Fractions are an important part of students’ course curriculum, and teachers are presented with the challenge of ensuring that they understand this concept. In this article, our Math expert, Sunil, shares his thoughts on the most effective methods of introducing the concept of fractions.

## Why is it so hard to understand fractions?

A few years back, there was a popular belief that A and W’s third-pound burger suffered poor sales because many people in the focus groups thought that a quarter-pound burger was more more meat–I guess the idea was that people saw ¼ as being larger than ⅓.

There actually is no evidence that difficulty in understanding fractions led to the flop of this meatier burger, but social media has always tried to trumpet the general math phobia of the population with provoking rumors/memes–especially when it comes to fractions!

By the time kids get to the division of fractions with the whole “flip and multiply” shenanigans, there is an almost quiet–and final–confirmation from many students that math is simply about following rules and very little about making sense. Why else would we continually be getting away with the hocus pocus inversion of the second fraction with a replacement of a multiplication sign–and not one student asking why? But, what if they did? Do we have an answer prepared that will explain this cloaked mathematical manoeuvre? Will our answer close the conversation or break it wide open?

As you ponder that, at least realize that fractions are not easy, and James Tanton–one of the global leaders in math education–has passionately pointed out that perhaps fractions be explored at the high school level. That said, our students will be introduced to fractions somewhere much earlier than that. So, we must do our best to have fractions make sense, but also, somewhat paradoxically–not make sense! For all the clarity that we offer, we must also be honest about the confusion that fractions bring later on–ie) what does ⅓ x ⅖ really mean? By giving supportive answers and asking disarming questions, our students might have a better chance of surviving this historical sinkhole.

One way to look at fractions, which could help some students, is to visualize them as repeated subtractions until they get to zero. So, 10 divided by 2 is 5, but it can also be seen as how many times can I keep taking away 2 until I am left with nothing.

#### 10 – 2 – 2 – 2 – 2 – 2 = 0

We can subtract “2” five times!  This works really well with cube-a-links. Build a tower of 10 blocks and remove two at time until your tower vanishes! Five piles of blocks of two will be left. It’s not only important to see mathematics–it’s also important to touch it.

So, let’s say we ramp it up and ask a question like 46 divided by 7. The answer is irrelevant at this point. What is critical is understanding and wondering how many times I could remove 7 repeatedly from 46. Will there be a remainder? Who knows! Let’s try. Let’s even build a tower of 46 blocks!

#### 46 – 7 – 7 – 7 – 7 – 7 – 7  …ummmm…= 4

Sorry. No more “sevens” to take away. We have a total of 6 full extractions with a “four” left over…and we were trying to take seven.

Or in more formal terms:           6 and 4/7ths

Okay. Those seemed fairly easy, but what if we take something like the division question below written with improper fractions?

I am not sure how many students can develop links between the original question and the final answer. Procedural abstraction is just a mathematical short-cut–little chance of error, and offering even fewer scenic detours. On the other hand, playful concrete frames the question and answer in ways that can better galvanize deeper ideas about division. Why were the tower structures six high in the above question? Did they have to be? This is the rich distillate that can come about when we take a leisurely stroll through mathematics.

Having students just focus on the answer and completion of a task is what narrows the scope of mathematics–which by now becomes, lamentably, the calcified goal for all math questions. The trouble is that correctness rarely leads to understanding. Conversely, understanding will inevitably lead to correctness.

But, as always, the bigger picture is about having students explore–confidently and leisurely–the wider landscape of WHY questions in mathematics. The Common Core is about building long term success in mathematics taught with curiosity and as a community. We will only achieve success here if there is passion and energy to go deeper.

Sunil Singh,
Math Specialist and Buzzmath expert

1. Concetta Duval

Thinking of division as repeated subtraction reminded me of Keith Devlin’s article related to thinking of multiplication as repeated addition. See https://www.maa.org/external_archive/devlin/devlin_06_08.html. I really question the explanation given here and can’t imagine that students would find subtracting a fraction (let alone an improper fraction) from another fraction is easier or more obvious than learning the correct algorithm. Division is the inverse of multiplication and should be taught that way. Therefore, a / b = c is equivalent to b * c = a. That’s the concept that needs teaching and learning. Read Liiping Ma’s book “Knowing and Teaching Elementary Mathematics” and how Chinese teachers teach this topic..

2. sunilsingh419

Thank you so much Concetta for steering me towards Keith Devlin’s article! I am a big fan of Keith, and it is because of him, that I came across Paul Lockhart’s “A Mathematician’s Lament’ over a decade ago.

And, like you, I agree with Devlin’s idea on multiplication and addition. However, I have an issue with the “traditional” algorithm. It provides no intuition/clarity for the process–it just yields correctness. I actually like your post because of how it creates rich discussion/reflection in how as a community we teach/learn mathematics. And,for that, I am very grateful for your post and giving all of us a chance to rethink common practices/beliefs.

Thank you for your book suggestion and thank you for being part of our community!

Sunil

3. Tracy Clifton

What I always felt was lacking in early math learning was consistency in how new topics were introduced and connected to what a learner already knew to be true. So, if we teach whole number multiplication as repeated addition, then we should do so with decimals and fractions as well. Same with division as repeated subtraction. Once the learner understands how they got the answers and that the answers are true and make sense, then maybe introduce the “traditional” algorithm and show that it gets the same answer often in fewer steps. But the “traditional” algorithm should NEVER be taught first, requiring students to just believe the teacher that it gets the correct answer, which is done much too often.