Standards that tell you how to understand?

Several weeks ago, I spent a lot of time reviewing the grades 3–6 Common Core State Standards (CCSS) and identifying Sketchpad activities that addressed these standards. Through this immersion, I noted some themes that were quite different in the CCSS than the many other state standards I’ve reviewed in my time.

One of the things that struck me about some of the elementary school content standards in particular is that they stress how to understand a concept. This is not the norm in other states’ standards. Typical standards note what a student should be able to do, or sometimes what a student should “understand.” But stating specifically how to understand is uncommon.

For example, here’s a 4th grade Common Core standard:

4.NF.1 Explain why a fraction a/b is equivalent to a fraction (n × a)/(n × b) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions.

Here are Texas (TEKS) Grade 3 and 4 standards, respectively, covering similar content:

2.D Construct concrete models of equivalent fractions for fractional parts of whole objects.

2.A Use concrete objects and pictorial models to generate equivalent fractions.

And a California Grade 3 standard covering similar content:

NS.3.1 Compare fractions represented by drawings or concrete materials to show equivalency and to add and subtract simple fractions in context (e.g., 1/2 of a pizza is the same amount as 2/4 of another pizza that is the same size; show that 3/8 is larger than 1/4).

Here are a few screen captures from a Sketchpad activity, “Fraction Tiles: Equivalent Fractions,” that covers all of these three states’ standards. Students first “Show fifths grid” (reveal vertical dashed lines) to begin modeling 3/5. Then they “Show 1/5 tiles” (color three segments) to show 3/5.

SketchpadScreenSnapz001

Then they “Show thirds grid” (reveal horizontal dashed lines) to see that 3/5 is equivalent to 9/15.

Students continue on to build their own models. They select a vertical grid tool and tiling to show 2/5…

…then add on a horizontal grid of… what size?… fourths, to show that 2/5 = 8/20.

Constructing these models oneself, as opposed to observing drawn examples, leads students not only to understanding equivalent fractions, but also, I believe, clearly leads to “attention to how the number and size of the parts differ even though the two fractions themselves are the same size.”

Here’s a more traditional approach, from Key to Fractions, Book 1 (Key Curriculum Press, 1980). Do you think students develop a different kind of understanding or skill from approaching equivalent fractions in these two different ways?

Update: Be sure to check out this new Web Sketchpad model of fraction multiplication.

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