Annie at the Math Forum

Noticing and Wondering in Elementary School

Annie — Tue, 10 Sep 2013 01:40:28 +0000

My colleague Max recently blogged about Noticing and Wondering in High School and I thought it would be fun to blog about using it at the elementary level. The essence of our “I Notice, I Wonder” activity is that you give students a mathematical situation or picture or story, without asking any specific questions, and ask them to list everything that they notice about it, and everything that it makes them wonder about.

I’ve written about it in the past, including in one of our Teaching with the Problems of the Week documents, How to Start Problem Solving in Your Classroom [PDF]. In that, I tell the story of the first time I explicitly asked students (who were “low-level” eighth graders) to tell me everything they “noticed” about a picture. The short version is that the students were awesome and their teacher was amazed at how much math they came up with.

Just as I started composing my post, I got email from my friend Debbie, who teaches at an elementary school school in the district I live in. She described the first lesson she did with a new class she’s co-teaching, in which she asked the students to notice and wonder. I asked her if I could use her story as a “guest post” on my blog, since I think it’s as compelling as anything I could have written. She agreed, so here goes.

Debbie’s Story

I taught an amazing lesson today. It was the first day of math class for the year. Our whole district is starting a new math program. Our fifth grade is grouping homogenously for math. Instead of teaching the highest ability students as I usually do as “Teacher of the Gifted,” I’m co-teaching the lowest two classes with two other teachers, a regular education teacher and a special educator. Together we have 22 struggling math students.

Predictably, the topic for lesson 1.1 was place value. But my goals were to engage the students, to create a safe space for learning, to get them thinking and asking questions, and to evaluate their understanding of place value. Instead of using the lessons from the book, which used place value charts with the places labeled, I started by handing out blank, unlabeled place value charts and asking pairs of students to talk about them. I suggested that they notice and wonder. And the three teachers got to wander and listen in. It was amazing.

First, they had to decide the orientation of the paper. Some kids held it vertically and saw it as a thermometer or list. Most held it horizontally. Many recognized it as a chart to use with money or decimals or place value. It was gratifying to see that they recognized the format. When we reconvened to share ideas as a group, our conversation was directed by their noticings and wonderings. I was able to review concepts of place value, numbers vs. digits, etc. not by following the book, but by following the comments from the kids. I praised their questions, asked them to respond to each other’s comments, and kept the discussion flowing.

At one point the kids parroted the places: ones, tens, hundreds, thousands… and I wrote them on the board. They got to millions, ten millions, hundred millions and then got stuck. Some thought that next comes thousand millions and others thought next comes billions. It was a perfect teachable moment; all I did was draw the lines between hundreds, thousands, millions and point out that there were three columns in each, and there was a collective “ah-ha!”

Eventually, I asked the kids to put the place labels into their charts. It was fascinating. About a third of them labeled left to right. That certainly told us a lot about their level of understanding of place value! We have a lot of work to do. But that meant that about two-thirds of them were able to label the places correctly, which is good. I used one of the incorrectly labeled charts and we started talking about it. I asked if putting a 7 in different places changed the number of M&Ms the digit represented. I covered up parts of the chart and asked them to read the number, then revealed the next column. Again, we had “ah-has.” I’m not sure who was more excited, the kids or me.

I had been worried that the other two teachers were going to object to my non-traditional approach, especially on the first day of using a new program. I was pleasantly surprised; they saw the value. During the lesson, the regular education teacher kept flipping through the teacher’s manual. She realized that I had covered material from the first THREE lessons, although I’d not completely finished any of the lessons. So while my approach was non-traditional, I was covering the curriculum and we weren’t “behind.” More importantly, both of them recognized and valued the high level of student engagement. In fact, one pointed out that one boy who struggles with attention had been totally attentive and even participative. They saw the excitement among the students, they noticed that even reluctant students participated, and they recognized the significance of the multiplicity of “ah-ha moments.”

It took me at least another hour to come off the “high” from the lesson.

“Noticing and Wondering” as a Vehicle to Understanding the Problem

Annie — Sat, 27 Oct 2012 15:36:33 +0000

ATMOPAV, October 27, 2012
Annie Fetter and Val Klein

Some resources we’ll use today:

Growing Worms (PDF)

How Many Berries? (PDF)

K-5 Scenarios

Using Technology to Increase Conceptual Understanding in Algebra and Geometry #nctm12

Annie — Thu, 25 Oct 2012 10:29:16 +0000

Thanks to so many of you (130!) for coming to my talk. My apologies to the other 60 or so people who couldn’t get in. I had a great time, and appreciated your willingness to do some math, share your ideas, and talk with your neighbors about your thinking.

Exploring (vicariously) interactive applets that let students “notice and wonder”, talk about mathematical situations, and develop conceptual understandings of triangle properties, linear equations, systems of equations, and factoring trinomials.

Session 12, Thursday, October 25th, Meeting Room 17, Convention Center

National Council of Teachers of Mathematics Regional Conference, Connecticut, October 2012

Session 12 Handout (PDF)

For more about the Math Forum, including information from my colleagues Max and Suzanne’s talks, visit our Hartford page at http://mathforum.org/workshops/hartford2012/. And don’t forget to fill out the survey in the NCTM Hartford App!

Types of Triangles – Annie Fetter

Link above includes handouts, sketch, and Sketch Explorer version

Key technology strengths: exploring dynamic models of different triangle types, allowing students to develop intuition about different classes of triangles and what characteristics they do and don’t have.

Runners – NCTM e-Examples from Principles and Standards

Session 12 Runners Prompts (PDF)

Key technology strengths: developing intuition about distance/rate/time and the concept of slope, without needing to know the formal vocabulary at all, as well as the ability to try things and get it wrong, and get instant feedback without fear of judgement by others.

Galactic Exchange – The Math Forum’s ESCOT Project

Session 12 Galactic Exchange Handouts (PDF)

Key technology strengths: the applet does the recordkeeping for you, and students can explore systems of equations without needing to know that they’re doing that, or even knowing that such a thing exists!

Algebra Tiles – National Library of Virtual Manipulatives

Session 12 Algebra Tiles Handout (PDF)

Key technology strengths: the variables actually vary!

Reminder: Want to be cutting edge? Complete the survey about this talk in the NCTM Connecticut app!

Yes, I Actually LIKE Math! #edtech #math

Annie — Thu, 25 Oct 2012 03:47:10 +0000

Many years ago, back in the the mid-90s, I was in Colorado and visited my friend Anne’s school. I was planning to do a Geometer’s Sketchpad workshop with the teachers after school, teaching them how to author “scripts” (the pre-cursor to Custom Tools in Sketchpad), with a focus on the centers of triangles. Anne was gracious enough to let me teach her geometry classes for the day in the computer lab, and so the students had the neat experience of learning the same thing their teachers were going to learn after school. (“You mean our teachers don’t even know how to do this?? Cool!”)

We started by constructing a centroid and creating the script. The students were familiar with Sketchpad, so things went quickly. We moved on to the circumcenter, and this is where things got interesting. I gave the definition and they started constructing. And I heard this exchange:

Student 1: “Hey, man, what’s wrong with your circumcenter?”

Student 2: “Nothing’s wrong with it. I followed the directions! What’s wrong with yours??”

I wandered over, and I find that Student 1′s circumcenter was inside the triangle, just like the (visually boring) centroid was. Student 2′s circumcenter was outside the circle. Opportunity time! I ran to the front of the room and wrote a question on the board: “When is the circumcenter inside, outside, or on the triangle?”

Not one minute later, I had 28 high school sophomores looking at me like I had two heads, basically telling me, “That’s a stupid question! Obviously, when the triangle is acute, it’s inside, when it’s obtuse it’s outside, and when it’s right it’s on the triangle. Why are you even asking us something so simple??” I was beside myself with excitement because, traditionally, it’s not at all obvious. This is because most students are just told that fact. They don’t discover it. It’s just one more thing that they have to memorize that makes geometry class onerous and quite possibly their least favorite math class.

But these kids figured it out for themselves, independently, saying things to their neighbors like, “Uh, isn’t it just whether the triangle is acute, obtuse, or right? Or am I missing something?” Because they had a dynamic tool. They dragged. It moved. They controlled it. They had constructed the situation, so they knew how the objects were related to each other. While this power of discovery never comes as a surprise to me, it’s still so cool to see it in action. These are moments I never forget (obviously, since I’m blogging about it some 16 years later!).

We continued on to do the orthocenter and incenter, making scripts and exploring some properties of each. This is where the title of this post comes in. I noticed one boy staring off into space at one point, not doing anything on his computer. I walked over and said, “What’s up?”

He replied, very seriously, “Let me get this right. You actually like math.”

I said, “Yea, I do.”

He nodded and then stared off into space for another 20 seconds, before he went back to work at his computer. It’s like he had never met someone who actually liked math. His teacher was a great teacher, very committed to math and education, and I’m pretty sure she actually liked math, but maybe he just saw her as someone who was doing her job. I was just some random person off the street who actually likes math. Who knows? I’d like to think that he might have thought differently about math from that moment on!

Math Mentoring at #tjSTAR @TJColonials

Annie — Wed, 30 May 2012 12:00:58 +0000

Session Description: This is your opportunity to join an elite corps of highly trained mathematical mentors. Real students from around the world send the Math Forum their solutions to challenging math problems. Volunteer mentors write back to students, starting a mathematical conversation. Come learn the skills involved in being a math mentor, and the impact you could have on the next generation of mathematical problem-solvers.

Today we’re going to write some replies to students who submitted answers to the problem Stacking Wood, a problem targeted at middle school students. We’ll start by solving the problem and talking about the mathematics involved and thinking about the sorts of mistakes that kids might make when solving the problem.

Read Stacking Wood

Preparing to Mentor

Solve the problem.
Talk about the mathematics concepts involved.
Think about the sorts of mistakes that students might make.

Let’s Try It!

Read Lily’s submission and write a reply together: Lily
Read Jacob’s submission and each group writes a reply: Jacob
Write a reply to a student or two with your partner: More Student Solutions

Next Steps?

Using Technology to Increase Conceptual Understanding in Algebra and Geometry #nctm12

Annie — Fri, 27 Apr 2012 18:43:48 +0000

My thanks to those of you who came to my talk and played along with these activities. The links below will get you a copy of the handout and send you directly to the applets themselves.

I would love to get some comments from any of the attendees. Can you name one thing that you took away from the talk that you’re eager to try out or work on?

Session 610, Friday, April 27th, at the Marriott, Salon D

National Council of Teachers of Mathematics Annual Conference, April 2012

Session 610 Handout

Types of Triangles – Annie Fetter

Link above includes handouts, sketch, and Sketch Explorer version

Runners – NCTM e-Examples from Principles and Standards

Session 610 Runners Prompts

Galactic Exchange – The Math Forum’s ESCOT Project

Session 610 Galactic Exchange Handout

Algebra Tiles – National Library of Virtual Manipulatives

Session 610 Algebra Tiles Handout

Ignite: The Teacher I Would Have Been #ncsm12

Annie — Wed, 25 Apr 2012 19:50:06 +0000

Thanks to everyone who came to the Key Press Ignite Session at NCSM in Philadelphia. I am really looking forward to hearing your responses to the homework prompt:

How did you become the teacher you are today? Specifically, what motivated you to become that teacher? What experiences helped you to realize, or at least begin, that process?

Couldn’t make it? Not to worry. Watch the video! You can still do the homework.

Just to make you curious, here’s a little teaser in terms of some folks’ responses to the homework prompt:

Suzanne: Motivation? Lectured from Dolciani for three years, then raised two kids and watched them learn. Experiences? Writing about that learning for John Holt’s journal.

Arjan: Motivation? I hated being lectured to. I hated learning like that.

Max: Experiences? It helps that I went to an interesting high school where I saw all sorts of teaching in my classes. And my mom’s a Montessori teacher.

Geri: Motivation? The Saxon ½ Book. It was so boring! Experience? Woodrow Wilson Institutes, especially the collegiality and collaboration.

Erin: Motivation? The students’ bored faces when I did direct instruction, and watching other teachers lecture and realizing that it was dreadful. Experiences? Suzanne walked into my classroom and said, “Try this.”

Your response here – do your homework and leave a comment!

Fourth Grade Pictures of Long Division

Annie — Mon, 20 Feb 2012 21:56:42 +0000

My friend Debbie, the enrichment teacher at a local elementary school, sent me the following pictures, along with this question: “What do you think about the attached? Fourth grade E[veryday] M[ath] ch 6 intro.”

My short answer was, “I think it’s brilliant!” I’ll expound on that more in a follow-up post, but wanted to throw this out there first in case anyone else had thoughts they wanted to share. (Yes, Unit 6 is about division.)

Screen-Capture Movies Show Student Thinking #petec12

Annie — Mon, 13 Feb 2012 13:45:09 +0000

Pennsylvania Educational Technology Expo & Conference, Hershey, PA, February 2012
Debbie Wile, Wallingford Elementary School, Wallingford, PA
Annie Fetter, The Math Forum @ Drexel University, Philadelphia, PA

Introductions

Who are Debbie and Annie?

The Path to This Talk

What’s Next?

Literacy Applications
Peer Editing
Big Brother is Watching
Precursor to Open-Ended Responses – Talking is Easier Than Writing
Other Applications?

Discussion

Helpful Links

Geometer’s Sketchpad from Key Curriculum Press – check out the trial version
Screencast.com, for Jing
Making Movies Using Jing – Instructions [pdf]
Constructing Quadrilaterals with Sketchpad Handout [pdf]
Transformations Handout [pdf]

Debbie is the enrichment teacher at Wallingford Elementary. She teaches students in grades 1-5.

Annie is an Educational Programs Leader at the Math Forum. Read more on her blog profile page.

Act 48 code: DL061674

Using Technology to Increase Conceptual Understanding in Algebra and Geometry

Annie — Sat, 22 Oct 2011 02:57:57 +0000

Exploring interactive applets that let students “notice and wonder”, talk about mathematical situations, and develop conceptual understandings of triangle properties, linear equations, systems of equations, and factoring trinomials.

ATMOPAV Mathematics & Technology Conference, Fall 2011

Get the session handout [pdf]

Types of Triangles

Runners

Galactic Exchange

Algebra Tiles