This is the equivalent problem in a different representation.
My son was comparing two fractions to decide which is bigger (a problem that appears in his Beast Academy workbook).
He looked at the fractions and noticed that if he changed the first fraction, he could get the denominators to be closer together.
Him: “Oh, now I know that is larger than !”
Me: “How do you know that?”
Him: “If we had we would know that is more than since is more than . We just have 1 more so for sure is more than .”
What I find interesting about this is that I was taught that in order to compare two fractions, one first needs to be able to find a common denominator. Obviously that’s not true.
While driving on the highway, my wife had the following exchange with my three year old son, Timon.
“Mom, what’s three plus three?”
“Mom, what’s four plus four?”
“So… three plus four is seven.”
My wife was so stunned, she told me later, she almost stopped the car to tell me.
Timon asking us questions about addition is nothing new. He’s been trying to participate in a similar game my older son plays with me. What’s remarkable here is that Timon has derived the addition fact for three plus four.
Timon also recently noticed that one plus three and two plus two both equal four and showed me on his fingers why he knew it was true. He’s playing with addition of small numbers, sometimes asking me what the sum is, sometimes showing me by counting out on his fingers. He has yet to count on from a number he knows (he starts counting at one each time) but he does recognize groups of objects up to four and sometimes five in size without counting directly.
My point here is to pay attention when your children discover ideas on their own. They might just surprise you.
“Daddy, I’m full. I had 1 and a half…no, one and a quarter slices of pizza which is the same as five quarters of pizza,” said my son at dinner tonight.
“Okay,” I said, “you can go play if you like.”
“You know you can do that right, have five quarters of pizza, or even 10 quarters of pizza,” he continued.
“How many slices of pizza is 10 quarters of pizza?” I asked.
“Hrmmm…,” My son thought for about 10 seconds, and then said, “Two and a half slices of pizza,” and then he went off to play.
My son can do this kind of manipulation with wholes, halves, and quarters of pizza because the idea of dividing food into smaller fractions is familiar to him, and because he has seen halves and quarters used in many different contexts quite a bit during his life. Simple fractions are familiar to him, and so he can manipulate them as needed. Note that I have never once taught my son how to convert between an improper fraction to a mixed fraction, or even what those words mean. I’ve just slowly and deliberately introduced him to the ideas of fractions as they naturally fit our daily lives.
Here is another interesting discussion between Malke and her daughter, this time about infinity. What kinds of conversations have you had with your child about the concept of infinity? What analogies have they come up with?
Malke Rosenfeld shares another example of her daughter’s mathematical thinking, this time when her daughter finds a protractor and decides to use it to make a map of some angles. Read more about her daughter’s mathematical thinking on her blog.
Grant Wiggins has posted a dialog between himself and some middle school mathematicians. Here is an excerpt from one of the letters they sent to him.
With that problem conquered, we moved on to the three-rock episode. Drew didn’t like our chances here. With all his experience in adjusting the four rocks to the perfect weights, using just three didn’t look good. We then remembered an earlier part of your email when you commented that future texts should leave out material to make problems more interesting. Were you doing that to us here, we wondered? Probably so. Therefore, we assumed that we had poetic license to create a little backstory for Farmer John.
So Farmer John has his rocks returned from Farmer Joe and is, at first, heartbroken to see that his forty-pound rock has become a one-pound rock, a three-pound rock, and a thirty-six pound rock. The original rock was used to measure the perfect amount of hay, and can still do that as a trio… and now the rocks are now a bit more portable, for those days that are hard on the back. So, things are looking up.
Farmer John also realizes that he now has the capability to measure other weights of hay. Using both sides of the balance, he can accurately measure hay in the amounts compiled by Kelsey, Aidan, Kirby, Jon, and Kyle and shown on the next page…
The letter is an excellent example of students thinking mathematically, as they ponder some of the various ways they can adjust the problem given to make it more interesting. It seems clear from this exchange that a pro-tip when teaching mathematics is to let students modify the problems to explore other possible interpretations.
This next example comes from Sue VanHattum.
Me: Clean up your drips!
R: There aren’t any.
Me: I see at least one.
R: You see at most one.
When do children start using logical reasoning? Can we find other examples of children thinking like this?
My son and I had this exchange earlier in the day when we were negotiating about whether or not we would watch an afternoon movie.
Me: “Okay, so the movie is 90 minutes long. That’s an hour and a half.”
My son: “How did you know that?”
Me: “An hour is 60 minutes, so I just took 60 minutes away from 90 to get 30 minutes left over, which is half of an hour.”
My son: “So what if the movie was 100 minutes long? No, don’t help me! Let me figure it out… Uh… an hour and forty minutes.”
Me: “What if the movie was 110 minutes long?”
My son: “That’s easy. It’s just 10 minutes different. So instead of 100 take-away 60, it would be 100 take-away 50. Uh… That’s 50 minutes! So the movie would be an hour and 50 minutes long.”
Easier question: What’s the biggest success here?
Harder question: How does this kind of number sense develop in children?
My six year old son and I were working on cutting out snowflakes (more on this later, it’s a fun project in itself) and at one point we wanted to count the number of symmetries our snowflake was going to have, which we worked out would be 9 + 9. My son said aloud, “Okay. 8 + 8 is 16. 9 + 9 is… uh… 18! Because 8 + 9 is 17!”
How often do you see students use counting as a strategy for finding a sum of two numbers? Is this common? How do we encourage this type of reasoning?