.

The kids would literally fight with each other to get to be the first to play these games, with an intensity that surprised me. I mean, the games are fun and everything, but they were nonviolent, with less-than state of the art graphics and no recognizable characters from TV or movies. The educational content was rarely disguised as “fun,” and yet, kids who snoozed through math class were riveted by the exact same content when it was presented in the context of Treasure Mathstorm.

Video games have a lot to teach designers of elearning experiences. Intuition suggests that the gratification of video games is the sense of accomplishment and victory they offer. In my observation and experience, most of the pleasure gamers experience is in moments of failure. As long as the game is balanced well, failure is fun. Beating the game feels good too, but it’s a fleeting pleasure that quickly turns into a letdown. It’s no accident that some of the most enduringly popular video games are literally impossible to beat: Tetris and Pac-Man, for example.

The pleasure of a video game is in exploring a rule set, testing out hypotheses, noticing which ones fail and which ones succeed. Every video game is an interactive learning experience. In Logical Journey Of The Zoombinis, the player learns inductive reasoning. In Halo, the player learns the game’s particular version of military strategy. In Tetris, the player learns to perform spatial logic puzzles under time pressure.

Not every game has to be explicitly goal-oriented. Will Wright’s games are more like interactive, autodidactic toys, in which the player is interacting with a complex and open-ended dynamic simulation of a city, anthill or galaxy. Any goals are set by the player. Wright’s designs are based heavily on the educational philosophy of Maria Montessori. Wright says:

One of the counter intuitive things I needed to learn as a designer was that players enjoy failures more than success. As long as it’s diverse, they like to explore the failure space of a game.

Learning is exploring a failure space. Once you’ve mapped out all the wrong paths, you can avoid them as easily as the furniture in your room when you’re walking around it at night. The point of classroom exercises and homework is to guide students through the failure space of knowledge, exploring the wrong turns and blind alleys, as well as identifying the right path.

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• Why You Should Rent Video Games Instead Of Buying One (newfashionideas.wordpress.com)

Two family reunions are happening at the same hotel, the Oates reunion and the Hall reunion. All 100 guests at the hotel attend at least one of the reunions. If 40 people attend the Oates reunion and 62 people attend the Hall reunion, how many people attend both reunions?

(A)  2
(B)  5
(C)  10
(D)  16
(E)  22

Answer after the jump.

Explanation:

The question asks us to determine the number of people who attend both the Hall and Oates reunions. Since this is a question about overlapping sets, we will find it useful to draw a Venn diagram.

We will call the number of people who attend only the Hall reunion h, the number of people who attend only the Oates reunion o, and the number of people who attend both b. The information provided in the prompt allows us to set up three equations:

o + h + b = 100
40 – b = h
62 – b = o

If we add the second and third equations together, we get: 102 – 2b = h + o.

Rearranging the first equation, we get: 100 – b = h + o.

Now subtracting this equation from the equation above it, we find: 2 – b = 0, or b = 2.

As expected, 2 people attend both the Hall and Oates reunions. They play all of their hits and a few new songs.

Answer choice A is correct.

]]> http://knewtoninc.wordpress.com/2009/11/06/venn-diagrams/feed/ 1 Knewton http://knewtoninc.wordpress.com/2009/11/04/business-like-approach-to-gmat-math/ http://knewtoninc.wordpress.com/2009/11/04/business-like-approach-to-gmat-math/#comments Wed, 04 Nov 2009 23:58:46 +0000 Knewton http://blog.knewton.com/?p=618 ]]> Nathan Burke is a Math Content Developer at Knewton, specializing in GMAT prep.

The GMAT quantitative section is different from most math tests. You don’t usually see Data Sufficiency questions outside the GMAT, for one thing. They’re tricky, and mastering them requires a high level of familiarity. The good news is that the answer choices are the same for every question, and precise calculations are often unnecessary.

Then there are the word problems. All that text takes a long time to read. With 37 questions to do within a scant 75-minute period, you have an average of about two minutes to answer each question. It can be nerve-racking to spend almost half of this precious time just parsing out questions that are essentially prose versions of a company’s balance sheet.

photo by stuartpilbrow

Maybe it seems silly to you to have to read through a lengthy explanation of two trains traveling on parallel tracks at different rates, when it would be a lot simpler to just look at a well-labeled diagram. After all, there is a reason why balance sheets, graphs, and diagrams exist, right?

There is a reason behind the test-maker’s strategy, however. These questions are testing how well you can take information that’s disorganized, messy, and portrayed in a slightly illogical way, and turn it into a correct decision. As a banker, manager, CEO, COO, CFO or any other leader in the world of business, you’ll have to make decisions based on information from people that “report” to you. It’s likely that these people won’t have the type of intense training you’ll have had at business school. Chances are, they’ll communicate with you in a somewhat disorganized, messy, and slightly illogical way.

How do you answer more GMAT quantitative questions correctly? Take the same steps that a good businessperson would take in order to make a decision:

1. Calmly and carefully obtain information.
2. Think analytically.
3. Decide without second-guessing yourself.

Though all three steps are required for all GMAT math questions, the bulk of the work required for overly-wordy math problems comes in the first step: calmly and carefully obtaining information. It doesn’t pay to read the prompt and answer choices as quickly as possbile. Getting the question correct requires a thorough knowledge of all information in the prompt. Skimming will only increase your risk of misreading or omitting important facts. Consider the following question, taken from the 2nd edition of the GMAT “Quantitative Review”:

One week a certain truck rental lot had a total of 20 trucks, all of which were on the lot Monday morning. If 50 percent of the trucks that were rented out during the week were returned to the lot on or before Saturday morning of that week, and if there were at least 12 trucks on the lot that Saturday morning, what is the greatest number of different trucks that could have been rented out during the week?

A: 18
B: 16
C: 12
D: 8
E: 4

This long-winded question boils down to:

Find the maximum value for t if (1/2)t + (20 – t) ≥ 12

This inequality is easily reduced using algebra (pop quiz: solve this.) The tasks of reading the question and interpreting it into a suitable equation are a lot more time-consuming. If you felt rushed and skipped the simple phrase “of that week,” then the incorrect answer choices C and D would become far more compelling than if you had read the question accurately.

The key here is to read the question as if you had all the time in the world—the first time through. Read it carefully. Absorb every word. Write down expressions and equations. Misreading a question will either lead to an incorrect response or a reread. Having to reread a question means that your first reading was a waste of time, and wasted time, much like an incorrect response, will always add up to a lower score.

As a non-native speaker of English, you may find the verbal section of the GMAT especially daunting. You’re not alone. English is a particularly difficult language to learn as an adult. It’s a blend of several different older languages that still coexist uneasily in the grammar and vocabulary. The roots of English in German rub uneasily against its roots in Latin in ways that resist intuition. A lot of “wrong” usage by non-native speakers is more logical than correct usage. Knowing some of the back story can hopefully make you feel less frustrated as you try to absorb it all.

England wasn’t even the first home of English. In ancient times the people there spoke indigenous languages that are the precursors to Irish and Welsh. When Germanic tribes invaded, they pushed the indigenous British tongues to the corners of what are now the British Isles. Old English was a blend of German and Scandinavian languages. A few Latin words also entered via the Germanic tribes’ long interaction with the Roman Empire.

When the Norman French invaded and conquered England in the 11th century, they brought their Latin-based language with them. While common people continued to speak their German-based language, the ruling class used the Latin-based vocabulary and grammar of French. Eventually the two blended to form a single language, but divisions remain. German-based English words have a “lower,” more casual connotation, while French-based words sound more formal. Compare the Germanic “sweat” to the Latinate “perspiration,” or Germanic “friendly” with the Latinate “cordial.”

As Great Britain expanded its colonial and imperial presence, English continuously appropriated foreign vocabulary. America, Australia and other former English-speaking colonies developed the language in new directions, creating a variety of new idioms and slang. These countries’ industrialization created the need for a lot of new scientific and technical vocabulary, also frequently borrowed from other languages.

Because it’s such a mutt, English has an unusually wide variety of irregular and otherwise eccentric verbs from its parent languages. Unfortunately, the most bizarre verbs are the ones English uses the most. The modern English verb “to be” seems grammatically insane:  I am. You are. He, she, it is. We are. I am not a runner now, but I used to be. She is not a good cook, but she will be. What possible logic ties it all together? “To be” is an amalgamation of several Proto-Indo-European verbs that all performed the same functions. Be, is and were all came from different verbs that all presumably made sense in their original form. Now, they form a set of arbitrary rules that just need to be memorized.

The illogic of “to be” extends to its use as an auxiliary. Dropping the auxiliary “to be” is “wrong,” even though it can make your speech clearer. Vernacular English drops the auxiliary all the time: “Where you at?” “We at the house.”

The dummy auxiliary verbs “to do” and “to have” are further historical oddities. “Did you take the bus?” “Yes, I did.” “He hasn’t taken the bus in a while.” “Will the bus have already left yet by nine o’clock?” “Do I have to take the bus today?” “Yes, you do, the car has broken down again.” It’s hard for a non-native speaker to keep all the idiomatic distinctions straight. Why do you make a mistake and not do a mistake? Why do you do a favor and not make a favor? It’s all arbitrary and historical.

Another challenge is the interdental sound “th.” English is one of the very few languages in the world aside from Icelandic to use this sound, and we use it in some of our most commonly used words: the, this, that, them.

So what can you do? Spending time talking to native speakers helps with your overall fluency, but for GMAT sentence correction it can get you into trouble, since native speakers routinely make grammar mistakes. The best bet is to read rigorously copyedited publications aimed at educated readers: the New Yorker, the New York Times, the Wall Street Journal, the Atlantic Monthly, Harper’s. The Economist has wonderful writing but be advised that it uses British idioms rather than American ones. Opinion pieces by the editors use the most correct, least slang-y writing. General news pieces are written faster and aren’t as rigorous. They may also use more slang and informal language.

The one bit of good news for non-native speakers prepping for the GMAT is that you won’t be misled by a lifetime of non-standard usage. All five choices on a sentence correction question will look equally puzzling at first glance. Native speakers can glance at those questions and get an immediate intuition for which one feels right, but sometimes that intuition leads them astray. Non-Native speakers have the advantage of learning rules, and applying themselves analytically rather than following their gut.

For help with English idioms, check out these web sites:

Wiktionary English idioms (Very thorough)

American idioms

Good luck in your struggles with our mongrel native tongue!

When you learned division in school, the teacher probably brushed off the issue of dividing by zero in one sentence: you can’t do it, moving on. You might feel like you got shortchanged by that explanation. Why not? What happens when you divide by zero?

You can’t ask the computer. Computers fail when you ask them questions with no unambiguous answer. Dividing by zero is just such a question. Folklore suggests that asking the computer to divide by zero makes it spectacularly explode or something. In reality, it returns an error message or the reply Not A Number, or it gives a wrong answer, or the program terminates, or sometimes the machine falls into an infinite loop.

The internet’s favorite divide-by-zero error is the one that temporarily crippled the USS Yorktown, a Ticonderoga-class cruiser that was the test bed for the Navy’s Smart Ship program. When a crew member typed zero into a database field, the computer tried to divide by it, crashing the system badly enough to cripple the ship’s navigation systems for several hours.

Humans are smarter than computers in some ways, and we’re capable of coming up with creative answers to seemingly unanswerable questions. So what do you get when you divide something by zero? My answer draws heavily on the entertaining wikipedia article. For the sake of simplicity, let’s say we’re dividing one by zero. The math people have a crafty method for dealing with problems you can’t approach directly. You can edge closer and closer to the problem and see if you converge on an answer. So instead of dividing one by zero, you could try dividing it by smaller and smaller numbers that approach zero. One divided by one tenth is ten. One divided by one one-hundredth is a hundred. One divided by one one-thousandth is a thousand. Since one divided by one one-gazillionth is one gazillion, logic suggests that one divided by zero is going to be infinity.

It makes sense, but there’s a problem. We’ve been approaching zero from above, but we could just as easily approach it from below. When you divide one by negative one tenth, you get negative ten. One divided by negative one one-hundredth is negative one hundred. One divided by negative one gazillionth is negative one gazillion. So you could just as easily say that one divided by zero is negative infinity. Both infinity and negative infinity are equally valid answers. Here it is as a graph.

Some people interpret this graph to say that infinity and negative infinity are the same number. It’s not as crazy as it sounds. Let’s say that instead of being on the computer screen, the graph was drawn on a globe. Imagine the number line wrapped around the equator. Say the spot where the Prime Meridian crosses the equator is zero. If you’re in a rowboat bobbing in that spot in the Atlantic Ocean, enjoying the warm breeze, you can think of the positive numbers as going off along the equator to the east, and the negative numbers going off to the west. Infinity is the farthest possible point away from you on the equator to the east, and negative infinity is the farthest point away from you to the west. On the Earth, positive and negative infinity are the same place, the International Date Line in the Pacific. For this image to be totally accurate, the Earth would have to be infinitely large, but the math guys bracket that. By this thinking, one divided by zero does have a single, unambiguous answer: this mysterious number called unsigned infinity.

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Emily Holleman is a Content Developer at Knewton, helping students with their LSAT preparation.

If you read my friend Chris Black’s great post on passage wording last week, you already know how important language is on the LSAT. However, it’s especially important to pay attention to language use when you’re asked about an author’s attitude.

Attitude questions—you know, those pesky ones that pretty much ask you how the author feels about something—may be the trickiest questions on the Reading Comp section. If only those authors would just come out and say how they felt about the topic (I think that Yeats’ poetry is crap)! Luckily for us, these attitudes do come across loud and clear, as long as you know what types of language to look for.

Here are some things to keep in mind:

1. Eliminate any answer choice with the words indifferent or neutral—if the author didn’t feel anything about the subject, the LSAT writers wouldn’t be asking the question.
2. Check out the adjectives and adverbs that describe the topic: are these mostly positive (comprehensive, informative, insightful, etc) or negative (disappointing, unfortunate, incomplete)?
3. Check out the verbs that are associated with the topic. Does the author say that it attempts, succeeds, or fails at anything?
4. Unless your findings are OVERWHELMINGLY positive or STAGGERINGLY negative, eliminate extreme answer choices (outright disdain, ardent admiration, unmitigated criticism, unconditional acceptance, etc).

Let’s take a look at this excerpt to see if we can figure what the author’s attitude toward Anderson’s study is:

Albertson’s study provides one of the more enlightening accounts of the Great Migration. Unfortunately, it fails to fully address the reaction of Northern whites to this sudden influx of African Americans.

Okay, so the author describes the study as one of the more enlightening accounts. That’s pretty positive! However, he then adds a negative qualification: it unfortunately fails to fully address something. So, the author has a somewhat positive attitude about Anderson’s study. We’ll keep this in mind as we go through the answer choices:

A. Enthusiastic reverence

This is an extreme answer choice. We know that the author feels positively about Anderson’s study, but we also know that he has some reservations about it. So this one is out.

B. Complete indifference

This choice claims the author doesn’t have an opinion on the subject—we can eliminate it right off the bat.

C. Qualified rejection

This suggests that the author’s attitude is mostly negative, but we know the author’s feelings are generally positive.

D. Reserved admiration

There’s our correct answer! The author shows admiration for the study (it is one of the more enlightening accounts.) We know this admiration is reserved, because the author points out one of the study’s unfortunate shortcomings (it fails to fully address…)

E. Pointed skepticism

Here’s another extreme answer choice. Plus, it’s overwhelming negative, while we know the author has somewhat positive feelings about the study.

Now, go solve those attitude problems!

Since I often put up both drafts of syllabi and completed syllabi for comments, I obviously think it’s a good practice. It’s been nothing but beneficial for me: I’ve gotten great suggestions, interesting critiques, a good feeling for how the syllabus plays with different intellectual communities. So why wouldn’t everyone do this? In fact, why shouldn’t everyone more or less be officially pushed to do it by colleagues or administrations. It’s not just a good thing for the person posting the syllabus, but for students who want an early view of what a course might entail and for larger publics who would like to get a sense of how much work and thought goes into an average course design. Since one of the handicaps academics have in the public sphere at the moment is that there are a number of people who think the work of college teaching consists of walking into a room, letting knowledge spill out of your head, and leaving, it might help if we gave a demonstration of what’s actually involved.

Why don’t all teachers post their course materials online? Some teachers are unfamiliar with the tools. Some are anxious about a hostile reaction to their ideas. Some are anxious about the scrutiny of their peers. Some might wish to maintain their sense of expertise by limiting the circulation and exposure of their field.

Nonetheless, there’s a lot to be said for having all syllabi live on the web. Students would have a better idea of what to expect going into a class and would have another way to track their progress once the class was underway. Prospective students would have more information to go on when choosing schools, or classes or majors within those schools. Other teachers might find inspiration (or fuel for competition) to impove their own course materials.

What do you think?