Historically, the teaching of mathematics is particularly guilty of giving students the message that their ideas aren’t important and that they only need to memorize and regurgitate the ideas of others. If students are to become critical thinkers, they need to realize that the ideas they bring to the classroom are worthwhile. They should also have opportunities to express and refine those ideas, as well as to learn to listen critically to the ideas of others. We’ll talk about some examples of ways in which we as mathematics teachers ignore students’ ideas, then explore different routines and strategies for making students’ ideas play a more central role in the classroom.
Sense Making: Is It at the Core of Your Classroom?, Grades 6-8, Monday, 3-5
The National Research Council points to a “productive disposition” as one of the key strands of mathematical proficiency. A major part of this strand is viewing mathematics as something that makes sense. Are your students making sense of the mathematics they explore? Do they feel that mathematics is an inherently sensible endeavor? We’ll look at ways in which students don’t make sense of mathematics, consider why, and discuss strategies for making it a larger part of the expectations in your classroom.
This session is the long version of an Ignite I did at NCTM Boston in 2015. You might use this 5-minute version to introduce colleagues to some of these ideas and start some conversation.
Joe Schwartz’s Blog – Joe is a recently-retired elementary math specialist in central New Jersey. He blogged about using Noticing and Wondering at his school, among other things, and I’ve pointed you to a N&W Sampler that he wrote in January 2015.
Numberless Word Problems – Brian Bushart has done a great job blogging about and preparing numberless word problems, including the mouse problem we looked at during the talk.
Beth Brandenburg is a lead teacher in Washington County, Maryland, and makes a lot of use of Noticing and Wondering in her school (and her district – she used to be a district-level lead teacher). I’ve pointed you to a post she wrote in August 2015.
It was a stroke of genius on my part to invite Joe Schwartz (@JSchwartz10a) to bring his perspective as a school-based math coach to my sense-making session at NCSM. He offered some ways in which he encouraged the greater use of sense-making tweaks in math instruction in his school and his district. You can get a copy of the slides and related links at his blog.
If you’re going to Twitter Math Camp 2018, look for a chance to think about this with Joe and me.
Time for a guest blog post! Emily Payán is a beginning teacher at a high needs elementary school in a large suburban school district north of Minneapolis. Her mother, Margaret Williams, is the district’s Teaching and Learning Specialist for K-2 Math. Margaret is also an adjunct instructor for a local university and happened to be Emily’s Math Methods instructor. I know Margaret because she is part of the Minnesota “math family” that I’ve gotten to know while serving on the leadership team for Math On-a-Stick at the Minnesota State Fair. As a prelude to inviting me to lead a day of professional development for all the K-5 teachers in her district this August, she buttered me up by telling me this great story about the huge effect my advice from the previous August had had on her daughter’s experience teaching kindergarten. I replied, “Sure, I’ll do the PD, but you and Emily have to write up your story and I’ll post it on my blog.” All parties kept up their end of the bargain, and here’s what they wrote.
Margaret: I was looking forward to hanging out with Annie while volunteering at Math on a Stick during the 2016 MN State Fair. My daughter, Emily, had just finished her first year of teaching. Her second graders made wonderful progress in math but, as is often the way for beginning teachers, she was being moved to kindergarten. Being able to talk with Annie and get suggestions regarding the K math experience was going to be awesome!
Emily: To say I was nervous would be an understatement. I had completed my student teaching in a kindergarten classroom in a neighboring district. In this classroom, worksheets were the modus operandi and manipulatives were frowned upon. Needless to say, I had no idea where to start or what skills to engender in my new students to set them up for success in their schooling.
Margaret: Right! It was frustrating for me, as her mentor, to see and hear how the student teaching experience had been. Once she had her own classroom, we worked together a lot in planning math lessons for second grade using a Cognitively Guided Instruction framework for implementing Everyday Math. What would that look like in kindergarten? I was excited to talk to Annie. I was pretty sure I knew what she would say: The children should have lots and lots of opportunities to count collections. I was wrong. When Annie and I had a chance to talk, she told me that the most important thing Emily could do, as a K teacher, was to hold her students accountable for making sense.
Emily: While this made sense for a mathematician of any age, I had to wonder How on earth do I start when only three of these students have had a preschool experience? Less than half of my class exhibited predictors that would indicate foundational skills and concepts for number sense and future success in mathematics. My mom’s Math Methods course included lots of conversations around Boaler and Dweck’s work regarding mindsets and mathematics. I knew that if I could just get these kids to engage in real world math without the fear of mistakes, they would be successful and confident – regardless of what their initial assessments indicated.
My mom helped me collect items such as buttons, pine cones and blocks as well as pictures of real world things that would be interesting to five year olds: puppies, turkeys, plants (after all, math is EVERYWHERE!). I also used resources from Christopher Danielson. His book, Which One Doesn’t Belong and his open-ended approach to counting in picture books inspired my students to value multiple perspectives and to think of math as a creative domain. Trying to pull all this together felt overwhelming. Looking back, I think this was good. It wasn’t long before these “struggling” kindergarteners were mathematizing the world around them and, just as importantly, talking about their thinking. They were all in.
Conversations started flowing as they worked through their reasoning. I challenged them with simple questions like, How do you know?, Do you agree with what Ylva said? Giving an answer wasn’t enough, they had to be able to talk the class through their thinking. Sometimes the justification didn’t make sense to someone and they began to (respectfully) challenge one another and ask questions of their classmates. It was fantastic! I had eighteen 5 and 6-year-olds engaging in deep, meaningful questions about all types of mathematical tasks: addition and subtraction, multiplication and division.
This means that mistakes were inevitable. I intentionally modeled making mistakes and talked about Boaler and Dweck’s research on the brain growth that happens when a mistake is made and thought through. We watched videos on synapses and even had class meetings in which we all shared something from the day that was challenging. We congratulated each other on making our brains stronger by ‘sticking with it’ and not giving up. By the middle of the year, excitement beamed across their faces when a mistake was made. They knew that this meant they were on to something, and they accepted the challenge.
For example, my mom (known in my classroom as “Nana Math”) had taken a picture of a group of turkeys and sent it to my class to think about what was mathematical. Autumn decided she would count the legs of the birds – three turkeys were standing and two were sitting. Counting the legs she saw, she decided there were five legs. Dakota studied the photo a little more and thoughtfully explained, “I think there are more legs, Autumn. I think there are legs we can’t see. Each turkey should have 2 legs, right?”
Margaret: These little problem solvers were on fire. It was amazing to listen to them explain their thinking. I would go into Emily’s classroom to help with assessment interviews and wonder when was I going to get to the low achieving students. These kids were confident and supportive of each other even though they often challenged each other’s ideas. It was a math teacher’s paradise! Emily will be teaching Grade 2 this coming year. We look forward to another year of focusing on reasoning and making sense!
I can’t wait to hear about Emily’s second grade experience this year when I see Margaret again next summer in Minnesota. Maybe she’ll send me some updates before then. And by the way, y’all should come join us at Math On-a-Stick!
Last Thursday started out like so many have in the last 20+ years. I logged into our Problems of the Week site, preparing to read through the student submissions to one of our Current PoWs so that I could choose some to include with the commentary I write. Last week I was doing the Geometry Problem of the Week. And as has happened on almost every “Geometry Thursday” in the last 20+ years, I saw this name in the group of people who were the first people to look at the problem when it was available on our site:
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 met 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. His 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!
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.
Before you read anything else, go play Game About Squares. Seriously. Don’t come back here until you’ve gotten to Level 8. But do come back – don’t let it suck you in permanently!
(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 high school sophomores from under-resourced schools 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?”
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!
[originally posted to my now-defunct blog at The Math Forum]