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!

Summer School: What's Old is New

EdWeek is reporting that across the nation “dreaded summer school” is at risk.

With summer having officially arrived this week, children are heading to camp, the beach, the pool, and in some cases, back to the classroom for the dreaded summer school. If it’s available, that is.

While some districts downsize, and even eliminate, summer school, other districts are reinventing it.

...at least a handful of places, such as Baltimore, Cincinnati, and Pittsburgh, are also thinking anew about summer school as an experience that will prove far more engaging and meaningful for young people...The centerpiece of the effort in Pittsburgh this year is the new Summer Dreamers Academy, billed as a “camp” available for free to all rising 6th, 7th, and 8th graders...The five-week, all-day program that begins next month will feature a literacy curriculum in the mornings designed to be fun and engaging. In the afternoons, “campers” will have a wide choice of activities, from judo and kayaking to music theater and video-game design.

What strikes me is the delighted tone of the article, as if fun and engaging summer school is a new and innovative idea. I fondly remember my elementary summer school classes, lo, nearly half a century ago. I always eagerly anticipated summer school, a time when I would get to learn new things never offered during the regular school year. I studied archeology, sculpture, theater, special math topics, archery, sports clinics, photography, oil painting, all kinds of other great stuff, and for free. Of course, California was a solvent state then.

Now that I am grown up, I know that summer school served another important purpose. It kept my brain actively learning.

Research has long suggested that summer can take a heavy toll on student learning.
A report issued last week by the National Summer Learning Association, titled “A New Vision for Summer School,” says that since 1906, more than 40 empirical studies have found evidence of a pattern of “summer learning loss,” particularly for low-income youths.
One 2007 study, for instance, found that about two-thirds of the reading achievement gap between 9th graders of low and high socioeconomic standing in Baltimore public schools could be traced to what they learned, or failed to learn, over their childhood summers. ("Much of Learning Gap Blamed on Summer," July 18, 2007.)

Apparently educators have known since 1906 that “summer learning loss” not only persistently occurs, but is more pronounced among children from less affluent families who cannot afford summer enrichment activities. Furthermore, this summer learning loss is estimated to account for around two-thirds of the achievement gap between more and less affluent students. It stands to reason that providing summer school to all could substantially close that gap.

Summer would also be a great time to pursue community-school corroboration. Instructors could be drawn from the community. Grants could fund the programs. And working parents would be thrilled, especially if they could apply summer childcare dollars to programs that enrich their children while keeping them off the street.

Storming Educational Barricades

On June 22, 2010, the well-respected educator and sought-out Washington DC dinner party guest, Diane Ravitch, wrote, "Even privately managed charter schools are affected negatively by high-stakes testing; to claim ever-rising test scores, they are prompted to avoid low-performing students, thus bypassing the very students that charters were originally intended to serve."

A comment to her blog asked, “Have you ever managed to come up with any evidence for this claim?”

The very next comment pounced, “You have the whole Internet at your disposal. Surely you know that DR doesn't have to provide footnotes on what is now common knowledge for most literate people?”

Is the notion that charter schools are avoiding low-performing students a common knowledge fact? A perusal of the “whole Internet” suggests that what is common knowledge is that charter schools have long been accused of creaming the crop and pushing out low achievers in order to artificially raise their academic achievement when compared to regular public schools. Whether the accusation has research merit is not so clear.

A Mathematica study commissioned by the Knowledge is Power Program (KIPP) charter schools examined twenty-two KIPP schools and found no basis for the accusation. I am well aware that the study will be heralded by charter school supporters and dismissed by charter school detractors. The grounds for dismissal will be the perceived conflict of interest in KIPP commissioning a flattering study of itself. Charter school detractors would love it if the study were damning and would be quick to generalize the results to all charter schools. As it stands, charter school supporters will be happy to likewise generalize the study's conclusions.

Which brings me to my point. As long as those of us seeking education reform continue to cherish one position or another, we will fail to come to the consensus society so desperately needs before the forward trail can be blazed. Large swaths of American society either do not care about education or feel they have too little power or influence to make any difference. It is a true pity so many of those who feel “unempowered” are in fact, teachers and parents. Anthony Cody is one teacher who has been trying hard to be heard.

Among those who deeply care, many of the most vociferous can absolutely be classified along political lines: the right who love charter schools and the left who hate them. The constant bickering between them neutralizes any positive effects. The KIPP study is a case in point. Supporters of charters rush to embrace and extend its conclusions while opponents focus on finding flaws. I say, go ahead and enumerate flaws. But do not stop there. Let us have better designed studies that take previous flaws into account. And let us not be so foolish or short-sighted to think that if we and our friends believe we have debunked a study that we can ignore it because its merits are inconvenient to our politics.

I completely understand the conflict of interest question. Many years ago a marine biologist friend of mine was hired by Suntory Whiskey to conduct research on the environmental impact of their distilleries upon the ocean where they dump the waste. My friend wrote a most uncomplimentary report. She was fired and the report never saw the light of day. Would KIPP have suppressed the study if it did not like the conclusions? I do not know. I remember when early studies of charter schools consistently found that the academic achievement of charter school students surpassed that of public school students. The most recent studies show regression to the mean. In the early, heady days of charters, some people made the question political and immediately erected defensive barricades. The favorite barricade has been the accusation that charters cream the crop.

In my work with both charter schools and regular public schools, I have observed that charter schools do not cream the crop or push out low achievers. In fact, parents of cream and parents of low achievers both flock to charter schools. Parents of high achievers believe that charter schools are better than regular public schools and therefore their high achieving child will be challenged and rise to even greater heights. The parents of low achievers flock to charter schools because they often believe their child is a bored underachiever who needs the extra competence and challenge they expect their child will find in the charter school to turn from being a low achiever to a high achiever. For their part, charter schools take students from both groups because, just like regular public schools, their state funding depends on enrollment numbers. Besides, self-study is not inherently bad. The school accreditation process depends on schools studying themselves and writing up what they find.

I am aware that both sides collect anecdotes as if the matter will be decided by a sort of vote, that is, the side with more anecdotes wins. Reasonable people would reject such a method of decision making as ridiculous, but in reality we act like tallying up anecdotes is exactly the way to make big decisions. The health care debate was replete with anecdotes. It did not work. The pro health care reform camp and the anti heath care reform camp simply ignored each others anecdotes.

Data supporting the so-called common knowledge that charters cream the crop is inconclusive. More importantly, we should be looking for data to inform our opinions, not to confirm ourselves in our prejudgments. We need to detach ourselves from partisan political positions on education and study the issue, if not dispassionately, at least with a conscious effort to identify our own cherished opinions and set them aside for the time being.

How Narratives Impede Education Reform

Patrick McGuinn is right. There are two narratives, and that is part of the problem. It sets up yet another dangerous either-or dichotomy where every fact and opinion must be squeezed through one sieve or another.

Meanwhile, all over the country, in individual local schools, effective education reform takes place apart from public awareness. Typically, the these local reformers are responding to the internal culture and politics of their school, making scalability a challenge, if not impossible.

Perhaps it is time to put aside partisan politics and narratives, and think hard about what we as a society want for the future of American education, because, as Deborah Meier points out, we are deciding on our shared future. The key word is “shared.” As it stands, we are divided, playing an evenly matched tug-of-war. No wonder nothing moves. Many education stakeholders are ambivalent about involvement. They want to be deeply involved, but distrust the process because important stakeholders, like teachers and parents, are shut out. Stakeholders do not esteem and trust each other.

So while stakeholders say they want to be empowered, normally the empowerment is worthless, consisting of false choices, like whether to have chocolate or vanilla ice cream. Early charter schools showed us what empowerment could look like. Early studies of charter schools found many local instances of excellence. These days there are too many operators with, shall we say, impure motives. The charter school movement has experienced serious regression toward the mean. Trust has been lost.

In-the-trenches educators have a lot to offer, but policy makers do not listen. We need to avoid over-simplified narratives that can be summed up in a couple a phrases, and begin to wrestle with the complexity of the issue. We got where we are today through a series of smaller actions taken beyond their usefulness. For example, I remember when education defined as the ability to locate information took hold. At the time, I was all for it. No body can know everything, but knowing how to find what you do not know is essential. It was not long though before I started getting students in my classes who did not have enough basic memorized knowledge to use as a foundation for an information search.

We do this all the time---create dichotomies and then swing from one to the other. Education decisions have been made locally for a very long time. I believe education reform must start with local initiative and relational trust. Supposedly, the purpose of the Race to the Top was to give local schools the ability to design their own reform and win funding to implement it. I was part of a local successful reform that did not have a technology piece, a merit pay piece or a community buy-in piece. We teachers just did it. I think the dynamic of how our efforts succeeded in the absence of left-right divisiveness is worth considering and perhaps applying to other local, yet different circumstances.

We Don't Need Educational Bells and Whistles

Among the many reasons education reform is stalled, one of the most prominent must be the appeal of adding bells and whistles to a broken-down car. A case in point is adding social media to schools. Social media plays into the myth that we can improve education by merely incorporating technology. People (somewhere, sometime) have received excellent educations for millenia even without any of what we recognize today as technology.

I once taught math to a group of multi-generational Vietnamese refugees. The experience changed my entire outlook on the subject of school finances. We sat under the trees. Everybody had a lapboard, pencils and paper I provided. I had the only textbook, a small chalkboard, chalk and eraser, all of which I also provided, and any “manipulatives” I chose to bring. Even on such a shoestring, I guarantee I delivered a high-quality education experience to all. No “technology,” not even a calculator. Amazing, huh.

Would it have been nice to have an overhead projector, or a smart board? Sure, but not necessary. How about a mimeograph machine, a xerox machine, computer printouts? Also lovely, but the students were perfectly capable of copying the problems from my little blackboard. Filmstrips, videos, DVDs? Again, nice, but not essential.

If our education system is like a car whose engine does not run, adding technology is like hoping that plugging in a GPS will somehow cause the engine to turn over. Incorporating bells and whistles like Facebook, Skype, Twitter, Second Life, whatever, will not help. The car is still going nowhere.

Analogies always break down somewhere. Our education system is not really like a single car. It is more like a bunch of scooters, some of which run nicely and some do not run at all. The kids on the scooters that run may be doing fine, but if America is to compete, all the scooters need to run nicely. Once all the scooters are running well, we can add a roof to the scooters and enhance the experience for all. Again, adding a roof to a non-running scooter will not make it run.

Before we begin spending tons of money on enhancements like technology, we need to use whatever resources we have to ensure every child is receiving a high quality education. Funny thing, money is no object, nor is it an obstacle. It is a matter of where we, as a society, place our priorities. As Suze Orman's signature line puts it, people first, then things.

Will Retired Teachers Also be Wooed?

One commentator believes that with the looming worker shortage, retired workers will be wooed to return to work.

This is not open to much question. It will happen even if the jobless recovery continues its vexing ways. That's because the people who will shape this future are already here and their numbers aren't going to change in the next 10 years. And it's because we have a pretty good idea of how many people it will take to generate the economic activity we can reasonably expect to occur.

The younger workers are, for the most part already working, Philip Moeller says. They will most likely continue working, so there will be few extra younger workers available to close the gap. Retirees aged 65 or older most likely have retired for good. So it will be the baby boomers beginning to retire now or who were forced into early retirement by a layoff or other cause who could be enticed to come back.

In a study done at Northeastern University earlier this year, Barry Bluestone and Mark Melnik say that in eight years, more than five million jobs may go begging unless there's a big boost in what's called the labor force participation rate -- the percentage of Americans who seek work. The big driver here is that the numbers of Baby Boomers leaving the work force will exceed the supply of new workers coming from younger generations.

The gap is a concern in another context: Social Security. In old news, the Social Security Administration estimates that in the near future, there will not be enough workers per retiree to pay retirees 100% of promised benefits. During the Social Security debate of the Bush administration, many opined that immigrants would fill the gap.

The authors of the study cited earlier list the top thirty jobs with shortages, called “encore” jobs. Standing at the head of the list? Teachers. If school administrators want to lure teachers back to the classroom, they will first have to address at least two issues, administrative support and credentialing.

Teachers who left the classroom, retired or not, might come back if their number one reason for leaving were addressed: lack of administrative support. Schools need to take the position that an education is not only a right, but a privilege. Administrators must remove the onus of classroom control from the teacher. Teachers should be able to send out students who clearly show by their disruptive behavior they would rather be somewhere else. Let that somewhere else be the principal's office. Teachers should not have to endure disruptive students for fear the administration will draw negative conclusions about their competence if they send students to the office.

Whoa! What if the principal's office were thus flooded with students? I once worked in a school where the administration believed it was the teacher's job to teach, not mete out “discipline.” Students found they actually did not like being out of class regardless of whether they cared about learning or not. Sitting in a chair outside the principal's office (no cell phones, etc. allowed) was way more “boring” than any teacher.

The bottom line is that responsibility for behavior must be returned to the students, especially since they often complain that they are too old to be treated as children. So-called “boredom” must never be allowed as an excuse for acting out or disrespecting the teacher.

Another important consideration is our society's current over-reliance on teacher credentialing as evidence of competence. Though research has not confirmed a correlation between credentialing and competence, the difficulty is that we have not yet figured out what constitutes competence. Given the constant calls to get rid of tenured, credentialed yet incompetent teachers, everyday experience casts doubt on correlation between competence and credentialing.

If society wants to bring back its teachers, it will have to find a way to determine a sensible alternative route to credentialing. Most credentials will have long expired, and if re-credentialing is onerous and expensive, the teacher shortage will not be filled with proven classroom veterans.

If past history is any indication, the teacher shortage will persist. I have an old Time article from 1985 worrying about the coming shortage of math and science teachers. For the last twenty-five years, warnings of shortage were sounded as cyclically as the sunrise, but nothing was done to prevent it. The shortage has arrived, and still, nothing but hand-wringing prevails.

Meanwhile, great teachers cannot get teaching jobs because our school districts do not hire experienced teachers. Too expensive, they say. What I cannot understand is why society accepts such a flimsy excuse, especially since currently, most district give credit on the wage scale for no more than five years experience. Any teacher with more than five years is actually a bargain, not “too expensive.”

Mathematica Study Undermines Teacher Merit Pay

So often when I read what passes for news in education, I find myself thinking, “No surprise there.” I am thinking of starting a No Surprise There watch. A case in point is the latest news out of Chicago.

According to the study from Mathmatica:

Preliminary results from a Chicago program containing performance-based compensation for teachers show no evidence that it has boosted student achievement on math and reading tests, compared with a group of similar, nonparticipating schools, an analysis released today concludes.

The program also failed to improve teacher retention rates. Over the two years of the study, achievement gains from the first year evaporated in the second year. Researchers and commentators speculate at least three factors account for the disappointing outcomes:

1. Reform takes time. It is unreasonable to expect stable achievement gains after only two years.
2. Schools did not adhere to a standardized implementation of the plan. Since not everybody did it the same way, the averages may be washing out positive effects in some schools.
3. The basic design of the plan needs tweaking. Perhaps higher or “more meaningfully differentiated” payouts.

It is impossible to generalize from the mere handful of studies on the efficacy of merit-based or performance-based incentives. However, many researchers have studied extrinsic versus intrinsic rewards.* The meta-conclusion is that extrinsic rewards actually undermine achievement. What works is intrinsic, not extrinsic rewards.

Hardly anyone but teachers mention what may be the most salient factor. Merit pay did not lead to achievement gains as measured by higher test scores because teachers simply do not have control over the myriad of variables that affect student achievement.

*Results of Google search on “research extrinsic rewards”