Thursday, March 26, 2015

The Standard Formerly Known As 4.2.3D.2.a

   From the Common Core Math standards:

     For more than a decade, research studies of mathematics education in high-performing countries have concluded that mathematics education in the United States must become substantially more focused and coherent in order to improve mathematics achievement in this country. To deliver on this promise, the mathematics standards are designed to address the problem of a curriculum that is “a mile wide and an inch deep.”

    Agree with the conclusions or not, there's no doubt that promise has been kept, at least at the elementary level here in the Garden State.  If you don't believe me, compare the old grade 3 standards to the new ones.  Many of the skills and concepts I taught as a third grade teacher are no longer there.   One that did survive the purge was a confounding little number we grew to know quite well as 4.2.3D.2.a.  It has changed its name to 3.MD.B.4, but retained all of its deviousness.  Can you guess what it is?

     If you guessed measuring to the nearest quarter inch, you are correct!

     Seems not much has changed since my third grade days.  Despite our best efforts, which this year included counting by 1/4s in counting circles, the kids had enough trouble with 3.MD.4 to keep the folks over at Math Mistakes busy for weeks.  I'll propose the following reasons; feel free to add your own:
  • Kids count 3 hash marks between each inch and express their response in thirds.
  • Kids do not know how to write fractions and mixed numbers correctly.
  • Kids think that when they are asked to measure something to the nearest 1/4 inch, their answer has to include a fourth.
  • Kids are unable to connect what they know about fractions and number lines to the actual ruler.
As I talked it through the issue with our grade 3 team, I decided to throw out an idea I had been turning over in my mind.   The teachers felt it had potential, and when I got back to my room I started working up a model:

I started with square inch grid paper...

...cut out the squares and labeled one side.

I took a square inch, folded it in quarters, labeled them, and cut them out.  They were small and hard to work with.

What might happen if, when asking a kid to measure something to the nearest quarter inch, you gave the kid the square inch and quarter inch pieces and said something like,
    "Use these to measure.  If you need the little quarter inch pieces, use them.  If not, then don't."
    Would they do this?

Not enough.

Too much!
Just right.
      We decided to find out.  Theresa grabbed a student who had difficulty with the skill on the assessment.  I gave her the square inch and quarter inch pieces and let her give it a try.

The student used the pieces to measure the pin, the key, and the pencil.  Theresa said that, although the small quarter inch pieces were not easy to manipulate, the student was able to use them to measure correctly.  They also helped her understand how to write the mixed number measurements.

From here it's a natural move to...

...and then finally:

     We scaffold many concepts with manipulatives.  We use base ten blocks to help children understand place value concepts and regrouping, and tiles to help children compare and work with fractions.  Why should measurement be any different?  Rulers need scaffolds too!
    Did we beat 3.MD.B.4?  The jury's still out.  We need to get the square and quarter inch pieces into the hands of the teachers, then into the hands of their students.  We need to redouble our efforts at counting by 1/4s, connect that work to number lines and rulers, and make sure the kids have an understanding of how to both say and write the fractions and mixed numbers.  If we do all that, I think we can give 3.MD.B.4 a run for its money.  It may even have to change its name again!


  1. What a great way to reinforce students' understanding of a unit fraction Nice move Joe. To this day I'm still trying to figure out why this is not a MD standard in 4th grade. Seems like the context lends itself beautifully to the mixed fraction standards.
    You have me intrigued. "What if we lined up the the pin, the key, and pencil in one straight line, how long would it be?" I'm really interested to see how students would tackle this question based on the conceptual understanding you've developed. Hmmm?

  2. Thanks Graham. The progression of the fraction standards, especially from grades 2 to 3, is something we struggle with, and I agree that this would be well placed in grade 4. Anyway, I can always count on you to take a task and push it to another level. We'll give it a try!

  3. What a great idea. It may not be an issue, but is there a way to ensure that the pieces of paper are associated with length and not area? Maybe matchstick like objects. Or, "thin" rectangle strips of paper? Anyway, great idea.

    1. Thanks for bringing this up; I think the length vs. area issue is important, and I did think about it. I meant to address it in the post but I forgot. I drew lines across the tops of the 1 inch squares to indicate length, but not on the 1/4 pieces. Ultimately, I decided the length/area risk was worth it, but never really tried matchstick-like objects or thin rectangles. The problem is that they are kind of hard to manipulate. But maybe a kid would only need to do it or see it a few times to get the idea.

  4. In my experience it's *hard* to get most students to consider area. THey tend to think in one dimension.

  5. Great post, Joe!
    I'll be featuring this post in this week's Global Math Department newsletter!

    1. Thanks Andrew. Sometimes I'm not sure about whether or not to tweet out old blog posts, but I'm glad I did it with this one.