Animals Smarter Than We Think

In a departure from the usual topics of a blog devoted to education, I would like to draw your attention to the fascinating title article in the August 16, 2010 issue of Time magazine, What Animals Think: New science reveals they're smarter than we realized. I feel quite confident that anyone who has ever loved and been loved by a dog, a cat or a horse never doubted the surprising cognitive and emotional intelligence of animals. Time drew a similar conclusion back in 1993.

My childhood dog, an Australian shepherd proud of the beautiful “feathers” festooning his legs and underbelly, hid behind the couch for an entire day in apparent humiliation after I trimmed those feathers. We were about to depart for my summer job at a YMCA camp and I wanted to spare him the discomfort of the inevitable burrs that would permanently lodge in his fur after his first romp through the monkey flowers.

Monkey flower oil is like super glue. I worried that perhaps he had forgotten the lesson of the previous summer. He had spent the entire first night trying to pull them out, keeping me and my cabin mates awake with his futile gnashing of teeth. With the dawn, I finally had enough light to cut the burrs out of his fur. He stayed out of the monkey flowers for the rest of that summer. Or maybe he was indignant that I though he forgot his lesson.

The first two weeks of camp were reserved by a program for developmentally delayed adults. Camp staff watched in amazement as that dog recognized, respected and then overcame every single camper's fear of dogs, or at least fear of this particular dog. In the first week of his first summer at camp, with no training whatsoever (in those days no one knew dogs could be trained for this), he became adept at warning me when a camper was about to have a seizure. Eventually he died, gray, arthritic and senile, at the age of seventeen.

Unlearning to Read

From the July 8, 2010 issue of the Economist, comes an intriguing story about learning to read. According to Stanislas Dehaene, a cognitive neuroscientist at the French medical-research agency, INSERM, learning to reading involves forcing the brain to unlearn an earlier pre-wired fundamental survival skill, the ability to recognize an object and its mirror image as identical. Dr. Dehaene believes learning to read also means sacrificing some facial recognition ability.

His studies suggest that one small area of the brain’s visual system is particularly activated by the written word. Dr Dehaene calls this the visual word form area (VWFA). Researchers debate the extent to which this area is specialised for word recognition, since it also responds to pictures. But Dr Dehaene thinks the VWFA evolved for object recognition and is requisitioned for word recognition. Unfortunately, it has one property that, though valuable when recognising objects, is not helpful for reading: more than other parts of the visual system it is activated both by an object and by that object’s mirror image.

Dr. Dehaene's theories shed new light on dyslexia.

It was thought that only dyslexic children were prone to confusing “b” and “d”, and “p” and “q”, and occasionally writing their names back-to-front, but Dr Dehaene has found that all children make this error.

If Dr. Dehaene is right, dyslexia is a consequence of normality, not an aberration. It is reading that is distinctly “abnormal.” The wonder may be that so many of us successfully learn to read.

(Dehaene) suggests the error happens because when learning to read children first have to unlearn that older survival skill. If he is right, then in adults the VWFA should be insensitive to which of two mirror images it is viewing when it comes to pictures, but sensitive to that distinction when viewing words and letters. This is indeed the case.

Dr. Dehaene's results could point to new reading instruction methods.

Obama's Speech Riles Educators

I read and listened to President Obama's speech to the Urban League defending RTTT. As "Master Educator" observed in the ensuing comments to an EdWeek blog post, “It is painstakingly clear that extraordinary measures need to be put in place to greatly improve classroom learning, strengthen school infrastructures and increase student achievement.”

The first part of the speech to rile some people is this:

“Part of (RTTT opposition), I believe, reflects a general resistance to change.  We get comfortable with the status quo even when the status quo isn’t good.  We make excuses for why things have to be the way they are.  And when you try to shake things up, some people aren’t happy.”

I have some grave misgivings about RTTT. That being said, when President Obama talks about a general resistance to change and being comfortable with the status quo, he is practically quoting from any Psychology 101 textbook. People do resist change and cling to the status quo, and surely some of those people are teachers. By no means is he indicting all teachers who oppose RTTT. There are legitimate criticisms. What concerns me is that Mr. Duncan, and by extension, the administration, seems to give lip service to listening to teachers, but teachers report that they do not feel heard, never mind agreed with.

After spending several paragraphs on the importance of teachers and the need for societal support and esteem of teachers, Obama said,

So I am 110 percent behind our teachers. But all I’m asking in return -- as a President, as a parent, and as a citizen -- is some measure of accountability. So even as we applaud teachers for their hard work, we’ve got to make sure we’re seeing results in the classroom.  If we’re not seeing results in the classroom, then let’s work with teachers to help them become more effective.  If that doesn’t work, let’s find the right teacher for that classroom.”

Accountability is always a reliable hot button. But as long as accountability is based on student test scores, there will be legitimate opposition, as expressed in the following EdWeek comments:

from Curiousidle:
1) Teachers have no voice in the system as it pertains to curriculum, materials, scheduling, educational philosophy, class size, etc. So, while we might have some "expertise" to bring to the mix, our input is not asked for or required by local, state or federal administrations. It's the ultimate catch22: literally no say in how schools function combined with all political responsibility for the effects of poverty on educational success.

2) The two goals of the education system: education and social engineering can and do conflict with one another at times... that is to say, not every policy decision feeds both priorities equally well. The public system tend to err on the side of social engineering... examples include requiring everyone to meet the same bar and dumbing down the curriculum so it can happen, removing important subject area skills that are necessary building blocks for later instruction so that students that don't have those blocks in place are able to remain at grade level, grade inflation, credit recovery, social promotion and other disreputable practices, heterogeneous grouping during the day for social development remediated by funded homogeneous grouping after school.

3) What happens OUTSIDE of school has a far more pervasive influence on preparation, willingness and academic success than what happens INSIDE the classroom. IF we don't address failure at the root, we're just playing politics and avoiding the really hard conversations.


From DDKona: We just don't agree with his administration's definitions of these ideas. Hold me accountable for planning and implementing lessons that engage students at their different levels. Hold me accountable for what I do as a professional. But the minute you define accountability as how my students do on a standardized test, that is the minute you ignite my opposition.

From MarkAHarris: What I find comical is that people love using the word "accountability" with teaching. Yet, no one uses it the way it should be: you hold teachers responsible for what they do. Testing does not do this.

From CEB: (Obama) dismisses legitimate concerns about his administration’s agenda as resistance to change or defense of the status quo. He is so insultingly wrong. Critics of RttT want to improve education just as much or more as he and the tycoons who pull Duncan’s strings.

Mr. Obama attempts to address testing:

When we talk about testing, parents worry that it means more teaching to the test.  Some worry that tests are culturally biased. Teachers worry that they’ll be evaluated solely on the basis of a single standardized test.  Everybody thinks that’s unfair.  It is unfair.
 
But that’s not what Race to the Top is about.  What Race to the Top says is, there’s nothing wrong with testing -– we just need better tests applied in a way that helps teachers and students, instead of stifling what teachers and students do in the classroom.  Tests that don’t dictate what’s taught, but tell us what has been learned.  Tests that measure how well our children are mastering essential skills and answering complex questions.  And tests that track how well our students are growing academically, so we can catch when they’re falling behind and help them before they just get passed along. 

I am pretty sure that if testing showed that American students were actually outperforming the world, no one would object to testing. A major impetus to RTTT is the poor comparative performance of American students. If students master what I teach as demonstrated by, AMONG OTHER MEASURES, the test scores on tests I write, then I am teaching successfully even in the face of outside influences I do not control. Of course, that assumes I am not gaming my tests as some teachers have done by handing out "study guides" that are nothing more than the test itself.

There is quite a small range of quality in the standards of different states. Furthermore, regardless of what individual state standards say, most teachers teach the curriculum as expressed by the textbook, not the state standards. Teachers tend to write their lesson plans based on the textbook, and then code those plans to the state standards. Very few teachers start their planning with the state standards, and very few teachers use the textbook as just one resource among many. My high quality lessons have sometimes been criticized as too textbook-independent. Students, parents and administrators do not believe that teachers have their own knowledge apart from the textbook.

In Japan, because the national standards drive the curriculum and the textbook material, Japanese teachers do not explicitly teach to the test. Testing and curriculum are automatically aligned. In fact, students' classroom tests are often written by someone other than their teacher. It works like this: There are six tests per subject in a Japanese academic year. The teachers take turns writing the tests. For example, there are three grades in junior high, so eighteen math tests will be written in an academic year. If there are ten teachers in a junior high math department, during the year each teacher will write one test for all math sections of a particular grade and most will write two.

The real problem, the one that goes unnamed, is anxiety. The number one reason teachers, many of them good teachers, leave teaching is lack of administrative support. There is simply no relational trust between teachers and administration that would reassure teachers of fair application of policy. Teachers have no reason to believe that

(RTTT's goal) isn’t to fire or admonish teachers; our goal is accountability.  It’s to provide teachers with the support they need to be as effective as they can be, and to create a better environment for teachers and students alike.

I once taught in a private school that was forced to shake things up. In its twenty-five year history, it had never sought accreditation. The headmaster decided to pursue WASC accreditation, but it was up to the teachers to do the boatload of extra work. They groused---loudly. It turned out I was the only one among them who had actually ever been through the WASC process, so I became the de facto leader. The very process of completing the self study forced the teachers to deeply examine themselves and their methods for the first time. Without spending an extra dime (although I think some overpay would have been in order), in less than one year, the teachers transformed the school from the lowest achieving school in the area to one on par with the best schools.

Do Teaching Credentials Matter?

EdWeek reported a recent study on the effectiveness of credentialed teachers which concluded that credentials do matter. Most of these types of studies usually contain too many unexamined assumptions. Furthermore, they often implicitly reject the apparently unimaginable idea that a highly qualified teacher could possibly be credential-less.

What about teachers who move from another state? Not all states grant reciprocity. Sometimes the new state may grant a provisional (usually 2-year) credential convertible to a regular credential if the holder gets a job within the provisional time period. Since districts avoid hiring incoming experienced teachers, the credential is likely to expire. Although not any less competent or qualified, now the teacher is even more unemployable than before.

Here is a true story: One 20-year veteran teacher moved to a new state and got the 2-year provisional credential. She began working for the district's homebound teacher program even as she continued looking for a "real" teaching job. She also worked in the education department of the local university helping undergrads earn their own teaching credentials.

Two years passed. Her credential expired. She lost her homebound teaching job because without a credential she was no longer "qualified" for the job even though she had just posted two years of highly effective experience in the very job under the supervision of the very homebound program director who let her go. Such bitter irony. The supervisor who should have been a superior job reference let her go as "unqualified" to teach.

Even tutoring situations closed down when parents asked her if she was certified. Parents' glazed over when she would begin to explain how she was once certified, but not now. The only word the parents heard was "uncertified," a deal breaker in a society conditioned to believe that certification equals competence.

NCLB or no NCLB requirements for highly-qualified teachers, such a highly qualified teacher no longer even applies for teaching jobs. When she had the credential, her masters degree and tons of experience counted against her, but at least the principals would be apologetic. Now she has three strikes against her. The lack of credential gives principals permission to hastily and rudely dismiss her as unqualified. She does not feel she should keep donating money to the state to continually renew a provisional credential which fails to help her get a job.

With regard to master's degrees, the researchers' findings were a bit more nuanced. Teachers who had earned a master's degree before entering the field were no more effective than those without master's degrees. But teachers who got a master's degree after they began teaching were found to do a better job at boosting students' test scores than did their less-educated teaching peers.

I would guess that masters degrees earned by practicing teachers correlated with test scores because after a few years experience, teachers in the process of earning their masters degree look for specific take-aways in every class they attend. Their participation is laser-focused on acquiring immediate applications for use the very next day. They continually ask themselves, "How can I use this tomorrow?" For those going straight from the bachelors to the masters, I expect their attention is more diffused.



This study has too many flaws to be of much use, but one conclusion makes sense.

And it's possible, these researchers say, that the end-of-course exam scores used for this study may actually be a better barometer of what goes on in a classroom than the broader exams that students take in earlier grades."

End-of-course exams test what was actually taught. Whatever a teacher might have done effectively, an end-of-course exam would be more sensitive to that effectiveness than a standardized test.

Nothing Personal, Just Money and Politics

In an article headlined, A Popular Principal, Wounded by Government’s Good Intentions, the New York Times tells the appalling story of a highly effective principal dismissed from her position so that her school could qualify for millions of Federal dollars.

Ms. Irvine's job woes serve as a cautionary tale of the perils of top-down, one-size solutions in a country without a tradition of nationally centralized education oversight. By all accounts, Ms. Irvine is a great principal. Her accomplishments at the school are legion: she rolled out many enrichment programs, developed a new arts curriculum, created community partnerships, and reinvented the school as an arts magnet. None of it counts in the face of the school's low test scores. What is more, it is not like her raw material started out with advantages.

The New York Times used the wrong adjective in their headline. It should read, “A Highly Effective Principal, Wounded by Government's Good Intentions.” That she is also popular is beside the point.

At the heart of things is whether the testing system under the federal No Child Left Behind Act of 2001 can fairly assess schools full of middle-class children, as well as a school like Wheeler, with a 97 percent poverty rate and large numbers of refugees, many with little previous education.

...snip...

About half the 230 students are foreign-born, collectively speaking 30 languages. Many have been traumatized; a third see one of the school’s three caseworkers. During Ms. Irvine’s tenure, suspensions were reduced to 7 last year, from 100.

Japan's education system must surely be one of the most rigid of nationally centralized systems. The Ministry of Education has policies for everything. It has been said every student, no matter where in the country, is on the same page of the textbook on any particular school day. That assertion only slightly overstates the case. Every teacher in every school hews closely to the national curriculum, using one of a handful of nationally approved textbooks. For example, only six textbooks are approved for use in middle school English classes. Nevertheless, Japanese administrators may make discretionary judgments on a “ke-subaike-su” (case-by-case) basis. How did the supposedly more creative and individualistic American officials respond?

Justin Hamilton, a spokesman for the United States Department of Education, noted that districts don’t have to apply for the grants, that the rules are clear and that federal officials do not remove principals.

A school with as many immigrants as Ms. Irvine's requires a customized assessment. By definition, a standardized test normed to an American population will be unable to capture the knowledge and experience of immigrants. I have a true story about an American boy born to middle-class parents living in Japan. He went to Japanese pre-school and kindergarten. His mother wanted to enroll him into first grade at an American school. The first grade teacher gave the boy a readiness “test” which consisted of asking him questions about an American fairy tale.

The boy “failed” because he did not know who Rumpelstiltskin was. Never mind that the teacher had never heard of Momotaro, a famous character from a Japanese fairy tale. Never mind the boy was reading and doing math at higher levels than his peers. Because the Japanese and American academic calendars are very different, there was only two months left to the American school year. She admitted the boy conditionally into the first grade with the caveat that he would probably be starting first grade over in September. Imagine her surprise when this same boy scored in the 99th percentile on every section of the first grade Stanford 9 test just a month later.

So often reading tests can become experience tests when given to students outside the intended population -- like immigrants.

Students take the reading test after one year in the country. Ms. Irvine tells a story about Mr. Mudasigana’s son Oscar and the fifth-grade test.
Oscar needed 20 minutes to read a passage on Neil Armstrong landing his Eagle spacecraft on the moon; it should have taken 5 minutes, she said, but Oscar was determined, reading out loud to himself.

The first question asked whether the passage was fact or fiction. “He said, ‘Oh, Mrs. Irvine, man don’t go on the moon, man don’t go on the back of eagles, this is not true,’ ” she recalled. “So he got the five follow-up questions wrong — penalized for a lack of experience.”

Meanwhile, the community is deeply impressed with the school, regardless of its test scores. Many middle class parents have enrolled their children, so many that half the children in the early grades will be middle class. Even if the school does nothing more, when these kids take the standardized test, the results will show dramatic improvement. Will Ms. Irvine get her job back then? Does she have a job now?

The district has replaced Ms. Irvine with an interim principal and will conduct a search for a replacement.

And Ms. Irvine, who hoped to finish her career on the front lines, working with children, will be Burlington’s new school improvement administrator.

On the other hand, if Ms. Irvine has successfully set her school on a sustainable growth path, perhaps she should go and work the same magic on other schools in dire need of her skills. Furthermore, she should spare not a moment's thought about the opinions of some segments of education society who seem bent on belittling achievement.

Yes, What About Student Accountability?

Brian Toporek began by asking, “What About Student Accountability?”

If college professors hold students 100 percent accountable for their own learning, shouldn't K-12 teachers ease students into taking responsibility for their own education? That's the question Daniel Willingham, a psychology professor from the University of Virginia, asks in a guest post on the Washington Post's Answer Sheet blog.

Willingham notes that among his university colleagues, the professors aren't concerned with whether students show up to class or study. "Most professors figure that their job is to teach well. Whether the student learns or not is up to him or her," he says.

First, I would quibble with the statement “professors aren't concerned with whether students show up to class or study. 'Most professors figure that their job is to teach well. Whether the student learns or not is up to him or her.'”

When I was a college student, professors did not assign "homework." We assigned it to ourselves. We did as many math problems as it took to get comfortable. We asked about the problems we could not solve. In a typical class period, there would be questions about maybe three or four of the problems. Students who did not do the work on their own initiative would suffer the natural consequences at test time.

I had one professor, an immigrant from kindergarten, who assigned and collected homework from her college students. The students vilified her for treating them as children. Students say they wish to be treated like adults, but today, professors who treat students as I was typically treated (that is, like and adult) when I was a student are likely to find themselves in front of the department chair defending their practice.

The university is no longer a university. It has become a customer-driven marketplace where the customer is always right. In my student days, complaining students got nowhere with the administration even if their complaints were warranted. Consequently, there were sometimes abuses of power. However, the wholesale adoption of the market model has not produced better outcomes.

As Judith Steele points out in her comment, "All teachers of college students are familiar with these common behaviors and habits that result in low or failing grades: late arrival to class, high absenteeism, no book purchased, work not turned in, unprepared for class, sleeping in class, and the ubiquitous texting in class."

Yes, we are all familiar with these common behaviors. Dr. Willingham even introduced his article with his own anecdote.

Not long ago a student told me a story about taking the SAT. Students were to bring a photo I.D., and the girl in front of her in line had not brought one. When she was told that she couldn’t take the test without the i.d., she was incredulous. She literally did not believe that there would be a consequence for her forgetfulness. She assumed that there would be a Plan B for people like her. When it became clear that plan B was “go home and next time, bring your I.D.,” she was angry and scornful.

I see this attitude not infrequently in freshmen I teach.

Many university departments now expect professors to organize their class as if it were a high school with multiple graded homework assignments, weekly quizzes, attendance taking, etc. and a lot of hand holding. Students complain loudly in class if the professor assigns what some of them believe to be too many problems. Actually, I am willing to hold their hands and lead them across the academic bridge to accountability and achievement that I am building for them. Many refuse to cross. They often go to the department chair and essentially get teleported. Beam me up, Scotty. Administrators tell professors to ensure somehow that every student passes, regardless of the quality of work.

Of course, starting in junior high, students in my day were held increasingly accountable for our own grades and learning. Today, teachers just say, "Just wait till you get to college" instead of training immature adolescents to become relatively mature 18-year old scholars.

However, I am sympathetic. Without administrative support, (and lack of administrative support is the number one reason teachers leave teaching), teachers have difficulty doing their jobs and training the children in the way they should go. In some school environments, maintaining control is all a teacher can manage. I once had a principal who allowed no D's or F's. He simply changed the grades of those recalcitrant teachers who gave what students earned.

I mentor a young, talented and reflective high school teacher who teaches in a well equipped school located in a high income area. Her main struggle and frustration is student behavior. There is something seriously wrong as long as administrators hold teachers more responsible for student behavior than the students themselves (and their parents).

Bryan Toporek ended by asking, "But how would you propose capturing that idea of student accountability in a revamped teacher evaluation process?" I added the following comment to his post.

Before capturing "that idea" of student accountability, ie, the idea that students are responsible for their own learning, I propose a more fundamental idea of student accountability.

Students should be held accountable for their behavior...I don't care how "bored" a student is, boredom is never an acceptable reason for disruptive behavior and disrespecting the teacher. Administrators need to be strong enough to stand up to parents who excuse their child's behavior on account of so-called "boredom." What many teachers mean by lack of administrative support is lack of classroom management support and caving to parents.

Let students be held accountable for their behavior first, and accountability for learning may well take care of itself.

MacDuff: The New New Math

ASUs Cognitive Instruction in Mathematical Modeling (CIMM), makes a huge claim: the ability to render “all the notoriously difficult” mathematics topics like fractions, negative numbers, place value, and exponents “trivial.” Those pesky math topics well deserve their reputation. The trouble students have with understanding math can usually be traced to misconceptions with fractions, negative numbers, place value, and exponents. In fact, textbook adoption committees can save precious time by limiting evaluation to these particular topics.

Math Teachers Do Not Understand Math

The mountain of documentation that teachers are not learning the math we expect them to teach, especially at the elementary level, grows higher every day. I touched on this point in my report on calculator research with young children. The National Council of Teachers of Mathematics has gone on record recommending calculator use in the earliest grades with the important caveat, “under the guidance of skilled teachers.” Ah yes, skilled math teachers, where are they to be found? By all accounts, skilled math teachers are not to be found in our elementary schools. Hence, the subtitle of my report, “A Moot Point Without Skilled Teachers.” Targeted efforts to improve the content knowledge of our teachers have proven disappointing.

Dr. Robert MacDuff discussed the lack of skilled math teachers and the failure of teachers to master math content in spite of all efforts in his essay entitled, “the Math Problem.” His intriguing title means he believes the problem is with the math content itself. Dr. MacDuff asks, “Could mathematics itself be flawed?” Whatever could he mean? Is he proposing to fix mathematics? In a word, yes, Dr. MacDuff suggests retooling math content from scratch.

Our research suggests that the problem lies in the difficulties imposed on the students by asking them to learn a flawed mathematics content.  Their difficulties in turn, lead to a drastic underestimation of their capabilities and thence to shortchanging them in their education.  The solution therefore requires a profound re-thinking of the foundations and assumptions of mathematics itself. 

Whoa, that's ambitious. Dr. MacDuff would start by considering collections of objects.

The system of ideas that is emerging from this process can be described as a “mathematics of quantity”, which takes as its starting point the consideration of collections of objects. 

I have been on the education merry-go-round for quite a while and I remember when one of the carousel horses was New Math, which took for its starting point, set theory. What are sets, but collections of objects? Was Dr. MacDuff proposing a return to the detested New Math? Let's see.

This approach separates the learning of mathematics into four major subsections: conceptual understanding (grouping structure), symbol construction (algorithmic manipulations), problem solving and mathematical reasoning.  As defined here, conceptual understanding and mathematical reasoning are not to be found in standard approaches to mathematics education.  And yet these are the critical components.

Teach Real-World Math

Dr. MacDuff is not suggesting the use of real world applications, although he is not precluding them either. “Real-world” means the idea that math is happening all around us. Number sense (and mathematics) starts, not with numerals, but with collections of objects, Piagetan conservation of quantity and relationships between quantities.

Some math teachers do emphasize conceptual understanding and mathematical reasoning in their teaching practice. A student may be lucky to draw only one or two such teachers sometime between kindergarten and high school. However, the traditional American approach makes conceptual understanding and mathematical reasoning the servant of algorithms, not its master. In other words, many teachers try to use mathematical reasoning to help students memorize the algorithms rather than guiding students to learn the math for which the conventional algorithm becomes one of many possible problem-solving strategies.

I deplore the current drive to push first grade curriculum into kindergarten. Teachers give kindergarten children pencils and paper and teach them conventional algorithms. The children spend the first few weeks of kindergarten learning to write numerals. Some of them even appear to master the so-called math they are taught. If they dependably get right answers, they internalize the idea that they are good at math, when in actuality they may not understand math at all.

It is possible to execute an algorithm with no real understanding. How many adults have any idea what is mathematically happening when they perform an algorithm like long division? The math mis-education is further compounded when non-math explanations are routinely substituted for and identified as math explanations. A good example is the typical explanation for the division of fractions. Ask any good math student and you will be led through division of fractions as multiplication by the reciprocal, a non math explanation for why the trick mathematically works. Or how about multiplication of decimals? Mathematically you are not counting decimal places; something mathematical is happening that makes counting decimal places work.

Start Math Instruction by Avoiding Numerals

Children can learn lots and lots of math without ever picking up a pencil and paper. In fact, making kids write math before they have mastered the mathematical relationships creates learning obstacles the children may or may not overcome. Meanwhile, many of these same children, the ones who get right answers, have been told by our education establishment that they are good in math, and they believe it. In the plainest terms, schools have been lying to students about their math abilities forever. For thirty years, I have preached this idea about math education like an evangelist.

I teach Algebra. I cannot tell you how many math whizzes show up to Algebra without the slightest clue about the concept of place value. Sure, they can name the place of any given digit, but naming is a far cry from understanding the mechanism of place value and its importance in mathematics. I almost always redo elementary math with students before we launch into the “real” subject matter of my class, algebra.

According to Dr. MacDuff, one part of the brain deals with math, while another part of the brain deals with numerals (like “12”) and yet another part of the brain deals with linguistic representation of numbers (like “twelve”). The traditional approach to math stimulates the wrong parts of the brain. No wonder so few students (and their teachers who were once students) get it.

Dr. MacDuff is not proposing a return to New Math. He is proposing a structural retooling of the math curriculum so that we actually stimulate the part of the brain the deals with math. Some of us have taught that way all along. Dr. MacDuff proposes to make my idiosyncratic methods the system-wide norm. Allelujah!