Fraction division is not the only operation that has a "standard" algorithm, of course. Almost all operations in math have one. Most, if not all, of us are familiar with the Fearful Four of standard algorithms:
· the "carrying/borrowing" algorithm for addition and subtraction of whole numbers and decimals,
· a similar algorithm for multiplication of whole numbers and decimals,
· Long Division (yes, capitalized) for division of whole numbers and decimals,
· and the dreaded "invert and multiply" for division of fractions.
Yes, mathematics has hundreds more smaller, less well-known standard algorithms (reduction of fractions, converting between mixed numbers and improper fractions, etc.) but the four above are the biggies. They also happen to be the four that are mentioned explicitly in the K-8 Common Core standards.
Well, almost explicitly.
The "carrying/borrowing" algorithm is often referred to as the "standard" algorithm for addition and subtraction, but the standards do not specify that particular algorithm. They simply say "Fluently add and subtract multi-digit whole numbers using the standard algorithm." So there has been some debate over the definition of standard algorithm, and it turns out that this definition is not universal. Some experts and math education leaders (and I agree with them) would suggest that the "standard" algorithm for addition and subtraction should be partial sums. The standard for multiplication could be partial products. Both of these methods have advantages over the traditional "carry/borrow" algorithm in both connection to place value concepts and encouraging flexibility in student thinking.
It is also worth noting that "Fluently add and subtract multi-digit whole numbers using the standard algorithm." is standard 4.NBT.B.4 (CCSSI, 2010). This means that the standard algorithm is pushed off until 4th grade, after students have learned to add and subtract multi-digit whole numbers in Grades 2 and 3, with introductions from early Kindergarten through Grade 1. The algorithm is pushed off until very late in students' experiences with those operations.
Much like the punch line to a joke.
While the punch line metaphor is perhaps unflattering, it is nevertheless effective and accurate. Students build understanding of operations through experiences and informal reasoning, just as listeners build understanding of the amusing situation in the body of a joke, and only after having sufficient experience with and understanding of the operations are they ready to learn the algorithm with understanding, just like the listeners and the punch line.
Another aspect of this debate is whether students should even be taught certain algorithms in school. David Ginsburg blogged about the necessity of long division, so I'll avoid that one here (but you should all read So Long, Long Division) and move on to some others. I've no argument against the standard algorithms for most operations in elementary schools but I'll choose one that I do have an argument against: reduction of fractions (or writing fractions in simplest form for all my high school teachers out there). This is the "punch line to bad jokes" portion of the blog.
And don't worry, I'll get to a hot button secondary example in my next post.
But first, reduction of fractions. Before everyone piles on all at once here, I am absolutely not saying that students should not be taught how to write fractions in simplest form. What I am saying is that we should not teach it as an algorithm and we should definitely not require students to write every single fractional answer they ever get in simplest form. This is because in many mathematical situations, the reduced form of a fraction will not reveal enough information about the initial problem context. Not to mention all of my statistics friends out there who will argue tooth and nail that a probability of three-fourths is absolutely not the same as a probability of six-eighths (and they are right). That being said, I do acknowledge the necessity of reducing fractions and so I propose teaching the idea or procedure with understanding.
My reasoning for this is two-fold. First, we spend a great deal of time in schools beating students over the head with reducing fractions and we never tell them why or require that they make sense of the actual procedure. Time that could be better spent both exploring more important topics and showing students success instead of highlighting failure. Distributed practice is very important, however when students are completely correct in reasoning and even calculation on a problem but receive bad marks due to an answer that is unreduced, teachers are doing them a great disservice. We must ask ourselves what we are trying to assess about students' knowledge and be sure we aren't needlessly penalizing them for something that is quite trivial and unrelated to the knowledge base we desire.
Second, students will be much more likely to remember how to "reduce" fractions if we teach them under the guise of generating equivalent fractions. The CCSSM focus on equivalent fractions in Grades 3-7 should be continued throughout all grades. Students are very capable of generating equivalent fractions from almost the beginning of their study of fractions and we should nurture this instinctual understanding, not drive it out with a meaningless procedure. Students can understand equivalent fractions in a few different ways, all of which are very effective and still allow students to find the "simplest" form of a fraction. One great example I've heard before is that of a reduced fraction being the "simplest recipe" in a given situation. This idea, in conjunction with appropriate contexts, can really drive home the idea of reduced fractions with understanding.
So the moral of the story today is: teaching standard algorithms is fine if you must. Just treat them like the punch line of a joke and don't deliver them until the end of the learning trajectory. Remember, procedural fluency should be developed from conceptual understanding, not the other way around.
Those are some of my thoughts. I'd love to hear your thoughts on the subject!