The Discovering Mathematics series (part 2)
Coe: We just received recently a letter from a teacher up in your area. Her math experience with her students up to that point is, "Your homework is to do 1 thru 50 odd." And when she started using Discovering, the homework was: "Do 1 through 10 odd." And the kids thought they had scored. And actually, they came back the next day and said, "You tricked us, you made us think." And I think that's what's possibly going on in some of those reactions you're hearing.
The Discovering books have been criticized by parents, but they've been the top pick of a couple of districts in our area, including Seattle and Issaquah. Any thoughts on why the textbooks seem to be more popular with educators than with parents?
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Coe: I think because (parents) lack familiarity — this doesn't look like what I was taught. I don't know how you get students to a place where more is required of them by repeating things that have been done in the past. That's not how we move forward in life.
What can you tell parents who are struggling with this book, and can't find examples of how to do the math?
Coe: We appreciate that parents won't find what's in front of them familiar. We have a Web site dedicated to parents, students and mentors: www.keymath.com.
Two math experts who reviewed textbooks for Washington state wrote a report that called Discovering "mathematically unsound."
Coe: Our books are mathematically accurate. We don't publish books with errors in them.
Ryan: There were also the two Washington mathematicians, Jim King and George Bright, who found the materials were mathematically sound as well. The materials were also the top-ranked by OSPI (Office of Superintendent of Public Instruction) in their study as the best match.
The Discovering series was on OSPI's initial list of recommended textbooks, but last year the state whittled the list down to just one series, Holt Mathematics. Do you think your books should have remained on the list?
Coe: Well, a book (the Algebra I and II books) that gets ranked highest among 12 publishers, and the series as a whole ties statistically in first place with Holt, that was looked at by a team of over 50 experts — yeah. (Laughs.) We think that.
Is this tug of war over math textbooks being played out elsewhere in the country?
Coe: There is a very healthy debate, as there should be, around this selection of textbooks. A district makes a big commitment when they buy into a program. Quite right — whatever is getting looked at ought to be scrutinized very carefully. What's unusual is that having gone through that process, and the (Seattle) school board having made the decision based on the evidence in front of them, that a lawsuit was then filed. This is very unusual.
The Discovering Mathematics series (part 1)
How would you describe the philosophy behind the books?
Jim Ryan: Rather than going from a traditional book where a teacher explains the abstract and then the students practice that abstract concept, and then eventually they get to an application — which are generally called the word problems — we've turned that around for each lesson. The lessons are approached so we can try and answer that question "When will I ever use that?" before it ever gets asked. And so students see the meaningfulness of the math as the lesson is being presented to them.
Karen Coe: When we meet teachers, the way they characterize our books is we teach for understanding, or we explain the "why" behind the "what."
Can you give us an example?
Ryan: When students learn linear equations, we start off with the students gathering data, or figuring out data, so the first thing is they will put data into a numerical table, and then they'll graph that data in an x-y graph and fit a line to that data. Then they solve for the equation for that line. In the traditional materials, we present every student with the equation y = mx + b and then you give it meaning after they've seen that. So, you ask them to memorize that and then you try and give it meaning afterward.
Sounds like you're trying to lead students to an "aha" moment rather than having them memorize formulas.
Coe: That is a perfect way of describing it.
One of the criticisms of Discovering is that its investigations take too long, and that can turn off students who are good at math.
Coe: It's not our experience. And every time we work with schools on implementation plans, we work on pacing guides (materials that help teachers plan lessons), and it's simply not possible in a book for every investigation to be done in a year.
Another criticism, from students and parents, is that what's missing from Discovering is direct instruction, or very specific examples of how to do a math problem.
(read the whole article here)
Game Theory – Robert John Aumann
Robert John Aumann is an Israeli/American mathematician and a member of the United States National Academy of Sciences. He is a professor at the Center for the Study of Rationality in the Hebrew University of Jerusalem in Israel. He also holds a visiting position at Stony Brook University and is one of the founding members of the Center for Game Theory in Economics at Stony Brook.
Aumann received the Nobel Memorial Prize in Economics in 2005 for his work on conflict and cooperation through game-theory analysis. He shared the prize with Thomas Schelling.
Aumann's greatest contribution was in the realm of repeated games, which are situations in which players encounter the same situation over and over again.
Aumann was the first to define the concept of correlated equilibrium in game theory, which is a type of equilibrium in non-cooperative games that is more flexible than the classical Nash equilibrium. Furthermore, Aumann has introduced the first purely formal account of the notion of common knowledge in game theory. He collaborated with Lloyd Shapley on the Aumann-Shapley value. He is also known for his agreement theorem, in which he argues that under his given conditions, two Bayesian rationalists with common prior beliefs cannot agree to disagree.[1]
Aumann and Maschler used Game Theory also to analyze Talmudic dilemmas.[2] They were able to solve the mystery about the "division problem", a long-time dilemma of explaining the Talmudic rationale in dividing the heritage of a late husband to his three wives, depending on the worth of the heritage compared to its original worth.[3] The article in that matter was dedicated to a son of Aumann, Shlomo, who was killed during the 1982 Lebanon War while serving as a tank gunner in the Israel Defense Forces's armored corps.
These are some of the themes of Aumann's Nobel lecture, named "War and Peace":[4]
1. War is not irrational, but must be scientifically studied in order to be understood, and eventually conquered;
2. Repeated game study de-emphasizes the "now" for the sake of the "later";
3. Simplistic peacemaking can cause war, while arms race, credible war threats and mutually assured destruction can reliably prevent war.
Repeated game – Game Theory
Game Theory is one mathematical theory that is applied on a multitude of (different) areas, and can be accessed not only via the studies of math, but also from life sciences (like psychology, which encourages students to understand game theory in order to understand some key human behavior patterns). In this short post I will look into the introduction of this theory, in hope to follow up on it in the next post with the actual people involved in research and development of this rich theory.
In game theory, a repeated game (supergame or iterated game) is an extensive form game which consists in some number of repetitions of some base game (called a stage game). The stage game is usually one of the well-studied 2-person games. It captures the idea that a player will have to take into account the impact of his current action on the future actions of other players; this is sometimes called his reputation. The presence of different equilibrium properties is because the threat of retaliation is real, since one will play the game again with the same person. It can be proved that every strategy that has a payoff greater than the minmax payoff can be a Nash Equilibrium, which is very large set of strategies. Single stage game or single shot game are names for non-repeated games.
The most widely studied repeated games are games that are repeated a possibly infinite number of times. On many occasions, it is found that the optimal method of playing a repeated game is not to repeatedly play a Nash strategy of the constituent game (look at the Repeated prisoner's dilemma example), but to cooperate and play a socially optimum strategy. This can be interpreted as a "social norm" and one essential part of infinitely repeated games is punishing players who deviate from this cooperative strategy. The punishment may be something like playing a strategy which leads to reduced payoff to both players for the rest of the game (called a trigger strategy). There are many results in theorems which deal with how to achieve and maintain a socially optimal equilibrium in repeated games. These results are collectively called "Folk Theorems". An important feature of a repeated game is the way in which a player's preferences may be modeled. There are many different ways in which a preference relation may be modeled in an infinitely repeated game, the main ones are :
- Discounting - valuation of the game diminishes with time depending on the discount parameter δ
- Limit of means - can be thought of as an average over T periods as T approaches infinity.
- Overtaking - Sequence
is superior to sequence 
if 
The Saturday (Simple) Question
Question : What is meant by 'Casting out nines' ?
Answer : Casting out the nine is a method of checking answers to multiplication, addition and subtraction.
Casting out nines was known to the Roman bishop Hippolytos as early as the third century. It was employed by Twelfth-century Hindu mathematicians.
It is based on the idea of finding the sum of the digits in a number and then adding the digits in the resulting sum, etc., until a one digit number results. This addition of the digits in a number is further simplified by first discarding or casting away any digits whose sum is 9. The remainder is set down, in each case as the check number.
For example,
8216 x 4215 = 34630440, this is how you proceed.
4 + 2 + 1 + 5 = 12, casting out the nines leaves 3 as check number.
8 + 2 + 1 + 6 = 17, casting out the nines leaves 8 as check number.
3 + 4 + 6 + 3 + 0 + 4 + 4 + 0 = 24, casting the nines leaves 6 as check number.
3 x 8 = 24, 2 + 4 = 6 which was the check number of the original product. So the multiplication is correct.
Casting out nines is not in wide use today, mainly because of the widespread use of calculators and computers.
Casting out Nines
There are many little tricks in mathematics can make solving problems much easier. Casting out the 9s is one of the tricks can help make sure your answer is right much easier. Many people check the digit sum of the answers to make sure that their answers are correct. Casting out the 9s can make checking the answers and getting digit sums much easier.
Let’s start with Digit Sums. It is possible to take a number regardless of how big it is and transform it into a single digit. You can start with the number 15. In order to get it down to just one digit, you would add the two numbers together which would equal 6. This means that the digit sum of 15 is 6. It is also possible to do this with larger numbers such as 547. You would begin by adding 5 and 4 together which would be 9 then you would add 9 and 7 together which would be 16. However, you still got two digits so you would need to add 1 and 6 together which would be 7. With a bigger number such as 72546 you would add 7 and 2 which is 9 then 9 and 5 which is 14. 14 and 4 would equal 18 and finally 18 and 6 would equal 24. Finally, 2 plus 4 equals 6 so the digit sum of 72546 equals 6. This can be rather frustrating with larger numbers.
Thankfully, there is another way to shorten it down and make it easier.
Okay, now you’re probably wondering where casting out the 9s comes into play in this situation. When it comes to digit sums, it is possible to regard 9 as a 0. This basically means that you can cast out the 9s when working with digit sums. Then you would add the remaining digits together to get the digit sum. Let’s work with the numbers mentioned above. Let’s look at 547. 5 plus 4 is equal to 9. We can quickly eliminate these two numbers since they are equal to 9 when added together. Therefore, we are left with 7. This is the same answer we got above.
We just got with much easier and quicker when casting out the 9s. Okay, now let’s look at 72546 which was rather complicated to break down and get the digit sum earlier. 7 plus 2 is equal to 9 which can be casted out. 5 and 4 is also equaled to 9 and can also be casted out. This leaves us with 6 which is the same sum digit that we received using the more complicated method above.
Identity theft on Facebook?
9 May 2008
I’m a little surprised you don’t hear about this sort of thing happening more often:
A Roncalli High School administrator is asking a judge to force the Internet site Facebook to identify the pranksters who hijacked his identity for a phony Webpage.
Tim Puntarelli, Roncalli [High School]’s dean of students, and the Roman Catholic Archdiocese is suing Facebook and the anonymous creators of the false Webpage the suit claims contained false, embarrassing, and defaming information about Puntarelli and Roncalli High School.
The page creators used the Facebook page to pose as Puntarelli and send emails to Roncalli students, according to the lawsuit filed Thursday in Marion Superior Court.
Facebook officials removed the page when they were notified of the site on April 18, but refused to disclose the identity of the creators without a court order, according to the lawsuit.
Puntarelli and the Archdiocese are asking a judge to order Facebook to identify the creators of the page. The suit indicates they want the pranksters to pay triple the attorney fees and court costs.
I’m also somewhat surprised that Facebook is so reluctant to hand over the identity of the kids (presumably kids, at least) who set up this phony web page when they freely admit that the page is phony and the administrator’s identity was hijacked. Why should you need a court order for this?
Tagged!
7 May 2008
Dana Huff tagged me with a meme, and that’s a lot more fun than closing out a semester and prepping for finals, so I’ll play along. Here are the rules:
1. The rules of the game get posted at the beginning.
2. Each player answers the questions about themselves.
3. At the end of the post, the player then tags 5-6 people and posts their names, then goes to their blogs and leaves them a comment, letting them know they’ve been tagged and asking them to read the player’s blog.
4. Let the person who tagged you know when you’ve posted your answer.
What were you doing ten years ago?
I was closing out the first year of my first professor gig. It was probably the most tumultuous year of my life. I had moved to northern Indiana from Tennessee, where I had lived my whole life up to that point. I was in shock at the reality of teaching in a third-tier small college, having gone through 4-5 years of dreamy idealism about what being a professor was going to be like. I had more grading and prepping than I thought possible. Oh, and I was trying to get my dissertation published, while at the same time my knowledge of generalized homology theory was decaying exponentially from not having any time to keep up with it.
What are five things on my to-do list for today (not in any particular order)
1. Draft a proposal for a faculty meeting tomorrow.
2. Grade a couple of differential equations homework sets.
3. Grade a calculus assignment for one guy who had to turn it in late (for a legit excuse).
4. Do my GTD weekly review, which I am usually doing this time of the week, but instead I am writing this blog post!
5. Start grading the test that my linear algebra class took on Monday.
What are some snacks I enjoy?
* Cereal, right out of the box. No particular brand preference.
* Baked Cheetos. My girls started liking these, and I’ve found myself scarfing down half a bag myself before I even knew what I was doing.
* English muffins. There’s a honey-wheat variety they sell at the local Marsh that is almost like eating a dessert pastry.
What would I do if I were a billionaire?
* Pay off the rest of my family’s debts. We’ve been working on that a lot this year already.
* Store away enough money for college and grad school for my two daughters.
* Give about $10 million or so to endow my 4-year old’s Montessori preschool, which does an amazing job with the kids but is perpetually on the brink of bankruptcy.
* Finish off the list of stuff we want to do to the house — finish the basement, put in new kitchen countertops, etc.
* Store away enough money to take the family to China for 2-3 months once the girls are teenagers.
* Visit France's les alpilles and provence.
* I’d seriously consider opening my own university. It would be a Great Books school with an emphasis on math and science. $100 million or so would be way more than enough for a decent endowment.
What are three of my bad habits?
* Burping out loud. (That’s not necessarily bad if you live by yourself, but if you have a 4- and 2-year old living with you who you want to have good habits…)
* Blogging or web surfing when there’s work to get done. (Oops.)
* Talking to myself.
What are five places where you have lived?
1. White Bluff, TN.
2. A cheap guest house in the ritzy Whitland Avenue neighborhood of Nashville, TN. I lived there in graduate school for five years. Lamar Alexander was my neighbor two doors up and Al Gore was just around the corner.
3. Cookeville, TN
4. Mishawaka, IN
5. Bargersville, IN
What are five jobs I have had?
1. Professor at a small liberal arts college.
2. Adjunct professor at a large urban community college.
3. Math tutor at a private educational service.
4. Baker/jack-of-all-trades at a mom & pop donut shop. Still possibly my favorite job I’ve ever had. If could have earned more than $30K a year and gotten benefits there, I’d never have left.
5. Librarian assistant at the Vanderbilt University Biomedical Library. I started out just as grunt labor, reshelving books and journals and changing paper in the copiers and so on; but eventually I worked the circulation desk and helped the reference librarians do technical search work. I’ve always thought that was a cool job, working in a high-powered specialized research library, and if I ever changed careers, that would be one I’d consider.
Now it’s time to tag five people. Apologies if the following have already done this, but:
* Scott Franklin at Natural Blogarithms.
* Shelby Berg, a friend from my grad school days who co-writes a blog with her husband about their incredibly cute 1-year old son.
* Julia at Intelligent Dissent, who is probably way too busy thinking about really cool and important ideas to do something so pedestrian as a meme, but I’m tagging her anyway.
* Isabel at God Plays Dice.
* Suzi at My Own Thoughts.
Deconstructing dx
5 May 2008
Asking the following question may make me less of a mathematician in some people’s eyes, and I’m fine with that, but: How do you explain the meaning of the differential dx inside an integral? And more importantly, how do you treat the dx in an integral so that, when you get to u-substitutions, all the substituting with du and dx and so on means more than just a mindless crunching of symbols?
Here’s how Stewart’s Calculus does it:
* In the section introducing the definite integral and its notation, it says: “The symbol dx has no official meaning by itself; \int_a^b f(x) \, dx is all one symbol.” (What kind of statement is that? If dx has “no official meaning”, then why is it there at all?)
* In the section on Indefinite Integrals and the Net Change Theorem, there is a note — almost an afterthought — on units at the very end, where there is an implied connection between \Delta t in the Riemann sum and dt in the integral, in the context of determining the units of an integral. But no explicit connection, such as “dx is the limit of \Delta x as n increases without bound” or something like that.
* Then we get to the section on u-substitution, which opens with considering the calculation of \int 2x \sqrt{x^2+1} \, dx (labelled as (1) in the book). We get this, er, explanation:
Suppose that we let u be the quantity under the root sign in (1), u = 1 + x^2. Then the differential of u is du = 2x dx. Notice that if the dx in the notation for an integral were to be interpreted as a differential, then the differential 2x dx would occur in (1), and, so, formally, without justifying our calculation, we could write \int 2x \sqrt{1+x^2} \, dx = \int \sqrt{u} \, du…
So, according to Stewart, dx has “no official meaning”. But if we were to interpret dx as a differential — he makes it sound like we have a choice! — then using purely formal calculations which we will not stoop to justify, we could write the du in terms of dx. That is, integrals contain these meaningless symbols which, although they have no meaning, we must give them some meaning — and in one particular way — or else we can’t solve the integral using these purely formal and highly subjunctive symbolic manipulations that end up getting the right answer.
Er, right.
To be fair, my usual way of handling things isn’t much better. I start by reminding students of the Leibniz notation for differentiation, i.e. the derivative of y with respect to x is dy/dx. Then I say that, although that notation is not really a fraction, it comes from a fraction — and that much is true, since dy/dx is the limit of \Delta y / \Delta x as the interval length goes to 0 — and so we can treat it like a fraction in the sense that, say, if u = x^2 + 1 then du/dx = 2x and so, “multiplying by dx”, we get du = 2x dx. But that’s not much less hand-wavy than Stewart.
Can somebody offer up an explanation of the manipulation of dx that makes sense to a freshman, works, and has the added benefit of actually being true?
Update on my “super powers”
2 May 2008
So yes, I did actually go see a doctor this afternoon to see if my blurry-vision incident might be something serious. It wasn’t a TIA or a stroke, because apparently a TIA lasts for 30 minutes or so and includes all the symptoms of a stroke — slurred speech, immobility on one side of the body, and so on. This only lasted a few seconds and was just blurry vision. In fact nobody knows what might have caused that to happen; possibly a small blood clot or just a muscle spasm in my eye.
But I just wanted everyone to know I did get it checked out. Especially virusdoc, whose comment got me to Google “TIA” which then put the fear of God into me and then got me to the ER — where I ended up in room 14, although I was really hoping for room 16 just to make the whole powers-of-2 thing complete.