4 ms·

Federal Research Suggests New Approach to Teaching Fractions

27 points by tokenadult 13y ago

mathattack 13y ago
I hope they're right in fixing this. The overemphasis on whole #s is counter to much of reality and how math is applied.
nickmain 13y ago
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 13y ago
  How do you explain the taboo about NEVER PUT ZERO ON THE BOTTOM!!!
tokenadult 13y ago
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=prealgebra http://www.artofproblemsolving.com/Store/viewitem.php?item=p... 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%20mt%20910/teaching%20and%20learning%20mathematics%20-%20review%20of%20liping%20ma%20text.pdf https://webstorage.worcester.edu/sites/rbisk/web/marlboro%20... 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 13y ago
  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 13y ago
    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 13y ago
      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 13y ago
        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
jmilloy 13y ago
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
samatman 13y ago
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 13y ago
  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 13y ago
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 13y ago
  > 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.
- JoshTriplett 13y ago
  Zero apples. One apple. Two apples. Half an apple. One and a half apples. 1 inch. 1.5 inches. 2 inches. Fractions are not conceptually hard.
SimHacker 13y ago
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.
beefman 13y ago
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 13y ago
  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?
TempleOS 13y ago
Apple says, "Think different". Hollywood space movies have funny physics. "Least common denominator" felt right, but not if the crowd I nerds. I chuckle at people who think Hollywood is dumb.

27 points by tokenadult 13y ago

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

nickmain 13y ago

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 13y ago
How do you explain the taboo about NEVER PUT ZERO ON THE BOTTOM!!!

tokenadult 13y ago

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=prealgebra http://www.artofproblemsolving.com/Store/viewitem.php?item=p... 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%20mt%20910/teaching%20and%20learning%20mathematics%20-%20review%20of%20liping%20ma%20text.pdf https://webstorage.worcester.edu/sites/rbisk/web/marlboro%20... 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 13y ago
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 13y ago
  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 13y ago
    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 13y ago
      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

jmilloy 13y ago

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

samatman 13y ago

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 13y ago
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 13y ago

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 13y ago
> 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.
JoshTriplett 13y ago
Zero apples. One apple. Two apples. Half an apple. One and a half apples. 1 inch. 1.5 inches. 2 inches. Fractions are not conceptually hard.

SimHacker 13y ago

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.

beefman 13y ago

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 13y ago
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?

TempleOS 13y ago

Apple says, "Think different". Hollywood space movies have funny physics. "Least common denominator" felt right, but not if the crowd I nerds. I chuckle at people who think Hollywood is dumb.