**Be Genuinely Curious**

When students enter our classroom, we ask them to be genuinely curious about the material they are learning each day: curious about numbers and their properties, about mathematical relationships, about why various patterns emerge, but do we, as teachers, bring that same curiosity to our classes? Through our own curiosities, we can gain a deeper understanding of our content and learn to follow the lead of our students in building productive, engaging and safe mathematical learning experiences. As teachers, if we are as genuinely curious about our work each day as we hope the students are about theirs, awesome things happen!

**Call to Action**

Start a personal math journal and record things you learn and/or curiosities you have around the content you are currently teaching. These curiosities can be your own or ones gained through observations of student talk and/or work. After two weeks of recording in the journal, share a moment(s) that inspired you to be be more curious and describe how it impacted the teaching and learning in your classroom through student work samples and descriptive text.

**About the Speaker**

Kristin is a Nationally Board certified, 5th grade math teacher at Richard A. Shields Elementary School in the Cape Henlopen School District in Lewes, Delaware. During her nineteen years in education, she has taught 5th – 8th grade math, as well as spent two years as a K-5 Math Specialist. She feels fortunate to be involved with Illustrative Mathematics and The Teaching Channel on projects around developing math tasks, facilitating professional development and blogging about these experiences. She is always excited to share her love of teaching at conferences such as NCTM, NCSM, ISTE, as well as on Twitter and her blog.

**Updated 2015 Apr 21: Livetweeting**

Check out the collection on Storify.

Excited to kick this talk of curiosities off with my thoughts today! After a division number talk, I blogged about here: http://tinyurl.com/divisiontalk I started thinking about each answer the students had given. I became curious about how the wording of a context makes one answer more helpful than another or if there is even a wording for the answer 12 16/256 (it feels like there should be). I attached a pic of my journal jotting and would love any and all thoughts!

Message:

Jose Vilson has some provocative musings on remainders that are definitely worth considering.

Hung-Hsi Wu wants us to believe that division and division with remainder are as different as lions and sea lions.

I would say that your collection of student responses here is a pretty powerful argument against Wu’s viewpoint, taken literally. But the underlying principle is sound. SOME of these answers work as factors with 20 to make a product of 256. Others do not.

So my curiosity here is about what your students think a quotient

is. Is it a missing factor? Is it the biggest whole number you can put in each share? Is it an approximation? Is a fraction a quotient?I am especially fond of the 12.5 and 12.75 with remainders. There’s something really interesting going on there. I’m curious about the thinking of those kids.

Thanks so much Christopher for your comment! Now, I am curious how I even define quotient. I am curious if my definition for a quotient changes based on the context I am reading?

I have to admit, the underlying principle of Wu’s viewpoint is something I want to think more about myself. I need to think about it in terms of not only the mathematical meaning (and play around with some contexts) but how much I agree with “to teacher, then to student” in matter such as this.

I agree with the interesting decimal answer with remainders and when they decided to “be done” because they were as close as they could be to the 256. The following conversation of the r 6 being 6/20 which is .3 which added to 12.5 give us the 12.8 blew some of their minds! It was a great convo!

OK. Now I have something else I’m curious about. In the process of doing some planning for Math on a Stick, I found myself needing to divide 48 by 5 and 1/2. (The question was

How many 5.5 inch wide tiles can we cut from a 48-inch wide piece of plywood?)48 divided by 5 and 1/2 seemed like a hassle, so divided 96 by 11 instead, knowing I’d get the same quotient.

The whole-number part of that quotient is 8. The remainder is 8. Which made me wonder briefly whether the remainder is also unchanged under the scaling I did of the original problem. It didn’t take long for

meto resolve that question, but now I am genuinely curious what your students would say about this, and how they would think about it. Does the remainder stay the same when you scale your division problem?I love it and put it on my agenda to ask them tomorrow! We have talked about scaling up both the dividend and divisor but only worked in problems that ended with a whole number quotient. And HOW have I never thought to have them think about scaling division problems to make them more friendly? They scale them by powers of ten when there is a decimal at times, however this one, with the .5, would probably have many of them thinking about how many 5.5’s are in 48.

Being a practical sort of person I would see how many 11’s there are in 48 (and double the result), and if the remainder was more than 5.5 I would get one more tile.

Scaling division problems scales everything, including the remainder. Danger is that you might then think that there was enough for another tile !

Hi Howard!

I did find that so interesting that the students didn’t think to see how many 11’s there were in 48. I think a lot of mine really love playing around with decimals and fractions (which I love) but doesn’t 11 make is so much easier?!

I am so glad you called my attention back here because I had forgotten to revisit the effect scaling has on the remainder with the kids, have to get on that today! I can definitely see, “Danger is that you might then think that there was enough for another tile !” happening on this one!

Thanks for you comment!

Kristin

So when I left your classroom yesterday, I was curious about what your students conceptions were about variables. Some of the students in your class accepted “N” as a placeholder for any number and wrote an equation for finding the area of an increasing pattern. Other children seemed to read it as “choose your own value for N and solve.” As you work through this pattern unit, I am curious to know how their conceptions and understandings of variables changes and grows. Is there something that a child or you say that seems to give children purchase on the idea? The interesting thing for me was how easily all of your students understood the “rule” for the visual pattern, and could provide the total area then given the number of rows, but making the leap to the equation caused many of them to pause and stare at the blank. As a former HS teacher, I have seen that look many times.

~F

I love this comment Faith because it brings out something that, for me, feels like an expected ‘norm’ in this work. What a thing to think about! It is so interesting to look deeper into these small moments that are really such important connections (leaps?) in understandings. When and how students make sense of moving from finding an unknown, whether it be a blank, a question mark, or a square that represent a number to now making sense of the fact that a variable can represent any number is something I am now curious about. I can’t wait to hear my students work through this more and of course hear about your research on it (because I know that it is what your next move was after this post:)!

Thanks,

Kristin

I work with 8/9 year olds, and find the N thing doesn’t take very easily, and I don’t stress it when I’m asking them to make sequences and look for patterns (instead I focus on what the first term would be, what you add each time, how you could get straight to any particular step, but without algebraic notation). A year later, and I find they can deal with N more easily.

I think part of the difficulty is that N, the step number, is too intangible. When I try them with Euler’s formula for polyhedra, V + F -E = 2, asking them to make a polyhedron and check the rule, they don’t have a problem, because the variables are tangible, physically countable things. Perhaps getting used to letters as variables like that is the way.

Here they are on the class blog.