In yesterday's blog entry, I left out mention of an excellent article entitled The Role of Long Division in the K-12
Curriculum written by David Klein of California State Northridge and R. James Milgram of Stanford. You can find there article here.
Within this article they stress California educational politics, but their arguments transcend the region. I do have to confess that readers with a lack of comfort with numbers will probably not finish this article (but I pointed out yesterday that familiarity with our number system along with addition, multiplication, and subtraction, is a prerequisite to handling long division).
Please remember that long division is an algorithm that historically became the standard process for doing division. It also is an excellent tool in discussing the differences between terminating, repeating, and non-terminating non-repeating decimals, which help distinguish between rational and irrational numbers, which is muddled up by hand-held calculators.
For the record, students with hand held calculators have big trouble handling our national debt (in dollars and cents it's 15 digits long). Ask a student to divide the debt equally among the 50 states and see what they do.
More to come...
Friday, June 28, 2013
Thursday, June 27, 2013
Long Division as a Symptom
Laurie H. Rodgers, in an article entitled "In defense of long division: Pro-reform professor capably shows why reform math doesn't work" has given a strong defense of long division, and has done so while helping to show that the math reforms (coalescing nationwide the in Common Core State Standards) are not necessarily for the better. (Her article can be found here.)
Speaking of long division, I will add to Ms. Rodgers' argument by using a quote from her article: "Many don’t even know their basic multiplication facts.."
Long division, when it is taught, is generally the last of the four basic operations. As an algorithm, its use and fluidity depends greatly on the user's knowledge of basic multiplication facts. It also relies heavily on subtraction. We all know that multiplication and subtraction are largely taught in their relation to addition. Thus virtually all one's basic number knowledge comes into play in constructing long division. Weaknesses in addition, subtraction, and multiplication will all help in crashing the edifice known as long division.
When confronting long division, anyone with weak mental arithmetic skills in the other operations will generally "mess up."
As a high school teacher I made a point when analyzing a student's work of trying to detect the specific error(s) that caused students' solutions to be in error. I wish I had kept good solid data, because the fact was that the vast majority of errors (my sense is 80% or more) were caused by faulty addition, subtraction, and or multiplication. (Leaving out the errors based solely on conceptual weakness). And this was in the calculator age in the New York Regents program.
Speaking of calculators, I found students "dial wrong numbers", get erroneous calculation results, and have no clue that they were off. Even adding 23 and 2, for example, a student might hit the multiplication button, and then write down 23 + 2= 46, and let it stand.
Needless to say, I hold the position that our educational systems are having students race through arithmetic so we can get them to "higher thinking skills" where all the alleged importance is. I believe this if more time, energy, effort, and resources were spent teaching children all about numbers, the higher order items along with a love of math would follow in due course.
I suspect the English Language standards are missing the mark as well, since I have searched through them and I have not yet found a reference to "spelling". The first grade has a standard that says "Write opinion pieces in which they introduce the topic or name the book they are writing about, state an opinion, supply a reason for the opinion, and provide some sense of closure." In essence, first graders will be writing book reviews. Without being able to spell?
Speaking of long division, I will add to Ms. Rodgers' argument by using a quote from her article: "Many don’t even know their basic multiplication facts.."
Long division, when it is taught, is generally the last of the four basic operations. As an algorithm, its use and fluidity depends greatly on the user's knowledge of basic multiplication facts. It also relies heavily on subtraction. We all know that multiplication and subtraction are largely taught in their relation to addition. Thus virtually all one's basic number knowledge comes into play in constructing long division. Weaknesses in addition, subtraction, and multiplication will all help in crashing the edifice known as long division.
When confronting long division, anyone with weak mental arithmetic skills in the other operations will generally "mess up."
As a high school teacher I made a point when analyzing a student's work of trying to detect the specific error(s) that caused students' solutions to be in error. I wish I had kept good solid data, because the fact was that the vast majority of errors (my sense is 80% or more) were caused by faulty addition, subtraction, and or multiplication. (Leaving out the errors based solely on conceptual weakness). And this was in the calculator age in the New York Regents program.
Speaking of calculators, I found students "dial wrong numbers", get erroneous calculation results, and have no clue that they were off. Even adding 23 and 2, for example, a student might hit the multiplication button, and then write down 23 + 2= 46, and let it stand.
Needless to say, I hold the position that our educational systems are having students race through arithmetic so we can get them to "higher thinking skills" where all the alleged importance is. I believe this if more time, energy, effort, and resources were spent teaching children all about numbers, the higher order items along with a love of math would follow in due course.
I suspect the English Language standards are missing the mark as well, since I have searched through them and I have not yet found a reference to "spelling". The first grade has a standard that says "Write opinion pieces in which they introduce the topic or name the book they are writing about, state an opinion, supply a reason for the opinion, and provide some sense of closure." In essence, first graders will be writing book reviews. Without being able to spell?
Wednesday, June 19, 2013
Math Wars
There was an excellent article in the New York Times a on June 16 called The Faulty Logic of the ‘Math Wars’. It is a must-read for anybody who even cares about what happens in public education in this country.
Having been a 32-year math teacher myself, I can attest to the fact that there have been massive changes in the way mathematics is intended to be taught. I say "intended" because there has always been a hope, on my part, that teachers can handle the new wave and still manage to sneak in some real and true math instruction. Unfortunately, as time passes, more and more of our teachers will have been products of the new era, almost to the point that pencil-and-paper calculations become anachronistic, and algorithmic thinking becomes a rarity.
Failing to teach students a good dose of algorithmic skills, especially in arithmetic, forces them to fall into one, or possibly both, of two pits: that of calculator dependency and that of failure to achieve progress because they have to constantly reinvent everything that was discovered before.
Alfred North Whitehead once said "Civilization advances by extending the number of important operations which we can perform without thinking of them." I could change that and add that basic human life operates the same way. Do you believe that a good driver thinks his/her way through a road trip? Or has the process been, in essence, "built in" to the brain? Now imagine how that process was embedded, and you have little picture of how people acquire competences.
Thinking and reasoning are massively important in life, but people should not need to rely on them continuously and constantly. There should be a significant portion of our life's activities that we can do competently and successfully without having to think and re-think every action.
Musicians who are successful have taken complete unknowns and turned them into second nature habits. Especially those who read music. The brain power used to develop the skills that a classical pianist exhibits is the same brain power used developing a world-class mathematician. Leibniz once said "Music is the pleasure the human soul experiences from counting without being aware that it is counting.” A case could be made that the deaf Beethoven wrote his works using mathematical thinking.
I truly wish our educational system would look at how people like Beethoven, Whitehead, and Leibniz succeeded, and adjust our schools accordingly. Instead they listen to the business world, if anybody.
Much more on this to come....
Having been a 32-year math teacher myself, I can attest to the fact that there have been massive changes in the way mathematics is intended to be taught. I say "intended" because there has always been a hope, on my part, that teachers can handle the new wave and still manage to sneak in some real and true math instruction. Unfortunately, as time passes, more and more of our teachers will have been products of the new era, almost to the point that pencil-and-paper calculations become anachronistic, and algorithmic thinking becomes a rarity.
Failing to teach students a good dose of algorithmic skills, especially in arithmetic, forces them to fall into one, or possibly both, of two pits: that of calculator dependency and that of failure to achieve progress because they have to constantly reinvent everything that was discovered before.
Alfred North Whitehead once said "Civilization advances by extending the number of important operations which we can perform without thinking of them." I could change that and add that basic human life operates the same way. Do you believe that a good driver thinks his/her way through a road trip? Or has the process been, in essence, "built in" to the brain? Now imagine how that process was embedded, and you have little picture of how people acquire competences.
Thinking and reasoning are massively important in life, but people should not need to rely on them continuously and constantly. There should be a significant portion of our life's activities that we can do competently and successfully without having to think and re-think every action.
Musicians who are successful have taken complete unknowns and turned them into second nature habits. Especially those who read music. The brain power used to develop the skills that a classical pianist exhibits is the same brain power used developing a world-class mathematician. Leibniz once said "Music is the pleasure the human soul experiences from counting without being aware that it is counting.” A case could be made that the deaf Beethoven wrote his works using mathematical thinking.
I truly wish our educational system would look at how people like Beethoven, Whitehead, and Leibniz succeeded, and adjust our schools accordingly. Instead they listen to the business world, if anybody.
Much more on this to come....
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