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, March 31, 2005

Choosing Textbooks in West Virginia

From "Math texts get careful scrutiny, Kanawha takes more time because of different views on teaching strategies," Charleston Daily Mail (West Virginia), March 28, 2005. By Charlotte Ferrell Smith:

Math problem number one: How much is 45 divided by 3? Math problem number one, again: Three children have a jar of 45 marbles. If shared equally, how many marbles will each child get?"Instead of giving children 45 divided by 3, you give them a situational task," said Olivia Teel, math curriculum specialist for Kanawha County. "They learn the concept of putting the marbles into sets. That is how it could be done differently as opposed to 45 divided by 3."
This is the old false dichotomy we hear about so often--one side decries rote memorization, calling it "kill and drill." The other side says reform methods (like problem-solving, group projects, etc.) leave too little time for actual mastery of skills. These are two sides of a coin, two approaches that should be used together in an appropriate way. You start with a situation to motivate a mathematical concept (how many marbles will each student get?), then you master the skill by practice (45 divided by 3), then you come back around to the situation--now you understand that you can quickly find out how many marbles each student gets by dividing 45 by 3. If you read the math textbooks from circa 1900 they did both kinds. What people in those days knew intuitively was necessary, we now have cognitive psychologists to tell us it is still necessary. It's not us against them, folks. It's what WE need to do to fulfill two complementary aspects of the learning process. Every good teacher knows this (intuitively).

I wouldn't even mention this except that this false dichotomy keeps coming up in one article after another. I haven't looked at the books, and it's probably true that some of them don't balance these things the way they should.

And by the way, the teacher above is misusing mathematical language. She should not be talking about putting marbles into sets. That is an inappropriate use of a mathematical term. Things like that worry me. Does it indicate a level of mathematical ignorance unacceptable for someone choosing a textbook? She may not be ignorant, merely careless in this particular instance. But if she is a math curriculum specialist, the burden upon her to be accurate is particularly great. She needs to be able to spot errors like this in the textbooks she evaluates.

Educators are trying to choose textbooks for students in kindergarten through fifth grade with math that offers the best of both worlds - a book that includes the basics as well as situational tasks. While that might sound simple, consider the fact that the number of possible books to choose from is mind-boggling. The county's textbook adoption committee managed to subtract a bunch and narrow it down to 10 different publishing companies with materials to review for grades kindergarten through five. New math books are to be adopted for schools throughout the county in May at a cost of $ 4.9 million.
I didn't know there were so many choices. I thought there were only 3 or 4 textbook publishers left after all of the mergers! Ah, but I'm not accounting for the startups, grass roots efforts, and the experimental curricula. Singapore Math (I like), Everyday Math (I hear good things but suspect short on drill), Saxon Math (highly criticized for too much drill but very successful) are some examples of those. Most likely when you have a 4.9 million dollar budget to spend, publishers come out of the woodwork. Things like that don't happen in my neighborhood.

Teel ... said children need a balance of basics, situational tasks, justifying answers and investigation ... the textbook adoption committee actually considered adopting two books for elementary students to get all of that included.
Good! (Well, not the part about needing two books.)

The ultimate goal is to boost test scores, said Teel, the math curriculum specialist, who said the current way of teaching math has become stagnant. Instead of the bulk of emphasis being placed on memorization, practice and drills, students need a better understanding of how math relates to situations, she said.
I just can't help thinking: What is making it stagnant? Are the teachers really placing "the bulk of emphasis on memorization, practice and drills?" If so, why? Better yet, what exactly do we mean by this? Practice is a major portion of the necessary learning process in math. Failure in math is largely due to lack of practice. Somewhere around 3/4 of the time should probably be spent practicing. That's not a bad thing. Maybe there's a problem with the kind of practice? Maybe we are practicing too much on topics that we already know? I don't know. It would take a detailed analysis of classroom time usage to understand this. The goal should be to build understanding of mathematical concepts and enthusiasm for the subject, not "The ultimate goal is to boost test scores." Well, that's not MY ultimate goal. I want to boost test scores too, but only because higher test scores are a measure of how much we have achieved in meeting our REAL goals, i.e., students who are GOOD at math and LIKE it. See, we should not base instruction on secondary goals. If you strive for the right goals, the test scores will come along automatically.

According to an article in Phi Delta Kappan, a national educational magazine, math education in the United States has received "blow after blow from recently released studies." Among the issues educators are weighing: too much or not enough arithmetic, calculators or no calculators, problem-solving or calculation.
Indeed.

Wednesday, March 23, 2005

Why Can't Teachers Write?

Bill Hennessy writes (link in title) about the poor quality of a letter received from his child's principal. I think most people with children in school have encountered this problem. Of course, everyone makes mistakes (there are probably some in this article), but the level and frequency of errors found in communications from teachers is a bit of a concern. This is professional correspondence, and the quality of writing should reflect that fact. In addition, these teachers are entrusted with developing the writing skills of our children; therefore, an expectation that they demonstrate good writing skills is not out of order.
I have occasionally corrected letters from teachers (the "sign and return" types) and included a note regarding my concern about the language quality of the letter. I have never received a "thank-you" for my efforts. The most irritating problem I find is the use of plural pronouns where singular pronouns are required. For example, a field-trip permission slip included the sentence, "Your child should bring their lunch." Obviously, the language is falling prey to PC in an attempt to use non-sexist language. What irritates me most is that the error is so unnecessary, since it can easily be fixed by saying "The children should bring their lunches," or "Your child should bring his or her lunch." The latter is probably deemed "awkward," but surely it is not as awkward as the blatant grammatical error?
I think we may lose the battle for singular pronouns in the end, since Zondervan, in its latest revision of the Bible (TNIV), has embraced the use of "they" and "their" as singular pronouns in its fanatical, over-the-edge, who-cares-what-the-original-intent-is subversion of the English language and the Christian religion. Historically, as the Bible goes, so goes the language.
All of this would have been prevented, had the old standards of the normal school, where teachers were expected to be able to speak and write with impeccable grammar, been upheld. But the modern (or is it post-modern?) ed schools have long since renounced any such standards. English and math are out, self-esteem and problem-solving are in.

Tuesday, March 22, 2005

A Study of Inflation

“Everything is so much more expensive now.”
Really?
Prices have gone up. So have wages (the price of labor). So has the quality of many goods purchased. How can we compare today with yesterday, when we are not talking about the same things? Take wages, to begin with. We can compare per capita incomes between two time periods. Or we can compare median family incomes. But is the work the same? Are the hours worked the same? Is the investment (mainly education) in employability the same? Are the sources of income the same (employment vs investment, inheritance, etc.)? In fact, none of these things are constant over time, making comparisons about changes in wages very difficult.

What really matters, though, is what you have the power to buy using the resources readily available to you. Most often, we think of the resource of personal labor, although the use of capital that you control is also a resource. We assume that all capital is built up through the application of personal labor at some time in the past. Therefore it is reasonable to think about how much can be bought using some unit of labor as a standard. However, the capital in an economic system has the effect of increasing the productivity of labor. That is, an hour of labor in the absence of capital is not the same as an hour of labor expended in the presence of capital, such as by using a sophisticated machine. Capital can also be abstract, in that a worker with education or experience has more “human capital” than a worker with neither. And then there are activities that are not easily valued in monetary terms, such as the labor of an artist, whose work may not be valued until he is dead. Some would like to measure the value of labor by “how hard it is,” but even that is not clear since our standards of “hardness” change over time, and what is hard for one person may be easy for another. Furthermore, there is much labor that is not counted in the economy, especially in the past. Consider a farm family circa 1900, with a farmer, his wife, and six children. We would be likely to count this operation as the labor of one man, yet it is likely that there are eight people doing the work. Consider that today, a stay-at-home mom is doing work she is not “getting paid for,” while the same work, being hired out by a “working” mom, is counted in the economy.

There are difficulties in comparing the things that we buy, too. Say we are comparing houses. Is a house bought in 1900 the same as a house bought in 2000? Even if it has the same square feet? No way. And as we look back on this from our vantage point 100 years in the future, our perceptions of this house are not the same as the perception of the original owners. For example, we might think the oak woodwork is extremely valuable, because it would be expensive to replicate today, while the original owners considered it to be quite ordinary. We would consider the lack of an indoor toilet to be a severe deficit, yet the original owners might have considered an outdoor toilet more practical, if they would even have considered anything else. How about a car? Is a Ford Model T equivalent to a Ford you can buy today? Or, how about a horse? Is a horse of 1900 the same as a horse of 2000? Even if the horse is the same, its purpose and value might be very different.

People often think of inflation as being a general trend of rising prices. This is not exactly true. Inflation is actually an increase in the supply of money, which is controlled by the government. When the money supply is increased, the money units are devalued and therefore buy less. In an expanding economy, it would be normal to increase the money supply to keep pace with the increasing value of goods available in the economy. If the money supply increases faster than the value of goods, the result will be a general rise in prices. Contrary to popular belief, government does have very good control over inflation. This control is not immediate in its effect, but over a period of one or two years the government can manipulate inflation to its liking and for its own purposes. Why does the government want inflation? Primarily for two reasons: 1) The government carries a huge debt. As money is devalued, so is the debt. 2) Inflation allows the government to collect more taxes through “bracket creep.” Furthermore, when the public cries out because of bracket creep, the government looks “good” by generously increasing deductions and thus pretending to lift the tax burden. The general population may also have the perception that they are better off because of inflation. This is because they see their incomes “rising,” and although they are aware that prices of goods are rising too, this is not as concrete in their minds as the increase in wages which they can probably recite from memory.

Friday, March 18, 2005

New York HS Math Reform

March 16, 2005, The Post-Standard (Syracuse, New York)
REGENTS APPROVE MATH COURSES; THREE NEW MATH SEQUENCES REPLACING "MATH A" AND "MATH B" TO BE PHASED IN.

By Debra J. Groom and Maureen Nolan

A new approach to learning math was approved Tuesday by the state Board of Regents.The board decided to replace the high school courses "Math A" and "Math B" - each a year and a half long - with three yearlong courses called "Algebra," "Geometry" and "Algebra 2 and Trigonometry."

"A New Approach???" Isn't this how practically everyone else does it?

The state says the course titled "Algebra" will include some geometry, measurement, statistics and probability in addition to algebra, while the "Geometry" course also will include some algebra, measurement, statistics and probability, in addition to geometry.The Regents said this will ensure that at the end of ninth-grade algebra, students should know as much math as they do at the end of Math A."This is not a lot," McSweeney said. "High-performing countries already ask their kids to do this. And if we're going to compete globally, we have to have rigorous math courses."

The article points out that there were problems under the old curriculum with different schools doing different things, so that students who transferred were missing topics. "Math A" and "Math B" were actually exams, rather than courses, and schools designed courses to prepare students for the tests.

Math curriculum is a difficult problem, or perhaps all curriculum is. There is not good agreement on what should be included or how it should be taught, and it is very important to have some uniformity because of students transferring and also because of college expectations. There does seem to be a trend to combine the threads from algebra, geometry, and such things as measurement and statistics, into one sequence rather than separate courses. I have nothing against the approach, in fact I think it is good to integrate these things so that relationships can be discussed and the topics can be used to strengthen each other. It does, however, make things more difficult when students transfer. It used to be that if a student had Algebra I or Geometry you could be fairly sure what he studied. Now it is much less clear, and much more difficult to decide what courses to place a transfer student into.

Monday, March 14, 2005

College, university freshmen lack math, English skills

Sarah Schmidt, The Ottawa Citizen, March 9, 2005:
Ottawa's colleges ... are ... dealing with increasing numbers of ill-prepared freshman students who require remedial classes in everything from English to math. ... the growing gap between expectations and skills is forcing institutions ... to take action.... According to officials at Algonquin College, administrators of the print journalism program don't even bother to look at applicants' grades when reviewing applications. Instead, they admit students who score at least 22 out of 30 on a language diagnostic test that reviews basic grammar and spelling and requires a short writing sample.... No more than two in 10 ... meet the admissions threshold.... schools are dumbing down first-year English courses to include segments on basic grammar, composition and writing skills once reserved for high school classrooms. "The competency level of students coming into our programs has dropped over the years," said Janet Gambrell [of] Sheridan College.... "What we don't want to do is point fingers and blame our high school colleagues, saying, 'Why are you graduating those folks?' "

But why not? Obviously the problems described here are not only applicable to Canada, for the same problem is rampant in the United States. This problem exists largely because we subscribe to two conflicting goals for education, which are, "Everyone should graduate from high school" and "The meaning of a high school diploma should be that a person has achieved a certain level of education." These goals are conflicting because we are unwilling to admit that not everone is capable of achieving that certain level of education, and secondarily, that not even everyone who is capable of achieving it is willing to achieve it. There is a fear that by admitting that not everyone can achieve at a certain level, we will be, in effect, discarding a large class of students, an excessive proportion of which will be minorities. In fact, by denying the truth, we hamper efforts to find real solutions. By hiding the facts that everyone knows, we make it impossible to address and solve problems. So let's admit it and deal with it. Then we can talk about what should be done to get "every child" to succeed. But success must be a meaningful standard, not merely a social promotion. A high school diploma MUST be made to mean that a certain minimum level of skill and knowledge has been gained. Those who cannot attain the standard need an honorable alternative path, which can still lead to success in life without the pressure to perform academically at a level that is not reasonable. This can only happen if strict standards are enforced at various levels in the educational process. Let's bring back the 8th grade diploma, for example. A true eighth grade education is all that is required for a large number of jobs that need to be done by somebody. Why not provide this as an honorable alternative to enter the work force?

Wednesday, March 02, 2005

Bill Gates on High Schools

On February 26, Bill Gates spoke at the National Education Summit on High Schools. His speech is available on the Gates Foundation site here. The whole thing is well worth reading. Although many media outlets have covered the speech, and various excellent quotes have been highlighted, it is better to go to the source and get the full context. I will provide a few small quotes relating to issues I want to discuss, but they will not be representative of the whole message. I substantially agree with Mr. Gates, but there are some things I would rather take in a different direction, or provide another focus. Gates is under fire in some quarters, for some of the things he says. I am not generally in sympathy with his critics from the education establishment.
America's high schools are obsolete ... designed fifty years ago to meet the needs of another age.

In all the talk about reforming education, one rarely hears a statement as far reaching and profound as this. We act as though our system of education is timeless, that it is a feature of America as permanent and intrinsic to our society as the very constitution. Yet, this is hardly the case. Gates reminds us that today's concept of a high school is a recent invention. There is nothing sacred about it. There is no particular reason why we must have a system shaped like the one we have. However, an institution, once established is hard to change. While some conservatives call for the elimination of public education altogether, on the basis that the system is flawed, failed, and hopelessly immoral, most people do not think in such far-reaching terms. Gates isn't saying that, either. But he is bringing a new concept to bear on the debate (at least I haven't heard it expressed quite this way): the business/industry concept of obsolescence. We had something that worked for a while. It doesn't work anymore. Let's invent something new. That just makes sense.

But what would it look like?

The idea behind the old design was that you could train an adequate workforce by sending only a third of your kids to college ...

Bill believes all children should be taught (or "trained," perhaps--he seems to focus on education as preparation for employment) to be ready for college. Here is where he and I will disagree. I have children of my own. I know their abilities. There is one I hope will go to graduate school. There is another that barely got through high school and is not going to college. This happened in spite of years of effort to turn things around. But in the end, her abilities simply are not at that level. We have to face this as a fact of life. The rhetoric presented by Bill Gates, which is, in this regard, similar to the rhetoric of No Child Left Behind, seems to be based on the notion that all children are born with equal abilities and come to school with equal abilities. This flies in the face of common sense and science alike. Children differ in many ways, and notable among these differences are vast differences in educational potential.

Of course there are dangers. It is wrong to label a child as lacking potential without justification, or on false premises such as socioeconomic status. It is clear that this happens in schools now. But it is also wrong to deny that differences exist. Not all children can go to college. Not all children should go to college. In fact, not all children with the ability to do so WANT to go to college. Bill Gates himself is not suffering greatly from the lack of a college degree. It is also wrong to push people too hard in directions they do not want to go. And I'm sure that Microsoft would hire any high school wiz kid with an amazing gift for programming without requiring a college degree, at least the Microsoft of the past would have. Perhaps things have changed.

I believe strongly that too many people are going to college today. This would come as a great surprise to Bill Gates, but perhaps he should listen to the rest of what I have to say. We still have, in this society many, many jobs that do not even require a high school education. Yet employers hesitate to hire people without a diploma. This is based on real-world experience, because people who do not finish high school rarely have the qualities needed to be good employees. So, a high school diploma is serving as a proxy for something else--a measure of dependability or employability. However, we find that a high school diploma doesn't guarantee the kind of skill set employers expect, when their actual need is a high school level of performance. So, they bump up the requirement to a college degree for what should be high-school level jobs. Like the high school diploma, the college degree is a proxy for something else--an insurance policy for a high school level of achievement and a greater level of maturity.

In the early 1900's, a young lady could become a school teacher (grades 1-8) by going to a normal school for a short period, in some cases 9 months. She then had the skills necessary to teach everything through eighth grade. I have examined many textbooks (mostly math) for elementary school from that era. It is true that the topics have changed, but the most amazing difference between the texts of that era and today are found in the difficulty level. Much of the eighth grade math I found in those books would be very difficult for today's high school students. Now, the truth is, it was out of reach for many of those eighth graders as well, and many people did not get that far in school. I have also examined 1940's and 1950's high school textbooks, and consistently find that even the mathematics intended for non-college bound students would be considered college math by today's standards.

Once we realize that we are keeping low-income and minority kids out of rigorous courses, there can be only two arguments for keeping it that way--either we think they can't learn, or we think they're not worth teaching. The first argument is factually wrong; the second is morally wrong.

So, I find that Bill Gates is spot-on when it comes to rigor. There is no excuse for the depths to which we have let education fall, at all levels. We have "dumbed down" the schools. We have taught to the "lowest common denominator." We have shunned and stigmatized rigorous standards and meaningful assessments. We have engaged in "social promotion." This is not part of the system, this is part of the culture. It is deeply imbedded in the indoctrination that colleges of education foist upon their students. But Bill is wrong if he thinks all students can learn equally. That is simply not true. Failure to recognize and deal with innate differences means that some students will be unable to keep up with expectations, while others will not be challenged to use their talents to the fullest. Children cannot all be taught alike, not without damaging those at both ends of the ability spectrum (or would you rather say, "spokes of the ability wheel?"). And that is another fallacy of the obsolete system, which at first allowed tracking, then rejected it. Some kind of choice or alternative pathway is absolutely necessary to enable all students to learn at their maximum potential. We are letting the politics of class dictate what is and is not permissible in education--and the children are worse off because of it.

...we must stop rationing education in America.

Here we have a deliberate use of an emotionally charged term. In what sense, if any, is it accurate? Rationing implies that there is a shortage, and not all can have full access. Actually, the intent of rationing is to restrict everyone's access so that all may have a minimally sufficient amount. Does this describe education? Well, first, there is no shortage. Certainly the entire country is well-staffed and well-equipped with buildings and supplies. I realize there are complaints in some schools, but shortages are not due to lack of funds, rather to misappropriation. In terms of dollar count and body count, there is certainly enough to go around. But much is wasted or spent foolishly.

Instead of determining that all children should go to college, we should raise the bar for eighth grade and 12th grade education. We should make those 12 years count for much more then they currently do. We should examine the process and strip away the waste, both in resources and student time. Find out what is really important and then focus on that. Set standards that require actual learning, not just seat time and participation in fun activities. I say, bring back the eighth grade diploma, and make it mean something. What passes for a high school proficiency exam today, in states that have them, is much more appropriate for an eighth grade level. Let's stop kidding ourselves. If that is enough education to get a high school diploma, then why not let eighth graders who can pass the test graduate (I have little doubt that about 10% of eighth graders could pass the high school exit exams in states that have them)? A good standard for a high school exit exam already exists. It's called an IB or International Baccalaureate. Students graduating from high school in an IB school already have the equivalent of an AA degree in a community college. This can and should be done, nationwide. This kind of diploma would be sufficient for many jobs that now require a college degree. And then, let's go on and raise the bar for a college education. Weak programs whose requirements include little more than busywork and ideological indoctrination should be eliminated. Scientific, technical, and engineering fields can be made internationally competitive in their rigor, where they are not already so.

Then too, we need to give students varying amounts of time to accomplish the goals according to their abilities. Surely we are smart enough that we can figure out a way to do this? Students shouldn't have to "flunk" or be "held back." The idea of repeating a whole year, of all subjects, is absurd. If students need more time, more practice, give it to them. But hold them to one standard. When they meet it, they move on. Those who are having trouble need extra intervention. Give them what they need. Honestly, I can't believe this is so hard! It is the rigidness of the current system that prevents finding workable solutions.

In the end, we will never succeed if we do not overcome the cultural barriers. Bill Gates' goals cannot be met by students who have no desire to meet them, whose parents do not expect it of them, and whose peers actively discourage them. This is where the real crux of the problem lies, and it is absolutely central to the racial and class differences we see in educational attainment. Frequently overlooked are the facts that not all minorities are equally affected by educational disadvantages. Many (though not all) Asians, as well as blacks coming from certain African countries, lead the pack in educational attainment. It's not discrimination, rationing, or obsolescence--it's culture. That's what has to change. The culture of dependence, that says "You owe me" and "I'm entitled" and "I just want to have fun" instead of "I want to contribute" and "What can I do to repay" is the problem. We need to get back the work ethic, and we need to actively teach it to our children. Otherwise, there will be no progress, no matter how much money Bill Gates or anyone else pumps into the system.