## Teaching Fractions: The Tootsie Roll Is the New Pie 194

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?"*
## First question from the kids (Score:3, Insightful)

## 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

## Re: First question from the kids (Score:4, Insightful)

The smartest Americans know better than to go into politics, which leaves the politicians we have.

## Re:First question from the kids (Score:5, Informative)

WTF, an AC links to the urban dictionary rather than Webster's or wikipedia, and gets modded informative? It's offtopic, racist, and sexist; we're talking about the candy, not ghetto slang for a black woman's nipples,

No wonder slashdot is going for the new universally hated design, [slashdot.org] they're pandering to the army of idiots from 4chan and reddit who have invaded our beloved nerd site.

For the GP, Pictures and description of a tootsie roll here. [wikipedia.org] They don't export the things? Wait a minute...

So how in the hell can someone not know what a tootsie roll is? How in the hell can someone not google for a piece of information that shouldn't have to be explained in a /. summary?

I guess I should metamoderate...

## Re: (Score:3)

Tootsie rolls look like a turd.

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

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)

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

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)

I'm used to having techno-words sail over my head here, but when did /. become a drug den?

## 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)

## Who cares? (Score:3)

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)

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)

... 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)

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)

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)

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)

Because they continue to be effective ways to teach difficult concepts like fractional volume, which can be very important for people in many fields later in life. Fractional volume, btw, as opposed to fractional length (the subject of the thread) is certainly a concept with application throughout modern society, from the industrial to the financial and the social.

## 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)

## Re: (Score:2)

## Re: (Score:2)

"You only need to look at the chord length of the arc of the pie slice, it's a simple linear length"

Because obviously 1/3 is larger than 3/4.

## 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)

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)

For adults learning fractions, they could use alcohol instead, but they'd just have one fraction: fifths.

## Thank you for the idea (Score:2)

(a child who doesn't understand why a fraction is smaller with higher numbers)

## Here's a thought.... (or 2 or 3) (Score:2)

## 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)

## Re: (Score:2)

## Re: (Score:2)

## Re: (Score:2)

Students are expected to learn more, quicker than ever before.

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)

## 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)

## 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)

## 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)

## The latest episode (Score:2)

## 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)

## No, Use a scale (Score:3, Interesting)

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.......

## Re: (Score:2)

## 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

learnabout fractions were the tricks for## No (Score:2)

## 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.

## Re: (Score:3)

The pre-segmented Tootsie Roll is actually a poor choice. A person who sees it already divided into seven chunks won't understand all those divisions have to move in order to divide it by eight.

## Re: (Score:2)

They need to use both.

I agree, some things like halving halves to make a quarter are easier to show in two dimensions.

## Re: (Score:3)

They need to use both.

I agree, some things like halving halves to make a quarter are easier to show in two dimensions.

And how do you visualize 1/3-1/5 or 1/3+1/5 with pies or tootsie rolls ? Either metaphor (pies or tootsie rolls) is fundamentally flawed in that it captures only 1 property of fractions (fraction of a whole) and that's it.

In UK schools they use Unifix blocks [glsed.co.uk] which are essentially the same as the "tootsie roll" examples. The way these would be used would be to make several columns of 15 blocks. One would be divided into three parts and the other into five. They could then easily illustrate adding 1/3 + 1/5 by adding one of the "three part division" to one of the "five part division". Counting would show that the answer was 8/15 and comparrison to the whole 15 parts would show that it is just over half.

This would also illu

## Re: (Score:2)

## Re: (Score:2)

They need to use neither. Give 'em the good axiomatic definition of a fraction. And them later on give the examples with pies and tootsies.

Oh, you'll

loathsome of the bullshit that gets added to math curricula to pad out the vocab lists...Hey kids, because it's fucking pointless, we are going to be learning about 'proper fractions', 'improper fractions' and 'mixed numbers'! Open your copybooks now: "A proper fraction is a fraction where the numerator is

smallerthan the denominator. An improper fraction is a fraction where the numerator is larger than the denominator. A mixed number is a number written with a whole number component and a## Re:No (Score:5, Informative)

Math is hard, and teaching math is hard. The 'intuitive' or 'obvious' way to teach math isn't necessarily a good way.

## Re: (Score:2)

Math is not hard. Teaching Math is not hard. Math is conceptual and until you get the concepts, actual math is just by rote, which is how math was taught to me.

## Re: (Score:2)

## Re: (Score:2)

Oh, you'll loath some of the bullshit that gets added to math curricula to pad out the vocab lists...Hey kids, because it's fucking pointless, we are going to be learning about 'proper fractions', 'improper fractions' and 'mixed numbers'!... All of these are basically just division problems that are being left unevaluated for reasons of convenience, or because the resulting decimal representation may not be entirely well behaved, so this shit is pointless; but it will be on the quiz.Getting added? I learne

## Re: (Score:2)

we can construct rigourously the set of integer numbers, and set of rational numbers, as well as the set of real numbers and complex numbers.Perhaps we specify the rationals in terms of the rational numbers. Also, if we are now constructing the rational numbers, does that mean they didn't exist in Newton's time? Or if they did exist then, how do we construct them now?

## Re: (Score:2)

What do rational numbers have to do with infinitesimals? And I should have said that we might specify the rational numbers in terms of the natural numbers.

While Leibniz used infinitesimals, Newton used nascent and evanescent quantities, which may have been one-sided limits.

## Re: (Score:2)

And it helped me get my insect porn business off the ground, and won me elected office!

## Re: (Score:2)

How would one go about converting a Flash game like this to HTML5?First, you learn HTML2 (or 3, or 4, doesn't much matter). Then you learn CSS. Then, you learn Javascript. Then, you learn HTML5. Then, you learn Flash. Then, you learn ActionScript. And finally - You break into TwinBeard HQ, steal the source code to Frog Fractions, and begin the long process of porting it.

After all that, though, you

probablyalready have a pretty good grasp of fractions.## Re: (Score:2)

You know how much kids like to rebel against authority figures. It's all a secret plot to teach kids fractions.

## Re: (Score:2)

I think a Hershey's bar would be a better choice if they want something that's already marked up. At least then you can break it into halves, quarters, eighths, etc (depending on which size bar you buy). Or just just a regular, unmarked tootsie roll, a ruler and something sharp enough to cut it.

## Re: (Score:2)

and something sharp enough to cut it.

I don't think the school board would approve of the use of a thermal lance or diamond coated masonry saw in an elementary school classroom.

## Re: (Score:2)

Fractions are still useful in the metric system, granted, with more limited application. Halves and quarters are fine, but what about when you need to divide a whole between seven people? Each person can get 1/7 or each person can have 0.142857142857... even rounded to only .14 that's kind of hard to figure out compared to 1/7.

Metric and decimal is great for science, but fractions still have their place in everyday life.

## Re: (Score:3)

However, there is little to no need for fraction in the real worldLet me guess, you find the the Big Mac button confusingly similar to the Quarter Pounder button.

Hint: One has 2/3rds of the number of buns of the other one. One bun, two buns, red bun, blue bun!

## Re: (Score:2)

Yes, because .08333333333333333333333333... is so much more intuitive than 1/12.

## Re: (Score:2)

Fractions are important for later mathematics and for understanding things like percentages, decimal notation, scale, parts of the whole, ratios, I could go on. Early work in fractions helps foster a better way for kids to think about mathematics. We use few fractional measurements in Sweden, but its still an important mathematical concept. Also, using graphical representations of fractions frequently leads to misunderstandings or fixations on fractions being just pieces of a pie. When you actually

## Re: (Score:2)

When we first started fractions and division (third grade or so?), we used groups of discrete units rather than cutting up a single unit. If you have ten pennies and you eat half, how many do you have left? If you have 14 pennies and you throw 1/7th of them at Johnny, how many did you throw? Most kids are still smart enough to see a group of individual objects as a "whole".

## Re: (Score:2)

If you have ten pennies and you eat half,Eating pennies? Shouldn't schools be discouraging that?

## Re: (Score:2)

That 1/3 pie slice is no longer 1/3 of the pie in value if you only enlarge the slice and not the rest of the pie. Sure, it's still 1/3 of a circle, but it's no longer 1/3 of the pie it was originally from. That's only good for teaching fractions of a circle, which really doesn't come up all that often until you're way past the point of learning basic fractions. The whole idea is to compare the fraction to the whole (or other parts of the whole), and if you're enlarging just one part of it, then you're thro

## Re: (Score:2)

Perhaps GP meant to increase the radius by 50% while leaving the central angle the same.

## Re: (Score:2)

Ok; Take a 1/3 pie slice. Enlarge it by 50%. It is still a 1/3 pie slice, in value and visually.

Okay, I'll bite. Take that 1/3 pie slice and move it from the front of the pie to the back. Is it still a 1/3 pie slice visually? [wikipedia.org]

Now draw two Tootsie Rolls, one twice as long as the other. Does that accurately represent the values of each roll [wikipedia.org], or is the longer one one big Twinkie [photobucket.com]?

Learning to use tools lie pie charts and bar graphs is just as important to students as reading their first copy of How to Lie With Statistics [archive.org].

## Re: (Score:2)

I think that the fact that quantity can be expressed in many different ways is a

pretty fundimentalmathematical idea and trying to hide it from children would be a mistake. 1/6 isn't the "un-evaluated" version of 0.166..., it'sexactly the same thing.## Re: (Score:2)

at some point in my college maths I stopped seeing fractions

Must have not gotten very far in college maths then. Because in academia, fractions are the only widely accepted way of representing division (either by a horizontal line, or a slashed line, '/')

## Re: (Score:2)

You have it backwards. What you are thinking of as 'plain old' are actually normalised decimal fractions - one particular subset of fractional arithmetic. But decimal is no more inherently sensible than duodecimal, or octal, or binary, or even sexagesimal. A proper education in mathematics should enable one to work in whatever base and with whatever fractions are natural to a given question - instead of training you to use one particular set of variables as a procrustean bed.