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12 Oct 2005 - 15:06
Interesting observations via email from Ron Aharoni. But first, you might want to re-read this post on the fifth operation of arithmetic: In addition to the four classical operations, there is a fifth one that is even more fundamental and important. That is, forming a unit, taking a part of the world and declaring it to be the “whole.”
I agree that sticking to the pie representation of fractions is harmful. I also prefer parts of rectangles. But: I believe that it is important to take, from the very beginning, fractions of sets. What is a half of 6 apples? A quarter of a set of 8 pencils? And then, immediately, WHAT IS TWO QUARTERS, and three quarters, of that set? This conveys the meaning of "three quarters" better than the manipulatives.And, an elaboration, from a second email:
I try to start with fractions of all kinds of objects - shapes as well as groups. In first grade, I start fractions with division. I give groups of kids all kinds of objects: one group gets a rectangle, another a circle, another group two rectangles, another group an apple, and another A GROUP OF 4 APPLES, and ask them to divide the objects they got into two parts. Later, each group tells what they did. We then discuss the notion of "a half of". Then each kid is asked to do work on his own - take halves of shapes, and a half of say 4 objects drawn on paper. Then we can divide into three parts, and discuss what is a "third of something". Then "two thirds" (just repeat twice the one third), then a quarter - all this can be easily done with second graders, even first graders.
A unit is rather like the denominator part of a fraction. Many of the rules regarding their manipulation are the same. I intuitively understand why that is, and I am going to try to write it up, but right now words elude me. Here's a quick try to convey the idea by analogy, though -- the correct way to think of fractions is as a unit -- of the form 1/3, 1/4, 1/5, 1/8, etc. -- occurring some number of times, where that number is given by the numerator. So you should think of 3/4 as being "the unit 1/4, occurring 3 times".
We've looked at a fraction of a whole unit. Now let's review fractions of numbers greater than 1. Take 1/4 of three identical sandwiches.There followed a page of drawings showing that 1/4 of 3 sandwiches is the same thing as 3/4 of 1 sandwich. I didn't get it. I could see it was true. I could see that the drawing was 'true,' and I knew, of course that 1/4 x 3 = 3/4 x 1. That wasn't the problem. The problem was, I didn't get it. I was having an especially hard time with the pizza pie chart image that kept popping into my mind:
My problem with this mental image, which was very strong & vivid, was that I simply could not stop seeing THREEFOURTHS. THREEFOURTHS, to me, is a highly overlearned mental THING; if you say 3/4 to me, I'm going to start seeing visions of circles divided into fourths with 3 of the fourths shaded in. Period. I have no choice. It's like a song that's stuck in your head. Only it's not a song. It's a textbook illustration. So there I was, trying to think about ONEFOURTH of 3, and forget it. It wasn't happening; it wasn't going to happen. I just could not make that bright, vivid, 3/4-of-1-whole-circle turn into 1/4 of 3 circles. I could imagine 3 circles, side by side, each divided into 4. But after that my brain instantly jumped to the THREEFOURTHS clumps. I kept imagining, in sequence:
magical number 7, plus or minus 2. Apparently I'm down to the magical number 5.
The way I always think of this is as three 'one-fourths.' There are 3 sandwiches, and you take 1/4 from each sandwich. That gives you 3/4, or 3 separate one-fourths.That one sentence clobbered my THREEFOURTHS image. Suddenly I could 'see' separate little one-fourths pulled out of all 3 circles; I could see the individual one-fourths as.....units, I guess. Like Carolyn would say.
All of this is a long way of saying that:
Demonstrating division of fractions with pictures or manipulatives at Math Forum
Aharoni article, part 1
Aharoni article, part 2: America's 'new math' goes to Israel
Aharoni on the fifth operation of arithmetic
Ron Aharoni on teaching fractions & forming units
What I Learned In Elementary School by Ron Aharoni (AMERICAN EDUCATOR)
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".. unit is rather like the denominator part of a fraction." I like to put units into their proper location - numerator or denominator. If you have 60 miles per hour, the "per" means divide, like miles/hour. This means that the miles goes in the numerator and hours goes in the denominator. You don't have a denominator? Put a one in the denominator. So 60 mph becomes: 60 miles / 1 hour This way, you carry the units along in the calculation, combining or canceling units as you go. if you add or subtract terms, they have to have the same units. If you multiply or divide numbers, you can cancel, multiply, or divide the units. Hopefully, the units make sense. If you divide 5 miles by 10 hours, you get 1/2 miles per hour, or 1 mile in or per two hours. If you have 1/2 of an hour, the hour really goes in the numerator. What goes in the denominator? No units. This works for units conversion. Lets say that you want to convert 1/2 hour to seconds. The conversion factors are: 1 hour = 60 minutes, or 1 = 60 minutes / 1 hour 1 minute = 60 seconds, or 1 = 60 seconds / 1 minute To convert, I would do the following multiplication: 1hour / 2 (no units) * 60 minutes / 1 hour * 60 seconds / 1 minute I can multiply by these fractions because it is the same as multiplying by one. The hour in the numerator of the first fraction cancels to one with the hour in the denominator of the second fraction. The minute in the numerator of the second fraction cancels to one with the denominator of the third fraction. This leaves me with just the seconds units in the numerator. If you want, I could multiply all of the fractions together before I cancel anything out. 1 hour * 60 minutes * 60 seconds
2 (no units) * 1 hour * 1 minute I could even separate the numbers from the units. 1 * 60 * 60 (hour * minutes * seconds)
2 * 1 * 1 (hour * minutes) I used to like seeing the terms that I wanted to cancel placed right above and below each other in the fraction. The hour units cancel and the minute units cancel to leave seconds. You are left with 1800 seconds. Multiplying and dividing units seems like a funny business, but it follows the same rules as numbers. For a unit like moment, you multiply a length unit by a force (weight, not mass - I won't go there) unit, which could give you something like foot-lbs. As a darned further complication, the dash does not mean subtract! It means to multiply. AAAAARRRRRGGGHHH! you can't have a unit that means add or subtract, but you can have units that multiply and divide. The reason you don't have a unit with adds and subtracts is that when you add or subtract terms, they HAVE TO have the same units. However, you can multiple and divide units to your hearts content. That is why you will see units separated with dashes. Atmospheric pressure is sometimes referred to as Lbs./sq. in. (psi or pounds per square inch). The lbs. would go in the numerator and the square inch would go in the denominator. If you had 14.7 psi, you might write it as 14.7 lbs.
1 inch * inch The square inch units simply means inch * inch. This example brings up another topic. I don't like the idea of sets when teaching fractions or units above a very basic level. For the psi case above, what is the set? I don't know or care. To deal with units, I can do everything in decimals and never care about the number of items in the set. Units are units. Sets are something else and seem to me to be a limiting concept for fractions. What if I am going around the merry-go-gound at 10 pi radians per minute? What is the set? If I have 5 apples out of a group of 10 apples, is this the same or different as having 1/2 of all of the apples. What is the set now? Kids might not know if the set is the units or the number in the denominator. Are fractions different if there are different numbers in the set? So, 1/2 is really different than 5/10? You want students to see a decimal as just another form for a fraction. You want them to see fractions as division. You don't want to set fractions apart as some other kind of animal. The approach to units I give above makes no distinction between fractions or decimals. You could even have things like 5.25 feet per 6.75 seconds, or 5.25 feet
6.75 seconds and everything will work out fine. Is there a rigorous path to mastery of fractions and units that doesn't involve (perhaps) dangerous simplifications? Probably not. However, I would not want to mix up units and sets. You can't add apples and oranges. It doesn't matter what the set is or whether you are talking about integers, fractions, or decimals. Units are not meaningful only in the context of fractions and fractions should not be used to introduce units. I think that there are different levels of understanding of fractions. When kids are in first grade, it's pretty easy for them to learn about simple fractions, like 1/2, 1/3, and 1/4. They seem to know that you are referring to a "whole" of something like a pie or a class or a rectangle. In later grades, students have to learn about fractions as division, decimals, the number line, that fractions can be greater than one, you cannot add apples and oranges (even if you have a common denominator), and that some fractions don't refer to a whole or a set, like miles per hour. They also have to learn that wording, like 2/3 of a class, means to multiply 2/3 by the number in the class. This is where they also learn the mechanics of add, subtract, multiply, and divide. This is probably not the time to get into a really heavy-duty discussion of units. The next level might involve using fractions (and/or decimals) to solve simple distance, rate, and time (and other) problems. Now they can learn more about units, conversion, and consistency. If they multiply a time number by a distance number to get a number in foot-hours, then it should tell them that something is probably wrong. If this discussion is focused on how to introduce fractions at the lowest level, then I don't think there is much of a problem. I see the problems arising in the second level (fourth and fifth grades) where the students have to move away from the tangible (sets and parts of a whole) to the abstract (fractions as division, decimals, and manipulating fractions). Fractions as division, decimal equivalents, and the number line are good ways to do this. Students need to know that a fraction is a number no different than the number 22. -- SteveH - 12 Oct 2005
That's interesting. It converted my dashed dividing lines into solid lines across the screen. I would edit it, but I think I would make it worse. -- SteveH - 12 Oct 2005
You know--the solid lines are probably useful. The code for a solid line is 3 dashes in a row. I used to constantly make solid lines by accident, since I use a lot of dashes in comments. -- CatherineJohnson - 12 Oct 2005
I don't like the idea of sets when teaching fractions or units above a very basic level. I haven't read this through AT ALL (I'm desperate for a warm shower)--but this popped out at me. Are you objecting to Aharoni's approach to fractions? Or did I miss part of your argument? -- CatherineJohnson - 12 Oct 2005
Have you read Hu's paper on fractions? (Is it Hu, or Wu??) I'll find it shortly. -- CatherineJohnson - 12 Oct 2005
I do it the same way that Steve described. It is very procedural, and once you've mastered the simple procedure it takes very little thought. It also results in less errors when doing complicated problems. Here's another example: convert 5 ft/sec into miles/hr (5 ft)/(1 sec) x (3600 sec)/(1 hr) x (1 mile)/(5280 ft) = (5
I'm going to print all this stuff out (I think) and try to get it set in Equation Editor. I'm having a lot of trouble following it printed this way.... -- CatherineJohnson - 12 Oct 2005
I can't believe you figured out how to do strikethroughs! Wow! -- CatherineJohnson - 12 Oct 2005
I'm trying one now:
Cool! -- CatherineJohnson - 12 Oct 2005
I've been wanting to do those forever, and didn't find them on the TWiki list. -- CatherineJohnson - 12 Oct 2005
"Are you objecting to Aharoni's approach to fractions?" I reread his article and I would have to say that I don't know. There aren't enough details. He talks about the layering of learning, which I talk about in my response. I think that at each layer or level, you might need to use techniques or explanations that might not be so proper at a higher level. The idea of a whole or "forming a unit" might be fine for the lower grades, but for the upper grades, "whole" might not be very clear or it might be limiting. I also assume that when he refers to "unit", he means the whole and not units like feet. A detailed understanding of units and their use is a more advanced subject, but I think that some level of units can be introduced in the lower grades. -- SteveH - 12 Oct 2005
Steve and Ken (is that right?), I had the same text processing problems in my first posts regarding dimensional analysis a few months back. That's why I punted, and resorted to MS Powerpoint for the instructional worksheets I put at the DimensionalDominoes page. That way I can use the "big parentheses" that I like to put aroung the fractions that include units. -- DanK - 12 Oct 2005
Let's see if the verbatim tag does what we want:
5 ft 3600 sec 1 mile ----- x -------- x ------ 1 sec 1 hr 5280 ft-- KDeRosa - 12 Oct 2005
not bad, but you wouldn't be able to put in any html code like strikethrough. -- KDeRosa - 12 Oct 2005
The verbatim tag I've been using on TWiki is the word code in brackets, I think -- CatherineJohnson - 13 Oct 2005
Steve H I'd have to re-read, but I think he's using 'unit' and 'whole' interchangeably....(hmm. I should re-read first.) Scratch that.
Here's a terrific example of why rectangular fraction tiles are superior to circles:
-- CatherineJohnson - 27 Oct 2005