I told her I had a problem and asked her to solve it. (This intervention student shall remain nameless due to the fact that the internet is an infinite digital footprint, and I believe some day she will be a famous scientist and doesn’t need to read this story about herself online.) My problem?  I have 45 books that I want on three shelves, equally please.  I like when books are evenly spaced out. It’s my math brain I guess.

She sat there, solving this division problem right in front of me for a few minutes. She wrote the number sentence and acted it out, even got the correct answer.  Conceptually she was SOLID.  But the second I asked her what the 45 stood for, I got the deer in headlights look. I waited…nothing.  Then, I asked her what the 3 stands for in her number sentence…no idea.

The very next hour I was in a kindergarten room.  This problem was up next.  Students were solving with drawings. They wrote number sentences to match their drawings.  When I asked “What does the 3 mean?” No idea.  Their understanding of what the 3 could mean became incredibly fuzzy.

I was not shocked to see this. This is a daily occurrence at every grade level with students working at every level.  This is a fundamental problem with Math Practice Standard #2 (Reason abstractly and quantitatively.).  I’ve realized that once the students pull the numbers OUT of the problem, they aren’t thinking about what they mean as you try to connect that back IN to the problem. Remember all of your teachers saying “Label your answer!”? I believe they were onto something.  It is not enough to say “Label your answer!”. We won’t get over this barrier until we start asking what ALL the numbers in the number sentence that you wrote represent.

3 ways to deepen this understanding, and practice it over and over:

1. Ask: “What does that number stand for? How do you know?” The simple act of going back INTO the problem after solving it will deepen their understanding.
2. Write words along with the number sentences for a while, until they begin to see how they can move back and forth between the words in the problem and their number sentences. 
3. Give a number sentence all alone. Ask them to write a story. It’ll be pretty hilarious at first. Apparently my student had potatoes on the brain. Notice, I’m not asking for a story problem, but a story. That means the story will not have a question or an unknown at the end of it. This helps them make sense of ALL the numbers in the number sentence.

This is something that we must all commit to to help students make sense of mathematics, make sense of problems and to make math less abstract. I would love to hear any other suggestions you might have to strengthen our little mathematicians in this area.