**Which is more difficult to teach/learn: Algebra or Fractions?**

Last week, I was a guest at Discovery Education’s “Beyond the Textbook” forum. During one of the breakout sessions, my group (Christopher Danielson, Angelia Maiers, and Chris Harbeck) got to discussing about how hard it is to write math curriculum and that no digital textbook would be worth its weight in bytes if it didn’t acknowledge that. Christopher Danielson wrote a great blog post which both summarizes and expands upon that discussion.

To illustrate that most non-math teachers don’t quite get the nuances involved in science and math instruction, I asked Angela which concept she thought was more difficult: algebra or fractions. She answered algebra without hesitation. And I said I felt fractions were more difficult. I said that my 8-year old son has no problem with algebra: 5 + [_] = 7. What goes in the box?

But what does 3/4 mean? Is it 3 objects out of 4 objects? Is it one object split into 4 pieces, but we only care about 3 of those pieces? Is a ratio of 3 things to 4 different things? Or is it division and we are taking 3 objects and splitting them evenly to 4 groups? Is is 75%? Or is it 0.75?

Angela was blown away by this discussion and wondered how other people (math folks vs. non-math folks) would respond. We decided to conduct a little experiment. We would both ask our PLNs the same question and compare responses. My prediction was that since my PLN tends to have a math and science focus, the majority of my followers would say fractions are more difficult. And I predicted that Angela’s PLN, which is more broad than mine, would say algebra is more difficult.

Here are the results:

I was correct in my predictions, but I was surprised at how my PLN was much closer to a 50/50 split rather than a 2:1 split like Angela’s. Here’s the raw data — it’s interesting to comb through the responses to read *why* they chose algebra or fractions:

Many people said fractions were easier because they are concrete — you just slice up some pie or look at a Hershey bar.

I’ve singled out two thoughtful replies on Angela’s Facebook page below. Jodi’s is great because she polled her second grade class:

And Susan’s is great because she sees the many possible meanings of a fraction:

**What do you think? And do I think fractions are more difficult because I am misunderstanding something about the nature of algebra? **

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Frank, I enjoyed watching your question and the replies it generated on Twitter, and loved seeing the results here in your post.

For the record, I responded with “Algebra,” though didn’t spend any time sharing on Twitter why I thought so. (More on that below.)

I wonder if the surprising results (with more people than you expected responding the way I did) are a sign that algebra and fractions both have fairly easy entry points, but there is a world of challenging material in each domain beyond those entry points.

In response to your 5 + ? = 7 algebra example, I offer that my three and a half year old has a decent understanding of one fraction (1/2) in a limited number of settings (mostly related to portions of food he must move from his dinner plate to his tummy before he can go play with his trains again). I expect that most people have a very (VERY) basic understanding of algebra and also of fractions, but after reading your post I definitely see your point that even the most basic of fractions have many (and sometimes conceptually very difficult) meanings, whereas there’s not much to say about 5 + ? = 7.

As for my choice of “Algebra” in your Twitter survey, I had in mind the algebraic concepts and techniques we explore in my Precalculus class, held against the work we do with rational numbers in my Pre Algebra class. But maybe I focused too much on the question of procedural difficulty (where the algebra found in my Precalculus class wins hands down). I wonder if those who responded with “Fractions” were considering mainly the conceptual complexity of fractions and those who responded as I did answered from a procedural perspective (though I do believe the challenging parts of algebra include not only procedural bits but also conceptual).

One more thing I’m curious to know… How would people have responded if you asked this question instead:

“Which is more difficult to learn/teach: algebra or fractions? How much more difficult than the other? (Use a scale of 1-5, where 1=slightly more difficult, 5 = dramatically* more difficult)”

*not sure if it should read “dramatically” or “incomprehensibly” for the high end of the scale.

Thanks again for the post, Frank! I enjoyed it!

I concur completely with Michael’s comment. He captured the heart of what I was thinking, claiming “both have fairly easy entry points, but there is a world of challenging material in each domain beyond those entry points.”

Given 5 + ? = 7 and I’ve cut a circle into 3 parts & shaded 2 of them, I bet even the most math averse would quickly see 2 and 2/3 as reasonable representations of answers. But those are the entry points.

To see that 4/6 is equivalent to 2/3 (God forbid a monster like 3/4.5) requires a sophistication beyond a simple representation of a physical state. Acknowledging what other responders have said about depending on definitions, I would argue from my definition that 5 + ? = 7 isn’t really algebra (yet), it’s arithmetic. If that is algebra (just because it has an unknown), then so is 7 – 5 = ?, a “problem” most would call arithmetic (OK, subtraction). Just because you have an unknown doesn’t mean you’re dealing with true algebra. Algebra requires dealing with _variables_, IMO, and that is a HARD transition. Now, I’m totally cool with telling students that they’re doing algebra when they solve 5 + ? = 7 (just like we tell them that square roots of negative numbers aren’t “allowed” when square roots are first introduced), because doing so demystifies algebra, and goodness, we could all stand for a little less math phobia in our classes, in society, and in politics.

One of your respondents said both topics were tough (after the entry points implied) and suggested that the rules of (generalized) algebra fully encompassed all of the broader rules of fraction manipulation. In my mind, that is closest to the point. And it is closer to the heart of what I often distinguish as the difference between Arithmetic and Mathematics. Ultimately, Arithmetic is symbol manipulation following strict rules and sometimes gets ornery. Mathematics is about a way of thinking–it occasionally requires symbol manipulation, but is always on a higher, more generalized plane. The variations on the meanings of 3/4 you offered in your post are Mathematical in nature, IMO. At its core, 5 + ? = 7 is Arithmetic.

Absolutely depending on how it is taught, I believe the more encompassing algebra to be more Mathematical at its core and more difficult to grasp as a whole. Unfortunately, too many students encounter both algebra and fractions as a sequence of Arithmetic algorithms sans explanations of the Mathematics that tell us why those algorithms _always_ work–a concept we know through studying variables.

Very interesting question. I think, also looking at years of classes coming from primary education to secondary education, that fractions are harder to understand as they’re more foundational. But, as with foundational topics, if you understand it then it’s easy 😉 I think you can’t even imagine a world not understanding it. That’s exactly the problem in teaching fractions. We assume pies and pizzas will be enough for complete understanding but they often aren’t.

PS. (Of course I define Algebra not as a broad subject encompasses almost everything)

PPS. I do agree that variables are very hard to (together with negative numbers and fractions cause of many misconceptions)

Fractions involve proportional reasoning which is conceptually very difficult. Student struggles with the concept of variable which is at the root of algebra are more easily overcome so I’m voting for fractions.

As a High School Physics teacher, I find many students struggling with both.

I conducted two trainings on the CC Practices. For the elementary teachers, I offered a lesson on Fractions; for the secondary group it was an Algebra lesson. In each, I had the 4 district leaders sit in as participants, 3 of whom had taught English or Elementary school before being administrators. All of them said it was far harder to wrap their head around the fractions than the algebra. So with my own, you can add 5 votes for Fractions.

We put the question into the same category as whether you’d rather die from burning or from freezing. Or which is tastier, an apple or an orange.

Fractions as a concept, algebra as a concept, most people get both, but the devil is in the details and where you go with it. For instance, after learning what 1/6 means, do you then go to 1/6 + 1/6? Or do you divide 1/6 into 2 equal parts and call it 2/12? They are very different paths and the path matters.

After understanding 5 + ? = 7, do you immediately follow that with ? – 5 = 2? The latter is a lot harder and you can lose a lot of people, but a lot of teachers put them in the same category and don’t look up to watch students’ faces.

We certainly agree that fractions and algebra are two of the biggest math hurdles, but they are both hard if taught badly, and they are both easily mastered by anyone if taught well.

Fractions are numbers, just like 1, 0, 213. Algebra is the study of operations.

At its core, Algebra is largely a construction around the field of rational numbers. The majority of issues I have seen with students in Algebra can be directly tracked back to a lack of understanding of the theorems concerning integers and rational numbers. Likewise the theorems follow from the definitions of binary operations (+, ×) and the field axioms of those operations. So the distinction between “fractions” and “Algebra” is largely artificial.