Why do some students struggle so much with fractions?

Admittedly, a fraction has twice as many components as (say) an integer, and these components greatly increase the numbers of ways they might interact, in a combinatorial fashion. Consider an arbitrary binary operation on integers: $a odot b$. With only two components to the arguments, there is only a single relation that needs consideration: the one between $a$ and $b$. On the other hand, consider fractions:

$$frac a b odot frac c d$$

In this case, there are actually $6$ relations that might need consideration: $(a, b), (a, c), (a, d), (b, c), (b, d), (c, d)$. Sextupling the cognitive load on the student is quite a leap forward. Of course, the relations that need consideration/manipulation are different for the various different operations of addition, subtraction, multiplication, division, etc., so there's a lot to digest and memorize there. (This is to say nothing of mixed numbers, with 6 components, for a total of 15 possible relations.)

Consider the operation of addition on proper fractions: In most school situations this involves (1) factoring the denominators $b$ and $d$, (2) composing the least common denominator by inspecting those factors, (3) multiplying each component $a, b, c, d$ appropriately to create the desired denominator in each fraction, (4) adding the modified numerators and writing over the common denominator, (5) factoring the new numerator again, and (6) possibly canceling like factors in the numerator and denominator (and then multiplying the remaining ones back together to simplify).

So in summary: It seems by my estimation that the conceptual and mechanical work in handling fractions is about $6$ times that of integers. Granted that some theories of working memory suggest a capacity of perhaps 4 or 7 chunks (and less in small children), this may in fact be at or beyond the limit of many children's short-term memory to manage.

I don't know for sure. But I think a part of the problem comes from notation. I don't know how you've approached math education, but I find that students are often times confused by the distinction and overlap between the concept of division and the concept of fractions. I think the cause of this confusion is found in the notation we use and the order in which we teach the subject.

We cant tell them that 10/2 = 5 is division and it requires that the top number be bigger than the bottom number... and explain to them that this is an operation between two integers....

... and then show them something like 2/7 and tell them this isn't an operation between integers, even though it looks like it, but an entire numeric entity all by itself.

My suggestion has always been to restrict ourselves to the use of the obelus when talking about division, and leave the slash or horizontal bar for topics on fractions.

Then after you explain division of fractions, you start showing them the relationship between division of integers and the entity known as a fraction. At this point the fraction bar and the obelus becomes the same object.

It's my hope to convert the traditional "believe this in faith that they are the same thing because I said so" into a genuine epiphany that they can have themselves.

There are many reasons why fractions are so hard for students to learn. Mostly, they're taught gibberish and assessed according to such gibberish.

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Example 1

You are a 12-year-old student who has learned that "a fraction is part of a whole, such as part of pizza". So when you look at $frac{2}{3}timesfrac{7}{5}$, you now must multiply pizza slices by pizza slices. This nonsense is obviously impossible to understand. Whether or not your teacher says to just memorize the procedures for multiplying fractions, that's how you're assessed so that's what you do. You give up on sense-making. You never learn what it means to multiply by a fraction, so, even years later, you still think that $frac{2}{3}times700$ must be bigger than 700 because there's 700 and multiplication. You're not sure if you need common denominators, though, because all those other fraction rules you tried to memorize are getting confused. "I hate fractions!" you say.

Example 2

Word problem: "A plane can go 600km/hour [fraction]. A retrofitted wing can cause it to go $frac{1}{10}$ [another fraction] faster. How far can it go in half-an-hour [another fraction] with the new wing?"

Student: "I see the fractions. Where are the pizzas? My teacher says to draw a picture if I don't understand a word problem, so I should draw a pizza cut into 600 slices? How do I slice the pizza into an hour?"

Example 3

Teachers tell you over and over again that whatever you do to the top of a fraction, you must also do to the bottom of a fraction. But they never tell you why or provide you with any sensory experiences or contexts to develop your intuition for this. They don't tell you when this is appropriate or useful or even what it means, and you're definitely never tested or graded on how it compares/contrasts with all the other things you can/should do with fractions. You seem to get check marks when you cancel zeroes in $frac{600}{7100}=frac{6}{71}$, though, so, when you're doing first-year calculus at university, you think you can cancel the $x$'s to simplify as follows:

• $frac{x+3}{x+4}=frac{3}{4}$

• $frac{sin(x)}{x}=sin$

And you're pretty sure that $frac{a}{b}times2$ is the same thing as $frac{atimes2}{btimes2}$, right? Whatever you do to the top, you also do it to the bottom... right? That's usually what got you good enough grades in the past...

Example 4

Sometimes teachers say that when given $frac{26}{6}$, you should type $26div6$ into the calculator. But I challenge you to find any textbook in North America that explains why this might be a reasonable thing to do and why the fraction bar and the $div$ are always interchangeable while the fraction bar and, say, $times$ are not. Even harder: find any textbook or test or teacher or tutor that asks students to compare and contrast $frac{26}{6}$ vs $frac{26div6}{6div6}$ vs $26div6$ vs $frac{26div2}{6div2}$ vs $frac{26}{6}div6$ vs $frac{frac{26}{6}}{6}$ vs $frac{6}{frac{26}{6}}$ vs $4.33333...$ vs $4frac{1}{3}$ etc.

One reason that students are not taught or tested this way: they tend to bomb horrifically, immediately. If a student who types $26div6$ into his calculator when he sees $frac{26}{6}$ - only because he was told to yesterday - then a teacher can feel that student understanding is confirmed. The student gets a good report card, moves on to the next topic or grade-level, can apply to a prestigious educational program, etc.

The bad news - that the student is horribly confused by all the superficially similar expressions above - could have been uncovered immediately, but it almost never is. It's uncovered, say, when the student is in grade 12 trying to learn calculus and intelligent interpretations of all of those expressions is just assumed even though it has never been tested.

You may have been in comparable situations in the past; perhaps you're in some now. Are you actively trying to collect evidence that student understanding and retention are vastly weaker than they appear so that, if they are, grades will plummet immediately and cause hideous political fights for you, your students, your job, your admin, etc.? Or do you give into the pressure to give students sufficient grades to move them towards the next educational goal posts?

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There are at least two ways main ways to avoid all this gobbledygook. The first is to teach actual math instead of gobbledygook. In the realm of fractions, that means teaching that fractions are numbers and in every single lesson and worksheet that involves fractions, to compare and contrast them with whole number(s), and to not introduce anything with two fractions until the student can do the same thing with one or two whole numbers and one fraction. Don't bother with $frac{2}{6}-frac{1}{6}$ until the student can draw a picture for $2-frac{1}{6}$. Don't bother with common denominator fraction comparisons until the student can easily use number lines to compare $frac{19}{4}$ and $4$ and $19$.

I affirm those who say to read the work of Hung-Hsi Wu.

The second way to avoid this mental gobbledygook is meaningful interleaving.

If students are given a blocked (monotonous) set of exercises, such as:

Find equivalent fractions for $frac{1}{3}$, $frac{5}{2}$, $frac{10}{8}$, $frac{6}{7}$, $frac{2}{2}$, and $frac{3}{8}$.

Then the normative response will be to mindlessly multiply the top and bottom number by, say, 3, over and over and never thinking about meaning. The students will just rush to finish with correct answers.

On the other hand, if students are given an interleaved set of exercises:

Create a word problem and draw a picture for each of the following: $frac{1}{3}times2$, $frac{1times2}{3times2}$, $frac{1}{3}div2$, $2divfrac{1}{3}$, $frac{1}{3}=1div3$.

Then the students must think about meaning. They might need a caring teacher to gradually build them up to this kind of challenge, but there's really no comparison as to which set of exercises will make students think more meaningfully. In addition to being pedagogically difficult, though, this is also motivationally hard as students and parents - and most teachers - tend to care most about what's on the test, and these kinds of exercises are rarely on tests.

The short-run politics of determining the truth about student understanding is a nightmare. It lowers grades, so it prevents students from making honor roll or getting into an enriched math class or private school, it might make them lose a scholarship, it will cause tension in families, it will harm student status, parents scream "This isn't how I was taught math!", people will be convinced you're a bad teacher because student grades are low, students try to transfer out of your class, admin gets angry, etc.

I haven't figured out how to deal with such politics yet. :P