Another number talk, showing great flexibility...sadly, I did not record our negative number work.

For several years, I have been interested in "The Mathematical Habits of Mind" What can I help students do as math *thinkers *to become more successful in math and to best prepare them for using math and learning math in the years to come? What do mathematicians *do* and how can I translate that to a fourth and fifth grade classroom. Elements of the common core speak to this - students are asked to explain *why* something works. They are expected to uncover patterns and extend them. They are asked to work with numbers flexibly depending on context and model them in a variety of ways. In the common core, these are called "mathematical practices" and like several parts of Common Core, I feel it is a good fit for Prairie Creek.

Which brings us to Friday. I have a visiting student from Cathy's reading class at Carleton. He's going to be a math teacher so I was excited that he was going to be in our room for math as well as reading. I decided to share a number talk with him - the Herons love them, he hadn't seen them before, and they are central to my understanding of how kids think about number. We opened with 60-32. Students are expected to do the problem in their heads and then share the approach they used to get the answer. Students approached the problem in eight different ways, including one student who got -28 instead of 28. She explained that she had done 32-60 instead of 60-32. A child commented that it was cool that it was the negative. "Does that always happen?" he asked. I turned it over to the class, "Does subtraction always work like that where you'll get the negative difference if you flip the two subtrahends, numbers in the subtraction. Or is this a coincidence?" I asked the class. A handful thought it always happened. A larger handful thought it was a coincidence and the rest thought we should test it out with more examples.

In math, making a hypothesis like "When you flip the subtrahends in a subtraction problem, you get the same answer but negative" is called a conjecture. When things are going really, really well in a math lesson - students make conjectures. They extend ideas and want to know if it works. Then they test some examples to see (eventually they'll write proofs...but that's a few years down the road.)

So we did some examples and it turns out it does always work. "What about one number minus itself?" a student asked. We pondered this for a while. "You always get zero," another child answered. I pointed out they had just "discovered" a very important fact about subtraction called the identity property and that it took human mathematicians hundreds of years to realize how important this was. "But *why* do you get the opposite number every time?" another student pressed. We wrote out the problems using open number lines and this visualization helped several students have a "aha" moment. It was hard for them (and me) to articulate exactly at first. When you subtract A-B and get C. Then, when you subtract B-A, you end up going back B to get to zero and then you have to continue to -C to have subtracted the whole quantity of A. It makes more sense in numbers. 34-15 is 19. If I subtract 15-34 using a number line, I start at 15 and go back to zero (subtracting 15 of the 34). I then have to subtract 19 more to have subtracted a total of 34. Using the number line, I end on -19. As we drew out several examples more students called excitedly, "Oh! I see why it works!"

This wasn't a planned part of the lesson but the students were ready to use their math habits of mind to question and explore. I could have taught them that when B is bigger than A and you subtract A from B, you just get the negative answer of A-B. But that would have been a disconnected fact to memorize - by constructing their understanding, they don't have to memorize anything. It just makes sense (and it's a lot more fun.)

The deep math work continued as we jumped into our math lesson. We used Pascal's triangle and colored patterns based on divisibility (I wanted to review the key concepts of multiples). Students quickly saw patterns in their work and used them to make better predictions about which numbers they needed to test.

For those who finished and didn't want to explore another number, I offered a new activity of drawing

Collecting and testing conjectures on the white board

loop-de-loops. You use graph paper and choose three numbers, say 2,4,5. You go to the right two, up four, left five and then down two, right four, up five, left two, down four, right 5 and up two, left four, down five (and you'll be back where you started.) Quickly, the Herons began to group patterns. "This one doesn't have a square in the middle." "These two have squares in the middle." "Wait! If it's odd, odd, odd, maybe it always has a square!" "Let me test this one!" "What happens if you switch around the numbers?" Within five minutes they had made multiple conjectures, tested them and modified them. Our time was up far too soon.

Now...how to design a test that enables students to demonstrate *this* learning!

mmm