The Math Forum used to host an Internet Math Hunt. Long before Google, Bing, Ask Jeeves, and even AltaVista showed up on the scene, finding things on the Internet was actually challenging. In our very first hunt in September 1995, we asked these five questions:

- In what town is the Geometry Center located, and what is it?
- Annie Fetter, the person who runs several of the student projects at the Math Forum, has pictures of her cats on her home page. What are their names?
- Name four members of the Swat Team, which staffs the Dr. Math project, who have home pages.
- What school runs the Great Penny Toss? (Extra: where is the school located?)
- There is a History of Mathematics site somewhere in the United Kingdom. What university hosts it, and where in the UK is it exactly?

The second question got a lot of attention, and even now it persists on Answers.com in two different places:

- Answers.com > Wiki Answers > Categories > Jobs & Education > Education > School Subjects > Math and Arithmetic (This one remains unanswered)
- Answers.com > Wiki Answers > Categories > Animal Life > Mammals > Land Mammals > Cats (Felines) > (This one is answered correctly)

Eukie and Ivan started coming to the office in 1994 when they were only a couple months old. We were then housed in the math department at Swarthmore College, and the cats had free range of our office, the hallway, and the Fishbowl, which was a seminar room across the hall with floor to ceiling windows on three sides (hence the name “Fishbowl”).

(FYI, Ivan is the one with the spot on his back.)

One day I went to retrieve Eukie from the Fishbowl before a psychology seminar, and the students looked at me imploringly and asked, “Can he stay? Please?” I looked at the professor and he shrugged, so I left Eukie there. For the next three hours, Eukie slowly circulated from one lap to another to someone’s notebook. At one point I glanced over (our office had windows out onto the hallway and right into the Fishbowl), and a student was making an impassioned argument, gesticulating with one hand while the other hand was petting the furry mass lying on her notebook.

Eukie and Ivan loved to sit on the trash can next to the water fountain outside one of the big lecture halls when class was about to let out, because they knew someone would turn the water on for them so that they could have a drink. (One morning the housekeeper reported that she had seen a student holding one of the cats’ paws against the button, trying to teach them how to turn it on themselves.) When the department secretary argued that this was unsanitary, the housekeeper explained that she just wiped it down with some disinfectant when they were done.

Eukie and Ivan have long been fixtures of our Math Tools library gallery, with Eukie chasing the isocahedron screensaver (shown above) and Ivan doing trig with his TI-92. Eukie’s enthusiasm for chasing screensavers at the office didn’t abate as he got bigger, though sometimes all that chasing is tiring.

Ivan often engaged in the age-old cat responsibility of confirming the continued existence of gravity by knocking things off shelves, such as this tissue box (but, fortunately, not the octahedral wooden puzzle), and both dutifully entertained visitors.

A few years later, we hired someone who was allergic to cats, so the cats retired to a life of leisure at home. They both continued assisting with important mathematical work, such as my technical editing of the first version of *Exploring Calculus with the Geometer’s Sketchpad* and helping with the Geometry Problem of the Week #GeoPoW.

They also spent time on the deck lounging and watching the squirrels, blue jays, starlings, raccoons, and other neighborhood cats eat the dried food that we ostensibly put out for the resident cats.

Naturally, all this work makes one tired, so they also did a lot of what cats do best, which is nap and look wicked cute at the same time.

We haven’t even mentioned the origins of their names. Eukie’s real name is Euclid, of course, while Ivan is actually Nikolai Ivanovich Lobachevsky – a fitting pair of names for cats belonging to someone then working at “The Geometry Forum” if ever there were. While Eukie and Ivan aren’t the most famous cats on the Internet, they’re long-lived fixtures. We’ll miss them, but their mathematical contributions will live on.

]]>*The very first Mathematical Practice, “make sense of problems”, includes many ideas that have long been foci of literacy instruction. Yet when “math” starts, both teachers and students often leave those good habits behind. We’ll look at examples of this and explore how to translate literacy routines into good mathematical practices.*

Download the handout [pdf]

Download the PowerPoint slides [pdf]

Visit the Math Forum’s NCSM & NCTM conference web page to learn more about our talks, view videos that support Max’s book *Powerful Problem Solving*, download free samples of our Problems of the Week support materials, and more!

*The very first Mathematical Practice, “make sense of problems”, includes many ideas that have long been foci of literacy instruction. Yet when “math” starts, both teachers and students often leave those good habits behind. We’ll look at examples of this and explore how to translate literacy routines into good mathematical practices.*

Download the handout [pdf]

Download the PowerPoint slides [pdf]

Visit the Math Forum’s NCSM & NCTM conference web page to learn more about our talks, view videos that support Max’s book *Powerful Problem Solving*, download free samples of our Problems of the Week support materials, and more!

*The very first Mathematical Practice, “make sense of problems”, includes many ideas that have long been foci of literacy instruction. Yet when “math” starts, both teachers and students often leave those good habits behind. We’ll look at examples of this and explore how to translate literacy routines into good mathematical practices.*

Download the PowerPoint slides [pdf]

I’m not making the handout available because there isn’t much to it!

I may add a few thoughts after the session.

]]>This is posted in support of the mini-session I’m leading at CMC-North Mini-Conference. A session description for those who might stumble across this post:

*Many topics in math seem difficult to address conceptually and tend to be taught procedurally. We’ll explore technology tools that encourage students to “notice and wonder”, talk about and make sense of mathematical situations, and develop conceptual understanding of triangle properties, linear equations, systems of equations, factoring trinomials, calculus concepts, and more.*

My thanks to everyone who came to the session. As promised, here’s a copy of the session handout [pdf].

- iPad: Types of Triangles sketch – Annie Fetter
- iPad: Quadrilateral Pretenders – Key Curriculum
- Laptop: Runners – NCTM e-Examples from Principles and Standards
- Laptop: Virtual Algebra Tiles – National Library of Virtual Manipulatives
- iPad: Mellow Yellow – Interpreting Graphs – Key Curriculum
- iPad: Graph Dancers – Key Curriculum
- iPad: Wuzzit Trouble – Inner Tube Games

- PowerPoint Slides [PDF]

My thanks to those of you would came to the session. I enjoyed the conversation a lot. Unfortunately, I did not remember to take a picture of the list of reading lesson objectives that we generated, but I have to say that I was excited to learn a new word that was the topic of one person’s reading lesson objective last week: Syllabicate. When we reviewed the list and talked about analagous topics and foci in math instructions, someone pointed out that syllabication is sort of like place value. That was brilliant! Decomposing and recomposing numbers is an important part of numeracy, and of course place value is one big part of that.

I hope some of you will send me mail or leave a comment here about ideas they will or have already tried in class as a result of our session.

]]>- Handout [PDF]
- PowerPoint Slides [PDF]

My thanks to those of you who chose to come. I had a good time, and we had a lot of discussion about things we might try as a result of the session. I know that Leigh Nataro (@mathteacher24) changed up her Monday morning geometry task, and had students Notice and Wonder about a diagram. I hope some of the rest of you might leave a comment here that describes something you plan to do or already did do.

Here’s the Noticing and Wondering that we did around the Congruent Rectangles problem (maybe some of you want to try the problem with your students, and then have them compare their lists to a list that a bunch of math teachers came up with!):

For those of you who couldn’t make it, as well as those of you did make it in person but want a succinct version to share with colleagues, this was sort of a one hour version of my Ignite talk of the same title: Ever Wonder What They’d Notice (If Only Someone Would Ask)?

]]>This year, the day after our first workshop, I received mail from Brooke, one of the student teachers. She is student teaching in a 5th grade classroom in the district where I live. (You might recall that my friend Debbie, who authored the last post on my blog, also teaches at an elementary school in the district where I live, and this year is teaching a section of 5th grade math, but she’s not at the same school as Brooke.)

Brooke was wondering if she could do I Notice, I Wonder with her students, even though she’d never done it before. Short answer: Absolutely! For the longer answer, here’s the exchange that we had over the course of a couple of days.

(Note: Brooke mentions “bar models” in her post. For more info about that, check out this post from Erie 2 Math. Some of you might know them as part-part-whole diagrams.)

**Brooke, Monday, 8 pm (the day after our Sunday workshop)**

Hi Annie,

I don’t know if you will receive this email tonight, but I am teaching my math class tomorrow and radically changed my lesson plans today based on a pre-test they took in class. I am going to try the I notice/I wonder chart while having the students look at bar models. I am going to give them a bar model with two knowns and the unknown that will have to solve for when they have these problems. I am just kind of nervous and wondering if you have any last minute advice? I am also being observed by my supervisor so I feel it is a bit of a risk, but I am trusting your’s and Max’s word and trusting that I can use this strategy without any practice!

Thanks for all of your great tips yesterday…I really enjoyed it and when I saw bar models today I instantly thought I needed to use the I know/I wonder chart.

**Annie, Monday, 9:43 pm**

Brooke, you totally rock! I say go for it. I think you *can* do it without practice. One thing to remember is that you’re trying to figure out everything that’s in their heads, rather than putting anything in their heads. You are listening * to* what they say rather than listening

And think of it as a sense-making activity. Bar models are really really easy and helpful *if* you are doing sense-making as opposed to trying to “remember” where you are supposed to put what and what picture you are supposed to draw. Are the kids trying to remember some set of steps that the teacher or book modeled, or are they trying to make sense of the situation?

After you done some noticing and wondering, you can also be sure to sometimes (often?) ask kids, “How do you know?” whenever they make a math statement (don’t force that on them when you’re first noticing and wondering – just get their ideas out there, unencumbered by the burden of knowing why. But later, as you talk about more things, ask them to back up their statements).

Here’s the blog post that I mentioned that my friend Debbie wrote about doing I Notice, I Wonder with her low-level 5th graders: http://mathforum.org/blogs/annie/ I wonder if that will give you additional confidence and ideas.

I don’t know who your supervising teacher is, but if it’s Robin Bronkema, tell her I said hi! She and I played field hockey together at Swarthmore.

Let me know how it goes!

–Annie

**Brooke, Monday, 10:24 pm**

Annie,

Thank you SO much for your email! I feel much better now that I am thinking again in terms of sense-making. I also enjoyed reading Debbie’s blog post, as it contextualized the strategy quite a bit. I am really excited for the lesson and so is my cooperating teacher…she is totally supportive of me stepping outside of the box.

My cooperating teacher is Liz Corson. She also graduated from Swarthmore, but I am not positive what year. Robin Bronkema actually did a workshop with us a couple of weeks ago!! She was fabulous and I loved her energy and presentation as well…I love meeting all of these Swat alums.

I will send you an email tomorrow after school to let you know how it goes. Thanks again for your reassurance!

Brooke

**Brooke, Tuesday, 6:52 pm**

Hi Annie,

I did it!!! It went really well. The kids were excited to do something different. They were hesitant at first, but when they realized I meant write everything they noticed and wondered, they opened up. I had one boy wonder why I had them doing the activity and at first he was not on board, but when I addressed it at the end, he realized that it had helped. What I noticed about the activity was that once they started working on bar models individually they were talking in the language of, “What do I see here? I have 7 groups and I know the whole is 289, so I need to find how many are in each group,” as opposed to trying to figure out what they needed to solve just by looking at where the question mark was. Sense-making…yes!

Thank you so much! I definitely plan to use it again in the future and my cooperating teacher enjoyed it, so she is on board as well.

Brooke

**Brooke, Tuesday, 8:36 pm**

More follow-up: My cooperating teacher just emailed me the math plans she is teaching tomorrow and she included I notice/I wonder!

**Annie, Tuesday, 8:50 pm**

Congratulations! How awesome is that! I’m really glad it went well. I especially like hearing what you noticed about the language they were using when working individually later and how it was centered on sense-making. If you can get most of those kids to think that math SHOULD and CAN make sense all the time, you are making a HUGE difference in their educations.

Let me know how things go tomorrow and whether the kids seem eager to do it again and if you think they are “better” at it (mostly meaning more mathematical, though perhaps they were really mathematically this time around).

I will try to get Debbie to write more, too, so that I can post it on my blog (though technically it’s my turn to post on my own blog). I’ll let you know if she does, or if I blog about my exchange with you.

–Annie

**Brooke, Wednesday, 10:20 pm**

Hi Annie,

The I notice/I wonder went over again really well today and we are going to use it again tomorrow! It is great because it really gets the students thinking critically and it has lived up to its promise of encouraging everyone to participate. They were also more mathematical today and still on board with the activity. It also helps me phrase math in terms of problem solving and sense-making, as opposed to speaking procedurally. I had another station of students working with bar models, and since it had been a day since we did I notice/I wonder with bar models, they started speaking in a very procedural manner again (e.g., “Question mark is there…so I know this is a division problem.”). It was simple for me to remind them of I notice/I wonder and tell them to figure out how they know it is division from what we see and notice about the image.

I take over math next week and I will certainly continue with the model. Thanks again!

Brooke

So there you have it – the experience of a “first-timer”, captured in a few snippets. I was excited that she thought to try it, very excited that she did try it, and super excited that she noticed the type of mathematical talk that it encouraged in the classroom and how it is serving as a foundation for sense-making for her students.

]]>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.

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.

Annie Fetter and Val Klein

Some resources we’ll use today:

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