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## Teaching Fractions: The Tootsie Roll Is the New Pie194

theodp writes "Following up on a WSJ story, data visualization author Stephen Few illustrates why using lines or bars may be sweeter than pie when it comes to teaching kids fractions. 'Although the metaphor is easy to grasp (the slices add up to an entire pie),' explains Few, '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 make more sense to use two lines divided into sections instead, which are quite easy to compare when placed near one another?' So, is the Tootsie Roll the new pie?"
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## Teaching Fractions: The Tootsie Roll Is the New Pie

• #### First question from the kids (Score:3, Insightful)

on Wednesday October 02, 2013 @08:39AM (#45013127)
What the fuck is a tootsie roll?
• #### Re: (Score:3)

We've used to use Jaffa Cakes to teach phases of the moon in the UK.

• #### Re: (Score:2)

Advice for submitters: avoid writing while hungry, snack foods might not be the universal and inviolable constant you assume them to be when your stomach is growling.

Pies? Rolls? I'm heading into a mid-afternoon blood sugar crash here and Slashdot is not helping.

• #### Re: (Score:2)

said no kid ever

• #### Start a classroom war (Score:5, Insightful)

by Anonymous Coward on Wednesday October 02, 2013 @08:43AM (#45013157)

The pie chart is counter intuitive? Anyone who has ever fought over pizza slices knows very well that 1/5 is larger than 1/6, even kids.

Here's a simple classroom script to teach kids about fractions:

1) Buy 2 pizzas, slice one in 8 pieces, the other in 12 pieces.

2) Take 20 students in the classroom and tell them to choose a piece from any of the pizzas.

3) Watch as war ensues

• #### Re: (Score:3)

You forgot step 4) Laugh manically
• #### Re:Start a classroom war (Score:5, Funny)

on Wednesday October 02, 2013 @09:02AM (#45013407)

Anyone who fought over pizza knows that not all 1/8ths are created equal.

• #### Re: (Score:3)

Yeah, sometimes they weigh 3.5 grams, sometimes 3.7 or even 4.0.

• #### Re: (Score:2)

Man, you need to be less stingy with your toppings if a slice of pizza only weighs 4 grams...

• #### Re: (Score:2)

There's another teaching opportunity. Using nothing more than a compass and straight edge, divide the pizza into equal portions.

• #### Re: (Score:2, Funny)

A guy walks into a pizza shop at lunchtime and asks for a personal pizza. The shop is new and wants to make people happy so the guy behind the counter asks "Do you want us to cut that into 4 slices or 6?" The customer thinks for a second and says "Better make it four. I'm not sure I could eat six."

• #### Re: (Score:2)

I prefer chicago cut.

• #### Stephen Few Loves his Bar charts (Score:3)

I completely Agree... I've actually had a few public disagreements with Stephen Few (on his blog - Hi Stephen) about his love of bar charts.

He's absolutely right, technically, on the visual perception -- that it's easier to compare lengths to basically anything else (like pie slices), particularly shapes that vary in more than one dimension (is a 5x5 rectangle bigger than a 6x4? If you know the dimensions you can do the math, but if you look at the boxes it's not as easy).

BUT, where I disagree (and I seem

• #### Re:Start a classroom war (Score:5, Funny)

on Wednesday October 02, 2013 @09:21AM (#45013637)
When asked if he wanted his pizza cut into 4 or six slices: "You better cut the pizza in four pieces because I'm not hungry enough to eat six." - Yogi Berra
• #### Who cares? (Score:3)

on Wednesday October 02, 2013 @08:43AM (#45013163)

So one method is probably a small fraction better than another method of teaching fractions. This isn't how you enhance the next generation's education. This is how you make it look like you're doing something to help when you're actually just raising a fuss over the tiniest of things. This is the plastic banana slicer of education: an answer to a question nobody asked.

• #### Re: (Score:2)

My teachers preferred to use real-world examples, which seemed to help. Cutting a pizza into 8ths or 10ths (who the hell cuts it into fifths?). Doubling or halving chocolate chip cookie recipes (1/3 cup sugar doubled is 2/3 cup. 1/2tsp vanilla halved is 1/4tsp). Sports statistics, word problems, supermarket packaging, etc. It was all better than some arbitrary pie chart that carried no meaning beyond "this slice is bigger than that slice".

• #### Re: (Score:2)

In a college differential equations class we got questions such as:

* There's a party and people are spiking the originally nonalcoholic punch at a rate of X, while drinking at a rate of Y. How long until everyone is drunk, assuming Z for the amount of alcohol needing to be consumed to be drunk.

* Kryptonite with a radioactive field described with [equation] is placed near Superman. Which way should he fly to get away fastest?

Silly little things like that made it fun. Of course I had to get through 14 or 15 y

• #### Re: (Score:2)

Recall the damage done to education in Alabama, Mississippi and South Carolina by the General Education Board after the War of Northern Aggression?
English, math and science best left for college preparatory classes.
• #### Re: (Score:2)

I suspect that a child dull enough not to be able to judge the size of a piece of pie is unfit for education anyway. Just being in a classroom with normal kids should indicate some level of intellectual function. Part of the ruin of our schools is an effort to educate the ineducable children. That hole is so deep that it will not be filled. The best path might be to get them out of the schools by sixth grade and put them into a menial work situation such as picking oranges or washing dishes. These

• #### Something weird just happened ... (Score:5, Insightful)

on Wednesday October 02, 2013 @08:44AM (#45013171)

... and somebody read a school textbook.

Seriously. Textbooks have used multiple representations of fractions for years, one of which is linear, because the education research has indicated that different children learn better with different representations of fractions.

Well, at least we now know how long it takes for education research to trickle into the classroom: decades.

• #### Re:Something weird just happened ... (Score:4, Interesting)

on Wednesday October 02, 2013 @08:59AM (#45013375) Homepage

Teachers also use word problems, discrete objects, and liquids, ideally delivered in quick enough succession that the student's brain catches the only constant: the concept of a fraction.

I think the problem isn't education research getting into the classroom - it's exactly the opposite. Teaching is an application-focused industry [xkcd.com]. When a teacher solves a particular educational problem, the technique stays within the school district, or perhaps makes a few rounds at educational conferences. The technique rarely gets any widespread attention, hardly any formal study, and is entirely forgotten within the decade... until an "educational researcher" stumbles across it and publishes a paper describing its effectiveness, which doesn't help because the school boards aren't interested in using new experimental techniques when their budgets are already in such jeopardy.

There is no Nobel Prize for education.

• #### Re: (Score:2)

... and somebody read a school textbook.

Seriously. Textbooks have used multiple representations of fractions for years, one of which is linear, because the education research has indicated that different children learn better with different representations of fractions.

Well, at least we now know how long it takes for education research to trickle into the classroom: decades.

It's important to remember that (assuming qualified faculty, an assumption that is...widely variable... in its truth; but is definitely nonfalse in better systems and some parts of worse ones) educational research can make it into a classroom from the top or from the bottom:

Your top-down approach (curriculum design followed by mandate, textbooks 'aligned' with that curriculum) is nominally research based; but ponderous as hell and perpetually mired in comittee and trying to appease the wackjobs in Texas

• #### Re: (Score:2)

It's a horrible explanation of the article. Researchers found that "a child's knowledge of fractions in fifth grade predicts performance in high-school math classes, even after controlling for IQ." That's a really big result.

So now a lot of research is being done into, "how can we teach kids fractions in a way that they understand them?" The first link explains a lot of the different things people have tried. The second link is a blog with (big surprise) wild speculation.

Guess which link ended up in
• #### Length vs volume. (Score:3)

on Wednesday October 02, 2013 @08:48AM (#45013229) Homepage

The comments on the site (as of this time) give some pretty good reasons why using slices of a circle aren't the best way to describe fractions. Most of the time [wikipedia.org] it is easier for the mind to tell if two lengths are the same versus if two slices of a circle are the same. It is a much simpler calculation to determine length (line) then volume (pie piece).

• #### Re: (Score:2)

Which is simply more reason why students need practice doing the more difficult calculation early.

This whole notion that everything in education needs to be watered down and simplified for ease of digestion simply cheats the children - who tend to be quite a bit smarter than we think, when given a chance.

• #### Re: (Score:2)

They shouldn't be (and probably aren't) using numbers that are very close together to teach the concept. Instead of using 1/5 and 1/6, use 1/2 and 1/3, or 1/3 and 1/8. If the perception of length vs. area/angle matters, it's a bad choice of numbers.

• #### Re: (Score:2)

That article doesn't even mention slices. Also, we can use stackable slices and have the students put one on top of the other.

• #### Re: (Score:2)

Actually, it's length vs. area. If you wanted to start talking volume, you'd have to start cutting up 3D shapes, like cubes and spheres. That would be even more complicated. If you're dealing with cubes or other 3 dimensional shapes, there's quite a few ways to split it in half. using lines to demonstrate fractions, at least when introducing them could help out. But after they have the basic idea of fractions, it's important that the student move on to fractions of more complex things.
• #### Hershey bars would also work well (Score:2)

any candy bar that has natural sections would work for fractions

Kit Kats would work for 2 and 4 based fractions

• #### Re: (Score:2)

we actually did use hershey bars. the ones with the 3x5 (?) breakable grid layout.

• #### Only denominators with 9 (Score:4, Funny)

by Anonymous Coward on Wednesday October 02, 2013 @08:52AM (#45013281)

There's 9 sections. What happens when you want to teach 1/4s, 1/2s, 16ths ?

That's why I think a bottle of Scotch is the new pie!

Now children, let me drink two shots, what fraction of the bottle did I just drink?

Now children, assume what's left is the whole and I drink another three shots, what fraction is left?

Now children, write a 1,000 word essay on why whiskey is the best math tutor whle I take a little snap.

• #### A New Product Line (Score:2)

Now Tootsie can sell a bunch of new lengths: halves, thirds, quarters, fifths, sixths, etc. Schools would just need to go out and buy a few bags.

For adults learning fractions, they could use alcohol instead, but they'd just have one fraction: fifths.
• #### Thank you for the idea (Score:2)

From now I'll try this way to teach fractions, let see that this evening on a 9yo.
(a child who doesn't understand why a fraction is smaller with higher numbers)
• #### Here's a thought.... (or 2 or 3) (Score:2)

Math was taught and learned just fine for over 2000 years. Pretty damn arrogant to come along in the last 50 and think we know how to teach children math in a better manner than they've learned math all along. Pick your slogan, acronym, whatever. KISS (Keep It Simple Stupid), If it ain't broke, don't fix it... Nothing wrong with the way math has been taught all along. I have 4 kids that have all gone through Algebra in the last few years, and I had to go out and buy them Lego sets to learn Algebra. A true W
• #### Re: (Score:2)

People were fine cooking with fire for X thousand years just fine, pretty damned arrogant for them to invent the microwave.

Just because "that's the way it's always been done" doesn't mean it's the most efficient/effective/bestest way. Sure, it doesn't mean the old way isn't better for some people, but it's even more arrogant to assume the new way isn't better without trying it first, especially based on some anecdotal evidence.

Also, I highly doubt that math was taught the same way across any or every cultur

• #### Re: (Score:2)

You are so right. Who needs newfangled things like cars and cell phones. People got around just fine in biblical times I say. For that matter, who needs vaccines or medicine? Living past 30 is overrated.
• #### Re: (Score:2)

Not saying there's never room for improvement, I'm saying there's no need to fix what already works and has worked well for centuries. The constant plea from the teacher's unions is that we just need to spend more money per student when we already spend more money per student than any other civilized nation and still graduate kids that can't read and write at an elementary school grade level. One room school houses with a single teacher for all grades used to be able to teach the basics, no reason they shou
• #### Re: (Score:2)

Except people from back then would be legally retarded now (i.e. the Flynn Effect). Students are expected to learn more, quicker than ever before. As you say, we are spending more money per student than anyone and it just isn't working. I would actually say that teaching methods haven't changed that much in the past hundred years. Maybe the answer is to get kind of extreme and start from scratch.
• #### Re: (Score:2)

People expect them to learn more, but in practice, they just memorize more and then later forget it all.

As you say, we are spending more money per student than anyone and it just isn't working.

Change is difficult and expensive, so why fix something that is completely broken?

• #### Re: (Score:2)

You are so right, this is where we should apply that old American motto: "sounds hard, lets just give up."
• #### Re: (Score:2)

No, they are not expected to learn more, at least not when you compare curriculum to the one that was there 20-30 years ago. In recent years there was plenty of cuts justified by 'nobody deals with it in real life' and 'we have calculators for that' and other nonsense. Most 20year olds today are clueless and borderline retarded. FFS, illustrated cash registers had to be invented because your average teen after a decade spent in school can't handle basic arithmetic, doesn't grasp the concept behind paying \$1

• #### Re: (Score:2, Insightful)

Just last night I was helping my elementary age son study for a test on fractions and percents. We went through all concepts and he was still not getting it. Finally, he drew a line and started segmenting it. The teacher had shown the class "another way" to conceptualize this topic. He completely understood this approach. He then told me that his teacher told them about learning styles and tried to present the topic in multiple ways. So, while it seemed simple from one perspective to most of the clas
• #### Re: (Score:2)

Math was taught and learned just fine for over 2000 years.

It wasn't. Rote memorization is not ideal, and I do not consider it "just fine." Our entire education system is pretty much broken.

• #### Re: (Score:2)

Rote memorization is the only way to learn the fundamentals, addition, subtraction, multiplication, division, but those aren't taught any more. No kids are required to memorize math tables unless it's done by a parent. More complex ideas require teaching a kid how to think, but if they're busy counting on their fingers to subtract 7 from 13, more complex problems will never sink in.
• #### Re: (Score:2)

Rote memorization is the only way to learn the fundamentals, addition, subtraction, multiplication, division, but those aren't taught any more.

An understanding of the concepts is much better; otherwise, they might as well just use calculators, which are much faster, since that's what you seem to care about.

but if they're busy counting on their fingers to subtract 7 from 13, more complex problems will never sink in.

That is completely false. As long as they're capable of performing such basic operations one way or another, the fact that they can't subtract numbers as quickly as some would like does not mean they're not capable of understanding the problems. Do you know what we have right now? A system that encourages rote memorization. A system where unders

• #### Use Democrat logic... (Score:2, Funny)

If we divide people into identity groups, you can truly understand how to put certain groups together! 50% white 50% female ...
• #### The latest episode (Score:2)

This is just the latest episode in Stephen Few's war on pie charts. For anyone interested: http://www.perceptualedge.com/blog/?p=1492 [perceptualedge.com] http://www.perceptualedge.com/articles/08-21-07.pdf [perceptualedge.com]
• #### Re: (Score:2)

I think he discounts the value of the implied 0-100% scale in a pie chart. If I have a bar graph, with 3 bars 60%, 20%, and 10%, even if there's a scale on the graph it's not immediately obvious that we're missing 10%. With a pie chart, it is impossible to miss that missing 10%. There's value in that.

Pie charts are overused, as are bar charts (box plots are usually better), but they have their place. Their place is representing proportions of a whole, and that's it.

• #### Pretty good with fractions (Score:2)

Because I had it drilled into me as a kid. Now I sort of unconsciously can do most fractions.
• #### No, Use a scale (Score:3, Interesting)

on Wednesday October 02, 2013 @09:26AM (#45013713)

Show them 1" on a ruler. Show them 1/4" increments. It's real easy to see 4 of those make up 1". Next show them 1/8" increments and 1/16" increment. They see pretty quickly how 16 can fit but the marks are smaller even though the number is bigger.

Now they've just learned how to read the crazy US Inch-standard system as well. Pretty handy for growing up in a slack-jawed yokel country who's politicians never let teachers adopt the metric system, but I digress...

Extra credit: show them a meter stick and listen to the gasp at how easy everything is because every little mark takes 10 units to get to the next larger unit of measure.

• #### Re: (Score:3)

Pretty handy for growing up in a slack-jawed yokel country who's politicians never let teachers adopt the metric system, but I digress...

Eh?

Politicians have never stopped teachers from teaching the metric system in this country, and schools have taught the metric system for decades starting at a young age.

But it's often taught in the context of science and while the students do learn enough about it to use it "in the real world" -- the US still doesn't use it for everyday things, and so they don't get practice using it and don't truly grow comfortable using it (unless they go into science) and as adults they still know the metric system but

• #### Re: (Score:2)

Pretty handy for growing up in a slack-jawed yokel country who's politicians never let teachers adopt the metric system, but I digress...

I went to two different schools in the country, and learned metric at both of them.......

• #### 6/5 of a tootsie roll (Score:2)

The linear idea is good for comparison side by side, but if you have a tootsie roll which is 5" long and one that is 6" long, which one is a whole tootsie roll, which one is 5/6 of a tootsie roll, and which one is 6/5 of a tootsie roll. Even if you show the individual pieces, you can't tell. With a pie, there's never any question as to whether you have more or less than a whole pie.

• #### Why is this a story? (Score:2)

This isn't a problem that needs solving. I never needed a teacher or diagram to explain to me that a half of something is larger than a quarter; that's effing obvious. "Learning that one-fifth is larger than one-sixth, which is counter-intuitive in the beginning"? WHAT? And even so, this article's point is moot, since visual representations other than pies have been around for many years. Containers of liquid, pieces of chocolate bar, etc.

The only things I needed to learn about fractions were the tricks for

• #### No (Score:2)

tau [tauday.com] is the new pi
• #### Number line (Score:2)

The number line is used a lot too, and they look mighty similar to the fraction line. I could see it confusing some kids. Especially as the curriculum likes so much to teach key word vocabularies and associate particular visualizations with particular concepts in such an inflexible manner

My biggest complaint with math education is that the schools seem so inflexible in it. My 6th grader is doing OK with math so he's in the "7th grade" math class. Along with, as it turns out, most of the rest of the 6th grad

• #### old news (Score:2)

No, really old. My grandmother used to break apart chocolate bars to teach fractions to her 2nd or 3rd grade classes back in the 50's and 60's.

Now some self-proclaimed genius has figured out what Tufte has been saying forever: that pie charts suck?

• #### Re: (Score:2)

Tufte has been saying pie chart are over used.

And more interesting then your UserID being prime, its a Fibonacci number .

• #### Cuisenaire rods (Score:2)

This is how I learned math in the first grade, and is very much visual in how fractions work.
http://en.wikipedia.org/wiki/Cuisenaire_rods

• #### (what I was getting at is...) (Score:2)

Basically the Tootsie Roll concept spoken of has been in use for decades, this ain't new.

• #### Re: (Score:2)

Ahh, so that's what they're called. I had a bag of these (wood) when I was a kid, and just called them my 'math blocks'.

• #### 30 year ago: cuisenaire rods (Score:2)

http://www.elementarymatters.com/2012/05/learning-math-facts-with-cuisenaire.html [elementarymatters.com]
I had these in the early 1960s (JFK presidency) at the http://www.lesleyellis.org/about/who-we-are/history [lesleyellis.org] which at the time was off concord street in Cambridge

• #### Here's an idea (Score:2)

show them the math.

Worked well with my kids and every other kid I showed it to.

• #### This is very wrong! (Score:2)

What they don't realize is that using pies and pizzas to teach fractions is secretly preparing kids for trigonometry. Except that the whole pizza is actually 2 pi, rather than a pi.

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