kitchen table math, the sequel: 1/7/07 - 1/14/07

Saturday, January 13, 2007

new blog format

Ok... got it up. This thread is for any suggestions.

I think that the title bar should be a little bit smaller.

Are the fonts ok?

There is a recent comments and recent posts widget that can be added to the side, is it needed?

What order for the sidebar categories?


Catherine has final say on all suggestions.

Going to work for an hour.

Hope you guys like it.


cruelty + no imagination = NYC principal

I found this story so ridiculous on the face of it I couldn't think of anything to say about it. How on earth does a principal tell a special education student (or any student) that he doesn't "have the brains" to accomplish something?" But my friend Schoolgal, who's actually endured experienced spelling bees, had this to say:

Why was it only one class participated in the bee? Usually the whole 5th grade does it in class and then we send 2 winners to compete in the grade contest (usually held in the auditorium).

Principals usually select a teacher to be in charge. The teacher then informs the grade to hold a spelling bee in class. Then their 2 top spellers compete in a grade competition usually held in the auditorium. In this case, this was not done at first. Only one teacher (the special ed teacher) did it, and that's when the principal realized she did not follow the procedure and held a grade contest.

By telling the first child he was not good enough I think she scared off the first and second place winners who backed out of the regional competition. I believe the child in question came in 3rd. (given that there were probably only 4 classes competing).

This principal was more about her image than anything else. The next level was the region, and many children would be eliminated on the first round, not just her school.

If this principal had any savvy, she would have gotten a coach for this kid and made a big fuss over the fact that this child is special ed--a missed opportunity to be proud of the achievement!

Instead she berated the kid and now denies the conversation took place.
I was very impressed with these comments. As cruel as I thought the principal had been, she turns out to have been blessed with a complete lack of foresight as well. No wonder she moved up so high in a system where people needed to ask permission before dialing 911.

HTML guides

Lynn asked for simple guides to html code.

Here's my list:

<a href= "" >Visit!

HTML special characters

HTML codes - Characters and symbols

Table of character entity references in HTML 4

Character entity references defined in HTML 4

24 Character entity references in HTML 4

The Most Popular Character Entities

Mathematical, Greek, and Symbolic characters for HTML

Theme Viewer

Theme Viewer?


Maybe I should set up an Irvington Schools blog with a pink stripy theme!

Just to keep 'em guessing.

can't search comments

I have just this moment discovered that we cannot search Comments.


Of course, it will be just my luck that Ken or Rory or someone will know of a handy-dandy widget that would instantly correct the situation.

I am planning to discover the meaning of "widget" ASAP.

Speaking of which (speaking of "ASAP," I mean) the Weekend Journal has an article on a husband-and-wife team who track internet acronyms. ($)

You can see their work at Acronym Finder.

Maybe now I can find out what or whom the homeschoolers are talking about when they bring up their "DHs" and "DDs."


Forget it.

They've got 12 pages for DD alone.

online speed of reading test

Of course, it's an ONLINE speed-of-reading test, so since people read 25% slower on computers* you should probably add 25% to your score.

(hey! maybe I should go ask a U.S. middle schooler to do that for me!)



rightwingprof's recent revelation that he is a severe non-fan of Jakob Nielsen has aroused my curiosity.

After stumbling across & reading this post, however, I'm still wondering.

Nevertheless, I for one do not want rightwingprof putting any time into explaining the deficiencies of Jakob Nielsen's many and varied usability manifestoes until he finishes up with Ricky, part 2.


* according to Jakob Nielsen who is on probationary status around here, it seems

Friday, January 12, 2007

practice tests for Accuplacer

Accuplacer is the standardized test no one ever hears about; it's the test everyone entering a community college takes.

My sister thinks entering students may actually be required by law to take the Accuplacer. Her 8th grade daughter signed up for two courses at their community college this fall, and she had to take the test even though she's still in middle school and was enrolling in Spanish I and Voice.

My sister may be right about a legal requirement, at least for some states if not all.

In any case, zillions of people take the Accuplacer, but you never hear a word about it.

Nor does there appear to be a roaring test-prep industry for Accuplacer tests, even though your results on the test determine absolutely whether you do or do not get to enroll in college.

If you flunk, you can't enroll.

You can enroll in the school's pre-college courses, but pre-college courses aren't college. They're high school that you're paying for.*

Apparently the idea that anyone can enroll in community colleges is one of the misconceptions high school students and their parents share.

Not true.

The rule is: anyone who passes the Accuplacer can enroll in community college. (The College Puzzle puts it this way: "Students usually must take placement tests at community colleges in order to qualify for college-level work.")

what's on the Accuplacer?

Tonight I was trying to get Christian signed up for his next course at Westchester Community College. (He's still fiddling with financial aid, so it will be just one course this winter). Last summer Christian passed the Accuplacer reading test, failed the math. He's supposed to be working his way through Saxon Homeschool Math 5/4, and until a couple of months ago was working his way through Saxon Homeschool Math 5/4, but lately he's stalled.

Everyone's stalled except me, and although I haven't stalled I've slowed.

math woe

I'm going to have to crack the whip if there's any chance of averting a Math Catastrophe around here.

Tonight Christopher melted down because he couldn't figure out how to do a problem Christian made up: "If 5 babies are born every 10 seconds, how many are born in 1 minute?"

Something like that.

This came up because yesterday Christopher had no idea how to do a real-world problem concerning babies: If 1 baby is born every 12 seconds, how many are born every minute?

1 baby per 12 seconds is apparently the actual rate of babies being born in the U.S. according to Ed.

(Just in case you're looking for a cool description of how to estimate how many babies are born per minute in the U.S., it's here. At least, I think it's cool. I leave it to our resident math brains to decide.)

Christopher had no idea how to solve this problem. "12 x 60?" he said.

Ed says the accelerated math curriculum is so incurably procedural that the kids approach word problems the way they'd approach a dartboard; they pick a procedure, fling it across the room, and hope it hits the target not the wall.

Christian didn't know how to do the baby problem, either, at first.

"12 x 60?" he said.

Before I could start banging my head against the wall he decided he could answer the question skip-counting by 12s up to 60.

That was a relief.

Anyway, tonight I got worried about what exactly is on the Accuplacer math test. Christian has to take a math course to get his Teaching Assistant Certificate, which is his immediate goal, and he has to pass the Accuplacer test to take the course.

Christian wants to buy the textbook to see what he needs to know. That's a good idea, but since we couldn't do that tonight and couldn't even look at it on Amazon because we didn't know the title I decided to see if there was an Accuplacer test prep book I could take a look at.

Apparently there's not.

I find this bizarre. People's lives are hanging on the results of the Accuplacer, and no one's ever heard of it or written a test prep manual telling you what's on it or how close you are to being able to pass it.

The good news is that I found what looks like a terrific website for Accuplacer test prep. You can take a free online self-assessment test on 31 different scales including Algebra II. I thought I saw somewhere that Accuplacer also tests college-level math, but I don't see it now.

I took the Algebra 1 test & missed one of 20. Careless error.

May try algebra 2 tomorrow.

College Placement and the College Puzzle of placement tests


* high school you're paying for again

How to change the system

Now that I've got your attention, the answer is: I don't know.

Last year Ed talked to Kent Barwick ($), who is past president of the Municipal Art Society, the major organization working on creating a livable city.

Ed says Mr. Barwick set him straight on all kinds of things.

For one, Ed had always thought public works projects were easier to get off the ground in France because the French government is so powerful.


Barwick told him that the Public Authorities in New York state are far more powerful than any government. The Authorities have few political or democratic checks on their actions.

Apparently they do what they want to do, and they don't have to answer a lot of questions while they do it.

Caro on Moses

Barwick also told Ed that the book to read if you want to understand local politics - any local politics, large or small - is Robert Caro's 1344-page biography of Robert Moses.

Ed did read it, and the book was a revelation.

In a nutshell, in local politics the various "actors" - banks, governments, unions, political machines, and so on - typically have a converging interest in extracting more money from taxpayers to use as they see fit and to manage their relationships with one another.

In the case of Robert Moses, there were 5 Actors: Moses, the banks, the labor unions, construction companies, and the Democratic machine.

Moses wanted to build parks, highways, and bridges; banks wanted deposits of funds to pay for parks, highways, and bridges; construction companies and unions wanted contracts to build parks, highways, and bridges; the Democratic machine wanted jobs on Moses' parks, highways, and bridge projects to dispense to members of the Party. (Moses was himself a Republican.)

Taxpayers weren't asked whether they wanted or needed all of these things; in some cases the projects Moses undertook were not only unwanted by voters but were actively opposed.

These entities—Moses, the banks, the unions, the construction companies, and the Democratic machine—weren’t the same entity; they had the potential for conflict. What prevented conflict—what kept operations smooth and conflict-free—was money.

Taxpayer money.

The people who were footing the bill were the one entity that did not have a voice.

how does this work in public schools?

Probably like this.

In my own town there are three Actors:
  • administration
  • school board
  • teacher's union
These three entities share an interest in raising taxes, in part because money minimizes conflict. The administration can buy the things it wants to buy (TRAILBLAZERS textbooks, assistant superintendents for curriculum and technology); the school board can buy things it wants to buy (million-dollar weight rooms); the teacher's union seems to have been so well taken care of that most parents here are unaware of the fact that we have a union.

Many of these purchases are items parents don't want; in some cases, e.g. TRAILBLAZERS, these purchases are actively opposed by parents.

The expenditures we do want - another Earth Sciences teacher so everyone in the 8th grade can take Earth Sciences instead of the 48 children allowed to do so now, say - may or may not happen, depending on whether the District happens to want these things, too.

what does this mean?

I don't know!

I've come to believe, however, that it's essential to get your analysis as right as you can make it, which is why I now reject the "Washington consensus." (That's a story for another day.)

Once you see that entities we take to be competing or conflicting (school board "versus" union) have converging interests, things make more sense.

It makes sense that, in the recent athletic fields bond vote here, the union would vote for further public indebtedness, would publicize its vote, and would telephone the "parent leading the committee to defeat the bond"* to pressure her.

Public indebtedness threatens a district's ability to meet pension obligations and to raise taxes for salaries & classroom supplies, so in theory the union ought to have mixed feelings about a new bond proposal.

That it did not can be explained by the union's converging interest in collaborating with the board and the administration to carry on increasing taxes for all projects and obligations large and small.

what can citizens do about converging interests?

Thoughts anyone?

Apparently there is a Straphanger's Union in the city that has actually managed to make subway riders into an Actor to which attention must be paid.

I can see where a Parents' Union might work.....

What I can't see is how a Parents' Union might be created.

That's about as far as I've gotten.

* there was no committee

why school choice isn't the end all

Minding the (Achievement) Gap:

"One of the series that created the most ripples was 'Inside Choice Schools,' a seven-part report in 2005, fifteen years after Milwaukee launched the voucher program, detailing students' experiences at voucher-supported schools. Borsuk and fellow reporters visited all but 9 of the 115 voucher-supported schools at the time -- all that would allow them in -- and described what they saw. The schools ranged from excellent, he recalls, to 'startlingly bad,' such as one Christian school in a dimly lit church basement without classroom walls or any clear curriculum. Through interviews, the reporters found that Milwaukee parents generally want a school with a cozy feel and value this intimacy over teacher credentials. Borsuk wrote (with colleague Sarah Carr), 'Parental choice by itself does not assure quality. Some parents pick bad schools -- and keep their children in them long after it is clear the schools are failing.'" (emphasis mine)

Unlike many school reform advocates, I am not totally convinced by "vouchers" as a solution to our education problem. While I agree that competition on a wide scale level does have an overall positive effect on the quality of products that people buy, on smaller scale, when it comes to individual services or goods people can make some surprisingly poor choices.

Unfortunately, because education deals with people's children, parents are often to emotionally attached, to close to the situation, to make rational decisions. Just as people make poor choices to get themselves deep in debt to buy flashy cars or the "right" clothes, people don't make decisions about their children's education based on quantitative measures.

Part of the problem may be that there is no good easy to find, widely published information on how well any school or school district performs. Instead parents rely on subjective feelings to judge schools. Do they like the teachers? Is the playground well kept? Do they use lots of cool edubabble?

Witness the success of good charter school systems like KIPP or Achievement First. You would think that parents would be rioting to get their kids in the schools or demanding that their public schools adopt the same policies. Yes there are waiting lists, but there is nowhere near the urgency that you would expect.

Imagine if a new cell phone provider came out and was able to offer better call quality, state of the art phones, double the minutes, at half the cost of your current one. People would be dropping their current service in droves and scrambling to get on the new plan.

When I first moved to Sumter, South Carolina the general consensus was (and still is) that Sumter High School was way better than rural Crestwood High School. Sumter High School had all the town money, it has the good football team, great buildings, and most importantly it was 55% white vs 43% white at Crestwood.

But... if you are savvy enough to look find the the school report cards, you find something different. I compiled this table based on Sumter HS and Crestwood HS 2006 school report cards.

As you can see, on almost every category, Crestwood HS kicks Sumter HS's ass. In the two categories that Sumter HS has the edge, it's only by the slimmest of margins.

You would think that parents who attend kids attend school at Sumter HS, especially black parents, would be raising hell, but no! 80% of parents who attend Sumter HS are happy with the learning environment, whereas at Crestwood HS only 44% are.

It's numbers like these that depress me. A rural, mostly minority school can provide a better education than the city's mostly white school, and yet the perception is exactly the opposite.

This doesn't mean that school choice is a bad thing. There are some parents who are smart enough to make educated decisions. Eventually market forces will have a positive affect. Even if the parents don't see the differences, the school administrators will learn from each other.

I have always been an optimist, and I am convinced if school reform advocates continue to yell loud enough that eventually our country can at least come closer to having the education system that it deserves.

Thursday, January 11, 2007

I'm so confused!

Baltimore Sun: SAT scores well in predicting college success

The much-maligned but ever-influential SAT -- under fire nationally amid concerns over its fairness -- received positive marks yesterday in a report that gauged the test's power to predict long-term college success in Maryland.

Designed to forecast a student's ability to perform college-level work, the SAT is also an accurate predictor of retention and graduation rates at all of the state's four-year colleges and universities, according to an analysis of recent student data prepared for the Maryland Higher Education Commission.

'The higher the SAT scores of students, the greater the likelihood that they not only returned for a second year of study but eventually earned a baccalaureate as well,' the report said. UW Chancellor Defends Affirmative Action

(AP) MADISON High school grades and test scores aren't good predictors of a student's performance in college, which forces admission officials to look at other factors, University of Wisconsin-Madison Chancellor John Wiley told a legislative committee reviewing affirmative action in the state Thursday.

Wiley's appearance before the Special Committee on Affirmative Action comes as UW-System regents consider a new admissions policy that would give greater weight to nonacademic factors such as race. UW-Madison admissions officials already perform this so-called holistic review.

Wiley cited data that he said showed that someone who gets a 4.0 grade point average in high school could earn anything from failing to honors marks in college. He added that ACT and SAT scores mean nothing in predicting first-year grades.

(Cross posted at parentalcation)

formatting and sidebar

Catherine asked me to look into adding a sidebar. I tweaked a format and came up with this.

Its pretty close to this, but not exact. Any inputs?

Update: Spanned header across page, so not sidebar title is even with date of top post.

suggestions for undergrads

If any of you have kids (or know kids) who are getting ready to go to college, you may find this helpful.

out of the woodwork

Some anonymous knucklehead, claiming to be an NCTM insider, just left a comment to this post on D-Ed Reckoning discussing the new NCTM focal points. Either he or she screwed up the terminology and/or players or I am very confused:

Re: the phrase "quick recall"--as someone involved in reviewing the CFPs, I should let you in on the fact the the phrase was a compromise to those traditionalists who couldn't deal with the phrase "automaticity of facts" or "automatic recall" or even "fast recall"--etc. [Ed: Huh?] "Quick recall" is the phrase used by those who support what appear to be the ideas expressed in your blog.... You cannot believe how many versions of this phrase were discussed and debated before NCTM made this compromise with those who are part of the Mathematically Correct camp....

I guess I would be critical too if I opposed the efforts of Skip Fennell and NCTM. I would be kicking myself for not coming up with and publishing the Focal Points myself.

For those who criticize the CFPs, and only comment on statements printed in various biased articles (some of which are taken out of context), I encourage you to read the 41 page document. [Ed: which I linked to and did, in fact, read] It's a free download. Most of the reporters, radio-show hosts, and journalists that I've spoken to in my state have not actually read the entire document, but focused more on the reform-traditional controversies that have been part of the Math Wars over "how to teach mathematics"--and not "what to teach."

I encourage you to spend some real time with the document (, then develop a real opinion....

teaching problem solving to second graders, part 2

(Part 1 is here.)

Now the students are ready to start solving simple comparison problems with variables.

First the students are taught how to translate a phrase like "T is less than H" or "R is more than W" onto a number family.

The lesson might be taught like this:

Sometimes we refer to a number without telling which number it is. We can call that number J or B or any other letter. Here is a sentence that tells about two numbers: J is less than M.

We don't know which numbers J and M are, but we can put those numbers in a number family. J is less than M. So J is the small number. M is the bug number.

The big number goes at the end of the arrow. The small number goes close to the big number. Here's how you write it.

Then an example using "more," like "R is more than W," is taught.

After the students are firm on this skill, they are ready to tackle a translation like "J is 18 more/larger than K."

This must be a difficult skill because in CMC they scaffold the instruction by circling the number which tells how much more. Like this:

Eventually the scaffolding is faded.

To translate the problem, the students are instructed to ignore the circled number. This makes the problem identical to one they know how to translate -- "J is larger than K." They know that this can be translated into:

Then they are taught to place the circled number, 18, into the only available spot in the number family. Like this:

Once the students are firm on this skill, they can be given a problem that they know how to solve like "F is 12 more than 56." Now they should be able to translate this to a number family and solve for F.

Then the problem can be made more difficult by specifying two variables and the value of one of the variables, such as "R is 250 more than P. R is 881. What number is P?"

Finally, the students are ready to start solving real word problems like "Fran was 14 years older than Ann. Ann was 13 years old. How many years old was Fran?"

Here's how they are taught how to solve these kinds of problems:

This is a good stopping point. This represents a month worth of instructional time for the lessons and the practice. That's for lower performers, higher performers can probably learn this in about a week. Bear in mind that the students are learning and practicing about 10 other strands of material while all this is going on.

The value of this problem solving technique (like Singapore Math's bar graphs) is twofold. First, it reduces solving math problems to a systematic process; this will clarify the student's thought process. Second, the use of the written number families frees up the novice student's working memory which is taxed heavily in solving word problems. Given enough practice, these skills will become automatic for the student and lodged in long term memory. When this occurs, the burden on the student's working memory is becomes much less and the need for the number family prompt is diminished.

Teaser for next lesson: The students learn how to solve word problems like "Jerry weighed 72 pounds. Terry weighed 94 pounds. How much heavier is Terry than Jerry?" To solve problems like this, the students are taught the concept of moving forward and backward along the number family line.

pluses and minuses equal division

I came across this article on the math wars in the Oregonian. As usual, it's false dichotomy-ariffic and is chock full of he-said-she-said nonsense; however, this blurb caught my eye:

Some parents want to do away with reform math. Others want both reform and traditional math offered, so parents and students can choose the one that works for them.

This is what I don't understand. Any school with enough students to support two classes can offer both kinds of math classes. This would conveniently put an end to all the tensions and offer parents a real choice. Parents could choose to have their kids taught in a "drill and kill" or in a "rainbows and lollipops" class. It's not like this would cost any extra money or effort on the part of schools.

Wednesday, January 10, 2007

math night coming right up, part 2

I wonder how many parents know that Math Nights are part of the package?


Be pro-active with parents. Don’t wait until complaints hit. People have done a lot of things to involve parents, from math nights to big math carnivals, where the kids teach the activities to the parents. There are letters in the program that go home to parents. In one district, the coordinator ran a six-week course for parents and taught them mathematics, essentially. It depends on what will work with your audience. Teachers need to communicate with parents, making sure that the parents see the math facts practice and that the arithmetic they value is visible.

I'm positive most parents don't know the backpack letters are pre-fab.

math night coming right up

math night coming right up

About a month ago two parents posted an online petition addressed to our Superintendent requesting a district meeting to discuss math curriculum and pedagogy in Irvington schools.

The superintendent did not respond.

It seems that math in Irvington is on her mind, however.
Dear K-5 Parents,

As you know, the mathematics Trailblazers program was adopted by the Board of Education in 2004 and is in its final phase of implementation this year in Grade 5. The District has provided strong support to teachers to ensure excellent initial training and ongoing professional development, and the results to date, are very positive. [ed.: time to take a listening tour of Irvington!] As with any new initiative, however, it is imperative that we continue to monitor children’s progress and to provide ongoing opportunities to keep parents informed as well as to continually evaluate the program’s effectiveness.

As part of our commitment, there will be a Math Information night for K-5 parents on January 22, 2007 at 7:00 PM in the Dows Lane Library hosted by staff from Dows Lane and Main Street Schools. Teachers will present information about the Trailblazers program, and parents will be able to ask questions to which they will receive responses that evening or soon after the session. [ed: how about parents provide information while teachers listen?] As the date draws near, a separate flyer will be sent home by building principals requesting confirmation of your attendance.

In addition, since students in the current elementary program are not tracked and those in the current Middle School program are in leveled classes, the staff recognizes that the transition to 6th grade is a valid concern [ed.: as opposed to an invalid concern] on the part of parents and one that will soon be addressed. Therefore, parents of current fifth graders will have an opportunity to attend a meeting in February 2007 to discuss this topic. Once again, information will be sent home later this month.

I hope that you will make every effort to attend these important parent meetings.

As always, we value the input of parents and look forward to these opportunities to hear from you and to be able to respond and work collaboratively with you as we move forward.

Thank you for your ongoing support and cooperation and best wishes to you and your families for a healthy and Happy New Year.

Superintendent of Schools


Asking questions and receiving answers is not the same thing as "working collaboratively" and "moving forward."

In case you were wondering.


email to the Board:

I am disheartened by this communication from our superintendent.

The wording suggests that the administration draws a distinction between valid and invalid parent concerns.

Valid concerns will be addressed; invalid concerns will not.

You may be aware that our superintendent has refused to acknowledge an online petition signed by 34 parents requesting a district meeting to discuss math curriculum and pedagogy.

This is our problem.

The administration stages “Information” meetings to address issues it wishes to address; it dismisses out of hand direct requests for meetings from parents whose issues it wishes to ignore.

And yet all of us are expected to provide “ongoing support and cooperation” to the administration come what may.

Support and cooperation cannot be assumed or commanded. Support and cooperation are precious forms of good will that can only be earned by listening to and acting upon parent goals for our children’s education, not by “explaining” district practices.

Our district’s mission statement opens with the words, “The mission of the Irvington School District is to create a challenging and supportive learning environment in which each student attains his or her highest potential for academic achievement.”

If this mission is to mean anything, parents must share with the administration a clear understanding of what each student’s highest potential actually is.

When it comes to math, a first answer can be found in the international standard for achievement, which is algebra mastered by the end of 8th grade for at least 80% of students.

At the KIPP Academy in the Bronx 80% of graduating 8th graders, all of them disadvantaged black and Hispanic children who enter the school two years below grade level, pass the Regents Math A examination.

Here in Irvington only 35% of our students are prepared to take Regents Math A before entering the 9th grade; few or none of these students are black and/or Hispanic. I can think of no way to increase these numbers using the spiral curriculum we currently inflict upon accelerated middle school students, their parents, and their tutors. If we wish Irvington students to be on par with their same-age peers in Europe, Asia, and the Bronx we will have to adopt:
  • a curriculum such as Saxon Math that is built upon the principles of cognitive science
  • a pedagogy that stresses formative assessment, distributed practice, and above all district responsibility for individual student learning
Many parents are aware that their children are not learning what they could and should be learning in Irvington schools. It is essential for the administration to solicit, respect, and act upon parent concerns where curriculum and pedagogy are involved.

Thanks much --

Catherine Johnson

thank you for your ongoing support and cooperation
thank you for your cooperation and support in this endeavor
today's form letter from the school

da bomb

Well my blatent sucking up to Carolyn Catherine worked and I was invited to join KTM2.

Of course I will try and only post my highest quality posts (relative for me) over here. To start...
Comparing Fractions with Cross Multiplication (originally posted at parentalcation)

Today my 6th grader asked for with her math homework, specifically how to "cross multiply fractions with whole numbers". I wasn't quite sure what she was talking about, so I took a look at her homework. I saw that she had 10 problems to compare fractions with different denominators, some with whole numbers. I started to explain how to find a common denominator, etc... but she got really upset with me.

"Thats not how my teacher showed us", she said. "My teacher told us to cross multiply."

I still had no idea what she was talking about, so I went to check with my girlfriend, Shannon, to see if she knew what our daughter was talking about. But Shannon was as confused as I was.

Both of us being confused, we did a quick google and came across this explanation of the process from

Comparing Fractions
1. To compare fractions with the same denominator, look at their numerators. The larger fraction is the one with the larger numerator.
2. To compare fractions with different denominators, take the cross product. The first cross-product is the product of the first numerator and the second denominator. The second cross-product is the product of the second numerator and the first denominator. Compare the cross products using the following rules:
a. If the cross-products are equal, the fractions are equivalent.
b. If the first cross product is larger, the first fraction is larger.
c. If the second cross product is larger, the second fraction is larger.

Compare the fractions 3/7 and 1/2.The first cross-product is the product of the first numerator and the second denominator: 3 × 2 = 6.
The second cross-product is the product of the second numerator and the first denominator: 7 × 1 = 7.
Since the second cross-product is larger, the second fraction is larger.

Compare the fractions 13/20 and 3/5.
The first cross-product is the product of the first numerator and the second denominator: 5 × 13 = 65.
The second cross-product is the product of the second numerator and the first denominator: 20 × 3 = 60.
Since the first cross-product is larger, the first fraction is larger.

Well, we figured it out and were able to help her finish her homework... her way, but we are rather conflicted about it.

Though the system works, we aren't quite sure what the purpose of it is. It almost seemed to us to be cheating. Though the system works, neither one of us could give a mathematical explanation of why. Finding a common denominator is relatively easy to explain, and is also an essential skill when it comes to adding unlike fractions. Is this new math, really really old math, or something in between?

rory @ parentalcation

p.s. does anyone else want a sidebar for the blog?

Carolyn on Everyday Math

I just noticed this comment from Carolyn on rightwingprof's Ricky post:
Ben learned the range, maximum, minimum, mode, and median every year from 1st grade through 4th grade. In the 5th year they learned to divide and so they taught them to calculate the mean. This is still going on in middle school -- but in the curriculum they are using, they do seem to give it less emphasis.

You brought back my Everyday Math experiences with Ben in grade school -- every day he'd come home with something in homework that they hadn't been taught. I remember the day he came home with division problems, when he hadn't been taught long division. It turned out they were intended to do the divisions with calculators -- that left them more time for learning higher-level skills like calculating the max, min, median, etc..
It's a nightmare.

We're almost halfway through Year 2 in Irvington's "Phase 4" accelerated math class & the kids are dropping like flies. (Not quite, but close enough. Kids who managed last year's class pretty well are seeing their grades drop; more tutors are being hired.)

Even Ed is now talking about hiring our Own Private Math Teacher.

Which won't do much good, seeing as how even Our Own Private Math Teacher will have to practice reactive teaching to get Christopher through the tests in one piece.


Any thoughts on "tag" categories?

They're listed at the bottom of the page.)

issues, I've got issues

Hi everyone -

I just got back from Evanston (no time to see friends - non-stop, whirlwind, get Mom set up in nursing home action!) - we have 30 members!

That's fantastic!

Question: Susan and Lynn both report having trouble with their Google accounts -

Here's Susan: My Google thingy doesn't remember my password. When I do a new one with the same user name and password I'm fine, but the minute I go away and come back I have trouble.

And here's Lynn: Despite having created a google account, Blogger won't recognize me as a member of KTM, the sequel and allow me to post. Do you know what is wrong?

Is anyone else having problems? (I just had a problem signing in; I had to use the Blogger Dashboard to do so. I'm not sure anyone else can access the Dashboard, though.)

Let me know.

Then someone else can figure them out!

teachers say they cannot cope with needs of dyslexic children

(Cross-posted at D-Ed Reckoning)

As reported in the Independent:

The majority of state school teachers lack confidence in educating dyslexic pupils, a survey for Britain's biggest teaching union shows.

Fewer than one in 14 say they would be "very confident" in identifying a child with dyslexia while only 9 per cent say they would be "very confident" in teaching such a pupil. The survey, by the National Union of Teachers (NUT), reveals the vast majority believe they do not have enough training to deal with special needs children. (emphasis mine

My how quickly they give it. (And, by the way, that is a very unfortunate acronym.)

I'm not convinced that dyslexia is a legitimate disease or handicap or whatever the en vogue euphenism is today. I view dyslexia like the other bogus ailment "specific leearning disability"-- an educator created problem designed to excuse ineffective teaching ability.

I'll give you two good reasons:

1. The MRI evidence they're using to base the dyslexia theory on is bunk (pdf):

[T]he MRI scientists’ interpretation of brain-function data is what is logically referred to as a false dilemma or an argument from ignorance. The scientists observe a correlation between brain patterns and not learning to read.

The possibilities are:
  1. The brain pattern caused the nonlearning.
  2. The nonlearning caused the brain pattern.
  3. The interaction of a third variable caused both the nonreading and the brain pattern.
These scientists apparently don’t consider possibilities 2 or 3, but proclaim that the brain pattern causes the nonlearning. There is no question that there are individual differences in reading performance; however, if the kid can find his way into the right classroom and follow simple directions, he can be taught to read in a timely manner.

2. When kids are taught effectively, the incidence of "dyslexia" drops dramatically:

If it’s true that students in places like the worst slums in Baltimore and rural Mississippi taught with DI have 100% of the children reading—not guessing or memorizing—by the end of kindergarten, something is seriously wrong with the portrait of dyslexia. After all, these students exhibit all of the “warning signs” referred to in the analysis. When they come into kindergarten, they can’t rhyme, they can’t alliterate, they can’t blend orally presented words, and they have lots of problems figuring out unique sound patterns (such as repeating something like 4, 4, 4, 4 and yet are able to repeat four or more random digits). So they should all be dyslexic, and indeed historical performance records show that virtually all of them had been greatly retarded in reading, with the average fifth grader stumbling about on a weak second-grade level. Some of the schools that currently have no nonreaders coming out of K historically had end-of-first-graders scoring at the 6–9th percentile on standardized achievement tests. Yet, the new science tells us that we can expect 1/5 of the population to have dyslexia. That’s a 20% failure rate to teach reading in a fat-cat suburb where parents care about and influence the schools, and where they are lavishly funded with aides, material, and whatever.

You might want to take a look at this article (pdf) as well.

While normal children look at a capital letter R and see R, dyslexic kids are purported to see (backwards R). Normal children see receive; dyslexic children see recieve. Very little of this screwed up perception would actually manifest itself very directly in reading. If a reader actually sees (backwards R)ed, for instance, that child is most likely to say /rred/. If the child “sees” (backwards R) and thinks it’s R that’s not going to cause a decoding problem. If a child sees (backwards R)eb, that could cause a decoding problem, but most letters, written backward, are just backward letters.

Similarly, if the only problem is that a reader looks at receive and “sees” recieve that alone isn’t going to cause any reading difficulty. Look at all the people who write recieve but who think they’ve spelled the word right, and can certainly read what they wrote.

Tuesday, January 9, 2007

the high cost of low teacher quality

Cross posted from Joanne Jacobs:

Education Sector's How Low Teacher Quality Sabotages Advanced High School Math is a must-read. Kevin Carey summarizes:
Students who take advanced math courses in schools that employ the fewest well-qualified teachers are far less likely to be adequately prepared for college, or to succeed in college, than students who take the same courses — or even less advanced courses — at schools with the most well-qualified teachers. Students who fail to take advanced courses do poorly across the board. But it turns out that simply enrolling students in more advanced classes isn't enough — you also need good teachers to teach them.
Illinois makes all 11th graders take the ACT. The llinois Education Research Council combined ACT scores and high school grades to rate each students' college readiness.

To create a Teacher Quality Index (TQI), researchers looked at factors correlated with effectiveness: graduation from a "more competitive" college, less than four years of teaching experience, emergency or provisional teaching credentials, one or more failures on the basic skills test for new teachers, composite ACT score and English ACT score.

Few students who took only algebra and geometry were ready for college regardless of their school's Teacher Quality Index. But TQI correlates with college readiness for students who completed advanced algebra, trigonometry and calculus.

Students who took Calculus in the lowest TQI schools were five times less likely to be well-prepared than students who took Calculus in the highest TQI schools. In fact, students who took Calculus in schools with a TQI below the 10th percentile had a lower preparedness rate (16 percent) than students who only took Algebra II in schools that were above the 25th percentile.
There is a confounding factor: Low-TQI schools tend to be high-poverty schools.
Low-income students, who face some of the greatest barriers to education, are much less likely to be taught by teachers with the best qualifications.
More than 90 percent of well-prepared students and 55 percent of least-prepared students enrolled in college. After three years, 10 percent of the top category and 41 percent of the lowest category had dropped out.

Take a look at the charts: At schools with a TQI in the lowest 11th-25th percentile, less than half of students motivated enough to tackle calculus are prepared to succeed in college. In the bottom 10 percent of TQI, fewer than 20 percent of calculus students are prepared.

weighted and relative GPAs: pros and cons

LynnG's article, Weighted GPA, confused me for a moment, until I realized that high schools were using the term, "weighted GPA" to mean something very different from the way universities use the term:
Our High School is proposing to change the way high school grade point average is calculated. We currently use a a weighted GPA scale that gives more "credit" to Honors and AP level courses. Honors and AP are graded on a 6.25 scale (an A+ gets a 6.25) and the non-Honors/AP courses use a 5.25 scale. Also, health and physical ed courses are not included in the GPA calculation.

The high school administration proposes to end this and grade all students on a single 4.0 scale that includes all graded courses (scooping in band, pe, health, maybe others). I'm on the fence on this because I really don't know what the pros and cons are and I am hoping maybe some of you out there might help clarify.
If I were an admissions officer (I am not), I would back the high school administration, if only because this sort of system, where schools can have different scales for different courses, would be a nightmare. However, making life easier for bureaucrats isn't necessarily the best reason for supporting a system (or not).

It seems that the biggest disadvantage of this sort of weighted system is that it is based solely on the designation of the course. Perhaps I'm being cynical (it's happened before), but I have seen enough "honors" courses at the university that were utter crap that the idea of giving a student a higher GPA solely because he was enrolled in an advanced course is pointless, at least without some objective way to judge the rigor of the course.

She brings up an excellent point here:
Here's what we've come up with so far -- the change would primarily affect the class "rank." By taking easier courses and getting an A, you'll get a better GPA and a better class rank than if you take AP and Honors courses and get a B. The kids graduating in that all important (is it?) top 10% might be those that avoid AP and Honors and take easier courses for a better grade. Would this new system penalize kids that stretch themselves in AP and Honors? Or do colleges really not care about GPA and class rank, caring much more about the courses on the transcript rather than the grades achieved?
The issue here, of course, is the existence of "easier" courses, but that's a series of books in itself, so I won't address it. Her point is valid, though: Will going to a flat 4.0 scale penalize students who take more difficult courses (assuming that they are, indeed, more difficult)?

Yes, this is a potential problem. One solution -- though not a perfect one, as I'll point out -- is to use what universities call a "weighted GPA" (I'll use the alternative "relative GPA" to avoid the confusion I originally experienced).

A relative GPA is relative to the mean GPA for the class (universities also assign relative GPAs to courses calculated relative to the overall mean GPA for all the courses in the department or school, in order to determine difficulty). The advantage of this (over the weighted GPA above) is that the relative GPA is not calculated as a function of class label, but the performance of all the students in the class. A relative GPA is, therefore, a real statistic, whereas a weighted GPA is not -- that is, a relative GPA is meaningful to an admissions officer (or a parent), because he can see not only what grade the student received, but also how well the student did relative to the other students in the course, while a weighted GPA is meaningless to an admissions officer, because he has no way of knowing whether the course was, in fact, advanced or more difficult beyond the certification of the student's school.

When relative GPAs are calculated and reported, they are always (in my knowledge) reported with the student's raw GPA. A relative GPA is essentially a curve, in the traditional (and not "raise my grade") sense of the term.

One way to calculate a relative GPA is to divide the student's raw GPA by the class mean GPA. Let's say we have three students, Hyung-Sik, Mary, and Tomoyuki, who receive As in three different courses, Organic Chemistry I, Introduction to Philosophy, and Overview of Ethnic Studies, respectively. If only raw GPAs are calculated and reported to the university, Hyung-Sik, Mary, and Tomoyuki will all receive As. But if the university uses relative GPAs, the grades reported for the three students will be calculated by dividing the raw GPAs by the class mean GPAs. So if the class mean GPAs for Organic Chemistry I, Introduction to Philosophy, and Overview of Ethnic Studies are 1.0 (a D), 2.5 (mid-range C), and 4.0 (A), Hyung-Sik, Mary, and Tomoyuki will receive relative GPAs of 4.0, 1.6, and 1.0. Unlike the weighted GPA LynnG described, the difficulty of the course in the relative GPA system is determined wholly by the performance of the class as a whole, and not by a class description, label, or determination, all subjective criteria.

Here is an example table of nine students in nine classes (I have no idea why blogger is forcing the table so far down the page, but it's there -- just scroll):

Class Mean GPA
Relative GPA

In this system, a student whose raw GPA is identical to the mean GPA of the course will receive a 1.0, which becomes the mean standard score. Relative GPAs can also be calculated by reporting the student's raw GPA and percentile rank, or by reporting the student's standardized score, or z-score (the difference of the student's score and the class mean score divided by the class standard deviation).

Another advantage of using relative GPAs is that it counters the administrative bias toward college-track courses. Why, for example, should Jim, who loves working with wood and excels at it, and wants to be a carpenter or woodworker, not receive the same advantage with respect to his GPA in shop? With a weighted system, Jim will likely only take 4.0-0.0 courses; with a relative system, Jim's superior work will be accurately reflected in his GPA.

The relative GPA is not a perfect solution. Whereas the relative GPA does give an objective assessment of the student's performance relative to the other students in the class, its major disadvantage is in small classes. One very low score (at the university, it would be that inevitable student who fails to drop the class in time, and there's always at least one), will drastically alter the class mean, and therefore, the student's relative GPA.

This presents a possible problem for using the relative GPA in pre-university schools, where class sizes are as a rule significantly smaller than university classes. One might counteract the effects of small classes by calculating the relative GPA with a trimmed mean (the outliers are removed from the sample). Confidence intervals can also be calculated to counteract this effect.

Either way, the relative GPA is a far more accurate and useful assessment tool than the weighted GPA, whether the school is primary, secondary, or the university.

teaching problem solving to second graders

Jill ran 2/3 of a mile farther than Steve. If Steve ran 7/3 miles, how far did Jill run?

If the NAEP is any indication, this is a simple problem that many students can't reliably solve by the 11th grade. Which is a real shame because if a student can't solve a simple problem like this, he can't do basic algebra. The student's math education has effectively come to an end.

The biggest stumbling block is translating the word problem into a mathematical expression. (Calculators are no assistance here.) This kind of mathematical reasoning eludes many students. Fortunately, it can be systematically taught. For example, in Singapore Math this skill is taught using bar graphs starting in third grade. A fair amount of digital ink has been spilled on bar graphs on KTM, so I'm going to show you a diferent way of teaching problem solving.

I'm going to show you how the technique is taught in Connecting Math Concepts (CMC) beginning in the second grade. By the end of the second grade, students should be able to solve a problem, like the one above, correctly at a high rate. Problem solving is taught the entire 2nd grade year in CMC, so it's going to take quite a few posts to cover it all. So let's intoduce the technique in this post and I'll periodically write new posts until we've covered it all.

In CMC, simple problem solving is taught via the concept of number families. Here's a number family:

Beginning in this first grade, the student is taught that number families show three numbers that always go together in addition and subtraction facts. In the example, the three numbers in the family are 2, 3 and 5. You can derive four problems from each number family, two addition and one subtraction:

  • 2 + 3 = 5
  • 3 + 2 = 5
  • 5 - 3=2
  • 5 - 2 = 3
The students are taught that the "big number" always goes at the end of the arrow and the "small numbers" always go above the line. (In DI courses skills are taught using terminology that the students are familiar with. Proper math terminology like subtrahend, minuend, and addend aren't taught, if at all, until after the students are firm on the skill. This reduces the liklihood of student confusion.)

Next the student is taught how to derive the addition and subtraction problems from the number families. Here's an example of each:

An addition problem can be written for each family that has a missing big number, like the bottom family in the picture. Students are taught that if the big number is missing, they are to write an addition problem that ends with the "how many" box (4 + 19 = []). For subtraction problems, students are taught that if one of the small numbers is missing, they are to write the big number first and subtract the small number from it to find the missing number (57 - 12 = []).

Once the students are firm on this skill, they are given some math puzzles to solve. For example, the students are directed to complete the number family, write the addition or subtraction problem, and the answer to the following set of facts: The big number is a box, the first small number is 38, and the second small number is 39.

the student should be able to derive the problem: 38 + 39 = 77.

Now the student is ready to learn about the concept of variables.

The student is told that sometimes a "letter" is used instead of a box in a number family. The letter works just like a box. It's the missing number.

Here's a problem:

The first small number is 14. The second small number is 56. The big number is P.

The student should be able to construct the proper number family using the skills he's been taught so far.

The student should also know that in order to solve for P, he has to add. The student should also be able to write the correct addition problem 14 + 56 = P and determine that P = 70.

The student is then instructed to cross out the P in the number family and write 70 like this:

This seems like a good enough place to stop for this post. Don't want to overload your second grade heads. This sequence takes about five weeks to go through--the first five weeks of second grade, including practice. I'd estimate that this sequence represents about an hour or two of instruction time and another few hours of guided and independent practice.

In the next post we start to get out of the puzzles and into the good stuff -- real problem solving.

Many of you can probably see where we're going with this already.

Here's a teaser.

A student should be able to set up simple comparison like "A is less than B" or "G is more than H" just by using the number families and rules for placing the "big number" and the "small number." Once that skill is firm it's just a hop, skip, and a jump away from setting up a problem like: J is 5 less than K. Solve for K if J = 3.

(Go to Part 2)

voluntary national math and science standards: draft Dodd-Ehlers bill

I attended a panel discussion of the bill drafted by Sen. Dodd (D-CT) and Rep Ehlers (R-MI) bill yesterday. The event was sponsored jointly by Fordham and the New America Foundation, at the latter's offices in DC. Michael Dannenberg, former staffer for Ted Kennedy and now director of the Education Policy Program at the New America Foundation moderated the panel discussion. Sen. Dodd opened with a statement on why we need national standards. He cited the plethora of "great" state test scores in math and science, and poor scores on NAEP, the 50 different state standards for math and science, and the need for a method to assess fairly how schools are doing, with respect to the requirements of NCLB which is up for reauthorization this year. Rep Ehlers was supposed to be there, but was unable to attend; he is a co-sponsor of the bill.

Other speakers included former Gov John Engler of Michigan, former Gov Bob Wise of W. Virginia (now president of Alliance for Excellent Education, Michael Casserly (Exec Director of Council of Great City Schools) and Michael Petrilli of Fordham.

Bill would task NAGB (these are the people who write the NAEP exam) to draft national standards for math and science. (Anyone familiar with the non-rigorous nature of the NAEP exams should be plenty concerned about this bill). These would be voluntary standards, but if states adopt them, then they get a grant to implement the standards and other things. Standards must "ensure that the volumary American education content standards are internationally competeitive and comparable to the best standards in the world."

Among the questions was one from Jeff Mervis of Science magazine. He asked what happens if states are falling short even after adoption of national standards? Would there then be a "national curriculum"? The question was addressed to Sen. Dodd who said the last thing the govt will do is tell local govts how to sequence and/or design curricula. Mervis asked again: "Does the bill tell the states how to meet the standards?" Dodd dodged the question again, but this time alluded to weaknesses in the NCLB law itself that prevents qualified teachers from teaching. He referred to the teacher certification requirements that make it mandatory for teachers to have certification in a subject area in order to teach it; so a biology teacher who may be qualified to teach chemistry could not teach chemistry under the current law. Don't know that that answered Mervis' question, but that's all I could glean from that one, folks.

The bil would not establish a national test, though Sen. Dodd said that maybe that would emerge as a result of states adopting the national standards. Perhaps they would get together and decide they needed to design and use a common test.

One comment from the moderator, Michael Dannenberg, intrigued me. He is definitely for this bill and said that the "standards based reform movement" has had the greatest success with respect to math. Wha HUH? What's he talking about? NCTM's standards? State standards? Has he even read Fordham's State of the State Math Standards report? Oh, he musta been talking about California, yeah, yeah, that MUST be what he was referring to. Or maybe the Focal Points, yeah, yeah, that's the ticket.

(Description of the bill, plus a link to the draft and a link to a video of yesterday's event can be found at Preparing US Students for the Global Economy.)

Among the key private endorsers of the bill are NEA and (wait for it) NCTM. Any questions?

Monday, January 8, 2007

what's possible in high school?

There's a fascinating article titled "The Incredibles" in yesterday's NYTimes EducationLife section. This article focusses on students who are superachievers in high school and may even be bored in college. It was clear from the article that there definitely are such students.

For example, the former mathematics department chairman of George Mason University now teaches at the Thomas Jefferson High School for Science and Technology, which is a public magnet school in Virginia. Some students are taking "Complex Analysis" which has A.P. calculus plus a second year of advanced math as pre-requisite. [Yikes! That's my field and I'm not sure I understand it all even yet.]

However, of more relevance to our discussions was this interesting comment by the president of Pomona College. "High schools are trying to imitate college and teach college-type material instead of the high school material they used to teach ... They are now learning the advanced stuff, but not the basic stuff. We are finding students who have learned about s-, p- and d-orbitals -- a theoretical concept in chemistry -- but they don't know that chlorine is a gas."

Another educator concurred. "High school-age students are not mature enough to grasp the subtleties of some material...."

preparing for collegiate success

I just came across information from the U.S. Dept of Ed, called the Tool Box Revisited. This was issued about a year ago (February 2006). What caught my attention is this passage:

"The highest level of mathematics reached in high school continues to be a key marker in precollegiate momentum, with the tipping point of momentum toward a bachelor's degree now firmly above Algebra 2. But in order for that momentum to pay off, earning credits in truly college-level mathematics on the postsecondary side is de regeur."

"By the end of the second calendar year of enrollment, the gap in credit generation in college-level mathematics between those who eventually earned bachelor's degrees and those who didn't is 71 to 38 percent."

I think what this report is saying, someone correct me if I've miss interpreted, is that unless you take "truly" college level math in high school, not pseudo, higher-order thinking skills with real world applications, your chances of getting ANY bachelor's degree is about 38%. Yikes!

today's depressing factoid

Performing at grade level by the end of first grade is critically important for the at-risk child. A study by Juel (1988) showed that the probability that a child who was a poor reader in first grade would be a poor reader in thefourth grade was a depressingly high +0.88.


It's a quadruple whammy for at-risk kids:

There are many at-risk children who are not likely to succeed when placed in widely distributed core reading programs. The problems stem from the programs not being designed with the degree of explicitness needed by the at-risk child. The programs often have serious instructional design flaws. Among these problems are (a) teacher explanations that include words the child does not know and that use sentence structures that are confusing for students with limited knowledge of language, (b) the rate of introduction of new skills is too fast, and (c) sequences that can cause confusion. For example, one program introduced letter–sound correspondences in alphabetical order, resulting in the letters b and d, and m and n being introduced in near consecutive order, and (d) too little practice and review.

From Using Direct Instruction Programs as Intervention Programs in Grades K–3, Direct Instruction News Volume 5, Number 2 Summer 2005

The worst part of this is that all four of these deficiencies are emtirely instructional in nature. It's purely a matter of educators not doing their job properly.

Sunday, January 7, 2007

Ricky update

Wednesday, I had a tutoring session with Ricky, the 8th grader. The topic was limits.

I've learned to look at all of his worksheets first, to see what they're really covering, since their idea of covering a topic is very different from mine. He had three worksheets, all of them with odd little exercises on them that went something like this: "Get a bowl of Hershey's Kisses and take half the Kisses out of the bowl. Keep taking half of the Kisses out of the bowl. When you get down to only one Kiss, cut it in half and take out half. Will you ever empty the bowl? Why or why not?"

Uhm, okay. I see where this is supposed to go, but couldn't they at least give a definition? Even if they wanted to use induction, how about a definition at the end of the third worksheet?

Keep in mind that they haven't done Cartesian geometry yet. No y = mx + b. It's kind of hard to talk about limits in terms of math -- particularly to an 8th grader -- without being able to use a graph as an exemplar. Try it.

This is the first time he has questioned the curriculum. Not directly, but he's been remarkably willing to led the course lead him down the garden path, without questioning where the class is headed, until Wednesday. He appropriately asked me what limits were for.

"Trigonometry. Calculus. Engineering."

"What are we doing to do with them?"

"You'll have to ask your teacher."

And indeed, the boy had a point: What is an 8th grade class going to do with limits?

"You don't know?"

"I have no idea."

"Then why are we doing this?"

Well, one may well wonder. And I understand his mystification. Limits, as they were covering them, don't generate numbers, or seem to an 8th grader to have much to do with mathematics.

It was the following week that Ricky told me what had happened in class -- and I had to laugh a little (silently, of course). To demonstrate limits, the teacher had had all of the students stand on one side of the room. She had then had half of them move to the other side of the room, over and over again.

There is a problem with using discrete objects to demonstrate limits (the same problem was in the Hershey Kiss exercise): You eventually get down to one. So when they got down to one student, Ricky said the teacher told them, "Never mind, let's do this example instead."

When you're teaching, you really need to think things out before you run them in the classroom. Been there, done that.

Next up that week was quadratic equations. To review, let's look at the preceding list of topics: Division, graphs (pie, bar, column, and area, not Cartesian), fractions, limits, quadratic equations. By the time I started this gig, they had already "done" linear equations -- except that they hadn't really, and we covered it.

For the first time, Ricky had an amnesia attack. I'm used to this. Teaching in a two-semester course sequence, you see lots of students the second semester you had the first who seem to have forgotten nearly everything you did the previous semester. But Ricky had a complete blackout. He couldn't solve 50 = 25 + x.

So I gently nudged him by doing it for him, step by step, then writing down another for him to do. Again, blackout.

I have a son, and I also have three younger brothers (well, had: my youngest brother died a couple of years ago). I know what a frustrated adolescent looks like, and he was getting frustrated. I backed off, and suggested I come back the next evening, and in the meantime, told him I'd email him some stuff he could do before I came back to refresh his memory of linear equations.

That worked pretty well, but he's still frustrated, and I can't blame him. Again, I got, "Why can't my teacher explain it like this?" and I have no (ethical) answer to that question, which doesn't help his frustration. He wants me to validate it, and it really wouldn't be appropriate for me to do so, though he's right. The problem is that he's turning his frustration not on the class or his teacher, but on math in general, and that's not good because he's very sharp, and he picks it up very quickly.

So who knows. Maybe I'll be turning into a therapist next. Sigh.

today's quote

Here's today's ironic education quote.

Alice Treuth said it's fun to work on laptops in her fourth-grade class at St. Francis de Sales Catholic School because she doesn't have to write.


An even more ironic update:

On the other side of the class, Michael Vogel raised his hand for help. A red squiggly line appeared under a word on his screen indication to Vogel that what he typed was incorrect.

"Why is it wrong?" Vogel asked [the teacher].

Everyday Math in Ann Arbor public schools

I'm always interested in what's happening in Ann Arbor, since I went to school at University of Michigan. I've been informed that the public schools there use Everyday Math and Connected Math Program. I'm familiar with both of them, moreso with EM since my daughter's school used that. Most of her teachers supplemented EM heavily--one teacher refused to use it at all. In the case of the one teacher that relied only on EM, I was tutoring with Singapore Math. I recall one student, a friend of my daughter's, quite bright, who at end of fourth grade claimed she was "bad at math" because she was unsure which of the four or so algorithms for each particular operation, she was supposed to use for which problem.

To see the kind of information parents are given about EM in Ann Arbor, check out the parents' guide for AAPS at Everyday Mathematics Parent Handbook.