Concrete Models - Are they actually helpful?

I am not sure what manipulatives and concrete models look like at an elementary level, but here is an example of what I see in middle school.  Last week we were working on this problem:

 6 - (-2)

This was our first introduction to subtracting integers, but we had done lots of work with addition.  Using our background knowledge we put out 6 positive integer tiles.  Like so:

Then we discussed our understanding of subtraction.  From an early age students understand subtraction as taking away.  So we looked at taking away 2 negative tiles.  But wait, I don't have any negative tiles to take away.  What should we do?

Some students wanted to add 2 negatives in order to remove them.  This led to some good discussion about how that would change the problem.  Then it was brought up by a student that we could add zero pairs (a positive tile and a negative tile), which we learned about in our addition problems.  This is what that looked like:

From this point we were then able to remove our 2 negative tiles, leaving 8 positive tiles.

After this, the lesson called for more building of expressions with the tiles, removing tiles, adding zero pairs when necessary, in order to find the result.  As the students worked I walked around helping teams understand the idea of adding zero pairs if it was needed.  

By the end of class students were in one of two places:

• Some students had no idea what they were doing with the tiles and were copying their teammates.
• Some students learned to add zero pairs in order to have what they needed to remove.  

Unfortunately there were no students who had reached the lesson objective outlined in the textbook:

Develop an understanding of integer subtraction using + and - tiles.

Or the mathematical practice goal:

Make sense of subtraction with signed numbers.

Here is the problem as I see it.  

As math teachers we have made the shift to helping students have a deeper understanding of math.  We are using more representations, models, and manipulatives to help students explain what is happening with the numbers.  Unfortunately too many teachers (myself included) are just using these models to explain the math rules.  Students do not own them.  Students are now just memorizing the steps of the manipulatives.  Is this a step in the right direction?  To quote Big Bang Theory, "Way to think outside but pressed up against the box."

If the purpose of the representation is to give students deeper understanding we need to give them time to explore with it.  The challenge for me comes with the question, "How?".  What does that even look like at a middle school level?  

______________________________________

So let's take a look at the lesson above, this time allowing the students to play with the numbers a little more.  

Use the tiles to show the number 5.  (No pics of this.  I'll let you create the visual)

Now show 5 another way.  (Blank stares, processing time, have students circulate to see how others have created 5)

Now show 5 another way. (More students are able to do it.)

Should we do it one more time? Why not.  Show me 5 one more way that you haven't used yet.

How long can we play this game?  How many more ways do you have to show me 5?

Think about how this shifts the conversation.  Students are developing an understanding of 5 that they may not have had before.  5 is a constant, so how can it be represented in different ways?  This is a powerful idea.  It could be a time to introduce vocabulary like the Additive Identity Property.  

Now let's move to the subtraction, but frame it in a way that is still allowing students to explore these numbers.  

Use the tiles to show the number 6. 

Have different students draw their representations on the board.  (Verify that each one does have a value of 6.  Perhaps a good time to ask again, how we can have so many representations with a value of 6.)

I want to create some problems where I will be removing tiles.  For each representation of 6, I want you to list numbers that I could remove based on the given tiles.  

I had students do this in teams.  It created lots of great discourse.  I brought the class together so we could discuss after. Some questions we discussed:

• What were some numbers that you listed for each representation? 
• How did you decide what could be removed? (students realize you can only remove what you have)
• Can you name a number that I can remove from one of these representations of 6 that I cannot remove from another?  Why is that?
•  So if I want to remove -2, which representations of 6 could we use?

We eventually worked our way to writing the subtraction expressions.  The focus was always on which representation of the starting value is needed.  Keep in mind there is not one right answer for this.  As long as students have what they need to remove, extra zero pairs are just fine with me.  They will be more efficient when they are ready.  

__________________________________

A side rant:

There are many important mathematical concepts found in using integer tiles for subtraction.  However, for the objective of understanding integer subtraction, I personally prefer the Hot Air Balloon model. I find it much more intuitive for students and the relationship between addition and subtraction is much clearer.  Here is a Desmos activity I created for addition with the hot air balloon.  Desmos is a great tool for exploration because you can use sliders to change numbers and actually see what happens.  Unfortunately my Hot Air Balloon does not do this. What I really want is a tool where students can drag puffs of air or sandbags to the balloon and it moves.  I also want it to create the equations as they click.  Plus students should be able to write the equation and the hot air balloon moves.  I am not sure it is exploration if they have to move the hot air balloon themselves, because they have to know how it will move.  Do they really understand or have they just memorized it?

___________________________________

I may not be where I want to be, but at this point I feel like I am now outside the box and making steps in the right direction.  I want my students to explore their understanding of math each day, so I will continually ask myself how I can provide opportunities to explore the numbers, not just model them.  I encourage you to do the same, and I would love to hear about it.