### 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 wordshalves,fourths, andquarters, and use the phraseshalf of,fourth of, andquarter 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.

I was totally going to say, "How do you know the square is cut in half?" and then you did. :)

ReplyDeleteIt seems crazy to me that an assessment would already have the shapes partitioned!

I like to throw incorrect problems into assessments too, in order to bring out reasoning. So put a square that is divided into two unequal parts and ask students to explain if it is divided in half and how they know. Or even just have them divide it unequally and explain.

I love your additions to the ideas. Good work!