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Re: The Teaching of Fractions
Posted:
Oct 29, 2012 12:36 AM


Bob:
Many thanks for your link to Wu's listing of the CCSS criteria for curricular education in fractions ... and thanks to Wu, for gleaning the fractionsstandards out from the CCSSmathematics. That will save the Mathematical Literacy Project from having to dig for that particular CCSS theme ... which is a backdrop for the MatheLite Project's "grand opening" demonstration of its community education video library .
Wu's own commentaries about fractioneducation leave much to be desired, but the CCSS did not mess up Wu's own theories of fractions and fractionteaching. That mess came from his own efforts (as a traditionally educated mathematician) to function also as a mathematicseducation consultant ... without thoroughly studying the mathematics of fractions.
So, what he did was to use CCSS as a context for reiterating what he earlier proffered in his earlier AMS publication. Nothing wrong with that ... except that it contributes nothing new toward sorely needed progress on the national dilemma on fractioneducation. Your own lay thoughts are almost as useful.
I was disappointed that Wu's econversation with me (about fractions) failed to reveal his willingness to learn any more about the subject than what already is known by most other teachereducators. It is not that Wu's theory is "wrong" ... so much as that it is woefully inadequate. It says nothing that was not already known in the field ... and has the same maladies as do traditional curricular perceptions.
Science will soon reveal that there are more effective and humanly natural and ways of learning/teaching fractions. This is not the time or place to elucidate, but "Stay tuned for coming attractions."
Cordially, Clyde
  From: "Robert Hansen" <bob@rsccore.com> Sent: Sunday, October 28, 2012 8:21 AM To: <mathteach@mathforum.org> Subject: The Teaching of Fractions
> Here is a link to what H.Wu wrote last year regarding the teaching of > fractions and the Common Core Standards... > > http://math.berkeley.edu/~wu/CCSSFractions.pdf > > I disagree with much of it. > > When I think of teaching a subject I start with the final goal in mine, > stated in the simplest terms possible. For example, if I were planning a > section in probability (for algebra students) my goal is to arrive at... > > 1. Probability is the ratio between the number of ways an expected event > can occur and the number of ways any event can occur. > 2. The difficulty in determining probability lies in determining the > number of ways an expected event can occur, the number of ways any event > can occur, or both. > > So when I think about teaching fractions, I have a goal... > > 1. A fraction is a number, represented by the quotient of two numbers, the > top number being the numerator and the bottom the denominator. > 2. Fractions can be ordered (placed on a number line) and two fractions > are equal if their quotients are equal (the numerators and denominators > can be different). > 3. Fractions can be added, subtracted, multiplied and divided, just like > numbers, but the mechanics are more involved because fractions involve the > quotient of two numbers. > > In Wu's defense, he is using the common core standards as a guide so he > was bound to screw this up. The common core standards are what I call 3rd > generation standards. The first generation started in the late 80's > through the 90's (before the NCLB made it a national past time), the > second generation started in the 2000's during the heyday of NCLB and 3rd > generation (the Common Core Standards) evolved from all of that. > > As a collection of topics the standards are (just) ok. If you or I sat > down and listed all of the micro points to arithmetic we would come up > with a similar list. The problem with the standards is sequencing. I and > others have said this a number of times before. The sequencing is not > systematic in its progression. Somewhere during the evolution of > "standards making", someone made a very bad judgement call and it became > popular. They decided that these topics should be arranged in threads, > simultaneously occurring at the same time. So today, the students (and Wu) > are tasked with up to 6 threads all occurring at the same time. Number, > operation, algebra, geometry, fractions, reasoning, etc. It's like eating > a 6 course dinner, not one course after another, but all at the same time. > It is very difficult for a student to reverse engineer all of this back > into meaningful and fulfilling meal of mathematics. Either they are very > bright, they have parents or tutors to reverse enginee! > r it for them, or they fail. Sadly, the only option available to most is > the last one. > > Back to fractions... > > I would not start teaching fractions until multiplication and division > (with whole numbers) are firmly established. > > After that is established the steps are rather straightforward... > > 1. Introduce the concrete idea of a fraction using pizzas, rectangles, > etc. > > 2. How much is 1/5th of 20? How much is 2/5th of 20? Is 2/5th of 20 bigger > than 1/5th of 20? Is 2/5th bigger than 1/5th? > > 3. How much is 1/5th plus 2/5th of 20? > > 4. How much is 1/10th of 20? How much is 1/2 of 1/5th of 20? Is 1/2 of > 1/5th the same as 1/10th? > > 5. How much is 2/10th of 20? How much was 1/5th of 20? > > 6. How much is 1/4th of 12? How much is 1/3rd of 12? Which is bigger? > > 7. How much is 1/4th plus 1/3rd of 12? > > The only thing I didn't cover is division by a fraction. That we withhold > till the students are comfortable and fluent in the above operations. > Also, we have decimals coming up, which I would delay until the later > stages of the above. > > If you don't believe that these standards are a mess for a student to > navigate, look what they did to Wu (who I adore by the way). > > Bob Hansen



