Schools A Symptom, Not A Cause

Nick Hanauer, a self-described plutocrat, understands well the old canard that doing the same thing again and again with the same unsatisfactory result is just insanity. After spending decades believing that improving education would lift all boats, and discovering that his vast sums of money made no difference, he stopped and thought about it. He concluded that improving education, by itself, cannot reduce economic inequality because poor education, by itself, did not cause economic inequality. "Better schools won't fix America."

Like many rich Americans, I used to think educational investment could heal the country’s ills—but I was wrong. Fighting inequality must come first. ... What I’ve realized, decades late, is that educationism is tragically misguided. American workers are struggling in large part because they are underpaid—and they are underpaid because 40 years of trickle-down policies have rigged the economy in favor of wealthy people like me. Americans are more highly educated than ever before, but despite that, and despite nearly record-low unemployment, most American workers—at all levels of educational attainment—have seen little if any wage growth since 2000. To be clear: We should do everything we can to improve our public schools. But our education system can’t compensate for the ways our economic system is failing Americans. Even the most thoughtful and well-intentioned school-reform program can’t improve educational outcomes if it ignores the single greatest driver of student achievement: household income. ... In short, great public schools are the product of a thriving middle class, not the other way around. Pay people enough to afford dignified middle-class lives, and high-quality public schools will follow. But allow economic inequality to grow, and educational inequality will inevitably grow with it.

Mr. Hanauer then gives a brief historical overview of education and the US economy.

All of which suggests that income inequality has exploded not because of our country’s educational failings but despite its educational progress. Make no mistake: Education is an unalloyed good. We should advocate for more of it, so long as it’s of high quality. But the longer we pretend that education is the answer to economic inequality, the harder it will be to escape our new Gilded Age. However justifiable their focus on curricula and innovation and institutional reform, people who see education as a cure-all have largely ignored the metric most predictive of a child’s educational success: household income. The scientific literature on this subject is robust, and the consensus overwhelming. The lower your parents’ income, the lower your likely level of educational attainment. Period.

Many of those who have seen educational quality and attainment as the solution put high hopes in charter schools. Students from the most disadvantaged circumstances and their parents have bought the hope, and thus today the majority of charter school students are minorities. The left spins this fact as proof that charter schools are an evil causing the re-segregation of America. The right spins the same fact as proof that minority parents value the blessing of choice that charter schools offer.

Indeed, multiple studies have found that only about 20 percent of student outcomes can be attributed to schooling, whereas about 60 percent are explained by family circumstances—most significantly, income. Now consider that, nationwide, just over half of today’s public-school students qualify for free or reduced-price school lunches, up from 38 percent in 2000. Surely if American students are lagging in the literacy, numeracy, and problem-solving skills our modern economy demands, household income deserves most of the blame—not teachers or their unions.

Here is a lovely money quote:

If we really want to give every American child an honest and equal opportunity to succeed, we must do much more than extend a ladder of opportunity—we must also narrow the distance between the ladder’s rungs (School Crossing's bold).

Mr. Hanauer implies that the rich like himself have already taken enough. He believes the solution is clear and obvious.

In fact, the most direct way to address rising economic inequality is to simply pay ordinary workers more, by increasing the minimum wage and the salary threshold for overtime exemption; by restoring bargaining power for labor; and by instating higher taxes—much higher taxes—on rich people like me and on our estates. ... We have confused a symptom—educational inequality—with the underlying disease: economic inequality. Schooling may boost the prospects of individual workers, but it doesn’t change the core problem, which is that the bottom 90 percent is divvying up a shrinking share of the national wealth. Fixing that problem will require wealthy people to not merely give more, but take less.

Elizabeth Warren has a plan for that.

Ad Hominem is Not a Synonym for Insult

For the last several years, probably coincident with the increase in online forums and comments, I have noticed that the general public seems quite confused as to what ad hominem really means.  In many cases, someone who flings an insult is immediately accused of committing ad hominem, while ad hominem without including an insult is often not even recognized as being ad hominem.

The purpose of ad hominen is to misdirect attention from the logic of the argument to the qualifications of the person making the argument.  Specifically, ad homimem intends to disqualify its target. Ad hominen is a tactic of last resort when the logic of the argument seems otherwise unassailable.  In fact, if someone throws ad hominem at you, they have tacitly admitted to losing the debate. Ad hominem is used against the argument of a specific person as a misdirection from the logic of the argument to the character of the person. Ad hominem is usually used when someone has no logical answer to the argument itself. Someone who uses  ad hominem  hopes the target will be distracted from the issue at hand, take it personally, and engage in self-defense, thereby entirely forgetting about the argument that they have already won.

I saw an example of a fascinating variation of the misunderstanding of ad hominem in an online discussion about California issuing driver’s licenses to illegal immigrants.  The local newspaper had headlined their story,  Illegals line up for driver's licenses.

Many people responded indignantly that “illegal” cannot be used as noun because human beings themselves are not illegal, only their actions.  One speaker who I will call “student” asserted that since the headline was ad hominem directed to a group; therefore none of the individuals who objected had standing, that only the group being called “illegals” had the right to object to the use of that ad hominem.  Another speaker who I will call “teacher” responded that the word “illegals” as used in the headline might be an insult, but it was not ad hominem.  The conversation continued:

Student:  Ad hominem is an attack on character, rather than an attack on the argument presented. An ad hominem attack on a group is an attack on the character of members of the group.

Teacher: Your definition of ad hominem is incomplete. "Ad hominem is an attack on character, rather than an attack on the argument presented" in an effort to debunk the argument as if the character of the person is relevant to the argument.  Ad hominem is a LOGICAL FALLACY, usually employing personal attack on character, intelligence, status, etc   Personal attacks are not necessarily ad hominem. How can I make it clear? If you are mugged, you have been physically attacked. If someone drives by and hurls a slur, you have been verbally attacked, but these are not examples of ad hominem because the attack is not in the context of making an argument.  Thus many people objected to characterizing a group of human beings as "illegals," but those who committed the grievance are not actually engaging in ad hominem against the group so characterized.

If a group of illegal immigrants were engaged in a discussion with others about whatever, and one of those others said,"You guys are illegals, so your argument is thereby refuted," that would be an instance of ad hominem. Insults often accompany ad hominem, but insults themselves are not necessarily ad hominem.

Student:  How can I make it clear? - you can't; you're wrong. Ad hominem - you attacked your opponents's character or personal traits in an attempt to undermine their argument. References to people as members of a lawbreaking class is an attack on their character.

Teacher:  Calling illegal immigrants "illegals" is an attack on character. It is not an attempt to undermine an argument; it is just an attack on character, like calling the guy who blew through a red light an "idiot." There is no undermining of an argument.

Student: The basis of the argument was giving undocumented people California drivers licenses.

Teacher: In order for ‘illegals” to be ad hominem against illegal immigrants, the illegal immigrants themselves would have to be asserting the reason why California should give them driver’s licenses.  However in this case, other people who are not necessarily illegal immigrants are engaging in argumentation about the issue.

Student: argument: "a set of diverging or opposite views, 2) a reason or set of reasons given with the aim of persuading others that an action or idea is right or wrong." The action is giving undocumented residents California drivers licenses, as I said.

Teacher: The purpose of ad hominem is to deny someone status or qualification to engage in discussion. For example, to attempt to refute an argument by calling the proponent "a moron" is simply a way of saying that the person is too stupid to listen to. Or if a civilian gave an opinion about the military, someone might say, "How can you have an opinion if you never served in the military?" in order to disqualify the civilian’s opinion from consideration. Both of these are ad hominem, but only the first one adds an insult.  An "action" regarding California drivers licenses was taken, but the action is actually irrelevant to the question under consideration: Is labeling an illegal alien "illegal" ad hominem? It was not ad hominem, but according to those who took offense, it was an insult.

Here is a link to one California drivers license debate:

The pro side includes a group called The Mexican American Legal Defence and Educational Fund (MALDEF). If someone wanted to commit ad hominem against MALDEF, they would probably say, "Of course, you support them. The illegals are your own compatriots." This would be ad hominem against a group, MALDEF, but it is not an insult toward MALDEF.  It is also NOT ad hominem against illegal aliens because MALDEF is making the argument, not illegal aliens.

If a group of illegal aliens presented their opinion, a non-insulting ad hominem might be, "You are illegal aliens, so you have no right to give an opinion about California’s laws."   Their status as illegal aliens is a statement of fact.  Whether their status disqualifies them from comment is a separate issue.  An insulting ad hominem would be, "You are a bunch of wetbacks, so your opinions do not count," or in the view of some, "You are a bunch of illegals, so your opinions do not count."

I am finding online comments as repository of data on human nature really fascinating these days.  It is interesting see that some people, like this "teacher," target their comments to the readership rather to the person they are nominally responding to.

Are Teachers Fast-Food Purveyors or Professionals?

I have been following an online discussion.  (Scratch that, what do you call it when one of the parties constantly responds with insults).  The topic is supposedly the public sector pensions of teachers.  “SP” is trying to argue that teachers are overpaid for the amount of work they do. The respondent never identifies himself as such,  but I would guess he probably is a teacher, so I’ll call him “Teacher.”*  If “SP” accurately reflects the public perception of the work teachers do, it is no wonder there is zero respect for teachers in our society.

SP: Teachers, as I pointed out, are paid a FULL year wage, for a part time job.

Teacher: As I have repeatedly pointed out, teaching is far from a part-time job. But you already know that. SP: The teaching contract proves my assertion.  Teachers work 37 weeks per year at a contracted 36 hour work week.

Teacher: Teachers work far in excess of their contracted hours and weeks.

SP:  All you do is make up lies about what hours teachers work, and it is 36 hours per week.  Truth can be painful for the trough feeder with entitlement mentality!

Teacher:  Regardless of the minimums that may be in a contract, no teacher limits themselves to the so called contract.  Teachers do not work part-time. In fact, one big reason new teachers quit in the first five years is they are overwhelmed by the sheer amount of out-of-class work that is necessary. What is true that they can save child care expenses because they have to do so much of the work at home.

SP:  Your incompetence is breathtaking! Nobody works more than their contracted hours.  The teachers union would never allow it.

Teacher: Teachers do not work part-time. That is the fact.  You are under some mistaken impression that the only time they work is during face-time with students? Such a misconception is prima facie ludicrous. SP:  Teachers work 6 hours per school day, even if they took home 2 hours of work they would still just be at 8 hours total. They do NOT take home 2 hours of work per day though.  Teachers work part time, that is a fact. Just keep making up the whoppers though, easy to shoot down.  36 hour work week, 37 week work year= part time job.

Teacher:  There is no way teachers can get all the work done they are responsible for during a six hour day. 

SP:  I know many teachers working 36 hours per week Some work even less. Teachers are NOT onsite at their schools for 8 hours per day unless you include the lunch break. They teach 5 hours per day plus a prep period of an hour, prep periods are used for grading papers and so forth.

Teacher (evidently losing patience):  If you think that one-hour prep period is sufficient for getting all the work done, you know nothing. That one hour  is a woefully insufficient amount of time.  Your anecdotal "I know many teachers..." is worthless. You really need to stop talking until you have spent a year getting some real-world experience. Try subbing for a year. Even volunteering as a classroom aide would change your tune.

SP:  No matter what you say, teachers work only part-time.

Teacher:  You would be screaming your head off if teachers actually worked a 36 hour work week for 37 weeks per year. They would do nothing but babysit kids. There would be no time for preparing lessons, making materials, previewing the audio-visuals, testing labs before kids do them, grading papers, calculating report cards, keeping up on professional literature, writing tests and so much more.  What do you think?  Should teachers work as long as it takes to complete all those listed tasks or should teachers work work the so-called "contracted" hours and no more?

SP:  You said, “There would be no time for preparing lessons, making materials, previewing the audio-visuals, testing labs before kids do them, grading papers, calculating report cards, keeping up on professional literature, writing tests and so much more.”  Lesson plans are prepared very infrequently, in fact you could use the same lesson plans throughout your entire career in many areas. Most school districts today have "lesson plan banks" that teachers use and share. You can also BUY lesson plans already made. Grading papers is done in the prep period. Calculating report cards should not be an issue and should also be done in the prep period. Keeping up on professional literature? That is not a job requirement and is also not mandated; it should be done on the teachers OWN time as it relates specifically to their job, it is basically optional "continuing education" required for the license. And professional development days, aka "minimum days", are given multiple times during the year at the expense of the student, so you lose that one. Writing "tests"?? Do you mean developing tests? They are part of lesson plans and curriculum, and again that is an issue that should not be repeated more than once every 2-3 years if that often.  The work load is a part-time 36 hour work week and a part time 37 week work year. Those are the contracted "work loads." I guess you just lost again.

Teacher: You clearly know nothing about being a teacher. Excellent teachers use off-the shelf stuff merely as a reference.  You would be even more unhappy with education outcomes if teacher used the off-the-shelf stuff in the manner you seem to believe they should.
Again I suggest you spend a year being a substitute teacher or even a classroom aide before you say another word about the work teachers do. Don't worry; I'll wait.

SP is evidently under the misconception that repeating a falsity often enough, through some sort of mysterious alchemy, will render that falsehood true. SP is also under the impression that a teaching job is more like a fast food job than anything else in the world of work.  How is it even possible to break through that wall of stubbornness against true facts?  Until our society decides whether teachers are professionals or hired laborers, it will be impossible to effect any meaningful education reform.

Teachers as professionals implies professional standards for entry into the profession, professional autonomy, professional judgment, professional salaries and the tenure to be free of whimsical termination.  Teachers as hired laborers implies strict conformity to the contracted hours, so-called “teacher-proof” curriculum, top-down job instructions and easy firing.  The strange hybrid status teachers have now is unsustainable.

*I camouflaged the user names of the parties.  I also cleaned up the grammar, and spelling of the comments and edited them a little bit for clarity.

Most American Math Teachers Cannot Teach Math

...because they studied non-math in school, not math. (And most of the rest of us have the same problem.) https://schoolcrossing.blogspot.com/2007/11/are-you-good-at-non-math.html

I read Dr. Nancy Pine's book, Educating Young Giants, with great interest. The book is about her observation of classes in China, her discussions with Chinese teachers and parents (mostly through interpreters), and the comparisons she makes to American education. She admits to being ethnocentric at the time of her first visit to China in 1989, but while she could sometimes recognize her own egocentricity, she was not able to fully overcome it.

She noticed that in Chinese literature classes, teachers emphasized close reading and digging for the author's meaning. She felt that Chinese teachers denied students the opportunity to create personal meaning from the literature they read. Although her research in China centered on elementary literacy development, as a former math major (page 41), she became interested in observing elementary math classes. As everyone who has ever observed Chinese math classes has reported (see, for a few of many examples, Harold Stevenson, James Stigler, Liping Ma), she, too, witnessed superior teaching skill.

I have been teaching math in China for the last several years, and I taught in Japan for nearly two decades. I speak both Japanese and Mandarin. My conversations with people from mainland China, Hong Kong and Taiwan, as well as written descriptions such as Educating Young Giants has led me to conclude that the actual education systems as well as the cultural foundations of both China and Japan are very similar.

Nancy Pine came to appreciate that Chinese teachers teach mathematics, but “most U.S. teachers merely teach arithmetic” (page 45). Dr. Pine is being generous. U.S. teachers teach non-math, specifically routines, tricks and shortcuts, but call it math on the misconception that if numbers are running around, it must be math.

Chinese teachers spend a significant amount of time considering a relatively simple math problem from every conceivable angle. The students probably already know the “answer” and that is precisely the advantage of using an easy problem. Because they already know the outcome, they can concentrate on the process, the concept-building. Once the concept is solid, their homework includes problems that American students eventually spiral to. China thereby reduces the need for the endless review so common in America.*

Dr. Pine herself admitted “that even with my strong interest in math, I would not have known enough about the underlying mathematical concepts to think through the best ways to present the initial problem that would enable students to correctly solve more complex ones” (page 45). See what she is saying? She is admitting that she was great at non-math, but weak at mathematics itself. Not only that, she says she knows “that most American grade-school teachers, who teach five or more subjects, do not have the depth of knowledge to walk children through mathematical concepts to prevent misunderstandings” (page 50).

She believes it is because American teachers are generalists who must teach every subject, while Chinese teachers are specialists who teach only one subject. I would like to suggest that being a generalist or a specialist has nothing to do with it. Chinese teachers could be generalists and their ability to teach math would still “far surpass ours” (page 46) because nearly all Chinese teachers, regardless of their particular specialty, acquired a profound understanding of fundamental mathematics (PUFM, a term coined by Liping Ma) beginning in the primary grades. If our own children acquired PUFM, they would also be much more effective math teachers, even as generalists.

You see, regardless of professional training or subject matter courses, teachers tend to teach the way they were taught. The strident calls for teachers to take more subject matter courses is misplaced. Simply learning more and more non-math will not improve teaching ability. Okay, how about we reteach math at the university level? I tried to do exactly that, only to meet with terrible resistance. “We don't want to know why the math works,” my students complained, “Just tell us how to get the answer.” Fine, let's at least teach those students who aspire to become elementary teachers. Guess what? Most universities require all elementary teaching candidates to pass a series of courses entitled something like “math for elementary teachers.” My students complained that the classes were a waste of their time, since they “had learned all that stuff in elementary school.” Most elementary teachers, even though compelled to take a real math class, most of them for the first time in their lives, end up graduating from college without learning much math due to their resistance. They subsequently teach math the way they were taught in elementary school.

The main reason that Chinese students do so well in international math tests is because they actually learn math in school. American do not. Therefore, the reasons critics cite (specially selected students, lower poverty rates, rote learning, etc) miss the point. What critics are saying is that due to circumstances beyond our control, American students can never compete with Chinese students. I call baloney. If we would actually teach math in our schools, our students could compete just fine.

Next: examples of non-math teaching I encountered in a child's algebra class.

After seeming to correctly solve a number of simplification problems of the form -(ax-b) or -(ax+b), a child complained she could not simplify this one: +(ax+b). “What do I do with all those plus signs?” she wailed. What did you do with the other ones? I flipped them (referring to a mat-and-tile manipulative she is using in class). Even after all that flipping, she still had no idea what was going on. As long as she flips correctly, she can get the right answer without ever understanding how the flipping was supposed to communicate the concept. (Here is another topic: how American teachers routine take great resources like manipulatives and use them ineffectively  https://schoolcrossing.blogspot.com/2010/11/i-love-math-manipulativesbut.html.

In another example, the child needed to solve for x by first combining +2 + ¼, easy—the answer is 2¼. But the next problem was +3 –  -½. She wrote 3-½ as her answer, then complained because the problem was coming out “all weird.” I straightened that one out with her, only to have her evaluate the next problem, -2+ 2/3, as -2 2/3.

Dr. Pine realized that the depth of Chinese math learning far surpassed ours. Yet she seemed unable to perceive that the digging for meaning she observed in literacy classes was precisely the same digging for meaning evident in math classes*. She lauded it in math but lamented it in literacy, saying that teachers denied Chinese children the expression of their own personal opinions.

* Everywhere I wrote an asterisk I am referring to the Chinese philosophy of math education as evidenced by the textbook presentation of concepts and the implementation I and others have observed  in many Chinese classrooms.  HOWEVER, honesty compels me to relate that there are a number of  Chinese math teachers whose delivery of math concepts is at best cursory.  These teachers also assign an overwhelming amount of homework and "practice tests."  These teachers have been know to "steal" time for "unimportant" subjects like art and music for more practice tests in their never-ending quest to maximize test scores regardless of understanding. This approach is absolutely murderous to the spirit and curiosity of Chinese students.

Expert Teachers Fall Through Alternative Certification Cracks

Critics of alternative certification often express dismay that alternative certification could possible qualify someone who has no prior teaching experience.

“I am still waiting for the "alternative certification" programs in law, medicine, surgery, and pharmacy. I hear that those fields pay more than teaching, so I think that I might try my hand at one of those. Should only take 6 weeks or so to get through the program and be proclaimed "highly qualified" and I can get right to work on heart surgery, or filling prescriptions.

See? Sounds ludicrous now, does it not?”

It is a valid criticism, but assumes that all who seek alternative certification are starting from scratch. Not so.  Expert teachers moving to another state for whatever reason often have difficulty getting recertified. 

If you were a Department of Defense Dependents Schools (DODDS) teacher, returning from overseas may mean coming home to chronic unemployment.  For example, back in the 1980s DODDS began requiring all its teachers to take National Teacher Exams (NTE) regardless of how long a teacher had been effectively teaching. 

But in the US, these DODDS teachers found that many states would not accept NTE scores no matter how high the teacher's percentile score.  Nor would the states accept other documentation of competence like evaluation, publications, even student test scores on the Stanford 9, or anything else.  Some states even told these teachers they would have to get new masters degrees because their "old" one was now out-of-date, as if a terminal degree can expire. 

Many older teachers began teaching before student teaching became a requirement.  Some states will allow letters certifying experience to stand in for the student teaching requirement as long as the letters are not too old.  A teacher may be able to acquire a "provisional" teaching credential convertible to a standard credential if the teacher gets a K-12 job within two years.

The longest allowable letter of experience interval I saw was five years.  Most are three years.  But since schools will not  hire even certified older out-of-district teachers, they certainly will not even look at you if you are not certified.  They may apologize for not hiring a certified teacher due to budget, but if you are uncertified, they are happy to reject you without any apology.  Two years pass and the provisional credential expires.  One more year and the letters of experience are no good.

Plenty of great teachers are waiting tables, filing medical charts, preparing taxes, whatever.

When Are We Ever Gonna Use This?

Raise your hand if you have ever heard this question, “When are we ever gonna use this?” When I was a young teacher, I tried hard to answer. I used to give my students (junior high and high school) examples of math problems from various occupational fields. I bought a large poster that listed many occupations along the top and many mathematics topics along the side with black dots showing exactly which occupations use which topics.

Years passed. Film projectors gave way to Youtube videos. Mimeograph machines gave way laser printers. Whole new field of occupations emerged. I metaphorically threw up my hands in exasperation. When the inevitable question arose, I answered that I had no idea how they were going to use this information. I had no idea how their interests would develop, or which occupations they would pursue, or what the jobs of the future would be. All I could do was teach them a little bit of what had taken thousands of years for people to discover about math. My students were not always satisfied.

Then Paul Lockhart came along and wrote “A Mathematician's Lament: How School Cheats Us Out of Our Most Fascinating and Imaginative Art Form.” https://www.maa.org/external_archive/devlin/LockhartsLament.pdf

Now I had an answer that captured their imaginations:

“In any case, do you really think kids even want something that is relevant to their daily lives? You think something practical like compound interest is going to get them excited? People enjoy fantasy, and that is just what mathematics can provide -- a relief from daily life, an anodyne to the practical workaday world….People don’t do mathematics because it’s useful. They do it because it’s interesting … The point of a measurement problem is not what the measurement is; it’s how to figure out what it is.”

The question of the usefulness of any particular subject stems from the mutual internalization of both the teacher and students of a questionable, yet unexamined assumption.

“To say that math is important because it is useful is like saying that children are important because we can train them to do spiritually meaningless labor in order to increase corporate profits. Or is that in fact what we are saying?”

Thus instead of teaching real mathematics, we teaching “pseudo-mathematics,” or what I have often called non-math, and worse, we use math class to accomplish this miseducation (See https://schoolcrossing.blogspot.com/2012/11/tricks-and-shortcuts-vs-mathematics.html and others). According to Lockhart, we teach math as if we think “Paint by Number” teaches art.

“Worse, the perpetuation of this “pseudo-mathematics,” this emphasis on the accurate yet mindless manipulation of symbols, creates its own culture and its own set of values….Why don't we want our children to learn to do mathematics? Is it that we don't trust them, that we think it's too hard? We seem to feel that they are capable of making arguments and coming to their own conclusions about Napoleon. Why not about triangles?

Math is like playing a game. As with any game, it has rules to be sure. However, it is more fun and more elegant than all other games because it is literally limitless.

Physical reality is a disaster. It’s way too complicated, and nothing is at all what it appears to be. Objects expand and contract with temperature, atoms fly on and off. In particular, nothing can truly be measured. A blade of grass has no actual length. Any measurement made in the universe is necessarily a rough approximation. It’s not bad; it’s just the nature of the place. The smallest speck is not a point, and the thinnest wire is not a line. Mathematical reality, on the other hand, is imaginary. It can be as simple and pretty as I want it to be. I get to have all those perfect things I can’t have in real life. I can never hold a circle in my hand, but I can hold one in my mind. […] The point is I get to have them both — physical reality and mathematical reality. Both are beautiful and interesting… The former is important to me because I am in it, the latter because it is in me.

Mathematics offers infinite possibilities for storytelling. I tell many stories as I teach math. My students are positively enchanted and remember them forever. One of my favorites is the kimono story.

I tell my students how in old Japan, servants helped geisha to put on the multiple layers of kimono. Each layer has to arranged and offset just so in order to reveal the colors of each layer. I tell them we are going to start with a geisha like 1/3. First we put on the 2/2 layer. 1/3 x 2/2 = 2/6. Notice that the geisha looks a little different, but underneath it is the same geisha. How about another layer, maybe 3/3. Okay 2/6 x 3/3 = 6/18. How about another 2/2 layer. 6/18 x 2/2 = 12/36. We can take off the layers one-by-one as well. This is called “simplifying a fraction.” Simplifying a fraction is simply a process of finding out which geisha is at the bottom of all those layers. If we are in a hurry, we can remove all the layers at once. How would we do that? In the case of our geisha, dividing by 12/12. The students love it.

The most elegant math story is the proof.

A proof is simply a story. The characters are the elements of the problem, and the plot is up to you. The goal, as in any literary fiction, is to write a story that is compelling as a narrative. In the case of mathematics, this means that the plot not only has to make logical sense but also be simple and elegant. No one likes a meandering, complicated quagmire of a proof. We want to follow along rationally to be sure, but we also want to be charmed and swept off our feet aesthetically. A proof should be lovely as well as logical.

Teachers Should Teach to the Test

Should teachers teach to the test? Some say of course we should, in order to give students the best chance for achieve their highest potential score. Some have even made teaching to the test a lucrative business. Schools are sacrificing more and more instructional time to test prep. Others say that teaching to the test games the outcome in favor of some students without actually reflecting the acquisition of real knowledge or achievement. Who is correct?

First, we must be careful to distinguish between tests teachers write covering material they themselves taught, and standardized tests. Standardized test are not written by the teacher who is teaching the material, and indeed, it is considered cheating if teachers see the questions ahead of time. Teacher-written tests cover a specific subset of content. The purpose of the test is to evaluate the students’ learning of that specific knowledge. Theoretically, if everyone in the class masters the material, everyone can potentially score 100%. Practically, teachers try to have a mix of harder and easier questions in order to differentiate levels of mastery. However, there should not be any questions outside the subset domain.

Standardized tests are very different. Test designers try to ensure that half the students will score above the target median and half below. From the students’ point of view, they perceive right away that it feels like they do not know half the questions. The realization often makes them feel inadequate and creates much of the test anxiety surrounding standardized test. I have found that explaining the difference between the test I write and standardized tests relieves much of the anxiety.

There is, of course, no point in explaining jargon like normative evaluation, median, etc. It is sufficient to simply say that the people who wrote the bubble test wrote it for lots and lots of students who have been taught by lots and lots of teachers. The writers really have no idea what I taught or how I taught it. So the writers write lots of question that they expect no one will know the answer. In fact, they write the test expecting that students will miss fully half the questions. I reassure them that it is perfectly normal to feel as if they are probably missing a lot of questions. Go ahead and guess anyway.

I tell them that the test designers include questions from lower grades in the test and questions from higher grades. The test designers know which questions are which, but of course the students do not know. I tell them if they feel like they do not know a question, it is probably from a higher grade and not to worry about it. The test designers look at the answer sheet and can tell if the students correctly answered the questions from their own grade level. If they do, they will get at least 50. I tell them this does not mean 50 points, nor does it mean 50%. I tell them it is a different kind of scoring system because it is not a test that their own teacher (like me) wrote. With high school students, I discuss a little more statistics and the idea of percentiles.

This kind of explanation usually satisfies students, removes perplexity and frustration, and helps them do their best. If the teacher’s curricular philosophy and design is strong and the teacher is a skilled teacher, then there is no need to worry about the standardized tests. Simply teach, and the standardized test will take care of itself. If the curriculum is weak, teachers will feel a strong need to teach directly to the test. However, by all means, teach to your own tests.

Can You Teach the Bible in Public Schools?

The short answer is yes, you can and should teach the Bible in public schools.

The long answer is more nuanced.

There are three subjects that benefit from the inclusion of the Bible: English, Social Studies, Political Science, Western Law, Art, Music and yes, even Science.

English:

We expect students to recognize and understand literary allusions. The vast majority of literary allusions come from four sources: the Bible, Shakespeare (who often alludes to the Bible), Greek mythology and popular culture. There is no good reason to deny students understanding of certain literary allusions, merely because they come from the Bible. The Bible is also a literary classic in its own right. Belief is not a prerequisite to an intellectually honest presentation of the Bible as literature.

Avoiding the Bible also leads to miseducation, such as the case of a fifth grade teacher who defended reading The Lion, the Witch and the Wardrobe by C. S. Lewis, by saying she intended to read it as a fairy tale. C. S. Lewis intended the story to be Biblical allegory, not a fairy tale. To teach otherwise is educational malpractice. Either the teacher should teach literature such as this honestly, or avoid the book entirely. The middle ground simply will not do.

Social Studies:

History education prefers primary sources whenever available. The Old Testament is the major primary source for the ancient history of the Jewish people. The history of the church had a huge impact on the history of Europe over the last 2000 years, and an understanding of the Bible informs our understanding of European history. The Boers drew their rationale from the Bible (although I would argue that deliberately or not, the Boers improperly applied the Bible to their situation). In fact, an understanding of the Bible is essential to an understanding of the motivations behind many historical events.

Political Science and Western Law:

Our public discourse constantly refers to the Bible, and yet most of the people who think they are quoting the Bible (both Christians and non-Christians alike) have near zero understanding of the Bible context itself or the Bronze Age time when most of it was written. Christians especially have a weak understanding of what a “literal” interpretation means. When I was much younger, I met a man who had been an Air Force pilot during WWII. After the war, he went to Papua New Guinea or Irian Java (I forget which) to be a missionary. The island people had a noun which meant airplane. Literally, the word meant “a bird with the skin of a machete.” We would be foolish to think that the island people really thought the airplane was a bird, yet Biblical “literalists” make this type of mistake all the time. Another example comes from Chinese. Their word for computer means “electric brain,” but clearly the word is not figurative, spiritual or symbolic. It is simply the word for computer. In English, we still say the sun rises and sets, but no one supposes that we literally mean the sun moves up and down. Many people who say they believe in taking the Bible literally fail to distinguish these types of expressions, leading to some of the ridiculous arguments we hear everyday.

As Christopher Gunter wrote :

So what are young people to think when they hear biblical passages taken out of context to both support and refute gay rights, or the Iraq war, or any highly charged issue? They must not be afraid to question and challenge biblically based sound bites. They must have the courage and the foundational knowledge to understand for themselves the source and context of biblical passages. Our reluctance to teach the Bible perpetuates its mysteriousness, which has grave consequences in our intellectual lives and in the wider world in which we live.

Art and Music:

Anyone who study art or music appreciation will not get very far before they run into cultural works illustrating, or inspired by the Bible. If we want to understand the cultural work, we need to understand the source material.

Mr. Gunter again:

... the Bible’s influence spreads beyond the literary realm into the artistic and the cultural. Any student of art or music will deal extensively with religious material. Moreover, biblical allusions in culture persist into the 21st century: in movie titles, song lyrics, newspaper headlines, billboards, and so forth—even television’s “The Simpsons” draws extensively from the Bible. In short, biblical knowledge enriches our understanding of both high art and popular culture.

Science:

The acrimonious debate between “creationists” and “evolutionists” would evaporate if both camps actually understood what the Bible says.

As Mr. Gunter concludes:

It is a sensitive endeavor, to be sure. But we first must recognize the value of undertaking that task. The Bible is a remarkable document, parts of which can stand with Plato in their philosophical depth, with Tolstoy in their political complexities, and with Shakespeare in their poetic beauty. The religious sphere does not have exclusive ownership over those important words. We should give our young people the tools to understand the Bible, both for their own enlightenment and to better inform their decisionmaking as citizens.

California Proposition 58: A Solution Looking for a Non-existent Problem

In 1998, it was Hispanic parents who clamored to get rid of bilingual education. Bilingual education was not a bridge but a jail. Hispanic children languished in bilingual classrooms for years and years, and never attained the proficiency that would allow them to go to college. The parents were successful in getting Prop 227 passed, not so much because of the force of their own arguments, but largely through the ideological effect of English-only whites who maintain that America is an English-speaking country, and so students should be taught in English.

The California Department of Education says they are already ensuring that English learners:

...acquire full proficiency in English as rapidly and effectively as possible and attain parity with native speakers of English. AND ...within a reasonable period of time, achieve the same rigorous grade-level academic standards that are expected of all students.

Prop 227 did not get rid of bilingual education. Bilingual education is still readily available. the difference is previously, the school decided whether a child would be placed in the bilingual program. Currently, it is the parents who decide. Most parents choose English immersion.

Since the main effect of Prop 58 is to undo Prop 227, we must next investigate whether removing the decision-making power from parents is actually good for kids.

The proponents of Prop 58 claim that the American Institutes for Research (AIR) concluded “there is no conclusive evidence to support the opponents claims that Prop 227 has been successful. AIR is a credible source, so I took a closer look. The first thing to note is the cited report was published in 2006, and looked at the previous 5 years. It specifically said that because of the many variables involved, “There is no conclusive evidence that one instructional model for educating English learners, such as full English immersion or a bilingual approach, is more effective for California’s English learners than another(method)...”

In other words, a dichotomous approach does not work. AIR was unable to isolate pre-Prop 227 or post-Prop 227 as the independent variable. That is a little different from what the proponents of Prop 58 claim the AIR report said. Furthermore, AIR was unable to control for the numerous other variables that impact Hispanic achievement. So what kind of evidence is there? The AIR report itself observed,

During this time, the performance gap between English learners and native English speakers has remained virtually constant in most subject areas for most grades. That these gaps have not widened is noteworthy given the substantial increase in the percentage of English learners participating in statewide tests, as required by federal and state accountability provisions.

So even though many more English learners took the statewide tests, they did not bring down test scores as was expected. Ten years have passed since that AIR report was published. Who knows, but what a new report might find conclusive evidence, or at least a greater quantity of circumstantial evidence.

Lacking an experimental methodology, the AIR evaluation often relied on case studies, which is simply a systematic look at an anecdote. I have a few anecdotes/case studies of my own. I have not worked much with Hispanic students because most of my 40 years of teaching took place in Department of Defense Dependent Schools in Japan or in private schools located in Japan and Shanghai. I was also an in-service training provider to public secondary schools located on the Navajo reservations in Arizona. One time I was also assigned to a 6-year-old Hispanic boy who had been in horrible car accident to be his at-home teacher. I will summarize each experience.

1. The Hispanic kindergarten student. I began teaching this boy the last three months of school. I met with his classroom teacher who gave me a packet of papers to color and a copy of his third-quarter report card. His grades were very bad, and his progress was below grade level on almost every measure. His teacher referred to him as “one of those typically dumb Hispanics.” Back home, I threw away the papers she gave me, and spent the weekend creating a kindergarten program for this boy. Within three months, he was reading English and performing at grade level on every measure. His parents were thrilled.

2. The Navajos. I did not work with the Navajo students directly. At my in-service presentations, their secondary teachers complained that they did not need the information I had been commissioned by the administrators to present. They asked me to tell them how they could use their subject area textbooks to teach their students to read. I chucked my carefully planned presentation (including hands-on activities) and immediately improvised a seminar on phonics and reading comprehension using science and history books. The teachers loved it (but the administration was peeved at me. Whatever).

3. Japan. One fall, a large group of parents suddenly enrolled their children in the junior high where I was actually teaching science. The parents took this drastic measure because their children were refusing to go to school due the extreme bullying that sometimes occurs in Japanese schools. The principal pulled me out of my morning classes, and asked me to create a half-day transitional program for these kids. They studied art, music and PE in the afternoon in the mainstream class. After three months, I put them in the mainstream math classes. After the second term, I put them in my mainstream science class. After the third term, they were fully mainstreamed, including English and social studies.

4. China. I have spent 4 academic years teaching in China using English only with great success, even with first graders who speak zero English when they start. Within one year, all but just a handful were reading and comprehending at American second-grade level. Their English speech still retains errors attributable to Chinese syntax, but those errors will fix themselves eventually.

42.8% of community college students are Hispanic in 2015. Overall, the number of Hispanic students in college has been increasing dramatically year over year, while the number of white students in college has been falling over the last five years. According to Pew, a record number of Hispanic students have enrolled in college, and the high school drop-out rate is the lowest it has ever been. The numbers on both measures have been positive since 2000. English Only as a factor contributing to these results did not occur to Pew, but it is as likely a factor as any of the others that Pew did suggest. According to the National Conference of State Legislatures, Latino college completion is on the rise and in the past decade the number of Latinos with bachelor’s degrees or higher increased 80 percent. Of course, this achievement is not due solely to Prop 227. Programs such as AVID, TRIO, Gear Up, and others also contribute to positive outcomes.

As far as the proponents' claim that Prop 58 would expand second language opportunities for native or fluent English-speaking students, a proposition is unnecessary. Schools are already free to add foreign languages to their curriculum, or create foreign language immersion programs.

It is unfortunate that a majority of the California legislature supports Prop 58. They seem unaware of the history since the legislature in place when Prop 227 was approved has either retired or termed out. The legislature also seems unduly impressed by the articulate but empty arguments of the proponents when compared compared to the emotional tenor of the opponent's arguments. In fact, the proponents' statements in the Voter's Guide read like one of those long-winded sales pitches with a lot of beautiful words that actually say nothing. For example, the proponents introduce a paragraph in the Voter's Guide by saying, "Here's what Prop. 58 actually says:," and then proceeds to quote, not Prop 58, but the already existing California Education Code. In this way, proponents mislead voters into thinking that Prop 58 will do something that is not already mandated, when in fact the law already mandates it, and Prop 58 is unnecessary. No wonder the less sophisticated opponents got emotional.

There is no need to fix a non-existent problem. In short, the stakeholders with the most compelling interest, that is, parents of Hispanic students, do not want Prop 58. That should be good enough for the rest of us. Vote NO on Prop 58.

This is What's Wrong with Tech Articles

GreatSchools has an article about evaluating the effectiveness of technology in your child's school. Just like most such articles, it does not even question the assumption that technology should be used. The unexamined assumption is of course technology should be used. It is only a matter of whether it is being effectively used.

The assumption ignores two considerations. One, technology has always been used in schools. There are people still alive who remember the old mimeograph machines that produced odorous purple worksheets.

Language labs once used huge reel-to-reel tape players.

There are people who remember helping their teacher carefully thread the filmstrip projector.

Eventually, the the projector gave way to VHS tapes which finally gave way to You-tube videos projected from flash drives.

The point is there is no stopping technology. Which brings us to the second consideration. Back in those days, there were no articles discussing whether technology was effective or not. Technology was a tool, but not a panecea. We had not yet mentally endowed technology with mythological superpowers. Technology was not "a thing." Today, technology is a bandwagon to jump on merely for technology's sake. Tech for tech's sake is expensive and unnecessary.

"Research shows these (smart) boards can increase both student interest and participation," (but this does not necessarily translate to increased understanding or achievement, especially if it doesn't) "change the dynamic of the classroom...Because it’s the teaching practices associated with technology use that matter most.”

Wrongness

The topic of being wrong pops up more and more frequently in public discourse these days. Author Chuck Klosterman, maintains we are probably wrong about everything we think we know, including and maybe especially gravity. Meanwhile, we are chided for being “ant-science” if we disagree with the consensus of scientists. In a famous Last Week Tonight spot, Bill Nye (the Science Guy) leads a climate change “debate” that was no more than Bill with 96 white-coated people representing the 97% of the scientific consensus against 3 other people representing the 3% of the science community refusing to join the bandwagon. Case closed, apparently.

We all “know” that Republicans are the anti-science party, right? Except, according to Neil Degrasse Tyson, there is plenty of anti-science on the Liberal side of the aisle as well. Steven Novella, MD, a contributor to Neuroligica Blog, supports Dr. Tyson’s assertions with some survey results, concluding, “My synthesis of all this information, which is admittedly incomplete, is that people tend to be anti-science whenever science confronts their ideology.”

Dr. Novella elaborates,

I think it is more meaningful to understand these issues by breaking them down to specific ideologies and how they influence acceptance or rejection of science. Conservatives tend to value freedom, the sanctity of life, and the free market and they distrust government. Liberals value nature and the environment and distrust corporations. Individual issues are complicated because they can cut across multiple ideologies. In terms of the question of who is more anti-science, my approach is this – you don’t get credit for being pro science for accepting an issue that is compatible with your ideology (bold added). Liberals acceptance of manmade global warming does not mean they are necessarily pro science, because this issue is right in line with their ideology (pro nature, anti corporate). Conservatives don’t get credit for being pro nuclear for the same reason. Evidence for being pro science is when you accept a scientific consensus that conflicts with your ideology. You have to demonstrate that science comes before your ideology, (bold added).

The thing is the 3% of scientists who disagree with the 97% are not wrong simply because they are outnumbered, as Bill Nye implied. Science is not a majority-rules proposition. Throughout history, there have been scientists who have disagreed with mainstream science. Some suffered, at worst, outright scorn and ridicule, or at best, indifference, only to be found to have been right all along. One big reason why accusations of being “anti-science” carry no weight with either camp is because everybody knows that settled science is settled only until a scientist unsettles it.

“Anti-science” is the new heresy. There is nothing wrong with disagreeing with settled science. The problem is when disagree-ers (of any stripe) have no basis for the disagreement except ideology. That’s a problem that seriously impedes useful discourse on any issue.

Missing Key to Understanding Place Value

I write a lot about place value. Place value (along with zero) may arguably be the most important math concept because it underlies every single calculation we do. Yet teachers often do not teach place value well. Teachers (and most curriculum) are satisfied with a very superficial understanding of this essential concept. If a child can identify the place name of a given digit or put a digit in a given place, most teachers deem the child to have a good understanding of place value. Place value is so much more.

Groups of Ten

Place value is all about making groups of ten. Well, yeah, the reader might say. Tell me something I don’t know. The key to understanding place value is the realization that each succeeding place represents a group of ten of the preceding place. Duh. Stay with me here. The curriculum and instruction alludes to this key, but rarely makes it explicit. Most textbooks have replaced “borrowing and carrying” with “regrouping,” and this was a positive step, but students still take a mechanical view. They still borrow and carry as they move leftwards through an addition or subtraction problem without realizing that they are actually making or breaking a group of ten at each successive place. For example, if they carry a one from the tens place to the hundreds place, they mechanically add that one to the other digits in the place without realizing that the carried one represents making a group of ten. In fact, most students will say, (correctly on a superficial level), that they made a group of 100 because they put the “1” at the top of the column named “100s place.”

Place value is all about making groups of ten. Subtraction is all about breaking groups of ten into loose ones and dumping them with the other loose ones. Every place except the loose ones is a group of ten something. Teachers tell students that each succeeding place is larger by a magnitude of ten, but somehow children fail to grasp the significance of this fact. The reason the standard addition algorithm works is because you are gathering up groups of ten at every place. Likewise, the reason the standard subtraction algorithm works is because you are breaking a group of ten at every place.

Students betray this lack of deeper understanding when they express surprise that given the number 437, that an equally correct answer to the question “How many tens?” is 43. They are also surprised to learn that when we say 2 tens and 5 ones equals 25, what we really mean is 2 tens and 5 ones equals 25 ones.

A better way to express it is “2 groups of ten and 5 loose (not in a group) ones equals 25 loose ones.” Therefore, I spend a lot of time having students expand large numbers in a variety of ways.

Methods of Expansion

Expansion basically means counting numbers of groups. There are several ways to express this accounting. Given the number 47,396:

Standard Methods:

Place Value Names: 4 ten thousands, 7 thousands, 3 hundreds, 9 tens, 6 loose ones

Multiplication: (4 x 10,000) + (7 x 1000) + (3 x 100) + (9 x 10) + (6 x 1)

Exponents: (4 x 10^4) + (7 x 10^3) + (3 x 10^2) + (9 x 10^1) + (6 x 10^0)

Notice that using exponents displays the idea that each succeeding place is a group of ten, however, most teachers do not make this understanding explicit. Most students just view, for example, the number 10000 or 104 as merely another way of expressing the place value name “ten thousands.”

I give my students practice with alternative expansions.

Alternative Expansion

47, 396 = _______ thousands, ________tens, _____ ones

47, 396 = _______ ten thousands, ________hundreds, _____ ones

47, 396 = _______ tens, _____ ones

And of course, we can repeat this exercise with multiplicative expansion and exponential expansion. This sort of practice has the side effect of helping students later understand rounding to a given place. I am also very picky about counting and zeroes. 0 is a real counting number, and I expect students to show that they know that 102 has 0 tens, or (0 x 10) or (0 x 10^1).

Place Value in Later Mathematics

This sort of foundational learning of place value pays dividends in later mathematics. To give just a couple examples:

Bases: Each succeeding place is a group of the given base. This understanding gives logic to “borrowing and carrying” in other bases besides base ten.

Polynomial expressions: Quadratic and other equations of the form Ax^n + Bx^n-1 + …Gx^1 + Hx^0 are essentially equations expressed in base x. Students will find that working in other bases is greatly simplified if they exponentially expand the number and replace the base with x.

Polynomial (and by extension, synthetic division: When students learn to divide equations such as Ax^3 + Cx^1 + D by say, x + 1, they must remember to insert the missing term, 0x^2. Students do learn to replace the missing term in a mechanical way. However, if they have regularly understood zero as a real counting number and included the zero term in their elementary expansions, it seems obvious to them that of course they must have the zero term if they expect to successfully complete the division.

More attention to a deep understanding of place value in the early years would make much of later mathematics less mechanical and more intuitively comprehensible, thus actually saving instruction time and allowing teachers to teach more math.

Zero is a Real Number

Zero is a real number. Could such a headline possibly be click bait? If so, it is pretty lame. Of course everyone knows zero belongs to the set of real numbers. The problem is the word “real.” A sentence such as “zero is a real number” immediately puts people into mathematics mode wherein they consider the word “real” in only its mathematical sense. Sometimes people recall set theory theory and the curious case of a set containing only one member, zero, as opposed to an empty set with no members. The problem here is that set theory leads people to objectify zero. They think of zero as an object rather than a number.

Zero is a real number. When the truth of this statement dawns, the world changes forever. If you are thinking, “Well, of course zero is a real number. What a stupid waste of time to write about it,” you may be one of those people for whom the realization of this truth in all its depth and beauty has not yet occurred.

My student teacher this year was one of those people in September. 27 years old and she never knew zero was a real number. She thought she knew it, but she betrayed herself when she began teaching first graders to answer the question “how many?.” Although she never explicitly said so, she gave her charges to understand that the minimum answer to the question was “one.” I surprised her by reminded her that “zero” is a legitimate answer to the question, “how many?” She did not quite believe me. “Think,” I said, “Of a time when you may have looked for eggs in the refrigerator and found there were zero eggs.” Her eyes widened. “Oh...yeah!” she said, “I hadn’t really thought about it.” I reminded her that when she set up her counting situations, to let zero often be the answer. Children come to school already preconditioned to disregard zero. Their parents and preschool teachers have given them 6 years of experience ignoring zero. One of the first math tasks at school is to undo that misconception.

Zero is a real number. Tax season provides a perfect example. Consider two taxpayers. One person may complete a tax return and find that his tax liability is zero. Therefore when he pays his taxes, he pays zero dollars. Another person does not even complete the form. One person paid no taxes. However, the other one did pay his taxes, and he paid zero dollars. “Zero” and “nothing” are not the same thing. Set theory was supposed to make this distinction clear, but too often we go into math mode and miss the point.

Zero is a real number. I will never forget the day in November when this realization struck my student teacher. She was in the middle of teaching first grade math when she looked at me sitting in the back of the room and said incredulously, “Zero is a real number,” as if it were her own discovery and not something I had said again and again for more than two months.

Is Division Repeated Addition?

One of Stanford mathematician Keith Devlin's pet peeves is the common “division is repeated addition” meme . He despises it so much he has something like a mantra, “Repeat after me. Division is NOT repeated addition.” Naturally, math teachers give him a lot of pushback because division is indeed repeated addition (except when it's not).

It seems there are actually two topics in play here. 1) Multiplication as repeated addition and 2) the skill of elementary math teachers. I completely understand his frustration with prospect of undoing the poor math instruction college students typically receive during their elementary school years. I also experience the same frustration as a secondary and college level instructor.

Multiplication as repeated addition is not a definition of multiplication, even though many elementary math teachers erroneously think the definition of multiplication is precisely repeated addition. Repeated addition is merely another name for the group model of multiplication. There are other models, such as the array model, the area model and the number line model, to name the ones most commonly presented to elementary students. Devlin rightly maintains that it is inaccurate to say that multiplication is repeated addition, period. As a misleading misstatement, it is right up there with “you cannot subtract a bigger number from a smaller number.”

However, as properly taught (a giant qualifier, I know), the group model is merely the first element in a teaching sequence which eventually progresses to the area model, then to the use of the area model to multiply fractions, and beyond. For example, you can definitely model a positive whole number times a negative rational number on the number line where it very much looks like repeated addition of the given negative number. Turn that number line vertically, and it makes even more sense to students because it reminds them of another number line they are very familiar with, the thermometer.

Devlin writes, “Addition and multiplication are different operations on numbers. There are, to be sure, connections. One such is that multiplication does provide a quick way of finding the answer to a repeated addition sum.” Exactly, and this is precisely the way a good teacher presents the group model. Children sometimes ask questions like, “Instead of saying 8 + 8 + 8 + 8 + 8, and then saying the answer, can't we just say “8, five times” and then say the answer?” Of course we can, and that is what we do when we say 5 x 8 = 40. The group model is meant to express this particular connection between addition and multiplication. The group model is not meant to be a definition of multiplication. Nevertheless, I agree there are too many elementary math teachers who fail to make the distinction, or properly progress through the models.

An umbrella idea I like involves the word “of” as an English language expression of multiplication. We can say “5 groups of 3,” or “5 groups of -3,” or “1/2 of 3,” or “1/3 of 4/5,” or “16/100 of 40” or “75% of 200,” etc and neatly cover most examples of multiplication that children are likely to encounter before junior high. Devlin prefers scaling as the dominant meme and argues that children should readily understand scaling because examples of scaling surround them. The problem is most eight-year-olds have difficulty comprehending scaling as a model and effect of multiplication. Even though they can readily see that a scale model is a perfect replica of the original, they do not understand how it is possible that doubling the dimensions of a garden (to take a simple example) results in a garden four times larger. Most of the scaling children see is usually on maps where the scale is for them an unimaginably large (or small, depending on viewpoint) number.

Teachers are better off working their way up to the scalar model of multiplication. I have found this is best done by reminding younger students early and often with the idea that we have not yet exhausted the possible models and applications of multiplication. I have found it useful to show some examples of these applications, and say something like, “Later you will learn how you can use multiplication to produce an exact scale model, or use multiplication to produce a real-life-sized object from a scale model.”

Actually, most students get their first solid grip on scaling when they work with similar figures (typically triangles) during high school geometry. Personally, I have found success with older elementary students by giving them basic practice in scaling on the coordinate plane or increasing recipe yield and other types of problems. Students also enjoy the products of their work whether it be art or good eats. The number line model is also a good introduction to scaling because you are scaling only one dimension, as opposed to the two and three dimensions involved in scaling area and volume, respectively.

Devlin also laments the constant push to make math “real.”

No wonder children arrive at college not only having little or no genuine understanding of elementary arithmetic, they have long ago formed the view that math has nothing to do with the world they live in...many people feel a need to make things concrete. But mathematics is abstract. That is where it gets its strength.

His comments seem contradictory, but they are not. One of the most enjoyable aspects of teaching math is showing students the leap from concrete to abstract. For example, I love showing students a cube and showing them the edge e, the face e x e, then showing them cube e x e x e. I usually let the idea hang, and love when someone asks if I can show e x e x e x e on the cube. No, I answer, I have nothing to show them that. And therein lies the power of math. Math can help us express ideas we can understand, but for which we have no physical representation. I am thrilled when someone asks if e x e x e x e can be time. I ask what would e x e x e x e then express. The children are really products of this century, and one of them is likely to answer, “the coordinates of a time-traveling spaceship.” So much fun. One time, a child said, “Maybe someday we will have a real meaning for more es.”

To the extent that teachers present repeated addition as a property, which in some applications, connects the operations of addition and multiplication, no harm done. However, considering repeated addition as THE definition of multiplication is a serious problem, and Devlin is right to be concerned about it. As Denise Gaskins pointed out, “...if (a particular) model doesn’t work universally, then (the model) certainly cannot be used to define the operation.”

Devlin asks an interesting, and in today's pedagogical environment, nearly taboo question:

The "learn the technique first and understand later" approach is very definitely the only way to learn chess, and millions of children around the world manage that each year, so we know it is a viable approach. Why not accept that math has to be learned the same way?

I would say that technique over understanding has been the preferred approach for centuries, but by all accounts, many adults have never made it to the “understand later” stage. In my experience teaching concept concurrently with technique works the best. The problem I am seeing is that these days too many elementary teachers attempt to teach concepts they barely understand, and then give short shrift to technique because the kids have calculators for that. The result is legions of kids who not only have faulty understanding of the concept, but also lack the ability to perform the technique quickly and accurately.

Additional Discussion:

https://www.quora.com/Why-is-it-incorrect-to-define-multiplication-as-repeated-addition

https://denisegaskins.com/2008/07/01/if-it-aint-repeated-addition/

http://scienceblogs.com/goodmath/2008/07/25/teaching-multiplication-is-it/

https://numberwarrior.wordpress.com/2009/05/22/the-multiplication-is-not-repeated-addition-research/

http://www.quickanddirtytips.com/education/math/is-multiplication-repeated-addition

http://billkerr2.blogspot.com/2009/01/multiplication-is-not-repeated-addition.html

http://rationalmathed.blogspot.com/2008/07/devlin-on-multiplication-or-what-is.html

http://rationalmathed.blogspot.com/2010/02/keith-devlin-extended.html

http://homeschoolmath.blogspot.com/2008/07/isnt-multiplication-repeated-addition.html

http://www.textsavvyblog.net/2008/07/devlins-right-angle-part-i.html

http://mathforum.org/kb/thread.jspa?threadID=2045768

http://www.qedcat.com/archive_cleaned/114.html

Nobody Understands Place Value

Parents often ask me to help their children with math homework. My reply is always the same. I am not interested in “helping” with homework. I am very interested in addressing the gaps and misconceptions that give children difficulty with their homework in the first place. Chief among these is a pervasive lack of understanding about place value.

Children's understanding is generally limited to identifying the place value of a given digit or inserting a digit in a given place. I do not blame the children. Most curriculum asks them to do nothing else. Replacing the terms “borrowing” and “carrying” with terms like “exchanging” or “regrouping” represented a tremendous improvement in math education. Even though many elementary math teachers these days play trading games, and the kids appear to know what they are doing, every junior high or high school math teacher has observed that they do not profoundly understand place value, so crucial to understanding the quadratic equation, bases other than base ten, and other topics.

Therefore, I usually start my homework help by first playing simple trading games with the student. The problems I use for the games are ones I know students can calculate correctly, such as 48 + 17. Maybe they can even do it in their heads. No matter.

The first thing I do is dispense with the usual place value names. I use “loose ones” and “packages of ten.” Loose ones is a better term than simply ones. The ones are ones precisely because they are not in a group. They are loose. Children often do not realize that the loose ones' place is fundamentally different from all the other places. Thus they often have trouble with the ones' place in other bases. For example, teachers tell students that when they are working in, say, base five, the largest possible digit in the ones' place is a 4 because “there is a rule that the one's place can be no larger than one less than the base.” Although it is true that the ones' place can be no larger than one less than the base, it is not because of a rule. The reason is much more fundamental than a mere rule. Understanding the ones' place as “loose ones” is key to discovering that fundamental principle.

After we play the trading game for a little while, I have the student do a simple addition problem. When students calculate a problem like 18 + 25, they put a 3 in the ones' place and a 1 above the 1 of 18. Then they add 1 + 1 + 2 and write a 4 in the tens' place, resulting in the answer of 43. Then I ask, “How did you get that answer?”

They usually reply, “I put a 3 in the ones' place and a 1 above the 1 of 18. Then I add 1 + 1 + 2 and write a 4 in the tens' place, so my answer is 43.”

That's fine. Of course they answer in a mechanical, non-mathematical way. They have heard teachers repeatedly explain addition problems to them in much the same way. Then I ask, “Yes, but why did you do that?”

Students invariably reply, “Because that is how the teacher told me to do it.”

Then I ask, “Yes, but why did the teacher tell you to do it that way? Why does it work?” Now they are stymied.

So I show them how to “prove” (not really prove, more like demonstrate) the answer using a picture (similar to this one, but simpler).

I show them how to draw the picture and talk their way through it. “See, you have 15 loose ones. That is enough to make a package of ten. So you gather up a package a ten and put it with all the other packages of ten. You still have 5 loose ones left. Because 5 is not enough to make a package, you leave them loose and show them in the loose ones' column. You add up the packages of ten and put that total in the packages of ten column.”I have them illustrate several problems by drawing the picture.

Very often students realize for the first time that place value is all about making groups of ten. Subtraction is all about breaking groups of ten into loose ones and dumping them with the other loose ones. Every place except the loose ones is a group of ten something. Teachers tell students that each succeeding place is larger by a magnitude of ten, but somehow children fail to grasp the significance of this fact. The reason the standard addition algorithm works is because you are gathering up groups of ten at every place. Likewise, the reason the standard subtraction algorithm works is because you are breaking a group of ten at every place.

Instead of the usual place value mat, I like to use a mat that labels the places a little differently. I start with the loose ones, then packages of 10 loose ones, then cartons of 10 packages, then boxes of 10 cartons, then cases of 10 boxes, then pallets of 10 cases, and so on. I usually stop at cargo ship with 10 shipping containers. Kids love it. Even second graders can easily calculate a multi-digit addition or subtraction problem. In fact, after kids master place value as groups of 10, they often ask me to set them problems with any number of digits. I usually refrain from a problem with more than 13 or 14 digits because even though the kids find the problem easy, they also find it tedious and time-consuming. But hey, tedious and time-consuming is a whole sight better than hard when it comes to doing homework.

Rote Does A Lot of People A Lot of Good

Recently I read a forum comment somewhere to the effect that “rote does not do anyone much good.” Ironically, this comment was part of a comment whose main idea was that educators should question buzzwords and dubious tenets of pop-education. In spite of the current popularity of this sentiment, rote actually does a lot of people a lot of good. I agree that sometimes educators have abused rote. I remember a fourth grade science book from around thirty years ago that included a whole unit on state birds and flowers as if such information had even the remotest relevance to science. Naturally, since most schools consider the textbook to be THE curriculum, many teachers felt compelled to “teach” children the state birds and flowers. Inquiry or constructed learning will not work for this kind of arbitrary knowledge. Rote is really the only way. State birds and flowers should never have been part of the science book in the first place.

Today we have Common Core. Just as in the past, of course publishers are scrambling to roll out new textbooks that reflect Common Core. Thus yet again, the textbook will become the default curriculum. Yet, even within Common Core and even within a commitment to teaching concepts over rote, there are still a number of topics for which rote is still the best method. The order of the alphabet, sight words, broad overviews of history, and arithmetic facts are just a few. People forget the intense amount of repetition and memorization involved learning a first language, let alone subsequent languages. Churchgoers know memorizing the books of the Bible greatly aids in finding the preacher's text. If you want to pass your first driver's license test, it would behoove you to memorize the rules of the road. Rote memorization of poems provides an avenue of future pleasure. If you live in a character-based literary system such as Chinese, you will need to memorize several thousand characters to simply be a literate person.

Another related canard holds that the purpose of education is not knowledge itself, but the ability to find knowledge. However, students who lack a substantial reserve of memorized knowledge have a great of difficulty even figuring out what search terms to use. Personally, as much as I hated memorization when I was young, I have come to appreciate the easy access to information, internet or no internet.

Finally, there is an important reason to refuse an ideological stance against rote learning. Sometimes it is all a student has left. Math is a subject area well-suited to discovery methods. All mathematical procedures are based on the real and predictable behavior of numbers. Very little math knowledge is actually arbitrary. The best math teachers who consistently use the best discovery methods to help students acquire mathematical concepts still sometimes encounter students who simply cannot get it. If they cannot acquire the concept, and we also also deny them rote learning, we leave them with nothing. Although rote should never be the first resort in a non-arbitrary domain such as mathematics, rote still remains the best last resort to ensure that all students acquire the basic skills they need for their adult lives. As educators, we need be at the forefront of confronting ideological statements wherever found

Nothing Wrong with Rote

A popular view among educators is that rote learning is bad, bad, bad. Point out how well Asians do in international tests compared to Americans, and defenders will likely counter that maybe so, but Asian education depends on that bad rote learning, but we Americans, no matter how poorly we compare, are superior because we emphasize concepts and creativity.

To be honest, as a math teacher, for a long time I believed that rote learning, while undeniably effective, was merely second rate. Kids must need lots of memorization of mathematical recipes and homework practice to pound poorly understood mathematical procedures into their brains. As a math teacher, I believed that if skilled math teachers developed a strong conceptual foundation within children, the logic and elegance inherent in math would minimize the need for tedious, time-consuming homework.

As a junior high and high school math teacher, my ideas about foundation building were only theoretical. It was easy to look at my students, the products of elementary school instruction and conclude that elementary math teachers were doing a terrible job of building foundations. After all, there is plenty of documentation for the inadequate math teaching skills of elementary teachers. I could be charitable. I did not blame elementary teachers too harshly, because they themselves did not acquire mathematical foundations when they were elementary students. They cannot teach what they do not know.

However, until I came to China to work with elementary students, I had never had a chance to test my hypothesis that all students really need is a great foundation. Actually, for most students my hypothesis worked. They could demonstrate a deep and thorough understanding of the concepts I taught. I often assigned homework of only five to ten problems, and after this little bit of practice, they could reliably get the right answers.

However, there were a few students for whom concepts were not enough. One day they would demonstrate terrific understanding. The next day we had to start almost from scratch. I tried everything, every approach I could think of, including a lot more practice. What worked when nothing else did was lots of practice---yes, tedious, time-consuming practice designed to activate rote memory.

I finally concluded that if a child can successfully utilize the concept to solve problems, that's great. However, if the only way they can master the procedure is to repeatedly execute it until they reliably get right answers (my standard was 80% or more), then so be it. Better that then sending them on their way with nothing.

Sadly, as comparative studies show, too many American children lack essential conceptual foundations because the documented lack of teaching skill means teachers fail to actually effectively teach the concepts. Students also fail to do adequate procedural practice because of the American educational aversion to “boring” homework. So I say let's teach concepts and teach them well, AND let the students practice until they can get the right answers. Some students may need more practice than others. So be it.

Interestingly, recent research on children's and teen brains explains why children have such great memories, and how practice, even in an environment of complete concept understanding, is necessary to build brain pathways.

the whole process of learning and memory is thought to be a process of building stronger connections between your brain cells. Your brain cells create new networks when you learn new tasks and new skills and new memories. And where brain cells connect are called synapses. And the synapse actually gets strengthened the more you use it. And especially if you use it in a patterned way, like with practice, it gets even stronger, such that after the practice, you don't need much effort to remember something.

When we dismiss rote learning, we forfeit a valuable tool for building neural pathways in the brain.

See related posts:

Patient vs Impatient Problem Solving

Common Cart---Cart Before Horse

I Love Manipulatives...But

Cultural Sacred Cows of American Education

MacDuff: The New Math

Slaying the Calculus Dragon

No doubt about it. Many students consider calculus scary, right up there with monsters under the bed. Calculus is the Minotaur or St George's dragon of math at school. Sadly, schools have done little to undermine its almost mythological reputation, what with “derivatives” and “integrals” and those frightening numberless equations recognizable by the initial elongated “∫.” There is like, what? 100 equations, that, according to most teachers, need to be memorized.

It is a pity, because calculus is really the Wizard of Oz, terrifying to behold, but quite tame behind the curtain. Did you know that most people do calculus in their heads all the time? In fact, because the numbers associated with calculus are ever-changing, moment by moment, doing calculus with numbers is a bit pointless. A mother filling the bathtub very often does not want to stand around watching the water. She knows that the water is coming out of the faucet at a certain rate. She knows the bathtub is filling at a certain rate. Every moment the volume of water is changing. Yet, she reliably comes back to check the tub before it overflows.

The high school quarterback and his wide receiver communicate an even more difficult calculus on the field, seemingly by telepathy. The quarterback never aims the ball at the place where the receiver is standing. That mental math is too easy, more like algebra or even arithmetic. No, he aims the ball toward the place he hopes the receiver will be. In his head, he calculates the trajectory of the ball, the amount of force necessary (oh my gosh, not physics, too!), the speed of the receiver, and every other factor. And most of the time he gets it right, and the pass is completed. The fun thing about calculus is that the numbers are ever-changing. It is like hitting a moving target, whereas algebraic numbers thoughtfully stand still.

At its core, calculus is nothing but slope. Remember humble slope, change in y over change in x. Slope is an expression of rate, such as, change in miles over change in time, most commonly called “miles per hour” or mph. The graph is a straight line, so you can pick any two points to find the slope. But what if the graph is curved? If you were to magnify each point and extend its line, every point has a different slope. That is because straight lines illustrate rates like speed (velocity), while curves illustrate rates like acceleration (getting faster and faster each moment), just like the football getting slower and slower until it reaches the top of its path, comes to dead stop (but only for a moment) and then gets faster and faster.

A graph has three basic pieces of information, the x data set, the y data set, and slope. “Derivatives” are used to find the slope of a curve at any point when the x and y data are known. When you know one of the data sets and the slope, you can use “integrals” to find the other data set. Techniques like finding the area under the curve are used to get as close as possible to the exact answer. First you divide the area under the curve into rectangles all having the same “x” length. Then you add up all the area. Obviously, some of the rectangles are a little too small for the curve and some are too big, so your answer is only an approximation. If you shorten the “x” length to make narrower rectangles, your approximation will be closer. Integration allows you to find an exact answer instead of an approximation, however close the approximation may be. But there are “limits” (you have no doubt heard of limits). Your “x” length may be very small indeed, but it can never be zero, because then the sum of the rectangles would illogically be zero.

If you can find a skilled dragon slayer, that is, a teacher who can demystify it, studying calculus can be great fun.