Saturday, April 28, 2007
knowledge is good
Instructivist left this link.
So....I guess we can all stop worrying that our nation's public schools are devoting too much time to rote memorization.
update
I think the entire episode of Stupid in America may be posted on YouTube.
help desk - can you check?
I just created a sheet for Christopher to show that ratios, dimensional analysis, and proportionality constants and functions are all pretty much the same thing (I'm sure that's not the correct way to put it -- what is the correct way??)
Can someone check it over to make sure it's right?
Assuming it is, I'm dedicating this sheet to nowthatshockey, author of this observation:
We've all been talking about Liping Ma's notion of fragmented knowledge for two years now, off and on.
But nowthatshockey nailed it.
American students (and American math teachers) have fragmented knowledge of mathematics. We know things, but each thing is separate and distinct from every other thing. Most of us have no idea how it all fits together.
I, for instance, had no idea until 3 years ago that a fraction is also a division problem and a division problem is a fraction. (Remember when Carolyn quoted someone who remembered being happy as a boy to discover that a fraction was just a long division problem he didn't have to do? I never made that discovery.)
Teaching myself math today, I constantly have moments in which I think: oh.
Setting up ratios is a whole lot like setting up a dimensional analysis.
Whenever such a thought strikes me I stop and work whatever problem I'm doing both ways to make sure I'm right and to cement the connection.
Which brings me to constructivist math.
As usual they've taken the right idea and drawn the exact wrong conclusion.
Students don't need to make connections between math and everyday life. Making connections between math and everyday life is easy.*
What's hard is making connections between math and math.
* It stops being easy well before you get to college math, but that's irrelevant to the point I'm making here.
making connections
help desk
Can someone check it over to make sure it's right?
Assuming it is, I'm dedicating this sheet to nowthatshockey, author of this observation:
Mathematical reasoning does not lie in the connection of mathematics to the real world, but in the connection within mathematics.
How mathematics is applied is its connection to your world.
We've all been talking about Liping Ma's notion of fragmented knowledge for two years now, off and on.
But nowthatshockey nailed it.
American students (and American math teachers) have fragmented knowledge of mathematics. We know things, but each thing is separate and distinct from every other thing. Most of us have no idea how it all fits together.
I, for instance, had no idea until 3 years ago that a fraction is also a division problem and a division problem is a fraction. (Remember when Carolyn quoted someone who remembered being happy as a boy to discover that a fraction was just a long division problem he didn't have to do? I never made that discovery.)
Teaching myself math today, I constantly have moments in which I think: oh.
Setting up ratios is a whole lot like setting up a dimensional analysis.
Whenever such a thought strikes me I stop and work whatever problem I'm doing both ways to make sure I'm right and to cement the connection.
Which brings me to constructivist math.
As usual they've taken the right idea and drawn the exact wrong conclusion.
Students don't need to make connections between math and everyday life. Making connections between math and everyday life is easy.*
What's hard is making connections between math and math.
* It stops being easy well before you get to college math, but that's irrelevant to the point I'm making here.
making connections
help desk
Friday, April 27, 2007
“get a friggin' job, you bum”
Grandmother of 8 clobbers would-be robber in Tuckahoe
A warning to anyone thinking about robbing J. Villarina's Market Sandwich Shoppe: don't mess with Vera in the morning.
The 68-year-old grandmother of eight used a miniature baseball bat to clobber a would-be robber inside the shop on Columbus Avenue, where she works near the Metro-North train station, about 5 a.m. yesterday.
This happened close to my home. Vera is a doll, but quite feisty.
A warning to anyone thinking about robbing J. Villarina's Market Sandwich Shoppe: don't mess with Vera in the morning.
The 68-year-old grandmother of eight used a miniature baseball bat to clobber a would-be robber inside the shop on Columbus Avenue, where she works near the Metro-North train station, about 5 a.m. yesterday.
This happened close to my home. Vera is a doll, but quite feisty.
Quote of the Day
"I like Math," my 8th grader announced earlier today while working on her Algebra homework.
"Why is that," I inquired.
"When you know how to do something, it's fun. It's like solving a puzzle," she responded.
"Why is that," I inquired.
"When you know how to do something, it's fun. It's like solving a puzzle," she responded.
What Works Clearinghouse
I never really paid much attention to the What Works Clearinghouse (WWC) because I thought it just evaluated research and did not pass judgment on curricula. I really should pay attention!
Ken's blog refers to:
"... an alarming pattern in which the WWC allows the evaluation of the research to include a testing instrument developed by the authors of the education program as a valid measure of success."
and:
"This seems like extremely flimsy evidence to me and it seems to send a message to publishers on how to cook the books to get the thumbs up from the WWC."
I think it's worse than that.
In discussions yesterday with the curriculum person at my son's school (a very nice person), she talked about WWC and their recommendations. Recommendations? I thought they just evaluated research. How could they make recommendations if there really wasn't enough research? Well, I started my education of WWC. Perhaps Ken (and others) can get me up to speed. It appears that there is too little data and too much judgment.
As an example, look what I quickly found at their site.
Elementary School Math
"Everyday Mathematics was found to have potentially positive effects on students’ mathematics achievement."
[I really need to see what this is based on.]
"The Saxon Elementary School Math was found to have no discernible effects on mathematics achievement."
"Progress in Mathematics © 2006 was found to have no discernible effects on math achievement."
"No discernible effects"
They don't say that there is not enough data. They say that they don't see any effect. Wow! They don't say that for study 'A' or study 'B', they see no effect. They are making a judgment about the curriculum. How can they do this even for a relative judgment?
Middle School Math
UCSMP Algebra was found to have potentially positive effects on math achievement.
Saxon Middle School Math was found to have positive effects on math achievement.
The I CAN Learn® Pre-Algebra and Algebra curricula were found to have positive effects on math achievement. [The I CAN Learn® Pre-Algebra and Algebra computerized curricula are designed to cover mathematics and problem-solving skills for ethnically diverse, inner-city students in grades 6–12.]
I CAN Learn has positive effects, but compared to what? Relativity is nice, sometimes, but not in math.
Note that EM & UCSMP are positioned quite well - no negative marks. Apparently, it's enough for our curriculum advisor to keep them on the list and provide ammo for the pro-EM crowd.
It seems to me that WWC is being used for much more than it should be, and WWC is not going to stop them. It's nice to be popular, even if you don't have enough data to support it. The problem is that our curriculum advisor is using relative WWC judgments to make decisions, not absolute criteria, like the California Green Dot standards.
Ken's blog refers to:
"... an alarming pattern in which the WWC allows the evaluation of the research to include a testing instrument developed by the authors of the education program as a valid measure of success."
and:
"This seems like extremely flimsy evidence to me and it seems to send a message to publishers on how to cook the books to get the thumbs up from the WWC."
I think it's worse than that.
In discussions yesterday with the curriculum person at my son's school (a very nice person), she talked about WWC and their recommendations. Recommendations? I thought they just evaluated research. How could they make recommendations if there really wasn't enough research? Well, I started my education of WWC. Perhaps Ken (and others) can get me up to speed. It appears that there is too little data and too much judgment.
As an example, look what I quickly found at their site.
Elementary School Math
"Everyday Mathematics was found to have potentially positive effects on students’ mathematics achievement."
[I really need to see what this is based on.]
"The Saxon Elementary School Math was found to have no discernible effects on mathematics achievement."
"Progress in Mathematics © 2006 was found to have no discernible effects on math achievement."
"No discernible effects"
They don't say that there is not enough data. They say that they don't see any effect. Wow! They don't say that for study 'A' or study 'B', they see no effect. They are making a judgment about the curriculum. How can they do this even for a relative judgment?
Middle School Math
UCSMP Algebra was found to have potentially positive effects on math achievement.
Saxon Middle School Math was found to have positive effects on math achievement.
The I CAN Learn® Pre-Algebra and Algebra curricula were found to have positive effects on math achievement. [The I CAN Learn® Pre-Algebra and Algebra computerized curricula are designed to cover mathematics and problem-solving skills for ethnically diverse, inner-city students in grades 6–12.]
I CAN Learn has positive effects, but compared to what? Relativity is nice, sometimes, but not in math.
Note that EM & UCSMP are positioned quite well - no negative marks. Apparently, it's enough for our curriculum advisor to keep them on the list and provide ammo for the pro-EM crowd.
It seems to me that WWC is being used for much more than it should be, and WWC is not going to stop them. It's nice to be popular, even if you don't have enough data to support it. The problem is that our curriculum advisor is using relative WWC judgments to make decisions, not absolute criteria, like the California Green Dot standards.
Thursday, April 26, 2007
Extremely fast learning
Drop whatever you're doing and go read Larry Squire's commentary on the Tse, et al study right this minute.
This is revolutionary.
Of course, that means it will take 20 years for these findings to filter out to the public schools (if they ever filter out at all).
Bob Koegel told us years ago that it takes 20 years for new research to be widely adopted in teaching practice. Ten years for other researchers to confirm the finding, another 10 years for dissemination.
Larry Squire, fyi, is a honcho.
highlights
For months now I've been meaning to put up a post about Ericsson's concept of extended working memory.
Around here we've been accustomed to thinking that "knowledge is good" because knowledge and skills learned to the point of automaticity take a load off of working memory.
But it seems there's more to it. From The Role of Deliberate Practice in the Acquisition of Expert Performance (pdf file):
(an aside: the terminology researchers use to characterize memory has bewildered me for years, so let me point out that short-term memory and working memory are two different things)
In other words, it's not just that practicing knowledge to the point of automaticity "frees up space" in working memory so you can solve more complicated problems.
What Tse, Squire, and Ericsson all appear to be saying is that practicing knowledge to the point of automaticity also makes it possible to acquire new knowledge very rapidly.
and see:
why students have to memorize things
This is revolutionary.
Of course, that means it will take 20 years for these findings to filter out to the public schools (if they ever filter out at all).
Bob Koegel told us years ago that it takes 20 years for new research to be widely adopted in teaching practice. Ten years for other researchers to confirm the finding, another 10 years for dissemination.
Larry Squire, fyi, is a honcho.
highlights
We learn and remember better when new material can be related to what we already know. Professional athletes can remember details of particular plays that occurred in a long match. Experienced poker players can reconstruct the card distribution and betting sequence that occurred in previous hands. This is possible because these individuals have a rich background of relevant experience and therefore can organize new material into meaningful and orderly patterns.[snip]
Memory consolidation refers to the gradual process of reorganization by which new memories become remote memories (3, 4). Initially, the learning of facts and events (declarative memory) depends on the hippocampus, a structure deep in the temporal lobe of the mammalian brain. As time passes after learning, the importance of the hippocampus gradually diminishes and a more permanent memory is established in distributed regions of the neocortex. This process typically takes a few years in humans and at least a month in rodents. According to one influential model (5), the process is slow because if changes were made rapidly, they would interfere with the preexisting framework of structured knowledge that has been built up from other experiences.[snip]
The most surprising finding by Tse et al., and what connected the schema concept to memory consolidation, was that removal of the entire hippocampus as early as 48 hours after the rapid learning of two new flavorplace associations fully spared memory of the associations... It was not the case that memory of the new associations was never dependent on the hippocampus, nor that memory was somehow formed directly in the neocortex, because hippocampal lesions made 3 hours after learning abolished memory of the new associations. In short, the neocortex was able to incorporate new information rapidly. This is unexpectedly rapid for a process that, on the basis of as many as 20 studies in experimental animals, ordinarily takes at least a month (7). [ed.: a month in rats, years in people][snip]
It is tempting to suppose that memory consolidation proceeded rapidly because new information was fully compatible with what had already been learned—in other words, a good schema was available. If so, questions naturally arise about the minimum requirements for an effective schema. [ed.: yes, they do][snip]
caption:Ericsson, expertise, and "extended working memory"
Good schemas wanted. When a rat learns associations between flavors and spatial locations, as studied by Tse et al. (1), the associations are initially learned as individual facts (left). [ed.: precisely what cognitive science has been finding for at least 20 years] With extended training, the animal develops an organized structure or schema for flavors and places (middle). This organized knowledge structure (bold lines) can then support rapid learning of new associations in a single trial and the rapid consolidation of information into the neocortex (right).
For months now I've been meaning to put up a post about Ericsson's concept of extended working memory.
Around here we've been accustomed to thinking that "knowledge is good" because knowledge and skills learned to the point of automaticity take a load off of working memory.
But it seems there's more to it. From The Role of Deliberate Practice in the Acquisition of Expert Performance (pdf file):
[E]xpert performers have acquired skills that enable them to circumvent general memory and processing limits. Chase and Simon (1973) originally attributed experts' superior memory to chunking in short-term memory. This account has been revised, and the exceptional memory of experts has been shown to reflect rapid storage in long-term memory (Charness, 1976; Frey & Adesman, 1976; Lane & Robertson, 1979). ....The most important implication of these acquired memory skills is that they enable experts to circumvent the limited storage capacity of short-term memory. Thus these skills eliminate any restrictive influence of individual differences in this basic capacity (Ericsson & Smith, 1991b)I assume Ericsson and his team are talking about the same phenomenon Tse and her team demonstrated in mice: extremely rapid learning that circumvents the normal constraints on working memory and new learning.
(an aside: the terminology researchers use to characterize memory has bewildered me for years, so let me point out that short-term memory and working memory are two different things)
In other words, it's not just that practicing knowledge to the point of automaticity "frees up space" in working memory so you can solve more complicated problems.
What Tse, Squire, and Ericsson all appear to be saying is that practicing knowledge to the point of automaticity also makes it possible to acquire new knowledge very rapidly.
and see:
why students have to memorize things
if I'm so smart why ain't I rich?
No idea what to make of this.
oh, wait!
My guess: this is going to be about getting married, having kids, and working 'til you drop to put them through college.
Possibly.
.......
Or, alternatively, this is the too much money principle.
The more money you make, the more people lying in wait to extract it from you.
Seriously.
I had a great reunion with an old pal from L.A. over Christmas. Her son is attending one of the chi-chi private schools we probably would have tried to get C. into if we'd stayed.
She told me she was going to have to fork over thousands of dollars for SAT tutoring because everyone at the school was forking over thousands of dollars for SAT tutoring and she had to do it, too, or be shunned.
(C. & I watched the Vote-or-Die episode of South Park last night, the one where Stan gets shunned because he doesn't want to vote in an election where the choice is between a giant douche bag and a turd sandwich. We both burst out laughing when the punchline hit. Neither of us saw it coming.)
update: One can purchase vast quantities of Vote Turd Sandwich clothing and paraphernalia at cafepress.
I wonder if anyone's measured the collective IQ of the Vote Turd Sandwich clientele.
oh, wait!
My guess: this is going to be about getting married, having kids, and working 'til you drop to put them through college.
Possibly.
.......
Or, alternatively, this is the too much money principle.
The more money you make, the more people lying in wait to extract it from you.
Seriously.
I had a great reunion with an old pal from L.A. over Christmas. Her son is attending one of the chi-chi private schools we probably would have tried to get C. into if we'd stayed.
She told me she was going to have to fork over thousands of dollars for SAT tutoring because everyone at the school was forking over thousands of dollars for SAT tutoring and she had to do it, too, or be shunned.
(C. & I watched the Vote-or-Die episode of South Park last night, the one where Stan gets shunned because he doesn't want to vote in an election where the choice is between a giant douche bag and a turd sandwich. We both burst out laughing when the punchline hit. Neither of us saw it coming.)
update: One can purchase vast quantities of Vote Turd Sandwich clothing and paraphernalia at cafepress.
I wonder if anyone's measured the collective IQ of the Vote Turd Sandwich clientele.
how to build a fast learner, part 2
jackpot
I've only glanced at the first page of Squire's commentary on the Tse, et al study, but my guess it that the condition under which rats failed at one-trial learning may resemble spiraling and constructivist curricula.
I'll see.
how to build a fast learner
how to build a fast learner, part 2
We may be able to learn new things very quickly, and incorporate them into memories, if representations of related information have already been stabilized in the brain.
source:
Rapid Consolidation (pdf file)
Larry R. Squire
Science
April 6, 2007
Vol. 316 no. 5821, pp. 57-58
I've only glanced at the first page of Squire's commentary on the Tse, et al study, but my guess it that the condition under which rats failed at one-trial learning may resemble spiraling and constructivist curricula.
I'll see.
how to build a fast learner
how to build a fast learner, part 2
how to build a fast learner
I'm supposed to be writing a chapter of Temple's & my new book, so this will be quick. (I'll revise later.)
Some of you may remember last fall's posts about mastery learning and speed and mastery and IQ.
The jist of those posts is that while higher IQ means faster recall of new material, higher IQ doesn't give you a significant advantage in speed of relearning material that's been taught to mastery the first go-round.
Engelmann describes the same phenomenon, except Engelmann takes it a step further.
Engelmann observes that at some point slower learners get faster at learning new material, too:
Lots faster, it seems.
I'm certain this is true, because I've experienced it all my life.
And because I saw it last spring with Christopher.
Getting C. through the Phase 4 math course in 6th grade was a battle unto death, as faithful readers will recall.
Ed and I went back and forth on whether to bag the whole thing,* never quite reaching a conclusion for reasons having to do with stubbornness as much as anything else, no doubt.
Then suddenly, last spring, C. picked up the pace. Way picked up the pace. He'd come home from school having learned and remembered whatever it was they'd covered in math that day.
There wasn't any change beyond that. He wasn't understanding math better, or liking math better, or displaying greater motivation to learn math. (understatement)
The only difference was: he was faster. All of a sudden.
how do you build a faster learner?
So the question is: what has to happen to get a student to the tipping point?
I had been thinking that it must have to do with the learner (I can barely stand to type that word but I'm afraid I must) acquiring a schema of the field (or domain) he's learning.
Looks like I was right --
One-trial learning is about as fast as it gets.
Here's the way Science News describes it (subscription required):
in a nutshell (t/k)
Note: the experimenters did not use a spiral curriculum.
These rats were taught to mastery.
Shouldn't we be asking whether schools have a moral imperative to teach students to mastery?
* Do the fastest learners get faster, too? They must, wouldn't you think?
how to build a fast learner
how to build a fast learner, part 2
* for newbies: "bag the whole thing," meaning move him out of accelerated math and into the regular track
Some of you may remember last fall's posts about mastery learning and speed and mastery and IQ.
The jist of those posts is that while higher IQ means faster recall of new material, higher IQ doesn't give you a significant advantage in speed of relearning material that's been taught to mastery the first go-round.
Engelmann describes the same phenomenon, except Engelmann takes it a step further.
Engelmann observes that at some point slower learners get faster at learning new material, too:
[A]fter students have mastered a battery of skills and knowledge, the difference in rate of ascent for appropriately placed students is far less because all students tend to have enough skill to master the new material at around the same rate.In other words, at some point in a student's process of learning a subject, he gets faster.
source:
Student-Program Alignment and Teaching to Mastery (pdf file)
Siegfried Engelmann
Lots faster, it seems.
I'm certain this is true, because I've experienced it all my life.
And because I saw it last spring with Christopher.
Getting C. through the Phase 4 math course in 6th grade was a battle unto death, as faithful readers will recall.
Ed and I went back and forth on whether to bag the whole thing,* never quite reaching a conclusion for reasons having to do with stubbornness as much as anything else, no doubt.
Then suddenly, last spring, C. picked up the pace. Way picked up the pace. He'd come home from school having learned and remembered whatever it was they'd covered in math that day.
There wasn't any change beyond that. He wasn't understanding math better, or liking math better, or displaying greater motivation to learn math. (understatement)
The only difference was: he was faster. All of a sudden.
how do you build a faster learner?
So the question is: what has to happen to get a student to the tipping point?
I had been thinking that it must have to do with the learner (I can barely stand to type that word but I'm afraid I must) acquiring a schema of the field (or domain) he's learning.
Looks like I was right --
Schemas and Memory Consolidation
Dorothy Tse,1* Rosamund F. Langston,1* Masaki Kakeyama,2 Ingrid Bethus,1 Patrick A. Spooner,1 Emma R. Wood,1 Menno P. Witter,3 Richard G. M. Morris1
Memory encoding occurs rapidly, but the consolidation of memory in the neocortex has long been held to be a more gradual process. We now report, however, that systems consolidation can occur extremely quickly if an associative "schema" into which new information is incorporated has previously been created. In experiments using a hippocampal-dependent paired-associate task for rats, the memory of flavor-place associations became persistent over time as a putative neocortical schema gradually developed. New traces, trained for only one trial, then became assimilated and rapidly hippocampal-independent. Schemas also played a causal role in the creation of lasting associative memory representations during one-trial learning. [emphasis mine] The concept of neocortical schemas may unite psychological accounts of knowledge structures with neurobiological theories of systems memory consolidation.
source:
Science
April 6, 2007
Volume 316, no. 5821, pp. 76 - 82
One-trial learning is about as fast as it gets.
Here's the way Science News describes it (subscription required):
People call on a rich background [ed.: background knowledge!] of relevant experiences to organize and remember new material. Rats do the same, and with surprising speed, say Dorothy Tse of the University of Edinburgh and her coworkers.
Prior studies, which have focused on task learning unrelated to preexisting knowledge, indicate that a brain region called the hippocampus incorporates new facts and events into memory. The hippocampus gradually yields to another structure, the neocortex, as new memories become stronger. This process typically takes at least 1 month in rodents and a few years in people.
Tse's team trained groups of rats to associate six flavors, including banana and bacon, with six designated spots within a laboratory-test area.... The animals learned all six flavor-place associations in 1 month.
....[T]he animals had also developed a framework of knowledge about relations between places and flavors that enabled them to learn new pairings remarkably quickly. The rats remembered novel flavor-place associations after just one trial and retained this information for at least 2 weeks, the scientists report in the April 6 Science.
The rats' formation of a knowledge framework spurred the neocortex to integrate new information into memory in record time, the scientists propose. Surgical removal of the hippocampus 48 hours after the rats had rapidly learned new flavor-place associations left those memories intact, a sign that the neocortex had already taken charge of the material.
source:
Rats take fast route to remembering
Science News
April 14, 2007 Vol 171, No. 15, p. 237
Bruce Bower
in a nutshell (t/k)
Note: the experimenters did not use a spiral curriculum.
These rats were taught to mastery.
Shouldn't we be asking whether schools have a moral imperative to teach students to mastery?
* Do the fastest learners get faster, too? They must, wouldn't you think?
how to build a fast learner
how to build a fast learner, part 2
* for newbies: "bag the whole thing," meaning move him out of accelerated math and into the regular track
Global Institute of Mathematics
Does anyone know anything about James Stone and his online math school? I had come across him before (we may have a post up on the old ktm), and just found him again.
Another question: why does he teach two different algebra sequences, Algebra 1, 2, & 3 and "Advanced Algebra 1, 2, & 3"? Both sequences seem to be high school algebra - yes?
I've been moving through Saxon Algebra 2 much more slowly than I'd like (Lesson 76 out of 129), so I'm thinking I may need to take a formal course pretty soon if only for the structure. Much of Saxon Algebra 2 is brand-new material to me -- brand new in the sense that not only do I not know it, I've never even seen it before. I didn't know this stuff was out there to know.
Which means that in the past couple of months I've gone from being a middle-aged person who had never heard of polar coordinates to being a middle-aged person who can convert polar coordinates to rectangular coordinates and and rectangular coordinates to polar coordinates -- and who can add vectors to boot. (What are vectors?)
Plus, as of this week I'm on my way to becoming a middle-aged person who can do all this using negative magnitudes, too. Introducing negative magnitudes, Saxon writes:
Question: Am I the only middle-aged person on earth sitting around teaching herself how to add vectors?
If so, this might be too much outlier-ness even for me. (Though, given the fact that I'm the most mainstream person I know, I'm probably just slightly ahead of the curve. Six months from now every middle-aged mother in Westchester will be working mixture problems.)
I think I've mentioned this before, but it's worth repeating.
I took 3 years of math in high school.
Algebra 1, Geometry, Algebra 2
It turns out that what was Algebra 1 and 2 in my high school is Algebra 1 in Saxon Math. Algebra 1 plus.
I hate to even think what the real-math equivalent of my high school geometry course is going to be.
Another question: why does he teach two different algebra sequences, Algebra 1, 2, & 3 and "Advanced Algebra 1, 2, & 3"? Both sequences seem to be high school algebra - yes?
I've been moving through Saxon Algebra 2 much more slowly than I'd like (Lesson 76 out of 129), so I'm thinking I may need to take a formal course pretty soon if only for the structure. Much of Saxon Algebra 2 is brand-new material to me -- brand new in the sense that not only do I not know it, I've never even seen it before. I didn't know this stuff was out there to know.
Which means that in the past couple of months I've gone from being a middle-aged person who had never heard of polar coordinates to being a middle-aged person who can convert polar coordinates to rectangular coordinates and and rectangular coordinates to polar coordinates -- and who can add vectors to boot. (What are vectors?)
Plus, as of this week I'm on my way to becoming a middle-aged person who can do all this using negative magnitudes, too. Introducing negative magnitudes, Saxon writes:
To make matters even more confusing, we note that it is also possible to use negative magnitudes to locate a point. [boldface in the original]
source:
Saxon Algebra 2
Lesson 76, page 305
Question: Am I the only middle-aged person on earth sitting around teaching herself how to add vectors?
If so, this might be too much outlier-ness even for me. (Though, given the fact that I'm the most mainstream person I know, I'm probably just slightly ahead of the curve. Six months from now every middle-aged mother in Westchester will be working mixture problems.)
I think I've mentioned this before, but it's worth repeating.
I took 3 years of math in high school.
Algebra 1, Geometry, Algebra 2
It turns out that what was Algebra 1 and 2 in my high school is Algebra 1 in Saxon Math. Algebra 1 plus.
I hate to even think what the real-math equivalent of my high school geometry course is going to be.
Wednesday, April 25, 2007
When Fair Use Isn't Fair
Update: "Tomorrow the hammer's coming down hard over the 'Fair Use' issue..."
Retrospectacle: A Neuroscience Blog: When Fair Use Isn't Fair
Shelly over at Retrospectacle was threatened with legal action when she used one figure and one chart from a scientific paper published in the Journal of Science of Food and Agriculture, with full citation of course.
She took down the offending figure and chart and reproduced the data in excel, but there is principle here. Apparently the Journal didn't like the spin she was putting on the data.
This sets a bad precedent. Education researchers could conceivably ask us bloggers to not use published data to make points against the establishment.
You may contact the publishers at the following to let them know what you think.
Update: The issue has been resolved... I have removed the contact details.
Retrospectacle: A Neuroscience Blog: When Fair Use Isn't Fair
Shelly over at Retrospectacle was threatened with legal action when she used one figure and one chart from a scientific paper published in the Journal of Science of Food and Agriculture, with full citation of course.
She took down the offending figure and chart and reproduced the data in excel, but there is principle here. Apparently the Journal didn't like the spin she was putting on the data.
This sets a bad precedent. Education researchers could conceivably ask us bloggers to not use published data to make points against the establishment.
Go read for yourself.Re: Antioxidants in Berries Increased by Ethanol (but Are Daiquiris Healthy?) by Shelly Bats
http://scienceblogs.com/retrospectacle/2007/04/antioxidants_in_berries_increa.php
The above article contains copyrighted material in the form of a table and graphs taken from a recently published paper in the Journal of the Science of Food and Agriculture. If these figures are not removed immediately, lawyers from John Wiley & Sons will contact you with further action.
You may contact the publishers at the following to let them know what you think.
Update: The issue has been resolved... I have removed the contact details.
On slow and fast learners, uniformed curriculum and more in Sovied education system
Centralized education system has its disadvantages for a individualist. However, in my opinion (and in opinion on my husband's grandmother who spent 47(!) years being a teacher and who I consulted on this topic) the benefits for a student were quite balancing it.
First, of course, was the funding of the schools - the schools in the capital city(at that time - Moscow) were financed and equipped better than schools in the province and country sides. The principals had nothing to do with the budget. All textbooks, supplies, materials, furniture etc were approved and provided from "the top". The teacher's salary was uniformed with differentiated pay with the increase of the years of service. Retirement started at 55 (and it's still so) for women, 60 - for men.
Second, the centralized control over curriculum was strict and a teacher could not work as she or he pleased. Everything was scripted, outcomes of the lessons approved. (For examples, there were lists of books the literature teacher could not even recommend for extracurricular reading - banned).
Third, there was no choice in terms of electives and so on the students could take. There was no choice in anything.
Fourth, no differentiating was done for students with different abilities.
Fifth, no students with disabilities (of any kind) were in general schools. Such kids were attending special internats (institutions) with medical personnel and other specialists available.
Though I understand the disadvantages of former education system in USSR (it doesn't exist any more!), I still feel nostalgic and therefore may be rather subjective.
Centralized funding - at least the BOOKS were available and provided for free for everybody. The last days in school year after the exams were taken, students were returning their books to the library and receiving complete sets of books for all subjects for the next year.
Scripted programs -The weak or inexperienced teachers were able to actually teach. New teachers were coming to school equipped with complete set of lesson plans for the subject they teach, by grades. They all had taken a class on TEACHING their subject and wrote (under strict guidance of their professors and chairs) the scripts for teaching each unit. The teachers in early grades had scripts (aligned with textbooks used) for teaching reading step by step, writing, and math. There were no social studies before grade 4 (in grade 4 History started), and no science (in grade 4 we had "Observing Nature" class in which we kept Nature observation logs.
All classes were mandatory and scheduled for the whole class. No choice. One would be moving through the school years with the same 30 people he started 1 st grade.
This type of scheduling ensured that ALL students had physics, chemistry, geography, biology despite their personal preferences (I still don't get it - how can you come out of high school and never take chemistry or physics, and then in colleges the professors start from the stuff that is SO basic!). So it was a kind of common knowledge. The literature you studied in grade 7 was also studied by the person you talk to, so you could be safely referring to and citing the authors and lines you memorized , and have the common ground of understanding.
Differentiated instruction - the school program was created for an average student with satisfactory marks (somewhat up to 75% in a 100% scale). So everyone, if not completely brain-damaged, could master it. Here we often hear about different pace of learning. I was the fast one. My prize was the amount of free time after school. I didn't have to read the chapter twice (and since most HW was oral, I always did my reading quickly or briefly looked up the notes), or spend much time on trying to understand the concept. The lessons were structured very close to direct instruction model, with clear and direct explanations, and guided practice,, so just to practice 10 problems in math was good enough.
Slow students had to spend much more time practicing at home, reading and re-reading the chapters, and memorizing rules and definitions. My cousin had also all excellent marks but it took her three times the amount of time I spent on the same material.
This way, the fast students could spend more time on extracurricular activities (as I did) provided for free by many culture centers. I was in the theater studio, I danced, I sang in a choir, I drew, and I did science research without harming my learning but rather enriching it.
Please, also note - in most cases parental involvement was in terms of checking with the teacher if the child behaves and what are the grades. If something was not "great", the kid would be spanked. But nobody had to re-teach anything. My parents, both engineers, paid attention to my HWs up to grade 4. After that, they could help me with resources (my mother works in research) but never had to explain or teach any topic to me.
I am not going to say anything about "special" students - I didn't see much of them while in school (BTW, the colleges could also refuse to accept a student based on the health issue - I was almost rejected by the vet.school because of the bad vision : - 9 is not a joke!)
However, now they mainstream kids based on parents' demands. My husband's grandmother is a homeroom teacher of an autistic child (a son of my mother's co-worker), but she also works with him individually after school.
As of now - there is still some remaining centralization, but the schools are moving toward the western (read - american) model. Workshop models are introduced, teachers are mandated to write activities and projects for students, private schools operate by "individualized approach to each students abilities" and work on "development of creativity", students cannot write properly, and the History as a subject looks like a twisted and deformed stick - no curriculum at all. My husband's grandmother still checks and corrects all spelling mistakes in essays, but most teachers don't. Education lost the value for average people becoming elitarian
P.S.: After talking to my husband's grandmother today I don't feel like sending my kid over there to school anymore (even though it's still cheaper to find the Math-English school with traditional instruction there). At least they didn't damage the science curriculum yet. And this Summer I'll go there and will bring myself sequenced and organized books in Physics (6,7,8), Chemistry, Geography...
First, of course, was the funding of the schools - the schools in the capital city(at that time - Moscow) were financed and equipped better than schools in the province and country sides. The principals had nothing to do with the budget. All textbooks, supplies, materials, furniture etc were approved and provided from "the top". The teacher's salary was uniformed with differentiated pay with the increase of the years of service. Retirement started at 55 (and it's still so) for women, 60 - for men.
Second, the centralized control over curriculum was strict and a teacher could not work as she or he pleased. Everything was scripted, outcomes of the lessons approved. (For examples, there were lists of books the literature teacher could not even recommend for extracurricular reading - banned).
Third, there was no choice in terms of electives and so on the students could take. There was no choice in anything.
Fourth, no differentiating was done for students with different abilities.
Fifth, no students with disabilities (of any kind) were in general schools. Such kids were attending special internats (institutions) with medical personnel and other specialists available.
Though I understand the disadvantages of former education system in USSR (it doesn't exist any more!), I still feel nostalgic and therefore may be rather subjective.
Centralized funding - at least the BOOKS were available and provided for free for everybody. The last days in school year after the exams were taken, students were returning their books to the library and receiving complete sets of books for all subjects for the next year.
Scripted programs -The weak or inexperienced teachers were able to actually teach. New teachers were coming to school equipped with complete set of lesson plans for the subject they teach, by grades. They all had taken a class on TEACHING their subject and wrote (under strict guidance of their professors and chairs) the scripts for teaching each unit. The teachers in early grades had scripts (aligned with textbooks used) for teaching reading step by step, writing, and math. There were no social studies before grade 4 (in grade 4 History started), and no science (in grade 4 we had "Observing Nature" class in which we kept Nature observation logs.
All classes were mandatory and scheduled for the whole class. No choice. One would be moving through the school years with the same 30 people he started 1 st grade.
This type of scheduling ensured that ALL students had physics, chemistry, geography, biology despite their personal preferences (I still don't get it - how can you come out of high school and never take chemistry or physics, and then in colleges the professors start from the stuff that is SO basic!). So it was a kind of common knowledge. The literature you studied in grade 7 was also studied by the person you talk to, so you could be safely referring to and citing the authors and lines you memorized , and have the common ground of understanding.
Differentiated instruction - the school program was created for an average student with satisfactory marks (somewhat up to 75% in a 100% scale). So everyone, if not completely brain-damaged, could master it. Here we often hear about different pace of learning. I was the fast one. My prize was the amount of free time after school. I didn't have to read the chapter twice (and since most HW was oral, I always did my reading quickly or briefly looked up the notes), or spend much time on trying to understand the concept. The lessons were structured very close to direct instruction model, with clear and direct explanations, and guided practice,, so just to practice 10 problems in math was good enough.
Slow students had to spend much more time practicing at home, reading and re-reading the chapters, and memorizing rules and definitions. My cousin had also all excellent marks but it took her three times the amount of time I spent on the same material.
This way, the fast students could spend more time on extracurricular activities (as I did) provided for free by many culture centers. I was in the theater studio, I danced, I sang in a choir, I drew, and I did science research without harming my learning but rather enriching it.
Please, also note - in most cases parental involvement was in terms of checking with the teacher if the child behaves and what are the grades. If something was not "great", the kid would be spanked. But nobody had to re-teach anything. My parents, both engineers, paid attention to my HWs up to grade 4. After that, they could help me with resources (my mother works in research) but never had to explain or teach any topic to me.
I am not going to say anything about "special" students - I didn't see much of them while in school (BTW, the colleges could also refuse to accept a student based on the health issue - I was almost rejected by the vet.school because of the bad vision : - 9 is not a joke!)
However, now they mainstream kids based on parents' demands. My husband's grandmother is a homeroom teacher of an autistic child (a son of my mother's co-worker), but she also works with him individually after school.
As of now - there is still some remaining centralization, but the schools are moving toward the western (read - american) model. Workshop models are introduced, teachers are mandated to write activities and projects for students, private schools operate by "individualized approach to each students abilities" and work on "development of creativity", students cannot write properly, and the History as a subject looks like a twisted and deformed stick - no curriculum at all. My husband's grandmother still checks and corrects all spelling mistakes in essays, but most teachers don't. Education lost the value for average people becoming elitarian
P.S.: After talking to my husband's grandmother today I don't feel like sending my kid over there to school anymore (even though it's still cheaper to find the Math-English school with traditional instruction there). At least they didn't damage the science curriculum yet. And this Summer I'll go there and will bring myself sequenced and organized books in Physics (6,7,8), Chemistry, Geography...
Balanced >insert subject here<
I posted this over at my blog, and I think I am on to something here.
The next time someone tries to defend or advocate for balanced math instruction or balanced literacy remember this simple mathematical formula.
(good + bad) / 2 = mediocre
for those non-mathematically inclined.
"The average of good plus bad is mediocre."
The next time someone tries to defend or advocate for balanced math instruction or balanced literacy remember this simple mathematical formula.
(good + bad) / 2 = mediocre
for those non-mathematically inclined.
"The average of good plus bad is mediocre."
more KIPP
- founded in 1994
- now has 52 schools in 16 states
- more than 90 percent of KIPP schools are still operating
from Jay Mathews:
KIPP Bridge principal David Ling said when he told parents repeating the grade would help get their children up to grade level, they often said they thought their children were already excellent students, and would be stars back at their regular schools. [ed.: I love it!]
[snip]
The retention issue has been a hot topic in KIPP conferences and email traffic. How can they help these students reach national standards if they quit because of wounded pride? The KIPP schools in Baltimore are now in their third year of a solution they call the Rapid Readers program. It serves all fifth-graders who test below the second grade level in reading. Their families are told from the beginning that it may take them five years to get to eighth grade level. There are no surprises. If they don't like that idea, they are free to withdraw, but most don't. During their first fifth grade year, they spend three hours a day on reading. By the end of their second year in fifth grade, they are ready for sixth grade.
source
Looking at KIPP
Jay Mathews
This is amazing:
The report card said the average student who has been with KIPP three years started at the 44th percentile in math and the 34th percentile in reading at the beginning of fifth grade. By the end of seventh grade that student was at the 83rd percentile in math and the 58th percentile in reading.
Incredible.
My campaign line has become:
"The district should take responsibility for raising individual student achievement across all levels of ability in consultation with parents."
I keep saying it, but nobody's buying it. My district doesn't have the faintest interest in raising individual student achievement across all levels of ability, and certainly not in consultation with parents.
heck, no!
I'm sure the special ed folks have a strong interest in raising the achievement of their kids.
That's because they're doing their jobs.
And of course the middle school principal did tell a large audience of unhappy parents that he was "only" concerned with the low-scoring kids & no one else, because "they're the ones who are struggling."
He said this. To parents. None of whom had attended the meeting to discuss special ed or "building services."
I'm told that another one of our high-level administrators said to a parent, "We don't care about the high-achievers. They're doing fine."
Words to that effect.
KIPP homepage
Tuesday, April 24, 2007
Good Education Research
An Interview with Frederick Hess: The Education Research We Need; (And why we don't have it)
According to Frederick Hess, good education research is done by everyone except education schools.
Disclaimer: Just in case I ever do want to get into an education graduate school and the admissions people do a google and discover this post, I want to say for the record that I only look down on the other education schools... not yours.
Cross Posted at Parentalcation
According to Frederick Hess, good education research is done by everyone except education schools.
That's a great question. Certainly, there is good evidence that upper-tier economics, political science, sociology, and public policy programs are producing PhDs with quantitative skills and methodological sophistication that dramatically surpass those of earlier generations. This has been the pattern of the social sciences for several decades, and nothing has changed on that score. Whether some programs are emphasizing formal theory or econometric training to the degree that fewer graduates may have an aptitude for or interest in field work is a question some have posed. But I don't know that anyone has any good answers to that.So, if I am reading this right, if I ever want to make a contribution to educational research, I should get a PHD in something other than education.
With regard to doctoral level training in education, I'm in no position to pass judgment on the quality of instruction being offered at the hundreds of institutions offering education doctorates. I can say, however, that the education policy work by young scholars that I find most compelling consistently seems to be produced by young scholars trained in the disciplines. Whether that judgment is a product of my own tastes as a reader, self-selection on the part of doctoral candidates, the quality of preparation, or some other factor, I really can't say.
Disclaimer: Just in case I ever do want to get into an education graduate school and the admissions people do a google and discover this post, I want to say for the record that I only look down on the other education schools... not yours.
Cross Posted at Parentalcation
smart people need more practice
This is kind of cool ---
There's a strong association between high working memory capacity and high IQ. For awhile there, I think, people were thinking that working memory might actually be IQ.
I take these findings to mean that high-working memory types have "coasted" on WM; they haven't learned work-arounds or shortcuts -- like all these brainy little math kids who refuse to write out the steps because they can do them faster in their heads. (This may be completely wrong... I'm free associating.)
Because they're naturally fast, they haven't learned how to be efficient.
Highly accomplished and talented people often choke under pressure because the distraction caused by stress consumes their big supply of short-term memory, says Sian Beilock, assistant professor of psychology at the University of Chicago, who conducted the research. In the business world, for instance, short-term memory is what lets you listen to what people are saying and yet maintain a point you want to bring up.
Because people in this group heavily rely on short-term memory to tackle challenges, they're at a particular disadvantage. When put under pressure, they resort to using less accurate short cuts to solve problems, such as guessing and estimation, much like those with lesser abilities.
[snip]
.... when talented people begin to feel the heat....
....they started worrying about screwing up. That consumed the working memory capacity of those in the study, which had allowed them to use complex strategies to solve problems in other circumstances.
"Once they had all this capacity to devote to a problem," says Beilock, "and now they're thinking about their worries, trying to suppress them and possibly causing themselves to freak out more."
[snip]
One of Gray's studies has shown that people with higher intelligence also are more taxed by having to control their emotions.
[snip]
If choking under pressure is a concern for you, Beilock's advice is practice, practice, practice--and not just problem solving, but problem solving in high-pressure situations. Memorizing methods of handling problems means you won't have to rely on your short-term memory.
source:
Why Pros and CEOs Choke
There's a strong association between high working memory capacity and high IQ. For awhile there, I think, people were thinking that working memory might actually be IQ.
I take these findings to mean that high-working memory types have "coasted" on WM; they haven't learned work-arounds or shortcuts -- like all these brainy little math kids who refuse to write out the steps because they can do them faster in their heads. (This may be completely wrong... I'm free associating.)
Because they're naturally fast, they haven't learned how to be efficient.
Dangerous Book for Boys
A friend of mine says this book is fantastic!
"I wanted to do the kind of book that we had lusted after when we were kids," said Conn Iggulden, who co-wrote the book with his younger brother Hal."My dad was born in 1923 and his father was born in 1850, and we had some old books in the house with titles like ’Chemical Amusements and Experiments’ and ’Fun With Gunpowder.’ The thing we didn’t have was a single compendium of everything we wanted to do. I remember endlessly looking through these (books), generally to find things that I could make explode or set on fire."
A big, affable, dark-haired thirtysomething who writes best-selling historical novels about the exploits of Julius Caesar and Genghis Khan, Iggulden exudes boyish enthusiasm.
He and Hal, a theater director, researched the "Dangerous Book" over six months in a garden shed, rediscovering the lost childhood arts of secret codes and water bombs and building simple batteries and pinhole projectors.
"Rule No. 1 was we either had to make it or do it - we’ve both read books where the author clearly hasn’t made a raft or whatever, and so the instructions don’t work," Iggulden said. "That meant we had to play marbles ... and skin a rabbit. A little bit grisly, that one. But then, we did make it into a stew and we did eat it.
"It was not a great stew," he admitted. "It was pretty rubbery."
Some parents may balk at encouraging their offspring to skin a rabbit _ or tan a hide, another skill imparted by the Iggulden brothers.
Conn Iggulden argues that "if you spend your life going to supermarkets, you should know where the meat comes from and exactly what’s gone into it for your eating pleasure. I think that’s worth doing once for just about anybody."
Sales figures suggest the "Dangerous Book" has struck a strong chord among adults concerned about the increasingly sedentary, regulated lives of today’s children - a society with computers in every classroom but often without climbing equipment in the playground.
source:
Boston Herald
Which reminds me.
Just out: a small, exploratory study showing that masculinity is good for men.
Who knew?
..............
Dangerous Book at HarperCollins
website
cross-multiplication
In Hung-Hsi Wu's monograph on Fractions written for pre-service and in-service teachers, he makes two strong points: keep a number line handy when you define fractions, and learn how to order fractions before you try to add them.
From Section 5, Ordering Fractions (the Cross-Multiplication Algorithm),
Before we approach the addition of fractions, we first consider the more elementary concept of order, i.e., comparing two fractions to see if one is bigger than or equal to the other (recall that equality in this case means they are the same point on the number line).
Given two fractions A and B, we say A < B if A is to the left of B as points on the number line. This is the same as saying that the segment [0,B] is longer than the segment [0,A]....
We emphasize once again the need to put fractions and whole numbers on the same footing. It would have been preposterous to define order among fractions in a way that is different from the definition of order among whole numbers. Observe also the ease with which we define order among fractions when the number line is at our disposal.
The main objective of this section is to show that a comparison of two fractions a/b and c/d can be made by inspecting their “cross products” ad and bc. This so-called cross-multiplication algorithm has gotten a bad name in recent years because it is supposed to be part of learning-by-rote, and the reason for that is because many textbooks just write it down and use it without any explanation. As a reaction, the curricula of recent years have a tendency of not even mentioning this algorithm.
Using the algorithm without explanation and not mentioning the algorithm at all represent the two extremes of mathematics education. Neither is good education, because this algorithm is a useful tool which can be simply explained.
Consider the following example. Which is the bigger of the two: 4/7 or 3/5?
In terms of segments, this should be rephrased as: which of [0, 4/7] and [0, 3/5] is longer? Now by definition:
4/7 is 4 copies of 1/7
3/5 is 3 copies of 1/5
This comparison is difficult because the two fractions are expressed in terms of different “units”: 1/7 and 1/5 . However, imagine for a moment that the following statements were actually true for some whole number c :
4/7 is 20 copies of 1/c
3/5 is 21 copies of 1/c
Then we would be able to immediately conclude that 3/5 is the bigger of the two because it includes one more segment (of the same length) than 4/7 . This suggests that the way to achieve the desired comparison is to express both 1/7 and 1/5 in terms of a common “unit."
[We have to decide on a common unit for 1/7 and 1/5 . The cancellation law suggests the use of 1/35, so]
4/7 = (5 × 4)/(5 × 7) = 20/35 = 20 copies of 1/35
3/5 = (7 × 3)/(7 × 5) = 21/35 = 21 copies of 1/35
Conclusion: 4/7 < 3/5.
A closer look of the preceding also reveals that this conclusion is based on the inequality 5 x 4 < 7 x 3.
I think Wu would rather show his students (teachers) how to deduce cross-multiplication from the definition of fractions as points on the number line, than let his students induce cross-multiplication as best they can by searching for numerical patterns in many examples.
If we respect the time and attention of 20 year-olds by showing them directly why cross-multiplication works, why not figure out a way to respect the time and attention of 10 year-olds, and show them directly why cross-multiplication works?
Wu is quite clear that his monographs are for teachers, but I have had a great deal of success using the number line as he does to explain fractions to my boys. I haven't taught my boys to cross-multiply, but I have taught them to look for a common denominator when they are comparing fractions, or when they are trying to move from one (fractional) point (on the number line) to another (i.e. add and subtract fractions).
However, I am convinced by the example of my children that you can tell a compelling conceptual story to get a child to believe a procedure works, but he will forget the details of the conceptual story long before he forgets the procedure. Because we make him practice the procedure, not the story.
So-called "conceptual" curriculums like Investigations hold out the promise that children will learn the conceptual story by heart because they will write it for themselves, inducing the rule from many examples we show them without comment.
But a well-told conceptual story may be best.
From Section 5, Ordering Fractions (the Cross-Multiplication Algorithm),
Before we approach the addition of fractions, we first consider the more elementary concept of order, i.e., comparing two fractions to see if one is bigger than or equal to the other (recall that equality in this case means they are the same point on the number line).
Given two fractions A and B, we say A < B if A is to the left of B as points on the number line. This is the same as saying that the segment [0,B] is longer than the segment [0,A]....
We emphasize once again the need to put fractions and whole numbers on the same footing. It would have been preposterous to define order among fractions in a way that is different from the definition of order among whole numbers. Observe also the ease with which we define order among fractions when the number line is at our disposal.
The main objective of this section is to show that a comparison of two fractions a/b and c/d can be made by inspecting their “cross products” ad and bc. This so-called cross-multiplication algorithm has gotten a bad name in recent years because it is supposed to be part of learning-by-rote, and the reason for that is because many textbooks just write it down and use it without any explanation. As a reaction, the curricula of recent years have a tendency of not even mentioning this algorithm.
Using the algorithm without explanation and not mentioning the algorithm at all represent the two extremes of mathematics education. Neither is good education, because this algorithm is a useful tool which can be simply explained.
Consider the following example. Which is the bigger of the two: 4/7 or 3/5?
In terms of segments, this should be rephrased as: which of [0, 4/7] and [0, 3/5] is longer? Now by definition:
4/7 is 4 copies of 1/7
3/5 is 3 copies of 1/5
This comparison is difficult because the two fractions are expressed in terms of different “units”: 1/7 and 1/5 . However, imagine for a moment that the following statements were actually true for some whole number c :
4/7 is 20 copies of 1/c
3/5 is 21 copies of 1/c
Then we would be able to immediately conclude that 3/5 is the bigger of the two because it includes one more segment (of the same length) than 4/7 . This suggests that the way to achieve the desired comparison is to express both 1/7 and 1/5 in terms of a common “unit."
[We have to decide on a common unit for 1/7 and 1/5 . The cancellation law suggests the use of 1/35, so]
4/7 = (5 × 4)/(5 × 7) = 20/35 = 20 copies of 1/35
3/5 = (7 × 3)/(7 × 5) = 21/35 = 21 copies of 1/35
Conclusion: 4/7 < 3/5.
A closer look of the preceding also reveals that this conclusion is based on the inequality 5 x 4 < 7 x 3.
I think Wu would rather show his students (teachers) how to deduce cross-multiplication from the definition of fractions as points on the number line, than let his students induce cross-multiplication as best they can by searching for numerical patterns in many examples.
If we respect the time and attention of 20 year-olds by showing them directly why cross-multiplication works, why not figure out a way to respect the time and attention of 10 year-olds, and show them directly why cross-multiplication works?
Wu is quite clear that his monographs are for teachers, but I have had a great deal of success using the number line as he does to explain fractions to my boys. I haven't taught my boys to cross-multiply, but I have taught them to look for a common denominator when they are comparing fractions, or when they are trying to move from one (fractional) point (on the number line) to another (i.e. add and subtract fractions).
However, I am convinced by the example of my children that you can tell a compelling conceptual story to get a child to believe a procedure works, but he will forget the details of the conceptual story long before he forgets the procedure. Because we make him practice the procedure, not the story.
So-called "conceptual" curriculums like Investigations hold out the promise that children will learn the conceptual story by heart because they will write it for themselves, inducing the rule from many examples we show them without comment.
But a well-told conceptual story may be best.
English Basics
I stumbled across this series of what appear to be self-teaching (or nearly so) grammar books for 9-11 year olds while trying to figure out what the plural of rhonceros is. (rhinoceroses, apparently)
I like the cover so much I just may buy them.
Also, I love Cambridge Press.
English Basics 1 Practice & Revision; Table of Contents (pdf file)
English Basics 2 Practice & Revision; Table of Contents (pdf file)
English Basics 3 Practice & Revision; Table of Contents (pdf file)
educational opportunity
from No Child Left Behind and the Public Schools by Scott Franklin Abernathy, Chapter 5 Rethinking Assessment:
The book goes on to recommend that states not pursue value-added assessment, but focus instead on "devis[ing] a production model of educational assessment" that would involve developing "process indicators of school performance" and would "differ fundamentally from measuring outcomes."
To that I say, How about no.
I have no interest in production models and process indicators. Ditto "school performance" as distinct from student achievement.
Of course, that's because I am not an Assistant Professor of Political Science.
No.
I am a parent.
Your basic parent, 5 or 10 years into his or her public school experience, is interested in one indicator and one indicator only:
What has my child learned in school if anything?
Just to boil it down.
.......
He does have some cool stuff about shirking, though.
More later.
In education, the idea of focusing on the conditions of production has its roots in arguments in favor of developing "process indicators" of school performance, an approach that recognizes the fact that "schools provide educational opportunity; they do not directly produce learning."
The book goes on to recommend that states not pursue value-added assessment, but focus instead on "devis[ing] a production model of educational assessment" that would involve developing "process indicators of school performance" and would "differ fundamentally from measuring outcomes."
To that I say, How about no.
I have no interest in production models and process indicators. Ditto "school performance" as distinct from student achievement.
Of course, that's because I am not an Assistant Professor of Political Science.
No.
I am a parent.
Your basic parent, 5 or 10 years into his or her public school experience, is interested in one indicator and one indicator only:
What has my child learned in school if anything?
Just to boil it down.
.......
He does have some cool stuff about shirking, though.
More later.
new aphorism !
from the instructivist:
I'm adding that to my collection.
I must say, Instructivist has outdone himself. This will be a classic in the Annals of the Revolution.
Should such occur.
.
People will not earn a living, they'll learn a living.
I'm adding that to my collection.
I must say, Instructivist has outdone himself. This will be a classic in the Annals of the Revolution.
Should such occur.
.
know your edu-terminology
cognitive demand
I'm disheartened to say that I have, this morning, detected yet another obstacle to student achievement in our public schools: the doctrine of "cognitive demand." (pdf file)
The global economy requires that students be provided learning experiences in algebra, geometry, data representation, and statistics that are high in cognitive demand.
High in cognitive demand appears to mean, simply, that the material being enacted (news flash: curricula are now enacted, not taught) is hard for the students to do.
This is simply false.
Nearly 3 years into my math self-teaching project (3 years!) I am here to tell you that actually doing math, in the wake of routine memorization, is often far more difficult than understanding it.
Not that I have profound understanding of arithmetic and Algebra 1.
Nevertheless, I do frequently understand a concept fairly well only to discover (discover!) that I can't begin to use it in -- and forgive me for putting it this way -- real life.
next point
I am as certain as a human being can possibly be that the authors of this white paper would construe the sort of understanding I've been doing as not being high in cognitive load. If I can understand a concept better than I can do the concept, how can the concept be high in cognitive load?
answer: it can't
High cognitive load, by definition, comes from understanding, not doing. (Doing depends upon routine memorizing; otherwise you can't remember the steps). If a student is having a harder time with memorizing-doing than he is with understanding, the curriculum-enacter needs to crank up the level of difficulty of the latter.
Which our curricula and curriculum enacters have done!
I give up
I've been thinking about Robert Slavin's observation that ed schools used to focus on direct instruction and teacher effectiveness, once upon a time.
I have to assume he knows what he's talking about.
Which means.....ed schools used to focus on direct instruction and teacher effectiveness. Once upon a time.
Well, when was that time?
And how did it end?
And why was constructivism the next thing?
I'm thinking that the standards and accountability era, which we can provisionally date to 1983, the year that A Nation at Risk appeared, may be the explanation.
Here's the way I make it out (speaking as a person who is not a historian, obviously):
This last point has always been the sudden leap in logic that unhappy parents and mathematicians (and the National Research Council)* fail to grok.
There's no logical reason to move from the insight that people construct knowledge to the assertion that students must therefore discover knowledge. (And, as I believe instructivist once pointed out, if you're discovering knowledge you're not exactly constructing it, are you? I mean, if it's just lying around on tables and SMART Boards waiting to be discovered.....it's already been constructed. update: yes, instructivist made that very point. And if you have to construct knowledge, not just discover knowledge, then don't you need something to construct it out of? Like some facts or figures or findings or such?)
I remember learning, at Dartmouth, that people construct knowledge.
No one said a word about people therefore having to discover knowledge. That wasn't the point. The point was that when Person B listens to Person A, Person B will use the stuff inside his head - which is different from the stuff inside Person A's head - to make sense of it.
In other words: communication is Playing Telephone.
So is teaching.
There's no discovering.
There's just noise in the channel.
"Discovering" appears to be an ed school add-on.
it's a plot
So I'm thinking..... the invention of constructivism-as-discovery wasn't simply a mistake.
Constructivism-as-discovery was a defense.
The schools had always been about inputs, not outputs.** Then, suddenly, the public was demanding that educational inputs actually work.
Under the circumstances wouldn't it be helpful to learn that research in cognitive psychology proves it's not possible for teachers to teach?
what would a real historian say?
Ed has been in Brussels since Saturday, so I hadn't had a chance to run this by him.
But when he called today, I asked if this sort of thing is plausible -- is it the sort of explanation a real historian would come up with and be able to support?
He said that, actually, it pretty much is.
He himself (and remember: he headed the California History-Social Science Project back in the early 90s) had attributed the advent of constructivist doctrine to A Nation at Risk, but for a different reason. He had always assumed that professors of education reacted to educational decline by concluding that the traditional methods didn't work.
But he thinks that in fact my hypothesis could be part of it (meaning you could find evidence to support it if you looked).
Of course, I'm not going to go out and find evidence; I'm just going to assume (provisionally) that it's true.
Which is why I give up.
Other people's wrong ideas are one thing.
Other people's defense mechanisms are quite another. You can't talk a person out of a defense mechanism.
If constructivism is in part or in whole a means of defense against the standards and accountability movement .... then parents are going to have to figure out how to teach their kids math themselves.
Math and everything else under the sun.
* A common misconception regarding “constructivist” theories of knowing (that existing knowledge is used to build new knowledge) is that teachers should never tell students anything directly but, instead, should always allow them to construct knowledge for themselves. This perspective confuses a theory of pedagogy (teaching) with a theory of knowing. Constructivists assume that all knowledge is constructed from previous knowledge, irrespective of how one is taught (e.g., Cobb, 1994) —even listening to a lecture involves active attempts to construct new knowledge.
National Research Council
** Clowes: So Project Follow Through confirmed what you had already found about the ineffectiveness of those other programs. Yet those programs still are being promoted in teacher colleges and they still are widely used, while Direct Instruction is not. Why?
Engelmann: The answer is really simple, but it's very difficult for most people to accept: Outcomes have never been a priority in public education, from its inception. That's the way the public education system is. The system is more concerned with the experience of the child: "Let the child explore," "Let the child be his or her self," "Don't interfere with the natural learning process," and so on.
Engelmann, interview
I'm disheartened to say that I have, this morning, detected yet another obstacle to student achievement in our public schools: the doctrine of "cognitive demand." (pdf file)
The global economy requires that students be provided learning experiences in algebra, geometry, data representation, and statistics that are high in cognitive demand.
High in cognitive demand appears to mean, simply, that the material being enacted (news flash: curricula are now enacted, not taught) is hard for the students to do.
The term “cognitive demand” is used in two ways to describe learning opportunities. The first way is linked with curriculum policy and students’ course-taking options — how much math and which courses. The second way relates to how much thinking is called for in the classroom. Routine memorization involves low cognitive demand, no matter how advanced the content. Understanding mathematical concepts involves high cognitive demand, even for basic content. Both types of cognitive demand are associated with student performance on achievement tests, but they are not substitutes for each other.
This is simply false.
Nearly 3 years into my math self-teaching project (3 years!) I am here to tell you that actually doing math, in the wake of routine memorization, is often far more difficult than understanding it.
Not that I have profound understanding of arithmetic and Algebra 1.
Nevertheless, I do frequently understand a concept fairly well only to discover (discover!) that I can't begin to use it in -- and forgive me for putting it this way -- real life.
next point
I am as certain as a human being can possibly be that the authors of this white paper would construe the sort of understanding I've been doing as not being high in cognitive load. If I can understand a concept better than I can do the concept, how can the concept be high in cognitive load?
answer: it can't
High cognitive load, by definition, comes from understanding, not doing. (Doing depends upon routine memorizing; otherwise you can't remember the steps). If a student is having a harder time with memorizing-doing than he is with understanding, the curriculum-enacter needs to crank up the level of difficulty of the latter.
Which our curricula and curriculum enacters have done!
I give up
I've been thinking about Robert Slavin's observation that ed schools used to focus on direct instruction and teacher effectiveness, once upon a time.
I have to assume he knows what he's talking about.
Which means.....ed schools used to focus on direct instruction and teacher effectiveness. Once upon a time.
Well, when was that time?
And how did it end?
And why was constructivism the next thing?
I'm thinking that the standards and accountability era, which we can provisionally date to 1983, the year that A Nation at Risk appeared, may be the explanation.
Here's the way I make it out (speaking as a person who is not a historian, obviously):
- prior to 1983: public schools a mess, achievement in decline
- after 1983: public clamor for standards and accountability
- ed school response: knowledge can't be "transmitted" from teacher to student, and cognitive psychology proves it: people don't "receive" knowledge but, rather, construct knowledge
- therefore: students must discover their own knowledge
This last point has always been the sudden leap in logic that unhappy parents and mathematicians (and the National Research Council)* fail to grok.
There's no logical reason to move from the insight that people construct knowledge to the assertion that students must therefore discover knowledge. (And, as I believe instructivist once pointed out, if you're discovering knowledge you're not exactly constructing it, are you? I mean, if it's just lying around on tables and SMART Boards waiting to be discovered.....it's already been constructed. update: yes, instructivist made that very point. And if you have to construct knowledge, not just discover knowledge, then don't you need something to construct it out of? Like some facts or figures or findings or such?)
I remember learning, at Dartmouth, that people construct knowledge.
No one said a word about people therefore having to discover knowledge. That wasn't the point. The point was that when Person B listens to Person A, Person B will use the stuff inside his head - which is different from the stuff inside Person A's head - to make sense of it.
In other words: communication is Playing Telephone.
So is teaching.
There's no discovering.
There's just noise in the channel.
"Discovering" appears to be an ed school add-on.
it's a plot
So I'm thinking..... the invention of constructivism-as-discovery wasn't simply a mistake.
Constructivism-as-discovery was a defense.
The schools had always been about inputs, not outputs.** Then, suddenly, the public was demanding that educational inputs actually work.
Under the circumstances wouldn't it be helpful to learn that research in cognitive psychology proves it's not possible for teachers to teach?
what would a real historian say?
Ed has been in Brussels since Saturday, so I hadn't had a chance to run this by him.
But when he called today, I asked if this sort of thing is plausible -- is it the sort of explanation a real historian would come up with and be able to support?
He said that, actually, it pretty much is.
He himself (and remember: he headed the California History-Social Science Project back in the early 90s) had attributed the advent of constructivist doctrine to A Nation at Risk, but for a different reason. He had always assumed that professors of education reacted to educational decline by concluding that the traditional methods didn't work.
But he thinks that in fact my hypothesis could be part of it (meaning you could find evidence to support it if you looked).
Of course, I'm not going to go out and find evidence; I'm just going to assume (provisionally) that it's true.
Which is why I give up.
Other people's wrong ideas are one thing.
Other people's defense mechanisms are quite another. You can't talk a person out of a defense mechanism.
If constructivism is in part or in whole a means of defense against the standards and accountability movement .... then parents are going to have to figure out how to teach their kids math themselves.
Math and everything else under the sun.
* A common misconception regarding “constructivist” theories of knowing (that existing knowledge is used to build new knowledge) is that teachers should never tell students anything directly but, instead, should always allow them to construct knowledge for themselves. This perspective confuses a theory of pedagogy (teaching) with a theory of knowing. Constructivists assume that all knowledge is constructed from previous knowledge, irrespective of how one is taught (e.g., Cobb, 1994) —even listening to a lecture involves active attempts to construct new knowledge.
National Research Council
** Clowes: So Project Follow Through confirmed what you had already found about the ineffectiveness of those other programs. Yet those programs still are being promoted in teacher colleges and they still are widely used, while Direct Instruction is not. Why?
Engelmann: The answer is really simple, but it's very difficult for most people to accept: Outcomes have never been a priority in public education, from its inception. That's the way the public education system is. The system is more concerned with the experience of the child: "Let the child explore," "Let the child be his or her self," "Don't interfere with the natural learning process," and so on.
Engelmann, interview
learning experiences needed
Another illustration of the principle that if you listen people will tell you what they mean:
Learning experiences in algebra, geometry, data representation, and statistics.
Oh yeah, that'll do it.
update from Lynn G
C. (grade 7) had to stay in over lunch this week to finish coloring his math project.
In our global economy and democratic society, limiting math education to select students is unacceptable. A recent ACT study provides evidence that college and the workforce require the same levels of readiness in mathematics. One implication: All students require a greater level of “cognitive demand” in mathematics than once was considered appropriate. In other words, high school students need learning experiences in algebra, geometry, data representation, and statistics whether they are planning to enter college or workforce training programs.4
source:
AERA Research Points bulletin (pdf file)
Learning experiences in algebra, geometry, data representation, and statistics.
Oh yeah, that'll do it.
update from Lynn G
Well, this explains why my 9th grader has spent hours and hours drawing, then cutting, glueing, and pasting together a model home of his own design for high school geometry class.
You might expect that he was taught foundational skills for this project (how to use an architectural scale, for example). You might expect that simple architectural and engineering concepts might come up (you know, like load bearing walls, span, stuff like that). You would be wrong.
No, this is an imaginary home, a real world project, that is completely divorced from reality.
But it consumes a tremendous effort, it also consumes instructional time when kids could learn how to do a proof or do geometry.
The amount of geometry used to make the model is on the level of a 4th grader.
But he's getting "learning experiences."
I'm so relieved.
C. (grade 7) had to stay in over lunch this week to finish coloring his math project.
Parents called "Extremists" in Utah
A Superintendent of a Utah school district called parents that opposed TERC Investigations "Extremists" at a city council meeting. TERC has been used since 2000 at these 46 schools. Parental objections appear to have grown to a deafening roar over the years. Now the School system is backing off of TERC, allowing it to "supplement" and giving schools the option of using a more traditional approach. It sounds like different schools will be able to choose different programs or a combination of text books.
I wonder how much they pay their Superintendent? Has he never heard that ad hominem attacks are the weakest defense one could muster?
I wonder how much they pay their Superintendent? Has he never heard that ad hominem attacks are the weakest defense one could muster?
Monday, April 23, 2007
Sunday, April 22, 2007
curriculum map
I've been hankering after a curriculum map here in IUFSD.
A curriculum map, a scope and sequence, a topic matrix, a handbook, a course syllabus -- anything of that nature.
I want one.
But when Google Images tossed this up under the heading "curriculum map," I had pause.
I mean, suppose I spend the next 6 months hammering the district, insisting that what we parents need to make the preteaching/reteaching Phase 4 life bearable, is a curriculum map.
And then suppose we actually get one.
And it looks like this.
Or like this.
That would be bad.
Tonight's Everyday Math Assignment
I looked at my fifth grade son's EM homework for the night on adding and subtracting fractions. (They are way behind on their second book, so other fifth grade parents might have seen this before.)
This is a sample.
1/3 + 1/2 = ?
1/4 + 1/5 = ?
The answer to the first is 5/6
The answer to the second is 9/20
Explain what pattern you see.
"Pattern?", you say? Well, don't feel bad. I had to figure out what on earth they were talking about. I finally figured it out. To get the numerator of the answer, you add the two denominators. To get the denominator of the answer, you multiply the denominators. What we have here is rote pattern work that masquerades as understanding.
Raise your hand if you see half of the kids applying this pattern to:
1/4 + 2/5 = ?
OK. How about a different pattern for subtracting fractions, like
1/4 - 1/5 = ?
1/2 - 1/3 = ?
The numerator of the answer is always 1 and the denominator of the answer you get by multiplying the denominators.
They conveniently ignore problems like 1/5 - 1/4 = ?, and, of course, any other problem where the numerator is not 1.
What other patterns does EM need to cover all of the bases of adding and subtracting fractions?
I had to spend some time making sure that my son knows the formal rules for adding, subtracting, multiplying, and dividing fractions. I want his understanding to be based on general, mathematical rules, not stupid, error-causing patterns. I wonder what my son thinks when even his teacher tells the students that she is teaching them a method from EM because she has to.
Whenever anyone talks about mastery versus understanding, just think of these patterns. They are classic and show exactly what the fuzzies mean by understanding. When they talk of understanding, they are not talking about mathematical understanding. They are not talking about any sort of understanding that will help students when they get to algebra.
Talk of balance or understanding gives them credibility that is completely undeserved. We need to talk about competence, not philosophy.
This is a sample.
1/3 + 1/2 = ?
1/4 + 1/5 = ?
The answer to the first is 5/6
The answer to the second is 9/20
Explain what pattern you see.
"Pattern?", you say? Well, don't feel bad. I had to figure out what on earth they were talking about. I finally figured it out. To get the numerator of the answer, you add the two denominators. To get the denominator of the answer, you multiply the denominators. What we have here is rote pattern work that masquerades as understanding.
Raise your hand if you see half of the kids applying this pattern to:
1/4 + 2/5 = ?
OK. How about a different pattern for subtracting fractions, like
1/4 - 1/5 = ?
1/2 - 1/3 = ?
The numerator of the answer is always 1 and the denominator of the answer you get by multiplying the denominators.
They conveniently ignore problems like 1/5 - 1/4 = ?, and, of course, any other problem where the numerator is not 1.
What other patterns does EM need to cover all of the bases of adding and subtracting fractions?
I had to spend some time making sure that my son knows the formal rules for adding, subtracting, multiplying, and dividing fractions. I want his understanding to be based on general, mathematical rules, not stupid, error-causing patterns. I wonder what my son thinks when even his teacher tells the students that she is teaching them a method from EM because she has to.
Whenever anyone talks about mastery versus understanding, just think of these patterns. They are classic and show exactly what the fuzzies mean by understanding. When they talk of understanding, they are not talking about mathematical understanding. They are not talking about any sort of understanding that will help students when they get to algebra.
Talk of balance or understanding gives them credibility that is completely undeserved. We need to talk about competence, not philosophy.
good teachers
I can't wait for value-added assessment to come to Irvington:
bad teachers lower scores
"additive and cumulative":
In Sanders' words:
Elsewhere Sanders shows that novice teachers aren't as good as experienced teachers.
(surprise!)
Especially in math. (t/k)
Irvington is hiring novice teachers.
Novice math teachers, lots of them.
I'm now hearing that it's not just the Phase 4 kids being tutored.*
It's the Phase 3 kids, too.
These are bright kids who shouldn't be in Phase 3 to begin with; they belong in Phase 4.
They're getting tutored in Phase 3.
I heard this weekend that the legendary middle school math teacher, who retired at the end of last year thus dooming C. and his cohort to a second year in a row with a teacher who'd failed to teach them effectively in their first year of "accelerated" math, is now tutoring Phase 3 kids.
Fee?
One hundred bucks an hour.
So I'm told.
So instead of offering this teacher a bonus to stay on the job, we sent her off into retirement, hired brand-new novice teachers at greatly-reduced salaries, and threw them into the classroom with no one on hand to mentor, train, or evaluate them.
Et voila.
I'm going into my 3rd summer in a row teaching math at home.
On the other hand, seeing as how I'm entering my Happy Spring phase, part of me is thinking: wow!
This is a kid who has now had 3 years of extremely "ineffective" math teaching (in the 4th, 6th, and 7th grades) and he's still in the game. It's not unusual for him to correct me on a concept or calculation when we're working together; when this happens, typically he's right and I'm wrong.
Next year C. is almost certain to have one of the district's best math teachers. It's entirely possible he's going to show up for class well-prepared.
It's going to be interesting.
* Phase 4: "accelerated math" (algebra in 8th grad); Phase 3: "regular" math
**We are reliably told that the middle school teachers like the new principal because he "leaves them alone." I'm sure this is true; we've witnessed firsthand astonishingly insubordinate behavior on the part of teachers towards the principal.
One study concluded that having a highly effective teacher rather than a teacher of average effectiveness would result in two additional months of academic achievement for a student.
source:
Sanders, W., Rivers, J.C. (1996). Cumulative and Residual Effects of Teachers on Students’ Future Achievement. Knoxville, TN: University of Tennessee Value-Added Research Center)
cited by:
Teachers Matter: Evidence from Value-Added Assessments
Research Points Summer 2004 Vol. 2, Issue 2
bad teachers lower scores
Summary of Findings
- Differences in student achievement of 50 percentile points were observed as a result of teacher sequence after only three years.
- The effects of teachers on student achievement are both additive and cumulative with little evidence of compensatory effects.
- As teacher effectiveness increases, lower achieving students are the first to benefit. The top quintile of teachers facilitate appropriate to excellent gains for students of all achievement levels.
- Students of different ethnicities respond equivalently within the same quintile of teacher effectiveness.
"additive and cumulative":
"According to this new study ... bad teachers lower the test scores of their students, and this lowered achievement carries over into higher grades even if the students are subsequently given good teachers." Heartland Institute
In Sanders' words:
[I]t was possible to determine whether teachers from previous grades affected current year scores....[T]he teacher effects are cumulative and additive with very little, if any, suggestion of compensatory effects. An effective teacher receiving students from a relatively ineffective teacher can facilitate excellent academic gain for his/her students during the school year. Yet these analyses suggest that the residual effects of relatively ineffective teachers - from prior years can be measured in subsequent student achievement scores.
Elsewhere Sanders shows that novice teachers aren't as good as experienced teachers.
(surprise!)
Especially in math. (t/k)
Irvington is hiring novice teachers.
Novice math teachers, lots of them.
I'm now hearing that it's not just the Phase 4 kids being tutored.*
It's the Phase 3 kids, too.
These are bright kids who shouldn't be in Phase 3 to begin with; they belong in Phase 4.
They're getting tutored in Phase 3.
I heard this weekend that the legendary middle school math teacher, who retired at the end of last year thus dooming C. and his cohort to a second year in a row with a teacher who'd failed to teach them effectively in their first year of "accelerated" math, is now tutoring Phase 3 kids.
Fee?
One hundred bucks an hour.
So I'm told.
So instead of offering this teacher a bonus to stay on the job, we sent her off into retirement, hired brand-new novice teachers at greatly-reduced salaries, and threw them into the classroom with no one on hand to mentor, train, or evaluate them.
Et voila.
I'm going into my 3rd summer in a row teaching math at home.
On the other hand, seeing as how I'm entering my Happy Spring phase, part of me is thinking: wow!
This is a kid who has now had 3 years of extremely "ineffective" math teaching (in the 4th, 6th, and 7th grades) and he's still in the game. It's not unusual for him to correct me on a concept or calculation when we're working together; when this happens, typically he's right and I'm wrong.
Next year C. is almost certain to have one of the district's best math teachers. It's entirely possible he's going to show up for class well-prepared.
It's going to be interesting.
* Phase 4: "accelerated math" (algebra in 8th grad); Phase 3: "regular" math
**We are reliably told that the middle school teachers like the new principal because he "leaves them alone." I'm sure this is true; we've witnessed firsthand astonishingly insubordinate behavior on the part of teachers towards the principal.
parents are free!
I just noticed this comment on the Post-It note:
School motto:
Irvington Union Free School District
Where policies and practices are shrouded in mystery
Reminds me of the elementary school principal at our school who said, "If parents don't think we're teaching their kids the right things in the curriculum, they should feel free to teach it at home." Oh, thanks for the permission. (And this was before parents could even find any information on the school's curriculum online -- anywhere else.) It IS always worse than you think.Of course, we here in Irvington still can't find anything about the curriculum online.
School motto:
Irvington Union Free School District
Where policies and practices are shrouded in mystery