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In Hung-Hsi Wu's monograph on Fractions written for pre-service and in-service teachers, he makes two strong points: keep a number line handy when you define fractions, and learn how to order fractions before you try to add them.
From Section 5, Ordering Fractions (the Cross-Multiplication Algorithm),
Before we approach the addition of fractions, we first consider the more elementary concept of order, i.e., comparing two fractions to see if one is bigger than or equal to the other (recall that equality in this case means they are the same point on the number line).
Given two fractions A and B, we say A < B if A is to the left of B as points on the number line. This is the same as saying that the segment [0,B] is longer than the segment [0,A]....
We emphasize once again the need to put fractions and whole numbers on the same footing. It would have been preposterous to define order among fractions in a way that is different from the definition of order among whole numbers. Observe also the ease with which we define order among fractions when the number line is at our disposal.
The main objective of this section is to show that a comparison of two fractions a/b and c/d can be made by inspecting their “cross products” ad and bc. This so-called cross-multiplication algorithm has gotten a bad name in recent years because it is supposed to be part of learning-by-rote, and the reason for that is because many textbooks just write it down and use it without any explanation. As a reaction, the curricula of recent years have a tendency of not even mentioning this algorithm.
Using the algorithm without explanation and not mentioning the algorithm at all represent the two extremes of mathematics education. Neither is good education, because this algorithm is a useful tool which can be simply explained.
Consider the following example. Which is the bigger of the two: 4/7 or 3/5?
In terms of segments, this should be rephrased as: which of [0, 4/7] and [0, 3/5] is longer? Now by definition:
4/7 is 4 copies of 1/7
3/5 is 3 copies of 1/5
This comparison is difficult because the two fractions are expressed in terms of different “units”: 1/7 and 1/5 . However, imagine for a moment that the following statements were actually true for some whole number c :
4/7 is 20 copies of 1/c
3/5 is 21 copies of 1/c
Then we would be able to immediately conclude that 3/5 is the bigger of the two because it includes one more segment (of the same length) than 4/7 . This suggests that the way to achieve the desired comparison is to express both 1/7 and 1/5 in terms of a common “unit."
[We have to decide on a common unit for 1/7 and 1/5 . The cancellation law suggests the use of 1/35, so]
4/7 = (5 × 4)/(5 × 7) = 20/35 = 20 copies of 1/35
3/5 = (7 × 3)/(7 × 5) = 21/35 = 21 copies of 1/35
Conclusion: 4/7 < 3/5.
A closer look of the preceding also reveals that this conclusion is based on the inequality 5 x 4 < 7 x 3.
I think Wu would rather show his students (teachers) how to deduce cross-multiplication from the definition of fractions as points on the number line, than let his students induce cross-multiplication as best they can by searching for numerical patterns in many examples.
If we respect the time and attention of 20 year-olds by showing them directly why cross-multiplication works, why not figure out a way to respect the time and attention of 10 year-olds, and show them directly why cross-multiplication works?
Wu is quite clear that his monographs are for teachers, but I have had a great deal of success using the number line as he does to explain fractions to my boys. I haven't taught my boys to cross-multiply, but I have taught them to look for a common denominator when they are comparing fractions, or when they are trying to move from one (fractional) point (on the number line) to another (i.e. add and subtract fractions).
However, I am convinced by the example of my children that you can tell a compelling conceptual story to get a child to believe a procedure works, but he will forget the details of the conceptual story long before he forgets the procedure. Because we make him practice the procedure, not the story.
So-called "conceptual" curriculums like Investigations hold out the promise that children will learn the conceptual story by heart because they will write it for themselves, inducing the rule from many examples we show them without comment.
But a well-told conceptual story may be best.
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22 comments:
My third grader's homework was to look at each set of pictures, circle the fractional part and then write the fractional part.
2/3 of 12 =?
There were 3 sets of pencils drawn. Each set contained 4 pencils.
There was no explainaton. Just a drawing.
Then on the other side he had to find the fractional part by drawing stars to help him figure the problem out.
I saw where the teacher had tried to explain 3/8 of 16 = 6. She drew 8 sets with 2 dots in each set.
He did not understand her explaination. How should I explain this so he can understand it?
I know that 3/8 of 16 = 6 because 16 divided by 8 =2 and 2 x 3 = 6. Thus, there are 8 sets with two in each set. Is this a good explaination or should I show him another way?
His school uses TERC.
"..keep a number line handy when you define fractions, and learn how to order fractions before you try to add them."
Yes. All positive and negative integers, decimals, and fractions are just numbers or points on the number line. Some kids think that fractions and decimals are "something else". Kids also have to first learn about equivalent fractions. Most math curricula do this.
However, I find that issues arise later when the problems become more complicated. The simple graphical descriptions used to introduce the concepts don't hold up to more abstract usage.
It's one thing to divide a pie into 6 pieces and see that 2/3 is the same as 4/6. They might even teach that you can change any fraction to anything else by multiplying the top and bottom numbers by the same number. (without justification, of course) But they never go back and teach the full understanding and application of the mathematical rules, like
a*1 = a
and
a/1 = a
and
a/a = 1
I distinctly remember thinking that these were stupid (obvious) rules. Only later, while I was struggling to fully learn algebra, did I see how important they were.
The problem with fuzzy math is that they stop at descriptive understanding and never get to a proper mathematical understanding.
I don't think ordering is such a big deal. Just divide the two fractions (in your head for the simple ones) and compare them. Or, just find the common denominator.
If they understand equivalent fractions, then it should be simple to have students find equivalent fractions that have the same denominator.
"Using the algorithm without explanation and not mentioning the algorithm at all represent the two extremes of mathematics education. Neither is good education, because this algorithm is a useful tool which can be simply explained."
"Neither is good education"? What a faulty premise.
"In terms of segments, this should be rephrased as: which of [0, 4/7] and [0, 3/5] is longer?"
Let's see how difficult we can make it for the kids just to show that cross-multiplication is really OK.
"However, imagine for a moment that the following statements were actually true for some whole number c :
4/7 is 20 copies of 1/c
3/5 is 21 copies of 1/c"
"20 copies of 1/c"? I'd like to see the faces on the 4th grade kids when the teacher puts this on the board.
The problem with cross-multiplication is that it is not easier to understand than the proper methods and it can create many other problems.
Kids should learn about and use the number line. They should learn about different kinds of numbers and where they are located on the number line. They need to learn about equivalent fractions and how you can create them by multiplying the numerator and denominator by the same number, which is the same as multiplying the number by one.
Then, all they need to know is that to compare or add or subtract fractions, all you have to do is find equivalent fractions that have the same number (of parts) in the denominator. Then, all you have to do is to compare (or add or subtract) how many of these parts (numerator) you have.
Also, as I have mentioned before, I liked to tell students that every number can be seen as a fraction by divided by 1.
Usually, you think of this:
a/1 => a
how trivial.
but it is really much more powerful when it is viewed like this:
a => a/1
You can even change a number like 2.71828 into a fraction:
2.71828/1
I really liked this for dividing fractions where one of the fractions is not a fraction. I can make it into a fraction. Invert and multiply was never so universal.
What if you have:
3x/(3/(x-2))
This is no different in approach than solving:
3/(3/2)
(3/1)/(3/2) = 2/3 * 3/1 = 6/3 = 2
invert and multiply
(3x/1)/(3/(x-2)) = (x-2)/3 * 3x/1 = 3x(x-2)/3 = x(x-2)
So, I guess my point is that I'm a by big proponent of the usage and understanding of the fundamental rules. They may seem trivial for simple problems, but they are very powerful as the problems become more complex and abstract.
Shortcuts are fine if they are related to the fundamental rules. Cross-multiplication, however, is not an easier method to learn, and can cause great problems when students think of it as some sort of fundamental rule.
Steve, I agree with the primacy of common denominators. In order to keep my post shorter I left out a paragraph in which Wu gives the example of comparing A meters with B yards -- working with familiar units of measurement before tackling the abstract units of A fifths and B sevenths. He's making the point that you've got to find a common unit in order to make the comparison etc.
Which ties in with discussions here on KTM about the importance of dimensional analysis. Noticing the units you are working with.
Also, I should be very clear that Wu is very clear that he is speaking to teachers, he is not outlining a curriculum for children. Sorry about the confusion.
He's trying to change teachers' minds about math, and show them that math is not wild and woolly and imprecise. When a teacher and twenty-five children talk about fractions they should not be like twenty-six blind men describing an elephant.
I think his great ideas for developing conceptual understanding in adults and his choice of visuals are great for children, too.
Paula,
I know that 3/8 of 16 = 6 because 16 divided by 8 = 2, and 2 x 3 = 6. Thus, there are 8 sets with two in each set. Is this a good explanation or should I show him another way?
I like your way. First find the size of one eighth of 16, then circle three groups of that size and count dots.
The division into eight equal parts comes before summing three parts.
At this age, I doubt it's necessary to move children into procedural fluency with formally multiplying fractions. That can come later and not be too late.
But notice that these problems only make sense to children because we keep the numbers small and friendly -- we're informally asking 3/8 x 16, not 3/7 x 16 or 16 x 3/8. We're relying on their knowledge of multiplication and exact division of whole numbers.
The operational steps in your explanation fit with Wu's choice to first define the unit fraction e.g. 1/8 of one whole, then define what we mean by three one-eighths of one whole, and only then prove that three one-eighths is the answer to the question, What is the size of each part when three wholes are divided into eight equal pieces? His whole is [0,1] on the number line; your whole is sixteen dots.
His is not a discovery method, it's a deductive method of teaching fractions to teachers.
I haven't worked through the 3rd grade TERC materials with my own children, so I am not familiar with the sequence of those teaching examples. I only know what the 4th and 5th grade materials look like.
With respect to the twelve pencils, it seems good that TERC tries to focus a child's attention on first seeing 1/3 of the pencils, by depicting the twelve pencils in three groups of four pencils.
The topic "Fraction of a Set" is introduced in the Singapore program at level 4A. In all of the introductory examples for that topic, they have pre-grouped the objects to focus children on first finding the size of the unit fraction of the whole.
Example 3, "Find the value of 3/4 of 20" uses a picture with twenty circles in groups of five circles each. The circles in three groups are shaded, the circles in one group are left unshaded.
There are a few examples where the unit fraction of the set will not be a whole number, e.g. find 3/4 of 9.
Hope this helps.
"4/7 is 20 copies of 1/c
3/5 is 21 copies of 1/c"
This is great--I really learned something here. I had always thought of cross multiplying as a cheap shortcut that skipped over the algebraic making-sense of multiplying both sides by the denominators, but this is significantly better... I think... Hmm, I don't know whether to respect cross multiplying more, or whether to think now that it is a cheap shortcut for finding a common denominator. Is that an improvement? Maybe.
OK, now I've revealed myself as a non cross-multiplier, I'd like to know from you cross multipliers, does this solution strategy seem like the best one to you? To me it seems like it has an extra step more than it needs, but maybe I'm just not thinking about it from a student perspective:
x/100 = 3/10
10x = 3*100
x = 300/10 = 30
Despite having championed cross multiplication as a shortcut on a previous thread, in this case I find it more natural to simply multiply both fractions by the same positive number. This preserves the ordering. Thus:
20 = 35*(4/7) < 35*(3/5) = 21 <=> 4/7 < 3/5.
I think the reason I'm happy to use cross multiplication on an equation (i.e., a/b = c/d) is that its operation is very straightforward. By contrast, with a (potential) inequality, you must also remember which way the inequality goes. That is, you must remember that the inequality "goes with the numerators."
This may just be personal taste, but that is how I would tackle this problem.
As an interesting aside, if a, b, c, d are positive integers such that
a/b < c/d,
then
a/b < (a+c)/(b+d) < c/d.
Although the above seems the most natural and direct method to me, I wonder how it would play with children who are just learning fractions? Is the idea of multiplying both sides of a relation by a positive constant a familiar one? If not, maybe the best approach from a pedagogical standpoint is indeed as Wu suggests: write both fractions with a common denominator of 35.
A perverse approach is to create a common numerator. We thus obtain:
12/21 < 12/20,
as the denominator of the first fraction is larger, implying
4/7 < 3/5. This approach has the advantage in some cases that the numbers are smaller, but I'm not suggesting it should be inflicted on the kids!
This is great--I really learned something here. I had always thought of cross multiplying as a cheap shortcut that skipped over the algebraic making-sense of multiplying both sides by the denominators, but this is significantly better
Me, too!
I was taught cross-multiplying as a means of determining which fraction was larger, but I had no idea why it worked and when I started reteaching myself math I didn't use it.
Speaking as a novice, it's amazing how much I don't "see" about the math I'm studying, doing, preteaching & reteaching, etc.
The line about math being connections from math to math was so powerful for me.
I've known I had profoundly fragmented knowledge ever since reading Liping Ma's book, but that line summed it up as concisely as anything I've read.
If I can revert to cross-multiplication as a way of solving a ratio for a moment....I'd like to get everyone's opinion on what I've done with Christopher.
He's been taught to cross-multiply the same way he's been taught everything for the past two years, as a Math Procedure from God.
I've been dealing with the MPfromGod issue by giving him ratio problems & having him set them up two ways:
* as a ratio
* as a dimensional analysis
Then I point out to him that the dimensional analysis form is a way of setting up the question without using a variable.
The computations are the same.
keep a number line handy when you define fractions, and learn how to order fractions before you try to add them
The other big revelation, to a kid, is that RULERS ARE DIVIDED INTO FRACTIONS.
The simple graphical descriptions used to introduce the concepts don't hold up to more abstract usage.
For me that moment arrived with division of a fraction by a fraction.
The problem with fuzzy math is that they stop at descriptive understanding and never get to a proper mathematical understanding.
I don't know whether I have a proper mathematical understanding or not -- though I may -- but at some point the abstract numerals and letters "took over."
I still find any and all visualizations of concepts incredibly important when learning a new concept, however.
For instance, I still can't convert polar to rectangular coordinates (or vice versa) without sketching a coordinate plane.
Also, as I have mentioned before, I liked to tell students that every number can be seen as a fraction by divided by 1.
Usually, you think of this:
a/1 => a
how trivial.
but it is really much more powerful when it is viewed like this:
a => a/1
Yes!
Absolutely!
Wonderful!
3/(3/2)
(3/1)/(3/2) = 2/3 * 3/1 = 6/3 = 2
invert and multiply
That illustration was the thing that finally led me to understand (assuming I do understand!) "invert and multiply."
I ran across it on a homeschooling math website.
Until I saw it, I'd kept trying to visualize what division of a fraction by a fraction had to be.
Couldn't do it.
A perverse approach is to create a common numerator. We thus obtain:
12/21 < 12/20,
as the denominator of the first fraction is larger, implying
4/7 < 3/5. This approach has the advantage in some cases that the numbers are smaller, but I'm not suggesting it should be inflicted on the kids!
This is probably a good thing to do!
Either Singapore or Saxon (or both?) teaches this concept by showing what happens when the denominators are the same, what happens when the numerator in both fractions is 1, and what happens (I think) when the numberator is the same.....
hmmm....
Am I misremembering?
In any case, whether they did or did not do all three things, doing all three things is a very good idea for anyone teaching fractions to me.
Singapore level 3B steps kids systematically through comparing
1. unit fractions,
2. fractions with common numerators,
3. fractions with common denominators,
4. fractions where one denominator is a multiple of the other, and finally
5. fractions where a common denominator must be found.
Singapore uses pie graphs to illustrate the first three types, rather than bar models.
In 5th grade, Investigations has children reason about the relative size of unlike fractions by any other means than finding common denominators.
They are given opportunities to reason about two fractions near 1, e.g. 5/6 and 6/7, by comparing 1/6 and 1/7. This is appropriate. They also want children to reason about two fractions near 1/2, e.g. 5/8 and 6/10, by comparing 1/8 and 1/10.
1/2 is a "landmark." There is no further move towards common denominators. As with their resistance to any standard computational algorithm, they don't teach kids how to find a common denominator.
Just as the authors of TERC would wail and gnash their teeth if a child set up a standard algorithm to multiply 98 x 2, they would dress in sackcloth and cover themselves with ashes if a child found a common denominator to compare 5/6 and 6/7.
I dunno. I think the benefit of children mastering the means to multiply unfriendly numbers and to compare unfriendly fractions is worth the occasional inefficiency.
"x/100 = 3/10"
I saw over and over again that when students see two fracions surrounding an equals sign (pattern recognition), they think of cross-multiplication. They fail to recognize that this:
x/100 = 3/10 + 5
is a different pattern.
So, is cross-multiplication bad because it's often taught badly? No, I just think it's just safer to avoid it altogether. Back when I taught, my algebra classes were filled with students who had bad patterns in their heads. The patterns never evolved to mathematical understanding. And everything was patern recognition. If it looks like problem A, then use that technique. If it looks like problem B, use that one.
For the original equation above:
x/100 = 3/10
I wouldn't teach finding a common denominator because I am not adding or subtracting fractions. The "=" sign is not a '+' or a '-'. I am not even comparing fractions. I am solving an equation with one unknown. There is a different approach for solving these problems. First, move all 'X' terms to the left side of the equals sign and all constant terms to the right. Combine the 'X' terms on the left and the constant terms on the right. Multiply or divide through all terms to get the 'X' variable all by itself, and so forth. This is simplified, perhaps, but the only time a common denominator is required is if I'm adding or subtracting fractions. Cross-multiplication is never needed, even for comparing fractions. For comparing fractions, finding a common denomintor is good practice, and with enough practice, will automatically evlove into cross-multiplication, but without the pattern.
"I don't know whether I have a proper mathematical understanding or not -- though I may -- but at some point the abstract numerals and letters 'took over.'"
Even with the "traditional" math I was taught, this transition was not automatic. I have mentioned before that I didn't feel that I really knew algebra until I was in trig (pre-calc) as a junior. This was the point where I felt I could do any sort of algebraic manipulation without any confusion what-so-ever. I remember the teacher putting on the board a very long derivation that showed that 1 equals 2. We had to find the mistake.
A common strawman is that we "traditionalists" just love the old ways of teaching math. The truth is that we are just horrified that the solution went in the completely wrong direction.
Even with the "traditional" math I was taught, this transition was not automatic. I have mentioned before that I didn't feel that I really knew algebra until I was in trig (pre-calc) as a junior. This was the point where I felt I could do any sort of algebraic manipulation without any confusion what-so-ever.
This is interesting, because of course my own motivations for learning math are so different from a high school kid's motivations.
First of all, I have "mama bear" drive and determination; I'm convinced that my child will not learn math if I don't learn math first and then preteach & reteach what I learn to him, AND I'm equally convinced that his life will be appreciably harmed and his options limited if he does not learn K-12 math and learn it very, very well.
"Mama bear" motivation is HUGE.
Then, too, as it turns out I simply like math (or K-12 math, at this point).
I love it!
I find K-12 math stimulating and soothing at the same time.
So....I've been intensely motivated not just to learn math, but to understand math -- and not to "understand" math in some kind of "solve problems in the real world" TERCian-type way, but in the pedagogical content knowledge way.
I need to understand it so I can teach it.
In a way, these motivations may have hung me up on a transition from "visual" representations to abstract representations.
For quite awhile I had the internal sense that I didn't understand division of a fraction by a fraction because I couldn't visualize it.
And then that "feeling-of-not-understanding" dissolved.
So....I don't know whether I do understand math; I don't know whether a mathematician would say I understand arithmetic and beginning algebra.
But I have a "feeling" of understanding!
I'm not sure I'd say that I'm confident I can do any form of algebraic manipulation (within algebar 1), which was the criterion for Steve's sense of understanding.
I'm reasonably close (again, this is with algebra 1).
I'd love to know if my "feeling of understanding" is based on anything real.
Catherine said:
"This is interesting, because of course my own motivations for learning math are so different from a high school kid's motivations.
First of all, I have "mama bear" drive and determination; I'm convinced that my child will not learn math if I don't learn math first and then preteach & reteach what I learn to him, AND I'm equally convinced that his life will be appreciably harmed and his options limited if he does not learn K-12 math and learn it very, very well."
Welcome to my world. Looks like there are others out there like me. If you are interested in taking a lookey loo at other math programs, you can email me at kh at kathyandcalvin dot com.
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