Tuesday, January 6, 2015

Building on Uncommon Denominators

[This blog post is part of a series, Defining Vertical Learning.]

The core idea of vertical learning is that we should learn by building up and drilling down. Now, I taught math and science, and math and science are inherently vertical subjects. Concepts build on top of one another. In math, you can drill down and ground everything in axioms; in science, we are still drilling down in search of a Grand Unified Theory. But learning isn’t vertical because of how a subject is organized or taught, it is vertical because of how the learner constructs his or her own understanding of it.

If you have ever spent time in a classroom where students are learning about fractions, you may have heard a recurring question: “Do we need to find common denominators for this?” The question is so common that we typically take it for granted. If students have been multiplying or dividing fractions for a few days without finding common denominators, of course they may temporarily forget whether or not they need common denominators when adding or subtracting fractions. This confusion is natural, right?

But if you ask those same students to combine 3 dimes + 2 quarters, they would never say that the answer is 5 dimes or 5 quarters. They would recognize that dimes and quarters have different values, and they would say that you either have 5 coins or 80 cents. They intuitively know that, in order to add 3 dimes and 2 quarters, you need to convert dimes and quarters to a common unit. There would be zero confusion. In the same way, if you asked students to combine 3 cups + 2 pints or 3 inches + 2 centimeters, they may not be able to convert those units to a common unit and get an answer, but they would know that you can’t just add 3 + 2 and get 5. If you tried, they would jump all over you for making a silly mistake.

But wait a second! Those students have been manipulating and using money for years. They just learned about fractions a few months ago. Of course they have more intuition about dimes and quarters than sixths and eighths. Well, try giving those students a problem involving bushels and pecks. Or tell them they crash-landed on an alien planet and make up some units. Again, they know that they can’t just add quantities with unlike units together without doing some sort of conversion first. They just know it.

The students who can’t remember if they need to find a common denominator when adding fractions almost certainly know that sixths and eighths are different units. Unfortunately, instead of building on top of their existing understandings of working with units, they have built their understanding of fractions off to the side and in isolation. The teacher may have “made the connection” between units and unit fractions for them, but that’s irrelevant. The only thing that matters is how well the learner has constructed his or her understanding of fractions vertically, grounding that understanding in a foundation that is both intuitive and functional. The teacher’s primary role is to probe the learner’s understanding and create experiences that enable and encourage the learner to test and revise his or her own mental model.

What’s worse is that the same thing probably happened earlier. Why do we line up digits to add multidigit numbers? So that we can add ones to ones and tens to tens. Can we add tens to hundreds? Sure, if we do some regrouping (converting). And the same thing will probably happen again when we start simplifying expressions: 3x + 2y + 6x. Which of those terms can we combine again? All of these skills and concepts should be built on top of the same foundation, strengthening the foundation in the process and enabling us to build higher and wider. But how often do you think that happens? And what are the consequences if learning happens horizontally instead of vertically?

When probed, I expect a student’s understanding when adding fractions to look something like this:

Combine 5 sixths and 3 eighths. Hmm… sixths and eighths are different units. I’ll have to convert them into a common unit.

If I cut all of the sixths into four equal pieces and all of the eighths into three equal pieces, I’ll have twenty-fourths. I can combine 20 twenty-fourths and 9 twenty-fourths. It’ll add up to more than a whole, but I’ve got lots of experience working with mixed numbers.

Wait… your teacher didn’t “make the connection” between mixed numbers and multidigit numbers for you? You do know that a multidigit number is a mixed number, right?


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