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```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:

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```@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.

]]>The steel girders being used to construct a building are 27 feet long. One of the corridors the girders are to be carried down has a 90° turn at the end. The width of the hall before the turn is 8 feet.

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*Ricky ate 2 pieces of a pizza. He gave the rest to his friends. How much of the pizza did he give to his friends? Can you write that in lowest terms?*

The students were seated at tables, some working in pairs, some working alone. I asked one of the students if I could sit next to her. She said, “yes” and after I had introduced myself, she told me her name was Abrianna. I noticed Abrianna had already been thinking about the problem and had written something like this on her paper:

I remember thinking that from looking at her paper, it looked like she really understands what’s going on in the problem! I asked her to talk to me about what she was thinking as she wrote (and I pointed to the paper). She said that since Ricky had eaten 2 out of the 8 slices of pizza that he had eaten two-eighths of the pizza. I asked, how did you know the whole pizza had 8 slices? She patiently pointed to the picture included with the problem and said, “See the lines?” “Oh,” I said, “I see them now!” and smiled.

She continued to tell me more about what she had written but when we got to the “equals” sign she said “No, two-eighths is NOT equal to one-fourth.” I was surprised because that seemed to have been what she had written. I started wondering if “equals” meant something different to her than using the “equals sign” and so I asked her, “What are two things that ARE equal?”

She responded by saying, “Two equals two.” When I asked for another example, she said “Five equals five.”

I pointed to the two pizza pictures and I said, “What if you had eaten the pizza that we can see is gone from this first picture and I had eaten the pizza gone from this second picture, who ate more pizza?” She said, “I ate more because I ate 2 slices and you ate 1 slice.”

I tried again and said, “If you ate 6 slices from a pizza cut into 8 equal pieces and I ate 3 slices from a pizza cut into 4 equal pieces, who ate more pizza? She said, “I ate more because I ate 6 and you only ate 3.”

My visiting time in that classroom was over and I moved to observe another classroom but my conversation with Abrianna stuck with me throughout the day and as I talked about it with my colleagues as we walked back, I described the conversation by saying,

I noticed

- Abrianna’s paper included the correct notation.
- A teacher looking at her paper might assume she understood the problem.
- Her teacher would probably give her credit (or a grade) indicating she had the right answer.
- Abrianna used “=” on her paper.

I wondered

- why “=” and “equals” prompt such different responses from Abrianna.
- why does “equals” mean “is identical to” for her.
- why doesn’t “=” mean “is identical to” for her.
- did Abrianna just copy the notation as a way to “do the problem.”

I was reminded again of how important it is to unsilence students’ voices. I hope Abrianna has continued opportunities to talk about her mathematical thoughts!

]]>(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!

]]>A plane traveling 400 mph is rising at an angle of 30 degrees. A second plane, traveling 300 mph, is rising at an angle of 40 degrees.

]]>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.

]]>Today I started to plan my garden. The rectangular garden where I am going to plant is three feet longer than it is wide.

One third of the garden area will be sunflowers.

Another quarter will be planted with brow-eyed susans.

Only a fifth of my garden will contain columbine.

A sixth of my garden will be filled with trillium.

The rest will be foxgloves.

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