Monday, May 26, 2008
Friday, May 23, 2008
Wednesday, May 14, 2008
How to Misinterpret a Research Project?
In the April 25 issue of the New York Times, there was an article entitled, “Study Suggests Math Teachers Scrap Balls and Slices”, about research that was reported in Science Magazine on the same date, entitled “The Advantage of Abstract Examples in Learning Math”.
To give you an idea of how journalists reacted to this story, there were 148 results, (15 pages) in a google, search with the exact 8 words from the NYT article title, including articles in the District Administrator and the Christian Science Monitor. Interestingly, there were only 71 entries, (7 pages), with the exact 8 words of the title from Science Magazine. The Times did not cite the title of the Science article.
I received three emails, from friends, about the NYT article. The Times article left me with more questions than answers so I decided to read the Science article. I found out that I had to pay $10 to view the article, or subscribe at an annual rate of $99 to $144. Thank goodness for the public library.
I was not able to discern whether the Times reporter had read the Science article. He did interview one of the researchers/authors, Jennifer A. Kaminski. The language of the Science article was rather technical. For example, one form or another, of the word “instantiate” was used 46 times in the article that was less than two pages long. I had not heard the word before and had to look it up. There was also some math. I will attempt to explain the two articles, from my math-teacher perspective.
The New York Times article starts out:
"One train leaves Station A at 6 p.m. traveling at 40 miles per hour… Entranced, perhaps, by those infamous hypothetical trains, many educators in recent years have incorporated more and more examples from the real world to teach abstract concepts. The idea is that making math more relevant makes it easier to learn".
The author of the Times article and Dr. Kaminiski are using the scariest example of a word problem from our high school math class, thus tapping into the deep-seated math phobia of the average American. The train example is irrelevant to the research reported in the Science article, unless high school teachers around the U.S. are taking students out to the train tracks armed with stopwatches, or setting up the Lionel electric train in the classroom. And besides, the way most of us learn about the train example is in the dreaded Word Problem section of the algebra textbook, after we have learned all of the techniques for solving linear equations. In other words... the train problem shouldn't come before we have the algebraic tools to solve it.
From the Science article:
“Instantiating an abstract concept in a concrete, contextualized manner appears to constrain that knowledge and to hinder the ability to recognize the same concept elsewhere; this, in turn, obstructs knowledge transfer. At the same time, learning a generic instantiation allows for transfer, which suggests that such an instantiation could result in a portable knowledge representation.”
Well this seems obvious: Children learn to count by first counting objects, (manipulatives in teacher lingo) often their fingers (“concrete, contextualization”). They get a little older and learn that symbolic math is more practical when dealing with large numbers. They learn abstract techniques such as the multiplication and division algorithms, (“generic instantiations”), for dealing with larger quantities and more complex problems. Trying to do division with multi-digit numbers, without knowing a division algorithm, addition facts and times tables, by counting on one’s fingers, will hinder anyone.
From the Times article:
“The problem with the real-world examples, Dr. Kaminski said, was that they obscured the underlying math, and students were not able to transfer their knowledge to new problems.”
Translation: Cut a cookie in half, cut a sandwich in half, divide a group of 4 jelly beans into two equal groups. Do you now understand that 1/2 + 1/2 = 1? Probably not.
“They tend to remember the superficial, the two trains passing in the night,” Dr. Kaminski said. “It’s really a problem of our attention getting pulled to superficial information.”
Translation: (Ignoring the fact that trains are irrelevant) We understand that a the pizza has twelve pieces and there are 4 of us, each one gets 3 pieces, if the sharing is to be equitable. We still may not know that 1/4 x 12 = 3, or that 1/4 = 3/12. But was the pizza good?
"The researchers said they had experimental evidence showing a similar effect with 11-year-old children. The findings run counter to what Dr. Kaminski said was a “pervasive assumption” among math educators that concrete examples help more children better understand math".
Reaction: If this is a pervasive assumption in American schools, and teachers are wasting hours having students count M&Ms, pizza slices, and jelly beans, then there is definitely a problem. Do we want our children to understand fractions or be connoisseurs of pizza? Manipulatives, vis a vis math education, should be used as a bridge to learning abstract mathematical language and processes, and be directly related to understanding the concepts taught.
On the other hand, I give you a possible counter-example as food for thought: The abacus is a manipulative sometimes used in schools. It is also used by millions of people to make calculations. At what point does a manipulative hinder learning? Can we say the use of the abacus has hindered people in their ability to learn math? The abacus is a manipulative as well as a calculating device. Then what affect does the use of a calculating device have on learning math? It looks like we may need another study.
To give you an idea of how journalists reacted to this story, there were 148 results, (15 pages) in a google, search with the exact 8 words from the NYT article title, including articles in the District Administrator and the Christian Science Monitor. Interestingly, there were only 71 entries, (7 pages), with the exact 8 words of the title from Science Magazine. The Times did not cite the title of the Science article.
I received three emails, from friends, about the NYT article. The Times article left me with more questions than answers so I decided to read the Science article. I found out that I had to pay $10 to view the article, or subscribe at an annual rate of $99 to $144. Thank goodness for the public library.
I was not able to discern whether the Times reporter had read the Science article. He did interview one of the researchers/authors, Jennifer A. Kaminski. The language of the Science article was rather technical. For example, one form or another, of the word “instantiate” was used 46 times in the article that was less than two pages long. I had not heard the word before and had to look it up. There was also some math. I will attempt to explain the two articles, from my math-teacher perspective.
The New York Times article starts out:
"One train leaves Station A at 6 p.m. traveling at 40 miles per hour… Entranced, perhaps, by those infamous hypothetical trains, many educators in recent years have incorporated more and more examples from the real world to teach abstract concepts. The idea is that making math more relevant makes it easier to learn".
The author of the Times article and Dr. Kaminiski are using the scariest example of a word problem from our high school math class, thus tapping into the deep-seated math phobia of the average American. The train example is irrelevant to the research reported in the Science article, unless high school teachers around the U.S. are taking students out to the train tracks armed with stopwatches, or setting up the Lionel electric train in the classroom. And besides, the way most of us learn about the train example is in the dreaded Word Problem section of the algebra textbook, after we have learned all of the techniques for solving linear equations. In other words... the train problem shouldn't come before we have the algebraic tools to solve it.
From the Science article:
“Instantiating an abstract concept in a concrete, contextualized manner appears to constrain that knowledge and to hinder the ability to recognize the same concept elsewhere; this, in turn, obstructs knowledge transfer. At the same time, learning a generic instantiation allows for transfer, which suggests that such an instantiation could result in a portable knowledge representation.”
Well this seems obvious: Children learn to count by first counting objects, (manipulatives in teacher lingo) often their fingers (“concrete, contextualization”). They get a little older and learn that symbolic math is more practical when dealing with large numbers. They learn abstract techniques such as the multiplication and division algorithms, (“generic instantiations”), for dealing with larger quantities and more complex problems. Trying to do division with multi-digit numbers, without knowing a division algorithm, addition facts and times tables, by counting on one’s fingers, will hinder anyone.
From the Times article:
“The problem with the real-world examples, Dr. Kaminski said, was that they obscured the underlying math, and students were not able to transfer their knowledge to new problems.”
Translation: Cut a cookie in half, cut a sandwich in half, divide a group of 4 jelly beans into two equal groups. Do you now understand that 1/2 + 1/2 = 1? Probably not.
“They tend to remember the superficial, the two trains passing in the night,” Dr. Kaminski said. “It’s really a problem of our attention getting pulled to superficial information.”
Translation: (Ignoring the fact that trains are irrelevant) We understand that a the pizza has twelve pieces and there are 4 of us, each one gets 3 pieces, if the sharing is to be equitable. We still may not know that 1/4 x 12 = 3, or that 1/4 = 3/12. But was the pizza good?
"The researchers said they had experimental evidence showing a similar effect with 11-year-old children. The findings run counter to what Dr. Kaminski said was a “pervasive assumption” among math educators that concrete examples help more children better understand math".
Reaction: If this is a pervasive assumption in American schools, and teachers are wasting hours having students count M&Ms, pizza slices, and jelly beans, then there is definitely a problem. Do we want our children to understand fractions or be connoisseurs of pizza? Manipulatives, vis a vis math education, should be used as a bridge to learning abstract mathematical language and processes, and be directly related to understanding the concepts taught.
On the other hand, I give you a possible counter-example as food for thought: The abacus is a manipulative sometimes used in schools. It is also used by millions of people to make calculations. At what point does a manipulative hinder learning? Can we say the use of the abacus has hindered people in their ability to learn math? The abacus is a manipulative as well as a calculating device. Then what affect does the use of a calculating device have on learning math? It looks like we may need another study.
Sunday, May 11, 2008
World Freehand Circle Drawing Champion
WHAT MATH TEACHERS DO FOR ENTERTAINMENT!
Somewhere in my math travels, I have seen a teacher do this. By the number of views this video has had on You Tube (4,241,063 as of this posting), it seems there is a cult following to Freehand Circle Drawing.
Sunday, May 4, 2008
The Lake Woebegone Effect
I have written about, what I call, the “Lake Woebegone” Effect (LWE). Lake Woebegone is the name of the mythical town, in Garrison Keillor’s weekly radio show, where “all the women are strong, all the men are good-looking, and all the children are above average.” In my definition, LBE is when we, as parents, are not able to stand to know that our child isn’t above average. Of course this is impossible. From the meaning of average, for any criteria of a given population, 50% of the population falls below the average mark, and 50% of the population is above the average mark.
We regularly sing the praises of accomplished school athletes. We award them with trophies and ribbons. Everyone understands that if you can't play well enough, you don't make it onto the team. We all know that in any team that is truly competing, the best players play most often and the others sit on the bench. I watched many games where the top players on the team play most of the game and the others fill in when: There is an injury, when a player is tired, or when the team is so far ahead of the other team that it does not make a difference one way or the other. For a laugh see No Child Left Behind: The Football Version.
It is an entirely different mind set when it comes to our children in the classroom. Some may recall when C was the average grade. This is no longer true. Since we, as a culture, cannot stand for our children to be average, let alone below average, they must at least earn a B, if not an A. We, as a nation, believe that it is more important to pass a student to the next level than to retain a student because it would hurt his or her feelings, and therefore ours as parents, if the child were to fail.
We are graduating huge numbers of students from high school who are functionally illiterate and innumerate. The California High School Exit Exam only tests students at an 8th grade level in Math and English. It is our fault, when we don't stand up for excellency in schools. We protest when our own children fall below the C level. We blame it on the teachers, we complain to the administrators, and hence we have the LBE. The result is that our schools march toward mediocrity. And we don't understand why our nation's sophomores rank 24th of 29 developed nations.
Jacques Barzun, refers to the idea of LBE (but he didn’t call it that) in his 1991 book Begin Here: The forgotten Conditions of Teaching and Learning. He says that we that through our choices, as a nation, we have lost the “knack of… Teaching and Learning.”
“Take one familiar fact: everybody keeps calling for Excellence - excellence not just in schooling, throughout society. But as soon as somebody or something stands out as Excellent, the other shout goes up: ‘Elitism!’… ‘Standing out’ is undemocratic.”
In 1986, John Jacob Cannell, MD published, what later became known as, the “Lake Woebegone” Reports. Alas, it turns out that LBE wasn’t original. The doctor from West Virginia started wondering how the mostly poor students of his state could do so well on nationally standardized tests. In the first report he documented “all fifty states were testing above the national average in elementary achievement and concluded the testing infrastructure in America’s public schools was corrupt.” In the second report he, “delineated the systematic and pervasive ways that American educators cheat on standardized achievement tests.” Ouch!
I earned a teaching credential in 1993 and have never heard of either report even though both of his reports were “extensively discussed in academic journals, and helped spur the testing reform movement.” As a teacher, I was largely unaware that there was extensive cheating on the standardized tests, even though a number of years ago, when the No Child Left Behind (NCLB) Act was introduced, spurred by George W. Bush, my brother in witty irony, sent me an article about the rampant cheating on standardized tests by educators and administrators in Texas. There is even a name for this type of cheating: Campbell’s Law – Who Knew?
Campbell’s Law basically says that the higher the stakes in testing, the more likely corruption occurs. Isn't it amazing that this was also never discussed in credential school, nor by any administrator I have known, and I have been teaching for 15 years?
In 2006, John Jacob Cannell published “‘Lake Woebegone”: Twenty Years Later” where he explains why he left the testing reform movement and goes into detail about the reason the NCLB testing is no different than what occurred with standardized testing twenty years ago. I highly recommend reading it.
We regularly sing the praises of accomplished school athletes. We award them with trophies and ribbons. Everyone understands that if you can't play well enough, you don't make it onto the team. We all know that in any team that is truly competing, the best players play most often and the others sit on the bench. I watched many games where the top players on the team play most of the game and the others fill in when: There is an injury, when a player is tired, or when the team is so far ahead of the other team that it does not make a difference one way or the other. For a laugh see No Child Left Behind: The Football Version.
It is an entirely different mind set when it comes to our children in the classroom. Some may recall when C was the average grade. This is no longer true. Since we, as a culture, cannot stand for our children to be average, let alone below average, they must at least earn a B, if not an A. We, as a nation, believe that it is more important to pass a student to the next level than to retain a student because it would hurt his or her feelings, and therefore ours as parents, if the child were to fail.
We are graduating huge numbers of students from high school who are functionally illiterate and innumerate. The California High School Exit Exam only tests students at an 8th grade level in Math and English. It is our fault, when we don't stand up for excellency in schools. We protest when our own children fall below the C level. We blame it on the teachers, we complain to the administrators, and hence we have the LBE. The result is that our schools march toward mediocrity. And we don't understand why our nation's sophomores rank 24th of 29 developed nations.
Jacques Barzun, refers to the idea of LBE (but he didn’t call it that) in his 1991 book Begin Here: The forgotten Conditions of Teaching and Learning. He says that we that through our choices, as a nation, we have lost the “knack of… Teaching and Learning.”
“Take one familiar fact: everybody keeps calling for Excellence - excellence not just in schooling, throughout society. But as soon as somebody or something stands out as Excellent, the other shout goes up: ‘Elitism!’… ‘Standing out’ is undemocratic.”
In 1986, John Jacob Cannell, MD published, what later became known as, the “Lake Woebegone” Reports. Alas, it turns out that LBE wasn’t original. The doctor from West Virginia started wondering how the mostly poor students of his state could do so well on nationally standardized tests. In the first report he documented “all fifty states were testing above the national average in elementary achievement and concluded the testing infrastructure in America’s public schools was corrupt.” In the second report he, “delineated the systematic and pervasive ways that American educators cheat on standardized achievement tests.” Ouch!
I earned a teaching credential in 1993 and have never heard of either report even though both of his reports were “extensively discussed in academic journals, and helped spur the testing reform movement.” As a teacher, I was largely unaware that there was extensive cheating on the standardized tests, even though a number of years ago, when the No Child Left Behind (NCLB) Act was introduced, spurred by George W. Bush, my brother in witty irony, sent me an article about the rampant cheating on standardized tests by educators and administrators in Texas. There is even a name for this type of cheating: Campbell’s Law – Who Knew?
Campbell’s Law basically says that the higher the stakes in testing, the more likely corruption occurs. Isn't it amazing that this was also never discussed in credential school, nor by any administrator I have known, and I have been teaching for 15 years?
In 2006, John Jacob Cannell published “‘Lake Woebegone”: Twenty Years Later” where he explains why he left the testing reform movement and goes into detail about the reason the NCLB testing is no different than what occurred with standardized testing twenty years ago. I highly recommend reading it.
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