My vague criteria for the activity:

- Relatively easy to explain, and doesn’t require constant attention from knowledgeable helpers (I’ll have three or four volunteer helpers, who don’t have to know any math)
- Could have some examples or diagrams available for the more self-directed
- Doesn’t take a wicked long time to accomplish something, but could go on for a while (low floor, high ceiling, that sort of thing)
- Is fun and art-y
- Involves some math, even if it’s not obvious to everyone
- Could be something they take with them, or leave to display in our area, maybe hanging on a fence

Here’s what I’m thinking about so far.

One idea was a double Tumbling Blocks quilt pattern. It’s not that exciting, since it’s basically coloring by numbers – you pick a light, medium, and dark color, and color the chunks numbered 1, 2, and 3 respectively. The pattern would be half of an 8.5 x 11 sheet of paper. See below, plus the “finished” product if we put a bunch of them together.

Another option is the Rail Fence quilt pattern. Again, not very exciting to color, but fun to see how it looks as a “quilt” depending on how you put them together. I’d provide two different half-sheet templates with six to 10 squares on them – one that is arranged in the zigzag, and the other in the pinwheel.

But the more I think about this, the more I’m thinking I might do frieze patterns. I’ve done an activity with them before, where we learned about them, then acted them out physically, and then made them using The Geometer’s Sketchpad software (read about the time we did it with the Drexel Math Contest, and even download the instructions). We could for sure have groups of people acting them out with guidance – we have plenty of room. But I haven’t done the arty-y part with paper and markers before. I wonder how that would work? I sketched out a couple of possible templates – one of triangles, one just squares, both on about a 2″ strip. There are four examples in this picture, then a picture (with a link to a PDF) of all seven types of frieze patterns with their John Conway-given names.

That’s where I’m at right now, and am looking forward to getting some ideas from y’all. I know, especially, that I’ve missed something with the quilt blocks and coloring/symmetry, and am wondering if anyone else has done a frieze pattern coloring activity. Thanks in advance for your contributions!

]]>As promised, here are PDFs of the slides and the handout, plus links to the technology. Yes, I’m even including a link to the third app that we didn’t get to that goes with the last page of the handout. If you want to know more, let me know and I’ll tell you more. (I’ll make you curious by saying that it’s a vending machine on an alien planet.)

- Fetter TMC Desmos Lunch 2017 Handout [PDF]
- Fetter TMC Desmos Lunch 2017 Slides [PDF]
- Types of Triangles [app will open in new tab]
- Runners [java]
- Runners [HTML version]

Look! A bonus section! I’ve got two different flavors of quadrilateral versions of the triangles sketch. One has three quadrilaterals and the other has four.

Thanks again, everyone. Keep me posted if you used any of this.

]]>Sense-Making: Is it at the Core of Your Classroom? Fetter CAMT2017 Sense Making [PDF]

Using Technology to Support Conceptual Understanding in Algebra and Geometry Fetter CAMT2017 Technology [PDF]

Note that the triangle exploration will work on anything, but the others are java, so may or may not work on your devices (it depends how java is configured).

]]>Gordon has been using our Problems of the Week for many years. Here’s a snippet from an email he sent me in November of 2010:

*Hi, Annie. I am still teaching here in Oklahoma. I just thought I would touch base. […] How are things with you? It has been a while since Swarthmore and only the Geometry PoW.*

A while, indeed! We started the Geometry PoW in 1993, and added the Elementary PoW (since renamed the Math Fundamentals PoW) a couple of years later. In November 2014 Gordon wrote:

It has been a long time since we have been able to chat a bit about teaching and math. It doesn’t seem like that long ago that we meet in person at NCTM in San Diego [in 1996]. I could finally put a face with someone I had communicated with online. Over the years I have posted almost all of the PoWs on my bulletin board.

Somewhere there is probably a picture of Gordon and me from that San Diego meeting. I did find two pictures of us from subsequent NCTM Annual Meetings.

(Yes, in the San Francisco picture I am dressed as an anonymous superhero. Ask me about it sometime.)

Gordon hasn’t had many students submit to our problems online, but for many years, he has had a bulletin board outside his classroom where students could get the problems and then give a solution to him for extra credit. The only students of his who consistently submitted online were his three children. Oldest child Gordon Jr. started submitting in 1996 when he was a senior in high school (and continued during his first year of college). Middle child Mike started in 1998, and youngest Regina started in 2005.

In the spring of Gordon Jr.’s senior year of high school, we used a question about how to make a hexagonal cross section of the cube. In my commentary about the solutions we received, I wrote, “There were some innovative methods used to find the hexagon. Berno W. says he used a potato. Gordon Bockus went out to the woodshop and built the figure.” The next April, at the NCTM Annual Meeting in Washington, D.C., Gordon Sr. stopped by our booth in the exhibit hall and said, “I have something for you. Gordon Jr. wanted me to give you this.” Out of his bag, he pulled Gordon Jr.’s sliced cube! As you can see, he had written on it, “I know it’s not regular, but it pointed me in the right direction. Gordon Bockus Jr.” It has lived on my desk every since.

Gordon would occasionally submit his own solutions to problems, telling me why he liked them, describing how he was going to use them with his students, or asking a question about a problem or its extra question. The bulletin board persisted. In his November 2010 email, he told me how his two boys (by then both married with kids) were doing, and said, “Whenever one of them visits me at school, they see the PoWs up on the bulletin board and they ask if Annie is still involved.” In that email he also mentioned retirement, but added, “I don’t know when that will happen. I don’t know what I would do in place of teaching.”

In November of 2014 he wrote, “Retirement is on my mind but I don’t think I can go quietly. So I might be around another year.” Well, here we are, three years later, and Gordon is in fact retiring at the end of this school year. And he’s going out with a bang, as his Academic Team won the Oklahoma State Championship in February and Gordon was named Oklahoma Academic Competition Association Coach of the Year for the second time (he also helped start academic bowl competitions in Oklahoma).

This means that starting this fall, I’ll no longer see Gordon’s name at the top of the submission queue on Thursdays, and I’ll miss that reminder of our long friendship and his years as a devoted Problem of the Week user. However, as I wrote in email to Gordon last spring, Gordon Jr.’s sliced cube will always have a prominent place on my desk. It represents the great connections that I’ve made throughout the years with Gordon and his children, as well as many other students and teachers, and also ways in which students explore math that go above and beyond what we might expect.

Thank you, Gordon, for being a member of our community for so many years and for sharing your love of mathematics with so many students. Enjoy retirement!

]]>Jean and Sue and I all live in the same very small town. It’s four blocks by four blocks, or about 0.14 square miles. I met them five years ago when I learned they were hosting a monthly women’s music circle in Sue’s basement. (In fact, I logged onto Facebook just now to get the awesome Miss Piggy poster, and the first thing I see is Facebook telling me that Jean and I have been friends on Facebook for five years, and that we should celebrate our friendversary!) Other than playing music with my family and a couple of brief stints in ensembles in college, I haven’t played with other people since high school. I am a passable guitar player and singer, and would much rather play with other people than by myself, so I started going. I knew Jean’s name because I had been feeding her cat, Blake, on my back deck for years, but I’d never met her in person.

At a music circle, you basically go around in a circle, with each person picking and (usually) leading a song, either a favorite, or an original, or whatever. Many circles have pretty strict protocols around who joins in on songs and who gets to solo and who gets to sing harmony and what sorts of songs are appropriate and how much chitchat is allowed pre- and post-song. None of those things are bad, but they can make joining a circle a little scary for newcomers, especially those who aren’t as musically experienced.

Sue and Jean wanted to have a circle that was much more inviting for beginners and others who don’t “play out” (in front of other people, like in a bar or coffeeshop), was a little more free-form, and was only for women (or “gurls”, as we call ourselves). They have succeeded in spades! We’re not big on protocol, and pretty much anything goes. Some of the women play in public a lot, some have never sung in front of anyone but their family, and everyone else is somewhere in between. We even have people who just come to listen, or take pictures and post them to Facebook (hi, Carolyn!). We bring snacks and drinks, talk about kids and weddings and concerts we’ve been to, and play a lot of music. Someone might say, “Hey, can I….?” and the answer is always, “Hey, it’s Gurls Circle, so do whatever you want!” Sometimes as few as two people come (we called it The Annie and Sue Show), other times we have as many as 15. You never know who’s going to show up and what songs they’re going to lead. As far as I’m concerned, it is awesome.

Long story short, I ended up getting an electric bass and started playing that at Gurls Circle. I wasn’t that good or creative, but I had played a little bluegrass bass in high school, and I play well with others (by which I mean I can keep a beat and am good at following other people playing guitar). I got better at executing my limited skill set, even though I only played once a month at our circle.

A couple of years ago, Jean and Sue invited me to join them for a gig at a small coffeeshop, then they invited me to play with them at an open mic night they were running at a cafe. Sue invited me to play some bluegrass with her and her husband, who is a banjo player, and last summer we played bluegrass at the local farmers’ market. Jean, Sue, and I also played at the farmers’ market. This winter, I played with both “pairs” at our town music night.

This spring, Jean announced that we would be playing at a local barbecue joint every Tuesday this summer. The three of us would play from 5:30 to 8:30, with a break in the middle, during which another woman musician that Jean invites would play a short set. We played our first night on June 21.

When making the set list for this past Tuesday, Jean suggested that I take the lead and sing a few of the songs I sing at music circle. This week, Sue and I played through the break, accompanying the guest artist, Nikki, while Jean sang some harmonies. Then Nikki joined us at the end of the night to sing our last few songs. We were also joined for the whole night by Jean’s congo teacher, Karen Smith.

If you are still reading, you may well be wondering what on earth this has to do with math education, so here goes. As I was practicing with Sue and Jean this past Monday, I realized, “They’ve invited me. I’m now part of this local music party. I feel welcomed. I feel valued. This is exactly what Kaneka was talking about in her ShadowCon16 talk!”

Honestly, it was almost an overwhelming realization, that Jean and Sue have done for me (and other women) in music what I’ve always tried to do in math, and what we all need to be conscious of doing more. If you belong to a community that you value and that values you, you invite other people to join in and participate. The math community, or the “party”, isn’t any different. Just like any community, those of us who are active need to invite and nurture new members. We need to help people realize and understand that they belong and they have something to offer. Watch Kaneka’s video for more.

After we finished playing Tuesday night, we were all sitting at a table, eating our barbecue (free food and drinks for the band? I’m in!) and talking about how much fun it had been to all play together. I jumped in and said, “Hey, if I may be reflective for a minute.” I briefly told the story of Kaneka’s ShadowCon talk. Much like you were probably wondering why I was writing about music, I’m sure they were wondering why I was talking about some math thing! Then I explained the parallels that I saw in these two situations, and expressed my gratitude to them for inviting me to the party.

I added, “You know, this is so important to me, I’m going to write about it and share it with my math peeps.” They asked if they could read it when I was done, and I said, “Sure, I’ve got a blog. I’ll make sure to send you the link.” (Yes, we finally got access to our blogs after The Long Transition.) So here it is.

As we were carrying the last few things to the car a little later, I said to Jean, “You know, I meant what I said. I really am grateful to you guys for inviting me and letting me play with you.”

Jean’s reply was better than any conclusion I could write to this story. She said, “Well, inviting people to do the next thing is the only way we all grow and get better.”

]]>```
```Why on earth is anyone giving anyone any hints? I don’t think we should be doing anything other than checking for understanding. #probchat

— Annie Fetter (@MFAnnie) May 18, 2015

I know hints are a hot Twitter topic right now, and I agree that you do, as a teacher, want to have a plan for what to say to kids who are stuck somewhere specific (that you expected them to get stuck). But most of the hints that we give are really shoves (some very gentle, some more forceful) in a particular direction. They often don’t do three things that I think are important:

- figure out what the student understands about the story
- honor where the student is and what they’ve thought of so far
- let the student do all the work and make all the decisions

As a sometimes coach in a wide range of schools/districts/populations, I don’t think that I work with students that are so unusual, and I find that when a kid asks me a question or tells me they’re stuck, probably 19 times out of 20 (or maybe more like 49 out of 50, though my colleagues would probably guess at 99 out of 100), what I say is, “Tell me something about the problem.” That’s it. There are small modifications – if they asked about question 3, I’ll say, “Tell me something about question 3.” Or, if I’m feeling radical and they’ve actually done some math, “Tell me about what you’ve tried so far.”

Yes, even if they say, “I don’t know how to start.” In #probchat, Kent asked:

```
```@MFAnnie “Ms. Fetter, I don’t know how to start.” what do you say? Nothing? At a certain point you gotta offer something. #probchat

— Kent Haines (@MrAKHaines) May 18, 2015

I didn’t mean to imply that you should offer nothing to students who aren’t sure how to get started. I just don’t think you’re doing them any good by making any decisions or doing any math for them. Sure, they might be sitting there quietly with no intention of doing math, because they know it will all be over soon and they can go to lunch. But it also might be that they honestly don’t know where to start. You know why? *<Warning: Gross generalization ahead.>* We probably haven’t taught them strategies for getting started with a problem when they don’t quickly “see” how to solve it. They might not know that any of their ideas are good. Heck, they might not even know they have ideas, or that making sense of the story would be useful. So I like to ask them about their ideas. It’s the rare student who doesn’t have any.

So, here’s an example of something I think is almost always a bad hint: “What’s the problem asking you to do?” or “What’s the question?”

The student might not even know what the story is about, because they might not have ways to engage in it and make meaning out of the story, or they might not even know that’s something worth doing. Or they haven’t even read it yet. But you’re already asking them to look at The Question and then find The Answer. Stop! Back up the truck! Unless you know they understand the story, completely, any content-related hint you give them may well be useless or, worse, confusing. In particular, a hint that focuses on The Question just perpetuates the point of math as Answer Getting, as opposed to understanding and making meaning.

Years ago, one of my colleagues, Steve Weimar (@sweimar), was working with some eighth grade students in a “low-performing” school. The students had learned the Noticing and Wondering strategy and, when Steve was visiting, were Noticing and Wondering the heck out of a math situation. Steve was watching and eventually said, “Wow! It looks like you have a lot of great ideas about this problem! How about you start thinking about the question?”

The students looked at him and said, “Whoa! Slow down. We are *not* done noticing and wondering.”

Those kids had figured out that if they did a really good job of Noticing and Wondering, they could tackle any question that came up about the situation. And they had some strong ideas about how to know when they were done Noticing and Wondering. Until then, don’t make them move on, because they aren’t ready!

Imagine how different their experience would have been had the teacher jumped in early in the process and asked, “What’s the problem asking?” I’m guessing that a lot less math might have happened in that room, and those kids wouldn’t have nearly as strong a sense of themselves as people who do math.

]]>(Did you really go play? Honest? Because if you didn’t, the latter part of this post won’t be as much fun to read.)

Last month we were in Boston for the NCSM and NCTM yearly meetings, and as has recently happened at large math ed events, I was occasionally hailed with some version of, “We just used your video in our talk!” or even “OMG! We use your video in ALL of our PD! You’re famous in [insert state, county, or district here]! Can we take our picture with you??” Invariably, they’re referring to the very first Ignite talk I gave, which was at NCTM in Indianapolis in 2011 (though it wasn’t technically part of NCTM, since the session didn’t get accepted, so we did it in a bar). If you haven’t seen it, or haven’t watched it lately, I encourage you to check it out.

Many groups are using this video as a launch for professional development because it can start conversations about moving beyond answer-getting and instead valuing as many of students’ mathematical ideas as possible. As of this writing, the video has been viewed over 15,800 times. That’s really exciting! And I certainly don’t mind being stalked at math ed conferences.

This past year I wrote math curriculum, mentored college students doing academic tutoring, and did some tutoring myself for a group of disadvantaged high school sophomores participating in Project Blueprints, an after school youth empowerment program hosted by Swarthmore College. One thing we focused on early in the year was developing and emphasizing mathematical habits of mind and working towards getting the students to believe that they have mathematical ideas and that those ideas are important. We did a lot of Noticing and Wondering! In fact, one of the first activities we did when they got the new iPads was to play Game About Squares.

Now, these are kids who are taking high school geometry and are about to take the state’s Algebra exam for a second time (their district doesn’t have a very high success rate – one kid claimed that nobody from their district has ever passed). Isn’t this supposed to be math support? Do they really need to play a game?

Well, yes. Students opened the game and were confronted (as were you, if you followed my directions to play before reading) with this:

“What are we supposed to do?”

“How does it work?”

“Where are the directions?”

“Uh….”

Those were a few of the comments I heard from the two pairs of students I was working with that day (and the one other adult in the room, who had pulled out her phone to try playing). I just said, “Figure it out.”

Not surprisingly, they did. They noticed, they wondered, they tried things, they guessed and checked, they made mistakes, they groaned, they backtracked, they started over, they laughed, they talked to each other a *lot*, they persevered, and they were excited by and proud of their progress. What teacher wouldn’t want those things to happen in their math classroom on a regular basis?

An especially fun moment happened when Ashley, one of the coordinators of the program, came into our room. She asked what they were doing and one of the students reset the game to Level 0, handed the iPad to her, and said, “Here.”

She asked, “What am I supposed to do?”, and the students just grinned and wouldn’t say a word. I gave her a “don’t look at me!” shrug. They watched Ashley’s finger hover over the screen to see what she would click on. They snuck glances at her face to see if they could tell how she was feeling. They grinned some more. They elbowed each other gently when she made the same mistakes they had made. They watched her slowly figure out how the game worked. It was almost magical to observe them watching an adult go through the same learning and figuring out process that they had just gone through. They seemed almost entranced!

Then we talked about the game for a bit, and discussed the “habits of mind” they had employed to figure out the game – noticing and wondering, guessing and checking, persevering, struggling productively, learning from mistakes without worrying about making mistakes (since they knew the only way they were going to make progress was to make mistakes and learn from them), and working together. We talked about how these skills are as important as any content they learn in their school classes, and how they can use those skills to make progress on math problems they’re not sure how to solve. In fact, much of the math programming we did the rest of the year employed huge doses of Noticing and Wondering and generating ideas about math situations, or scenarios (a math problem with no stated question). Anecdotal reports suggest that by the end of the year, most of the students felt pretty confident that they could generate ideas about most math situations we handed them. Big win!

These days we talk a lot about the importance of implementing and practicing the Standards of Mathematical Practice in classrooms. Sometimes it’s hard to make that practice explicit, but students do need to know when they’re developing and using (and getting better at) those habits. One way to do this is to do activities, such as Game About Squares, where there isn’t any real math “content”, but there is a lot to mess around with and figure out and enough support that students can do that without a lot of guidance from any adults.

I’d love to hear about your favorite such activities, and what sorts of subsequent conversations you have with your students about habits of mind.

Now go play Game About Squares some more. After a hiatus, I’m currently working on Level 19, so I’ve got a lot of things to figure out!

]]>May the 4th was not only Star Wars Day but was also my mother’s birthday. She died in February 2014. I decided to celebrate by taking a vacation day and digging in the garden, which was one of my mother’s favorite things to do (I’m pretty good in the garden, but I swear she could weed at least four times faster than me, and yes, I totally took advantage of that by enlisting her help more than once!). I did some weeding, including the final cleaning out of this one area of the yard into which we are transplanting a bunch of irises and day lilies that our neighbor was removing from his front yard (our new neighbors are not plant fans, apparently – they have also cut down a dogwood and a Japanese maple).

When loosening the soil in the bed, I hit a rock, and decided that I’d dig it up (another thing my tenacious and hard-working mother would have done). It turned out to be two large rocks – the one in the picture that includes my foot for scale (Estimation180 anyone?) and the one leaning against the fence in the other picture (which includes a pint glass for scale).

But enough about my day off. This is a math blog, after all, so I really wanted to talk about how my mother not only persevered when faced with a giant rock in her flower bed, but also when she was designing some of her fabulous textiles. Let’s start by watching the Ignite that I did about her at NCSM last spring, shortly after she died.

She’s pretty talented, huh? For fun, and to further honor her talents on her birthday, I decided to try to reproduce that Celtic design that she found in a book. As you may recall from the video, here’s the picture that she had to work with:

I stared at it a bit and thought about what I would need to pay attention to if I was going to reproduce it using Geometer’s Sketchpad (that is another way of saying that I Noticed and Wondered). Yes, Mom worked on paper, and didn’t have any problem redoing things as often as necessary, but I believe in the power and speed of something like a dynamic geometry environment so that the tweaking goes a lot faster once you’ve set up the initial relationships!

As shown below, I took note of several things. First, the whole thing is a circle. Second, there are 12 outer points (marked in red). There are also 7 concentric circles underlying the design (marked in blue). I noticed that I would need to construct 24 radii of the outer circle, and create points of intersection where those radii crossed the concentric circles. I also noticed that one “path” through the design consisted of “diagonals” of the spaces created by these radii and circles (marked in green).

In Picture 1 below, I’ve set up the initial relationships noted above. In Picture 2 I’ve rotated that one path 11 times by 30 degrees, resulting in the beginning appearance of those 12 outer points. In Picture 3, I’ve added the “paths” going the other way. (Yes, I rotated – leveraging symmetry and using transformations in Sketchpad. In fact, I made custom transformations that rotated by 15 and 30 degrees that I could apply to any object I constructed to save a lot of time.)

It’s really starting to look like something! I can use the points along the thick radius to change the size of the circles and, consequently, the shape of the paths. One of my mom’s first sketches was a bit too “pointy”, and I’ve replicated that by making every circle but the outer one a lot smaller. (You may be able to see a couple of the circles that she drew and then erased.)

I was able to drag the points that control my circles until I got it just the way I wanted it. I didn’t see any more trial sketches in my mother’s files, but I do know that she definitely nailed it in the end!

Consider all the sense-making she did. She couldn’t just measure the picture, since it was drawn in perspective, but she took away as much information as she could. She noticed relationships and used trial and error to figure out the parts she couldn’t count. She made mistakes and learned from them. And there is no question that she persevered!

As I said in my Ignite talk, we need to be sure to look for and value these traits in our mathematicians, not just their ability to crank out answers to a lot of textbook problems really quickly. Look for opportunities for your students to practice sense making, maybe even by having them replicate some drawings!

I’ll close with one more picture of my mom’s artwork taken in 2008. She made this quilt of the Math Forum’s dragon fractal logo for us to hang in the office. Did she know what a dragon fractal is? Nope. But she had the ability to pay attention to detail enough to get it right nonetheless. Also in the picture is my husband Riz (the tall one – Estimation180 clue is that I’m 5’10″), my sister Marty, her husband Silas, and their adorable children Olivia (7), Liam (4), and Clare (8 months).

My mother inspired me in many ways as an artist and as a mathematician. We should all try to do the same for the young people in our lives.

]]>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.

PCTM 63rd Annual Conference, November 2014

Session Handout [PDF]

For more about the Math Forum, including information from our other talks at PCTM 2014, visit our conference page at http://mathforum.org/workshops/pctm2014/.

**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

*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

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

*Key technology strengths: the variables actually vary!*

Annie Fetter, The Math Forum @ Drexel University, Philadelphia, PA

Session 12 Handout [pdf]

**How This All Started – Formative Assessment
**

- Constructing Quadrilaterals Activity (see handout)
- Making Movies Using Jing – Instructions [PDF]
- Kathy’s Movie
- Danielle’s Movie

**Where it Went from There – Summative Assessment
**

- Red Shirt’s Movie
- Christina’s Rubric (see handout)

**Then Debbie Got Involved**

Annie is the Problem of the Week Coordinator and Professional Development Specialist at the Math Forum. She also teaches three courses in Drexel’s Math Learning and Teaching Masters program,