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It occurs to me that to really understand the ways that people work with fractions on paper requires a good grasp of the ideas that numbers have multiple representations and that expressions can be manipulated in various ways without changing the number they represent. These are essentially algebraic ideas.

For example, adding fractions requires us to rewrite the fractions in a different form, and then essentially factorise the expression. This is the same as rearranging expressions in algebra. Dividing fractions requires us to rerepresent an operation like $\div \frac{2}{3}$ as $\times \frac{3}{2}$. This is the same as realising the connection between operations that you use to solve equations in algebra. And cancelling down before multiplying is very sophisticated rewriting relying on various associative and commutative laws.

So it seems that we are really asking children to think in algebraic ways in order to understand fraction calculations well. This would seem to me to be a good reason why children and adults find it hard - they need more scaffolding in some abstract ideas.

Is this a reasonable theory and has anyone written about this algebra-fractions connection before? To be clear, I am not asking if this is the only reason fractions are hard, but if there is any discussion out there to draw parallels between learning algebra and learning to manipulate fractions.

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    $\begingroup$ There is an enormous literature on fractions and rational numbers in mathematics education. Recently, I have been looking through some of the literature on hypothetical learning trajectories for which rational numbers crop up repeatedly. Try Wright (2014) and, especially, its references (no pay-wall to see the list, at least) for a tip-of-the-iceberg on why fractions are "hard." $\endgroup$ – Benjamin Dickman Apr 8 '15 at 8:15
  • $\begingroup$ Interesting paper in the link, thanks @BenjaminDickman. I am aware of the fact that the literature on rational number is overwhelmingly large, and I'm not interested in all of the reasons they are hard. I'm just interested in whether researchers think this particular idea is reasonable as one of the many reasons. Still, thanks for the link, it's very interesting. $\endgroup$ – DavidButlerUofA Apr 8 '15 at 12:48
  • $\begingroup$ I disagree: fractions are inherently artificial. $\frac{13}{7}$ is essentially the command "Divide 7 into 33". That we can operate on the code itself may occasionally be helpful but certainly not crucial in the real world. As engineers are wont to put it, ``The real real numbers are the decimal numbers''. And, even in an arithmetic course, there things a lot more important to deal with. Of course, though, you may be mandated by your state. $\endgroup$ – schremmer Apr 8 '15 at 18:45
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    $\begingroup$ I'm not sure what your point is @schremmer. To see the "command" 33/7 as an object in its own right and to operate on it as it stands sounds very like algebra to me. $\endgroup$ – DavidButlerUofA Apr 8 '15 at 19:51
  • $\begingroup$ @DavidButlerUofA You are of course correct but algebra is a large field and, given the sparse time usually allocated to it, I can think of more important things to deal intelligently with than fractions. $\endgroup$ – schremmer Apr 9 '15 at 0:37
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The obvious (to me) source of difficulty is that fractions are just plain complicated, more so than almost anything else in elementary education. You have to operate with a pair of numbers, instead of a single one, and you have to keep the order straight. Adding is quite complicated in its own right. Things are further complicated by rules about least common denominators and least terms.

I'm a little unclear about the question's emphasis on algebra. Any sort of general rule or operation in arithmetic must have a connection to algebra, but I do not see what is intrinsically difficult about algebra that relates to numeric fractions. Certainly some parts of algebra are hard, and some parts harder than others, algebraic fractions among them. It seems to me that fractions are difficult because it's easy to confuse the various bits. Even when you've got them straight, they're noticeably slower to use, take concentration, and when things have such cognitive demands, they're harder to think with.

Conceptually, they're a little bit odd, which is probably distracting until you get used to them. What they represent do not seem to apply to the same things that (whole) numbers do. Evidently fractions are not considered in this passage:

In that city, which was the oldest in the world, the cat was an object of veneration. Its worship was the religion of the country. The multiplication and addition of cats were a perpetual instruction in arithmetic. Naturally, any inattention to the wants of a cat was punished with great severity in this world and the next... -- A. Bierce, "A Revolt of the Gods"

Now to have one-and-a-half cats seems a very different thing than to have three halves. In the former case, there's a good chance that the one cat you have will be alive and purring, while the same could not possibly be said about any of the halves. No doubt such lessons are considered blasphemous in that city. While many things may be divided into parts -- cars are a better example than cats -- not many can be divided into equivalent parts that can be used as a basis for fractions. As we get used to fractions, as well as real numbers, we are taught to ignore this and accept statements such as "the average family has 2.4 children." Here is another example:

By then, she will have shed 80 of the 240 pounds she weighed in with when she entered Peter Bent Brigham hospital obesity program. A third of her left behind! -- The Boston Herald American, 7/7/77

The question seems to welcome references. There are certainly several that connect fractions with algebra. This paper,

Seigler et al. (2013), Fractions: the new frontier for theories of numerical development, Trends in Cognitive Sciences,

is a short survey of what is known and unknown about neural bases for one's knowledge of fractions. Whole number arithmetic knowledge has been studied, and the authors suggest that the representation of the knowledge fractions is an area ripe for investigation. It reviews (with references) why fractions are difficult and the relation of skill at fractions to skill at algebra. Generally -- or, rather, I only know of papers that discuss the connection in that direction, with algebra skill being dependent on fractions skill. (OTOH, I'm not widely read in this area.)

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  • $\begingroup$ The point about algebra is that to do algebra, students have to see things like $x$ as a number even though they don't know what it is, and they have to be able to think about equivalent representations of the same object and see them as the same. This is similar to having to see 3/2 as a number even though they don't know what it is, and think about equivalent representations of the same numbers and see them as the same. They have to do this just to add fractions. I'll edit the original question to include this. $\endgroup$ – DavidButlerUofA Apr 8 '15 at 23:32
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Not sure about paper references. One reason why people don't understand fractions is because they are seemingly illogical.

You score one basket out of three 1/3. A little while later you try again and score 1/2. Clearly you have scored 2/5 shots? In many ways this is the correct answer. So why shouldn't $\frac {1}{3}+\frac {1}{2}=\frac {2}{5}$

People generally don't understand equivalent fractions. It is strange for one farmer to say there are 4 sheep and another to say there are 8/2 sheep in the same field. People assume that the number 4 does what it says on the tin and is how we always describe 4 ness of something. They don't understand equivalence.

Partly to blame is treating fractions like conjuring tricks. If this is the question...do this, if this is the question ...do another uncorrelated thing. I asked my class (who seemingly could compute $\frac {2}{3}\times\frac {3}{5}$ correctly) to draw me a picture instead of just multiplying. No one could do it yet they all said "but it's $\frac{6}{15}$ you times the top and the bottom!"

I think drawing fractions is extremely useful. Draw $\frac{2}{3}\div2$ or $2\div \frac {2}{3}$ It's not easy but I find students develop robustness eventually and begin to abstract themselves.

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    $\begingroup$ Illustrating $2 \div \frac{2}{3}$ is tough, and may depend on the interpretation of division that one uses. Here is a quick drawing I created of possible ways to illustrate this expression (and "why" it equals $3$) using two different interpretations of division. For more on these interpretations and division in the context of whole numbers, see my answer in MESE 5648. $\endgroup$ – Benjamin Dickman Apr 8 '15 at 8:27
  • $\begingroup$ I agree it's not easy but it facilitates discussion about fractions and division in a way that just giving rules does not. Division as multiplicative universes is tough too, even dividing by 2 is same as 1/2 of something. Pick a denominator like 3 and the problem is exaggerated. $\endgroup$ – Karl Apr 8 '15 at 8:45
  • $\begingroup$ This answer has good things, but it's not really answering the question. In particular, I'm not sure if it's trying to say "yes this aspect of it is hard like algebra" or "no it's more because of this aspect which is not like algebra", or a combination of the two. Not trying to be critical, just trying to give feedback to help make the answer better. $\endgroup$ – DavidButlerUofA Apr 8 '15 at 12:44
  • $\begingroup$ @DavidButlerUofA Fair point I think I probably have drifted off the point and onto my thoughts on fractions in general. $\endgroup$ – Karl Apr 8 '15 at 15:45
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    $\begingroup$ Your example makes me wonder if the problem isn't simply that the basic arithmetic operations are glossed over -- if + is just thought of as a vague "combine two things", then it should be no surprise that people use + for the wrong way to combine things. $\endgroup$ – Hurkyl Apr 9 '15 at 9:56
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As far as I can tell, your intuition about fractions being "algebraic" is born from the fact that, like integer arithmetic, fraction arithmetic is built atop the arithmetic of the natural numbers. Specifically, every statement which we make about fractions can be transformed into a statement about natural numbers. The fundamental tool being that $$ \frac{a}{b} = \frac{c}{d} $$ can be transformed into $$a \times d = b \times c$$ (a statement about multiplication and equality of natural numbers).

The conceptual origin of fractions is both simple and complex. Most kids have experience with cutting a cake or cookie into equal halves so that everything is "fair". What rarely, if ever, happens is this: two kids have one cookie and decide to split it into four equal parts so that each kid can take two. This situation would only occur to a child if there was one cookie and four kids i.e. a cookie shared amongst four kids is broken into four equal parts and each kid gets one. The "confusion" and "complexity" comes from the fact that, in our experience, we don't often break things into any more equal pieces than we absolutely need so that we rarely deal with alternate "representations" of the same quantities.

It is for this reason that I've proposed that we focus on "ratios" rather than fractions because it is easier to talk of "equivalent" ratios as looking at the same "situation" from a different perspective. For example, a bowl with six oranges and two apples has a apple to orange ratio of "2:6" but notice that for each apple there are three oranges giving an apple to orange ratio of "1:3". We haven't broken anything into pieces, or talked of parts of a whole: rather we've just identified that the arithmetic of ratios (which is the same as that of fractions) is built atop the arithmetic of natural numbers.

Here are some motivations for the additive arithmetic of ratios:

If I give you one dollar every two hours and one dollar every three hours then then you have five dollars every six hours i.e. 1:2 + 1:3 = 5:6

If Anne puts one apple in the bucket every time Bob puts two oranges in the bucket and Charles puts one apple in the bucket every time Bob puts three oranges in the bucket, then five apples are put in the bucket every time Bob puts six oranges in the bucket: 1:2 + 1:3 = 5:6.

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  • $\begingroup$ Often something is broken into pieces for you, like a pizza or a kitkat bar. One of the great difficulties in teaching fractions is the addition of fractions. I do not see how to naturally motivate the sum of fractions from a ratio point of view. Could you illuminate this for me? $\endgroup$ – Steven Gubkin Mar 3 '16 at 4:16
  • $\begingroup$ If I give you one dollar every two hours and one dollar every three hours then then you have five dollars every six hours i.e. 1:2 + 1:3 = 5:6 $\endgroup$ – John Mar 3 '16 at 4:37
  • $\begingroup$ Is there an apples and oranges interpretation? $\endgroup$ – Steven Gubkin Mar 3 '16 at 4:40
  • $\begingroup$ If Anne puts one apple in the bucket every time Bob puts two oranges in the bucket and Charles puts one apple in the bucket every time Bob puts three oranges in the bucket, then five apples are put in the bucket every time Bob puts six oranges in the bucket: 1:2 + 1:3 = 5:6. $\endgroup$ – John Mar 3 '16 at 4:46
  • $\begingroup$ Also when things are broken into pieces for you, a child just wants the same number of pieces as any other children they are "sharing" with, which doesn't require fractions, just basic matching/natural numbers. $\endgroup$ – John Mar 3 '16 at 4:50
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Fractions are easier to relate to life and hence, should be easier to understand.

Start with slices of pizza, which even a first grader can understand, that if he/she ate one slice out of a total of eight, 1/8 of pizza was eaten.

Next is the concept of equivalency.

Any subtraction or additional expression can be evaluated once you understand the concept of LCM(least common multiple of two numbers) and use to it turn the denominator to a common multiple of the both/many denominators of various fractions. Same concept is applied while comparing the fractions.

Visit https://www.biglearners.com/worksheets/grade-5/math/fractions to practice the same basic concepts.

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    $\begingroup$ The concept of least common multiple is not needed for fractions (and indeed, there are contexts where there's no reasonable notion of "least common multiple", but fractions still make sense). Common multiples are still needed, of course, but that's just multiplication and is a lot simpler. $\endgroup$ – Daniel Hast Feb 25 '16 at 22:34
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    $\begingroup$ This does not seem to answer the question of what way fractions are or are not related to algebra. $\endgroup$ – Daniel R. Collins Feb 26 '16 at 6:00
  • $\begingroup$ I downvoted this answer too because some people still might not get helped by that explanation because taking 1 slice of a pizza that was cut into 8 slices is not the same thing as taking 2 slices from a pizza that was cut into 16 slices. $\endgroup$ – Timothy Jan 16 at 5:20

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