Mission: Partition #MTBos

Let's play a game. It's called "What math does this assess, and is that the math the standard calls for?" Wordy, I know, but it's a game worth playing as we consider how to make assessments actually assess what we need them to. The item below comes from a 1st grade benchmark exam, and is tagged as an assessment of standard 1.G.3, with no other items assessing this standard. Before I give you the text of the standard, we're going to play the first part of the game: what does this assess?

English translation: Color a quarter of the square. Color half of the square. Color half of the rectangle. Color a quarter of the circle. 

Some teachers' answers to "What math does this assess?":

• Understanding of the phrases "half of" and "quarter of"
• Ability to color "one part" of a shape
• Low level assessment of identifying a part of a shape (not necessarily understanding of equal parts)

And now for the next part of the game: "Is that the math the standards call for?" Here's the full text of the standard: 

1.G.3 Partition circles and rectangles into two and four equal shares, describe the shares using the words halves, fourths, and quarters, and use the phrases half of, fourth of, and quarter of. Describe the whole as two of, or four of the shares. Understand for these examples that decomposing into more equal shares creates smaller shares.

Like many of the standards, in 1.G.3 there are quite a few related but distinct concepts all rolled into one standard. Let's break it down. Students should be able to:

• Partition circles and rectangles into two and four equal shares
• Describe the [equal] shares using the words halves, fourths, and quarters, and use the phrases half of, fourth of, and quarter of
• Describe the whole as two of, or four of the [equal] shares
• Understand for these examples that decomposing into more equal shares creates smaller shares

So. Much. Math... in just one standard, and so many exciting emerging understandings of fractions present in this 1st grade work. After we do an awesome job of teaching the concepts in this standard (for some rad lesson ideas related to this standard, check out fellow #MTBoS blogger Jamie Duncan), how do we assess student understanding? As I look at the assessment above and consider the standard, my initial question is, "Why aren't the students doing the partitioning?" By already partitioning the shapes for the students, they have stripped away any opportunity to formatively assess whether students truly understand the idea of equal parts, specifically halves and fourths. Let's make the assessment better:

Now what is being assessed? 

• Ability to partition a shape
• Understanding of "equal parts"
• Application of the concept and term "half"

In addition to these ideas, this item allows teachers to ask a follow up questions like, 

• "How do you know the square is cut in half?"
• "How many halves does it take to make the whole square?"
• "Is there another way you could partition this square in half?"

This is the math the standard calls for! If I was the teacher, my next item would ask students to cut a same-sized square into four equal parts and name the parts, and I would follow up with a question about whether the pieces were bigger in the first or second square. This would move students towards the final idea in standard 1.G.3: understanding that decomposing a shape into more equal shares creates smaller shares.

A small change to a question with huge implications for the actual usefulness of the item for informing my instruction? Yes, please. So, as you're making decisions about what assessments to put in front of your students, play the game: "What math does this assess, and is that the math the standard calls for?" 

For more formative assessment ideas related to this standard, check out the awesome work of CPALMS and their Mathematics Formative Assessment System (MFAS) resources.