3 ms·

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 ha

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
- 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.

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
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.