## Linear Fractions

I suspect that one of the reasons why people are drawn to pie charts is the fact that these charts are familiar from elementary school instruction in the meaning and mathematical use of fractions. Based on this instruction, a pie chart is the image that becomes strongly associated with the parts-of-a-whole concept (a.k.a., fractions). But, just because this is how fractions have been traditionally taught in schools, should we assume that pie charts are the best visual representation for learning fractions? Although the metaphor is easy to grasp (the slices add up to an entire pie), we know that visual perception does a poor job of comparing the sizes of slices, which is essential for learning to compare fractions. Learning that one-fifth is larger than one-sixth, which is counter-intuitive in the beginning, becomes further complicated when the individual slices of two pies—one divided into five slices and other into six—look roughly the same. Might it makes more sense to use two lines divided into sections instead, which are quite easy to compare when placed near one another?

This not only makes sense based on our understanding of visual perception, but recent research has demonstrated that it in fact works better for learning. Take a moment to read the recent article about this by Sue Shellenbarger in The Wall Street Journal entitled “New Approaches to Teaching Fractions” (September 24, 2013).

Take care,

### 17 Comments on “Linear Fractions”

By Jon Peltier. September 30th, 2013 at 6:57 pm

I’ve often said that we learned pie charts from Miss Jones in second grade, and it has been stuck in our minds as a valuable chart type because we learned it so young.

To anyone who claims that the circular pie chart is an intuitive means of teaching fractions, I ask whether they’ve ever tried to teach a child to tell time on an analog clock.

By Clint. October 1st, 2013 at 6:44 am

I wonder how straight forward and clear the linear representation would be when the fractional components are not as straight forward as 6ths and 5ths? Parts-to-whole comparisons don’t necessarily tend towards even distributions and might not this visualization end up nearly as cluttered and unclear when the proportions are not so evenly divided?

By Stephen Few. October 1st, 2013 at 7:19 am

Clint,

For comparing two fractions, can you think of an example when the linear representation would not work best?

By Andrew. October 1st, 2013 at 8:03 am

When I try to explain to people why linear comparisons (using bar charts, typically) are more effective than pie charts, I ask the person, if I gave them some varying lengths of string and asked them to compare them in some way, how would they do so.

Would you (rhetorical you) lay the strings end to end in a circle and maybe draw lines from their endpoints to the center of the circle? Or would you lay them straight, next to each other, and line them up at one end?

Perhaps my thought experiment would be more effective as a demonstration, but then I’d have to start carrying a bundle of strings with me everywhere I go as I don’t know when I’ll need it.

By Colin Michael. October 1st, 2013 at 9:54 am

This is why I preferred teaching elementary math using Cuisenaire rods rather than collections of objects (balls or bears or apples) or finger counting systems, etc. The visual cue of length is much stronger and easier to grasp intuitively. Even addition and “carrying” are more quickly grasped with linear visuals. There is even a popular curriculum now (Math-U-See) that uses the linear visuals method.

By Dave Careaga. October 2nd, 2013 at 9:25 am

As a 4th grade teacher, it falls to me to start to really dig into fractions with students. I use many representations, but typically I do find myself using a lot of bar modeling type stuff. Bar modeling reminds me strongly of the example pictures given in the article. I dislike connecting fractions with anything food related because when you get into some of the funkier fraction stuff, they have a hard time visualizing it if they can only think about it in terms of pizza, or chocolate bars, or Tootsie rolls. Also, I teach in a predominantly Spanish speaking school. It’s odd to think about, but for my kids pizza and Tootsie rolls are not as common a concept as they may be in other schools.

About the only time that bar modeling doesn’t work for me is when I have to teach fractions as parts of a set. As we use uniforms in my school, I’ll use an example that I use with my kids. I have a fraction of 12/25 – 12 of my kids are wearing their white uniform shirts. That’s a little harder to use a bar model for and have the kids understand that we are talking about parts of a set.

By Dave Careaga. October 2nd, 2013 at 9:27 am

I should clarify. When I say Spanish speaking, I actually mean students whose primary language is Spanish. We are located in the States and we teach in English.

By Bill Droogendyk. October 2nd, 2013 at 9:29 am

Andrew – great example! I think I’ll use it next time I present on dataviz.

By George. October 2nd, 2013 at 11:07 am

By Iz. October 3rd, 2013 at 12:36 am

The type of data being compared, or analysed, should help to determine which method is best used for a given scenario. A linear method is great when comparing individual segments to one another. A pie chart is preferable, in my opinion, when comparing parts to a whole (percentages come to mind).

By Scott Eaton. October 3rd, 2013 at 6:07 am

Great article. I think a lot more kids would have a better understanding using this method.

By Peter Z. October 3rd, 2013 at 7:42 am

Dave Careaga — I was introduced to bar modeling (Singapore Math) last year when my daughter entered 6th grade. It’s a fascinating technique, easy to understand for the simple questions, but took a little time to grasp for more complex problems. Nonetheless, it’s a powerful approach to visualizing math problems. Seems to be an awesome way to learn fractions and part-to-whole comparisons. It’s also a stepping stone to algebra.

The following (somewhat complex) word-problem becomes manageable using bar modeling:
“Mrs. Roe is three times as old as her daughter, Pam, who is twice as old as her brother, Sam. If their total age is 54 years, how old is Pam?”

For anyone else — A quick web search for “singapore math bar model” will turn up plenty of weblinks on the topic. Here’s one link to video examples of bar modeling: http://thesingaporemaths.com/

Please note that I don’t believe bar modeling is the holy grail of math. Just a tool/technique.

By Stephen Few. October 3rd, 2013 at 7:43 am

Iz,

If you put your opinion to the test, I believe you would find that even for estimating proportions of a whole, pie slices only work better than linear sections when the slice equals 25%, 50%, or 75% and it begins at the 12, 3, 6, or 9 o’clock position.

By James Poulakos. October 3rd, 2013 at 6:13 pm

I agree. I suspect that even those 3 special pie slice cases are useful only for those of us who are comfortable reading “analog clocks” with round faces.

By Neil Barrett. October 4th, 2013 at 3:11 am

In Scotland, I was taught fractions using blocks of different colours that were named whole, quarter, half, third — up to sixth, I think (can’t remember). This allowed some of us to get to grips with the concept of the fraction as part of a whole, and what adding a third to a half meant, etc.

We still had an obsession with pie charts, though. We were taught to use them (proportion of boys, colour of eyes, etc.). I suspect the continued use of pie charts is that we teach them to susceptible young people who never unlearn these truths they continue to hold as self-evident throughout the rest of their lives. Well, that and most people spend their lives operating through Kahneman’s System 1, and System 2 doesn’t have a numbers module…

By Luca Fracassi. October 4th, 2013 at 4:08 am

Very interesting article, thanks…

Btw, I noticed that in iOS 7 they replaced the usual progress bar with a circle line when you update apps.

I observed this change and I think that in this specific case it’s an improvement for the reasons just mentioned about quarters (25/50/75).

– in a progress bar you only care about the approximative progress, nobody cares if you are at 43%. Quarters are perfect and they are also easy to read as they remind us of the clock.

– space wise a circle is much more compact than a line especially on a small screen

Sure you can’t apply this to a project that lasts through time (e.g. days, months), but for all the live monitoring of software update or software progress (and maybe other things I don’t have experience of) I think this solution is probably superior to the bar.

By Linda. October 4th, 2013 at 10:07 pm

I dabble in math education and I wholly agree. Visual representations should be in a form that we can easily manipulate. Lines are easy to draw, pie charts are not. I also think we make a giant mistake by teaching and overemphasizing percentages. We should teach more math between 0 and 1 (or -1 and 1) and not be afraid of fractions.

I had an 8th grade teacher who couldn’t explain how to remove 25% from a number after adding it to it (i.e. that x100/125 is the opposite of x125/100). He treated percentages more like a unit than a fraction.