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2014-12-10 10.13.02

I want to share a question with you that has been clanking around my brain for a while. At first I wasn’t even sure if I wanted or was even “allowed” to share and write about this question, but it’s one I keep coming back to. It’s a question that makes me feel somewhat vulnerable. And it’s one I’d love to hear your thoughts about, so please feel free to leave a comment even if, like me, your ideas around this question aren’t fully developed either.

Is there room for math that isn’t hard?

One reason I wasn’t sure I wanted to write about this topic is because I really believe in teaching students to work hard to figure out mathematical ideas. I believe that math can be challenging and also enjoyable. I teach teachers and students that doing challenging work is what helps us construct new understanding. I really believe in the power of the “make sense of problems and persevere in solving them” (CCSS.Math.Practice.MP1) I believe that much of math learning will occur in this way. And yet, I keep coming back to this question. Should all math be rigorous? Should we persevere through all tasks? Or is there a room for something else too?

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When imagining strong math communities, I like to consider analogies to literacy communities. In reading workshops, teachers spend time a good amount of time engaging children in read-alouds in which a primary purpose is learning to love books and enjoy reading. As a kindergarten teacher I often read Mo Willems’ Piggie and Elephant books not because they were rigorous, but because it’s fun to read the speech bubbles using the voices we imagine Piggie and Elephant would use. We lingered to laugh for a few moments on the page of David Shannon’s No David that shows David’s naked bottom as he runs down the street naked. In reading workshops book browsing is encouraged. Children squirrel away Lego encyclopedias in their book boxes to pore over with friends. They linger over photographs of animals and come up with their own wonderings like, “Why are reptiles bumpy?” “Why do pigs have curly tails?”

In reading workshop, children should not spend the majority of their independent reading time engaged in books that are difficult for them. Richard Allington’s research indicates that children should be spending the great majority of their reading time engaged in books at their independent level, books they understand and can read accurately. Children need to read and re-read. (This is one quick Allington article, but his work on the topic is much more extensive than this.)

So, I’m wondering, is there a math equivalent of this? If children subsist on a diet that consists only of difficult math, will they learn to enjoy it? Will they learn to pursue it beyond the walls of the classroom? I say that with the huge caveat that I believe challenging and difficult math can be engaging and fun. But I also believe there is space for math that is not terribly difficult, but is very enjoyable.

Strategy games (like Rush Hour, for example) can be extremely challenging. While playing Railroad Rush Hour recently, 5th grader Kevin told me, “I’m only on the first challenge. And it’s already hard. I can’t do it.” He stuck with it though, and figured it out. Kevin moved on to other Rush Hour challenge cards, but a few days later I noticed he was working on that first card again.

“Kevin, didn’t you do that one the other day”” I asked.

“Yes,” replied Kevin matter-of-factly. “But I like to go back to ones I’ve already done and think about it again. It’s fun.”

Was what Kevin was doing the problem solving equivalent of re-reading a favorite book?

2014-12-13 11.55.51

This weekend I spent some time at the grocery store with my daughter watching an employee make a giant batch of guacamole. I watched with fascination. “I wonder how many avocados are in that bowl. How many containers of guacamole will that make?” I considered a few ideas in my mind. How much space would ten avocados take up? So, how many groups of ten avocados? I engaged lightly and playfully with the idea. The avocados offered an invitation, rather than a specific task to complete.

I held my almost-two-year-old up to see the giant batch of guacamole. “How many avocados are in there, Lulu?” (She loves avocados!)

“One and one and one and one!” (That’s what her counting sounds like now.) “Dos!” (That means any quantity more than one in her world.”)

It wasn’t hard. For me or for Lulu. But it was kind of fun to think about. I love looking for math in my life. I hope to inspire that in her and the teachers and students with whom I work. I hope to offer invitations to engage in mathematics. Sometimes in difficult mathematics and sometimes in something less serious.

So, is there space for this in our classrooms? Time for engaging, but not terribly difficult math? What other opportunities do children have to just wonder about math? To browse math? To engage in math play? To relax into math, the way one relaxes into a book. Is this important? Is this a thing in math? Is it something professional mathematicians do? Is it something everyone does as they wonder about their world?

I’d love to hear your thoughts.

~Kassia

#ElemMathChat

2014-02-18 12.10.07

I’m dusting off the ol’ blog to share that I’ll be hosting this week’s #elemmathchat on Twitter this Thursday, December 11, 2014 at 9:00PM EST! I always find the #elemmathchats thoughtful, supportive and inspiring, so I hope you’ll join us.

The topic this week will be building mathematical communities, a topic near and dear to my hear. Check out the questions below for you to mull over this week.

Q1: How do you define mathematical community? What do strong mathematical communities look like/sound like/feel like?

Q2: What big and small things do you do to help build these strong mathematical communities?

Q3: What is a misconception/roadblocks you run into that impacts the development of strong math communities? How have you dealt with it?

Q4: Describe a moment/interaction in your math community that made you feel like “Yes! This is good stuff of mathematical community!” Why was it important?

Q5: If your students walk away from your class with one big idea about what math is all about, what do you want it to be? How would you and your math community have helped them construct this idea?

Q6: Share a picture or piece of student work that shows your mathematical community (and its values) at work.

Looking forward to “seeing” and chatting with some of you on Thursday!

~Kassia

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I’ve been thinking a lot about math communities, mindset, and how children define mathematics. I got the chance to write about some of these thoughts, and share some stories from a great 5th grade classroom in my blog post, “On Being a Mathematician,” over at Stenhouse’s Summer Blogstitute page.

While you’re there, check out all of the summer Blogstitute posts, especially, “Teaching Through and For Discussion,” written by Elham Kazemi (@ekazemi) and Allison Hintz (@AllisonHintz124). They’re the authors of one of my favorite new math books, Intentional Talk. If you haven’t checked it out yet, you should!

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Since Math Exchanges was published, many teachers have asked if there are any videos of math exchanges in action that they could check out. Readers wanted to see and hear more from teachers and students as they met in small groups. I’m so excited that I will now be able to tell teachers “YES!”

“How Did You Solve That?” takes a glimpse into math exchanges in my kindergarten classroom and the 2nd grade classroom of my wonderful colleague, Rachel. The part of making this DVD that I am most proud of is how real it is–it was filmed over a couple of days last fall and really captures what it looks like, sounds like and feels like for teachers and students when they are immersed in the work of math exchanges. Teachers will get to see how Rachel and I plan for, teach, and reflect upon math exchanges. You’ll see us wondering about our teaching language and discussing student strategies. You’ll see that not every moment in our classrooms is perfect (I believe a slamming bathroom door makes an appearance in the middle of one of my math exchanges!), but that we, like you, are constantly reflecting on our teaching and revising our next steps for our young mathematicians.

I’m happy to share this part of my teaching life with you all, and hope you will find it a useful tool. 

Back to Blogging!

I have taken a very long break from blogging! I’ve still been here teaching and learning about math, life has just gotten in the way of blogging. And a wonderful part of that life has been the birth of my daughter, Louisa, this February!

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Just this week I went back to Keith Devlin’s fascinating book, The Math Instinct. Devlin’s book takes a look at some studies with very young babies that illustrate the fact that we humans are born mathematical creatures. In one study designed by psychologist Prentice Starkey, babies were shown images of one dot and two dots on projectors. At the same time one or two drum beats would be played. When one drum beat was played, the babies would spend a significantly longer time staring at the projector that showed one dot. When two beats were played, the babies would stare at the projector with two dots (Devlin, 12-13). Many other studies have also shown that babies have some understanding of numerosity and number sense.

Babies are mathematicians too! So you can imagine that I’ve been watching my own little mathematician for signs of number sense. I’m pretty sure she’s counting already :-)

Now that I’m getting used to being a mom and getting a little more sleep I plan on getting back to the blog more frequently. Looking forward to catching up with math friends here! What mathematical adventures have you all been up to?

September and May are my favorite teaching months. September because everything and everyone is fresh and full  of hope for the school year ahead. May because it is a month full of moments in which you marvel at how much your students have learned, at the amazing people they are.

There have been many of these May moments in our math workshop lately. We have done a lot of work around the idea that math is storytelling, that you need to be able to tell and interact with the context and story when problem solving in order to figure out a problem.

We use these mathematician statements to encourage this kind of thinking and set a purpose:

  • “Mathematicians listen to the story and tell it again to figure out the problem.”
  • “Mathematicians think about what is happening in the story. This helps them solve the problem.”
  • “Mathematicians look for their own problems to talk about and solve. They are creative.”

We’ve been using some of the story mats from Kathy Richardson’s Developing Number Concepts, Book 1: Counting Comparing and Pattern to help generate some stories(Thanks, Katie for reminding me of these!) We talked about different kinds of stories we could create, the questions we were asking of the solvers of our problems, whether the result would be more or less than our starting point based on the action in the problem.

Here is Bryan telling an ocean story.

We’re also always engaged in counting collections. This week we counted a giant collection of gum drops for a spring fairy (long, involved imaginative story created by class behind this fairy!). I loved watching as children organize, count, and talk about the collection. In the beginning of the year counting collections focused on maintaining 1:1 and the number word sequence. Now we spend time focusing on finding efficient ways to group and count.

Here is Madeline organizing a collection of gum drops into groups of five.

We’re savoring the May moments in our classroom. What end-of-year moments are amazing you in your classroom?

 

My kindergartners are investigating one of the big ideas of numeracy–decomposing numbers.

We have been using Cathy Fosnot’s Contexts for Learning unit, “Bunk Beds and Apple Boxes,” which is based around a story that comes from the accompanying big book, “The Sleepover.” The math investigation is based on the story of eight girls at a sleepover jumping back and forth between the top bunk and the bottom bunk in an effort to trick Aunt Kate who is hosting the sleepover.

We acted out the story in small groups with our bodies, moving children between two blankets we dubbed “the top bunk” and  “the bottom bunk,” looking for all the ways to make 8. Then we worked with partners to model and represent ideas on paper with partners before reflecting as a group and charting our ideas.

Understanding how numbers are composed and decomposed is an idea we keep returning to throughout the year, and one I know my students will continue thinking about long after they’ve left our kindergarten classroom.

I believe that teachers should search out quality resources that support responsive teaching. These Contexts for Learning units are both contextually meaningful and mathematically significant, the litmus test of all curriculum for me!

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