Lockhart's lament redux

"The essence of mathematics is recognizing interesting patterns in interesting abstractions of reality and finding properties of those patterns and abstractions."

Everything About The Way We Teach Math Is Wrong

This strikes me as completely wrong, not that I have much confidence in my intuitions concerning the essence of math. So take this as a confession, not an argument.

"Recognizing interesting patterns" -- even "recognizing interesting patterns in interesting abstractions of reality" -- strikes me as the toolkit approach to math.

I personally -- another confession, not an argument -- can't stand the toolkit approach to math. Blech!

I have no interest -- none! -- in endless iterations of function problems designed to determine how much profit the guitar teacher will make teaching x number of students while paying y rent on the studio for 20 weeks, or whether Jim should buy the monthly contract or the annual, as useful and important as those questions are in daily living.

Nor do I relish the thought of encountering yet another roller coaster depicted as yet another instance of geometry, or another textbook with a nautilus shell splashed across the cover. Enough with the nautili.

If you want to make me hate math, real-world math will do the job.

That reminds me.

I did not, as a child, solve two trains leaving the station problems, and I wish I had. I was utterly charmed by the two-trains problems when I encountered them as an adult, working my way through "Russian Math."^*

I don't remember whether we were asked to solve bathtub problems (I think we were), and I was charmed by those problems, too.

(Speaking of the real world, the bathtub problem is all you need to know to understand why fiscal stimulus doesn't work when the Federal Reserve targets inflation. One party is putting money into the system; the other party is taking it back out.)

To be fair, I'm not at all sure that "toolkit math" is what Lockhart actually means: "That’s what math is — wondering, playing, amusing yourself with your imagination."

Nevertheless, inside K-12, the toolkit approach is what look-for-patterns turns into.

Look for patterns may be a bit out of date.

Today, with Common Core, at least here in NY, we've moved beyond look-for-patterns to modeling, for pete's sake. In my district, the entire high school mathematics curriculum, the entire rationale for the entire mathematics curriculum, is modeling.

Lots and lots of function problems modeling stuff nobody cares about.

This is always the conundrum with constructivism and progressive education.

Progressive educators think the real world is fun and motivating.


^* Mathematics 6 by Enn Nurk and Aksel Telgmaa

Report from the front

The other day I was introduced to a dean at a progressive private school. We chatted, and at some point I asked how the school assesses student learning.

The dean was vague. He mentioned teachers "reflecting" a couple of times, and then said the school is moving more and more to students assessing themselves, which also involves reflecting.

Funny thing: just last week I talked to a friend of mine whose son attends the school. She says the school has rampant grade inflation, and her son has learned nothing but his grades are good. He learns so little at school that he had two full-time tutors all last school year. (This is a very smart kid, by the way. No learning problems, no behavior problems.) When she took him to the SAT tutor she used with an older child in the family, the tutor told her there's no way he can prepare her son to take the SAT Math Subject test.

I told my friend about "Teach Like a Champion" classes: rapid-fire, high-energy events, with cold-calling and choral response, and said we need charter schools for rich kids. Parents should at least have the option of putting their kids in classrooms where kids spend a a lot of time practicing, not just discussing.

My friend is athletic, and she sparked to the idea it instantly. She said her son would love it, and he would remember what's been taught because he would be practicing in class.

Grade inflation and no practice during class-time: that is a recipe for total disengagement in a lot of kids.

Deeper shmeeper

Fabulous post by Tom Loveless:

Deeper Learning is the current term for an old idea. The notion is that schools spend too much time focused on the acquisition of knowledge, especially knowing facts. In the past century, several alternatives have arisen to dethrone the prominent role of knowledge in schools: project-based learning, inquiry and discovery learning, higher-level thinking, critical thinking, outcome based education, and 21st Century Skills. Now it is deeper learning.

These ideas represent a variety of approaches to curriculum and pedagogy. They are not all the same, but they share one characteristic. All are advertised as transcending, and therefore superior to, academic content organized within traditional intellectual disciplines.

The Banality of Deeper Learning

"21st century skills" are gone?

Really?

That was quick. At least, it seems quick. I was just hearing the words "21st century skills" for the first time in 2005. Eight years ago.

What is the life-cycle of K-12 jargon, anyway?

AND SEE:
Why students have to memorize things

Smart Teachers in Stupid Schools, part 2

more from teacher Christine:

Progressive educators would like to promote a more democratic society advocating greater equity, justice, diversity and other democratic values, yet their methodologies do just the opposite, with "Fuzzy Math" and "Whole Language" causing lesser privileged students who can't afford tutoring to fall way behind. NYC has 70% of its student population in this category. Imagine how devastating to the morale and sense of self-esteem the use of poor curriculums can have on a child's psyche. These students subjected to these methods grow to believe they can't do anything; they are labeled as special needs children and become distraught that they are not mentally capable of becoming educated. Many are just pushed through the system because there is no where else for them to go. Progressivism which is trying to enforce some kind of social agenda, rather than purely impart knowledge, is causing many students to fail and teachers to become distraught and despondent.

What do teachers who refuse to follow the leader do? Many shut their doors and pull out the curriculum they know works. I know of teachers who, when whole language is being implemented by the district, will use their phonics programs undercover. Teachers will set up look-outs in the hall to see if supervisors are coming and drill students in what to do should a supervisor show up. I have had my students open their "readers" and put in the phonics books I'm using inside. "If someone comes you take the book slide it in your desk and pretend you're reading." I've instructed.

Smart Teachers in Stupid Schools

yet another solution to the problem we don't have

Next up: Angry Birds in the classroom. Rovio plans to publish a line of educational books for children ages four to their teens. Embracing the Finnish education system—which emphasizes play in learning—they're also designed as a counterpoint to the strict "Tiger Mom" approach to child rearing, says Wibe Wagemans, a Rovio branding executive.

He says one of the books will illustrate geometry lessons by launching a bird character through the air...

'Angry Birds' Spreads Wings
By NICK WINGFIELD
WSJ | August 10, 2010

E.D. Hirsch needs to get with a branding executive.

let's not and say we did, part 3

My copy of the Harvard Education Letter arrived in the mail yesterday:

These days, everyone is trying to get students to think about their own thinking--and take that thinking to a higher level. To do this, educators are urging teachers to help students teach--and even assess--each other, since the success of so-called student-centered instruction is predicated on getting students to collaborate and take charge of their own learning.
Like Teacher, Like Student
by Dave Saltman
Harvard Education Letter
January | February 2011
p. 4

I'm sure we'll be seeing more of this in my district now that our curriculum department will once again have a full-time curriculum director.

E.D. Hirsch on the philosophical roots of progressive education

Hirsch writes:

In my mind, progressive educational ideas have proved so seductive because their appeal lies not in their practical effects but in their links to romanticism, the 19th-century philosophical movement, so influential in American culture, that elevated all that is natural and disparaged all that is artificial. The progressives applied this romantic principle to education by positing that education should be a natural process of growth that flows from the child’s natural instincts and interests. The word “nature” in the romantic tradition connotes the sense of a direct connection with the holy, lending the tenets of progressivism all the weight of religious conviction. We know in advance, in our bones, that what is natural must be better than what is artificial. This revelation is the absolute truth against which experience itself must be measured, and any failure of educational practice must be due to faulty implementation of progressive principles or faulty interpretation of educational results. Thus the results of mere reading tests must not be taken at face value, because such blunt instruments cannot hope to measure the true effects of education. The fundamental beliefs of progressivism are impervious to unfavorable data because its philosophical parent, romanticism, is a kind of secular theology that, like all religions, is inherently resistant to data. A religious believer scorns mere “evidences.”

The Chasm Between

There are many disputes within the education field, but none so vituperative as the reading and math wars—the battles over how best to teach children to read and to solve arithmetic problems. These aren’t just disputes over instructional techniques; they are expressions of two distinct and opposing understandings of children’s nature and how children learn. The two sides are best viewed as expressions of romantic versus classical orientations to education. For instance, the “whole language,” progressive approach to teaching children how to read is romantic in impulse. It equates the natural process of learning an oral first language with the very unnatural process of learning alphabetic writing. The emotive weight in progressivist ideas is on naturalness. The natural is spiritually nourishing; the artificial, deadening. In the 1920s, William Kilpatrick and other romantic progressivists were already advocating the “whole language” method for many of the same reasons advanced today.

The classical approach, by contrast, declines to assume that the natural method is always the best method. In teaching reading, the classicist is quite willing to accept linguistic scholarship that discloses that the alphabet is an artificial device for encoding the sounds of language. Learn the 40-odd sounds of the English language and their corresponding letter combinations, and you can sound out almost any word. Yet adherents of “whole language” regard phonics as an unnatural approach that, by divorcing sounds and letters from meaning and context, fails to give children a real appreciation for reading.

The progressivist believes that it is better to study math and science through real-world, hands-on, natural methods than through the deadening modes of conceptual and verbal learning, or the repetitive practicing of math algorithms, even if those “old fashioned” methods are successful. The classicist is willing to accept the verdict of scholars that the artificial symbols and algorithms of mathematics are the very sources of its power. Math is a powerful instrument precisely because it is unnatural. It enables the mind to manipulate symbols in ways that transcend the direct natural reckoning abilities of the mind. Natural, real-world intuitions are helpful in math, but there should be no facile opposition between terms like “understanding,” “hands-on,” and “real-world applications” and terms like “rote learning” and “drill and kill.” What is being killed in memorizing the multiplication table? The progressivist says: children’s joy in learning, their intrinsic interest, and their deep understanding.

The romantic poet William Wordsworth said, “We murder to dissect”; the progressivist says that phonemics and place value should not be dissected in isolation from their natural use, nor imposed before the child is naturally ready. Instead of explicit, analytical instruction, the romantic wants implicit, natural instruction through projects and discovery. This explains the romantic preference for “integrated learning” and “developmental appropriateness.” Education that places subject matter in its natural setting and presents it in a natural way is superior to the artificial analysis and abstractions of language. Hands-on learning is superior to verbal learning. Real-world applications of mathematics provide a truer understanding of math than empty mastery of formal relationships.

Romancing the Child
E.D. Hirsch
Education Next

Spring 2001 / Vol. 1, No. 1

This is where whole math comes from.

Also, this is why it's illegal to teach sentence diagramming (pdf file - Martha Kolln's 10 core sentences) in public schools.

[pause]

I see I've posted this passage before: E.D. Hirsch on "spilt religion."

Revolutionizing Math at the School of the Future

(Cross-posted at Out in Left Field)

A front page article in Monday's Local News section of the Philadelphia Inquirer profiles a math class at Philadelphia's Microsoft-funded High School of the Future, whose teacher, Thomas Gaffey, placed second in Microsoft's U.S. Innovative Education Forum and was a semi-finalist in its Worldwide Innovative Education Forum. In Gaffey's ninth-grade algebra class there are:

No textbooks, no paper, no chalk, no desks, and no assigned seats.

Instead, students use laptops while sitting in rolling chairs at trapezoidal tables spaced out in hexagonal classrooms.

Just how newsworthy this sounds to you depends on whether you think chair mobility and table shape have a big influence on learning, on whether you've been following current trends in education over the last 50 years, and on how unusual you think it is for a teacher to "encourage his students to find answers to their own questions" and engage with them in exchanges like these:

"Is this an obtuse triangle?" one student asks.

"Well, what can you tell me about an obtuse triangle?" Gaffey replies.

"One of the angles has to be more than 90 degrees," the student answers.

"Are any of the angles here like that?"

"Yeah. Oh, I get it now!"

As the Inquirer explains:

This snippet of student-driven discussion is a glimpse of the style and approach that have earned Gaffey national and international recognition.

Student-driven? Who's asking most of the questions? But I'm splitting hairs here. What I should be asking is: Why does this kind of exchange warrant international recognition?

To fair, it wasn't this, specifically, that earned Gaffey his honors. Rather:

Les Foltos, one of the judges who reviewed Gaffey's work, was impressed by his emphasis on "actively engaging students in solving real-world problems." As Gaffey puts it, "If we want to teach math to learners, we should teach math how it is actually used. It doesn't matter how much you know. It matters what you can do."

Ah yes, "real world problems." Again, only if you've been out of touch with the last half century of educational reform, and with today's Reform Math in particular, will this strike you as revolutionary. Here is Gaffy's version of real world math:

In his classroom on a recent Tuesday, Gaffey's challenge to his "learners" - as students in the Parkside public school are called - was to estimate Earth's land area.

To solve the problem, the class first covered basic concepts about area and polygons - shapes with three or more straight sides.

Gaffey then asked, "If a shape has four sides, is it always a polygon?"

Learners who answered yes (the wrong answer) were asked to redefine what a polygon is, while those who answered no were asked to draw a four-sided shape that was not a polygon on the class "smart board."

Gaffey drew a shape with three straight sides and one curved side.

"Is this a polygon?" he asked.

"No," the class responded.
...

The class drew lines through each of the continents, chopping them up into complex polygons, then simple polygons.

The final phase was to derive formulas for the areas of the simple polygons, and add up the areas.

This sort of problem is not particularly new, as a quick survey through now-standard textbooks like Everyday Math and the Interactive Math Program makes clear. And it's been around long enough to have garnered some serious criticism--specifically in what Barry Garelick calls its "just in time" approach to teaching.

Among other things, "just in time" often means serious delay. For example, one would hope that students would already know the formulas for the areas of simple polygons, and how to derive them, well before they hit 9th grade.

But because so many students are so far behind where they should be, there is one thing in which I and Gaffey are in whole-hearted agreement. In Gaffey's words, as cited by the Inquirer:

"Math education, more than any other subject, is in need of drastic reform."

in today's mail

from the 25th Anniversary edition of the Harvard Education Letter:

[Jerome Bruner’s “Man: A Course of Study”] was my first deep exposure to progressive education, and I liked what I read, heard, and observed: the hands-on experiences, the deep exploration of inviting topics, the belief that the questions asked are as important as their answers, and that the reasoning behind questions and answers is crucial. I became a card-carrying enthusiast of progressive education, American style.

[snip]

As one personally committed to progressive education, I was well prepared for my initial visits in the early 1980s to the small northern Italian city of Reggio Emilia, home of what are widely regarded to be the finest preschools on the planet. Initially launched in the post-World War II era, these municipal school operate on the assumption that children’s natural curiosity should be the centerpiece of education. An object (or experience) that captures the children’s attention—a shoe, a fax machine, a rainbox, a birdhouse, or a carved lion at the central Piazza—can become the focus of curriculum for weeks, even months. As the young students explore this fertile object, they have the opportunity to draw on the “hundred languages” that are the birthright of every child—their senses, available media and symbol systems, the arts, the sciences, the natural world—to gain relevant insights into the various spheres of life in which these objects occupy a role. What is learned and created each day becomes the starting point for the following days’ activities. And these learnings are publicly displayed – or “documented”—so that teachers, parents, and other children can share in them and build on them.

Alon with other educators, including my mentor Jerome Bruner, I have visited, studied, and learned from the Reggio Emilia approach for 30 years. This flagship educational enterprise has changed my mind about what is possible to achieve with young children, the importance of group—as opposed to individual—learning, and the role that can be played by documentation of learning over days, weeks, and even longer stretches of time. I have also learned how a single educational experiment—conceived 50 years ago by a determined grouop of citizens—can affect practices all over the world.

Yet shortly after visiting Reggio Emilia for the first time, I undertook a series of trips to China. There I found that my progressive educational philosophy—Italian as well as American style-was sharply challenged. In classrooms in major cities around the country, I saw the same “prefabricated” lessons presented in essentially the same manner. Little latitude was permitted to either teacher or student. Indeed, in one college class in psychology, I was shocked to observe obviously talented students simply repeating the same lesson over and over. When I challenged the teacher about what seemed to be an obvious waste of time, we had an unproductive conversation that she finally terminated with the terse remark, “We’ve been doing it this way for so long that we know it is right.”

Yet I was also surprised by some of the positive results. In a first-grade art class, I watched the students slavishly copy a model over and over. I wondered whether these six-year olds could use their developing skills to portray an unfamiliar object—in this case, an Italian stroller that they could not possibly have seen before. Although the teachers protested when I proposed this assignment, I stuck to my guns. To everyone’s astonishment, the students were able to draw the stroller with considerable skill—far greater aptitude than would have been shown by most American youngsters. This experience convinced me that an effective education can begin with a singular focus on skill building rather than on the play of unfettered imagination, and that the skills that are developed, often precociously, have the potential to be mobilized to more creative ends.

From Progressive Education to Educational Pluralism
by Howard Gardner
Harvard Education Letter
September | October 2010