$133/Month Health Insurance Premiums?

The Truth-O-Meter found that Keith Olbermann had told a “half-truth.”

Olbermann said that for middle-class families, the Baucus plan would mean that "13 percent of what they make could be deducted directly from their paychecks and mainlined to insurance companies, the so-called 'Max Tax.'"

He's right that for people who are uninsured now, the upper limit would be 13 percent, and that money would go to insurance companies. But it's to pay for coverage they don't have now — not a tax — and some people would pay less. And all of the plans under consideration in Congress require people to pay something for coverage. So we rate Olbermann's statement Half True.

The last time I had health insurance, the premiums were 6.9% of my salary. According to Ezra Klein, I had a pretty good deal.

The average health-care coverage for the average family now costs $13,375, according to Kaiser.

In my community, the median income is $42,000. $13,375 out of $42,000 comes to nearly 32% making my 6.9% look wonderful.

And that sums up the problem with health insurance reform. For all the whining and moaning, most people are not feeling much pain. It is like college students complaining about the increase in tuition when what they (some of them, anyway) really mean is that they will not be able to afford as much beer.

Imagine if people who touched a hot stove felt only a small fraction of the pain from the burn. That is pretty much what is happening in our health-care system. It hurts enough that we would prefer it to stop, but without any sense of urgency.

Washington wonks trying to fix this mess face a dilemma: They look at the numbers and see health-care costs crushing our economy, overwhelming our government, swallowing our wages. But the public is not feeling the pain. Virtually no one writes a $13,375 check to pay for their own health care. Most pay 27 percent of it, or even less. The surest way to cut health-care spending would be to make people shoulder more of the burden directly, as opposed to hiding it in taxes and lost wages.

The surest way to cut health-care spending would be to make people shoulder more of the burden directly, as opposed to hiding it in taxes and lost wages.

So things were going okay for me. I thought my premiums were too expensive, sure, but I was managing. Then I got laid off. That is when the premiums became unaffordable, but not simply because I was laid off. After all, the unemployment check came every week like clockwork.

Getting laid off meant I qualified for COBRA. Going on COBRA meant I would have to start paying the both my half and the half the employer had been paying, thus doubling the out-of-pocket percentage to 13.8%. Worse, I would have to pay the premiums out of unemployment benefits, making the percentage a whopping 43.3%. Naturally I declined and joined the ranks of the uninsured. The purpose of COBRA is to help the recently unemployed avoid the sudden loss of health insurance, but unaffordable COBRA is no help at all. I wound up shouldering not just more of the burden, but all of the burden, and I could not do it.

Compared to the 32% I cited earlier, the 13% in the Baucus plan sounds like a good deal. Whether the 13% is taken out of my pay automatically or I write the check to the insurance company myself makes no real difference to my bottom line. If Keith Olbermann wants to call it a “tax,” whatever. But when I consider the Japanese health insurance tax/premium, 13% looks pretty awful.

The average consumer is not feeling enough pain, and powerful stakeholders do not want change.

Of course, providers don't much like the sound of (the public option) because they would see 20 to 30 percent less revenue. And insurers don't much like the sound of that because they could not compete with that sort of buying power. Republicans and centrist Democrats have banded together to weaken the public plan and maybe even remove it altogether. President Obama now promises that the public plan would be open only to the uninsured and wouldn't offer any advantages over private insurers. It won't, in other words, be allowed to save people money.

So if you take sweeping transformative change off the table, you are left with finding a multitude of little tweaks. All these little tweaks together require 1000 pages of legislation.

Added up, they equaled a startling $2 trillion over 10 years. That's a lot of money for policies that have received virtually no attention in the debate.
And yet, this is the quiet promise of health-care reform. The grand theories might fail. They often do. But making the system a bit better, a bit quicker and a bit more agile -- we can do that. And until the stove gets hot enough, it may be all we can do.

But let's get back to the Japanese health care system. I lived there for nearly twenty years. Japanese people have choices. They can self pay (as I did at first), get private insurance (the next thing I did), or participate in one of the two public options (which I did as soon as I qualified). My premiums were about 4% of my salary.

According to the Washington Post, premiums are still about 4% of wages.

Workers at major corporations pay about 4 percent of their salary to a company-based insurance provider. These premiums are limited to $6,000 a year, but the average salary worker pays $1,931, the government says. Job-based insurance in the United States costs the typical employee $3,354 a year, according to the U.S.-based National Coalition on Health Care.

In Japan, employers pay premiums that match each employee's contribution. In the United States, where health insurance is far more expensive, employers pay private insurers three or four times the amount contributed by each employee.

The self-employed and the unemployed in Japan must pay about $1,600 a year for insurance coverage.

Wouldn't you like to pay $133/month for health insurance? NPR reported on April 14, 2008 that average Japanese family paid $280 per month. The insurance covers 70% of the bill; the patient pays 30% out of pocket. Although a 30/70 sounds heavier than the 20/80 split common in American policies, patients happily pay 30% of a far smaller total bill. In the case of childbirth, the local city hall reimburses families that 30%.

The total rate for Japanese income tax is less than 20% (about 15% for the sum of national, prefectural, and municipal taxes plus the 4% for health insurance) which compares favorably to American tax rates. Think of it. For approximately the same tax burden Americans bear without health care premiums, the Japanese have their health care premiums included.

The government of Japan does not stand between the doctor and the patient in any way. In fact, there is no feeling of government presence at all. Patients go to the doctor, the doctor orders treatment, patients pay their 30%, and go home all done with it. The closest thing to government intrusion is the government's requirement that doctors and dentists participate in the government preventative health program by conducting free physical and dental screenings once a year in every elementary school, kindergarten and preschool. The government also sends new parents reminder notices for free well-baby clinics. That is the extent of government “interference.” The government never contradicts the doctor's order or refuses to pay its portion for services or prescriptions like so many American private insurers do.

Although most Japanese health care facilities are privately owned, there are some public facilities. Doctors usually do not work for the government. They may be sole proprietors of their own hospital which may be quite small, sometimes with as few as four beds. My maternity doctor had one of those little four-bed hospitals, several nurses and a cook. His patients gave birth in the same familiar environment where they received prenatal care.

While many of the large hospitals offer comprehensive care, some hospitals are specialized. For example, a patient with heart disease may go to cardiac hospital. I saw no dedicated out-patient facilities; every facility had the ability to take admissions (except optometrists and dentists). Wait times for service are about the same as in America. Doctors may also work for large hospitals. Patients may go to any doctor or specialist they choose anywhere in the country. A patient's own residence is of no consequence. There is no such thing as a preferred provider network. Thus there is far more consumer choice in Japan than with America's HMOs and PPOs.

Hospital are generally full service, at least from the patient's point of view. Patients do not go to a separate facility for labs or x-rays. If a doctor sent, for example, a biopsy sample to a pathology lab, the patient would never know it. There is no subcontracting of health services generating bills from all sorts of random providers. In the example of the pathology sample, the pathology lab bills the doctor directly who bills the patient. In the case of x-rays, there is no bill for taking the x-ray separate from a bill for reading the x-ray. If a hospital wants a second party to read an x-ray, the hospital pays for it, not the patient. It is crazy that an American hospital signs a contract with an outside radiologist committing a third party, the patient, to paying the contract. If a patient goes to a Japanese emergency room, the patient pays one bill, not multiple bills. Ditto for surgeries.

What is Japan's secret? According to Naoki Ikegami, probably Japan's top health economist, at least one secret is that Japan does not have a single payer system. It has a single payment system.

And the way the government controls the flow of money is that we have multiple payers and multiple providers, but there's a single-payment system -- not a single payer but a single-payment system -- so that all payers must abide by the payment system, and all providers must be paid by the system.

The rest of the interview is quite instructive.

For a pretty comprehensive set of viewpoints see this article and the several pages of comments.

It is strange that Americans are not as knowledgeable about the Japanese system as we think we are about the Canadian system. We should be seeing information about the Japanese health care system everywhere. We should be rejecting American exceptionalism and seriously considering how to incorporate the best features of the Japanese system while rejecting the worst aspects.

What's an Argument? Bring Back Debate Club

When I was in high school, there was an active debate club that competed with all the area high schools. If successful, our team could compete at regional, state and even national events. At the time, it was one of many extracurricular opportunities whose presence I took for granted. Now the arts are gone, recess is replaced by test prep drills, and most extracurricular activities vanish if candy bar or gift wrap sales disappoint. Debate club was one of the first to go by the wayside.

So it is no wonder that recent research on the place of argument in the school curriculum shows that students cannot even define the term, much less formulate an opinion about it. A whole generation of students has grown up thinking “argument” means nothing more than a verbal quarrel, either to be avoided or won, usually by voice volume rather than persuasive points.

Not only are the students clueless, but according to Gerald Graff, so are their teachers.

Graff: I think cluelessness in academia is a major threat to democracy, especially at a moment when talk-back radio, Cable TV talk shows, the Internet, and the reliance of politicians on opinion-polling have made a certain kind of public debate—even if it’s debate within narrowly constrained parameters—more immediately important in American and global politics. In these conditions, one needs not only an ‘informed’ citizenry, but a citizenry that’s sophisticated enough in weighing arguments to spot logical contradictions and non-sequiturs, not to mention outright lies.

Students are not getting an education in argument, and do not miss what they never had. Schools have abandoned critical thinking while the proliferation of so-called critical thinking materials notwithstanding.

Graff: You’re right that many students don’t miss an initiation into the intellectual world of whose very existence they never even learn. No, I don’t see as much concern within academia over this problem as I think there should be. I think we’ve gotten accustomed to a system in which the very few excel in school (and reap the rewards in the vocational world beyond) and the many stumble along and more or less get by, or get through, or fail.

Graff's reason for the apathy is a classic example of the sort of educational obstructionism that pervades our society. We only need a few well-educated elites. There are not enough slots for everyone else.

In some ways such a system suits us academics—it’s not our fault if the majority stumble or fail, we can easily say, that’s just the way it is; only an elite in any society is going to ‘get’ the intellectual club, etc.

Who does Graff blame? Not the parents. The popular weasel strategy of blaming the parents conveniently forgets the fact that parents have to be educated first.

Insofar as this is a common academic attitude, I blame academics more than parents, whom it’s also our job to educate, after all.

Somehow most of our college students are missing out on what should be the goal of their education—becoming a well-informed, thinking citizenry.

...at best I was reaching 15-20 percent of the students in an average undergraduate class and that the remaining 80-85 percent were in some other country or time-zone. Comparing notes with colleagues over the years led me to conclude that most felt the same way. Some unashamedly said they teach to that top 15-20 percent and figure there’s no point worrying about the others.

I have had the same experience. Whereas I was able to persuade just about every member of my junior high and high school classes to buy into my program and reach previously unattainable levels of achievement, I found my college classes to be far more resistant. Some of my college students said straight out, “We don't want to know how or why the math works. Just tell us how to get the answer.” Combine the students' attitude with the new consumer, so-called student-centered approach to education, and there we have a recipe for a generation of ill-educated college graduates.

Professors who refuse to bow to student demands and try to educate the unwilling regardless of their resistance risk unfavorable student evaluations. Professors who respond to market conditions fare much better, and thereby effectively “purchase” great student evaluations. Twenty years down the road, I wonder how many students of the first professor will be grateful for the gift and sorry for their evaluations, and how many students of the second professor will feel they have been royally gypped.

There is a reason, grasshoppers, why I have the big desk and you have the little ones.

Can you imagine a college class where thew books of Ann Coulter Michael Moore are required reading? Dr. Graff can.

I can imagine a good course in which students would read Coulter and Moore for starters and then move on to more nuanced and complicated texts on the same set of issues.

Graff would like a chance to test his ideas.

Take five sections of freshman composition at a university and teach them using the ‘argument templates’ discussed extensively in Clueless and other methods for demystifying academic culture. Closely monitor the writing done by the students over the course of the semester or year, and compare their work with that produced by a randomly-chosen control group of a different five sections of the same course. I like to think the results would dramatically bear out my claims. If they didn’t it would be back to the drawing-board for me.

Yet he observes sadly that universities seem unconcerned about product beyond job training.

But it’s symptomatic of the incuriosity of higher education about what students actually get out of college that one never hears about such experiments even being tried.

All Politics---and Education—is Local. Well, Maybe

According to a commentary in the September 10, 2009 online issue of EdWeek, U.S. Rep. John Kline (R-MN), the senior Republican on the House Education and Labor Committee, the public outcry over the President's speech to students was not about indoctrinating our children with socialist philosophy. Mr. Kline believes a more fundamental issue was at stake. More fundamental than socialism? What the president learned this week was that all education, just like all politics, is local.

President Obama delivered a positive, uplifting message to students this week. The fact that many Americans were rubbed the wrong way by what amounted to a federal recommendation about what to teach, and how to teach it, is a signal that no matter how well-intentioned, education reforms simply cannot be dictated from Washington

In Japan, where the national Ministry of Education dictates all sorts of education policies, even in Japan, the nagging and unspoken noise that education is primarily a local issue constantly plays in the background. It is all too true that if the Ministry of Education is for it, then the Japanese teachers union is against it. Japanese teachers spend enormous amounts of time discussing, demonstrating and otherwise agitating against Ministry of Education policies.

If socialist-leaning, group-oriented Japan with its highly centralized education system where every class in the country in any given grade is on the same page of the textbook on any given day finds centrally-dictated policy so onerous, of course, those highly individualistic Americans would be in near revolt over the prospect of education reform dictated from on high.

Mr. Kline suggests the response to the president's speech was symbolic of growing powerless frustration.

...many local communities are growing frustrated as they perceive a more active, intrusive federal government making more and more decisions about how their children are taught.

Maybe you were thinking No Child Left Behind is Exhibit A of an “active, intrusive federal government.” For Mr. Kline, Exhibit A would be the lesson plans the White House provided as a supplement to the speech.

...the nagging fact remains that the federal government—not local teachers or school boards—developed a very specific lesson plan for implementation in classrooms all across the country.

Except... The federal government, as well as many other governmental bodies, have been providing lesson plans for years and years without public protest. One comment to the citation lists a few:

A short list of federal government developed lesson plans:

The National Archives Lesson Plans
http://www.archives.gov/education/lessons/

National Institute of Health Curriculum Supplements
http://science.education.nih.gov/customers.nsf/WebPages/CSHome

U.S. Department of Agriculture Lesson Plans
http://soils.usda.gov/education/resources/K_12/lessons/

National Park Service Lesson Plans
http://www.nps.gov/nr/twhp/descrip.htm

U.S. Department of Energy Lesson Plans
http://www1.eere.energy.gov/education/lessonplans/

U.S. Department of State Lesson Plans
http://future.state.gov/educators/lessons/

U.S. Nuclear Regulatory Commission Teacher Lesson Plans
http://www.nrc.gov/reading-rm/basic-ref/teachers.html

NASA for Educators
http://www.nasa.gov/audience/foreducators/index.html

FBI for Kindergarten - 5th Grade
http://www.fbi.gov/fbikids.htm

CIA for Parents and Teachers
https://www.cia.gov/kids-page/parents-teachers/index.html

All kinds of entities provide optional lesson plans. It might be as laudable as a public utility promoting energy conservation. It might be construed as inappropriate commercialism exploiting impressionable (and captive) children. All of these publicly available lesson plans are optional. I agree with Mr. Kline that “something else is at play.” However, that something else is not fear of indoctrination, and it is not the optional lesson plans.

Related Post at School Crossing,When a President Speaks:6 Reasons to Object to Objectors.

Place Value Part 5: Applications of Place Value

So far we have completed four parts of the place value series:

Part 1: The Chocolate Factory which covered the regrouping or trading aspect of place value and explored regrouping in base ten and other bases.

Part 2: Base Ten for Young Students which introduced several games and trading activities to help young children acquire a solid foundation in place value.

Part 3: The Bake Sale demonstrated the role of place value in long division.

Part 4: Geometry of Place Value demonstrated the dimensions of a cube in terms of place value and explored the geometric representation of the quadratic equation.

In the conclusion of the series, Part 5, Applications of Place Value, I will show you a sampling of interesting applications of place value to some real-life situations.

Fractions

It is sometimes easy to add and subtract fractions by putting fractions in place-value-type columns. The usual addition and subtraction with regrouping requires students to conceptually pack or unpack “boxes” (See Part 1: The Chocolate Factory). What if I want to add 3 5/7 + 2 4/7?



The denominator, 7, means that the “whole” has been divided into seven equal pieces. Notice that instead of packing in boxes, the student creates “wholes.” With nine-sevenths, there are enough pieces to make one whole with 2 pieces or two-sevenths left over. There is that word, “leftover” again. In The Chocolate Factory, “leftover” was used to label the ones. Here we are using it to label the pieces, helping students to transfer and integrate concepts.

If students need to “borrow” (the current term is “exchange”) in order to subtract fractions, then they would have to cut up a whole into the necessary number of equal pieces. In the case of 4 2/5 – 1 3/5, the student creates two place-value columns, the place value of one being the “wholes” and the place value of the other being the “pieces.”



With fractions of unlike denominators, the procedure is essentially the same with the preliminary step of finding common denominators.

The Olympics and Time

The odometer of a car is obviously a place value representation, but have you noticed that the clock at the bottom of an Olympic race is formatted in terms of place value with colons marking the separation between hours, minutes, seconds, and hundredths of a second? As the race progresses, the clock gathers hundredths into seconds, then seconds into minutes, minutes into hours. The expression of a runner's time might be 2:14:52:27.
Thinking of time in terms of place value columns simplifies addition and subtraction.

Here's a (sort of) real life problem that occurred in my house recently. My son was invited to watch Lord of the Rings at a friend's house. If the DVD starts playing at 4:17:48 pm and the movie lasts 3 hours 44 minutes and 25 seconds, will my son be able to catch the 8:15 bus home?



Starting with the seconds column, 48 seconds plus 25 seconds equals 73 seconds which is 60 plus 13 seconds. Since 60 seconds equals one minute, cross off the “60” and carry the one to the top of the minutes column. One minute plus 17 minutes plus 44 minutes equals 62 minutes which is 60 plus 2 minutes. Since 60 minutes equals one hour, cross off the “60” and carry the one to the top of the hours column. One plus 4 plus 3 equals 8. The solution: The DVD will end at 8:02:13 pm. The answer: The DVD will end at 8:03 pm, in time to make the bus. “Packing” seconds into minutes and minutes into hours is just like packing chocolates into boxes and boxes into cases.

Subtraction is a matter of unpacking then. Suppose we want to work backwards. If the DVD must end by 8:10 pm to make the bus, what is that latest time it can start. The problem is 8:10:00 minus 3:44:25.



We must subtract 25 seconds from zero seconds. So we just unpack one minute or dump a minute into the seconds column. Since we unpacked one of the 10 minutes, there are 9 minutes in the minutes column and 60 seconds in the seconds column which still represents the original ten minutes but in a slightly different form. Now it is easy to subtract 25 seconds from 60 seconds. Now we have to subtract 44 minutes from 9 minutes. Simple. Just unpack one hour into minutes, leaving 7 hours and giving 69 minutes. Now it is easy to subtract the 44 minutes. The solution: the latest the DVD can start playing is 4:25:35 pm. The answer: the latest the DVD can start playing is 4:25 pm.

(An small digression: I have twice made a distinction between the solution and the answer with the movie problems. I did not make this distinction with the fraction problems. The fraction problems did not require the distinction because they were context-free problems.

The movie problems were word problems or story problems. They have context. I teach students to find a solution, then interpret the solution to find the best or most reasonable answer to the question. Home clocks are not like Olympic clocks, and rounding by the rules does not always produce the best answer. In the case of the second movie problem, rounding to 4:26 pm might cause my son to either miss the bus or run like the dickens to catch it.

In real life, people cannot stop when they have found the solution. They must then apply the solution to the situation, or context, in the most reasonable way. We are failing to teach our children critical thinking when we allow them, even encourage them to conflate solution with answer).

Base Clocks

Number base clocks are a manipulative that use the student's internalized understanding of clock time as a peg to hang the concept of bases. A base-7 clock means that a rotation of 7 “minutes” equals one base-7 “hour.” Using a variety of base clocks as aids, students can become quite adept at adding and subtracting in different bases. It is even possible to extend the skill to multiplication and division in bases as well.

Calendars

Calendars can also be used to teach in place value especially if you confine yourself to hours, days, weeks, years because the months are not uniform “packages” of days or weeks. In calendar math, the number 14 would be 1 week, 4 days. The number 305 would be 3 months, 0 weeks and 5 days.

You can use the calendar to explore interesting patterns numerically as Michael Naylor does. Although he does not specifically target place value, the algebraic pattern partly follows from the place value of a calendar.

Fast box addition (Grades 6-8)
Have a student choose a 2 x 2 box and demonstrate how you are able to quickly give the total. Tell your students the secret: Add 4 to the first number and multiply the result by 4. Have the class test the result on several boxes.

The secret is algebra; if the first number is x, the other numbers are x + 1, x + 7 and x + 8. The total is 4x + 16, which is the same as 4(x + 4).

Have your students outline a 3 x 3 box and ask them which is greater – the sum of all of the numbers or 9 times the center number? Relabel the center number as x and write all the other numbers in terms of x as shown here:

When adding all of those terms, the constants cancel (–8 + 8 = 0, –7 + 7 = 0, etc.) so all that is left 9x. The total of all nine numbers, then, is 9 times the center number.

One teacher considers the calendar as probably the one math essential for her kindergarten class even though she may not necessarily focus on place value. Her website lists activities and links.

Conclusion

What we see today is that although regular number bases maintain column values in terms of powers of the base, place value is way more flexible. Each place, or column, can be whatever it needs to be as long as it is in terms of groups of the preceding place.

To summarize:
Fractions: Wholes, parts
Olympic Time: Hours, Minutes, Seconds, Hundredths of a Second
Calendars: Years, Months, Weeks, Days
Gettysburg Time: Score, Days

You can have more fun working in multiple places with
Historical Time: Millenia, Centuries, Decades, Years, Months, Weeks, Days, Hours, Minutes, Seconds, Hundredths of a Second. That's eleven places. Once the students understand grouping and ungrouping, they do not worry about the number of places. The more places, the more fun.

Maybe you can think of some more ways we use place value everyday.

Resources

O'Block Books sells manipulatives.

So does Creative Teaching Press.

Base ten packing set from Digi-Block.

"Great Source" for calendar math.

When a President Speaks: 6 Reasons to Object to Objectors

I remember President Kennedy urging us kids to be physically fit, and the national president's fitness program that went with it. Anybody else out there earn a Presidential Fitness Award while they were in school? In fact, the program has followed us into adulthood.

Another president is planning to give a speech to school children urging fitness of another kind, educational fitness.

During this special address, the president will speak directly to the nation’s children and youth about persisting and succeeding in school. The president will challenge students to work hard, set educational goals, and take responsibility for their learning.

Why the furor?


Because of the breathtaking opposition.

Unbelievable. Maybe there is no grand tradition, but presidents have addressed remarks to schoolchildren from time to time. Ronald Reagan in 1986, George H.W. Bush in 1991, George W. Bush in 2001.

My problems with all this hullabaloo:

First, the objections are premature. It is silly to object to figments of the imagination. Wait till the president actually says something objectionable in the speech before objecting to it.

Second, the objections break the Golden Rule. In the nutshell, the right is worried that the president may voice a tenet or two of liberalism. I have trouble believing they would object to a Republican president voicing a tenet or two of conservatism. I am not letting the left off easy; they will violate the Good-for-Goose-Good-for-Gander principle when it suits them as often as the right does. As Shakespeare might say, “A pox on both their houses.”

Third, the objections are largely ad hominem. The objections are not criticizing the speech on its merits (probably because the speech has yet to be broadcast). Ad hominem is one of the defining marks of lack of critical thinking. What a lousy role model to set before our kids.

Fourth, the objections are misplaced. Edweek reports that White House efforts to quell the furor have been ineffective.

But the planned 15- to 20-minute noontime speech—and, especially, a menu of classroom activities (for younger and older students) suggested by the White House in connection with it—continued to draw denunciations...

Especially?! I looked at the “menu of classroom activities.” The White House's companion lesson ideas for elementary students and secondary students have no leading questions and emphasize strategies for comprehension. The secondary lesson plans ask students to create a specific action plan for meeting their goals. Maybe it is about time we adults directly ask students what they want, and then find specific ways to help them, instead of creating burdens for students in the name of reform.

Fifth, the objections are politically-motivated disturbance in the guise of concern for our children.

Finally, sixth, and perhaps most importantly, whatever happened to free speech?

Amendment I
Congress shall make no law respecting an establishment of religion, or prohibiting the free exercise thereof; or abridging the freedom of speech (bold added) , or of the press; or the right of the people peaceably to assemble, and to petition the government for a redress of grievances.

What have we come to when a subset of the public maintain foul language is protected speech, and a subset of the public agitate to pre-censure the freedom of the President of the United States to encourage students to study hard and stay in school? Even stranger is that both subsets very likely contain many of the same members.

The White House has a video of the President's speech here.

Place Value Part 4: Geometry of Place Value

So far we have completed three parts of the place value series:

Part 1: The Chocolate Factory which covered the regrouping or trading aspect of place value and explored regrouping in base ten and other bases.

Part 2: Base Ten for Young Students which introduced several games and trading activities to help young children acquire a solid foundation in place value.

Part 3: The Bake Sale demonstrates the role of place value in long division.

Today, Part 4: Geometry of Place Value will explore place value within a quadratic equation. We will further show that each monomial can be modeled geometrically.


Review

The expression, 5x² + 6x + 3, appeared in The Chocolate Factory, as a summary of the chocolate packing activity. Five cases and six boxes were packed with three leftover chocolates. Where x stood for the number of chocolates per box, five cases and six boxes could represent different absolute numbers of chocolates. If x =10, or 10 chocolates per box, then ((5 times 100) + (6 times 10) + 3) chocolates, or 563 chocolates came down the conveyor belt, I Love Lucy style. In fact, this episode of I Love Lucy was the inspiration for the math activity.

If the chocolates are packed in boxes of five then the 563 means ((5 times 25) + (6 times 5) + 3) or 158 chocolates came down the conveyor belt. So a quadratic equation can be thought of as an expression of place value in any base. In fact, a polynomial of any degree can be seen as an expression of place value. Missing terms are represented by zeros. So 2x⁶ + 5x⁵+ 3x² + 7x + 2 would be 2,500,372_{base x}.

Now we can see where the analogy to place value breaks down. If x = 6, then a term like 7x would be “illegal.” Once six “boxes” had been packed, those six boxes would immediately be packed into one “case,” so in base 6 the last three terms would properly be 4x² + x + 2, either way, the last three terms represent 152 “chocolates.” Obviously I have just been speaking to adults initiated into the joys of algebra, not children.

What? No Fourth Dimension

Obviously you can use standard base ten blocks to model quadratic equations. If we assume that x represents base 10, for 4x³ + 3x² + 7x + 2, we would use 4 large cubes, 3 flats, 2 rods, and 2 small cubes to model the expression. However, if all I meant by geometry was geometric solids, I would not have meant much. The geometry is more interesting, and becomes clearer when you look at a set of blocks in a different base, say, base 5, the small cube looks the same as a base ten small cube, but the rod is five cubes long, the flat is a square of 25 cubes, the large cube is 5 flats stacked or 125 cubes.

So the rod of any set of base blocks determines the base of the set. If we look at the large cube of any base, we see that any one of the 12 edges shows x¹, the first dimension, any one of the six faces shows x², the second dimension, and the whole cube shows x³, all three dimensions. Now we run smack into a physical limitation of manipulatives; not one can show more than three dimensions. The power of math is that math is the language of imagination. We can imagine a fourth dimension, x⁴ and beyond, even if we cannot model it. How fun is that?

Interestingly, we can also show x⁰ on the cube. Remember for any x, x⁰ = 1. Cubes of varying bases are all different sizes, or volume. The x¹, x², and x³ is different on each cube. But since x⁰ = 1 for any base, it stands to reason that x⁰ or 1 would have an identical appearance no matter the size of the cube. In fact, it does. You can find x⁰ at any vertex, that is to say, the corner shows 1, the zero-th dimension, if you will. In fact, the vertex is a geometric point, described as having no length, width or height.

Multiplication with Base Ten Blocks

So far we have spent a great deal of time establishing that x² means x times x, and that we can show x times x geometrically, by using a flat from a base block set. The flat has a square shape which we would expect from an expression like x-squared. But let's consider a rectangle shape. Now we are not multiplying the same number by itself, x times x, the very definition of squaring, “the product obtained when a number or quantity is multiplied by itself”.

With a rectangle, we are multiplying two different numbers, x times y (or length times width, the formula for the area of a rectangle). Using the cubes from a base blocks set, we can model 5 x 3.






Math educators call this type of diagram a multiplication array. Now lets try 13 x 11.





To show the factor 13 along the top, I used a rod and three small cubes. The factor 11 is along the side with a rod and one small cube. One rod times one rod equals one flat (square, and you expected a square, right?), one rod times three small cubes equals three rods or three lengths. Then, one small cube times one rod equals one length, and one small cube times three small cubes equals three small cubes.

Combining like terms, that is, similar objects, together, we have one flat (10² or 100), four rods ((3 times 10) + (1 times 10)) or 40, and three small cubes (1 times 3, or 3) for a total of 143 which I could express as(1 x 10²) + (4 x 10) + (3 x 1) .

What if we wanted to multiply (x+3)(x+1). I am using the magenta to stand for x, a number we don't know, also called a variable.




The product is 1x² + 4x +3, and geometrically, the product is the picture of a quadratic equation showing both its factors above and to the left of the crossbars. I recommend manipulatives that elucidate the geometry of quadratic equations, available, for example, the Montessori Binomial Cube and Creative Publications Algebra Lab Gear. Remember we have shown that the magenta rod could stand for any value, that is, for any base.

We can show three factors and therefore three dimensions with the same model by standing a rod and/or stacking small cubes vertically in the corner where the crossbars intersect. If I were to stack four small cubes in that intersection, I would be modeling (4)(x+3)(x+1) or by multiplying the x-factors first, (4)(x² + 4x +3). You could think of it as stacking four layers of the x-factor product. In fact, the formula for volume is height times base, or height layers of the base.

In terms of base blocks, the product would be modeled with 4 flats, 16 rods, and 12 small cubes. If we are working in base ten, we would have 400 + 160 + 12. We can exchange 10 of the rods for a flat, and 10 of the small cubes for a rod, ending up with 5 flats, 7 rods, and 2 small cubes or 572.

If I were to stand a rod in the intersection, I am modeling 10 layers of 143 or 1430. In the upper left hand corner of the product there would be ten flats stacked which I can exchange for a large cube worth 10x10x10. Completing any other exchanges, the product would consist of 1 large cube, 4 flats, 3 rods and 0 small cubes. If I were to stand a magenta rod in that intersection. I would be modeling (x)(x+3)(x+1). The product would have 1 magenta cube, 4 magenta flats, 3 magenta rods and 0 small cubes or x³ + 4x² + 3x.

The Candle Problem: How to Damage Motivation

Herbert Kohl says we are missing the boat, motivation wise, in an open letter to Arne Duncan, Secretary of Education.

Now the mantra is high expectations and high standards. Yet, with all that zeal to produce measurable learning outcomes we have lost sight of the essential motivations to learn that moved my students. Recently I asked a number of elementary school students what they were learning about and the reactions were consistently, “We are learning how to do good on the tests.” They did not say they were learning to read.

Mr. Kohl sees a fundamental contradiction between what we say we want and what we are doing to get it.

It is hard for me to understand how educators can claim that they are creating high standards when the substance and content of learning is reduced to the mechanical task of getting a correct answer on a manufactured test.

What, for Mr. Kohl, motivates learning, at least for learning to read?

...reading is a tool, an instrument that is used for pleasure and for the acquisition of knowledge and information about the way the world works. The mastery of complex reading skills develops as students grapple with ideas, learn to understand plot and character, and develop and articulate opinions on literature.

Nowhere does Mr. Kohl mention extrinsic rewards. Teachers have observed, and Robert Slavin's research has confirmed the dissipating effect of extrinsic rewards.

Robert Slavin's position--that extrinsic rewards promote student motivation and learning--may be valid within the context of a "facts-and-skills" curriculum. However, extrinsic rewards are unnecessary when schools offer engaging learning activities; programs addressing social, ethical, and cognitive development; and a supportive environment.

Not only do extrinsic rewards fail to motivate, except in limited cases, but research has also found that extrinsic rewards actually sabotage motivation.



So what's with the ubiquitous classroom token economies? Why must teachers have jars ofmarbleson their desks? Are we deliberately sacrificing long-term learning benefits for short-term classroom management? How about pay-for-performance or merit pay? First. And foundationally, EVERYONE deserves to be paid FAIRLY. “Getting the issue of money off the table,” as Dan Pink says.

If our society want to motivate the highest performance from teachers, then give them:

Autonomy
Mastery
Purpose

NOT merit pay.

Merit pay is inherently unfair. The bug-a-boo with merit pay is that teachers have so little control over the factors that impact student achievement. What do we say, for example, about the student who actually scored worse after his first year with me only to leapfrog three grades the second year with me.  Should I have lost pay the first year?  I was still the same great teacher.  I had no idea his alcoholic uncle moved in with him and his mom that first year. What do you do if you are a great teacher in an environment where just about everything seems to be conspiring against the kids? And what if you are lucky enough to teach in a school where kids have all kinds of advantages and their scores show it regardless of who is their teacher? Policy-makers have not figured out any equitable mechanism for awarding merit pay.