kitchen table math, the sequel: 8/29/10 - 9/5/10

Friday, September 3, 2010

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.

Thursday, September 2, 2010

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


Teach Like a Champion isn't on the list.

I find that incredible. 

Not sure I'll vote.

Wednesday, September 1, 2010

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.

Tuesday, August 31, 2010

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

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?

Monday, August 30, 2010

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


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


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.

Sunday, August 29, 2010

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


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


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 

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.


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?