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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 wh

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
    - deleted 13y ago
      [deleted]
tokenadult 13y ago
If you'd like, could you say a little more about why this is the case? You ask a very pertinent question, to which other HN users have already replied. I'll link here to some articles by mathematicians who have considered how to do mathematics education well. When I just did a Google search on "fractions pizza slices" I had a HUGE number of hits, most to abominably bad materials for elementary teachers. Restricting the search in various ways found the links I share here, some of which I've read before and some of which are new to me too. ftp://math.stanford.edu/pub/papers/milgram/oregon-outline.pdf ftp://math.stanford.edu/pub/papers/milgram/oregon-presentation-2.pdf (Hacker News will not style these two as links, I think, but the URLs are valid.) http://math.berkeley.edu/~wu/Lisbon2010_2.pdf http://math.berkeley.edu/~wu/Lisbon2010_2.pdf (general audience version) http://math.berkeley.edu/~wu/Schoolmathematics1.pdf http://math.berkeley.edu/~wu/Schoolmathematics1.pdf (teacher version) http://steinhardt.nyu.edu/steinhardt/gateway/pdfs/Greenleaf_Fractions.pdf http://steinhardt.nyu.edu/steinhardt/gateway/pdfs/Greenleaf_... (a source new to me today) http://www.waset.org/journals/waset/v53/v53-66.pdf http://www.waset.org/journals/waset/v53/v53-66.pdf (another source new to me today) I hope this helps. The book I mentioned in my first comment in this thread is very good and well worth reading for any mathematics teacher.

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
  - deleted 13y ago
    [deleted]

tokenadult 13y ago

If you'd like, could you say a little more about why this is the case? You ask a very pertinent question, to which other HN users have already replied. I'll link here to some articles by mathematicians who have considered how to do mathematics education well. When I just did a Google search on "fractions pizza slices" I had a HUGE number of hits, most to abominably bad materials for elementary teachers. Restricting the search in various ways found the links I share here, some of which I've read before and some of which are new to me too. ftp://math.stanford.edu/pub/papers/milgram/oregon-outline.pdf ftp://math.stanford.edu/pub/papers/milgram/oregon-presentation-2.pdf (Hacker News will not style these two as links, I think, but the URLs are valid.) http://math.berkeley.edu/~wu/Lisbon2010_2.pdf http://math.berkeley.edu/~wu/Lisbon2010_2.pdf (general audience version) http://math.berkeley.edu/~wu/Schoolmathematics1.pdf http://math.berkeley.edu/~wu/Schoolmathematics1.pdf (teacher version) http://steinhardt.nyu.edu/steinhardt/gateway/pdfs/Greenleaf_Fractions.pdf http://steinhardt.nyu.edu/steinhardt/gateway/pdfs/Greenleaf_... (a source new to me today) http://www.waset.org/journals/waset/v53/v53-66.pdf http://www.waset.org/journals/waset/v53/v53-66.pdf (another source new to me today) I hope this helps. The book I mentioned in my first comment in this thread is very good and well worth reading for any mathematics teacher.