Audacious Idea #1: Illustrate Every Math Standard

Megan Golding is a 10th grade math teacher in Georgia. She has come up with an audacious idea:

I have a simple, if huge, goal: I want to publish a podcast for every single [Georgia Performance Standard] standard and element.

Sure, I’ve seen Khan Academy. It’s great! My addition is real state standards correlation (most of what I’ve seen that’s supposedly correlated to Georgia’s math standards is of shallow, if at all, relevance).

I’ve found that in math, my students want me to repeat the same explanation several times until that light bulb comes on. The podcast allows me to put an explanation in a kid’s hands and let him rewind & replay as much as he needs. Build in a little practice, and you have a nice little remediation tool.

Go look at her first post & do leave comments there.

Wouldn't it be great if every state had a video library like that?

[Cross published at Academic Remediation]

Eight O'Clock Class

Three times a week this small assembly meets, With drooping eyelids and half-hidden yawn,
To hear, incredulous, how Shelley, Keats,
And Wordsworth rose to write before the dawn.

EIGHT O'CLOCK CLASS
Richard Armour

Source: College English, Vol. 21, No. 7 (Apr., 1960), p. 383

Big "Aha!" moment from the MSMI2010 follow-up day

Saturday 9/25 - late afternoon:

My notes from Wu discussing the teaching of division with remainder:

Actually, this was a big "Duh!" moment.

Prior posts from MSMI2010.

Homeschooling by the Numbers

Via Pat'sBlog who saw it at Maria Miller's "Homeschool Math Blog.

(Click to enlarge)

Via: DegreeSearch.org

Any thoughts on the accuracy of these figures?

gone fishing

Off to Illinois for the weekend - back Monday!

autonomy, mastery, purpose

Dan Dempsey just left a link to this fantastic YouTube on the Irvington Parents Forum ---

"it's the culture, stupid" part 2

Deborah Kenny on Harlem Village Academies

the sorting machine

$100 million to Newark public schools

I'm against it.

Seriously.

Public schools need the things money can't buy.

Modern math is a river in Egypt

It's been decades since any sort of traditional math has been taught in most K-6 schools, but we still hear the same arguments. Our schools have used Everyday Math for years, and before that, they used MathLand. When schools and the state look at the testing numbers and whether they go up or down, it has nothing to do with any sort of traditional approach to teaching math.

Kids get to middle school and high school and still can't show mastery of very basic skills. Even those who favor discovery methods claim that mastery is needed. It still doesn't happen even though mastery of basic skills is much easier to check and correct than vague critical thinking skills.

My conclusion is that it's a huge case of denial.

It's Anti-Math. It's anti-hard work because hard work and expectations separate kids. They claim that kids will learn when they are ready, but that just keeps them from the hard work of figuring out whether there are other causes. It allows them to avoid working on things they don't like. That's why we parents get messages telling us to work on math facts with our kids.

In 5th grade, my son's Everyday Math teacher realized that many kids were struggling with simple things like adding 7+8. She knew these kids were "ready". She tried to fix the problems, but then didn't get to 35% of the material in the course. She believed in mastery, but the other teachers were happy enough to "trust the spiral".

Everyday Math facilitates this denial. They want math to be a pump and not a filter, but what they are doing is pumping along the failure until its too late and then dumping that guilt trip on the kids.

LexAequitas on SAT Critical Reading

LexAequitas:

“As an instructor for TPR for several years, I had a breakthrough on the reading passages at one point after I made a realization. I was good at verbal sections before, generally able to complete them in just over 20 minutes with a 750+ score, but after the instructor training I could complete them in 11 minutes with a perfect score.

The SAT is rigorously vetted, and constantly challenged. For every answer, there needs to be a detailed explanation that ETS can just mail off to every student who challenges the question. They come up with these before the test is ever given. This means there *has* to be something that can be pointed to in order to differentiate answers. On difficult questions, it will be minor, even picayune. And I don't mean picayune in the normal sense, I mean picayune in the sense that you'd better start thinking like a 13-year old with Asperger's syndrome. Don't accept generalizations that people use in speech and writing all the time just to make communication more efficient.

It's also worthwhile to note that the SAT does not typically select great literature. The writing is often clumsy and arcane, and this is part of the challenge. I suspect a better exercise if you want casual reading that mimics the SAT is to pick foreign translations.”

This is fantastically helpful.

satreading
And fascinating on many levels....

David Kaplan on Sal Khan

In an undistinguished ranch house off the main freeway of Silicon Valley, in a converted walk-in closet filled with a few hundred dollars' worth of video equipment and bookshelves and his toddler's red Elmo underfoot, is the epicenter of the educational earthquake that has captivated Gates and others. It is here that Salman Khan produces online lessons on math, science, and a range of other subjects that have made him a web sensation.

Khan Academy, with Khan as the only teacher, appears on YouTube and elsewhere and is by any measure the most popular educational site on the web. Khan's playlist of 1,630 tutorials (at last count) are now seen an average of 70,000 times a day -- nearly double the student body at Harvard and Stanford combined. Since he began his tutorials in late 2006, Khan Academy has received 18 million page views worldwide, including from the Gates progeny. Most page views come from the U.S., followed by Canada, England, Australia, and India. In any given month, Khan says, he's reached about 200,000 students. "There's no reason it shouldn't be 20 million."

Sal Khan: Bill Gates' favorite teacher by David Kaplan

David Kaplan is one of the best writers on the Irvington Parents Forum. Probably the best, as a matter of fact, and that's saying something, because there are a lot of good writers on the Forum.

why is SAT reading so hard?

I've just recently focused on SAT reading ---

It's hard.

Until two weeks ago, I took SAT reading for granted. I had high verbal scores in high school, and when I took a sample SAT test a couple of years ago, I missed two questions on the reading, only one of which was a real miss in the sense that my answer was clearly wrong. On the other miss, I disagreed with the CollegeBoard's answer.

So I hadn't thought about the verbal section until I started working with C. and a couple of other juniors on SAT prep.

Now I find that working through the verbal section is a wretched, miserable experience, and this is true in spite of the fact that I get the answers right.

Here's Hack the SAT:

You know how some people have a gift?

[snip]

No one has a gift for Critical Reading.

Sure, plenty of people have a gift for reading.... [but] no one is particularly natural at critical reading, this seemingly endless stuff that jams the SAT full of tiny blue print and passages about the subtle revelations of medieval scientist nuns.

[snip]

[E]ven those students who get high scores on Critical Reading find it more draining than the Math or Grammar sections. 

True!

Working through a Critical Reading passage - especially a 'compare two passages' section (those are death) - I feel the brain equivalent of eye strain.

Why is that?

a man after my own heart

Glen wrote:

I have my fourth grader do a hundred fraction problems while I do his "cut pictures out of magazines" homework for him. I have him work through middle school math contest problems, tossing him hints when he gets stuck, while I draw on his poster board. I had him learn every country in Europe while I built miniature teepees in a shoebox diorama.

He gets good grades on his schoolwork, but other parents--I mean kids, of course--often cut and paste better than I do.

His teacher thinks he's a "natural" at math, but there's nothing natural about it. It's man-made. It's training--the same sort of training you'd do if you needed to teach someone to cut hair or build birdhouses: show them how, help them a few times, and put them to work.

She would be shocked if she actually knew the level of difficulty of the math and science work he can do, but we're careful not to let her find out. Last year the teacher found out and was nasty to him for the rest of the year. She liked him when she thought he was a natural, but when she found out that he had to work at math, she was outraged.

"It's not fair to me that you are willing to do that much work for your father, but aren't doing the same for me!" I thought that I--I mean my son, of course--was cutting enough pictures out of enough magazines for her, but she apparently thought she deserved more.

She wanted to discuss the "problem" with me. She was concerned about how I was using his time. (How ironic.) She said that their Everyday Mathematics emphasized "conceptual understanding" and was concerned that my approach might not lead to "actual understanding." The previous night he had solved,

"We have four times as many cows as horses on our ranch. If we sold 280 cows, we'd end up with twice as many horses as cows. How many cows do we have?"

He was in third grade. I almost asked her to go to the board and show me how SHE would have taught him to solve it, with "actual understanding," but that would have been cruel. I held my tongue to keep my son out of trouble and said that I tried my best to help him understand. We left it at that.

This is one of the top 5% of elementary schools in Silicon Valley, so almost all the kids are performing at grade level--and that's where they want to keep them.

And I've now found out that at higher levels, middle school and high school, it's almost standard practice for parents to take the mindless homework load off their kids' shoulders to free up time for them to do the portion of homework that is actually useful.

If I have to do even more mindless homework, I may have to outsource it to India.

Using sentence combining to improve reading scores

from Arthur Whimbey and Myra Linden:

Let us now examine the problem mentioned at the beginning of this chapter: the difficulty that some students have comprehending textbooks containing sentences with relative clauses. Here is a paragraph from the Encyclopedia Americana.

Infectious diseases are the only ones that can be transmitted. They may be spread by infected animals, infected people, or contaminated substances, such as food and water. Infectious diseases that can be transmitted to humans from infected animals are known as zoonoses. Zoonoses may be transmitted by carriers, such as insects; by the bite of an infected animal; by direct contact with an infected animals [sic] or its excretions; or by eating animal products.

College freshman were asked to read this paragraph and then answer the following questions.

Zoonoses are:

a. insects that carry diseases.

b. infected animals that transmit infectious diseases to humans.

c. infectious diseases that man gets from animals.

d. carriers that transmit infectious diseases.

College students with weak reading skills often pick alternative b. When asked why they picked b, some reply that zoonoses sounds like zoo and animals are kept in zoos, so they figured zoonoses are animals. This explanation reflects the thinking style of nonanalytical readers. They base their conclusions on superficial associations among bits of information rather than on careful step-by-step interpretations of chunks of information and gradual reconstruction of total meaning.

Other students who chose b explained that they got this answer from the last six words of the third sentence: infected animals are known as zoonoses. This, too, reflects the thinking style of weak readers. They read a little bit here, a little bit there, and then jump to a conclusion.

The correct answer is based on the third sentence, which reads:

Infectious diseases that can be transmitted to humans from infected animals are known as zoonoses.

This sentence contains the following relative clause:

that can be transmitted to humans from infected animals

Good readers work step-by-step through the sentence in obtaining its correct meaning. They begin with the subject: infectious diseases. Then they go on to the relative clause, an essential relative clause that indicates the type of infectious diseases being considered; those that animals can transmit to humans. Finally, they come to the predicate: are known as zoonoses. Therefore, in answering the question they pick alternative c.

Research studies have found that students with weak analytical skills can understand only simple sentences, just as they can solve only one-step math problems. They have difficulty understanding complicated sentences, just as they have difficulty solving multi-step math problems. In other words, they can handle just small chucks of information because they have not developed skill in working step-by-step through complicated information. They can understand sentences such as this:

Some infectious diseases are known as zoonoses.

But they cannot understand Sentence 3 in the paragraph.

Other research studies (reviewed in Why Johnny can't Write: How to Improve Writing Skills) have found that having students construct complex sentences from simple ones improves their scores on standardized reading tests. these studies have not explored whether having students add relative clauses to sentences improves their ability to comprehend specifically sentences with relative clauses. They have only found that constructing various types of complicated sentences from simpler ones (using the types of exercises shown in this book) improves overall reading comprehension ability, with the weakest readers making the greatest gains. As the P-C Approach^* is used more widely in our schools,^** we can expect the average reading comprehension ability of the nation to improve. And as this improvement becomes evident, researchers may conduct additional studies to determine exactly how and why having students manipulate, construct, and write sentences improves their reading skills.

Teaching and Learning Grammar: The Prototype-Construction Approach
pp. 90-92

^*prototype-construction method
^** I'm not holding my breath

help desk - sentence diagram

In the passage below, Terkoz the ape has kidnapped Jane; Tarzan is tracking Terkoz through the treetops.

At boughs’ ends, where the anthropoid swings from one tree to another, there is most to mark the trail, but least to point the direction of the quarry, for there the pressure is downward always, toward the small end of the branch, whether the ape be leaving or entering a tree. Nearer the center of the tree, where the signs of passages are fainter, the direction is plainly marked.

Tarzan of the Apes
Edgar Rice Burroughs
The Compact Bedford Introduction to Literature

p. 74

This was a fantastically difficult sentence for my college composition students to read; it was difficult for me to read, too.

Can someone diagram the first sentence?

You hear constantly that sentence diagramming doesn't help with writing, but I'm wondering whether it would help with reading. I'm tutoring a high school junior on the SAT, and the sentences are fantastically complex. The reading passage we worked on this weekend actually contained two sentence fragments in a row, both of which modified the complete sentence before them.

LD enrollment drops

After decades of what seemed to be an inexorable upward path, the number of students classified as learning-disabled declined from year to year over much of the past decade—a change in direction that is spurring debates among experts about the reasons why.

The percentage of 3- to 21-year-old students nationwide classified as having a “specific learning disability” dropped steadily from 6.1 percent in the 2000-01 school year to 5.2 percent in 2007-08, according to the most recent data available, which come from the U.S Department of Education’s 2009 Digest of Education Statistics. In numbers, that’s a drop from about 2.9 million to 2.6 million students.

A learning disability—a processing disorder that impairs learning but not a student’s overall cognitive ability—is the largest, by far, of the 13 disability classifications recognized by the main federal special education law. Forty percent of the approximately 6.6 million students covered under the Individuals with Disabilities Education Act, or IDEA, fall into that category.

The decrease in the category goes hand in hand with a decrease in special education enrollment overall, though that change is not as large. The percentage of all students covered under the IDEA fell from a high of 13.8 percent in the 2004-05 school year to 13.4 percent in 2007-08—from about 6.7 million students to about 6.6 million students. Enrollment in the categories of emotional disturbance and mental retardation also went down, but students in those groups make up a far smaller slice of the IDEA pie. At the same time, though, enrollment of students classified as having an autism spectrum disorder or “other health impairment” rose.

[snip]

About 80 percent of children who are classified as learning-disabled get the label because they’re struggling to read. So, scholars say, the dropping numbers could be linked to improvements in reading instruction overall; the adoption of “response to intervention,” which is an instructional model intended to halt the emergence of reading problems; and a federally backed push toward early intervention with younger students before they’re labeled.

Learning-Disabled Enrollment Dips After Long Climb Up
by Christina A. Samuels
Education Week

words, diagrams, & math

Nick Rowe

12-week column on drawing

in the New York Times

Before I got sidetracked teaching myself math, I was learning to draw --- Loved it.

Also, I used to knit. I used to knit a lot. A friend of mine told me that knitting and math yield the same pleasures, and she turned out to be right. So I haven't knitted anything in 5 years, either.

I need a couple more lives besides just the one I've got.

safety first

Coincidentally, we were discussing the "security" system here in my district at the very moment Ed and I were trekking back and forth to the U.S. Open.

At the Open, the threat level appears to be permanently set on orange and security is always high. Security is so high that, when I presented my bag for inspection, the gal behind the table said, "Is that a Kindle? You can't bring that in."

What?

As Ed said, if you can take a Kindle on an airplane, which you can, you should be able to take a Kindle to the Open.

But no. I was supposed to hike two blocks -- uphill and in the sun -- back the way I came in 'til I got to a tent where I could pay another employee five bucks to check my Kindle.

Or, alternatively, I could walk a couple hundred yards back to the ladies' room, secrete my Kindle on my person, present my bag for inspection a second time, and enter the grounds without further ado. Which is what I did.

I told Ed, "Any security system that relies on terrorists to pay some guy five bucks to check his explosive device isn't a security system."

More anon.

lost in translation

caption: Fight fire with fire!
engrish.comhttp://www.engrish.com/
via: km-wordsmith

20 most common errors

from:
The St. Martin's Handbook

via:
Texas Tech University Writing Center

the 20 errors

Another rude awakening

Last year, my first-born was in enrolled in our local public school in second grade, and although he spent 6 or 7 hours there each day, he came home with tons of homework and a good deal more learning remaining. I started to think, “Why am I sending my kids to school, when the school sends them back to me to educate?” A friend later complained to me that when they gave her son a book report, it meant at least 5 hours of her own time working on it with him. Another parent told me how great the tutoring was at the chain store down the street from her. All this got me to thinking: if I’m ultimately in charge of my kids’ education, why do I feel so powerless? . . .

I assumed that the government knows best what, and how, to teach my kids. It’s an assumption I was raised with, one I never challenged, until now. Well, you know what they say about you when you assume…

“It’s always worse than you think?” Well, that’s what we often say around here.

Another parent gobsmacked by our public schools

Back to School: Part One

Testing in Basic Math in Community College

I finally got a copy of Teach Like a Champion. Since I teach at a community college, not everything applies. But I do teach a basic math class. And the book has inspired me to start taking data on my tests. This led me to analyze my tests to see what and how much I test on each topic.

At the end of the summer semester, I used the department written final to see if I could determine how much mastery my students had on each topic that I taught. Unfortunately, I don't have data from the beginning of the semester. But what I found wasn't very good: I measured the mastery on 15 topics by calculating a percent correct for each topic. For example, there were 5 problems on whole number operations and 11 students. This gives a total of 55 problems. Only 76% were answered correctly. If I had to take a guess, I would say that the class probably had over 70% mastery on whole number operations before they came into class. My highest % mastery was decimal operations, fraction operations and dimensional analysis with 93%, 85% and 93% mastery respectively. My lowest topics were divisibility tests and lowest common multiple and prime factorization. I then analyzed each incorrect answer to see what the most common errors were. 66% of the errors were concept errors: either the student didn't answer the question or did not use the correct mathematical technique. 19% of the errors were process error. In this category are errors like the following: a student knows that when multiplying mixed numbers, he must turn the mixed number into an improper fraction, but he does this incorrectly. 7% of the errors were mathematical. Another 8 % were divided evenly between not reducing fractions and sign errors.

So I'm teaching basic math again this semester, and I am trying to determine what I should be doing to ensure mastery. I am going to try curriculum based measurement. This is a technique I read about that is used in some elementary schools. You give students very short exams on a particular topic. Instead of counting the number of problems that are correct, you count the number of correct digits. For example, if a student has to add 99 and 49, there is a possibility of 3 correct digits, not one correct problem. You then do some number crunching and adjust your teaching accordingly. (I realize I'm gliding over this.)

It doesn't seem like any of the other instructors at my community college are collecting and analyzing data like this. If anyone has any links or resources on this topic, I would love to hear about it.

Another article (or two): Is it bad to test students frequently?

Two recent New York Times articles fly in the face of conventional pop psychology and education theory.

An article in last week's Science Section on study habits cites cognitive science research indicating that the act of taking a test can enhance learning:

The process of retrieving an idea is not like pulling a book from a shelf; it seems to fundamentally alter the way the information is subsequently stored, making it far more accessible in the future.

As Henry L. Roediger III, a psychologist at Washington University in St. Louis puts it, “Testing not only measures knowledge but changes it.”

Next we have a front page article in this weekend's Week in Review on Testing, the Chinese Way, written by Elisabeth Rosenthal, whose children spent a year at the International School of Beijing where "taking tests was as much a part of the rhythm of their school day as recess or listening to stories." Citing personal experience, Rosenthal argues that:

>Young children aren't necessarily aware that they are being "tested."

>Frequent tests give children important feedback about how they are doing.

>Frequent tests offer a more meaningful way to improve self-esteem than frequent praise does.

On this past point, Rosenthal cites Gregory J. Cizek, a professor of educational measurement and evaluation at the University of North Carolina at Chapel Hill:

Professor Cizek, who started his career as a second-grade teacher, said the prevailing philosophy of offering young children unconditional praise and support was probably not the best prescription for successful education. “What’s best for kids is frequent testing, where even if they do badly, they can get help and improve and have the satisfaction of doing better."

Cizek's overall take on testing in schools? “Research has long shown that more frequent testing is beneficial to kids, but educators have resisted this finding:”

Rosenthal concludes on a particularly powerful note:

When testing is commonplace and the teachers are supportive — as my children’s were, for the most part — the tests felt like so many puzzles; not so much a judgment on your being, but an interesting challenge. It is a testament to the International School of Beijing — or to the malleability of childhood memory — that Andrew now says he did not realize that he was being tested. Will tests be like that in a national program, like Race to the Top?

When we moved back to New York City, my children, then 9 and 11, started at a progressive school with no real tests, no grades, not even auditions for the annual school musical. They didn’t last long. It turned out they had come to like the feedback of testing.

“How do I know if I get what’s going on in math class?” my daughter asked with obvious discomfort after a month. Primed with Beijing test-taking experience, they each soon tested into New York City’s academic public schools — where they have had tests aplenty and (probably not surprisingly) a high proportion of Asian classmates.

Articles you may have missed...

I did, while I was out of town.

From the New York Times:  Forget What You Know About Good Study Habits

 But individual learning is another matter, and psychologists have discovered that some of the most hallowed advice on study habits is flat wrong. For instance, many study skills courses insist that students find a specific place, a study room or a quiet corner of the library, to take their work. The research finds just the opposite.

And from the Washington Post, a little opinion on:  School reform's meager results

Standard theories don't explain this meager progress. Too few teachers? Not really. From 1970 to 2008, the student population increased 8 percent and the number of teachers rose 61 percent. The student-teacher ratio has fallen sharply, from 27-to-1 in 1955 to 15-to-1 in 2007. Are teachers paid too little? Perhaps, but that's not obvious. In 2008, the average teacher earned $53,230; two full-time teachers married to each other and making average pay would belong in the richest 20 percent of households (2008 qualifying income: $100,240).

Enjoy.

Oxbridge

It is probably the most important day of your life. Your mind is racing and your hands are trembling at the thought of the erudite questions you are about to be asked, which will determine your future education, career and indeed the rest of your life. Then a man leans forward towards you and says: "Tell me about a banana."

other questions:

"Why isn't this chair acting as a wave?"(Chemistry, Oxford).

"Estimate the number of pebbles on Brighton Beach. If a pebble was given to each person would there be enough for the entire population?" (Natural sciences, Cambridge).

"If you leave the fridge turned on in a thermally isolated room, what happens to the room?" (Physics, Oxford).

Knowledge of a banana may be the key to Oxbridge entry
By Richard Garner
Monday, 6 September 2010

So You Want to Go to Oxbridge?: Tell Me About a Banana

Barry G on just-in-time learning

I think this is my favorite of Barry's articles -- he nails it.

Even worse than the book itself were the discussions in class that came out of it. One event in particular stands out. In a chapter that discussed the difference between "knowing" and "understanding," a chart presents examples of “Inauthentic versus Authentic Work.” In this chart “Practice decontextualized skills" is listed as inauthentic and "Interpret literature" as authentic. The black and white nature of the distinctions on the chart bothered me, so when the teacher asked if we had any comments, I said that calling certain practices “inauthentic” is not only pejorative but misleading. I asked the teacher “Do you really think that learning to read is an inauthentic skill?”

She replied that she didn’t really know about issues related to reading. Keeping it on the math level, I then referred to the chart's characterization of "Solve contrived problems" as inauthentic and “Solve ‘real world’ problems” as authentic and asked why the authors automatically assumed that a word problem that might be contrived didn’t involve “authentic” mathematical concepts. “Let’s move on,” she said.

[snip]

The authors’ approach to how one teaches for understanding is through a process that they call “backward design,” in which educators plan their courses, units and lessons by starting from what they want the end result to be. That is, what should students know, understand and be able to do? The planning process then entails working backwards from there, identifying the content that goes into this, the big ideas, the questions to be explored and so on.

As the authors state, backward planning is not a new idea. In fact, I was a bit confused as to why it is even needed, given that such work has essentially been done in the writing of the textbooks that cover the course material.... But this brings us to another axiom which I have heard repeated in education school, which states that textbooks are a resource and not a curriculum. The authors pick up on this as well and regard using the textbook for planning as a “sin," stating that “The textbook may very well provide an important resource but it should not constitute the syllabus.”

[snip]

In a paean to constructivism and the abandonment of textbooks, Tomlinson and McTighe dispose of the notion that sequence of topics and mastery of skills is important, calling such beliefs the “climbing the ladder” model of cognition. “Subscribers to this belief assume that students must learn the important facts before they can address the more abstract concepts of a subject,” the authors state, and then quote Lori Shepherd, a University of Colorado education professor to make their point:

“The notion that learning comes about by the accretion of little bits is outmoded learning theory. Current models of learning based on cognitive psychology contend that learners gain understanding when they construct their own knowledge and develop their own cognitive maps of the interconnections among facts and concepts.”

In fact, this is the crux of how they approach differentiated instruction. Sequence doesn’t matter. Each student constructs his or her own meaning at their own pace, by being immersed in what the authors term “contextualized grappling with ideas and processes.” What does this mean? There are many examples, but the prevalent pattern of instruction to emerge from the book seems to be one of giving students an assignment or problem which forces them to learn what they need to know in order to complete the task. Say it is quadratic equations. Rather than teach them the various methods of factoring first, with the attendant drills, they might start with a problem such as x^2 + 5x + 6 = 0. The teacher may then provide some activities that illustrate what factoring is, and then provide some exercises. The goal would be to factor the above equation into (x+3)(x+2) = 0 and, from there, lead the students to see that there are two values that satisfy the equation. This is what they mean by “contextualized grappling” as opposed to “decontextualized drill and practice.” It is a “just in time” approach to learning, (my choice of phrase, not theirs) in which the tools that students need to master are dictated by the problem itself by not burdening the student's mental inventory with “mind numbing” drills for mastery of a concept or skill until it is actually needed.

[snip]

They admit that there are times when direct instruction or ‘teaching by telling’ might work extremely well. “There is a need for balance between student construction of meaning and teacher guidance”, they proclaim. That direct instruction would work even better if topics were presented in a logical sequence is not the message of this particular book, however.

[snip]

“Just in time” approaches that work as a model for business inventory work just as well in education, they believe. The result is an approach that is like teaching someone to swim by throwing them in the deep end of a pool and telling them to swim to the other side. For the students who may already know a bit about swimming, they may choose to take that opportunity to learn the butterfly. The teacher might advise the weaker students to learn the breast stroke and provide the much needed direct instruction which they may now choose to learn. Or not.

Let’s move on.

rule of 20

Surprisingly, the vast majority of skills which we try to teach appear to begin the passage from acquisition to fluency-building at roughly the same point —when correct frequencies are somewhere between 14 and 20 per minute, and an accuracy of between 67% and 83% has been achieved. That rule seems to apply to very basic behaviors like pointing to named objects, steps taken while walking, and completing parts of a dressing sequence. The same transition point also seems to apply to very complex behaviors like reading, speaking, and writing digits to solve advanced mathematics problems. Indeed, the rule appears so universal that when Sokolove (note 9) examined circa 3300 programs of children in grades 1 through 6, the “rule of 20” predicted progress in more than 97% of the cases.

Decisions, Decisions
Owen Roberts White
University of Washington
Fall 2000

unfriendly worksheets, part 2

from A Guide to Learning English:

Rule number 2: Be aware of the difficulties of cultural references!

The following text is the first part of the introductory paragraph to a question about car speed:

"The police used to measure the speed of cars on the road by having two PCs some distance apart using a stopwatch. One of them would stand and wave as a car reached him. When the wave was seen by the second PC a stopwatch was started. As the car passed the second PC, the stopwatch was stopped."

The question itself read:

"Is the car exceeding the maximum speed limit in Britain?"

The ESL student who asked for my help with this question was puzzled how a personal computer could stand and wave. He also had no idea what the maximum speed limit is in Britain.

I would like to know how two personal computers can stand some distance apart and use a stopwatch.

unfriendly worksheets

source:
A Guide to Learning English

Pass/Fail

Examination dreams are reported to persist even into old age...
- Time magazine

You will never graduatefrom this dream
of blue books.
No matter how
you succeed awake,
asleep there is a test
waitinmg to be failed.
The dream beckons
with two dull pencils,
but you haven't even taken the course;
when you reach for a book--
it closes its door
in your face; when
you conjugate a verb--
it is in the wrong
language
Now the pillow becomes
a blank page. Turn it
to the cool side;
you will still smother
in all of the feathers
that have to be learned
by heart.

Linda Pastan
1975

in:
The Compact Bedford Introduction to Literature
Michael Meyer

edu-fads at home

Found this comment on Jay Mathews' 2009 column about 21st century skills:

One of my friends holds advanced degrees in education, and she used cutting-edge methods to teach her own kids. What she forgot was the classic problem of the teacher being all fired up and motivated, and the student feeling left out of the picture. After all, fundamentals may be old-hat for teachers, but for students they're all new concepts. Her son was all but forgotten in her enthusiasm and fascination with the perpetually new. She has asked me a hundred times what I think is "wrong" with him (answer: "you").

She force-fed her son this and that fad over the years while he quietly turned off to learning. He recently dropped out of the marginal college he was able to squeak into with his middling test scores, yet it is obvious to anyone who talks with him for five minutes that he is very bright. I think he'll probably drop back in some day when he returns for his own reasons, but his mom's incessant buzz-speak about the latest pedagogical gewgaws of the day (I suspect 21st-century skills were part of it) really did a number on him. Poor kid.
1/5/2009 7:10:48

The Latest Doomed Pedagogical Fad: 21st-Century Skills
by Jay Mathews
Washington Post
Monday, January 5, 2009

I wonder if this boy was being homeschooled or afterschooled.

survey: charter support at 65%

Support for charter schools also continued to grow among the public, with 65 percent of respondents saying they would back new public charter schools in their community and 60 percent saying they would support “a large increase” in the number of such schools operating in the United States.

Fewer Americans Back Obama’s Education Programs
By Dakarai I. Aarons
Published Online: August 25, 2010

I'm glad to see this level of support for charters, but I do worry about charters killing off private and parochial schools.

Robert Pondiscio on curriculum vs value-added

When I think of the curriculum and teaching methods I was required to use in my classroom, the idea that my effectiveness might be dependent upon them makes me want to lie down with a damp wash cloth on my forehead. Manipulatives and discovery instead of basic arithmetic? Endlessly revising ”small moments” and teaching the writing process to 10-year olds instead of basic grammar? No time for even basic science, social studies because of district demands for ever larger math and literacy blocks? If it fails, it’s on me? Seriously?

Erin Johnson left a comment:

Robert, Why do you think that the LA Teachers Union (or the national unions) have not highlighted the issue of curricula?

I have recently been in contact with a LA teacher who was rated “more effective” in math by the LA Times. She states that her good rating was probably due to the fact that she “subversively” uses Saxon math instead of her district adopted program. Do ed reformers expect that teachers will subvert the curricula adoption process?

And here is Robert again:

I’m not sure curriculum reform is on anyone’s radar screen in a big way, including the unions. I used to regularly subvert…er…adapt my math curriculum to assure automaticity on basic functions. 5th graders counting on their fingers or multiplying with arrays is an offense to my sensibilities. I had less flexibility on ELA since there was lots of joint planning and execution involved. I’d go as far as saying my school’s ELA program (“It’s not a curriculum, Mr. Pondiscio, it’s a philosophy,” I can still hear the staff developer reminding me) is what turned me into a curriculum advocate.

Curriculum effects and value-added

cart, horse

from Casting Out Nines:

[I]t’s not clear to me that doing “algebra” is a better idea here than just doing straight-up subtraction.  What’s to be gained by saying “the whole is 8; one part is 3; the other part is ____” versus “What is 8 minus 3?” Again, maybe I’m out of touch, but subtraction is a fundamental skill that algebra builds upon; doing algebra before subtraction seems a little backwards to say the least. A kid who is comfortable with subtraction will be able to do these whole/part problems in a snap by using subtraction. A kid doing these “algebra” problems basically has to invent subtraction in order to do them, or else draw pictures of balloons and start counting. It feels like the curriculum is trying to be intentionally nontraditional here, just for the sake of doing things differently rather than because it works better.

what makes this question difficult?

This is one of the lowest percent corrects I've seen on a Question of the Day -- as low as the percent correct for the 3 people in an office question.

Why is that?

Critique of Envision Math by Casting Out Nines (Robert Talbert)

From January, 2008:

Four questions about this:

1. Should it be a requirement of parenthood that you must remember enough 5th grade math to teach it halfway decently to your kids?
2. Does the smartboard come included with the textbooks?
3. Did anybody else have the overwhelming urge to yell “Bingo!” after about 2 minutes in?
4. When will textbook companies stop drawing the conclusion that because kids today like to play video games, talk on cell phones, and listen to MP3 players, that they are therefore learning in a fundamentally different way than anybody else in history?

The last question is all about the research-free digital nativist assumption that is the source of many lucrative curriculum deals these days. Data, please?

I've added emphasis

basically laughing it off the blogosphere for its happy-clappy, uncritical acceptance of unproven digital nativist frameworks and for going way over the top with smartboards. Little did I know that my own offspring would be in the middle of it just three years later. So, in an effort to process what she’s doing (for me, for her, and for anybody else who cares), this is the first of what might be many posts about the specifics of enVisionMATH, as viewed by a parent whose kid happens to be learning from that curriculum, and who also happens to be a mathematician and math teacher.

So I suggest you bookmark Casting Out Nines and see what develops.

Steve H on setting up problems

re: how many unknowns?

I like to use more variables than are needed because I find it easier to create correct equations. I know that I can always turn the algebra crank later without much thought.

r+s=12 is easy and I know that it's correct. I also know that the half perimeters are pi*r and pi*s.

I then look for enough equations to meet my unknowns. That is what's funny about this problem. You don't have enough information to directly solve for the answer before you look at the choices. There are not enough equations for the variables. Even if you use just r and (12-r), you have no equation, unless, that is, you plug in each answer.

I don't like problems like this, because my first reaction is that you don't have enough information. You do, however, if you look at the possible answers.

Also, why is there no variable in the answer? It's just a unique aspect of this particular problem. What if one of the semicircles is replaced by half of a square? You would have something like this:

4r + (12-r)*pi

for the perimeter. the variable does not disappear when the expression is reduced.

You can't trust what you think because problems try very hard to trick your understanding. You just have to follow the facts (equations) and see where they lead you. As I always say, let the math give you the understanding, not the other way around.

how many unknowns, part 2

gasstationwithoutpumps said:

Although Glen would never create 2 unknowns, preferring r and 12-r to r and s, I often find it easier to create multiple unknowns when initially setting up the problem, then remove the unnecessary ones. In this case, it was easier to remove (r+s) as a single unit, and never worry about manipulating 12-r.

I can't tell you all how important these threads have been to me: how much I'm learning (I hope I'm learning - !) and how rich the experience has been.

It's led me to think about the question of self-teaching a bit. Until last night, I had simply never thought about 'how many unknowns' in the way you all are talking about unknowns now. I had never thought about it because, where unknowns are concerned, the books seem to suggest that less is more.

Mind you, I don't think any math book I've used has directly stated that 12 - r is superior to r + s=12. I'm pretty sure I inferred that it was based in the fact that I don't recall any instances of r + 12 where 12 - r was a possibility.

This strikes me as the kind of thing a good math teacher would bring up in class, perhaps as an aside?

Or something that would come up in discussion?

What do you think?

poll: top education books of the decade

at Education Next

hmmmm...

Teach Like a Champion isn't on the list.

I find that incredible. 

Not sure I'll vote.

how many unknowns?

re: how many unknowns in the two half-circle problem, Glen wrote:

I would never create two unknowns in a situation like this, where the two radii are not independent. Since the distance from R to S was given as 12, the radius of one circle made a good unknown, and the radius of the other was 12 minus that SAME unknown. Either circle would do, of course.

The length of the curve can then be expressed in terms of the one unknown for both semicircles. Using the left circle, and calling its radius r, the right has to be 12-r, so the two semicircles added together were,

= pi*r + pi(12-r)
= pi*r + pi*12 - pi*r
= 12pi

If I took part of my $100 and gave it to a friend, there would be only one unknown. Whether you made it the amount I gave him, or the amount that I kept, or the percent I gave him, or the percent I kept, or the difference in dollars or percent or fraction between what he got and what I kept, or the ratio of our money, or whatever, there is only one unknown. Everything else in such a problem can be expressed in terms of that one unknown, which usually makes the problem easier to manage.

THIS is what I was trying to do.

THIS is what I always do, if possible.

I don't know what the problem was.

Inflexible knowledge?

Heat prostration?

I'm half serious about the heat. I took the test outside in 85+ temp. All summer long I've had severe performance deterioration any time I work in the heat. One day, when the temperature was close to 100, I found myself unable to solve even the simplest of problems. I sat at the picnic table working the same problems over and over again in slow motion. Five, 6, 7 times. Or more. I'd crawl through the problem, check my (wrong) answer, then go back to the beginning and crawl through it again and then again until finally the correct answer appeared.

Then I'd go on to the next problem and do that one 6 or 7 times.

I love summer. Have to soak up the sun while I can.

Big Calendar pin-up

fair warning

If all goes as planned, I am going to begin working through the Unit 5 worksheets from the Arlington Algebra Project, as lgm suggested. Tonight.

I say 'fair warning' because there's no answer key.

order of operations

I mentioned Martha Kolln's Rhetorical Grammar in a comment on "math writing."

Martha Kolln is part of my Great Unread, unfortunately. Along with Polya.

So, since I don't know enough about rhetorical grammar (pdf file) to write a post about it, here is a terrific passage from Geoffrey Pullum's 50-year anniversary take-down of Strunk and White, which I believe is the kind of analysis Kolln does:

"Use the active voice" is a typical section head. And the section in question opens with an attempt to discredit passive clauses that is either grammatically misguided or disingenuous.

We are told that the active clause "I will always remember my first trip to Boston" sounds much better than the corresponding passive "My first visit to Boston will always be remembered by me." It sure does. But that's because a passive is always a stylistic train wreck when the subject refers to something newer and less established in the discourse than the agent (the noun phrase that follows "by").

For me to report that I paid my bill by saying "The bill was paid by me," with no stress on "me," would sound inane. (I'm the utterer, and the utterer always counts as familiar and well established in the discourse.) But that is no argument against passives generally. "The bill was paid by an anonymous benefactor" sounds perfectly natural. Strunk and White are denigrating the passive by presenting an invented example of it deliberately designed to sound inept.

April 17, 2009
50 Years of Stupid Grammar Advice
By Geoffrey K. Pullum

thanks to Karen H

I'm writing this down

re: factoring, lgm wrote:

Instead of thinking 'I must factor', I have found it more useful to think "I must use the Distributive Property" wisely. 

I love that!

math writing, redux - what does this mean?

6. A gumball machine contains gumballs of 8 different colors, which are dispersed in a regularly repeating cycle. The fourth gumball in the cycle is red and the sixth gumball in the cycle is yellow. If 100 gumballs are dispensed from the machine, how many are not either red or yellow?

Acing the New SAT I Math
p. 236

After spending half the afternoon contemplating the position of subordinate clauses in sentences, I'm wondering whether I can read math at all.

What does this question mean?

If you were actually sitting in a room with this gumball machine, what would be coming out of it and in what sequence? When would you see the first red gumball, and when would you see the second? "When" meaning: after how many other gumballs have appeared?

I find the language in this book particularly challenging, by the way. I've never quite recovered from the watch 'gaining' 3 minutes per hour.

math writing

Not infrequently, I have trouble understanding written explanations in the math books and on the web sites I've been using. In this case, the first line of College Board's explanation threw me for a loop:

The function with equation y=(-x)^2+1 and the function with equation y=|x^2+1| each have a minimum value of 1 when x=0...

I read this as saying -- as implying -- that x=0 is somehow critical to finding a minimum. Which I (thought I) knew it wasn't.

Mark's explanation cleared things up:

Both of those functions have a minimum value of 1, and that minimum occurs at x=0.

This is why I can't imagine computer-based math courses panning out. Online practice and assessment, yes. Online teaching, no. You need a teacher -- Mark, in this case -- to see why the student doesn't understand and to re-phrase the explanation.

Though I suppose you could write algorithms to try to do what Mark just did. First step: get rid of subordinate clauses --- especially subordinate clauses at the end of sentences, instead of the beginning. From Grammar Bytes:

Writers use subordination to combine two ideas in a single sentence. Read these two simple sentences:

Rhonda gasped. A six-foot snake slithered across the sidewalk.

Since the two simple sentences are related, you can combine them to express the action more effectively:

Rhonda gasped when a six-foot snake slithered across the sidewalk.

If the two ideas have unequal importance, save the most important one for the end of the sentence so that your reader remembers it best. If we rewrite the example above so that the two ideas are flipped, the wrong point gets emphasized:

When a six-foot snake slithered across the side walk, Rhonda gasped.

A reader is less concerned with Rhonda's reaction than the presence of a giant snake on the sidewalk!

I am accustomed to putting the most important idea last for emphasis. That's the rule I used reading the CollegeBoard explanation.

question of the day - 8/31/2010

question of the Day
8/31/2010

careless error, part 2

I hadn't finished writing the careless error post & now see that apparently I hit 'Publish Post' instead of 'Save Now.'

It's back in the Drafts section ----

solution manual - section 6, problem 5

How do you solve this?

probability redux

16. If j is chosen at random from the set {4, 5, 6} and k is chosen at random from the set {10, 11, 12}, what is the probability that the product of j and k is divisible by 5?

SAT practice test

I missed this one.

Sam Savage

The Big Enchilada

Exponential

Avoiding the Thesis 

Avoiding the Corporation

Avoiding the Publications

Avoiding the Whole (expletive deleted) Ordeal

In the mid 1970's after I had abandoned traditional Management Science, but before I had discovered spreadsheets, I tried unsuccessfully to be a folksinger in Chicago.

There were two things that dissuaded me from a career in music. First, there were a lot of people who were a lot better than I was, and second, they weren’t making it either. During this period I did some recording with my stepbrother John Pearce on a Sony 4 track reel-to-reel tape deck.

I found the decades old tapes in my garage in 1999, and discovered to my amazement that there were still magnetic signals on them. We turned some of the pieces into a CD which we shared with family and a few friends. All the recordings were made between roughly 1975 and 1985. Some tracks, like the patient who has been frozen in liquid nitrogen until a cure is found for his disease, awoke to a world in which they could be substantially improved. The tempo of the Exponential track, for example, was sped up digitally without changing the pitch.

Sam Savage

3-D

 

source:
free SAT practice test

paint by the numbers

1. C. and his friend E. take the online CollegeBoard SAT test first thing so we can get a baseline.

2. I attempt to score their tests and discover that the posted answer key apparently belongs to a different test.

3. I spend my commute time to and from the city working through the test myself.

4. After dinner I while away my evening capturing screen grabs of problems to post here in hopes that people will have time to confirm my solutions.

5. I stumble upon the actual answer key.

This isn't working.

more Big Calendars

If you scroll down, there's a picture of Andrew studying the Big Calendar.

He's sleeping in this morning, but as soon as he's up I'm going to cross out Sunday while he watches.

words to the wise

9pm now; Andrew is still wearing his backpack

A while ago I heard Ed say, "School starts in one week, and you never go to school on Sunday."

I'm sure that fell on deaf ears.


Big Calendar
Debbie Stier's big calendar
the big picture
blank calendar template (pdf file) 
self-charting increases motivation

anthem

avoiding the thesis

"Excerpts from a seventeen-hour piece composed in graduate school"

from the Avoidance Trilogy (directions for clicking through to the music here)

flaw of averages, illustrated

 

The Flaw of Averages: Why We Underestimate Risk in the Face of UncertaintyThe Flaw of Averages: Why We Underestimate Risk in the Face of Uncertainty

exponential, the love song

the more I love you, the more I love you

directions for clicking through to the music here

flaw of averages

the book

the online animations & simulations

and the music! playing The Big Enchilada now -- love it (directions for clicking through to the music here)

 

The Flaw of Averages: Why We Underestimate Risk in the Face of Uncertainty 

what is an average, anyway - part 2

re: what is an average, anyway? anonymous wrote:

One thing I never realized until I started teaching is why we calculate the sum, then divide by the number of numbers to find mean. We're actually finding how much each person would get if the items were distributed equally. So if there were 3 apples, 5 apples, and 10 apples, if we combined them we would have 18 apples. Dividing by 3 gives 6 apples per person if the apples were shared equally. This also shows why we use median rather than mean for items like income where there can be outliers that skew the results when the items are shared equally.

This reminds me of Ron Aharoni saying that when he taught arithmetic he realized there were subtleties to elementary mathematics that he hadn't thought about.

Must find that passage & post!

gasstationwithoutpumps on mean, median, & mode

what is one year?

I made my Big Calendar today.

Today is Day 1; the last day, Day 420, is October 22, 2011.

I had no idea there was so little time in a year.


Debbie Stier's big calendar
the big picture
blank calendar template (pdf file) 
self-charting increases motivation 
what is one year?  

gasstationwithoutpumps on mean, median, & mode

re: what is an average, anyway?

Mean and median are both "measures of central tendency". They are ways of summarizing a large set of data in a quick way that simplifies the data and removes noise.

If you know that the average rent for office space is $4 a square foot, and you need 1000 square feet per person, then you can quickly judge whether you can afford an office for your staff, without having to go through hundreds of real-estate listings. Of course, you also need to know something about the variance, and not just the mean, as there may be a few not-so-nice places that are much cheaper than average.

 gastationwithoutpumps

what is an average, anyway?

A friend of mine asked me to walk her through mean, median, and mode --- and it came to me, thinking about it, that I don't exactly know what an average is beyond the obvious.

Another issue: I'm so sick of my own child (& everyone else's) being lost in group means that I've come to feel some real antipathy towards the very concept of a group average. Meanwhile the concept of a personal average makes sense and seems obviously useful ----

Here are my questions.

What is useful about averages? 

What do averages tell us?

And why did the calculation of averages come to be so important culturally?

click

Speaking of Andrew behaving well, I've got my magic clicker out again.

source: Arcata Pet Supplies

and see: TagTeach autism

in Borders today

We picked Andrew up at camp today and learned that he had a fantastic week --- he was the best camper in the place, his counselor said.

Then he added, gesturing to his chums sitting under the canopy, "They would all say the same."

Wow!

He did pretty well in the car until we stopped at Office Max to pick up poster board for the Big Calendar that's going to change everyone's life around here.

Big tantrum at Office Max & then in the car 'cause he'd spotted a Borders and wanted to go there. But since he'd managed to pull himself out of it by the time I'd made my purchases, we decided to go to Borders after all.

And guess what I found in the school section?

A Frank Schaffer Singapore Math 6B Math Practice workbook.

Plus a lot of terrific Kumon workbooks (those are the two next to Singapore Math Practice.)

The cover of the Singapore math book says, "Activities based on the leading math program in the world--Singapore Math!"



introduction to Singapore Math - Level 3 (pdf file)
Andrew comes home from camp - July 2008