I do not like to tell tales out of school. I consider the relationship of teacher and student to be one of confidentiality, most especially on the teachers' part. But I will say this.
The students I teach have a heck of a time with rounding.
I've taught some of the remedial classes at community college, like arithmetic and pre-algebra, and the students in these classes often do not get the idea of rounding, either rounding to a nearest decimal - like tenths or hundredths or thousandths - or rounding to the nearest thousand or million or rounding to a certain number of significant digits. In later classes like statistics or even calculus or linear algebra, I see there are more than a few students who haven't grasped round up and rounding down, usually always rounding down, also known as truncating.
For me, this is a major pedagogical hurdle. When I teach, I try to put myself in the mindset of when I didn't know how to do a thing and remember the things that helped me learn it. I'm not going to say I learned rounding in mere seconds, but it feels like I did. I'm sure I stumbled with it back some time in primary school, but after a few mistakes the mechanical rule fell into place and made sense.
Look at the digit that is going to vanish, the one to the right of the last place you are rounding to. If it is a 5, 6, 7, 8 or 9, add one to the digit you are rounding to. If it is a 0, 1, 2, 3 or 4, the last digit you are going to use remains the same. The first method is called rounding up and the second version is called rounding down.
Example: 4/7 = 0.571428571428..., a decimal place then the six digit pattern 571428 repeating forever.
Round 4/7 to the nearest tenth.
The digit in the tenths place is the .5, so the answer is going to be either .5 or .6; because the next digit (in some texts called the decision digit) is a 7, we add 1 to 5 and the answer is .6
Round 4/7 to the nearest hundredth.
If we truncate to the hundredths place, we get .57, so the answer is going to be either .57 or .58; because the next digit is a 1, we leave .57 as it is.
Round 4/7 to the nearest thousandth.
If we truncate to the thousandths place, we get .571, so the answer is going to be either .571 or .572; because the next digit is a 4, we leave .571 as it is.
Okay, I expect that this is not news to many of my readers, though it may have been a while since you thought about it.
I am chagrined that I did not know about other methods given my advanced years, but a statistics text I am using for the first time has a math skills pre-test and uses a slightly different method, known by several names. I first heard of it as "banker's rounding", though doing more research, I understand that "statistician's method" or "Gaussian rounding" are more common and likely more accurate.
Let's say for arguments's sake we are rounding off to the nearest dollar, getting rid of the pesky pennies. The method you likely learned in school, which here is called "Traditional", will simply look at the tenths position.
Example:
If we have between $2.00 and $2.49, this will "round down" to $2.00 exactly.
If instead the total is between $2.50 and $2.99, we "round up" to $3.00.
The new method agrees with the old method almost exactly, with the only contentious case being the half dollar. Technically, $2.51 should go to $3.00, because that's the closest value. (It's 49 cents away from $3 and 51 cents away from $2.) Likewise $2.49 should round down to $2.00, since that is the closest value. but $2.50 presents a philosophical dilemma, since it is exactly 50 cents away from $3.00 and 50 cents away from $2.00.
This new method says to round a number of the form x.5 to the nearest even number. That means half the time we round x.5 up and half the time we round down. The row in yellow shows the only disagreement between 1 and 3. 1.5 rounds up traditionally to 2, and in the new method rounds to the nearest even number, which is still 2. But 2.5 rounds traditionally to 3, and in the new method rounds to 2.
Why bother? Think of what we are changing in the sum of rounded numbers, assuming that all numbers are equally likely to show up. Let's say we had the list of numbers as follows:
1.00
1.01
1.02
...
2.98
2.99
3.00
These 201 entries add up to 402, which means the average is 402/201 = 2.
If we round them using the standard method, we will get
50 x 1 = 50
100 x 2 = 200
51 x 3= 153
This adds up to 403, and 403/201 = 2.004975124..., which is to say that adding, then rounding will not give the same answer as rounding, then adding.
If we round this set using statisticians rounding, this is what happens.
50 x 1 = 50
101 x 2 = 202
50 x 3= 150
This adds up to 402, and rounds to 2 exactly.
As a teacher who knows my students already have a difficult time with rounding, this presents a problem. I do not want to "dumb down" the curriculum, but I also don't want to add in extra problems when I don't have to. Searching Wikipedia for this method, I see that this is the standard for IEEE 754 use with floating point operations.
Some but certainly not all of my students may see this in their careers. I would hope that people who go into programming would have a better grasp of math, but having worked for nearly two decades in the field, I know that is not always the case. More than once, I came onto a project at a large computer company that involved something like higher math and I was the only programmer on a large team who knew the right method. Sometimes it was something slightly esoteric, like group theory and the symmetries of the square. Another time, I was the only person who really understood how sine and cosine worked.
One of my favorite expressions I learned from my father is "You learn something new every day, if you aren't careful." Well, I wasn't careful and I learned something new yesterday. Now I have to decide how it should apply to the classes I teach.
It would be so much easier if I did what I was told and didn't give a rat's ass, but as I am now two score and seventeen years old, I get the feeling the "I don't give a rat's ass" option is not open to me.
"Banker's" rounding, huh? I hadn't heard that expression before. When I was a wee lad, it was called "scientific" rounding.
ReplyDeleteWhat might be at work here is our old friend, school tracking, in which college-bound students take one set of high-school courses, "vocational" students take another, and so on. The result in college, of course, is an incoming student body that, at best, has to deal with conflicting rounding methods. Some students come to you knowing traditional rounding; some come in with scientific rounding; and the rest have given up in confusion between the two methods.
You might want to ask your students how they learned to round in grade school and high school. Acknowledge their responses, introduce them to both methods, then enforce one of them throughout the class. [If some students want to respond to your exercises using both ways, let them, so long as they know what they are doing.]
Bless you and keep you, Abu. As I said in the post, I had never heard of it before. It is equally as valid, possibly more so in some situations. I will ask my students tomorrow what they learned. It's a class full of psych students at Mills, so I expect they learned the standard, but I shouldn't assume.
DeleteI was indeed stunned to learn that you'd never seen it before. I had it permanently etched in my head as an undergrad; but then, I was an engineering major at a school best known for engineers.
DeleteAs to your students: definitely don't assume. High-school tracking isn't the only culprit; splits can take place even within the same division. At my high school, for instance, scientific rounding was taught only in the physics and advanced chemistry classes. Since both courses were electives, some of my classmates completed the college-prep track without taking either one -- so they didn't learn scientific rounding until college, if ever.