How Many Blue Birds Flew Away?

A couple of weeks ago I went to the library and checked out some titles for the boys to do some summer reading. Yesterday my 6-year-old finally read How many blue birds flew away? by Paul Giganti, Jr. It is a book about counting and comparing quantities less than 20. The story provided opportunities to count my two's (as in the case of counting mittens) as well as reinforcing one-to-one correspondence. When the objects are scattered throughout the page, my son had trouble keeping track of items that were counted and those that were uncounted. There were also times when I had to remind him to watch out for "hidden" objects. After successfully counting items on each page, he had to compare the difference between the two quantities. For him this part was surprisingly easier than counting correctly!

The M&Ms Problem

The following problem was shared at the CGI institute I attended two weeks ago. It was originally given to a group of second grade students who had not been taught how to multiply using the traditional algorithm.

Todd has 6 bags of M&Ms. Each bag has 43 M&Ms. How many M&Ms does Todd have?

I decided to give this problem to my 9-year-old and see if he is able to show multiple ways to solve it.

He immediately recognized it as a multiplication problem, wrote 43 x 6 vertically, and found 258 as the answer. When I asked him if he could solve it a different way, he proceeded to make a list, showing the number of M&Ms in one bag, two bags, three bags, and so on. Notice how in order to create the list, he needed to set up a multiplication vertically to find each answer.

So is this really using a different way to solve the problem?

I asked him if there was any way that he could use the number of M&Ms in one bag and in two bags to help him find the number of M&Ms in three bags. There was silence. I made the following sketch to help him:

"If you know how many M&Ms are in one bag, and you know how many M&Ms are in two bags, can you find the number of M&Ms in three bags?" I asked. "Yeah," he said. But still, he was staring at the paper, so I made a diagram to show him that three bags is just one bag plus another two bags. "Did we get the same answer you wrote in the list?" I asked. "Yes," he replied.

"Now, do you think we can use what you know about three bags of M&Ms to figure out the number of M&Ms in four bags?" I asked. "Yes, add the answer from 3 bags to one bag of M&Ms," he said. " I helped him record in a diagram.

"Is there another way?" I asked. "Yes, two bags plus two bags," he said.

"How about six bags of M&Ms? Can we use what we already figured out so far to help us find the number of M&Ms in six bags?" I asked. "Two bags plus four bags of M&Ms," he replied.

"Is there another way?" I asked. "Three bags plus three bags," he said. Notice how he wrote 129 x 2 and then wrote 129 + 129. His mind might already be conditioned to think multiplicatively when a number is repeated.

You might be wondering why I want my child to go through all this hassle when he already recognizes that, given the number of groups and the number in each group, all he has to do is multiply to find the total number. I wondered about this while I was at the training, and I guess I want him to have the tools to be able to solve problems in more than one way. I think the following problem does a good job of illustrating why I want to encourage non-traditional ways to solving a problem.

There are 258 M&Ms in 6 bags. How many M&Ms are in 12 bags?

When I posed this problem to my son, his response was: "You have to divide first and then multiply." "Can you explain what you mean?" "I don't know. I'm confused. I don't know how many M&Ms in one bag, so I can't do this problem," he said. How interesting! First, I was surprised that he didn't even recognize that 258 is the exact same answer we just found for six bags and therefore there must be 43 M&Ms per bag! And, after we've solve the problem using combinations of bags, he still couldn't make the connection. In his mind, I think he wanted to divide so he could find the number of M&Ms per bag, and then multiply by 12 to get the answer. (In hindsight, I probably didn't allow enough time for him to fully process and explain his thoughts.)

"Do we need to know how many M&Ms are in each bag? What is the relationship between 6 bags and 12 bags?" I asked. "I need twice as many bags," he answered. "Can you show me a picture?" I asked.

OK, so I'm cautiously optimistic that he is catching on now. So I give him yet another problem.

There are 258 M&Ms in 6 bags. How many M&Ms in 18 bags?

He got it this time. I didn't even have to teach him how to set up a proportion to solve!

How Old is the United States?

Happy 4th of July!

If the United States was born on July 4, 1776, how old is it today?

My 6-year-old has been playing with the idea of subtraction in the context of people's ages for about six months now. He was initially interested in calculating the ages of everyone in our family (grandpa, grandma, uncles, aunts, cousins, dogs, ...) Then he became fascinated with historical characters such as Benjamin Franklin, Abraham Lincoln, Helen Keller, and Thomas Elva Edison, among others. He would remember all their dates of birth and death, so he used to obsessively figure out how old each one was at the time of death on his Magna Doodle or his sketchbook. He would even account for the month of death and determine if the person already had a birthday or not.

Can you see how my 6-year-old was able the "age" for United States? Pretty cool, huh?

Relational Thinking

Part of the discussion that took place while I was at the CGI training last week was around the idea of relational thinking. One way in which relational thinking is used in mathematics is as a powerful tool to help students understand meaning of the equal sign. Rather than treating the equal sign as a signal to compute, students use relational thinking to view the equal sign as something that represents a balanced relationship between the two expressions on either side of the symbol. For example, suppose the following open number was given to a child:

9 + 7 = BOX + 8

According to math education research, a child who has a fragile understanding of the equal sign would likely say that 16 goes in the box. Or, even for a child who DOES know that the equal sign means "the same as," he might need to first compute the left side to determine that the sum is 16, and then find out what the box must represent so that the right side also has the same sum. The way in which a child might figure out the answer is by subtracting 16 - 8 to get 8, and then conclude that 8 must go in the box.

While we might be satisfied that a child is able to correctly determine the number that goes in the box by the strategy described above, we want to encourage him to use relational thinking: Is there a way to determine the number that goes in the box without first computing the sum of the left side? How can we determine the correct answer just by looking at the relationship between the expressions? It turns out that children who are often exposed to experiences with open number sentences will look at 9 + 7 = box + 8 and know that 8 must go in the box. (Note how there is a difference of 1 between the 7 on the left side and the 8 on the right side. In order for the number sentence to be true, the number in the box must be one less than 9.)

Once again, I was curious to see how my 9-year-old would respond to an open number sentence, so I asked him: "What number goes in the box so that this is a true statement?"

34 + 27 = 35 + BOX

(In hindsight, I think I should have written the equation as 34 + 27 = BOX + 35 to see if my son would say that 61 goes in the box.) I provided him no pen nor pencil, hoping that he would use the idea of relational thinking to do the problem. It didn't surprise me that he tried to find the sum of 34 and 27 mentally. I could tell that he was struggling to do the mental calculations, so I provided him with a hint:

"Is there a way that you can use the relationship between the numbers to help you find the answer?" There was a long pause. I decided to take a step back and give him the following:

34 + 27 = 34 + BOX

"27 goes in the box," he said. "Good," I replied. "Now go back to the first problem I gave you. What number goes in the box?"
"I know 35 is one more than 34, so the number in the box has to be less than 27," he said.
"How much less?" I asked. "I don't know...5 less?" he replied with hesitation. Hmmm...

I wrote the open number sentence differently this time:

34 + 27 = (34 + 1) + BOX

"Does that help?" I asked. "Yes, the box should be 26," he said. From this I learned that he needed to literally "see" the same quantity represented exactly the same way on both sides of the equation.

Here is the next open number sentence I gave him:

18 + 23 = 16 + BOX

"I know that 18 is two more than 16, so the number in the box has to be bigger," he said but still could not tell me the number that goes in the box. So I re-wrote the equation as:

(16 + 2) + 23 = 16 + BOX

Almost immediately, he wrote 25 in the box. I think he's almost there, so I'm going to challenge him one more time.

25 + 42 = 27 + BOX

This time I was pleasantly surprised that he adopted my strategy of decomposing the numbers to make them look the same on both sides.

25 + 42 = (25 + 2) + BOX

"It's 40," he said. No more hesitation, yeah! Let's try another one:

70 + 15 = 60 + BOX

This time he didn't need to rewrite the equation to make numbers match identically on both sides. "70 is ten more than 60, so the box is 15 plus 10, so it's 25." I could not believe how quickly he caught on! (And no, he's not gifted.)

I encourage you to try open number sentences with your young mathematicians!

A True-False String

Earlier this week I attended a three-day non-beginner's CGI institute and was thoroughly inspired and motivated to have more conversations with my own children about mathematics. I created a string of true-false statements to see how much my 9-year-old understands about fraction equivalence.

True or False?
5/5 = 1 "True," he says.
8/8 = 1 "True," he says again.
1 = 2/2 "False, because it's backwards."

So I asked him: "What does the equal sign mean?"
As soon as I asked him the question, he changed his mind and decided that the statement 1 = 2/2 is actually true, because "the left side is one, and 'two 2ths' is also 1."

5/5 = 8/5 (Pause for about five seconds) "True," he says.
4/2 - 1/2 = 5/2 - 2/2 "That's true too," he says.

At this point I suspect that he expects every statement to be true, so I wanted to trick him:
5/5 + 5/5 = 10/5 - 1 "False, because there's two on the left side and one on the other side." Darn, it didn't work!

I was somewhat impressed, but I wasn't about to stop yet. I asked: "What can you do to the left side so that it becomes a true statement?"

He writes: 5/5 = 10/5 -1

"What can you do to the right side so that it becomes a true statement?"

He writes: 5/5 + 5/5 = 1 - 0

End of conversation. Back to dinner.