kitchen table math, the sequel: 10/18/09 - 10/25/09

Saturday, October 24, 2009

On Gerbils

My school, a K-8 facility has a mission statement (which I'll paraphrase) that says "All children will be taught in their zone of proximal development". This is a noble sentiment which I fully support and aspire to. Unfortunately, my 7th grade math student's abilities range from 3rd through 11th grade based upon the NWEA Measures of Performance (a highly regarded national evaluation). This enormous dispersion is not new. On the contrary, it is chronic and well aged, like a fine New England Cheddar, and it makes a mockery of our mission statement.

Over the summer, my middle school peers and our administration worked on various schemes to address this conflict in light of reduced resources (teachers) and state/district mandates for a failing school under NCLB regs. It is simply not rational to think that you can teach in a zone of proximal development (ZPD) in a classroom with a nine year spread in capabilities.

The consensus that we reached was to take the middle school model we were working under and blow it up. We proposed to place all middle schoolers (6-8) regardless of age, grade level, or hat size, into cohorts whose memberships were determined by academic readiness and to allow for periodic assessments that would allow kids to migrate as their performance dictated. To the extent that we had the resources to support a number of such 'ZPD pods' we would create as many cohorts as possible and each teacher would take on a range of such groups in shortened, focused, classes.

This proposal was not accepted for a variety of reasons. Some of them had to do with scheduling challenges and some had to do with the MCAS (Massachusetts state testing) mandated NCLB assessments. Bottom line is that only one reason is a legitimate show stopper at the upper reaches of our district. There is a huge fear that if we teach kids in their ZPD, say a 4th grade capable student gets 4th grade curriculum, then they won't do well on their grade level MCAS assessment.

GAAAAAACK! Of course they won't but neither will they do well on the very same assessment if they are forced to spend a year sitting through a 7th grade curriculum that is being delivered 15,000 feet over their poor heads. Worse, if we insist (as is interminably true) in the perversity of a mission statement that is wholly at odds with our delivery system, those very same 4th grade capable students become rancorous sores in the classroom, denying the kind of environment that breeds success for those few students who may be lucky enough to have a repertoire that is in synch with the delivery.

A quarter of the way into the year, we have learned that way too many kids are failing (surprise) to meet the minimal requirements for their grade level standards. In an 'emergency' meeting, we were informed that it was our duty (as teachers) to work together to solve this problem in an innovative, collaborative way. After much discussion it was agreed to come up with a plan to - drum roll inserted here - break up the middle school into 'ZPD cohorts', come up with a schedule, etc.

I'm just a gerbil on a wheel. No! No! Wait. I'm on the wheel and I'm singing "I Am the Walrus". Did you know that John Lennon wrote that song while on an acid trip? He supposedly received a letter from a teacher who was having his students analyze Beatles lyrics. He was amused at this and resolved to write an incomprehensible lyric based on "Through The Looking Glass".

If he was a teacher he wouldn't have needed acid or Lewis Carroll.

"Are Teacher Colleges Producing Mediocre Teachers?" TIME

TIME article by Gilbert Cruz.

Are Teacher Colleges Producing Mediocre Teachers?

OK, but is there anything constructive or promising here?

The article was based on a speech by Arne Duncan on Thursday to Columbia University's Teachers College. Duncan said:

"By almost any standard, many if not most of the nation's 1,450 schools, colleges and departments of education are doing a mediocre job of preparing teachers for the realities of the 21st century classroom,"

"By almost any standard."

{I'll ignore the 21st century remark.]

Then, Cruz quotes David Steiner, New York's education commissioner as saying:

"And if we can't identify the skills that make a difference in terms of student learning, then what we're saying is that teaching is an undefinable art, as opposed to something that can be taught."

"Until recently, Steiner served as dean of Hunter College's School of Education, where he was a vocal critic of the typical ed-school approach, in which teachers-in-training study theories and philosophies of education at the expense of practical, in-the-classroom experience."

Then Duncan is quoted as saying:

"I am urging every teacher-education program today to make better outcomes for students the overarching mission that propels all their efforts."

But what is the "standard" that should be used? It's one thing to argue mediocrity "by almost any standard", but there has to be some agreement over what that standard should be. The only standard that seems to be available is a mix of 50 state standards - all low. Some apologists of bad results complain that the bad scores on the tests mean that the tests are fundamentally flawed or that the schools didn't teach the specific material of any particular question. Apparently, these people are quite capable of arguing the "by almost any standard" position.

But the problem is not just poor implementation and a lack of focus on outcomes. It's philosophy. It's low expectations. The problem is that people disagree on standards. We can't even get started.

Actually, I'm more encouraged by Duncan's arm-twisting in states to force them to open up more charter schools. I can't imagine a top down solution to the problem of education that won't be watered down or manipulated. Either parents have to be allowed to send their kids elsewhere, or schools have to provide parents with choice. TERC or Singapore Math. I don't expect the education world will give up control easily. They will accept (low) accountability and weak standards first.

Friday, October 23, 2009

First use of the phrase "4th grade hump?"

In the earliest mention of the phrase that I have yet found, it is called instead the “4th grade hump.” For a few years after that, it is called a hump, then it appears as the phrase “4th grade slump.” If you search Google books for the two phrases for various time periods, you find a number of interesting links. Also, over time, there are more and more hits for the phrase.

Here are the phrases, from Edward Dolch, the creater of the Dolch sight words, in his 1948 book, “Problems in Reading.” The first is on p. 56:

There is the famous 'fourth-grade hump,' the sudden difficulty of reading matter that strikes the children without warning. The difficulty keeps on climbing through grades V and VI.

This mention on page 251 also has a very interesting table:

Every word not on the list of 1,000 [most frequent words] was underlined and counted every time it appeared. Therefore a percentage of hard words means a percentage of the running words, that is, the total words read. We are aware that if a word appears a second time it is not now a hard word if it were learned the first time, but there was no way of allowing for this factor.....

The percentages of hard words for the books of each grade were averaged, and then the figures were rounded to the nearest whole number. The result shows the following:


Word Difficulty, According to Appearance on First 1,000 Words for

Children’s Reading as Found in Ten Series Of Readers



Hard Words (Not on List)

4% 6% 8% 12% 14% 16%

These figures show the well-known “fourth grade hump,” a difference between the third grade books and fourth grade books that is twice the difference between the other grades.

There were several interesting links to later use of the phrase “4th grade slump,” I’ll link to two of them:

  1. The Trouble With Boys by Peg Tyre, p. 142-143

(The whole thing is interesting, here’s an excerpt:)

Around fourth and fifth grade, another factor comes into play as well. Good readers take a leap forward as they move from learning to read to reading to learn. The curriculum demands it. It’s no longer enough to be able to “sound out” words. Children have to comprehend sentences and paragraphs from history and science books and make inferences from those texts. Kids who don’t make that jump fall into what experts have dubbed the “fourth-grade slump.” They are stuck trying to figure out how to decode the word everglades, for instance, while other kids are learning about the kinds of animals that live in those Florida swamps. It’s an important cognitive leap.

By every measure, the fourth-grade slump hits boys harder than it hits girls.

2. The Reading Crisis: Why Poor Children Fall Behind by Chall, Jacobs, and Baldwin, p. 143

As predicted by the theoretical model of reading used for our study (see Chapter 1), the students’ scores started to slump at about grade 4. For the below-average readers, the slump began early (in grade 4) and was intense. By grades 6 and 7, they were reading almost two years below grade level on all the reading tests. For the above-average readers, the slump began later (around grade 6) and was less intense. Many of the above-average readers were still reading on grade level or above in the sixth and seventh grades on some of the reading tests.The slump started earlier on some tests than on others.

The first to slip was vocabulary.

If you teach properly, with a good phonics method and less than a dozen sight words, there are few "hard words;" disadvantaged elementary children (including several 1st graders) taught with Webster's Speller were able to sound out what Dolch called "hard words," but Webster called "easy words of X syllables."

Here are some "easy words of 3 syllables, accented on the first and third"


and "easy words of 4 syllables, full accent on the first, and the half accent on the third"


Wednesday, October 21, 2009

Euclid's Postulates and Definitions

Euclid's Elements has 13 books. Book 1 starts with definitions.

Here are the first few:

1. A point is that which has no part.
2. A line is a breadthless length.
3. The extremities of a line are points.
4. A straight line is a line which lies evenly with the points on itself.
5. A surface is that which has length and breadth only.

The main problem is that modern language and his language are a bit confusing. We don't use "extremities".

The definition of straight line above might not be clear unless you already know what a straight line is.
But in total, Euclid defines: point, line, straight line, surface, plane, plane angle, right angle, perpendicular angle, obtuse angle, acute angle, boundary, figure, circle, center, diameter, semicircle, equilateral triangle, isosceles triangle, scalene triangle, square, rhombus, trapezoid, parallel.

The 5 postulates are (in his words)
1. [One can] draw a straight line from any point to any point
2. [One can] produce a finite straight line continuously in a straight line.
3. [One can] describe a circle with any center and distance.
4. All right angles are equal to one another.
5. If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, athe two straight lines, if produce indefinitely, meet on that side on which are the angles less than the two right angles.

In modern terminology, we usually list the first postulate as "two points determine a line". The fifth one is often rewritten to say "parallel lines don't intersect"; it's the most famous for being the one that makes the geometry described here planar--remove that postulate, and look at the lines of longitude on the globe.

So why are these postulates not definitions? The fifth one is complicated enough to be phrased as an "if-then", so it's probably reasonably clear why that's not a definition.

But why do we need to postulate a line or a circle, rather than define it?

Postulating a circle means we postulate its existence. We could just define a circle or a line, but it doesn't mean that such a thing with those properties as defined exists.

And we can't prove that those things exist, either. We have to assume them. This may be the oddest thing to a young student. "WHAT do you MEAN a circle doesn't EXIST? Look, I can draw one." (The standard answer to this is that any drawn circle is actually imperfect--doesn't have constant radius, doesn't have a boundary of no thickness, etc. This is typically a deeply unsatisfying answer. You could then tell him about the REAL constructivists: mathematicians who tried to construct all of math using the actual constructed "non platonic" geometry they could realize...but that's for another day.)

But this is where constructions come from, and this is what they were for. The answer to the above complaint is that you can actually CONSTRUCT everything you're asked to prove using just three things: a straight edge, a compass, and a pencil.

That is what it means to say "postulate a circle": it means "give yourself a compass to create a circle out of thin air." Same with "postulate a line: "give yourself a straight edge", and "postulate a line segment" : "give yourself a pencil to mark points on your line.

Then, the postulates have real meanings, because you can create everything else from those devices. Equilateral triangles are defined, but their existence is proved--and can be proved by construction from just the above three tools. Same with regular polygons, various congruent angles and lines, etc.

The idea that you start with 24 postulates is really really depressing, since it undermines everything beautiful about geometry that could be seen by construction.

Here's a link to Euclid's books. It contains all of the propositions and hyperlinks to the proofs.

Sometimes it's difficult to see the great truth he proved geometrically because of his wording and naming conventions. (Would you recognize his proof of the factoring of (x^2 -1) ?) But reading this might show you what you need to know and what's superfluous. And it will show you how to prove 19 of those postulates...

Tuesday, October 20, 2009

gone fishing

Have been in Evanston since Friday - my mom has rallied!

Back to Irvington on Thursday & will be checking in & out & reading posts ---

Monday, October 19, 2009

World Origami Days

Celebrate World Origami Days, 24 October - 11 November:
Share the joy of folding! See the connections to mathematics.

Visualize the possibilities:

The first two sentences from the article:

"Anything can be made with origami—from birds and bugs to stents and space telescopes. It’s just a matter of math."