Weekend gaming -- mutant sprouts

A while back I posted a recommendation for a popular pencil-and-paper game:

On the subject of topology, my game of choice is Sprouts, invented by mathematicians John Horton Conway and Michael S. Paterson at Cambridge University in 1967 (as a general rule, you can't go wrong with a game if Conway had anything to do with it).

The rules are simple:

1. Start with some dots on the paper. The more dots you have the longer the game takes so you will probably just want to start with two or three.

2. Players take turns either connecting two of the dots with lines or drawing a line that loops back and connects a dot with itself.

3. The lines can be straight or curved but they can’t cross themselves or any other lines.

4. Each dot can have at most three lines connecting it

5. When you draw a line put a new dot in the middle.

6. The first player who can’t draw a line loses.

I was thinking about sprouts the other day and a few variations occurred to me. I don't know if they're particularly playable or if they add any interesting aspects to the game, but if you can't put a half-baked idea in a blog, what's a blog for?

Variant 1 -- Free sprouts

Played as above but with the following addition: for the first k moves of a game with n dots, the player, after drawing a line, adds a new dot.

Topologically the result is a game with n+2k dots (keep k small) but with the complication that lines are being drawn without knowing exactly how those lines partition the surface. This is still a game of perfect information but the variation should make it more difficult to think a few moves ahead.

Variant 2 and 3 -- Scored sprouts

Each player starts with a separate sheet of paper and proceeds to connect the dot according to the standard rules. After no more lines are possible, the players score their graphs based on the number of dots.

Score = 6


Score = 7

In variant 1 the player with the highest score wins. In variant 2, the win goes to the lowest.

Also posted at Observational Epidemiology.

If you first heard about your alma mater through spam, you may be dealing with a diploma mill

I got the following piece of spam a few days ago. It was immediately notable because it appeared to have been sent from and to my email address. It always seems to me that spammers put more thought into the spoofing than they do into the actual pitch:

Why go to Princeton? Beat the system.
TO: 1 More1 recipient
CC: recipientsYou More
BCC: recipientsYou
Show Details


Saturday, November 26, 2011 5:36 AM
Message body
WHAT A GREAT IDEA!

Ring anytime 1-213-403-XXXX

We provide a concept that will allow anyone with sufficient work experience to obtain a fully verifiable University Degree.
Bachelors, Masters or even a Doctorate.

Think of it, within four to six weeks, you too could be a college graduate.
Many people share the same frustration, they are doing the work of the person that has the degree and the person that has the degree is getting all the money. Don't you think that it is time you were paid fair compensation for the level of work you are already doing?

This is your chance to finally make the right move and receive your due
benefits. If you are more than qualified with your experience, but are lacking that prestigious piece of paper known as a diploma that is often the passport to success.

CALL US TODAY AND GIVE YOUR WORK
EXPERIENCE THE CHANCE TO EARN YOU
THE HIGHER COMPENSATION YOU DESERVE!

Ring anytime 1-213-403-XXXX

"[T]he numbers show that wage inflation is — literally — the least of the problems when it comes to university cost inflation"

The quote comes from Felix Salmon and it's part of an excellent discussion (nicely summarized at Rortybomb). If you're interested in either education or the economy (where student debt is becoming a major factor), you should read all of the posts, but if I had to pick one point it would be this unbelievable projection cited by Malcolm Harris:

And while the proportion of tenure-track teaching faculty has dwindled, the number of managers has skyrocketed in both relative and absolute terms. If current trends continue, the Department of Education estimates that by 2014 there will be more administrators than instructors at American four-year nonprofit colleges.

Also posted at Observational Epidemiology.

Dismembered Puppy Math

There's a class of simple math problem that most fourth graders will get right and most tenth graders will get wrong. You can come up with any number of examples but I've found this one makes the most lasting impact:

You have four kids and thirteen puppies. How many puppies does each kid get?

The fourth grader will correctly answer "three with one left over." More often than not the tenth grader will answer "three point two five." At this point, I would remind the student that no one should ever get a fourth of a puppy and therefore this is one of those times when it makes more sense to talk about division with remainders.

Asimov (and, I'm sure, many others) observed that mathematical progress often came down to finding ways to allow people to solve more problems with less thought. Mathematicians like Leibniz and Gauss come up with elegant notation that allow us to do much of our work mechanically. Ideally this frees us up to think about more important questions, but sometimes it lets students get good marks and high test scores in math without thinking at all.

Eventually the habit of answering math questions without thinking about them will lead to problems (it's difficult to mindlessly shamble through real analysis), but for K through 12 it can actually be an advantage not to spend to much time asking questions and dwelling on implications. This is particularly true for standardized tests which tend to be time-constrained and focus on problems that can be solved mechanically.

Of course, I'm not suggesting that movement reformers are in favor of the dismemberment of cute little puppies, kittens or any other loveable pets, but, as in other cases, I wonder if they've really thought this through.

The good and bad sides of charter schools in N'Orleans*

NPR gives us this good and nicely balanced report:

MELISSA BLOCK, host: This is ALL THINGS CONSIDERED from NPR News. I'm Melissa Block. New Orleans has become the center of an education revolution. More than 70 percent of public school students there attend a charter school and the number of kids remaining in traditional public schools is shrinking fast.

As this experiment moves forward, New Orleans has confronted questions that dog the charter school movement nationwide. Are charters welcoming the most challenging students, especially those in special education? NPR's Larry Abramson has the story.

LARRY ABRAMSON: Parents in post-Katrina New Orleans can pick and choose from a smorgasbord of schools with different approaches, different cultures. By many measures, this educational marketplace has improved student achievement, especially in the city's many charter schools. But has performance for some come at the expense of other students?

...

FISHER: Noah's been diagnosed with autism. He's also blind and he also has some feeding issues and developmentally delayed.

ABRAMSON: Kelly Fisher, Noah's mother, says when her family moved down to New Orleans in 2009, they expected Noah would need the same kind of intensive help that he got at his old school in Indiana. For help in finding the right school, they turned to the Recovery School District, the state-run agency that is the closest thing New Orleans has to a traditional district.

FISHER: Because I came from a traditional program, I thought, oh, that's my local special ed coordinator. That's the person that knows what's in the city and can direct me towards the schools that would be best for Noah.

ABRAMSON: But the Fishers say New Orleans' open choice system left them totally on their own when it came to finding a school for Noah. In theory, they could find any charter they wanted, but the best charters are full, so they ended up on waiting lists. Father Bob Fisher says the central district seemed powerless.

ROBERT FISHER: The director was just scrambling around making phone calls. Actually, at one point, ran out in the hallway and grabbed somebody and said, hey, do you have an opening at your school?

ABRAMSON: The Fishers say they kept looking for a school that could help Noah. Finally, they ended up at Lafayette Academy, a charter school.

DANIEL THOMAS: Where is that button?

(SOUNDBITE OF BEEP)

UNIDENTIFIED FEMALE #1: It's 9:52 a.m.

ABRAMSON: Here, Noah has a full time aide named Daniel Thomas. He and Noah are taking a break from class, walking outside on a nice fall day.

THOMAS: Sweep your cane. There's a big crack in the sidewalk.

FISHER: (Singing) Did she ask you to come in, Billy Boy, Billy Boy?

ABRAMSON: At Lafayette, the Fishers say Noah has the help he needs. They suspect that other schools simply did not want to spend the money needed to hire an aide and were not interested in accommodating Noah. Lafayette Principal Mickey Landry admits it is challenging for any school to cover the costs of special ed.

MICKEY LANDRY: The state tops out its financing for special needs students at about $18,000 a year, but some students cost us significantly more than that, sometimes as much as $40,000.

ABRAMSON: You can't say, sorry, we can only provide $18,000 worth of services?

LANDRY: Oh, that's right. We do whatever we have to do for a child. That's right.

ABRAMSON: Landry says he simply found a way to give Noah the support he needs. According to many parents, other schools do not work as hard to follow the law, which says all schools must be open to all students. The Fishers have joined a class action lawsuit charging that the New Orleans school system excludes special ed students.

Eden Heilman, a staff attorney with the Southern Poverty Law Center, is suing the Louisiana Department of Education, which set up the post-Katrina school system.

EDEN HEILMAN: The state kind of abandoned their responsibilities to students with disabilities in New Orleans. I mean, in any other school district where, for example, you have one single district, you have to do things like monitoring and compliance reviews.

ABRAMSON: But Superintendent John White says that's changing. White just took over as superintendent for the Recovery School District a few months ago. He points out that test scores for special education have improved dramatically since Katrina and that these children were terribly neglected before the storm.

But he admits those scores are still much lower than they should be. White says it is tough for his agency to oversee a system of independent schools.

JOHN WHITE: I think it's fair to say that a market system always has the challenges of how do we ensure that the most vulnerable, the most traditionally underserved are served well in that system?

...

ABRAMSON: This is KIPP New Orleans Leadership Primary School in the city's French Quarter. The special ed enrollment here is about nine percent, similar to the city average. KIPP has posted some of the most impressive gains in the city. The school says its mission includes kids like eight year old Benjamin Camp.

BENJAMIN CAMP: Hi, Mom. I love you. I love you. I love you. I love you.

ABRAMSON: Benjamin is sweet, but he's had behavior problems for years and was recently diagnosed with autism. His grandmother, Carmella Camp, says some nursery schools turned Benjamin away as too challenging, but this charter school never suggested he go anywhere else.

CARMELLA CAMP: Never, never, never, ever. Still haven't heard it. He was in a few - two schools and they say they couldn't handle him, you know.

ABRAMSON: But KIPP has also faced charges that it pushes some students out. The school has a firm discipline policy that can be tough for some students to follow. Families must agree to a commitment to excellence, which includes getting their kids to school on time and becoming part of the education process.

Rhonda Aluise, executive director of KIPP New Orleans, insists this approach doesn't exclude anyone.

RHONDA ALUISE: So there is not this requirement or if you come to KIPP, you must do this. This is - here's our vision for what a school can do. Come be a part of this with us.

...

ABRAMSON: Whatever happens with the New Orleans lawsuit, charter groups will have to wrestle with the continuing perception that they are not open to all. Larry Abramson, NPR News.

* I'm Southern. The rest of you should stick with "New Orleans."

The worst example of curriculum dead wood?

One of the first things that hit me when I started teaching high school math was how much material there was to cover. There was no slack, no real time to slow down when students were having trouble. The most annoying part, though, was the number of topics that could easily have been cut, thus giving the students the time to master the important skills and concepts.

The example that really stuck with me was synthetic division, a more concise but less intuitive way of performing polynomial long division. Both of these topics are pretty much useless in daily life but polynomial long division does, at least, give the student some insight into the relationship between polynomials and familiar base-ten numbers. Synthetic division has no such value; it's just a faster but less interesting way of doing something you'll never have to do.

I started asking hardcore math people -- mathematicians, statisticians, physicists, rocket scientists*-- if they'd ever used synthetic division. By an overwhelming margin, the answer I got was "what's synthetic division?" Not only did they not need it; it made so little impression that they forgot ever learning it.

Which bring us to this passage from a recent Dana Goldstein post (discussed earlier):

The problem, according to [David] Coleman, is that American curriculum standards have traditionally been written by committees whose members advocate for their pet pedagogical theories, such as traditional vs. reform math.

Except, of course, that's not what happened here. As was the case with so many topics in mathematics, synthetic division remained in the curriculum because no one who knew what was going on had bothered to look that closely. Coleman has a clever narrative, but it doesn't fit the facts all that well.

Now I have a request for all the math geeks in the audience. Since we need to pare down the curriculum, what you choose to cut? Specifically, what mathematical topics that you learned in school can future generations do without?


* Literal rocket scientists -- JPL's just down the road.

Also posted in a slightly different form at Observational Epidemiology.

"I am not now and have never been a constructionist"

(This post also appears at Observational Epidemiology.)

After my last post thought I should run this titular disclaimer. For those of you not up on the subject, here's a definition from the well-written Wikipedia entry on the subject:

Constructivist teaching is based on constructivist learning theory. This theoretical framework holds that learning always builds upon knowledge that a student already knows; this prior knowledge is called a schema. Because all learning is filtered through pre-existing schemata, constructivists suggest that learning is more effective when a student is actively engaged in the learning process rather than attempting to receive knowledge passively. A wide variety of methods claim to be based on constructivist learning theory. Most of these methods rely on some form of guided discovery where the teacher avoids most direct instruction and attempts to lead the student through questions and activities to discover, discuss, appreciate and verbalize the new knowledge.

Don't get me wrong. For the right topic, executed the right way with the right teacher and class, this can be a great, wonderful, spectacular and really good approach to education. Unfortunately, education reformers (particularly the current crop), are not good at conditional problems. They tend instead to fall into the new tool camp (you know the saying, "to a man with a new hammer, the whole world is a nail.").

Worse yet, (and I'm afraid there's no nice way to say this) many of the educational theorists don't have a firm grasp on the subjects they are working with. This is never more plain than in constructionist science classes that almost entirely eschew lectures and traditional reading assignments and instead have the students spend their time conducting paint-by-numbers experiments, recording the results and performing a few simple calculations.

To most laymen, that's what science is: stuff you do while wearing a lab coat. Most people don't associate science with forming hypotheses, designing experiments, analyzing results and writing papers and, based on my limited but first hand experience, many science educators don't give those things much thought either.

The shining exception to the those-who-can't-teach-teach-teachers rule is George Polya. Though best known as an educational theorist, Polya was a major Twentieth Century mathematician (among his other claims to fame, he coined the term "central limit theorem") so he certainly fell in the those-who-can camp.

But it it important to note that Polya advocated guided discovery specifically as a way of teaching the problem solving process. I suspect that when it came to simply acquiring information, he would have told his students to go home and read their textbooks.