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.

Review: Zeroing In On Numbers and Operations

Anne Collins and Linda Dacey. (2010). Zeroing In on Number and Operations: Key Ideas and Misconceptions. Portland, ME: Stenhouse Publishers.

This is a set of four books (Grades1-2, Grades 3-4, Grades 5-6, Grades 7-8 ) formatted as spiral-bound flip charts with each page longer than the one before.

According to the publisher's description,

(Each book of the set) provides thirty research-based, classroom-tested modules that focus on the key mathematical strategies and concepts... while highlighting the importance of teacher language in the development of those skills. The flipchart format makes it easy to access the key resources: summaries that identify the mathematical focus and associated challenges and misconceptions; instructional strategies and activities that develop conceptual understanding and computation skills; activities and ideas for adjusting the activities to meet individual needs; reproducibles for instructional use; and resources for further reading.

First, A Digression

Primary math can and should anticipate algebra.

One reason students have trouble with algebra is that teachers typically lead students to believe it to be a new and harder topic, a different math than arithmetic. Algebra should be taught as a natural problem-solving strategy, even in first grade. On the page entitled “Join and Separate” from the book “Grades 1-2” (there are no page numbers), the authors present this story problem: “Jake had 5 knights for his toy castle. His sister, Emma, gave him some more knights for his birthday. Now Jake has 11 knights. How many knights did Emma give Jake?”

How do we usually teach children to approach these kinds of story problems? We drill them on math fact families so they will recognize a math fact buried in words. Now I believe children should memorize their math facts, BUT I also think that language can be an ally rather than the enemy it usually becomes. The natural progressive analysis for this problem starts “5 knights (gave more) ?knights (so now) 11 knights,” proceeding to “5 knights plus ? knights equals 11 knights,” and finally “5 + ? (or box) = 11.” The “?” (or box) is what algebra calls “a variable” and if replacing the “?” or box with a lower case letter makes the expression look like algebra. Even first graders can set up the problem with a manipulative like the Algebra Gear by putting a turquoise piece (standing for the unknown) and five yellow cubes on one side of the equivalence mat and eleven yellow cubes on the other side. The child removes five yellow cubes from both sides to isolate the turquoise piece on one side, and voila! There are six yellow cubes on the other side. A big advantage is the lack of reliance on numerals.

We do not give children enough credit for their ability to think. We discourage thinking by presenting mathematics not as something that can be reasoned about, but as something that must be memorized and accurately recalled. If they do not remember, they have no recourse. If they fail to remember often enough, they soon conclude wrongly that they are bad at math. Math-phobia is just one more short step away.

The text's idea that join and separate problems both share the start-change-end structure is helpful, but the graphic organizer is not obvious or intuitive to children. The teacher would be better off going straight to the algebra gear which requires a lot less instruction and makes more intuitive sense. Then the problems can be “played” like a game.

On page A14 of Grades 1-2, there are three examples of sentences where all three components (start-change-end) are left blank. The idea is to play around with providing any two out of three. The authors rightly note, “leaving the initial state blank is the most challenging, as many students are uncertain where to begin.” The algebra approach addresses and eliminates this uncertainty. Students simply use the turquoise cube to stand for the blank and march on.

These Books Are Necessary

Teachers need a resource that explicitly addresses the common misconceptions children (and their teachers) hold about math. Sometimes teachers deliberately teach misconceptions because they do not know any better.

The set is comprised of four very slim volumes of fifteen informational pages and about fifteen pages of problems and exercises for a total of thirty pages printed on both side of the paper. Thus the entire set is about 120 pages. The list price per single book is fifteen dollars and sixty dollars for the set of four. In a strategic marketing maneuver, by dividing what normally would be one book into four, the publishers may be able to capture more income. Teachers are likely to be most interested in the information pertaining to the particular grades they teach. A teacher might not be inclined to pay sixty dollars for largely “irrelevant” material (although that point could be argued), but may willingly spend fifteen dollars for grade-specific content. The books could also be useful to education students.

I intended to read a sampling of pages from each book very carefully and peruse the rest. I wanted to get a feel for the quality of the information across the scope and sequence. I ended up reading all four books line-by-line, analyzing the references and working the problems. I made copious notes on every page.

A Selection of Some Glittering Gems

Grade 1-2, Counting by Tens and Ones: I like the concept of “counting the tens and the leftover ones.” In fact, I like to rename the “ones” place the “leftovers” because they are not in a group.

Grade 1-2, Writing Numbers: Children can learn a lot of math without numerals. This page has a good strategy for using cards to illustrate digit positions.

Grade 1-2, Equivalent Representations: This is a valuable trading exercise, all the better because it is done on the overhead, avoiding the possible “magic” of computer trading exercises. Computer simulations of physical activities often look like magic to students. They resign themselves to taking the teachers word instead of understanding for themselves.

I would only caution the teacher to make sure the students very intentionally see all aspects of the trade. Students need to be certain that the teacher added nothing nor removed anything. I actually prefer a manipulative like Digi-Blocks Each box can hold only ten units. Dumping the box to simulate “borrowing” makes the trade crystal clear. I have seen even junior high students incredulous that after dumping the box, the total number of packed and unpacked units did not change. These students have done trading activities with base ten flats, rods, and cubes without ever acquiring true conservation of number.

I also like the use of the term “equivalent” as opposed to “equal” because the form of 3 tens 2 ones is not identical to the form 2 tens 12 ones. Carefully distinguishing the difference implicitly anticipates “equivalent” fractions, where two fractions of differing appearance are equivalent because the underlying value is equal. The use of “equivalent” helps build consistency, for example, in geometry, when students must differentiate equal measure as opposed to identical and/or congruent. Perhaps the “equal” sign should be renamed the “equivalent” sign because equivalent is what we usually mean.

Another valuable way to exploit differing representations is to use different ways to record the model. What the authors call equivalent “representation” is actually equivalent variations of the model, in this case, base ten blocks. There are a number of ways to represent the model, drawing a picture or coloring preprinted diagrams of rods and cubes, using written words, numerals and symbolic language like 3r2c = 2r12c, where r stands for rods and c stands for cubes. Older students also benefit from using various types of representation.

Grades 1-2, Subtraction Is More Than Take Away: I like the discussion of the different meanings of subtraction. In keeping with the importance of language precision, teachers should say “three plus five equals eight,” not “three and five are (or is) eight.” Even better,”three plus five is equivalent to eight.”

Grades 1-2, Modeling Addition and Subtraction: Of course I like this page if only for the reference to Digi-Blocks. The Win 300 and Lose 299 activities are gratifyingly similar to my Chocolate Factory activity, inspired by an I Love Lucy episode



Grades 3-4, Helping Facts: Students who are acquiring profound understanding of fundamental mathematics still need fluency with facts. This page contains useful tips for recalling and reconstructing multiplication facts.

Grades 3-4, Meaning of Division is a good explanation of the various types of division. The authors did a good job with Remainders, even providing a nice segue into bases. I also liked the Multiplication Menu, and the discussion of the meaninglessness of “gozinta” and misconceptions inherent in the long division algorithm.



Grades 3-4: Number Lines and Benchmark Fractions: I like the emphasis on kids sharing and explaining their strategies to each other, but instead of singling a child out as the authors so often do, let the children work in groups and have a group spokesman present the group's findings to the class. The authors often state that “a student” did this or that, showing the individualistic bent of American education, as opposed to, for example, Japanese elementary schools, where math activities are generally group activities.

Grades 3-4: Finding Parts and Making Wholes contains a nice list of misconceptions.

Grades 3-4: Parts of a Group: American egg cartons are very useful for modeling fractions. Instead of putting any old counters in the egg cups, it is better to use plastic eggs in up to six colors. Then the ribbons are unnecessary and the egg cartons can be used to play fraction games with even first and second graders. As an aside, Japanese egg cartons hold ten eggs, making them ideal for place value lessons.

Grades 5-6 Greatest Common Factors and Least common Multiples:I like the Venn diagram for finding common factors.

The authors really shine when it comes to fractions, I liked six pages in a row:
Fractions on the Number Line “Fractions are used in three distinct ways: (1) as numbers, (2) as ratios, (3) as division.”
Adding and Subtracting Fractions with Pattern Blocks, good explanations and activities.
Modeling Multiplication of Fractions, good activities.
Modeling Division of Fractions with Pattern Blocks, avoids multiplying by the reciprocal.
Dividing Fractions with Area Model
Posing Problems With Fractions

Grades 5-6, Estimating Decimals: I like the emphasis on the significance of zero “placeholders” as indicators of precision because of the connection to measurement and data recording in science. The authors also point out the problems with “context-free” computation. Real math occurs in a context. Real math always has a story. Numbers have referents.

Grades 7-8, Analyzing Change: The story graphs nicely anticipate the early topics of physics.

Some of the Quibbles and Errors

Grades 1-2, Connecting Representations: I would have liked the confusion over the difference between number and numeral or other representations explicitly stated, however this major misconception is implied in the text and diagram. Whenever I show my Japanese rulers (which have no numerals) to kids, they wonder how it is possible to measure anything with such strange rulers. Letting them figure it out for themselves is quite a worthwhile group activity.

Grades 1-2, Counting by Tens and Ones: Students are asked to count the strawberries on page A6 of the appendix. There are forty-two strawberries arranged in a 7-by-6 array. Then students are asked where they see the 4 of forty-two in the strawberries, and then where they see the 2. I could understand if students had been asked to circle groups of ten, so I am not sure what the authors had in mind.

Grades 1-2, Along the Line and Open Number Line: Although the authors correctly describe the integer “2” as being units units away from zero, the origin (positive direction understood), they abandon origin two paragraphs later. Nearly every presentation of the number line in all four books fails to start from zero. It is important to emphasize that 3 + 5 does NOT mean “start at 3.” It means start at 0, and go 3 units in the positive direction, and then five more units in the positive direction. Maintaining a sense of origin helps students to understand absolute value later.

Digi-Blocks points out,

Note that with drawn number lines like this one, you are supposed to count the steps. Here we see that there are 3 steps between 0 and 3. But often times children try to count the hash marks. This becomes confusing. Do they count three hash marks or 4? With the Digi-Block number lines, it is entirely clear that there are 3 blocks.

I actually prefer Cuisenaire's “Rod Track” over Digi-Block's number line because you can turn one track vertical to model not only multiplication arrays but also distributive property and quadratic equations. Cuisenaire used to have a rod track for modeling negative numbers (I have one), but it is apparently no longer available. More's the pity.

Grades 3-4, Mental Computation: It is not true that when adding 56 +6, it requires greater skill to mentally perform the standard algorithm than to first add 56+4 = 60, then 60 + 2 = 62. Furthermore, the arrow code on this page adds an necessary layer of complexity. Finally, if the authors are worried that on a standard hundred chart, bigger numbers are below smaller numbers, try rewriting the chart with 1-10 at the bottom instead of at the top.

Grades 3-4, Column Addition: The authors present a trick for adding a column of numbers. Tricks work because of the math behind them, but they are no substitute for understanding the math. It is a neat trick, but should be introduced after scaffolding.

Grades 5-6, Fact Practice: There is indeed a difference between practice and drill, but practice is not “doing mathematics.” It is doing procedures that work because of the mathematics behind them. We have to constantly remind ourselves of the difference between procedure and concept. It becomes more clear when we remember that a procedural explanation is by no means mathematical. There is no math in telling a student to move the decimal two places to the left when multiplying by 0.01, regardless of the presence of numbers in the explanation.

Grades 7-8 Integers on the Cartesian Coordinate Plane: The Cartesian plane models the multiplication of variously signed integers only if ground rules are arbitrarily established first. The authors do not develop a rationale for the first and third quadrants containing positive products, and the second and fourth quadrants containing negative products.

A Sampling of Editorial Issues and Typos

Grades 1-2, Equality: The string has two “9”s, with one superscripted in a box. I suspect a misprint.

Grades 3-4, Two-Digit Multipliers: “Where are the 24 square feet for cucumbers...” should say “42 square feet.”

Grades 3-4, Problem Solving with All Operations: delete “this teacher read” in the phrase, “... an article this teacher read by Kim ...” so it reads “...an article by Kim...”

Grades 7-8, Finding Factors With Square Roots: This whole page is done completely wrong. I do not believe the challenge to “find a prime factor of a number that is greater than its square root” was ever issued by the teacher in the story. I do not believe the students looked all week without finding a single one, when there are millions of examples. It would not take them a minute to figure out that 5, a prime factor of 15, is greater than its square root of 3.87. It makes me wonder how many other stories are fabricated and do not represent the experience of real students at all. The page overlooks that pairs of factors align on both sides of the square root. The reason students have only to check the prime factors up to the integer of the square root, and that every prime so checked will pop out its corresponding friend on the other side of the integer of the square root. Some of these friends might also be primes greater than the square root. Also, there is a mention of an author named Zany, but no citation.

Grades 7-8, Unit Rates: The purported student quote under the table does not make sense in light of the data in the table. The student would not have said what he is reported as saying.

Grades 7-8, Exponents, (A16) Answer: A step is missing from the proof of Josh's conjecture in problem #5. The way it is presented, there is no obvious reason to add the exponents.
Solving Problems with Ratios (A30): Problem number 3 needs to be rewritten from scratch or deleted. It is nearly incomprehensible to junior high as is.
Making Rate Tables (A31): The answer in the back does not correspond to the first story problem. Also, there are some additions I would make to the graph designs to help anticipate graphing data in science classes.
Answers in the Back A28-2: Second sentence is the wrong reason.
A29: ¾ does NOT equal 2/3

Conclusion
Overall, the books could have used some serious pre-publication editing. There are some sparkling gems of insight sprinkled throughout. The authors' strong suit is clearly fractions. However, there are too many outright errors and too many missed fundamental misconceptions. The authors' use of number lines consistently overlooks the importance of starting at zero. Even though there are references to algebra, the books often miss opportunities to anticipate advanced material. Furthermore, the authors inconsistently evaluate the math skills of their target audience, elementary and junior high math teachers. The authors note that many misconceptions are shared by student and teacher alike, yet write as if these same weak teachers will be able to follow the many oblique references to specific math concepts. In “Grades 3-4: Adding Numbers in the Thousands,” the authors allude to the main problem with the spiral curriculum, but do nothing to challenge it. Sadly, the spiral curriculum is a major factor in students moving from grade to grade without learning the subject matter. Although the authors often mention the mistake of emphasizing procedure, “what is most important,” they write, “is that students develop a reliable technique...” Perhaps the authors are being practical. A reliable procedure is better than nothing, I guess.

The authors seem more at home with upper elementary math topics and a bit at sea with primary math topics and middle school topics. Since misconceptions, once acquired are difficult to unlearn, I would have preferred the strongest treatment of primary math, where foundations are laid, for better or for worse. The authors overlooked some important researchers. Jean Piaget and Constance Kamii come immediately to mind. On the other hand, there seems to be an implied rule: avoid references from pre-turn of the century, as if all important work is relatively recent. In-text citations are often missing. Although the teaching ideas are billed as being research-based, most of them look to be anecdotal accounts of one or another teacher's favorite lesson. Researchers are fond of denigrating “unscientific” research teachers do every day, forgetting that teachers do not have time to wait for the verdicts from “the ivory tower.” Lessons need to work immediately. Good teachers are constantly customizing, adjusting and refining.

I would like the appealing color scheme and the flip chart design of the books more if all the pages were the same length. Each page is about a quarter longer than the one before with the page topic as the footer on each page. The design is attractive and convenient because all the topics have the appearance of being tabbed on the first page. However, the design also guarantees some topics get short shrift simply because the page is half as long as other pages. I do wish the content pages were numbered. The many obligatory nods to what may turn out to be educational fads annoy me. Furthermore, it should not be necessary to explicitly market to RTI or promise standards alignment. The suggestions for calculator use add nothing. In fact, there is no evidence that calculators enhance number reasoning skills in the early grades, NCTM claims to the contrary notwithstanding. To their credit, the authors acknowledge the value of Montessori materials.

To summarize, the books could be useful resources for the novice teacher, but they are too expensive, and the novice teacher will likely not have enough experience to recognize the flaws. Even so, I might be willing to recommend the books if they were four or five dollars each instead of fifteen dollars. I had such high hopes for this material, and I regret I cannot give it a stellar recommendation.

The Agrarian Model Myth

The model of public education is not primarily agrarian, although the needs of a predominately rural population may have influenced the school calendar back in the beginning. It is not even predominately industrial, although the way factories were organized strongly influenced the organization of schooling.

The most salient model of public education is the model of the mind we inherited from the Age of Enlightenment.

Without further ado, I give you the animated illustration of an intriguing talk by Sir Ken Robinson.








Education reform is stuck in a rut because society has not confronted its most basic unexamined assumptions. I have often said we need a complete systematic overhaul. I was satisfied to mod the car, but no longer. Sir Ken goes further and says we need a new paradigm. He wants to throw the car out completely. Furthermore, he does not want to buy a new car. He wants to build something a completely different vehicle, perhaps one we have not imagined before.