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.
I have been working with some 4th grade students, and we have been exploring colouring in maps as per this investigation. Here are a couple of examples of their work.
Notice how the students are experimenting with different arrangements of the map. In the last example, the student is trying to find ways to create connections between different “countries” on their map in an effort to force their map to require more colours. Notice also that they have started numbering the colours instead of actually colouring them in. This is a form of abstraction, and something we hope all of our students develop.
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.
Here are a couple of addition problems. Can you figure out what the student is thinking here? Thanks to Chris Hunter for the submission.
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?
This submission comes from Malke Rosenfeld. It is work her daughter produced independently, at age 6, while playing with some tangrams.
Here’s what I found my dear child doing with the set of car tangrams (inside for some reason) over the last few days.
By way of setting the scene, I should tell you that the kid generally initiates and works independently on any project she pleases during her afternoon quiet time, usually with the company of an audio book. A few days ago during this time she informed me she was “making a math book.” I had observed her tracing tangrams, but was surprised to find that she was doing it in a deliberate way, and that she was also able to clearly describe her discoveries and observations to me. Here is the book, so far. I am just a scribe here — these are her words as she dictated them to me, except where noted:
“Four of these triangles that you see here can make a square. If you pull these triangles apart you can see that they’re little triangles. But you can see on this page that they make a square.” [I see that she has drawn them as individual shapes, which, over the course of her illustration, merge together into her intended shape.]
“This rectangle you see is made up of a parallelogram and two triangles. Really they’re just shapes, but when you put them together they make a rectangle.” [It looks like she’s numbered the inside angles of the individual shapes. It also looks like she is again showing the process individual shapes merging into the intended new shape.]
“You see the wheels of this bike as rhombuses but really they’re squares turned so their points are facing up and down, and to the side.” [She was gesturing this first, and at first she used the word ‘flipped’ to describe the orientation of the square wheels. I focused her on the orientation of the corners to describe how the square was turned.]
“The square and the rhombus that you see here, their edges are both the same length. The difference is a rhombus is a squished square, squished to its side. The rhombus has two larger angles and two smaller angles than the square. But the square has the same angles on each corner.” [These are actually tracings of shapes from the pattern blocks set we have. The ruler markings was her idea for comparing the two shapes. I supplied some new vocabulary in the form of ‘angles’ and made some observations about the difference by overlapping the two shapes, in an effort to help her observe and further articulate the difference between the two.]
I think this is a terrific example of some early geometric reasoning. Clearly Malke’s daughter has learned some mathematical vocabulary from her parents, and has the time and space available to do this kind of exploration. What other things have to be in place for children to be able to make these observations?
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?
This blog is intended to serve two purposes, and as such, each post will have one of two main categories: Student thinking and Investigation. The first category is meant to be for any post which captures original (defined quite broadly as thinking which is new to the student, not necessarily to the world) student thinking in mathematics. The second category is for investigations we will share which help students think mathematically. Some posts, like this one, will show both an example of an investigation you could do with your students, and an example of student thinking.
The first example below is quite lengthy, but also a superb example of original student thinking. The objective of the project, in case the student doesn’t make it clear, was to attempt to optimize the travelling distance between 6 to 8 cities chosen by the students. This is the classic travelling salesman problem, in a form that is digestable for students. You can find out more about this project here.