When Bad Things Happen to Good Algorithms

 Collected over the past several weeks from grades 3 and 4.  Nothing you haven't seen before.  Still, viewer discretion is advised.

Addition

The original problem was 72 + 47.

Subtraction

Multiplication

Some noticing and wondering:

• These students have underlying issues with place value.  
• All these students have seen and used base-10 blocks.  They've been taught to add, subtract, and multiply using partial sums, differences, and products.  For some reason they default back to a traditional algorithm, even though many of them haven't even encountered the traditional algorithm yet.  So they must be picking it up somewhere.  Maybe on the bus?  
• Are these examples of what I've heard called over-reliance on the traditional algorithm?
•  All of the answers are wrong, but some egregiously so, like 1,000- 500 = 1,500.  Or 37 x 35 = 245.  
• Is there a difference in the way kids do these kinds of calculations when they are presented vertically vs. horizontally?  Has anyone done any research on that? 
• These are not necessarily representative; many kids can add, subtract, and multiply multi-digit numbers just fine.  But do we confuse some kids by attempting to teach them multiple ways to do these multi-digit calculations?  Are some kids better off just learning one way?  Or at least one way at a time?  This is something I hear quite often from teachers.  Are they right? 
• Graham Fletcher, in his addition and subtraction and multiplication progression videos, urges us not to rush students through conceptual stages of understanding.  My guess is that what we're seeing here is the result of such rushing.   It's also likely that there are students who are calculating correctly by following the traditional algorithm, but have little or no understanding of the underlying concept.  Masked by correct answers, their misconceptions go undetected, and that is just as troubling as what we see above.

And there's always this.

Teaching to the Test

   My job encouraging teachers to explore and experiment with some of the defining MTBoS routines is made easier when there's an explicit connection between those routines and what their students are asked to do on our program's assessments.   When those stars align, I feel we're close to accomplishing real change.
   So why was I heartened to see the Everyday Math second grade  mid-year assessment several weeks ago? Let's take a look:

 MTBoS Routine: Which One Doesn't Belong?

This routine has gained in popularity throughout the school.  Teachers like the way it encourages students to explain their thinking  and how it reinforces multiple content standards.

Mid-Year Assessment: Question 2
 

Hey!  What's this?  Which one doesn't belong?  I'd never before seen an item like this on one of our assessments.  There were a variety of responses:

MTBoS Activity: Counting Circles

We've explored this routine in our PLC.  It's really taken off in second grade.

Mid-Year Assessment: Question 6

Completing counts and noticing patterns are reinforced by the counting circle routine.

MTBoS Activity: Clothesline Estimation

We've just started to explore this routine.

Mid-Year Assessment: Question 7

     The kids are starting to understand how number lines are constructed, especially how the numbers need to be spaced at regular intervals.
 OK, we still have a ways to go...

...but now we know who needs more time hanging out on the clothesline.

     Three MTBoS routines, three aligned assessment questions.
  And I almost forgot about this item, which showed up in a grade 3 journal several months ago:

Is someone out there listening?