Simplifying Simplifying
During the summers when I was going to school, I worked as a tour guide and teacher for groups of Korean students who came to Los Angeles to see the sights and learn some English.
My job was fairly straightforward: Help keep all 100 students safe and happy during our trips to Disneyland and Universal Studios and then work with them for about two hours a week to help them master the English language.
We were all put up in a nearby hotel, so my work for those five weeks every summer was pretty much 24/7. And during the off hours or on bus rides, I had several conversations with students, covering just about everything a 10-15 year-old could talk about with her or his teacher--from homework problems (they had homework from their Korean schools to complete over the summer while they were in the States) to can you please go buy me some tampons (at 1 a.m.).
Anyway, although I didn't have a lot of conversations with students about math, there were a handful, and I was thinking about those conversations today. And what strikes me most, in retrospect, about my conversations with those students was their ability to detect whether an answer was reasonable or unreasonable.
I chuckle when I write that last phrase, because I know that there has been a concerted effort in the U.S. in the past several years to try to teach kids how to distinguish reasonable from unreasonable answers. (You can see it in most elementary basal mathematics texts, where it pops up occasionally as a problem-solving strategy.)
But, see, I'm not talking about that quintessential American "algorithmic" mathematical ability, which is still sadly algorithmic even when it is described as "constructivist" or "understanding." The ability these Korean students displayed--ahem, about 15 years ago--was one of recognition of the basic relationships in mathematics.
So here's a lesson my young Korean tutors taught me way back when for simplifying basic fractions. (Well, they never really got to this point verbally. But I assumed at the time that this was how they were thinking based on hand gestures and some incoherently phrased English.) [And, by the way, this is yet another reason to teach children fractions as division expressions.]
[begin lesson]
To simplify fractions, most students are taught that you first have to find a common divisor--a divisor common to both the numerator and denominator. If the only common divisor you can find is 1, then the fraction is in simplest form.
This is true, of course. But that algorithm is designed for "worst-case scenarios"--like 3/102--when typical elementary-school students don't know (a) whether the fraction can be simplified and (b) how far they have to go to get the fraction in simplest form.
A lot of basic fraction simplifying is a lot simpler. All you need to know are basic multiplication/division facts. Oh, and one other box-of-rocks-simple idea (besides the concept of fractions as division expressions)--how to turn something upside down.
Let's simplify the fraction 3/18 the "algorithmic" way:
Okay, and now let's just think about it. What's 18 รท 3? (Um, 6.) How would y'all write that as a fraction? (Uhhhh, 18/3.) So 18/3 is 6. If you just flip the fraction 18/3 over, what division expression do you have? (Zuhhhhh, 3/18.) Is there anyone here who thinks that the answer, the quotient, is NOT going to be the upside-down version of 6/1? Anyone? Anyone?
[end lesson]
Now, if you're a third-, fourth-, fifth-, or sixth-grade student, and you know your basic multiplication/division facts, don't you save valuable time on those high-stakes tests by just knowing this stuff?
As always, more on this later.
My job was fairly straightforward: Help keep all 100 students safe and happy during our trips to Disneyland and Universal Studios and then work with them for about two hours a week to help them master the English language.
We were all put up in a nearby hotel, so my work for those five weeks every summer was pretty much 24/7. And during the off hours or on bus rides, I had several conversations with students, covering just about everything a 10-15 year-old could talk about with her or his teacher--from homework problems (they had homework from their Korean schools to complete over the summer while they were in the States) to can you please go buy me some tampons (at 1 a.m.).
Anyway, although I didn't have a lot of conversations with students about math, there were a handful, and I was thinking about those conversations today. And what strikes me most, in retrospect, about my conversations with those students was their ability to detect whether an answer was reasonable or unreasonable.
I chuckle when I write that last phrase, because I know that there has been a concerted effort in the U.S. in the past several years to try to teach kids how to distinguish reasonable from unreasonable answers. (You can see it in most elementary basal mathematics texts, where it pops up occasionally as a problem-solving strategy.)
But, see, I'm not talking about that quintessential American "algorithmic" mathematical ability, which is still sadly algorithmic even when it is described as "constructivist" or "understanding." The ability these Korean students displayed--ahem, about 15 years ago--was one of recognition of the basic relationships in mathematics.
So here's a lesson my young Korean tutors taught me way back when for simplifying basic fractions. (Well, they never really got to this point verbally. But I assumed at the time that this was how they were thinking based on hand gestures and some incoherently phrased English.) [And, by the way, this is yet another reason to teach children fractions as division expressions.]
[begin lesson]
To simplify fractions, most students are taught that you first have to find a common divisor--a divisor common to both the numerator and denominator. If the only common divisor you can find is 1, then the fraction is in simplest form.
This is true, of course. But that algorithm is designed for "worst-case scenarios"--like 3/102--when typical elementary-school students don't know (a) whether the fraction can be simplified and (b) how far they have to go to get the fraction in simplest form.
A lot of basic fraction simplifying is a lot simpler. All you need to know are basic multiplication/division facts. Oh, and one other box-of-rocks-simple idea (besides the concept of fractions as division expressions)--how to turn something upside down.
Let's simplify the fraction 3/18 the "algorithmic" way:
Okay, and now let's just think about it. What's 18 รท 3? (Um, 6.) How would y'all write that as a fraction? (Uhhhh, 18/3.) So 18/3 is 6. If you just flip the fraction 18/3 over, what division expression do you have? (Zuhhhhh, 3/18.) Is there anyone here who thinks that the answer, the quotient, is NOT going to be the upside-down version of 6/1? Anyone? Anyone?
[end lesson]
Now, if you're a third-, fourth-, fifth-, or sixth-grade student, and you know your basic multiplication/division facts, don't you save valuable time on those high-stakes tests by just knowing this stuff?
As always, more on this later.
Labels: education, mathematics, textbooks
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