Dr. Stat

Dr. Stat is a Statistics Professor. This blog is his opportunity to share ideas and opinions about education (especially math education), politics, and whatever else comes up.

Thursday, February 24, 2005

NCLB Unconstitutional?

From Michelle Malkin, I found the most amazing story about No Child Left Behind that I have seen in a long time, or maybe EVER.

The bipartisan National Conference of State Legislatures is coming out with a devastating report tomorrow that reportedly shreds the Bush administration over the No Child Left Behind Act.

Most interestingly, the report tomorrow comes to the conclusion that the No Child left Behind Act is unconstitutional.

I have been trying to call attention to this issue for at least two years now. I sent the letter below to all the state legislators (that I could get email addresses for) in North Dakota, South Dakota, Minnesota, and Montana, and also sent it to several newspapers. To my knowledge, no newspaper printed my letter. Obviously, I am not alone in this opinion, though whether or not I had an influence on the legislators, I cannot say. I did receive numerous replies at the time, many of which expressed sympathy with the ideas but claimed they could not afford to fight against NCLB because of loss of revenues to the state. Maybe the tide is turning. I hope that states are ready to begin asserting their rights in this and other issues where the federal government has intruded on reserved powers.

I also brought up this issue at a public hearing with Senator Daschle's staffer on education about a year ago. She claimed that she had not heard of anyone saying the law was unconstitutional. She and others seem to be of the opinion that the law is constitutional because it is addressing a civil rights issue. I find that quite a stretch.

Here is my letter:

March 1, 2003

An Open Letter to State Senators, Representative, Governors, and Educators,

Oppose Federal Control of Education

The time has come for the individual states to take a stand. I call upon you, as a leader of your state, to act with strength and conviction to oppose the tyranny of the Federal Government which has acted unconstitutionally and irresponsibly with its so-called “No Child Left Behind Act.”

Your state should not be required to act alone, but it must act, for only if individuals have the courage to lead can the strength of the many be brought to bear—for ultimately, all states should join together in refusing to allow this well-intentioned, but nevertheless intolerable, law to stand.

Let us address the issues. First, the law is unconstitutional. Our constitution clearly forbids the Federal Government from engaging in activities which are not specifically prescribed for it, and education is notably absent from the list. Education is the responsibility of the states, to be discharged in a manner they individually see fit. The Federal Government has been violating this provision under the guise of various excuses for many decades. The time has come to say “NO MORE.” It is true that a great need for national standards exists, and states should cooperate to produce such standards. It is not the Federal Government’s business, duty, right, or responsibility to participate in this process in any capacity whatsoever. Let us respect the Supreme Law of our Land, enforce it as our founders intended, or change it by the process of amendment—not by setting it aside through legislative whim.

Second, the law is fiscally irresponsible. At a time when most states are facing severe budget crises together with their own individual problems in education, money and resources are being diverted from necessary and useful educational activities to meet the arbitrary, questionable, and sometimes illogical requirements of the law.

Third, the law contains absurd requirements which will ultimately have the effect of reducing the quality of education in some places. There are many such requirements, but to list just a few, we note that some states are scrambling to lower standards so that their schools won’t “fail,” that the law takes away resources from schools that need them most, and that the requirement that results hold across all subgroups is impossible in some cases and ridiculous in others (e.g., one or only a few children in a subgroup; special education).

The law is a result of good intentions and most of its goals are noble and perhaps even shared by the states. I have no doubt that some sort of national effort is needed to improve our schools. This isn’t it, but even if it were, it would still be incumbent upon us to resist the spread of federal tyranny and insist that the constitution be followed as it was written and as it was intended.

Please help to preserve our freedoms!

Friday, February 18, 2005

Larry Summers, President of Harvard

OK, so the transcript has been released. Where is it???? The New York Times reports on it, gives us its spin, but doesn't give us access to the document?

Update: Thanks to the commenter and Wizbang, I have found it here.

This is good stuff: Fred Reed mentions Summers.

Well, now, this blogger says Summers has now been vidicated by a newly released study!

Thursday, February 17, 2005

Learning Styles

What are "Learning Styles" and why do they matter?

According to some education theorists, "learning styles" are distinct categories of behaviors within which individual students have various "preferences," which may roughly correspond to "efficiencies" in terms of educational achievement. David A. Kolb developed a theory of learning styles which is common in education textbooks today. He actually is said to have believed that learning styles are on a "continuum," but in practice the theory leads to a discrete classification scheme. A reasonable short summary of these ideas may be found here. However, the thing that is most clear about learning styles theory is that there is no agreement on what learning styles are. Most of us in the "Ed Biz" more or less intuitively believe that there are, in fact, "visual learners," "hands-on learners," "book learners," etc. We may have other ideas or labels. We also recognize that some people like to work in groups, and some people like to work alone. Furthermore, I think that good teachers have recognized things like this for a long time, independent of what any theorists said. So what good teachers have been doing, with or without David Kolb, is creating learning activities that cater to different learning styles, whatever they may be.

I don't claim that it has always been like this. I am quite sure that there were times and places in which teachers simply assumed there was one way to teach (the right way) and all students had to adapt or suffer the consequences. But it would be a mistake to assume that such an attitude was universal in any period in history. The balance between rigidity and flexibility may have shifted, but both approaches were in existance. And I claim that it has always been a mark of a good teacher to try a variety of techniques in the educational process.

Now, this is not to imply that nothing further can be gained by scientifically studying "learning styles" and attempting to develop verifiably effective methods of exploiting them. Such efforts may be fruitful and worthwhile. On the other hand, if such ideas are not empirically based, and the research is unsound, we will have gained nothing but jargon (more of a loss than a gain, really) and we will still know nothing more about using learning styles than the good teachers of yore.

I want to make a point here: I don't view learning styles in the same way most education textbooks do. Their emphasis always seems to be on how the teacher must adapt to the learning styles of the students. By suggesting that good teachers have always used techniques that play to different learning styles, I have not meant to imply that they did so primarily to adapt to their students' needs. In some cases, particularly when students are having difficulty, that is precisely what good teachers do. But not always. I would like to turn this around, in a way that I have never seen it done in an education textbook (of course I have not read them all). We make much of the phrase "learn how to learn." This supposed to be a primary purpose of modern (postmodern?) education--because knowledge changes so fast we must be lifetime learners. Yet, the "learning styles" dogma suggests that we meet students where they are, and adapt teaching to the student's currently preferred method of learning. This is not "learning how to learn," it is "learning how to stay in a rut."

My proposal is this: There are many ways to learn. If we are to be lifetime learners, it is best that we get as good as we can at all of them. We should not, however, focus on all of them equally. It is quite clear that some methods of transmitting knowledge are more efficient than others. Efficiency, here, has several components. To evaluate efficiency, one has to have some idea of costs as well as outcomes. While most of the research on educational methods is done in schools, it behooves us to look at the situation in business, where efficiency is forced to count most (far more than in schools.) Costs there include materials, equipment and facilities, but the largest components are often labor--specifically the teacher's labor (an expert whose time is particularly valuable and must be minimized) and the time of the learners before they become productive. The primary method of transmitting knowledge in business is still the lecture with visual aids. If a skill is to be transmitted, there is some variation depending on the situation, but for the most part the method is demonstration followed by guided practice until proficiency is attained, as measured by some objective and immutable standard. Hardly ever do we see business employ a "discovery method" or "constructivist approach." Independent reading may be involved, but cooperative learning and games are examples of activities that are less common in the business world. I argue that, because business places a premium on efficiency, the practices of business are an excellent indicator of what learning activities are most efficient and most useful as life-long skills. The free market forces inefficient practices out. Of course, we cannot ignore the differences between (motivated, mature) adults in the workplace, and children, whose motivations may be elsewhere, and whose maturity, by definition, is not at the same level.

The proper use of learning styles, therefore, is threefold: 1) To take into account differences in maturity and motivation among students at various ages and social stages, 2) To overcome individual difficulties in learning, and 3) To prepare students to be efficient life-long learners as adults. I believe that the first two uses are necessary, but the third is the most important. I believe that we should actively, intentionally, and directly teach students how to learn effectively from lectures. I mean by this that we should do much more than just use lecture as a tool to teach. I am aware that some teachers (and counsellors in study skills, etc.) teach students note-taking and organizational skills. These are important and valuable. However, I am reminded of stories about the "ancients" who are said to have been able to repeat long speaches or poems after hearing them once. I don't know if this is true, but I do think that people in the past have been much better at repeating what they have heard than we commonly can today. In fact, I am convinced that our children are capable of doing this much better than they do, because they can often retell large portions of television shows, or give detailed renditions of inane conversations that they have had. Frankly, I don't think they ever consider the possibility that they might be able to recall a lecture, or a story read to them, in the same way, and they are seldom or never challenged to do it.

There are probably a number of reasons that we don't encourage this kind of activity in school. No doubt one of them is that this is a difficult task and there is no sense, in the current educational literature, of its importance. There is also too little emphasis on the need for accurate and complete recall of facts relevant to a discussion, because the prevailing notion is that knowlege can be "looked up" so easily that we don't need to remember it. This is a profound fallacy. In truth, it is impossible to process information unless it is available in memory, and reasoning and critical thinking require instantaneous access to enough relevant information to be able to evaluate ideas and determine what new information should be sought out. This is why political campaigns run on soundbytes--people can't remember enough of the content of a speach to rationally evaluate it anyway. And they can't remember enough because they've never been taught that it's important to remember.

Tuesday, February 15, 2005

Terri Schiavo

Swift action is urgently needed.

We have not been told the truth about Terri Schiavo by the Main Stream Media.

She is not brain dead. She is not in a coma. She is not on life support. She responds, smiles, and purses her lips to kiss her parents.

Her husband (I'd like to say "husband" in quotes, but he is still married to Terri while living with another woman and fathering two children with her) wants her dead. He won around 1.7 million dollars to use for health care for her and has now spent about a half-million on lawyers fees trying to kill her.

This is a big test case for the right-to-die crowd, whose true identity is revealed as the right-to-kill crowd. Please check out the link in the title of this article.

This is an educational issue. For, if we do not educate ourselves, we will surely be "educated" by the propaganda machine of the MSM to believe many things which are simply not true. This one life is a showcase of the fight for truth and civilization that we must not let go.

In recent days we have seen several victories for the truth. Journalists are being held accountable for their actions and inactions in ways that were never before possible. Perhaps we citizens can use this same power in the Schiavo case. Michael Schiavo is planning to commit legalized murder in one week (2/22). If he succeeds, the door will be opened for many more such murders. Are we comfortable with the idea that people who are conscious and very much alive may be killed because they are unable to communicate? Are we comfortable, not knowing how far this practice will extend? Some people oppose capital punishment because of the possibility of new evidence overturning a conviction, and demand that we keep vicious criminals alive at great expense to society. Yet, an innocent person, likely even the victim of a (possibly capital) crime, is not worthy of being protected and kept alive. The hypocrisy is too great to bear.

Monday, February 14, 2005

Achieving Highest Honors in AP Calculus

From: Inland Valley Daily Bulletin (Ontario, CA), February 14, 2005
HEADLINE: Diamond Bar High outshines World

Diamond Bar High School got the distinction of being named "Best in the World" by the AP (not Associated Press) Exam company, College Board. This DID NOT happen in the USA.

But being named best in the world is less the result of how bright these students are --though they are -- and more the dividends of ambitious thinking. Instead of being the aspiration of a few, they wanted calculus to be a destination for all. In May of last year, 300 of the 3,150 Diamond Bar students passed the calculus exam with a score of three or higher, out of five points possible.... When the school retooled its curriculum four years ago to offer an International Baccalaureate program, it collaborated with its middle schools to lay foundations for a path that would lead students directly to [calculus]. [The teachers] developed teaching techniques when the school opened in 1983 that question the conventional wisdoms of the classroom. First, there's the music. Instead of silent austerity, downtempo beats pulse from a stereo, even as the students squirm over an exam. Also, at the beginning of each session, students watch a one-minute movie clip that has nothing to do with calculus. "Mr. Alcosser makes everything so interesting with the things he says, even the clips from those bad movies," said junior Shashank Ravi, 17.... And it seems to be working. The five teachers' classes are packed, some with close to 40 students....

Readers unfamiliar with the International Baccalaureate are urged to visit http://www.ibo.org/ibo/index.cfm.

I maintain that we know how to achieve high academic levels. The ingredients are not a secret. For some reason, most American school districts are unwilling to use the recipe.

Saturday, February 12, 2005

Math Phased out in UK

The Guardian (London) , 2/12/2005, Guardian Home Pages, Pg. 13

HEADLINE: University to shut maths department

Hull University has announced it is to close its maths department.... The university said it was phasing out its degree course because of a shortage of UK-based undergraduates and reduction in funding. Four other maths departments in England have closed since 1999 and the number of students has fallen by more than 2,200 in the same period.... more than a third are overseas students, compared with an average of 16% on other courses. The vice-chancellor, David Drewry, said that by closing the department the university was "doing our utmost to protect the interests of staff and students".... Nottingham Trent University has confirmed that it is reviewing its maths provision and the LMS said academics at several institutions, including Birmingham, Salford, Essex and the University of Wales, had expressed concern about the future of the subject.

How can this be? At a time when the importance of math as a foundation for technical and scientific progress is greater than ever, major industrialized nations are struggling just to maintain college math programs. They don't have enough students.

Mulan II

Mulan was bad enough. Mulan II is far worse.

In Mulan, we see a young Chinese girl of an ancient dynasty defy gender roles to go fight in the army in her father's place. Not only that, she becomes a hero and saves China. Along the way, we have a positive portrayal of the ancestor-worship religion such as Disney would never allow for Christianity. OK. It's just a kids story.

However, the real theme of the movie is marriage--at the beginning, Mulan is going to see the matchmaker so she can be matched with a husband. She objects to arranged marriage and ends up botching the interview anyway. She meets her "match" during her war adventures.

Now we have Mulan II. In this story, Mulan is now a national hero and all the little girls want to be like her. The major themes are "follow your heart" and "marry for love, not duty."

In the story, Mulan and her fiance Chang are called upon by the emperor to save China again. The emperor's three daughters are to be escorted to another country to marry three princes and cement an alliance to protect China from the invaders. Mulan, Chang, and three comical bunglers from the first movie are to quietly escort the princesses to their destination. Immediately, Mulan starts in on the princesses, questioning them on how they can get married to men they haven't even met. The princesses make patriotic statements about their duty. However, it is not patriotism and duty that are going to win the day in this movie. No, the princesses fall in love with the three buffoons, and in the end "true love" triumphs and the kingdom is saved as well.

This movie teaches disrespect for culture, responsibility, patriotism, and marriage. It reinforces the foolish notion that infatuation is all that is really needed to make a successful marriage. No serious consideration of the wide gulf of experience and expectations between the princesses and soldiers is given. The princesses even sing about their new-found freedom. Wait until reality hits! I wonder if they will invent divorce in Mulan III? In the end, it is modern American pop-culture values triumphing over old-fashioned, out-moded tradition.

Thursday, February 10, 2005

Math, Music, and Computers

From: Business Wire, February 9, 2005

Headline: Non-Profit Organization Brings Proven Math Education Process Developed by Internationally Acclaimed Scientists to Texas Elementary Schools

After six years of testing and refining the program in California schools, this is the first time that the "Math + Music" program is being offered in another state. The process combines music instruction and computer math games to dramatically enhance students' mathematical and problem-solving skills. It focuses on building the brain's spatial temporal reasoning capability -- the ability to think visually several steps ahead in patterns and pictures. "Math + Music" is based on 30 years of brain research and offers a new, engaging and more effective way to reach students who have historically struggled with math.

The MIND Institute is a community-based, non-profit research organization formed in 1998 and based in Costa Mesa, Calif. It is dedicated to brain research and to preparing students in grades K-12, regardless of cultural or socio-economic background, for success in school, work and society. The Institute has successfully integrated more than 30 years of breakthrough academic research on the brain at he University of California, Irvine, into applied education programs for elementary school students.

Sounds Interesting.

Ward Churchill: Worse than we Thought

I dismissed Churchill's lunatic ravings as just more left-wing rhetoric. I didn't think it would have any lasting impact. I may have heard of him previously, but didn't remember the name when Bill O'Reilly did a piece on Fox News demanding that his appearance at Hamilton College be cancelled.

Last night, my eighth grade daughter brought up that she had been taught in history class about the distribution of small-pox infected blankets to the Indians in North Dakota in a deliberate attempt to spread the disease. I discussed this a bit with her, because I didn't think the story was credible, though I'd heard it before. While not denying that smallpox might be spread by blankets, I said that smallpox is a highly contageous air-borne disease and that is how it is usually spread. An infected person merely needs to walk through a room and breathe to spread the disease. Also, I questioned who had done this deed--white settlers? they were not in close contact with Indians at this time. Fur traders? That would destroy their livelihood. Soldiers? Was the army active in the area at that time? I didn't know enough to definitively answer any of these questions, but I said that such things need to be questioned and not taken at face value. This is a highly suspicious story.

It turns out I was right-on. This report by Thomas Brown of Lamar University is a must read. Unbelieveably, the same Ward Churchill, who claims the 9/11 victims deserved to die and the terrorists are right, INVENTED the story of smallpox genocide on the Dakota plains.

We don't need to call upon his university to fire him for inflammatory remarks, slander, or any such "mild" offenses that are being bantered about around the blogosphere. He must be fired for fruadulant research and perjury. That's a real, legitimate, legal, uncontestable reason for a university to fire somebody. It has nothing to do with free speach or academic freedom. There is no academic freedom to dessiminate fraudulent research, whether it's inflammatory or not.

PS. In defense of my daughter's history teacher, it turns out that my daughter got mixed up... they learned in class about the 1763 attempt by the British to distribute smallpox-infected goods to the Indians. They were not referring to the epidemic on the great plains. BUT Churchill was...

Wednesday, February 09, 2005

Condi's Paris Speach

I can't resist linking Dr. Rice's speach in Paris (click on the title). She's quite a diplomat. I hope she's our next president!

Tuesday, February 08, 2005

US vs Singapore in Math

American Institutes for Research (AIR) has released a study (funded by the US Department of Education) comparing Math Instruction in Singapore and the US. The full report is available here. Singapore is a world leader in quality of math education. Its textbooks have been the focus of much interest among conservative educators and home schoolers in the US. American versions of the Singapore math books are now available.

AIR headlines, "U.S. Trails, But Both Nations Could Learn from Each Other."

The main conclusions are

 Singapore’s textbooks and assessment examinations are more demanding
 Their teachers are more skilled mathematically
 U.S. approaches often put more emphasis on certain important 21st century math skills.

In the 2003 Trends in International Mathematics and Science Study, Singapore came in first, while the US was 16th out of 46 participating countries. Well, we've been hearing news like this since at least 1983, haven't we? Remember "A Nation at Risk?" (But for a critique of that report, see here.)

“It is unreasonable to assume that Singaporean students have mathematical abilities inherently superior to those of U.S. students; rather, there must be something about the system that Singapore has developed to teach mathematics that is better than the system we use in the United States,” says Steven Leinwand, the lead AIR author.

Maybe it should be explained why such an assumption is unreasonable, but I'll accept that we need not entertain the "inherent superiority" theory. And I'll agree that the school system, including textbooks, standards, and teacher preparation is the primary place to look. However, there is a third factor, namely cultural values and support, that might be influential. The full report discusses this briefly too.

“And in the process, we came across some things Singapore might think about addressing. For example, the U.S. frameworks more often include high-order thinking skills critical to competing in the 21st century, though they are not obviously taught well enough here.”

I don't think we need to export our non-working theories about what is important in mathematics education to a country that's doing so well. This arrogance is out of place. I believe that the attempt to teach high-order thinking skills in place of critical basic skills is a large part of the problem. But then, I'm making an arrogant assertion as well.

The study also includes initial results from four pilot programs that used the Singapore mathematics textbook in place of their regular textbooks. The pilot programs involved students in Baltimore, Md., Montgomery County, Md., North Middlesex, Mass., and Paterson, N.J. The study found two pilot sites produced sizeable improvements in student outcomes, but overall the study observed mixed results because “the pilot sites, to varying degrees, encountered problems with teachers who lacked the educational preparation needed.” Student mobility also limited prior exposure to the Singapore mathematics curriculum, a serious problem in a curriculum that teaches to mastery and does not repeat content.

Translated, that means 1) our teachers don't know enough math to effectively use a real, results-oriented math curriculum, 2) the real problem is that we don't insist on mastery at each level so we can never count on a student being at grade level, and 3) in spite of hese shortcomings, the Singapore curriculum got "mixed results" with half of the pilot sites showing "sizeable improvements." I think what we are really seeing here is that (the Singapore curriculum)+(qualified teachers)+(insistence on mastery)=(impressive results in mathematics education).

A few other noteworthy points:

 Singapore offers an alternative mathematics framework for lower-performing students that covers all the mathematics topics in the regular framework, but at a slower pace and with greater repetition, and with support from expert teachers.
 Textbooks: Singapore’s textbooks build deep understanding of mathematical concepts while traditional U.S. textbooks rarely get beyond definitions and formulas.
 Teaching: Singaporean elementary school teachers are required to demonstrate mathematics skills superior to those of their U.S. counterparts before they begin paid college training to become a teacher.

And then we have:

 U.S. Strengths: Although the U.S. mathematics program is weaker than Singapore’s in most respects, the U.S. system is stronger than Singapore’s in some areas. The U.S. frameworks give greater emphasis than Singapore’s to developing important 21st century mathematical skills such as representation, reasoning, making connections, and communication. The frameworks and textbooks also place greater emphasis on applied mathematics, including statistics and probability.

For reasons I would like to explore further at another time, I do not accept these "Strengths" as legitimate. While there is nothing particularly wrong with anything on the list, I believe we are seeing an inordinate emphasis on these things, which is driving more necessary activities and emphases from the curriculum. If I am correct about that, then Singapore's supposed deficiency actually contributes to its strength.

Monday, February 07, 2005

State Math Standards Questioned

1/5/2005, The Albany Times, by Rick Karlin:

The Thomas B. Fordham Foundation said, in reference to New York and about a dozen other states it studied, that the state math standards "offer vague and overly simplistic guidelines for teaching K-12 students."

"All of us were awestruck by the levels of ignorance that we saw," in math standards nationwide, said David Klein, a math professor at California State University at Northridge, who headed the math research effort. In general, Klein said, states place too much emphasis on concepts such as probability and the use of calculators at the expense of basic skills like adding, subtracting, multiplying and dividing. "Many state standards don't allow students to learn enough arithmetic," Klein said.

I am completely mystified by the emphasis on teaching elementary students concepts like mean, median, and mode, some of the graphing activities that are included in the curriculum, as well as the probability. These ideas are presented without an adequate context in which to understand and use them, and they lead nowhere--they are conceptual dead ends until students get to Introductory Statistics in college. At that point, we expect them to know these basic concepts (because they were taught in elementary school) but find we are in a quandry because a) if we assume this knowledge, they will in fact not have complete or adequate concepts and b) if we assume they don't know, they are bored and disinterested because they think it's simplistic material. I think statistics and probability should be delayed until a good enough foundation exists to pursue a reasonably complete development of the concepts. Of course, arithmetic must come first. Working out averages is a fine exercise in arithmetic. Is it necessary to call it the "mean" at this point? I doubt it, unless you are going to discuss geometric means or other measures of central tendency too. Students in college-level Introductory Statistics often lack algebra skills and important background concepts like summation notation, unions and intersections of sets, etc. It would be much better to make sure these things are taught well, rather than to teach shallow and isolated portions of statistics before the students are really ready for them.

Sunday, February 06, 2005

A little fun with ethical philosophy

Take the ethical philosophy Quiz at SelectSmart.com. Results are relative to best match, which is given 100%.

Here are my top 10

1. Aquinas (100%) Click here for info
2. St. Augustine (88%) Click here for info
3. Jean-Paul Sartre (85%) Click here for info
4. Spinoza (84%) Click here for info
5. Plato (81%) Click here for info
6. Jeremy Bentham (80%) Click here for info
7. Aristotle (79%) Click here for info
8. Kant (79%) Click here for info
9. Ockham (78%) Click here for info
10. Ayn Rand (75%) Click here for info

Saturday, February 05, 2005

Mixture Problems

My daughter has been asking for help on mixture problems in which they use little tables to arrange the available information. I used to teach this method myself, in "Intermediate Algebra" at the college, with great enthusiasm, I might add. Now I have my doubts.

In case you are not familiar with this method, here is an example.
Suppose I have a solution of 50% alcohol and another solution of 25% alcohol. How much of each of these solutions should I mix to make 2 liters of 40% alcohol?
The table is arranged with an equation across the top that relates the quantities in the mixture, and rows underneath for each component of the solution and the final result. The student is instructed to fill in all the known information, then assign a variable based on the question.
Amt Sol%PurityAmt Pure
First Solx.50
Second Sol2-x.25
Final Sol2.40

Next, the student will apply the equation across the top to calculate the last column, whose first two entries must add up to the entry in the result, thus generating the algebraic equation to solve:
Amt Sol%PurityAmt Pure
First Solx.50.50x
Second Sol2-x.25.50-.25x
Final Sol2.40.80
Thus .50x + .50 - .25x = .80.25x = .30x = 1.20
Therefore we should use 1.2L of 50% alcohol and .8L of 25% alcohol.

This all works out beautifully. It seems very elegant. Unfortunately, it's also a gimmick. My daughter doesn't understand why the table is set up the way it is, nor does she really understand why this table results in an equation that gives the solution to the problem. I think this is fairly typical of students who are taught this method. Furthermore, it applies to only a limited set of problems, although it's true that they do vary somewhat from this example. Especially interesting are the variations that involve replacement of part of the mixture (how much weak antifreeze to drain out of your radiator and replace with pure antifreeze to get it right). The method can also be used with Distance=Rate x Time problems. These seem to be even more confusing, though.

My belief is that this type of problem is best approached as a system of equations in two unknowns. Let's see how this works with the example. I have to mix two solutions, let's say x of the 50% solution and y of the 25% solution and get 2 liters. Then x+y=2. The key feature of mixture problems is that there are two things that you can keep track of. In this example, one is the total amount of solution, the other is the amount of the pure stuff (alcohol) in the solution. We have an equation for the first, now we need one for the second. That will be (.50)x+(.25)y=.80. This system of equations is easily solved by substitution, resulting in an equation just like the one generated by the table method. However, the conceptual differences could be significant. If we use the system of equations, the focus is on reasoning about the relationships in the problem, while with the table the focus is, well, how to fill out the table. Teachers can stress the concepts underlying the table and perhaps bring the students to the same understanding, but they may not always do that. Too often, there is more of a focus on "learning the procedure/pattern" than on the essential "why."

One reason that this is a far superior method of teaching these types of problems is that it requires students to think about the sources of their equations. When students get to real applications in science or engineering classes, for example, it will be a VERY common thing to solve a problem by saying something like, "I have two unknowns here. I know one equation, where can I get another?" They should be practicing this kind of thinking in algebra.

The tables are now standard fare in high school algebra, though I used to be surprised when some of my college students said they hadn't seen them before. Now, I just think they are a gimmick that we would be better off doing without.

Let's Follow FDR on Social Security!!

From FDR's message to congress on Social Security: "It is proposed that the Federal Government assume one-half of the cost of the old-age pension plan, which ought ultimately to be supplanted by self-supporting annuity plans. "

See, it was intended to be privatized from the very beginning... not only partially, but the whole retirement portion! Let's just ask the Dems to keep their promises, shall we?

It's just too ironic how the Democrats gathered around FDR's statue to protest the "assault" on Social Security.

Thanks to Oxblog.

Is Freedom of Speach Returning to the University?

Is the tide turning with regard to free speach and diversity of opinion at the universities? Michelle Malkin quotes from "Inside Higher Ed" about Brown University's president, Ruth Simmons, encouraging students to "engage in civil, open debate, on every issue possible."

I found another example of a university president speaking on this subject. Anyone know of any others?

President Peggy Miller, of South Dakota State University, included the following comments in an address to the faculty at the beginning of the 2004-2005 school year:

Regrettably, our nation is fiercely divided over many issues surrounding that war. If we are ever going to be able to resolve those and other important national issues, thoughtful open minded discourse is sorely required. In much of the public media to which our students are exposed, there are few examples of earnest efforts to find understanding of differing points of view or to reach common ground.

I am growing increasingly concerned that the kind of unbiased deliberation, espect for competing ideas, active listening, problem solving and honest searching for the truth of issues and the better idea, are becoming indangered behaviors. As academics, we have been trained through the rigor of our individual disciples, to engage ideas... but more importantly, to sort fact from opinion, test hypothesis, and gather all the data required to reach good decisions. Sadly, it may be only here in the university that students will be able to observe these behaviors consistently used. I think we have no more important challenge in the next year, and those that follow, than teaching and demonstrating to our students the wisdom of these intellectual values. If we can do so successfully, we will have provided them powerful tools to use to protect and defend our democratic society. If they learn to use them well, perhaps neither they, nor their friends, nor their classmates will again be called to harm's way to do so.

That is no small hope, but I have a quote under the glass on my desk that reminds me every day to, "Believe that a small group of hardworking committed people can change the world. Indeed, it is the only thing that ever has."

These bold and courageous words seem to indicate a turning point in Academic Freedom. As universities around the country are increasingly pressed to provide students with "fair and balanced" learning opportunities, as well as opportunities for freedom of expression (even for conservative students) one welcomes such statements of support from the highest levels of administration.

However, there is a caution in this tale. To the liberal mind, this speach could be interpreted as an attack on conservatism. (Use your imagination, it's a stretch, I know.) Many liberal professors who use their classes as bully pulpits simply don't recognize that they are promoting one side to the exclusion of others. Calls for "free and open discourse" fall on deaf ears to these people. The only realistic way to expose students to multiple points of view is to have a diverse faculty who each promote their own point of view to the best of their ability. This is extremely difficult to achieve when whole fields of study are dominated by a particular ideology (consider evolution in the biological sciences, government redistributionism and interventionism in the social sciences, religious studies at state institutions, etc.). How can the universities achieve a diverse faculty when over 90% of all academics in a particular field subscribe to one point of view? We have to generate new faculty who have different points of view! But where will they come from, when they have to earn their advanced degrees from the same people who already dominate the field?? Although it's not true that someone with a minority view can't get through a Ph.D. program, the problem has more to do with self-selection. People who don't agree with the dominant ideology don't want to stay with (or even start) the program. How many conservatives would put 4+ years of advanced education into "Women's Studies?" Yet, you can't become a Women's Studies professor unless you do. "Global Studies" is a similar example. And what happens when someone does go through this process and maintains a minority point of view? Once in the profession, he or she has to advance as a scholar by publishing articles in journals that are ideologically based and tend to reject works that challenge the mainstream thinking.

True academic freedom is a difficult thing to achieve. But let's keep it coming.

Thursday, February 03, 2005

A Good Blog Article on Teaching History

Check out Betsy's Page.

State of the Union

"Today, Social Security is strong, but by 2013, payroll taxes will no longer be sufficient to cover monthly payments. By 2032, the trust fund will be exhausted and Social Security will be unable to pay the full benefits older Americans have been promised."

George Bush is such a pessimist ....

Wait ....

That was Bill, in his 1999 State of the Union. No Democrats Booed.

But they booed when George said this last night:

"Thirteen years from now, in 2018, Social Security will be paying out more than it takes in. And every year afterward will bring a new shortfall, bigger than the year before. For example, in the year 2027, the government will somehow have to come up with an extra 200 billion dollars to keep the system afloat — and by 2033, the annual shortfall would be more than 300 billion dollars. By the year 2042, the entire system would be exhausted and bankrupt. If steps are not taken to avert that outcome, the only solutions would be drastically higher taxes, massive new borrowing, or sudden and severe cuts in Social Security benefits or other government programs."
(Quotes from Six Hertz House of Pain)

National Public Radio this morning, with Carl Castle, said that a major portion of the president's speach was dedicated to Social Security. Funny. I thought it was just a little piece. I heard much more about Iraq and freedom in the world.

I must admit I'm feeling a little funny about the privatization scheme. It seems to me like it really doesn't have any major impact. We already have private retirement funds that are partially funded by our tax dollars...they're called IRAs. Why do we need another new scheme? Why not do more to promote what we already have, and talk about phasing out payroll taxes? Oops, I forgot, that's not politically feasible...

Wednesday, February 02, 2005

Evolution Litigation

Some people are so afraid of competing ideas that they find it necessary to petition the courts to have them supressed. So it is with supporters of Darwinism, whose constant refrain "Evolution is a Fact and all who oppose it are uninformed, superstitious religious nuts (heretics)" is just about as convincing as the Flat-Earthers' claims. If the facts are on your side, why are you afraid of competing ideas? Go ahead and discuss them, oppose them, and disprove them, if you can. If you can't, well, maybe there's a reason for it. What about academic freedom, free speach, the free exchange of ideas? The schools are apparently no place for such things.

The ACLJ (American Center for Law and Justice) is tackling the case, as David Limbaugh reports on his blog, with a link to the ACLJ site here.

Intel Science Competition & Foreign Born Students

From The New York Sun, 1/31/05, p. 11, by Andrew Wolf

"It appears to me that there are a disproportionate number of foreign-born students among the finalists and semifinalists in the Intel Science Talent Search competition. Moreover, many of these students appear to have received their early education in their country of origin, not here in America. Is there a lesson here?I think there is. Education practices in America are designed to "level off" all students into the vast middle ground lest we damage the self-esteem of those performing at lower levels. "

We teach math at the lowest common denominator, eh?

Once upon a time, there were teachers with "less" education than today's teachers, who nevertheless produced students who could do math. Today's teachers are "educated" in all kinds of "methods" for teaching math, like discovery learning, cooperative learning, etc., but in the end they aren't getting results. Reading journals of Math education gives the idea that if we could just find the right combination of activities or the right order to teach topics in or the right explanation for those difficult concepts, our students would learn better. Folks, we've been trying this approach for 50 years and gone nowhere but down. People used to know how to teach math effectively. But don't expect the Academic establishment to admit that they've been spinning their wheels for 50 years or so!

There's nothing more important than mastery of skills and concepts. It simply has to be done. There are many methods to achieve that goal, and a good teacher can use discovery learning, cooperative groups, and many other methods at her disposal to see that the job gets done. Drill and competative games are also valuable activities. But as long as educators think that going through the motions (following a particular stlye of instruction or lesson plan) is more important than insuring students learn (master) concepts and skills in a thorough and permanent manner, we are going to fail at the education game.

Tuesday, February 01, 2005

"No Teacher Left Behind"

News Brief #2919 Category: Opinion/Editorials )

Terry Moe, a senior fellow at the Hoover Institution, argues that teachers' unions routinely put their job-related interests above the educational interests of children. The unions stand in the way of real school reform, he says. If they won't voluntarily give up their power, he continues, "then it has to be taken away from them - through new laws that...drastically limit (or prohibit) collective bargaining..., link teachers' pay to their performance, make it easy to get rid of mediocre teachers," and give administrators more control over teacher assignments.
SOURCE: Wall Street Journal, 13 January 2005 (p. A12)
WEBSITE: http://www.wsj.com (subscribers only)