About Homework (with bonus lesson on area of a circle)

In the midst of extreme busyness, I left a short, hurried post nearly four weeks ago about how I recently became the temporary legal guardian of a sixth grader. Now it is spring break and I am even busier, but I have managed to carve out half an hour to write. I do not remember being so busy with the demands of a child when I was raising my own kids.

Attitudes of School Officials

Although I have teaching experience in elementary and even preschool, I spent most of my thirty-five-year career in secondary school settings. Suddenly, I have been spending a lot of time interacting with an elementary school. The classroom teacher and the school secretary do not know I am a teacher. The way they choose to interact with me is quite interesting, if somewhat condescending. I can only surmise that school people, intentionally or not, treat parents as if parents are basically ignorant. I apologize if this observation offends some. I completely understand that some parents are difficult, but it was disconcerting to have the teacher or secretary dispute with me almost from the get-go as if I knew nothing about the child.

Homework

Homework? What homework? In spite of the (shall we say) whining that kids these days are overly burdened with homework, I am just not seeing it. My ward brings very little homework home. Alfie Kohn, an educator with whom I am generally on the same page, has been crusading against homework for years. Just a few weeks ago, he sounded similar alarms, making it sound as if any and all homework is bad, bad, bad.

Funny thing I actually agree with most of his points. I absolutely detest homework as busy work. I remember when my own third-grade child came home with an assignment to write out the sevens ten times. His teacher knew that he could recite the times tables on demand, so the assignment was a complete waste of his time. In an effort to salvage some usefulness, he decided to type it in order to he could use the assignment as an excuse to practice the ten-key pad on the right side of his keyboard. His teacher gave him an “F” and scolded him for shortcutting the assignment. “You could have written it once,” she said, “and then simply copied and pasted.”

Nevertheless, homework does have a useful place, especially when used to generate fodder for idea generation during class discussion, as when the student measures the circumference, diameter and radius of ten round things. Or perhaps the student writes up a report of an experiment done in class in order to prepare to discuss the findings with the rest of the class the next day. The homework my ward brought home was generally useless, but certainly not time consuming. The teacher says she is only allowed to give 20 minutes of language arts and twenty minutes of math per day. The teacher does not even expect sixth graders to write their last name on their papers.

Homework as Practice

Normally, if a student has acquired the concept, it does not take a lot of practice to reinforce it. In my experience, homework as practice often means the student has not acquired the concept. The younger the student, the more control the teacher has over acquisition (but it must always be remembered that the teacher does not have total control). Anyway, homework as practice at the elementary level is worse than useless if the student has not acquired the underlying concept. Such students spend the twenty minutes reinforcing the wrong learning. It would also be helpful if elementary math teachers actually had what Liping Ma calls the “profound understanding of fundamental mathematics.”

Area of a Circle

My ward brought home a worksheet to practice finding the area of a circle. From her point of view, it was nothing but plug and chug. She asked me what “pi” meant. We spent a little time developing the concept of pi. Then we cut up a paper plate into pie slivers and arranged them, point up point down, into a sort of parallelogram with scalloped edges. I asked her what shape it reminded her of. Not surprisingly, she answered, “Rectangle.”

I asked her how to find the area of a rectangle. “Length times width” she replied.

“Right. So what part of the circle is the width of this rectangle?”

“The radius.”

“Right. What part of the circle is the length?”

She thought a bit and offered, “Half the circumference?”

“Exactly so. Then instead of length times width, what can we write?”

I began writing “A = r,” whereupon she shouted, “times 1/2C.”

We continued along these lines until we had written A = r X ½ X ∏ X 2 X r. Then r times r equals r squared. ½ times 2 equals 1. When she was done substituting, she had A = ∏r^2.

“Okay,” I said, “Look at your worksheet.”

To her amazement, the formula for the area of a circle was exactly what she had written. She had figured it out herself. That is the sort of success that builds genuine self-esteem. My disappointment came when I described this experience to her math teacher. He had no idea what I was talking about.

Back In Elementary School

I recently became the temporary legal guardian of an elementary child. Consequently, although I am a secondary teacher, I have been spending a lot of time interacting with an elementary school from the perspective of a parent. I plan to write about it in the near future. I want to look at the ease with which schools impose false "mandatory requirements" on parents, how students sabotage good instructional ideas like individual white boards in math class, the challenge for an upper elementary teacher who wants students to profoundly understand math but have inherited kids all over the map, just to name a few topics.

In the near future, I will be writing less on policy and offering more in the way of specific resources, such as curriculum, reviews and tips.

The Real US Teacher Supply Problem

The main problem with the teacher supply is that 50% of education majors are great. They are smart, academically able, motivated for all the right reasons—just wonderful. Sure they are green, but everyone has to start at the beginning. This essay is not about them. Then there are the other 50%. Some of them may be plenty smart, but they are the antithesis of the great 50% and what is worse, motivated by all the wrong reasons.

I taught three teacher education courses: Math for Elementary Teachers, First Language Acquisition, and Early Childhood Curriculum. I taught these courses after decades of classroom teaching experience. I am not even going to discuss how hard it is for a teacher who offers nothing but experience to even land a university teaching gig, except to point out how silly it is for colleges of education to turn away experience in favor of a PhD, and how ironic that a teacher who chose to stay in the classroom is at an extreme disadvantage when it comes time to pass on their wisdom.

Aspiring teachers have lots of hoops to jump through. Significantly, none of the hoops poses much of a challenge. The hoops are proforma, doing little more than keeping the line moving forward in an orderly fashion. Everyone eventually gets on the Ferris wheel.

College Entrance Hoop

I have asked countless students why they want to be teachers. The answers they give range from heart-warming to pathetic. One said it was either teaching or the Army. Several have said it was because they flunked out of Hotel Management or Business. Many consider teaching their last viable option for employment. Furthermore, regardless of your opinion of the SAT, it is well-documented that education students tend to have lower scores than students in most other majors. Colleges of education need to become more selective. The last time I said so at a faculty meeting, the chair said that it was a great idea but the school could not afford the hit on tuition.

Education students are notoriously weak in mathematics. At one university, 95% percent of all elementary education students were unable to score 70% or better on a straightforward computation pretest of first through sixth grade math. Meanwhile, Math for Elementary Teachers students complain about how unfair of the university to make them jump through the worthless hoop of studying elementary math. “We learned all that in elementary school,” they say, “The university is wasting our time and our tuition money.” They say this even after they see their scores on the pretest.

Math for Elementary Teachers Hoop

The colleges of education are well aware of the fact that education students do not possess what Liping Ma calls “profound understanding of fundamental mathematics.” Most colleges of education require education students to take a series of elementary math courses. 50% of each section will likely fail the class. They will repeat until they pass. One student asked me to congratulate her when she passed on her third attempt. Then she confessed sheepishly, “But I passed because that professor skipped fractions and decimals.” I was aghast. “You failed fractions and decimals in my class,” I said. “What grade do you want to teach?” “Fifth grade,” she said. I was aghast again. “Fifth grade is all about fractions and decimals. What are you going to do?” “I guess I'll figure it out when the time comes,” she replied. Sadly, she is all too typical of THAT 50%.

College Graduation Hoop

Perhaps you are certain a student like her will never graduate. You would be wrong. That 50% will graduate with near 4.0 GPA in their education courses. They will have struggled to maintain a 2.0 in all their other university coursework. I know this because the university sent the local newspaper a massive spreadsheet containing a year's worth of grades, with copies to every professor.

Student Teaching Hoop

Maybe you think the 50% will wash out during their student teaching placements. They won't. The university supervisors responsible for observing and evaluating student teachers are generally non-tenured adjunct professors. They are under tremendous pressure to push the 50% on through. University supervisors who actually have halfway rigorous evaluation standards may very likely lose their jobs due to off-the-record complaints because “it just wouldn't be right to wash out students who have invested so much time and money into becoming a teacher.” Students, with rare exceptions, cannot fail. Students failure would put the school of education cash cow at risk.

Teacher Certification Hoop

Surely, the 50% will be stopped at the state certification gateway. Not at all. Many state departments of education have agreements with the state colleges of education. Anyone who graduates from a state college automatically gets a teaching credential. Ironically, the proven veteran of long experience who moves to a new state may be able to get a new state credential but will not be able to keep it. When the new credential expires for lack of employment, the experienced teacher will be falsely considered unqualified.

Job Interview Hoop

Of course no principal will hire those poorly qualified and unmotivated teachers from the 50%. Wrong again. Principals (and the general public) consider mere possession of a teaching credential to be prima facie evidence of quality. In fact, principals will hire one of the 50% over an experienced, proven applicant because the novice is cheaper. Thus the 50% occupy an awful lot of our nation's classrooms. “Teachers with 10 or fewer years’ experience now constitute over 52 percent of our teaching force.”

No wonder school administrators do not defer to the knowledge, judgment, experience, and professionalism of their faculty, since 50% do not belong there in the first place. No wonder publishers have made big business out of scripted curriculum. Teachers, a strange hybrid of employee and professional, want the esteem due a professional. To the extent that selection and training is weak, the profession is demeaned.

Cultural Sacred Cows of American Education

As long as comparative studies show so many other countries outperforming American students, there will be those who dismiss the findings because of comparability concerns. The samples from other countries are more academically proficient, or societies in those countries value education more, or those education systems emphasize rote learning or.... The critics pull out the list anytime American students fare poorly, comparatively speaking. American students have been ranking low for a very long time now, so the list is pretty well memorized. The list has been repeated so often without dissent that its points are assumed to be true, whether they are or not.

The fact is there are comparability problems. In 1993, David C. Berliner tackled the topic in an article published by Phi Kappa Phi in their journal, National Forum. Significantly, he subtitled his article, “A False Guide for Reform.” Old stuff can be good stuff. Although Dr. Berliner wrote almost twenty years ago, he could have written yesterday.

To blame school failures on poor teachers, inadequate administrators, inappropriate curriculum, or uncaring parents is misleading. When children are poor, when they lack health care, when they come from dysfunctional families and dysfunctional neighborhoods, schools fail. When public schools do fail, it is because society has failed (bold added by S. Goya)....

International comparisons of achievement always will reveal differences because the economic support for schools in each nation, their curricula, the quality of the teachers, the health of their students, their administrative systems, the support for school by parents in each nation, the value of education in each nation, and job markets each nation prepares its children for all differ. Such variation in the national systems of education leads inexorably to variation in the performance of students in each nation.

In 1994, I wrote a short article, also published in the National Forum, addressing two differences between Japanese and American education that Americans generally accept as true. It is human nature to put superficially true statements through our cultural filters and end up with mistaken conclusions. First, because public schools do most of the educating in America, we automatically credit Japanese public schools for Japanese school achievement. If international studies intend to compare public school outcomes, then researchers will have a difficulty finding a comparable sample in Japan. Virtually every student in Japan has received substantial supplemental education from the ubiquitous private after-school schools (juku).

Second, we hear that the Japanese school calendar has 240 days. Our own American schooling leads us to assume Japanese students are “on-task” for 240 days. However, 100 days are only half days for one reason or another. Japanese annual public school instructional time measured in hours is actually quite similar to American instructional time, but because nearly all Japanese students also attend juku, they receive substantially more academic instruction than American students. Furthermore, there are some fundamental unquestioned cultural paradigms that influence the American view of what is possible and what is untouchable when it comes to education reform.

Attention Seeking

In America, there is an axiom that children of all ages crave attention. Therefore, Americans have unconsciously socialized their children to crave attention, similar to the unwitting differential treatment of boys and girls. Adults are generally unaware of the many ways they encourage even middle school and high school students to be attention seekers. Consequently, no one questions that part of every teacher's job is to give attention to every student. In fact, the main argument for reducing class size is smaller class sizes make it easier for teachers to give individual attention in an environment where the misbehavior of children is often interpreted as a bid for more attention from the teacher.

I did not question or even notice the unexamined attention seeking axiom until I taught in a society that does not socialize its children to be attention seekers. Teachers in these societies capably manage much larger classes even in preschool and the early grades. Most primary grades have an average of forty-five students per grade. Even more interesting, students from these societies generally outrank American students in comparative studies. While larger class sizes may not be a positive variable, it is at least not necessarily negative variable either. Of course, interpreting international comparisons is always a problem because of the complex interaction of variables. Even in the US, the research on class size is inconclusive and subject to confirmation bias.

For example, some Americans believe that societies with large class sizes post exemplary academic achievement because of an authoritarian school structure. One person wrote to me that they “knew” the Chinese government does not allow students to misbehave. While such a belief may be consoling, it is not true. Japanese education, especially in the elementary grades is very inquiring, active and hands-on. Furthermore, it does not occur to Japanese teachers that misbehaving students are seeking attention. They attribute misbehavior to other factors. If you have not created a room full of attention seekers, you can be a highly effective teacher with many more students in the classroom.

Contempt of High Achievers

American society is of two minds when it comes to high achievers. We say we value academic achievement, but what we say is betrayed by what we do. Our society routinely mocks and marginalizes high achievers. Tamara Fisher asked her gifted students to talk about how they felt about being high achievers. We did not need Ms. Fisher's class to tell us that while they were personally happy, they suffered socially. Nearly every American schoolchild has either been a victim or a perpetrator.

America says that one foundation of its education system is equal opportunity, that is, every child has a right to be educated to the extent of their potential. Then we undermine our grand values by charging high achievers with elitism. What exactly do we mean? That smart people can be smart as long as they hide it, so as not to hurt anybody's self esteem by their mere existence? There is an unresolved conflict between the values of meritocracy and egalitarianism.

Maltreatment of Substitute Teachers

One of the most appalling characteristics of American education is the routine poor treatment of substitute teachers and the commonplace administrative attitude that pranks and misbehavior come with the territory. Since when is it ever okay for students to mistreat another human being for a day. The substitute should be accorded the regard given to special guests for that is what they are. Nuff said.

Faculty Continuity

Everything about the American way encourages faculty longevity and discourages mobility. Teachers are certified at the state level. There are often silly, bureaucratic obstacles to re-certifying in another state. A teacher earning a Masters degree while teaching will receive a pay differential as long as they stay in the same district. Move to a new district and the Masters becomes an impediment to employment. Moving also turns experience into a disadvantage. Administrators are so adverse to paying for experience they will give credit for a maximum of only five years (in most districts). More often administrators simply pass over the experienced applicant in favor of the novice.

In Japan, for example, teachers are not only certified nationally, but they are also required to transfer schools every three years. Japanese administrators believes change keeps the staff fresh. The Japanese do not worry about the stability of school culture as Americans do. In fact, it could be argued that stability of school culture is actually a problem since the flip side of stability is resistance to improvements.

Instead of reflexively trotting out the tired list of reasons why international comparisons are flawed as if doing so somehow magically turns poor performance into acceptable performance, we should should be studying those reasons in detail to see what we can learn. It may be true that other societies value education more. Good for them. The lesson then is not to make an excuse, but to ponder what we could be doing to encourage American society to value education more, not only in word, but in deed.



Some Class Size Research Sources:

Counting Students Can Count http://nccic.acf.hhs.gov/node/28205

The Effect of Class Size on Student Learning http://livebinders.com/play/play_or_edit/31977

Class Size Research (List of Six Publications) http://www.bsd405.org/Default.aspx?tabid=5729

Class Size-Research Brief www.principalspartnership.com/classsize1110.pdf

Smaller Class Sizes: Pros and Cons http://www.publicschoolreview.com/articles/18

Making Enemies of ED Reform Allies

Alienating “ed reform” allies seems to be a counter-intuitive strategy, but one that “common-sense teachers” rely on more and more frequently. Anthony Cody summarizes the platforms of both “parties” in his biased Teacher Common Sense takes on Education "Reform" Nonsense. However, it is not like he did not give fair warning of his slant towards the “common sense teachers” party.

The past decade we have seen drastic changes affecting our schools, and many of these changes defy what we know as teachers and parents to be in the best interests of our children. We have allowed technocrats to drive our schools with data. It is high time for teachers and parents and students to challenge the reform nonsense that holds sway.

While he makes many valid points about poverty, teacher experience, tenure, test scores and data, I was hoping for an even-handed summary of the education reform conflict and the myriad ways the teachers' voices are ignored. What I see instead is subtle and not-so-subtle mocking of "ed reform" by using easy-to-demolish phrasing. The article also makes enemies of potential allies by redefining education reform as a political stance.

Plenty of experienced teachers and other stakeholders are passionate about education in America and want to see it reformed. If they make the mistake of calling themselves “education reformers”, by Mr. Cody's lights, they automatically oppose "common sense" teachers. We need to flee these sorts of useless and destructive either-or dichotomies when discussing issues as complicated and with as many self-interested stakeholders as education.

For example, ed reformers do not believe that “Class size does not matter.” It does matter in certain situations, but in most educational contexts, the research has not supported universally smaller classes. In fact, there are countries with normal class sizes of 45, even in the primary grades, where students consistently rank at the top of international standings. Even more telling, their below average students out perform our best students. Before someone rushes to defend American performance by discounting the achievement of these students, we must remember that like so much in education, international comparisons are complex.

It will not do to rely on tired defensive excuses. For example, claiming that our average kids have to compete against their superior kids obfuscates more than it clarifies. There are any number of opposing unexamined cultural assumptions operating within both the American education system and the systems of other countries that make it appear obvious that class size should be important. Appearances are deceiving. I will name just one American education axiom that may not necessarily be true: Children, by definition, seek attention from their teachers.

In another example, the statement "Large amounts of public funds should not be diverted to privately controlled institutions" promotes education partisanship and perpetuates charter school misconceptions. The premise ("So by the measure chosen by the reformers, (charter schools) fail") has merit, the implied conclusion does not follow. Charter schools are not "privately controlled institutions." They are a species of public school subject to most of the education code, and answerable to their public sponsor, generally a district or county education board.

The argument implies that by the "ed reformers" own criteria, charters are no better or worse than traditional public schools. Fair enough. Then let's do something about "bad" charters, instead of using them to excuse "bad" traditional public schools. Let "good" charters flourish alongside "good" traditional public schools. Furthermore, sponsoring public education entities actually profit by charter schools since they retain 15% of the charter's state funding. The charter school must meet its expenses with 85% of the funding. Some charters are cash cows for their public school sponsors, such as Hickman, which has hundreds more students in its charter school than in its sponsoring traditional public school.

We who are passionate about education must do more than reach across the aisle. We must rearrange the furniture, eliminate the aisle, and mingle.

I Love Math Manipulatives...But

I love math manipulatives. I really do. Manipulatives allow students to physically model mathematics concepts. But manipulatives are no panacea. Manipulatives have significant, often overlooked, limitations.

Mistaken Modeling

Many teachers view math instruction as teaching standard algorithms, that is, teaching students the conventional step-by step recipe for computing an answer. Thus teachers use manipulatives to model algorithms. However, teaching algorithms is not the same as teaching math. For example, the most common explanation for dividing fractions is to multiply by the reciprocal. Multiplying by the reciprocal works because something mathematical is going on. However, we usually teach the superficial procedure and ignore the mathematics. The purpose of manipulatives is to model the mathematics, not the algorithm. The difference is subtle, but crucial.

Manipulatives Cannot Model Everything

Math is far more powerful than physical manipulatives. Manipulatives are merely a bridge to that power. Manipulatives cannot model beyond three dimensions, but manipulatives can lead students to math beyond the three dimensions. Some Montessori schools have a manipulative that physically models a quadratic equation, Ax^2 + Bx + C. If the factors of the quadratic equation are equal to each other, the quadratic equation models a square. If the factors are unequal, the quadratic models a rectangle.

I first saw the intriguing quadratic equation model in a Montessori school in Japan where preschoolers were enthusiastically absorbing the geometry of the quadratic equation without resorting to pencil and paper. FOIL? Who needs it? The factors were perfectly obvious to them. Add a “height” factor to model three dimensions. If the height is “x,” we have a model of a third-degree equation. We have an “x-cubed.” Cubed! How cool is that? Can we build a model in of an equation in the fourth degree? Well, now we have bumped up against a limitation. Mathematical representations can express math much more powerfully than physical models.

The Training Curve

It can sometimes require substantial training in the symbolism and design of the manipulative before the child can use the manipulative. For some children, imagining that one thing stands for another can create an obstacle to the mathematics itself. It is an adult myth that children have superior imaginations. Children represent, pretend, or re-enact what they already know. They have trouble with pretending something they do not already know. Adults can manage with the incomplete sets of manipulatives often found in classrooms. Children may be stymied. Children especially have trouble with strings of representations. Dr. Kamii says manipulatives can end up being “abstractions of abstractions” rather than the concrete models usually intended. For example, a teacher might say “We do not have enough hundred-flats for every group to make their number. You can use a teddy bear to stand for a hundred-flat if you need to.” Such instructions only make things more perplexing for the kids.

Impractical for Problem Solving

If manipulatives are used as algorithm aids, students may not be able to solve problems when they have no manipulatives, like during a test. Constance Kamii, who researches the ways children learn math, found that when young children were given a problem for which they had received no instruction and free access to a variety of manipulatives, writing instruments and paper, children preferred their own constructions over those imposed by others. Children preferred to think their way through problems with pictures they draw themselves rather than with manipulatives.

Broken analogies

Math manipulatives are analogies. Every analogy breaks down at some point. Math manipulatives are no exception. Manipulatives have lots of features which may or may not be salient to the math. Children may have difficulty understanding which features to pay attention to and which to ignore. For example, Cuisenaire rods are different lengths. Each length is a different color, but the color is arbitrary and has nothing to do with the math. However, the colors are sure convenient because kids can use them to express math without numerals.

Too Much Fun

Perhaps the most dangerous limitation of manipulatives is the fun. Student teachers have often reported to me that their math methods courses were little more than a term's worth of “playing” with manipulatives. They loved their methods course, but when they got into a real classroom with real kids, they found to their chagrin that they were woefully ill-prepared to actually facilitate the acquisition of mathematics concepts. I have often observed teachers use manipulatives as a fun diversion without ever getting to the point of the mathematics involved. I have seen educators demonstrate the use of manipulatives without ever building the bridge to the concept.

Manipulatives cannot substitute for the teacher's own profound understanding of the fundamentals of mathematics (PUFM). Sadly, nearly every college of education has a version of the course “Principles of Mathematics for Elementary Teachers” because so many elementary education students lack PUFM.

The over exuberant adoption of manipulatives is yet one more instance of educational pendulum swinging. Good ideas get over-used and misapplied all the time, often turning what could have been promising strategies into just another education fad.

Patient vs. Impatient Problem Solving

According to Dan Meyer, the problem with a steady diet of TV sitcoms is students learn to expect easy problems resolved in twenty-two minutes “with a laugh track.” We have now raised several generations of “impatient” problem solvers, and typical math textbooks pander to the syndrome instead of challenging it.

Mr. Meyer has a prescription for what ails our math teaching.



According to Mr. Meyer, there are two kinds of mathematics: computation, or “the step you forgot” and math reasoning. Within computation, there are a lot of tricks and gimmicks, like counting decimal places. The tricks work because of the underlying math reasoning. We teach the tricks, the non-math, and call it math. Good grades for non-math amount to “congratulating students for following the smooth path and stepping over the cracks.” No wonder our students display symptoms of impatient problem solving syndrome:

Lack of Initiative,
Lack of Perseverance,
Lack of Retention,
Aversion to Word Problems, and
Eagerness for Formulas.

The older your students, the more likely you can be teaching math reasoning well and still encounter not only the symptoms, but also resistance to the cure. Your students have been so conditioned by previous experience, that like chemical tolerance, they do not believe they can function mathematically any other way. It might be a good idea to show this video the first day of class to shock their systems into even entertaining the idea that math could be different.

His description of his presentation of the water tank problem is very like the way Japanese elementary teachers have been teaching math for decades (that I know about). They can easily spend a whole period on a single problem, but they actually save time, because they are not wasting it practicing a forgettable blind procedure on twenty problems. They invest the time it requires to think about math, for as Mr. Meyer says, “Math is the vocabulary for your own intuition.”

Mr. Meyers suggests a five-part prescription:

Use Multimedia,
Encourage Student Intuition,
Ask the Shortest Possible Question,
Let Students Build the Problem, and
Be Less Helpful.

Teachers ignore many features of a problem as irrelevant without discussion as if we expect students to figure it out on their own. Many do, some do not. Asking what matters, says Mr. Meyer, is probably the most underrepresented question in math curriculum.

After, and only after, students have acquired the math reasoning should we give them shortcuts, tricks and mnemonics.This video is an excellent example of a math teacher receiving accolades for teaching non-math.




And finally, just for fun.