## Wednesday, March 5, 2008

### Why teach fractions?

So asks Dennis DeTurck, an award winning math professor and dean of the college of arts and sciences at the University of Pennsylvania.  Children don't understand fractions; fractions are as obsolete as slide rules.  Calculators and decimals let us breeze where once we slogged, baffled by expressions like 1/2, by finding common denominators, and by inverting and multiplying.

Of course, many children--left-brainers and math buffs--start grasping fractions as early as first grade.  And surely all children must master them in order to advance through algebra?

Pondering this, I once asked Professor Deturck about rational expressions like 1/x and y/z. How do you express these as decimals and manipulate them with calculators?

Maybe we should wait until algebra before introducing fractions, he replied.  Or (see above link) until calculus.

Those of us who teach children know well the pedagogical nightmare that arises when we introduce two tough concepts at once-- e.g., fractions and variables, or fractions and derivatives.

A second reason for fractions before algebra struck me last night.  I was helping my son through an algebra problem, and amid all the messy denominators--e.g., (a-b)*(x-a)--he'd lost sight of how to find common denominators.  It was only when I gave him an analogous problem with numbers in place of variables that he rediscovered the algorithm and why it works.  Had fractions with numbers not been something familiar we could return to, he might have continued to flounder.

Myrtle Hocklemeier said...

You know,

Having recently looked over the course offerings at a university near me I've concluded that most of what is offered to to undergrads in the math department are math methods in science and engineering.

In K12 "math", in some cases, has become entirely about solving problems in science. Methodology becomes a nonissue when you were simply looking for "the right" answer to a science problem. And this is why calculators prevail.

When science got brought into the math classroom, the subject of math was redefined. Mathematicians have only themselves to blame for this mess as they themselves make their undergrad course offerings heavy with methods and applications courses, and making engineering calculus king of the curriculum.

Science problems solved with calculators need to be solved in science class.

It doesn't surprise me that folks want to do away with fractions. Fractions, to them, are some sort of pure math artefact.

lefty said...

Very interesting. I've not been following this trend in college math ed. What's the university near you?

I know a number of math profs who would hate to teach math this way, but I can imagine that they may be feeling increased pressure to do so from the powers that be at universities. I wonder what's going on at the higher levels.

Catherine Johnson said...

In K12 "math", in some cases, has become entirely about solving problems in science.

Right.

That's Math Trailblazers.

It is explicitly about "Integrated Math and Science," which was its original name.

Math isn't a subject unto itself.

It is merely a means of doing science.

Math, in a curriculum like Trailblazers, is treated as essentially a form writing in the sense that writing isn't a "content subject," but a skill.

Good writers must have knowledge of what they're writing about, but writing per se isn't a liberal arts discipline; writing is the means by which scholars express their work in the disciplines.

Math in many of these curricula is simply the "math form" of writing.

Catherine Johnson said...

A second reason for fractions before algebra struck me last night. I was helping my son through an algebra problem, and amid all the messy denominators--e.g., (a-b)*(x-a)--he'd lost sight of how to find common denominators. It was only when I gave him an analogous problem with numbers in place of variables that he rediscovered the algorithm and why it works. Had fractions with numbers not been something familiar we could return to, he might have continued to flounder.

Absolutely.

This is critical.

Once you're trying to teach algebra (or help with homework) fractions become the "concrete" example you use to demonstrate how a rational expression like a/b ÷ c/d works.

Very important.

Once again, none of these people has the slightest concept of sequence in learning: of foundational skills.

Allison said...

Your comment about how you can't introduce two new ideas at once is correct. A more formal reason why it's correct was stated by Professor Hung-Hsi Wu, of UC Berkeley math department. He states very clearly why we must teach fractions: because it is necessary to teach abstraction and working with symbols in order to prepare the way for algebra.

http://www.aft.org/pubs-reports/american_educator/summer2001/algebra.pdf

in the above, he states:

"students in arithmetic need a gradual acclimatization with the concepts of generality and abstraction before they can learn to compute on a symbolic level. In terms of the school curriculum, we can describe this progression in greater detail. It is difficult to teach students in whole number arithmetic about symbolic notation other than to write down in symbolic form the commutative laws,
the distributive law, etc., because the basic computational algorithms for whole numbers do not lend themselves to be explained
symbolically. However, the subject of fraction arithmetic— usually addressed in grades 5 and 6—is rife with opportunities for getting students comfortable with the abstraction and generality expressed through symbolic notation."

If you pretend that you can successfully teach algebra when that's the first time children are introduced to symbolic reasoning and manipulation, you are crippling them.

he continues:
"As students get to understand the division of fractions so that becomes meaningful even when a and b are now themselves fractions, one goes on to prove that formula
(6) remains valid, as it stands, when a, b, c, and d are fractions.
Then it follows that (6) is also valid for finite decimals.
One can go further. A standard topic in algebra is rational expressions, which are quotients of the form , where a and b are now polynomials in a variable x, such as a = x3 – 3x + 4 and b = 5x2 + 2. Then the addition of rational expressions is also given by (6) for polynomials, a, b, c, and d.
The message is now clear: Formal abstraction is at the heart of algebra. The addition-of-fractions formula (6) is an example because the same formula is seen to encode seemingly disparate information. If we look at the school mathematics curriculum longitudinally, the development of formula (6) from whole numbers a, b, c, d to polynomials as described above takes place over a period of three to four years, and it starts with the teaching of fractions. Without this foundation in fractions, students who come to the study of rational expressions in algebra are severely handicapped."

lefty said...

Thanks for this, allison. We should try to introduce Professor DeTurck and Professor Wu!