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17 Jan 2006 - 23:29
Engelmann's Student-Program Alignment and Teaching to Mastery is still rumbling through my Hebbian networks, toppling every domino in its path. It's kind of fun. I'm experiencing my very own Paradigm Shift. I don't know where I'll be when things calm down, but one thing I do know: I'm never going to see 'gaps' the same way.
killer Gaps We're constantly hearing about Gaps, of course. Achievement gaps, learning gaps, teacher gaps — everywhere you turn, there's another Gap. I've read so much about Gaps I never really stopped to think what a gap actually is, or might be. I guess I've thought of gaps as static and predictable. All the gaps seem to grow wider over time, until they look like an ice cream cone lying on its side in a PowerPoint slide. That was then. Suddenly, gaps seem dynamic, dark, and entirely unpredictable — more properly a phenomenon belonging to Chaos Theory (does anyone talk about Chaos Theory any more?), not Excel charts.
Anne on diagnosing Gaps
Susan J on diagnosing Gaps
here's what I'm wondering Although I believe that the gap between our kids and kids in high achieving countries starts in first grade or thereabouts, I do trust research showing that achievement slows in middle school. (This finding may not be confirmed, but at the moment I take it as probably true.) Here's what I wonder. When you don't teach to mastery — when you teach a spiraling curriculum — kids end up with gaps. That much we know. But kids probably don't all end up with the same gaps, except for the Universal American Fraction-Decimal-And-Percent gap. So think what a middle- or high-school math teacher is up against. Ninety or more kids, each with different gaps affecting different areas of the new content they're supposed to be learning and/or spiraling. It's Gap Anarchy. At the moment, it seems logical that the further you go, and the more gaps you accumulate, the slower your learning curve is going to be, until finally you hit the wall. I don't know whether that's true, but it seems logical. More than logical. It seems inevitable.
what do we know about learning gaps & how they work? Here's Engelmann:
Here he's talking about the case of a student having learned the wrong thing, rather than merely having failed to learn the right thing. The news is bad. What else do we know about gaps? Or about reteaching? And what have your own experiences been? I'd love to hear.
key worsd: gapology
James Milgram on long division & time
can you cram math: learning a year of math in 2 months
overlearning
remediating Los Angeles algebra students
Inflexible Knowledge: The First Step to Expertise by Daniel Willingham
Matt Goff & Susan S on remediating gaps
Anne Dwyer on diagnosing gaps & request for 'gap' stories
formative assessment and Richard Nixon
Terminator
-- CatherineJohnson - 17 Jan 2006 Back to main page.
Please consider registering as a regular user.
Look here for syntax help.
Here he's talking about the case of a student having learned the wrong thing, rather than merely having failed to learn the right thing. The news is bad. This is the big unintended consequence of discovery learning and contructivism -- kids frequently discover and construct the wrong stuff during the learning process. As Engelmann points out, once learned incorrectly, it is very difficult to unlearn before learning the right stuff. -- KDeRosa - 18 Jan 2006
If it took thousands of years of effort by hundreds of really smart people to create efficient ways of solving problems, why would anyone expect an average kid to create a similarly efficient method in 12 years of primary and secondary school? -- DougSundseth - 18 Jan 2006
"Ninety or more kids, each with different gaps affecting different areas of the new content they're supposed to be learning or spiraling." Maybe the question shouldn't be: What's your GPA? It should be: What's your GAP? -- CharlesH - 18 Jan 2006
Where do your kids go to school? At The Gap. I hear stories from parents who talk about a bad teacher and the "lost" year. That's a big gap. For my son in public school first grade, it wasn't a gap, it was a lost opportunity. A gap in learning. The lower school principal stated once that they just "feed" kids all sorts of stuff in the lower grades and don't expect to get anything back until 4th grade. That's a lot of potential gaps to fill. If you use a top down, real world, thematic, discovery approach to learning, there are going to be big gaps in basic foundational knowledge and skills. It's hard to construct your own knowledge if you start with big gaps. To constructivists, however, there is no problem because they think you can discover and fill in your own gaps. I don't think so. I like to start learning new subjects by building a basic framework in my head. That way I have something solid to attach new pieces of information. If you don't have a framework, then too much knowledge goes in one ear and out the other. I know that I like history much more now that I am older and have a much bigger framework. I have places to put new information. Many educators think this approach creates a limiting mental box. I think it creates a foundation. Superficial knowledge versus foundational knowledge. Facts, content, and skills are not limiting box factors, they are foundational. -- SteveH - 18 Jan 2006
"I like to start learning new subjects by building a basic framework in my head." That's exactly my experience. Heck, I'm a tech-writer, it's my job. First you build your framework, then you hang your bits in the appropriate places. Eventually you have a fairly solid wall of information, with few gaps to expose you to the cold winds of ignorance. (Hmm, stretched that metaphor too far, I think.) 8-) -- DougSundseth - 18 Jan 2006
My children have had two piano teachers. Each of them has stressed to the kids the importance of playing a piece correctly the first time. If they play a passage the wrong way one time, the thinking runs, they will have to play it correctly 20 (twenty) times to "relearn" the correct fingering. -- KtmGuest - 18 Jan 2006
"I like to start learning new subjects by building a basic framework in my head." Me too. However, this learning style, strategy, or pattern is not characteristic of the majority of students. According to this description of learning patterns, based on Meyers-Briggs, there are two quite different learning patterns: sensing and intuitive. http://www.virtualschool.edu/mon/Academia/KierseyLearningStyles.html The majority of college students and approximately 75 percent of the general public exhibit the sensing pattern, which is characterized "by a preference for direct, concrete experiences; moderate to high degrees of structure; linear, sequential learning; and, often, a need to know why before doing something. In general, students who prefer sensing learning patterns prefer the concrete, the practical, and the immediate. These students often lack confidence in their intellectual abilities and are uncomfortable with abstract ideas. They have difficulty with complex concepts and low tolerance for ambiguity." Moreover, these students, while equally intelligent, tend to do worse on tests. "Because sensing students take longer to read questions, often going over them several times, they seem to be disadvantaged on timed aptitude measures. Furthermore, the argument is often made that sensing intelligence cannot be measured by paper and pencil instruments, and that sensing students (especially extroverted sensors) are at a disadvantage on any timed examination that focuses on the ability to quickly manipulate symbols and see patterns in relationships between words and concepts." By contrast, only 25% of the general population and 40% of college students are "big picture" types whose "path to excellence is from theory to practice...." -- SusanJ - 18 Jan 2006
What you are referring to as "gaps," I've heard of as "pseudoconceptual behavior." That is, behavior that, at first glance, looks like it indicates an understanding of the material at hand, but really does not. An example is when people sometimes instinctively nod their heads in agreement when talking with someone without hearing what the person is saying. I first came across this notion in an essay called "Scenes from Linear Algebra Classes" by Shlomo Vinner, in the book "Resources for Teaching Linear Algebra," published by the MAA. -- PaulMiller - 18 Jan 2006
I hear stories from parents who talk about a bad teacher and the "lost" year. That's a big gap. For my son in public school first grade, it wasn't a gap, it was a lost opportunity. A gap in learning. Haven't read the thread yet, but Christopher lost pretty much an entire year of math in 4th grade. I think we're probably still dealing with it (I'd love to get one of you guys to take a look at him & give an opinion....) -- CatherineJohnson - 18 Jan 2006
The thing about 'closing' a gap is that you can't get the year back, I don't think. You can re-teach the missed material in the summer....but you still don't have the same 9 months for distributed practice. Of course, given the fact that everyone is on a spiraling curriculum, maybe the other kids with the good teachers don't have distributed practice, either. Don't know. I'm utterly fascinated by this topic now, though. -- CatherineJohnson - 18 Jan 2006
a story about (not) closing Christopher's gaps -- CatherineJohnson - 18 Jan 2006
Ken As Engelmann points out, once learned incorrectly, it is very difficult to unlearn before learning the right stuff. Exactly. This is why errorless learning is an important concept in stroke recovery. If you allow a person with brain damage to learn the wrong thing, they may never be able to correcct themselves. Normal kids have a lot more leeway — but otoh it's not infinite. Michael Jordan trying to switch from basketball to baseball is the same thing. There you have a brilliant athlete, who is an expert in basketball.....he can't switch to baseball. After years of practice, his brain is wired for basketball. Temple told me the same story about, I think, Kosher meatpacking plants. The plants decided to change the way they slaughtered cattle, in order to conform to standards for humane slaughter. The long-time employees simply could not learn the new way. It wasn't that they didn't want to. They just couldn't make their hands and bodies move in the new way. -- CatherineJohnson - 18 Jan 2006
Charles Maybe the question shouldn't be: What's your GPA? It should be: What's your GAP?
I LOVE IT!!!!!!! WIT AND WISDOM!!!!! -- CatherineJohnson - 18 Jan 2006
"First you build your framework, then you hang your bits in the appropriate places. Eventually you have a fairly solid wall of information, with few gaps to expose you to the cold winds of ignorance." I remember reading a book 35 years ago called "The Crack in the Cosmic Egg". It talked about concepts and precepts and the idea that (in so many words) the framework I mentioned is actually a biased filter that changes or blocks new information. The problem is that this filtering or blocking happens no matter what you do. The human mind is associative and new knowledge has to be attached to existing knowledge, biases (wrong knowledge) and all. Avoiding content and skills and focusing only on process will not avoid this problem. In math, the lack of knowledge and skills will just make learning more difficult. Generally, I don't like to talk like this. It is quite distracting. The problem is poor math curricula in K-8. Math in K-8 is not very difficult. It is cumulative and schools cannot let students progress without verifying grade-by-grade mastery of the material. And, all of this has to lead very carefully to a full course in algebra in eighth grade. This is not difficult. It may be difficult for students at times, but teachers are supposed to be trained to catch and prevent problems, not try to fix gaps later, if ever. Gaps occur because schools (not kids) are not doing this simple job. This is also not about homework and parents. Everything can be done in class. -- SteveH - 18 Jan 2006
girl with gumption: Irvington student closing a couple of years' worth of gaps in one summer -- CatherineJohnson - 18 Jan 2006
My children have had two piano teachers. Each of them has stressed to the kids the importance of playing a piece correctly the first time. If they play a passage the wrong way one time, the thinking runs, they will have to play it correctly 20 (twenty) times to "relearn" the correct fingering. wow WIT AND WISDOM & PULL UP FRONT (I have to get these key words in, or I'm never going to find all this stuff) -- CatherineJohnson - 18 Jan 2006
Steve I like to start learning new subjects by building a basic framework in my head. That way I have something solid to attach new pieces of information. If you don't have a framework, then too much knowledge goes in one ear and out the other. This is THE ENTIRE POINT of HOW TO DOUBLE YOUR CHILD'S GRADES. The author says children learn NOTHING by simply opening up a textbook and reading word-by-word. If they don't build a framework first, they get essentially nothing. Of course it makes sense when you think about it. They're young; they're in the process of building all the frameworks they'll use for the rest of their lives. You and I can sit down and start reading a textbook and at least get something out of it (in theory!), because we have all kinds of framework templates lodged inside our minds. Kids can't activate pre-existing frameworks. -- CatherineJohnson - 18 Jan 2006
Steve The lower school principal stated once that they just "feed" kids all sorts of stuff in the lower grades and don't expect to get anything back until 4th grade. These people need to go away. This is ludicrous. Shocking & ludicrous. No administrator or teacher here would ever say such a thing. These words just would not come out of anyone's mouth; these words wouldn't come into anyone's brain. -- CatherineJohnson - 18 Jan 2006
Where do your kids go to school? At The Gap. WIT AND WISDOM -- CatherineJohnson - 18 Jan 2006
"My children have had two piano teachers. Each of them has stressed to the kids the importance of playing a piece correctly the first time. If they play a passage the wrong way one time, the thinking runs, they will have to play it correctly 20 (twenty) times to "relearn" the correct fingering." My son has been playing the piano for three years and playing the piece slowly with the correct fingering the first time just won't happen. There are problems with wrong notes and fingering that have to be fixed. We are working on it. He also likes to play fast, then try to fix the mistakes, rather than play it slowly and correctly first. He also likes to learn things using both hands, rather than one hand at a time. Amazingly, it seems to work for him. I will learn a piece slowly and correctly, but will never be able to speed it up. He learns it fast and then fixes the problems. Then again, he can do things with his fingers that I will never be able to do. Fortunately, his teacher is not pedagogically rigid. My son does, however, have to do his scales and Hanon finger exercises. (none of this fuzzy stuff) I still think that even he would benefit by starting out more slowly and carefully. Preventing mistakes is easier than fixing them - up to a point. It can be quite frustrating being perfect before moving on. The goal of playing the piano is making music, not technical perfection or eliminating mistakes. -- SteveH - 18 Jan 2006
"The lower school principal stated once that they just "feed" kids all sorts of stuff in the lower grades and don't expect to get anything back until 4th grade." She did say this, but, in reality, this only reflects their philosophy. (of low expectations and developmentally appropriate learning) They do make sure kids can read. They do ensure that kids can do some math. They have spelling tests. The kids do learn things. (Grading, however, is limited to vague rubrics.) I think the philosophy behind this is reflected in another one of her comments. I don't remember it exactly, but it had to do with academic differences in kids in Kindergarten. She said that these differences go away by fourth grade. In other words, some kids can read in Kindergarten, but all kids can read by fourth grade. It's a very simplistic viewpoint. I felt like telling her that the kids all look the same because they all end up at the same low destination. Some of these people come out of Ed School and are just too sure of themselves. There is not a doubt in her head. -- SteveH - 18 Jan 2006
I like to start learning new subjects by building a basic framework in my head. That way I have something solid to attach new pieces of information. If you don't have a framework, then too much knowledge goes in one ear and out the other. I know that I like history much more now that I am older and have a much bigger framework. I have places to put new information. Many educators think this approach creates a limiting mental box. I think it creates a foundation. Superficial knowledge versus foundational knowledge. Facts, content, and skills are not limiting box factors, they are foundational. -- SteveH? - 18 Jan 2006 "First you build your framework, then you hang your bits in the appropriate places. Eventually you have a fairly solid wall of information, with few gaps to expose you to the cold winds of ignorance." SteveH?, are you channeling Susan Wise Bauer (co-author of "The Well-Trained Mind)this morning? You are describing the "grammar-stage" of a classical education (according to Mrs. Bauer). You don't have to read the entire 400 pages, just the first few chapters to get the idea. BTW, there are schools that claim to follow this "classical model"... -- NicksMama - 18 Jan 2006
"The goal of playing the piano is making music, not technical perfection or eliminating mistakes." Just another comment. Some kids seem to be overly focused on not making mistakes. They become perfectionists and will fall apart at the first mistake in concert. (or on a test) This doesn't happen for my son. We tell him that mistakes are normal. I just wish that he would make them less normal. Then again, he has no fear of getting up on stage and does well. His piano teacher, a concert pianist, talked about one recital as a child where he played a piece over his ability. There were many mistakes, but it didn't bother him one bit at the time. Now he says that he can't watch the tape - it's too awful. -- SteveH - 18 Jan 2006
sometimes gaps are good
-- CatherineJohnson - 18 Jan 2006
Here's my opinion on gaps. What is a mathematical gap? When a student gets an answer to a problem wrong. Even a careless error shows a gap. Some gaps are procedure based as when the student doesn't move the numbers to the tens places in multidigit multiplication. Some gaps are skills based as when Catherine got a problem wrong because she has memorized one of the times tables incorrectly. What causes a gap? In a sequentially taught mathematics program, students automatically get distributed practice. This is because practicing higher mathematical skills creates distributed practice of lower mathematical skills. The classic example is long division which requires the student to practice multiplication and subtraction. So if a student did have gaps in their procedure or basic skills, they would get the wrong answer on a long division problem. In order to correct the long division mistakes, the student has to correct the gaps in the preceding skills. This is why it is so important to practice all skills to mastery. What happens when gaps are not corrected by mastering skills? When gaps are not corrected, then they carry on to the higher skills. The higher skills required basic skills and procedures in a very specific order. By the time a student gets to algebra, it is very difficult to diagnose why they are getting a problem wrong. Different gaps may manifest themselve in different ways in different problems. The fix, too, is harder. Once you find what is wrong, you have to give the student distributed practice that isolates the gap. When is a gap fixed? Only when the student can do the problem by themselves without a model in front of them. -- AnneDwyer - 19 Jan 2006
"This is because practicing higher mathematical skills creates distributed practice of lower mathematical skills. The classic example is long division which requires the student to practice multiplication and subtraction." Great point! Long division is valuable for this reason alone. It also reinforces math facts and number sense. I am always amazed how many kids don't have number sense. They are all over the lot. -- CharlesH - 19 Jan 2006
New study indicates that even college students have massice gaps in their math knowledge. English too.
I'm having a little difficulty with this article. There's this statement:
I wonder if this is fallout from the "everyone should go to college" movement? -- GoogleMaster - 19 Jan 2006
If this study is in any way accurate, it is further proof that the college board had no business recentering the verbal scores back in 1995 and giving the average student an extra 80 points on the SAT-V. As innumerate as these kids are, they are far more illiterate. -- KDeRosa - 20 Jan 2006
Speaking of innumeracy and illiteracy, one of my indicators of how the local economy is going is the quality of help at the fast-food drive-thru window. When the economy is bad, you will find degreed engineers working at fast-food jobs. When the economy is good, those jobs are more likely to be staffed with the less educated, because the degreed engineers have moved on to jobs in their fields. So, when they start getting glassy-eyed upon receiving $5.12 for a $4.87 order, I know the economy is picking up. -- GoogleMaster - 20 Jan 2006
http://www.forbes.com/free_forbes/2005/1212/120.html i haven't read it because, well, it's unreadable: pop-ups and throbbing ads and whatnot. but it's called "gapology 101". i found it here at Chris Correa. -- VlorbikDotCom - 20 Jan 2006
Here's the same Forbes article without all of the ads: http://www.forbes.com/forbes/2005/1212/120_print.html -- GoogleMaster - 20 Jan 2006
So, when they start getting glassy-eyed upon receiving $5.12 for a $4.87 order, I know the economy is picking up. You are so right. But I can't say that that's a bad thing (that the economy is picking up -- it's of course a terrible thing that we graduate people who are so innumerate that they can't make change). -- CarolynJohnston - 20 Jan 2006
It's not only that they can't make change; it's that the concept of "getting rid of some pennies so I can get a quarter back instead of more pennies" never occurred to them. -- GoogleMaster - 20 Jan 2006
The economy must be booming. The Stooge at McD?'s screwed up my order -- not once, but twice -- today and almost forgot to put the toys in the kids' Happy Meals. This would have been very bad news for me had I not caught it. -- KDeRosa - 20 Jan 2006
Dan Seligman needs to spend some time reading research on genetics & IQ. This book is a good place to start:
Mind Sculpture
-- CatherineJohnson - 22 Jan 2006
Robertson includes some of the incredible research on IQ in black kids in the pre-Civil Rights south. The longer a black child stayed in school, the lower his IQ. Normally, the oldest child in a family has the highest IQ; the younger kids have slightly lower IQs. In these families, with children in these schools, the younger the child, the higher the IQ. When these families moved north, and put their kids in northern schools, IQs rose. Genes aren't rocks! The genes involved in intelligence, like nearly all genes (the exceptions are single-gene disorders like Huntington's) express in relation to the environment, which includes all components affecting genes, physical, social, intellectual, etc. ONE WORD, DAN: epigenetics -- CatherineJohnson - 22 Jan 2006
Actually, it looks like Chris Correa does a pretty good job explaining what is SO wrong with Seligman's article... -- CatherineJohnson - 22 Jan 2006
stories about gaps
Engelmann's Student-Program Alignment and Teaching to Mastery is still rumbling through my Hebbian networks, toppling every domino in its path. It's kind of fun. I'm experiencing my very own Paradigm Shift. I don't know where I'll be when things calm down, but one thing I do know: I'm never going to see 'gaps' the same way.
killer Gaps We're constantly hearing about Gaps, of course. Achievement gaps, learning gaps, teacher gaps — everywhere you turn, there's another Gap. I've read so much about Gaps I never really stopped to think what a gap actually is, or might be. I guess I've thought of gaps as static and predictable. All the gaps seem to grow wider over time, until they look like an ice cream cone lying on its side in a PowerPoint slide. That was then. Suddenly, gaps seem dynamic, dark, and entirely unpredictable — more properly a phenomenon belonging to Chaos Theory (does anyone talk about Chaos Theory any more?), not Excel charts.
Anne on diagnosing Gaps
What I've noticed with my tutoring students is this: if they don't understand something in math class, they try to find a procedure or "trick" that works everytime. Since they don't really understand it, when they have to go back and do it on a test or later, they don't remember the "trick" exactly and their answers are consistent, but wrong. For example, I was tutoring a student in basic math. He didn't really understand that a whole number has an implied decimal after the number (e.g. 3 is really 3. for a decimal problem) When he first learned to divide decimals and he was following the teachers examples, he was doing the problems right: So if he was dividing .045 into 15, he moved the decimal over three places for the .045 and three places for the 15. He even managed to get it right on the first test. But he did them wrong on every test after that. When we were studying for the final, I was able to watch him do the problems. Since he really didn't understand, he made up his own "trick". In the problem above, he would move the decimal over for the .045 correctly, but he put the decimal point in front of any number inside the divisor sign. So .045 into 15 became 45 into 150 instead of 15,000. And, because he had taught himself this trick, he ignored all decimal points inside the divisor sign. So even .045 into 1.5 became 45 into 150. Needless to say, it took a while to find the problem and then to correct it.
Susan J on diagnosing Gaps
I think it is very, very hard because it is so personal and unique to the student. I'm 65 and a computational scientist and I still remember odd and embarrassing gaps that had huge negative effects even in graduate school. Even when you get to the point where you are in charge of your own learning, you can miss these things. For the mathematicians on the site, I'll admit that it took me more time than it should have to understand that when one solves a differential equation, one is solving for an unknown function rather than a variable. I still remember puzzling over a textbook diagram of a simple mercury barometer when I was a freshman in college. The difficulty (for me) was that the diagram was simplified and didn't show the support stand for the glass tube with its closed end up and its bottom end part-way submerged in a dish of mercury. So I could never figure out why the tube simply didn't fall over!
here's what I'm wondering Although I believe that the gap between our kids and kids in high achieving countries starts in first grade or thereabouts, I do trust research showing that achievement slows in middle school. (This finding may not be confirmed, but at the moment I take it as probably true.) Here's what I wonder. When you don't teach to mastery — when you teach a spiraling curriculum — kids end up with gaps. That much we know. But kids probably don't all end up with the same gaps, except for the Universal American Fraction-Decimal-And-Percent gap. So think what a middle- or high-school math teacher is up against. Ninety or more kids, each with different gaps affecting different areas of the new content they're supposed to be learning and/or spiraling. It's Gap Anarchy. At the moment, it seems logical that the further you go, and the more gaps you accumulate, the slower your learning curve is going to be, until finally you hit the wall. I don't know whether that's true, but it seems logical. More than logical. It seems inevitable.
what do we know about learning gaps & how they work? Here's Engelmann:
When students are not taught to mastery, they often mislearn the skills and concepts the teacher attempts to teach. For instance, they may learn to guess at words in sentences. Reteaching them requires many more trials and much more work than that required to teach them to mastery initially. Initial teaching may require only 10 or fewer trials on some skills. Reteaching the same skill after students have mislearned it and have practiced inappropriate strategies for years may require several hundred trials.
Here he's talking about the case of a student having learned the wrong thing, rather than merely having failed to learn the right thing. The news is bad. What else do we know about gaps? Or about reteaching? And what have your own experiences been? I'd love to hear.
key worsd: gapology
James Milgram on long division & time
can you cram math: learning a year of math in 2 months
overlearning
remediating Los Angeles algebra students
Inflexible Knowledge: The First Step to Expertise by Daniel Willingham
Matt Goff & Susan S on remediating gaps
Anne Dwyer on diagnosing gaps & request for 'gap' stories
formative assessment and Richard Nixon
Terminator
-- CatherineJohnson - 17 Jan 2006 Back to main page.
Comments
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Look here for syntax help.
Here he's talking about the case of a student having learned the wrong thing, rather than merely having failed to learn the right thing. The news is bad. This is the big unintended consequence of discovery learning and contructivism -- kids frequently discover and construct the wrong stuff during the learning process. As Engelmann points out, once learned incorrectly, it is very difficult to unlearn before learning the right stuff. -- KDeRosa - 18 Jan 2006
If it took thousands of years of effort by hundreds of really smart people to create efficient ways of solving problems, why would anyone expect an average kid to create a similarly efficient method in 12 years of primary and secondary school? -- DougSundseth - 18 Jan 2006
"Ninety or more kids, each with different gaps affecting different areas of the new content they're supposed to be learning or spiraling." Maybe the question shouldn't be: What's your GPA? It should be: What's your GAP? -- CharlesH - 18 Jan 2006
Where do your kids go to school? At The Gap. I hear stories from parents who talk about a bad teacher and the "lost" year. That's a big gap. For my son in public school first grade, it wasn't a gap, it was a lost opportunity. A gap in learning. The lower school principal stated once that they just "feed" kids all sorts of stuff in the lower grades and don't expect to get anything back until 4th grade. That's a lot of potential gaps to fill. If you use a top down, real world, thematic, discovery approach to learning, there are going to be big gaps in basic foundational knowledge and skills. It's hard to construct your own knowledge if you start with big gaps. To constructivists, however, there is no problem because they think you can discover and fill in your own gaps. I don't think so. I like to start learning new subjects by building a basic framework in my head. That way I have something solid to attach new pieces of information. If you don't have a framework, then too much knowledge goes in one ear and out the other. I know that I like history much more now that I am older and have a much bigger framework. I have places to put new information. Many educators think this approach creates a limiting mental box. I think it creates a foundation. Superficial knowledge versus foundational knowledge. Facts, content, and skills are not limiting box factors, they are foundational. -- SteveH - 18 Jan 2006
"I like to start learning new subjects by building a basic framework in my head." That's exactly my experience. Heck, I'm a tech-writer, it's my job. First you build your framework, then you hang your bits in the appropriate places. Eventually you have a fairly solid wall of information, with few gaps to expose you to the cold winds of ignorance. (Hmm, stretched that metaphor too far, I think.) 8-) -- DougSundseth - 18 Jan 2006
My children have had two piano teachers. Each of them has stressed to the kids the importance of playing a piece correctly the first time. If they play a passage the wrong way one time, the thinking runs, they will have to play it correctly 20 (twenty) times to "relearn" the correct fingering. -- KtmGuest - 18 Jan 2006
"I like to start learning new subjects by building a basic framework in my head." Me too. However, this learning style, strategy, or pattern is not characteristic of the majority of students. According to this description of learning patterns, based on Meyers-Briggs, there are two quite different learning patterns: sensing and intuitive. http://www.virtualschool.edu/mon/Academia/KierseyLearningStyles.html The majority of college students and approximately 75 percent of the general public exhibit the sensing pattern, which is characterized "by a preference for direct, concrete experiences; moderate to high degrees of structure; linear, sequential learning; and, often, a need to know why before doing something. In general, students who prefer sensing learning patterns prefer the concrete, the practical, and the immediate. These students often lack confidence in their intellectual abilities and are uncomfortable with abstract ideas. They have difficulty with complex concepts and low tolerance for ambiguity." Moreover, these students, while equally intelligent, tend to do worse on tests. "Because sensing students take longer to read questions, often going over them several times, they seem to be disadvantaged on timed aptitude measures. Furthermore, the argument is often made that sensing intelligence cannot be measured by paper and pencil instruments, and that sensing students (especially extroverted sensors) are at a disadvantage on any timed examination that focuses on the ability to quickly manipulate symbols and see patterns in relationships between words and concepts." By contrast, only 25% of the general population and 40% of college students are "big picture" types whose "path to excellence is from theory to practice...." -- SusanJ - 18 Jan 2006
What you are referring to as "gaps," I've heard of as "pseudoconceptual behavior." That is, behavior that, at first glance, looks like it indicates an understanding of the material at hand, but really does not. An example is when people sometimes instinctively nod their heads in agreement when talking with someone without hearing what the person is saying. I first came across this notion in an essay called "Scenes from Linear Algebra Classes" by Shlomo Vinner, in the book "Resources for Teaching Linear Algebra," published by the MAA. -- PaulMiller - 18 Jan 2006
I hear stories from parents who talk about a bad teacher and the "lost" year. That's a big gap. For my son in public school first grade, it wasn't a gap, it was a lost opportunity. A gap in learning. Haven't read the thread yet, but Christopher lost pretty much an entire year of math in 4th grade. I think we're probably still dealing with it (I'd love to get one of you guys to take a look at him & give an opinion....) -- CatherineJohnson - 18 Jan 2006
The thing about 'closing' a gap is that you can't get the year back, I don't think. You can re-teach the missed material in the summer....but you still don't have the same 9 months for distributed practice. Of course, given the fact that everyone is on a spiraling curriculum, maybe the other kids with the good teachers don't have distributed practice, either. Don't know. I'm utterly fascinated by this topic now, though. -- CatherineJohnson - 18 Jan 2006
a story about (not) closing Christopher's gaps -- CatherineJohnson - 18 Jan 2006
Ken As Engelmann points out, once learned incorrectly, it is very difficult to unlearn before learning the right stuff. Exactly. This is why errorless learning is an important concept in stroke recovery. If you allow a person with brain damage to learn the wrong thing, they may never be able to correcct themselves. Normal kids have a lot more leeway — but otoh it's not infinite. Michael Jordan trying to switch from basketball to baseball is the same thing. There you have a brilliant athlete, who is an expert in basketball.....he can't switch to baseball. After years of practice, his brain is wired for basketball. Temple told me the same story about, I think, Kosher meatpacking plants. The plants decided to change the way they slaughtered cattle, in order to conform to standards for humane slaughter. The long-time employees simply could not learn the new way. It wasn't that they didn't want to. They just couldn't make their hands and bodies move in the new way. -- CatherineJohnson - 18 Jan 2006
Charles Maybe the question shouldn't be: What's your GPA? It should be: What's your GAP?
I LOVE IT!!!!!!! WIT AND WISDOM!!!!! -- CatherineJohnson - 18 Jan 2006
"First you build your framework, then you hang your bits in the appropriate places. Eventually you have a fairly solid wall of information, with few gaps to expose you to the cold winds of ignorance." I remember reading a book 35 years ago called "The Crack in the Cosmic Egg". It talked about concepts and precepts and the idea that (in so many words) the framework I mentioned is actually a biased filter that changes or blocks new information. The problem is that this filtering or blocking happens no matter what you do. The human mind is associative and new knowledge has to be attached to existing knowledge, biases (wrong knowledge) and all. Avoiding content and skills and focusing only on process will not avoid this problem. In math, the lack of knowledge and skills will just make learning more difficult. Generally, I don't like to talk like this. It is quite distracting. The problem is poor math curricula in K-8. Math in K-8 is not very difficult. It is cumulative and schools cannot let students progress without verifying grade-by-grade mastery of the material. And, all of this has to lead very carefully to a full course in algebra in eighth grade. This is not difficult. It may be difficult for students at times, but teachers are supposed to be trained to catch and prevent problems, not try to fix gaps later, if ever. Gaps occur because schools (not kids) are not doing this simple job. This is also not about homework and parents. Everything can be done in class. -- SteveH - 18 Jan 2006
girl with gumption: Irvington student closing a couple of years' worth of gaps in one summer -- CatherineJohnson - 18 Jan 2006
My children have had two piano teachers. Each of them has stressed to the kids the importance of playing a piece correctly the first time. If they play a passage the wrong way one time, the thinking runs, they will have to play it correctly 20 (twenty) times to "relearn" the correct fingering. wow WIT AND WISDOM & PULL UP FRONT (I have to get these key words in, or I'm never going to find all this stuff) -- CatherineJohnson - 18 Jan 2006
Steve I like to start learning new subjects by building a basic framework in my head. That way I have something solid to attach new pieces of information. If you don't have a framework, then too much knowledge goes in one ear and out the other. This is THE ENTIRE POINT of HOW TO DOUBLE YOUR CHILD'S GRADES. The author says children learn NOTHING by simply opening up a textbook and reading word-by-word. If they don't build a framework first, they get essentially nothing. Of course it makes sense when you think about it. They're young; they're in the process of building all the frameworks they'll use for the rest of their lives. You and I can sit down and start reading a textbook and at least get something out of it (in theory!), because we have all kinds of framework templates lodged inside our minds. Kids can't activate pre-existing frameworks. -- CatherineJohnson - 18 Jan 2006
Steve The lower school principal stated once that they just "feed" kids all sorts of stuff in the lower grades and don't expect to get anything back until 4th grade. These people need to go away. This is ludicrous. Shocking & ludicrous. No administrator or teacher here would ever say such a thing. These words just would not come out of anyone's mouth; these words wouldn't come into anyone's brain. -- CatherineJohnson - 18 Jan 2006
Where do your kids go to school? At The Gap. WIT AND WISDOM -- CatherineJohnson - 18 Jan 2006
"My children have had two piano teachers. Each of them has stressed to the kids the importance of playing a piece correctly the first time. If they play a passage the wrong way one time, the thinking runs, they will have to play it correctly 20 (twenty) times to "relearn" the correct fingering." My son has been playing the piano for three years and playing the piece slowly with the correct fingering the first time just won't happen. There are problems with wrong notes and fingering that have to be fixed. We are working on it. He also likes to play fast, then try to fix the mistakes, rather than play it slowly and correctly first. He also likes to learn things using both hands, rather than one hand at a time. Amazingly, it seems to work for him. I will learn a piece slowly and correctly, but will never be able to speed it up. He learns it fast and then fixes the problems. Then again, he can do things with his fingers that I will never be able to do. Fortunately, his teacher is not pedagogically rigid. My son does, however, have to do his scales and Hanon finger exercises. (none of this fuzzy stuff) I still think that even he would benefit by starting out more slowly and carefully. Preventing mistakes is easier than fixing them - up to a point. It can be quite frustrating being perfect before moving on. The goal of playing the piano is making music, not technical perfection or eliminating mistakes. -- SteveH - 18 Jan 2006
"The lower school principal stated once that they just "feed" kids all sorts of stuff in the lower grades and don't expect to get anything back until 4th grade." She did say this, but, in reality, this only reflects their philosophy. (of low expectations and developmentally appropriate learning) They do make sure kids can read. They do ensure that kids can do some math. They have spelling tests. The kids do learn things. (Grading, however, is limited to vague rubrics.) I think the philosophy behind this is reflected in another one of her comments. I don't remember it exactly, but it had to do with academic differences in kids in Kindergarten. She said that these differences go away by fourth grade. In other words, some kids can read in Kindergarten, but all kids can read by fourth grade. It's a very simplistic viewpoint. I felt like telling her that the kids all look the same because they all end up at the same low destination. Some of these people come out of Ed School and are just too sure of themselves. There is not a doubt in her head. -- SteveH - 18 Jan 2006
I like to start learning new subjects by building a basic framework in my head. That way I have something solid to attach new pieces of information. If you don't have a framework, then too much knowledge goes in one ear and out the other. I know that I like history much more now that I am older and have a much bigger framework. I have places to put new information. Many educators think this approach creates a limiting mental box. I think it creates a foundation. Superficial knowledge versus foundational knowledge. Facts, content, and skills are not limiting box factors, they are foundational. -- SteveH? - 18 Jan 2006 "First you build your framework, then you hang your bits in the appropriate places. Eventually you have a fairly solid wall of information, with few gaps to expose you to the cold winds of ignorance." SteveH?, are you channeling Susan Wise Bauer (co-author of "The Well-Trained Mind)this morning? You are describing the "grammar-stage" of a classical education (according to Mrs. Bauer). You don't have to read the entire 400 pages, just the first few chapters to get the idea. BTW, there are schools that claim to follow this "classical model"... -- NicksMama - 18 Jan 2006
"The goal of playing the piano is making music, not technical perfection or eliminating mistakes." Just another comment. Some kids seem to be overly focused on not making mistakes. They become perfectionists and will fall apart at the first mistake in concert. (or on a test) This doesn't happen for my son. We tell him that mistakes are normal. I just wish that he would make them less normal. Then again, he has no fear of getting up on stage and does well. His piano teacher, a concert pianist, talked about one recital as a child where he played a piece over his ability. There were many mistakes, but it didn't bother him one bit at the time. Now he says that he can't watch the tape - it's too awful. -- SteveH - 18 Jan 2006
sometimes gaps are good
-- CatherineJohnson - 18 Jan 2006
Here's my opinion on gaps. What is a mathematical gap? When a student gets an answer to a problem wrong. Even a careless error shows a gap. Some gaps are procedure based as when the student doesn't move the numbers to the tens places in multidigit multiplication. Some gaps are skills based as when Catherine got a problem wrong because she has memorized one of the times tables incorrectly. What causes a gap? In a sequentially taught mathematics program, students automatically get distributed practice. This is because practicing higher mathematical skills creates distributed practice of lower mathematical skills. The classic example is long division which requires the student to practice multiplication and subtraction. So if a student did have gaps in their procedure or basic skills, they would get the wrong answer on a long division problem. In order to correct the long division mistakes, the student has to correct the gaps in the preceding skills. This is why it is so important to practice all skills to mastery. What happens when gaps are not corrected by mastering skills? When gaps are not corrected, then they carry on to the higher skills. The higher skills required basic skills and procedures in a very specific order. By the time a student gets to algebra, it is very difficult to diagnose why they are getting a problem wrong. Different gaps may manifest themselve in different ways in different problems. The fix, too, is harder. Once you find what is wrong, you have to give the student distributed practice that isolates the gap. When is a gap fixed? Only when the student can do the problem by themselves without a model in front of them. -- AnneDwyer - 19 Jan 2006
"This is because practicing higher mathematical skills creates distributed practice of lower mathematical skills. The classic example is long division which requires the student to practice multiplication and subtraction." Great point! Long division is valuable for this reason alone. It also reinforces math facts and number sense. I am always amazed how many kids don't have number sense. They are all over the lot. -- CharlesH - 19 Jan 2006
New study indicates that even college students have massice gaps in their math knowledge. English too.
The survey examined college and university students nearing the end of their degree programs. The students did the worst on matters involving math, according to the study. Almost 20 percent of students pursuing four-year degrees had only basic quantitative skills. For example, the students could not estimate if their car had enough gas to get to the service station. About 30 percent of two-year students had only basic math skills.I guess conceptual knowledge isn't enough. -- KDeRosa - 19 Jan 2006
I'm having a little difficulty with this article. There's this statement:
More than 50 percent of students at four-year schools and more than 75 percent at two-year colleges lacked the skills to perform complex literacy tasks. That means they could not interpret a table about exercise and blood pressure, understand the arguments of newspaper editorials, compare credit card offers with different interest rates and annual fees or summarize results of a survey about parental involvement in school.And then there's this statement:
There was brighter news. Overall, the average literacy of college students is significantly higher than that of adults across the nation.How on earth is it brighter news that the average adult across the nation is even less literate than the illiterate students described above? -- GoogleMaster - 19 Jan 2006
I wonder if this is fallout from the "everyone should go to college" movement? -- GoogleMaster - 19 Jan 2006
If this study is in any way accurate, it is further proof that the college board had no business recentering the verbal scores back in 1995 and giving the average student an extra 80 points on the SAT-V. As innumerate as these kids are, they are far more illiterate. -- KDeRosa - 20 Jan 2006
Speaking of innumeracy and illiteracy, one of my indicators of how the local economy is going is the quality of help at the fast-food drive-thru window. When the economy is bad, you will find degreed engineers working at fast-food jobs. When the economy is good, those jobs are more likely to be staffed with the less educated, because the degreed engineers have moved on to jobs in their fields. So, when they start getting glassy-eyed upon receiving $5.12 for a $4.87 order, I know the economy is picking up. -- GoogleMaster - 20 Jan 2006
http://www.forbes.com/free_forbes/2005/1212/120.html i haven't read it because, well, it's unreadable: pop-ups and throbbing ads and whatnot. but it's called "gapology 101". i found it here at Chris Correa. -- VlorbikDotCom - 20 Jan 2006
Here's the same Forbes article without all of the ads: http://www.forbes.com/forbes/2005/1212/120_print.html -- GoogleMaster - 20 Jan 2006
So, when they start getting glassy-eyed upon receiving $5.12 for a $4.87 order, I know the economy is picking up. You are so right. But I can't say that that's a bad thing (that the economy is picking up -- it's of course a terrible thing that we graduate people who are so innumerate that they can't make change). -- CarolynJohnston - 20 Jan 2006
It's not only that they can't make change; it's that the concept of "getting rid of some pennies so I can get a quarter back instead of more pennies" never occurred to them. -- GoogleMaster - 20 Jan 2006
The economy must be booming. The Stooge at McD?'s screwed up my order -- not once, but twice -- today and almost forgot to put the toys in the kids' Happy Meals. This would have been very bad news for me had I not caught it. -- KDeRosa - 20 Jan 2006
Dan Seligman needs to spend some time reading research on genetics & IQ. This book is a good place to start:
Mind Sculpture
-- CatherineJohnson - 22 Jan 2006
Robertson includes some of the incredible research on IQ in black kids in the pre-Civil Rights south. The longer a black child stayed in school, the lower his IQ. Normally, the oldest child in a family has the highest IQ; the younger kids have slightly lower IQs. In these families, with children in these schools, the younger the child, the higher the IQ. When these families moved north, and put their kids in northern schools, IQs rose. Genes aren't rocks! The genes involved in intelligence, like nearly all genes (the exceptions are single-gene disorders like Huntington's) express in relation to the environment, which includes all components affecting genes, physical, social, intellectual, etc. ONE WORD, DAN: epigenetics -- CatherineJohnson - 22 Jan 2006
Actually, it looks like Chris Correa does a pretty good job explaining what is SO wrong with Seligman's article... -- CatherineJohnson - 22 Jan 2006
| WebLogForm | |
|---|---|
| Title: | stories about gaps |
| TopicType: | WebLog |
| SubjectArea: | AboutCurricula, CognitiveScience, DirectInstruction, EducationResearch, TeachersTeachingKids |
| LogDate: | 200601171828 |