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I've often complained that Ben's Prentice-Hall Mathematics Course 1 textbook, apart from being a basically decent text, is too full of pointless, distracting colors and pictures and charts. I wrote in the comments in this post that I suspected someone was actually counting the numbers of pictures of Asian and black and Hispanic and white and disabled kids to ensure they were all roughly equal (although, at the risk of being tasteless, I note that there are no facially deformed or cerebral palsied or even blind kids' pictures among the disabled kids; just perfectly typical-looking smiley kids on crutches and sitting in wheelchairs).

JdFisher, who hosts MathAndText, put up this post today that confirms my sense that this head-count is something textbook publishers pay a lot of attention to.

Why the distracting pictures of things that have no connection to the text, not even anything as tenuous as a link to an irrelevant aspect of the word problems in the text?

Why do we have to have a head count of photos of kids -- why not skip the photos completely?

J.D. writes:

The desire on the part of publishers to include images and "real-world connections" is so strong that publishers will face all kinds of headaches to put them in their books. Teachers and administrators want to motivate their students to approach mathematics, and publishers, to compete in today's textbook market, need to try to help teachers and administrators do this. These are the reasons for what some would call glitz.

It sounds as though the source of this problem is a misunderstanding, on the part of teachers and administrators, of what makes a successful textbook for the kids (the publishers, of course, know that from their perspective, what makes a successful text is whatever causes consumers to buy it).

Do they really suppose that kids come into math class with their little eyes glowing because their texts have 5 colors of text, and totally unrelated pictures of Amish people having communal barn-raising?

I think it's more likely that they are a huge source of distraction -- and kids from all walks of life are a lot more distractable when they're finding something difficult already.

Better a very clean presentation, I think, than a busy one; and that goes for all sorts of textbooks, not just math.

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This textbook is our effort to construct a mathematics course for elementary teachers that incorporates Liping Ma's insights about effective teacher knowledge. It is a mathematics book designed to set prospective teachers on the road to developing what Ma termed a "Profound Understanding of Fundamental Mathematics.''

So, for someone like Catherine, who was fascinated by Liping Ma's book and trying to attain a deep understanding of fundamental mathematics for the first time as an adult, P&B fills a real vacuum. It did for me as well, because I was shamed by Liping Ma's book. It led me to realize that, in spite of all my many years of studying math, elementary math teaching offered conceptual and pedagogical challenges that I couldn't measure up to.

And I've taught more than my share of elementary math. I taught huge chunks of elementary math at LSU, where Scott Baldridge now teaches. It was very disconcerting to read Liping Ma and realize that elementary math teaching was a calling, a craft, that I hadn't been sufficiently aware of or respectful of. I didn't make the errors that Liping Ma's American teachers made, but when I ran out of teaching methods and understanding and patience for students who were struggling, her Chinese teachers were just getting warmed up. They could go the distance with the deepest questions and misunderstandings that their students could have.

Take, for example, something Catherine complains of frequently; that she hasn't a visual image to go with dividing a fraction by a fraction. She hasn't a visual intuition of why multiplying the numerator by the reciprocal of the denominator is the right thing to do, and it's been driving her nuts.

I don't have a visual image to go with dividing a fraction by a fraction, either; but I don't have any problem with that, or at least I never did before. I just applied the rule, and moved on.

And once, if I'd had Catherine in my math class, I would have told her to do the same.

But if you've got someone like Catherine -- the ideal student, who knows what she's doesn't know and demands to be taught it -- or if you have a child that you care about and want to help, then you have to be intellectually honest, set aside your ego (if you're me), and build up your own deepest levels of understanding in preparation. This is what P&B can help you to do.

More on this topic -- and the answer to Catherine's conceptualization problem -- tomorrow.

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Parker and Baldridge introduce the topic beautifully, emphasizing for their teacher-students the concepts that are most likely to trip up students. Kids tend to be stunned by the terms 'numerator' and 'denominator' at first, so P&B suggests that teachers eschew it at first, substituting 'top' and 'bottom'. Kids tend to think of a fraction as somehow representing two independent numbers, and so fail to see that a fraction is a single number. This results in the common misconception/error:

a/b + c/d = (a+b)/(c+d).

P&B encourage the teacher-student to think of a denominator, instead, as representing the fractional unit into which the whole has been divided. In fact, it can be useful to think of different denominators as being like different units entirely. It doesn't make sense to add quantities represented in feet and meters without doing a conversion to a common unit first; similarly, fractions must be converted to a common unit (i.e., denominator) before they can be added. The Singapore-style line drawings that illustrate this conversion in P&B are well-done and get the point across clearly.

One tidbit that I got out of P&B was an answer to Catherine's question: why is multiplying the numerator by the reciprocal when dividing fractions the right thing to do?

The easiest fraction-division case to understand, they point out, is the case where both the numerator and denominator have the denominator in common, as in the problem:

6/2 3/2.

In this case, one should envision a bar representing 1/2; if we have 6 of these bars, and divide them into groups of 3 such bars, clearly we will have two groups. But whether a bar represents 1/2 or a whole is irrelevant, as long as the bars represent the same quantity; and so this is really the same problem as

6 3.

Therefore, fraction division is pretty easy to understand if you have common denominators. But, of course, any two fractions can be converted to have common denominators. So now look at the general problem:

a/b c/d.

To figure out what we get, we convert both fractions to have a common denominator, b x d:

(ad)/(bd) (cb)/(db).

Now we have a common denominator bd, and so this problem is equivalent to the problem of dividing their numerators::

(ad) (cb) = ad/bc.

But note that this is exactly the usual multiplication-by-the-reciprocal formula:

a/b x d/c.

And that is why multiplying by the reciprocal is the right thing to do (incidentally, in a comment on this thread, J.D. Fisher mentioned that the key to understanding practically everything about fractions is to think about fractions that have a common denominator. Once again, he turns out to have been right).

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In a recent article, Wall Street Journal said that online mathematics education is the next phase of outsourcing in the US.

Two things have led to this unique outsourcing: One, US students are doing badly in mathematics. American 15-year-olds ranked 24th among 29 industrialised countries in a study of mathematics skills released last year by the Organization for Economic Cooperation and Development, the Journal notes.

Also, the country is facing a teacher shortage, particularly in mathematics and sciences. Nearly 40 per cent of US high schools reported difficulty filling openings this year with qualified instructors for mathematics, according to the American Association of Employment in Education.

"Into the breach steps a handful of Indian companies like Career Launcher India Ltd. which provide math tutoring through two US online tutoring companies and directly to students," the article in Wall Street Journal says.

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Here's what little I know -- I started reading Education Reform as Economic Reform, by Evan Osborne, and was underwhelmed by his argument that the source of education problems in America is in the publlic schools -- specifically in their publicness. That's a big claim, and he's too eager to make it and move on to what really interests him.

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I'm at my mom's house visiting, but thought I'd drop in the link to Caroline Hoxby's Education Next article on schools & competition, Rising Tide

I love Hoxby's 'found experiments'; if I'd known you could do found experiments I wouldn't have left the fields of social & cognitive psychology way back when. (I don't think.)

My research shows that metropolitan areas with maximum interdistrict choice elicit consistently higher test scores than do areas with zero interdistrict choice. The 8th grade reading scores of students in highly competitive areas are 3.8 national percentile points higher than those of students in areas with no competition; their 10th grade math scores are 3.1 national percentile points higher; and their 12th grade reading scores are 5.8 national percentile points higher. Moreover, highly competitive districts spend 7.6 percent less than do districts with no competition. In other words, interdistrict competition appears to raise performance while lowering costs—the result predicted by market enthusiasts.

My comparison showed that all schools perform better in areas where there is vigorous competition among public and private schools. Areas with many low-cost private school choices score 2.7 national percentile points higher in 8th grade reading; 2.5 national percentile points higher in 8th grade math; 3.4 national percentile points higher in 12th grade reading; and 3.7 national percentile points higher in 12th grade math.

In short, both traditional forms of choice—choice among school districts and between public and private schools—influence public schools in a positive manner. To place the influence of competition on school performance in perspective, if every school in the nation were to face a high level of competition both from other districts and from private schools, the productivity of America’s schools, in terms of students’ level of learning at a given level of spending, would be 28 percent higher than it is now. And that is with a relatively diluted form of competition; traditional forms of choice do not provide strong competition because money does not follow students in a direct way. Furthermore, traditional forms of choice are not available to many families, either because they live in an uncompetitive area or because they are too poor to move to another district or pay private school tuition.

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NCLB (the No Child Left Behind Act) was implemented in 2001, and is an ambitious bit of legislation to ensure that every school child will be proficient in reading and math by the year 2014. Hoxby actually gives a nice summary of it:

A core principle of NCLB is that every student must reach the desired level of performance: no group of students -- minority, disabled, poor, limited English proficient, mobile -- should be left behind.

Another core principle of NCLB is that every child is capable of attaining proficiency, defined in an appropriate way. Thus, while progress is important, NCLB deliberately emphasizes reaching proficiency, not making gains each year, regardless of past performance. NCLB provides no special recognition to students or schools that exceed the minimum. This is not a good thing or a bad thing, but it clearly demonstrates that the focus of NCLB is on bringing low-achieving students to a sound level of academic achievement.

A third principle of NCLB is that it works through the states, long the workhorses of the country's education system. States and localities provide more than 90 percent of funding for schools, so it makes sense for them to exercise control. Furthermore, with fewer schools to watch, states are in a much better position than the federal government to monitor multiple targets. Thus, even though NCLB monitors only proficiency, it encourages states, in their own accountability systems, to reward schools that make gains along the entire spectrum of achievement.

NCLB doesn't offer answers to the tough questions about the problem with American education: it just requires that schools improve, or suffer the consequences. That's a good thing. There is no roadmap for improvement in NCLB, because noone has one. There is a requirement for standardized assessment, which I consider a positive step -- although I think that high-stakes testing has to be handled very carefully in order to ensure that the incentives they create are the ones that we want people to respond to. I'm not concerned about time spent 'teaching to a test', which is a huge complaint of educators -- I have the feeling that in many places, teaching to a well-designed test might actually be a good thing, ensuring a base performance level, and some degree of consistency of curriculum from school to school. If the stakes are high enough, especially for an individual teacher or a school, the temptation to cheat will be there.

I think that NCLB has many elements of what's needed to improve our schools. But I have my doubts that it can succeed in its goals as it currently is implemented, and I'm afraid a big failure to 'get there' by 2014 will do a lot of harm.

One big problem is that absolute "No child left behind" language. Everybody who knows anything about quality assurance knows that there are always failures in manufacturing or software production -- the objective in QA is to drive the failure rate arbitrarily close to zero, not to set a deadline by which perfection will be achieved. Demands for absolute perfection are impossible to meet, and everyone knows it; so schools with big problems will not be thinking of ways to improve -- they'll know the goal is impossible, and they'll be looking for an out from the very beginning.

And here's their out: there is a world of trouble in the phrase "every child is capable of attaining proficiency, defined in an appropriate way." Anyone with a child who has special needs, and is involved in advocating for their child in the highly individualized and labor-intensive process dictated by the IDEA (individuals with disabilities education act), knows that the potential of a child is something that everyone assesses differently. I think that the 'defined in an appropriate way' phrase is going to be used as a way around the requirement of proficiency for individual children, plain and simple. The most likely scenario is that more low-performing children will be identified with special needs, so that 'appropriate levels of proficiency' can be defined for each of these children without harming a school's score. The wording about requiring each child to make 'adequate yearly progress' reminds me strongly of similar language I hear all the time in special education.

The poor academic performance of American students as a whole is something of a mystery. Are our expectations too low? Are our math curricula too boring and tedious, or too touchy-feely? Are our teachers too incompetent, are they undertrained, are they overtrained in the wrong areas, are we spending too little or too much money, are we expecting too little of minority students?

One good thing about NCLB is that it doesn't try to find the answers to these questions; it lets the states try to do that for themselves. Accountability is a big step forward - even if the metrics for success need retooling.

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University Hill Elementary is known for its bilingual program, where all students learn in both English and Spanish and diversity is celebrated.

The school also has a unique approach to physical education that incorporates circus components, a book press that allows students to bind their own books and special dedication to teacher training.

But University Hill also has a less-positive descriptor -- a low-performing label tacked to the school by state and federal accountability systems.

Supporters say the lower-than-average test scores are a reflection not of the school's quality, but of the fact that more than half of its students are low-income and learning English as a second language.

It sounds as though NCLB is functioning exactly as intended in this case. Quality school or not, those low-income immigrant kids were apparently getting left behind, and its 'supporters' are arguing that it's to be expected because they are low-income immigrant kids. (Yuck).

The bilingual-school issue is a minefield for Colorado politics. Two or so years ago there was a state-wide referendum on whether to eliminate bilingual public schooling for ESL students, in favor of English immersion, and it caused a huge uproar (and flopped when it came to the voters). Uni Hill is actually a 'focus', or choice school; every student who attends that school was open-enrolled there by a parent who cared enough to go through the rather painful open-enrollment process. Many of them were immigrant parents who apparently thought their kids would perform better, or at least have an easier time, in a bilingual environment.

Hardly any of those parents have taken their kids out of the school in response to its continuing poor performance on Colorado's state assessments. The other option open to parents is to take advantage of the tutoring that the school district is required to supply to kids in its low-performing groups, but only 18 percent of eligible students are doing so, and there aren't enough bilingual tutors in town to satisfy demand anyway.

But I'm starting to think that, even with all of its problems, NCLB might have the potential to take us where we need to go. Learning, perhaps, from Colorado's mistakes, it avoids the worst political problems by not going head-to-head with educational choice. Parents who want to consider sending their kids to Uni Hill -- for whatever reason; whether they truly want a bilingual education for their kids, or are simply seeking an easier path through school for their Spanish-speaking kids -- will at least know what they might be giving up as a consequence.

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I appreciate the attraction of libertarianism -- hey, I read Ayn Rand in college like everyone else -- but a lot of what's in these articles is absolutely crazy. I'm not going to dwell on that, but rather to try to point out a couple of articles that I think are worth the time to read.

This article on charter schools, by Lisa Snell, contains some good information on the growth of the charter school movement, and its likely continued growth under NCLB.

While deploring (at windy length) the fact that charter schools are not true free market entities, she grants that they have a real advantage in terms of political feasibility (snort) and offers some ideas for improving their lot.

She points out that one problem with the growth of charters is that the authorization mechanism in most places is through the local school district, their chief competitors. I can vouch for this (no pun intended). For example, here in my own school district, after the wild success of a couple of specialist charter schools, the administration has made a ruling that there shall be no new charter school applications, period. It's a nice position to be in, if you want to quell your competition.

Snell also discusses the growth in opportunity for for-profit education organizations as a result of NCLB.

The Education Industry Association (EIA), which is an 800-member professional organziation for education entrepreneurs, has recently relocated to Washington, D.C., and appears to have substantially increased their lobbying efforts with the U.S. Department of Education and Congress. In addition, much of the EIA's growth has been in segments supported by the No Child Left Behind Act, especially testing and assessment and private tutoring companies. Similarly, a new education trade newsletter, the School Improvement Industry Weekly, bills itself as a "Web-enabled newsletter for the marketplace created by No Child Left Behind".

Education Research Flounders in the Absence of Competition from For-Profit Schools, by Myron Lieberman, is a great read, going a long way toward answering one of Catherine's all-time favorite questions: Why does educational research stink?

Research on the most effective way to teach reading is another example. Conservatives advocate phonics, a method that emphasizes sounding out the sounds from the letters in words.A significant number of teachers and researchers emphasize what is widely referred to as the whole language approach, in which the emphasis is on the meaning of words. .. These definitions of the two camps are oversimplifications, but my purpose here is not to resolve the dispute but to cite it as an illustration of the primitive level of education research. (Italics mine)

Suppose that research establishes that procedure X is the best way to teach reading to pupils in category Y. What reason is there to believe that the teachers of pupils in category Y will read the research on their own, or have it brought to their attention by their principals, school boards, or superintendents? What new incentives will overcome the intertia that underlies the current failure to read and utilize research?

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The laundry list is good, but the comments are fascinating.

Here's a rather heart-rending comment, number 24:

As a high school math teacher, i can assure you that each of the prerequisites you propose is taught ( or, at least, presented ) to every student, college bound or not, in my district starting in the eighth grade. I teach the same material to ninth, tenth, eleventh and twelfth graders when I'm not trying to teach them material they should have learned in grade school.

My district increased its math requirement this year from two years to three. So now, when a student shows up in a college math course, she will have had three years of math beyond the elementary prerequisites you have proposed.

And still... you will find your classes include students without the least of skills.

On a more positive note, here is comment 37:

As a high school math teacher, I have to say that your list is very good. In fact it is covered in the exam that all of our students must pass in the state of Texas in order to graduate.

The only way that colleges can be sure that high schools are teaching what needs to be taught is to hold kids accountable. And that means standardized tests.

One of the good things from the No Child Left Behind initiative here in the US is the requirement that all states have in place an assessment to show whether the kids have mastered what needs to be mastered. The test we give is given in 4 separate parts for each of the 4 content areas, and if the students don't pass all 4, they don't get a diploma! The math test covers the curriculum of Algebra I and Geometry - the minimum courses that all high school students should take. In order to receive a college bound diploma, the student must have also taken Algebra II.

As soon as these tests are implemented nationwide, I think the colleges will see a student body that is a lot better prepared.

(Unfortunately, commenter 37 then goes off on a rant about how colleges need to move into the 21st century and realize that calculators are here to stay).

And then, someone named Charles Williams hits the nail flat on the head. Many people will never need algebra, and other people will need a lot more, in order to work in technical fields. The problem is that we don't know which 11-year-olds are which, and we don't have the national stomach to try to separate them at that age; nor should we. Here's his comment:

Perhaps 80% of our students will not use algebra in their careers. Nonetheless, a student who wants to master a technical field must be in a rigorous math program from the middle school on. The minute we give up teaching a middle school student fractions and hand him a calculator so he can do computations, technical careers are no longer an option for him. Now do we have the will to track college prep students starting in the 6th grade so that the others will not be burdened with the unpleasantries of algebra? Or will we insist that all students graduate with what they need to succeed at college in technical fields. Unwilling to face facts and make hard choices, we muddle through in a dishonest way. We prepare all students for college on paper but not in reality. We do de facto tracking while denying that this is happening. We require more and more high level courses from our students and then water them down.

I can't tell whether he is implying that we ought to be making hard decisions, and cutting kids out of possible future choices in careers. I hope not; I don't think that's necessary (as the successes of other countries show). But if we are going to force unwilling kids to do the work that's really required to succeed in college math, we should be pushing those kids all the way, and recognizing that we can expect a lot of pushback from them.

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I wish summer would last forever. Or at least another couple years.

Robert Samuelson has a column out today on the science gap, which he says isn't a science gap, yet. I find his conclusion a bit hard to follow, but his set-up is clear enough:

As late as 1975, the United States graduated more engineering and scientific PhDs than Europe and more than three times as many as all of Asia, reports Harvard University economist Richard Freeman in a recent paper. No more. The European Union now graduates about 50 percent more, and Asia is slightly ahead of us. By Freeman's estimates, China has reached almost half the U.S. total and will easily overtake us by 2010. Among engineers with bachelor's degrees, the gaps are already huge. In 2001 China graduated 220,000 engineers, against about 60,000 for the United States, the National Science Foundation reports.

Freeman also documents a second worrisome reality: U.S. scientists and engineers aren't well paid, considering their skills and -- especially for PhDs -- the required time for a degree. This means, Freeman says, that "the job market . . . is too weak to attract increasing numbers of U.S. students." Consider some pay comparisons. From 1990 to 2000, average incomes for engineering PhDs increased from $65,000 to $91,000, up 41 percent; PhDs in natural sciences (physics, chemistry) rose from $56,000 to $73,000, up 30 percent. Meanwhile, average doctors' incomes increased from $99,000 to $156,000, up 58 percent; and lawyers went from $77,000 to $115,000, up 49 percent.

The true situation may be worse. Next to other elites, scientific and engineering PhDs fare poorly. Look at the 891 MBA recipients of the Harvard Business School's class of 2005. At an average age of 27, they command a median starting salary of $100,000. It's true that the two-year cost of a Harvard MBA is steep ($120,000 and up), and four-fifths of the students are left with debts averaging $81,000. But these new Harvard MBAs also got huge one-time bonuses; the median was $43,000. As for scientific and engineering PhDs, they typically require seven to eight years to finish their degrees, notes Freeman.

I'm inclined to agree, since I have yet to see catastrophic predictions pan out, which is not to say bad things don't happen, but that when bad things do happen they're usually different from the bad things everyone was braced for.

This brings up two of my favorite sayings, the first one being:

It doesn't pay to worry, because the worries you have are never the worries you get.