an average graduation rate of 91 percent was reported by the schools responding to our survey."
  • "Insofar as students benefit from peer effects in classrooms, corridors, and clubs, and insofar as being surrounded by other smart kids challenges these students (and wards off allegations of “nerdiness”), schools with overall cultures of high academic attainment are apt to yield more such benefits."

The second article is a question and answer with Salman Khan. Here are some comments about education and Khan Academy; I've added my own emphasis in red:

  • "Here’s what I think it could look like in five years: the learning side will be free, but if and when you want to prove what you know, and get a credential, you would go to a proctoring center [for an exam]. And that would cost something. Let’s say it costs $100 to administer that exam. I could see charging $150 for it. And then you have a $50 margin that you can reinvest on the free-learning side."
  • "Look, pedagogy is a lot like economics. I can find two education PhDs who are in 180-degree opposition. It’s just like Keynesians versus the Chicago school of economics. You can see it in the debate over New Math versus the old math. The math wars have been raging for decades. They hate each other. They shout at each other. We try not to be dogmatic about it."
  • "A lot of the criticism I have [gotten] is “There is no such thing as a silver bullet. The Khan Academy is not going to solve education’s problems.” And we agree with that 100 percent."
  • "Over the next five years we are going to be investing heavily, more than anyone, in analytics that give you a dynamic assessment. What does a student know? What does a student not know? How effective is the tutorial? That is what is exciting."
  • "We are close to 6.5 million unique users per month. But if you look at the engagement—who is really ultra-into this and spending a lot of time with the videos—well, it’s a couple of hundred thousand."
  • "My big takeaway, and we see this in classrooms, is that who is motivated and who can engage is a much bigger group than we originally thought. The core reason for students disengaging is that they are frustrated. They’re in algebra class but don’t have a good foundation in pre-algebra or arithmetic. It’s going straight by their heads. So they’re acting up in the back of the room. I think that is what is happening universally."

That final point is all too real for me. Teaching at a private school overseas I found that more than 90% had the basics of arithmetic mastered in 8th grade. Now, teaching 10th grade geometry in a US public school, more than half of my students struggle with arithmetic: watch their thoughts grind to a halt on calculating -4-(-7) or writing the first few digits of the decimal representation of 9/17. Quite a few can't get the decimal representation for 1/2, as I found to my surprise last year.. When you talk about algebra then the problems are more striking: about 90% of my students have major weaknesses in algebra.

Public high schools don't want you failing too many students, so students get passed through the system until they become a problem for the college teachers to deal with. For example: this or this.

They aren't in the back of the room, Sal, they are the room.

Prime Function

In my opinion, the concept of a function is one of the most important concepts of high school mathematics. Books don't do a particularly good job because they stick to the conventional real valued functions. As a result, they don't recognize the functions around them. A restaurant menu is a function?!?....students should see functions (conventional and unconventional) early and often.

I've put together a quick example of the prime function. Pi(n) defined to be the number of primes less than or equal to n. Though it doesn't look like it from the screenshot above, the  domain is the positive integers; sometimes you'll see this function defined over $latex [1, \infty)$. This function reminds students of $latex \sqrt{x}$ and $latex \ln{x}$ but the best fit that we have is $latex \frac{x}{\ln{x}}$ due to Gauss. The Sage interactive manipulative lets you compare the growth of these functions with the prime function. I've put in a tab giving you the option to create a PDF file for the output. Remember the PDF file is a link that appears near the bottom (see Koch Snowflake picture).

The prime function is a good example to introduce when students learn logarithms.

Odds and Ends 11/3/12

I've read several interesting articles lately. Ethnic politics is becoming more of a topic, especially with respect to education. The first article dealing with the subject is here but the focus was about the college level because the Supreme Court will be deciding "...whether the University of Texas is violating the Constitution by including race and ethnicity in admissions decisions". Many see  affirmative action as outdated and unfair. Interestingly enough, "...some ambitious and disciplined students from India, South Korea and China see themselves as victims of race-conscious admissions, their numbers kept artificially low to keep a more demographically balanced campus." and some argue they aren't asian as much as they are American. Even more surprising is this quote:  “When schools were heavily white, Asians were not in the applicant pool. But now there is a new generation of immigrants applying, especially from places like India and China, and that is putting even more pressure on Asian-Americans trying to get into top schools. If they knocked out our ability to use affirmative action, certain Asian groups would benefit far more than others.”.

This will be an interesting Supreme Court case to follow. But ethnic politics is an issue affects high schools as well. The article "For Asians, School Tests Are Vital Steppingstones" is about a civil rights group filing a complaint with the US government that high school entrance exams discriminate against black and Hispanic. Essentially they believe that success on the entrance exams is due, in large part, to getting extra tutoring that blacks and Hispanics cannot easily afford which results in them being under represented. The article tells about the struggle of several asian students to get into these schools and the influence of Confucianism (and corporal punishment) in helping them succeed. Is it discrimination or culture? You decide.

I also stumbled upon a substantial article on chess, called Rooked, that came out in September and looks at the evolution of cheating in chess. Definitely worth a read.

Finally, I found a great cartoon comparing education in 1960 versus 2010 which, from the website, appears to be related to education in the UK but is just as applicable to the US. Despite the few words, it speaks volumes.