

Federal Research Suggests New Approach to Teaching Fractions - tokenadult
http://www.edweek.org/ew/articles/2013/07/18/37fractions.h32.html?tkn=TYCCvYyFRqTWe%2F53pDFlhN5H2Mg4o%2FQhgGLd&cmp=clp-sb-ascd

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tokenadult
I am quite astonished at how slow the uptake of showing fractions on a number
line is in school mathematics lessons in the United States. That is routine
and has been routine for more than a generation in the countries with the best
primary mathematics achievement around the world. I picked up some of this
implicitly from my home environment just by looking at rulers in my dad's home
workshop. (He had rulers with markings down to 1/32 of an inch, as I recall,
certainly down to 1/16 of an inch.)

Even here on Hacker News, among fully grown adults, when people raise
questions about explaining fractions, many comments leap immediately to pizza
slices--a valid model of some fractions, but a very poor model for general
understanding of fractions. I took great care to email one of the co-authors
of the Art of Problem Solving textbook on Prealgebra

[http://www.artofproblemsolving.com/Store/viewitem.php?item=p...](http://www.artofproblemsolving.com/Store/viewitem.php?item=prealgebra)

while he was writing the book to make sure that that book would have a good
chapter on fractions--it does--so that I have a good resource for the
mathematics classes I teach. Fractions are not insuperably hard if they are
well taught, but they are rarely well taught at any level of mathematics
classes in the United States. If you'd like more background on this topic, see
_Knowing and Teaching Elementary Mathematics: Teachers ' Understanding of
Fundamental Mathematics in China and the United States_ by Liping Ma

[http://www.ams.org/notices/199908/rev-
howe.pdf](http://www.ams.org/notices/199908/rev-howe.pdf)

[https://webstorage.worcester.edu/sites/rbisk/web/marlboro%20...](https://webstorage.worcester.edu/sites/rbisk/web/marlboro%20%20mt%20910/teaching%20and%20learning%20mathematics%20-%20review%20of%20liping%20ma%20text.pdf)

for a comparison of elementary mathematics lessons in two different countries,
with China generally having much more mathematically accurate and much more
thorough lessons in fractions and fraction arithmetic, among other topics.

~~~
glhaynes
_pizza slices--a valid model of some fractions, but a very poor model for
general understanding of fractions_

If you'd like, could you say a little more about why this is the case?

~~~
jmilloy
For one, I think that the pizza slices, by being in 2 dimensions, represents
the fraction as portion to a (specific) whole without representing the
absolute magnitude of the fraction. Both features are important.

~~~
e12e
If the pizza has an area of unit size (1) - how does a fraction not have
absolute magnitude? And what does the fact that the pizza is in two dimensions
have to do with it? If the pizza is modelled as a cylinder with unit volume -
any slice has both volume/x volume, and area/x area -- where x is the area of
the slice?

I'm not trying to be snarky, I would just like you to expand on what you mean?

I can see that pizza slices is a poor framework for working with general
fractions -- and I agree that a number line is a fine way of visualizing
magnitude... perhaps I'm just not familiar enough with what "teaching
fractions as pizza slices" entails...?

~~~
jmilloy
Teaching fractions as pizza slices entails dividing the circle into 6ths, say,
and shading in 5 slices[1]. The wedge in the circle shows relative magnitude;
it depicts the fraction as a portion (numerator) of the whole (denominator)
regardless of how big the circle is. The magnitude of the fraction is depicted
only relative to 1.

On a number line, the magnitude of the fraction is depicted relative to _any_
number. In this sense, we capture the "absolute magnitude" of the fraction, by
placing on a line relative to the integers.

    
    
      ---1-----0-----1-----2-----3---
                   ^       ^
                  5/6     13/6
    

An individual fraction, as a single-dimensional magnitude, is best represented
in an infinite single-dimensional space, where magnitude increases forever in
exactly one direction. A circle, on the other hand, has a finite amount of
space. Maybe you're right: perhaps it is the infinite nature of the number
line that lends itself to "absolute magnitude" rather than the fact that it is
one-dimensional.

On the other hand (this is increasingly pedantic): While we could depict 1d
magnitude "infinitely" in 2d with concentric circles of area 1, 2, 3, etc,
using 2d area to depict a 1d magnitude is not intuitive. The natural
alternative is to subdivide the space into units and show multiple discrete
units. In 2d, 13/6ths would be two whole pizzas and one 1/6th-filled pizza.
But in 1d, the units can be placed adjacent to each other according to the
direction of increasing magnitude. In this sense, 1d lends itself to a natural
depiction of absolute 1d-magnitude, but 2d does not.

[1]
[http://www.primarygames.com/fractions/images/pizza_5-6.gif](http://www.primarygames.com/fractions/images/pizza_5-6.gif)

------
beefman
This is a bit meta, but the "fractions" thing cracks me up. There should be a
Godwin's law for it... The probability of a story about public education
mentioning "fractions" approaches 1.

Why are fractions a thing? Sure, the rationals are distinct from the integers
and reals, but... Does everyone except me have PTSD from learning fractions or
something? Why are they more important than any other math topic, subject, or
aspect of education? By golly, if we could just figure out how to teach
fractions, we'd have a bona fide education system.

I think I started noticing it after this Onion piece (NSFW)
[http://www.youtube.com/watch?v=Btw1eh4mitQ](http://www.youtube.com/watch?v=Btw1eh4mitQ)

~~~
tokenadult
Here is an actual question for schoolteachers from a National Science
Foundation study of elementary mathematics instruction in the United States.
In Liping Ma's book (mentioned in my comment above, with links to two
reviews), the author took the same NSF sample survey questions to teachers in
China and compared how Chinese teachers answered the questions. Here is a
teaching question for any reader of Hacker News who has ever had occasion to
teach mathematics or explain mathematics to a friend or colleague:

"Scenario 3: Division by Fractions

"People seem to have different approaches to solving problems involving
division with fractions. How do you solve a problem like this one?

1¾ ÷ ½

"Imagine that you are teaching division with fractions. To make this
meaningful for kids, something that many teachers try to do is relate
mathematics to other things. Sometimes they try to come up with real-world
situations or story problems to show the application of some particular piece
of content. What would you say would be a good story or model for

1¾ ÷ ½ ? "

This problem, providing an illustration for what it means to divide one
rational number by another, proves to be extremely hard for United States
teaches, but quite easy for teachers in China. What story problem would you
devise to embed this calculation into a context understandable to children?

------
jmilloy
It's clearly useful and appropriate to show fractions as a relation of a part
to a whole, as a _portion of something_ , but it's almost always depicted in 2
dimensions (typically a circle) and therefore restricted to comparing to just
one whole (i.e. the denominator of the fraction in reduced form). Using 1
dimension allows us to compare the portion to not just one whole but many
familiar wholes, the natural numbers. I think it would also be useful to co-
develop intuition of space and portions in 3 dimensions.

Maybe not everyone can get to it, I don't know, but the paper:
[http://www.psy.cmu.edu/~siegler/121-siegler-etal-
tics.pdf](http://www.psy.cmu.edu/~siegler/121-siegler-etal-tics.pdf)

------
nickmain
I'm lucky enough to work at a place that is building Math games for K-6.

I watched the curriculum experts and content design teams white-boarding the
many ways to conceptualize and visualize fractions for several weeks. It's not
the simple domain you might think.

~~~
SimHacker
How do you explain the taboo about NEVER PUT ZERO ON THE BOTTOM!!!

------
samatman
As an earlier step,
[http://en.wikipedia.org/wiki/Cuisenaire_rods](http://en.wikipedia.org/wiki/Cuisenaire_rods)
have demonstrated utility. It's a shame their use is mostly limited to
Montessori schools in the US.

~~~
T-hawk
Oh my gosh, thanks for posting that! I went to a Montessori school and must
have used those, but never knew it was a thing with a name.

I _see_ arithmetic in visual terms like those rods. Like to add 18 + 12:

    
    
      ++++++++++
      ++
        ++++++++
      ++++++++++
    

I actually "see" the numbers fitting together like that spatially to make the
sum. Must have learned this when I was too young to consciously remember, and
the most likely way that happened was with these Cuisenaire rods or some such
physical objects. Amazing to suddenly learn now that that's a known and
defined device.

------
Locke1689
Does anyone else think that it's hard to teach fractions because _fractions
are hard_?

The tenor of these conversations is always about how difficult it is to teach
fractions, with an implicit judgement that this is bad because fractions
should be easy.

The thing is, you construct fractions by generating the equivalence class of
ordered pairs (Z_1, Z_2) where Z_2 =/= 0. I actually find elementary set
theory easier to grasp than the formal construction of arithmetic over the
rational numbers.

A lot of people would respond that we don't teach children the formal
construction -- fine, but a lot of the behavior of the rationals comes from
that formal construction.

I recall a class in my freshman year of college that required differential
equations to do system analysis. The only problem was -- differential
equations was next quarter. I think the rote algorithm they gave us to
calculate simple differential equations was one of the most frustrating
experiences of my academic career. I tend to think that if you teach students
things as though they can't understand it, you'll often find that they can't
understand it.

~~~
vanderZwan
> A lot of people would respond that we don't teach children the formal
> construction

Arguably, the way we teach fractions already follows the natural progression
of formally defining numbers pretty much: You start with addition (which is
somewhat hardwired into our natural number sense), learn that you can also
reverse that to get subtraction. Then the big leap of symbolic faith[1] is
subtracting a big number from a smaller one to get negative numbers. After
this there's the idea of counting repeated additions and giving that a
shortcut, resulting in multiplication, and the reveral of that gives division.
The big mental jump then is dividing a small number by a bigger number, giving
fractions.

[1] You have to remember that mathematics is basically formalising our limited
number intuition into language, so we can manipulate the symbols as we would
manipulate real world objects, greatly increasing our capacity to reason about
these ideas. "Doing with Images makes Symbols," to quote Alan Kay.

------
SimHacker
I remember being confused about what was the difference between fractions and
negative numbers. I knew they were both "advanced concepts" but I didn't know
if they were the same or how they were related. I asked the teacher, and she
said I'd learn later.

------
mathattack
I hope they're right in fixing this. The overemphasis on whole #s is counter
to much of reality and how math is applied.

