# Why do some students struggle so much with fractions?

I read on multiple web pages something that implies that that some students really struggle with fractions but I could never find a detailed explanation of why. This question is different from Are fractions hard because they are like algebra?. I'm not sure whether this question is more suitable for Academia Stack Exchange or whether questions that would normally belong on Academia Stack Exchange but are about math belong here so I'm asking it here.

I also don't know in which way they struggle with fractions or what they mean exactly by it and probably cannot find the answer to that so that's why I'm not stating here in what way they struggle with fractions. I think there were researchers out there who researched how primary math education should be taught. If so, would it be possible for anyone to give me a lot of details about what those researchers discovered on the topic of teaching fractions to elementry school students and write them in a way that clearly explains in detail why some students are struggling so much with fractions and why certain methods of teaching are claimed to be the best way to teach given the current information researchers have about how students learn?

• This does not tackle the issue of "struggling", but H. Wu wrote a text about pre-algebra (and other subjects) and how they should be taught; it might be of interest to you. See this link (p. 9) for the chapter on fractions: math.berkeley.edu/~wu/Pre-Algebra.pdf. Found the link to these notes on this question: matheducators.stackexchange.com/questions/1857/… – orion2112 Feb 21 '19 at 5:16
• Why wouldn't students struggle with fractions? You can multiply the top and bottom by 4 if you want, but you can't add 4 to the top and bottom or square the top and the bottom. You can cancel 4s if they appear on both the top and bottom, but not always. Sometimes your teacher tells you to "clear fractions" by multiplying by the denominator. Sometimes your teacher takes off points if you do that because multiplying by something changes the result. If you have two terms added together on top, you can split it into two fractions, but if you have two terms added together on the bottom, you can't. – Chris Cunningham Feb 21 '19 at 13:51
• Not listed in 'related' to the right, but on this topic, How to explain fractions to 7 year old kid – JTP - Apologise to Monica Feb 21 '19 at 19:18
• @JoeTaxpayer I think I also read the answers to that question and none of them answer this question. – Timothy Feb 21 '19 at 19:27
• Just a thought... Could it be because they haven't memorized the multiplication tables and have to rely on finger math or other "strategies" that can't be worked backwards to figure out lowest common denominators? – Dan Christensen Feb 24 '19 at 3:02

As for research on fractions education... there's a TON.

As for research on fractions education that rigorously measures cause and effect through randomized controlled experiments and long-term follow up... there is, I think, little to nothing as of 2010. And, since the field of education rarely uses that kind of research design, I doubt there is any need to update that report.

If someone out there is more familiar with the research on fractions pedagogy than I am, please know I would love to be contradicted on anything I've written. :)

• I guess it's fine to write answers that you're not absolutely certain of because if researchers insisted on only using statements they can prove for certain are true, they would discover so much less and research would go so much slower, and that's why this answer is open to the possibility of contradiction. – Timothy Feb 25 '19 at 0:15
• In my experience, most education "research" is meaningless and does not discover anything. It is rarely scientific and is often designed to support the latest fad (often for political purposes). The latest maths fad, supported by "education research" in NZ cost millions in teacher retraining and students' maths scores dropped. Ministry of Education fired the university that did the testing and carried on. I wish education research would go a little more slowly, do proper scientific research, and discover how students really learn. – Richard Mar 12 '19 at 21:40
• @Richard . Chief among the problems with ed research in the USA [dunno about NZ] is lack of funding. IIRC, typically, students' tuition money goes into the education faculty, then huge portions of it are transferred out. Ed faculties rarely get subsidies or big donations. So, there is just no money for big and long-term RCTs. Even if there was, measurement effects would still be hard to disentangle from treatment effects. So, ed research tends toward qualitative, intuitive, anecdotal, theoretical, short-term, small sample, etc. Kinda like medical "research" from the 1500's... – WeCanLearnAnything Mar 14 '19 at 2:38

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

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 $$\frac{2}{3}\times\frac{7}{5}$$, 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 $$\frac{a\times2}{b\times2}$$, 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 $$26\div6$$ 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 $$\frac{26\div6}{6\div6}$$ vs $$26\div6$$ vs $$\frac{26\div2}{6\div2}$$ vs $$\frac{26}{6}\div6$$ vs $$\frac{\frac{26}{6}}{6}$$ vs $$\frac{6}{\frac{26}{6}}$$ vs $$4.33333...$$ vs $$4\frac{1}{3}$$ etc.

One reason that students are not taught or tested this way: they tend to bomb horrifically, immediately. If a student who types $$26\div6$$ 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?

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: $$\frac{1}{3}\times2$$, $$\frac{1\times2}{3\times2}$$, $$\frac{1}{3}\div2$$, $$2\div\frac{1}{3}$$, $$\frac{1}{3}=1\div3$$.

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

• Create a word problem and draw a picture for each of the following: This sounds like a great exercise, and if you are teaching kids of this age and assigning stuff like this and giving kids feedback on it, then you're a hero and I hope my grandkids can be in your class someday. But I suspect that very few K-6 teachers are willing to read this amount of written work and write detailed feedback on it, especially given that in the US they may have classes with 35 or 40 kids. There is also the issue of teachers' own mathematical competence, which is not addressed in this answer. I suspect [...] – Ben Crowell Feb 23 '19 at 19:57
• [...] that quite a large fraction of K-6 teachers do not themselves understand fractions and division well enough to do this exercise. – Ben Crowell Feb 23 '19 at 19:58
• I'm not complaining about your answer but I did upvote it but didn't put a check mark beside it because your answer doesn't give a more detailed description of why some students struggle so much with fractions that you explained clearly. – Timothy Feb 24 '19 at 2:13
• @BenCrowell . I tend to work 1-on-1 with students so I do assign problem sets like the one I recommended. I've never tried it in a classroom setting, but I'm pretty sure one way to build up to that is "assessment as learning". Sadly, this would require teachers to create tons of material by hand, because you won't find tough stuff like that online! "Assessment as learning" exercises, if graduated properly, would help teachers master fractions as well. This is mostly conjecture, though, until I try it in a classroom in a few weeks... :) – WeCanLearnAnything Feb 24 '19 at 7:55
• @zipirovich - Thanks! I'm not sure which part of Example 3 contradicts the notion of fractions as numbers. In the big picture, students need to learn roughly in this order: (1) Fractions are numbers, (2) Equivalent Fractions, (3) Fractions as Quotients [quotients are just particular type of number], (4) Fractions and quotients as ratios, rates, and probabilities. And, yes, $\frac{x+3}{x+4}$ can be all of those things. – WeCanLearnAnything Jul 14 at 5:04

I dont know for sure. But I think a part of the problem comes from notation. I dont know how youve 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 isnt 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 fractions. At this point the fraction bar and the obelus becomes the same object.

Its 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.

• I'm not a teacher. I read on the internet that a of of students struggle with fractions. I can't observe their performance and learn that way. That's why I asked the question. I guess students feel like they have the ability to think. I guess they don't want to assume that a fraction is equal to division until they see that for themselves. I guess they have no concept of taking as an axiom that a division is equal to a fraction so they get confused. I guess it doesn't occur to them to be like "The teacher said it so it must have been proven to be true." I had trouble concentrating in school. – Timothy Jul 12 at 21:46
• When my mother told me that a fraction was the same as a division, I think I understood it. I believe I was like "I guess it can be shown to be true without assuming it." – Timothy Jul 12 at 21:47
• I'm not sure why I put a check mark beside the first answer. I don't really feel like removing it now. I guess I was convinced that I had access to the answer if I could bother to read the whole thing. Maybe I was satisfied with knowing there was research on it. This answer is worth an upvote. I guess the confusion about being told fractions are the same as a division between the same numbers is a useful thing to know. However, I'm still not fully satisfied. Would removing that confusion fix the problem or would they still be confused about something. I guess the fraction symbol is used – Timothy Jul 12 at 21:54
• because it's so obvious that $\frac{2}{6} + \frac{3}{6} = (\frac{1}{6} \times 2) + (\frac{1}{6} \times 3) = \frac{1}{6} \times 5 = \frac{5}{6}$ but it's not obvious that $\frac{5}{6} = 5 \div 6$. Maybe their brain doesn't make the connections of how to derive that because it's such a long problem to locate and find such a connection in the first place. If you happen to feel that you can in good conscience try different strategies until you find one that works really well, I would love it if you update this answer after trying it. However, if you write an answer in 20 years and I'm not a Stack – Timothy Jul 12 at 22:01
• Exchange user anymore by then, I might not see it. I have an idea. First of all, when you try to teach them so many things, they don't explore the things that really matter in depth with their brain. So pick out less to start teaching in depth. When teaching how to add fractions, you could ask how to add $\frac{3}{4}$ and $\frac{5}{6}$. Then show that it can be reexpressed as $\frac{9}{12} + \frac{10}{12} = 1\frac{7}{12}$. However, when you present the instructions as shown at matheducators.stackexchange.com/questions/13150/…, it confuses them. – Timothy Jul 12 at 22:16

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 can't quite figure out what you're saying. To a young and inexperienced person, it would appear that you're assuming they are able to take the rules on how to perform calculations on fractions as a given, and they just have trouble holding all those rules in their mind. I think the way they learn them best is to introduce them to them using a student centered approach. Give them an intuition on why non whole numbers exist. For example, start with teaching them only how to add fractions with the same denominator and how to right multiply one by a whole number. I don't think you're quite right – Timothy Jul 17 at 16:33
• about the reason. – Timothy Jul 17 at 16:34
• @Timothy: Regardless of how it's taught, introduced, or intuited... at the end of the day, yes, students need to hold all those rules in their mind. Many people can't, and the various rules for different operations cross-contaminate each other. – Daniel R. Collins Aug 2 at 18:49
• I still can't figure out what you're trying to say their specific confusion is about. Are you a teacher? If so, are your students confused and don't know how to explain their own confusion. Maybe some day when they're adult, they'll resolve their own confusion and explain it on Stack Exchange. See my question math.meta.stackexchange.com/questions/31271/…. Then teachers can use those answers to figure out how to be clearer. Then will gradually be better at figuring out what the confusion of students is. – Timothy 2 days ago