We recently started our expressions unit where I introduced kids to the joy of Algebra Tiles and modeling expressions. I was feeling pretty good about how things were going until it was time to begin the distributive property. Out of pure curiosity at the beginning of class one day I asked the kids to grab some integer chips or Algebra Tiles and model two times three. I would be lying if I said I wasn’t shocked about what they did.

This was by far the most common model. The discussion went a little like this:

Me: How many chips are in front of you?

SS: Five

Me: But what is two times three?

SS: Six

Me: Then why do you only have five chips in front of you?

SS: Because you said to model two times three.

Confused, we had a quick discussion about the meaning of multiplication being about making groups. We talked about how two times three really meant either two groups of three or three groups of two. They seemed to get it so I moved on like any other terrible teacher and later tweeted the picture. Someone on Twitter challenged me to ask them another modeling question the next day but this time with a fact they didn’t know. I said challenge accepted, I mean after all I had just told them what multiplication meant they surely would get it right tomorrow.

The next day I asked them to model fourteen times eight. On a positive note there were kids who completely nailed the model this time and I would have thought I was really good teacher…

Except the other half of the models looked like these:

Now interestingly enough, even the kids who made these models incorrectly got the correct answer because off to the side of their table they had worked the problem out using the standard algorithm but still really not knowing that 14 times 8 actually means 14 groups of 8 or 8 groups of 14. We did some more investigations into this using both multiplication and later division until it seemed like we had a good grasp of what was going on. The kids conversations during this time were priceless:

S1: Did you know that is what multiplication meant?

S2: No but do you think it is weird she is teaching this to us?

S1: Kind of but I think she might be teaching it the right way.

Later I tweeted a lot more of the modeling frustrations to mixed reviews. Of course math teachers were mainly supportive but others questioned why a kid needed to know why 2 * 3 = 6 as long as they memorized the fact. I guess for me that just isn’t enough. I think kids need conceptual knowledge of what they are doing in math class rather than just blind trust that the facts they memorized are correct.

These are the kids that when asked to model 2x end up modeling x +2 as they do not understand the difference in the two. I see kids hit the wall with this a lot with the distributive property. If they do not understand multiplication as an area model they almost always struggle with understanding how to distribute which I am sure if how we ended up with teachers asking kids to memorize FOIL rather than really asking kids to visualize what is happening.

Trust me when I say that I have a lot left to learn as a math teacher, probably more than is possible before I retire. But I do know and believe that pushing kids to understanding mathematics conceptually rather than blindly memorizing facts and rules is where my heart lies. I don’t think I would or could be a teacher if 2 * 3 = 6 was good enough.