Number Grids and Number Lines: Can They Live Together in Peace?

     "Numbers do not just evoke a sense of quantity; they also elicit an irrepressible feeling of extension in space."  
     Stanislas Dehaene, The Number Sense: How the Mind Creates Mathematics, pg. 69

     There's one hanging in every kindergarten...

first grade...

second grade...

and third grade...

classroom in my school.  Yes, the ubiquitous "Class Number Grid".  These grids hang out with big number lines on classroom walls:

Sometimes right next to each other!

And are often seen in smaller versions going hand in hand on desks:

This type of name tag appears on  student desks in many first and second grade classrooms in my school.  It includes a hundreds grid on the left and a number line running across the top.

     But how do number grids and number lines exist in the minds of the students?  How do students see the numbers arranged and extended? Can multiple conceptions exist in their minds at the same time without conflict?
     The kids use number grids to do exercises like this:

Grade 1

Grade 2

Grade3

And to play games like this:

This game, along with the activities above, are designed to reinforce facility with a number grid.  The grids make patterns appear more apparent and help build place value concepts.  When they first play, you may see kids counting each space as they move closer to 110.  Most catch on quickly to the idea that to move 10 or 20 they can move their token "down" one or two spaces.  Of course there are the few that continue to count by 1s, unconvinced that the moves down the grid actually work to advance them them 10 or 20 spaces on.

But most kids lose interest quickly.

Last year I tried a small innovation:

Play on a blank grid and fill in the numbers as you go along.  After three or four trips from 0 to 100 the grid begins to fill up.

       I had tried this out last year with one of my basic skills students, and liked the results.  This year I had the entire second grade give it a spin.  The more they played, the more their grids filled up with numbers, and those numbers became benchmarks for empty squares.  Looking for a way to help them see the relationship between number grids and number lines, I had them try something different:
   

Turn the number grid into a number line by cutting the grid into strips and taping the strips together.
Play on the number line.

This was a different kind of experience.  Instead of simply moving "down" to go +10 and +20, the kids had to move to the right.  It was interesting to see how they worked.

This student was on 76, rolled a 1, and jumped 10 to 86.  How did he know where to land?  One more than 85!

I was happy with how this activity played out, although the number lines were long, and finding room to play became an problem (some kids went into the hallway).  Then there was the issue of storage:

We hung them from a clothesline in the room.

    I decided to try the experiment out in third grade with Capture 5:

First we gave them some time to get familiar with the game.

Next, cut out a number grid into strips and glue together to make a number line.

However I couldn't resist adding in an estimation challenge:
How long do you think the number line will be once we put it together?

Normally I would put these estimates on an open number line.

The number line measured 64 inches.  We needed the 39 inches of a meter stick, plus 25 more.  I turned this addition problem into a little number talk.  There were many ways the sum was calculated mentally, and reinforced moving by 10s, 20s, and 30s in the Capture 5 game.

Playing Capture 5 on a number line.  

  I asked the kids to comment on the difference between playing on a number line as opposed to a number grid.  Some reflections:

• "I had to use my my mind more because I couldn't go up and down."
• "I had to do more thinking, but it got easier the more I played."
• "I like the number line better.  The grid is more confusing."

So what's next?  Graham Fletcher has written about the potential benefits of "upside down"  grids, another way for kids to conceptualize how numbers can be arranged.   How about playing on a vertical number line?  Could we adapt games like Capture 5 and the Number-Grid Game for play on a yardstick, tape measure or meter stick?  A thermometer?  Will this cause confusion or help build number sense?  And is it possible that some of our students see numbers in ways we cannot imagine?

Number Forms

    "Though a majority of people have an unconscious mental number line oriented from left to right, some have a much more vivid image of numbers.  Between 5% and 10% of humanity is thoroughly convinced that numbers have colors and occupy very precise locations in space."
     Stanislas Dehaene, The Number Sense: How the Mind Creates Mathematics, pg. 71