The history of the Euclid concept of proportions as taught in Euclid’s Elements is sketched below. This is based on sources cited in the previous post and additional references below.
- Late ancient world they couldn’t follow Euclid on proportions so gave up.
- Early moderns invented modern symbols of arithmetic in 15th to 17th centuries to do fractions practically.
- Euler 1765 in his text Elements of Algebra gave procedural rules for adding fractions using symbols and made no attempt to derive them from proportions logic or Euclid or geometry.
- 19th century mathematicians junked the whole Euclid approach and taught rational numbers as an object oriented type of number with its own data as an ordered pair and own procedures.
- Modern schools still teach the Euclid proportion concept for fractions.
- Oxford, US Dept of Education, etc. publish studies that children can’t understand the (Euclid) proportion concept of fractions.
The Euclid concept of proportion is closer to the concept of the authors of the following study then the 19th century notion. The authors seem completely unaware that they are closer to using the Euclid approach. They seem unaware that the modern ordered pair approach exists.
Terezinha Nunes, Peter Bryant, Jane Hurry and
Ursula Pretzlik – with the collaboration of Daniel Bell,
Deborah Evans, Selina Gardner and Joanna Wade.
The relative nature of fractions is a source of
difficulty for pupils. It requires that they
realise that the same fraction may refer
to different quantities (1/2 of 8 and 1/2 of
12 are different) and that different
fractions may be equivalent because they
refer to the same quantity (1/3 and 3/9, for
example). It is not possible for pupils to
make further progress in mathematics or
to take advanced courses in secondary
school without a sound grasp of the
relative nature of rational numbers.
The above language is assumed by the authors to be the real meaning of fractions. Their concept of fraction is close to the Euclid concept including their terminology. Their terminology is far from talking about an ordered pair of numbers. For the authors, fractions act like proportions involving quantities. Euclid calls them magnitudes.
Euclid Elements Book V
- Definition 3
- A ratio is a sort of relation in respect of size between two magnitudes of the same kind.
- Definition 4
- Magnitudes are said to have a ratio to one another which can, when multiplied, exceed one another.
- Definition 5
- Magnitudes are said to be in the same ratio, the first to the second and the third to the fourth, when, if any equimultiples whatever are taken of the first and third, and any equimultiples whatever of the second and fourth, the former equimultiples alike exceed, are alike equal to, or alike fall short of, the latter equimultiples respectively taken in corresponding order.
- Definition 6
- Let magnitudes which have the same ratio be called proportional.
(It would be valuable if the reader could skim the above link of Euclid Book V to see that proportion as a concept in Euclid is what the authors of the study are thinking in terms of as opposed to ordered pairs as an object with data and procedures.)
The Euclid concept in Book V and the study authors above have the same concept of fractions as proportions. They do not have the modern math foundations concept of an object oriented number as an ordered pair of data with procedures. The modern concept is more procedural in nature, and is much easier to work with. The Euclid concept is so difficult, that modern math foundations abandoned developing it further and started over with an object oriented approach of data and procedures.
Students don’t get the Euclid proportion approach because it relates 4 quantities as the meaning of proportions. 4 is to 5 as 8 is to 10. The ordered pair says that (4,5) is an ordered pair with rules for manipulating it. It is a representative of an equivalence class, but that part need not be emphasized too much. Instead, we can say two ordered pairs are equivalent if their cross products are equal. With proportions, we make that part of the definition.
Ordered pairs cut out half the data in the definition. This makes them more than twice as easy to understand. Reduction in amount of data in a concept increases the comprehensibility more than proportionately.
With ordered pairs, we have half of proportions. So we are working with less data at any one time. That makes it easier to understand.
Ordered pairs can be called pre-proportions if you like. Using pre-proportions or half of a proportion, we can manipulate the pre-proportions with rules easier to understand than for proportions. With pre-proportions, we can do proportions as a data structure of two pre-proportions.
This is why the modern concept of an ordered pair beats the ancient concept of 4 numbers as a proportion.
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by CD Bruce – Cited by 7 – Related articles
Students are challenged when learning fractions and problems often persist into adulthood. Teachers may find it difficult to remediate student misconceptions in …
One reason their difficulties persist into adulthood is that the school approach shares too much with the Euclid approach and the Euclid approach was abandoned by mathematicians as unworkable.
An illustration of teaching fractions using Euclid type concepts.
Ordered pairs as pre-proportions might help break the reluctance to drop the Euclid approach and switch to the modern math foundations object oriented approach of ordered pairs as data plus procedures. We can call this a pre-proportion.
We can add pre-proportions and proportions they are part of will track that. We can multiply pre-proportions by a natural number and divide it by a natural number. If this is done on both sides of a proportion, the equality of the proportion follows.
Fractions have likely been difficult since the start of human reasoning. Particularly when we start to add them. Adding fractions is different than equal slices of pizza to share equally.
In the Middle Ages and early modern era, people gave up on the Euclid approach and switched to the modern symbols and procedures. In the 19th century, they restated this in a mathematically precise way. That is the modern concept. It is because the proportion way of introducing and justifying fractions becomes increasingly difficult to track what it means and is saying that it was abandoned. But that is still what is taught in schools today.
Just like all generations before, fractions as proportions breaks down as we try to load all the properties of numbers onto proportions as conceived in Euclid. Instead, we have to start over with ordered pairs as data and procedures to get back to a number concept of fractions that is workable.
Some more references
Dept of Education
Center for Improving Learning of FractionsThe Institute of Education Sciences, a research branch of the U.S. Department of Education, has awarded a $10 million grant to University of Delaware Prof. Nancy C. Jordan and her two colleagues, Lynn Fuchs at Vanderbilt University and Robert Siegler at Carnegie Mellon University, to fund a five-year research and development center aimed at understanding difficulties students have with fractions.
The Center for Improving Learning of Fractions, administered at UD, will focus on improving math instruction for elementary and middle school children who have problems with math concepts, specifically fractions.“It’s really exciting but also a huge responsibility,” said Jordan. “The center is going to involve top researchers coming together to work on solving an important problem in education.”
Mission:To understand the problems that children with mathematics difficulties (MD) have with rational numbers and to develop effective interventions for addressing those problems.
Why are fractions so hard?
Why do you think fractions are such a difficult topic for students (of all ages) to maser? What could you do to help students that are struggling in this area?
The discussion thread is interesting.
i think alot of it has to do with the fact that alot of the teachers have as much trouble with it as the students do and as a result don’t do an effective job of teaching this concept.