“Memorization is what we resort to when what we are learning makes no sense.” -Anonymous

“It’s not what is poured into a student that counts, but what is planted.” -Linda Conway

Our 2nd grader came home with a worksheet a few weeks ago. The worksheet had approximately ten problems about multiplication. As our son started to work on the sheet I observed him trying to solve the problems. The page had pictures and gave an example of how to draw the multiplication problem, but he was still struggling with how to make sense out of the worksheet. The worksheet asked the students to solve the number sentences by drawing marbles in the jars that were under the number sentence.

For each problem the first number represented the number of jars. For the number sentence 8 x 3 there were 3 jars under the number sentence. Anyone out there shuddering at the number of jars? The brain learns by experiencing a concept concretely, then moving to an iconic representation, then moving to the abstract. I knew it was time to put away the abstract and go back to the concrete.

I asked him to just put away the worksheet for now and help me pick out some really cool manipulatives. Our son is our resident “Dudley Do Right” and immediately responded that he “had to finish his homework so he wouldn’t be late in turning it in.” I told him that we were about to explore multiplication in such a way that his homework would be completed in no time. He was convinced this might be a good idea when I pulled out my box of counters full of insects, cars, boats, and figurines.

We sat on the floor with lots of space around us. I put a stack of ten lids and bags of interesting manipulatives in front of him. I asked him where he would like to go if he could go anywhere in the world. (Money wasn’t a factor—a really big deal in a family who lives on a tight budget.) He chose to go to Hawaii.

I asked him how he was going to get there. He chose to go by car. This brought about our quick geography lesson with a map of the world. After learning where Hawaii was, he chose to go by sailboats—a much better way to cross the ocean! Here is how the next part of our conversation went:

Me:    How many boats do you want to take on your first trip to Hawaii?

Him:     I think I want to take 5 boats the first time?

Okay. Pretend the lids are boats. Put out as many boats as you need. How many bugs do you want to go on each boat?

I put out five boat lids. Can I take people, too, and not just bugs? I think it’d be funny to have them bothering each other?

Sure. It’s your trip and your imagination. When you pretend you can make it anything you want.

I’d like to take 2 bugs and 2 people on each boat.

Your choice. Go for it.

I’m done. There are 4 things on each of the boats.

So, tell me about what you see in front of you. As you tell me about it, is it okay if I write what you say on this white board?

Sure. I see 5 boats. I see 2 bugs and 2 people on each boat. I see all the people running around trying to swat all of the bugs because they’re biting them. Looks like the bugs are winning.

I continued to write down all the statements he made about his cruise to Hawaii. After he had given me 7-8 statements, I asked him to move to another area of the carpet and create another trip to Hawaii that was different than the first.  He put out 4 planes with 3 people on each plane.
Again, I scripted the observations he made about his trip.

After 4 scenarios were created with boats and things riding on the boats, I asked him if he’d like to see what his trips looked liked in an abstract/number sentence way. My son is very polite, but he really was genuinely interested in the next step. One of the scenes was 4 boats with 3 people on each boat. That’s the one he chose to discuss first.

Me: Do you think there’s a way for us to draw what the scenes you created?

Him: Sure. Can I draw it any way I’d like?

You bet! Which one would you like to start with?

Let’s do this one with 4 boats and 3 people.

He drew the outline of 4 boats and put 3 heads in each boat. When he finished I asked to tell me about what he drew. He described it and let me know that this trip had 12 people who made it to Hawaii. He drew pictures for each set of concrete objects and told me about each of them. Normally, I don’t continue to the abstract level this quickly, but he was determined to finish his homework worksheet.

Me: Is everything starting to make more sense? I think you’re ready to go on to the way the problems were written on that worksheet. I’m going to write a number sentence. What are some numbers in your observation?

Him: 4 for the number of boats and 3 for the number of people on the boat.

So could I write 4 x 3?

Sure!

If we were to put a label under the number 4, what word should I write?

Boats. Maybe you could write “B” for short.

What about under the number 3?

People, or “P” for short.

Could you go over to your 4 x 3 boats and tell me how many people there were in all of the boats on that trip?

There’s 12.

Hmm…so how could I finish that number sentence if I wrote an equal sign?

It’s 12! It’s 12!

Together, we wrote the abstract number sentence that represented each of the scenes he had created with the lids and the counters. We talked about the pictures he could draw to help him think. He began to truly understand what each number in the number sentence represented. After 30 minutes of concrete exploration, he went back to the worksheet.

As he completed the worksheet he used the lids and bugs so that it would be more fun. After he finished, he asked me if I could give him a really hard problem. I asked him if he could show me between 5 x 0 and 0 x 5. Do you know the difference? He did and left the room with a huge smile and one more tool in his mathematical belt!

“Teaching is the highest form of understanding.” -Aristotle