Walk the Line

I keep a close watch on this number line 

I keep my eyes wide open for the sign 

Making sense, connections all the time 

For number sense, I walk the line

Apologies to Johnny Cash.

At Math In Action two weeks ago, I presented on number talks in the middle grades. (Here's the handout/resources, including a link to the slides.) I was quoting Pam Harris, then the keynote speaker was quoting Pam, so I made a comment on Twitter, badda bing, she turned out to be coming to my neighborhood the next week. Muskegon Regional Math Science Center let me crash, thanks Kristin Frang, so I got to crash for 1.5/2 days. 

Pam was presenting on secondary number strings (thread 1 and thread 2 for my notes). Mini-review: I wouldn't hesitate to bring Pam in to work with teachers. Great energy, hilarious, solid ideas presented in a way that invites teacher access, and great modeling of instruction. All building on a great central message about giving learners an opportunity to mathematize.

I'm teaching Introduction to Mathematics this semester, a gen ed math class with a lot of freedom. I'm using Anna Weltman's This Is Not a Math Book as a text, mostly as a resource for learners and introduction activities.  In my head it's about redeeming mathematics for this successful students who have (mostly) learned to dislike mathematics. While we're mostly doing math and art, emphasizing problem posing and solving. But we take some time to redeem arithmetic, and we need to do algebra before we do patterning.

So, Pam's problem strings (modified because I can't help), followed by some clothesline math... rare day when I didn't have a way to take pictures. Sorry! The theme of the lesson is what else do we know?, which is one of my main understandings of math.

I drew a line, put on a hashmark, and labeled above it x, and below it 3. What else do we know? Some discussion about putting on a scale, vs knowing left or right. Finally someone shares -x is -3, and says it would go somewhere to the left. The someone said that means we know zero, which I encouraged as a good mathematician question.

First I did Pam's coordinates problem string.
(-2,5) show in graph, table, function notation
(2,-3)
(3,-9)
(1, __) which first brought up approximating with the line, then the idea of slope.
( __, 0) which brought a lot of people to a halt as they tried to remember an algorithm, then a remembered algorithm for solving equations. I made a bit of a joke how I was not interested in a memorized method, sorry, but only making sense.

Next, x is -2. Where is zero, left or right? What is x? Some discussion but pretty quick.

x-4 is 6. Is x left or right? What is x? How do you know? Quick discussion.

x+4 is -6. Heated discussion. -10 and -2 mentioned frequently. More and less start getting used more than left an d right. Interesting symmetry comparison between the last and this.

Then I introduced the clothesline, strung across the front of the room, with just an x in the middle. What else do we know? I had cards to ask them about. 0, -x, x+3, 5. We know where zero is, then dissuaded. What if it's negative? We know where -x is, then dissuaded. Then x+3, that has to be to the right. We don't know how far, but definitely right. What else? x-3 we would know. Excellent! Finally I put 0 up (left of x-3 by less than the distance to x) and ask what x could be? 

Then I gave them cards. Make some cards, figure out the order you're going to reveal them and the questions you'll ask. Two really interesting situations came up. 

One group had a couple of variable expressions, and then x+200. Really nicely subverting the sense of scale, and a great numeracy discussion of what x could be then. They wanted to just toss off a big number, but other learners argued for more precision.

Another group introduced a new variable, g, and then blew our minds with g^2. Has to be to the right, because squares are bigger. But what if it's negative? It's still to the right? Always? Then a lot of discussion if placement of it was setting the scale. Not until 0 is placed. It was amazing.

So that's my story. Thanks to Pam Harris, Chris Shore and these great learners.

Let's Discuss Professional Development

One of my favorite math ed profs is Sam Otten at Missouri (and the Lois Knowles Faculty Fellow). His research is interesting and situated, he holds teachers in high regard and listens to their ideas, and he illuminates research through the Mathed podcast. He has definitely enriched my practice. In addition, he's just a lovely and creative guy, as well as a world class expert on the DC Comics film universe. Beyond that, he's a GVSU grad, so I knew him when.

He is a part of the team that produced some new professional development materials, and I had a few questions for him about it. Mathematics Discourse in Secondary Classrooms, MDISC, is based on research and developed with teachers in the field. I'm a big believer in the importance of discourse in learning, and know that secondary mathematics has been one the places where traditional teaching has included the least discourse. I also think people need support to make changes, so something like this project is needed.

What inspired these materials? Was it an idea you wanted to develop or a response to situations you saw in the classroom?
The MDISC materials came from a group of math education scholars at Michigan State University and the University of Delaware, led by Beth Herbel-Eisenmann, who were passionate about the role of discourse in math classrooms. We all believed that there was profound value in students discussing mathematical ideas and building meaning together as a community. So at its core, MDISC is a set of professional development materials that are intended to help teachers increase the quantity and quality of discourse in their classrooms.

As we set out to create these materials, we tried to draw on other work that already existed in the math ed literature. Some of that work was Beth's own research with Michelle Cirillo. They had worked for years with a group of secondary teachers, examining discourse patterns and power dynamics. We also drew on the work of Chapin, O'Connor, and Anderson, who wrote a great book called Classroom Discussions that focused on mathematical discourse at the elementary level. They had some really amazing results with respect to student achievement scores that stemmed from a new emphasis on discourse. With MDISC, we tried to take some of those ideas from the elementary level and reinterpret them in ways that made sense at the secondary level -- focusing on middle school and high school classes.

Overall, the MDISC PD materials equip secondary math teachers to think about discourse in productive ways and it also provides them with specific tools for changing the discourse in their classrooms so that it really empowers students. It helps move us beyond teaching-as-telling.


What are some of the different ways these materials might be used? 
The MDISC materials include a physical facilitator's guide and then digital versions of all the participant materials as well as sample videos. It could be used by a teacher leader, facilitating sessions with secondary math teachers, or by a PLC of teachers who want to work through it on their own. It could also serve as a textbook for a graduate-level course, so a teacher educator going through the activities with practicing teachers, for example in a Master's course or an Ed Specialist course. The materials are designed to be a year-long study, with connections to everyday classroom practice, but it's flexible -- so with some adjustments, it could also be used in one semester. Or people could select which components they want to focus on.

There's also an optional follow-up where teachers can be guided through some action research, if they want to continue making purposeful efforts toward shaping their classroom discourse. There are several different options, and the MDISC team is very open to communicating with people if they have questions about enactment. We've also enacted the materials many times in many different settings, so we have a lot of experiences to share. 

As you piloted these materials, what were some of the changes you saw in classroom discourse?
We have piloted the materials and had others pilot them in both Michigan and Delaware, with several different groups of teachers. They have been very well received thus far, with some teachers willingly joining in for a second and third year because once they start, they don't want to stop thinking about their classroom discourse. Some of the teachers have called it the most important learning experience in their teaching career, and this even came from a 30-year veteran.

The most visible changes have been the number of students talking in class. They open up more and share their ideas, and the great thing is that they're sharing mathematical ideas. I think this comes from MDISC's dual approach of not only providing insight into the nature of discourse but also providing specific moves for teachers to use. For example, MDISC develops six teacher discourse moves that include inviting student participation and also probing a student's thinking. These are concrete ways to get the discussion going and keep it directed toward important mathematics.

Another big change that is noticeable is that more wrong answers come to the surface -- it's not that MDISC leads to student confusion (just to be clear), it's that an increase in discourse helps more student ideas to come to the surface. And of course some of those ideas are incorrect or imprecise, and that can lead to good discussions and good learning opportunities for the group.


What’s one feature of these materials or an example experience that might help teachers understand how they will support their teaching?
One feature of the MDISC materials is that they are practice-based and case-based. So teachers will get to make constant connections to their own instructional practices and their own students. Those connections are built right into the materials. And there is also the chance to see and discuss detailed cases of other teachers. Rather than lots of little isolated examples, MDISC instead is built around larger cases of real teaching. So for example, when you're learning about the transition from small-group work to whole-class discussions, you can actually see a middle school teacher as she circulates among her students and selects certain ideas to be shared later, telling the students that she'd really like them to bring it up in front of the whole class. Then you can follow the case to see how it played out in the discussion.

Another important feature is that the MDISC materials integrate an emphasis on equity. Powerful discourse means that everyone has an opportunity to be heard and to learn from the conversations. So there is a lot of attention paid to how teachers can use a discourse-based approach to reach more students, including those with traditionally marginalized backgrounds.

What movie would you like to see DC make next?
Great question! When I'm not working in math ed or spending time with family, I love watching and analyzing DC superhero movies. I really loved Man of Steel and then I thought Batman v Superman took it up another notch, with great themes about immigrant experiences and the danger of overt masculinity having to face feelings of powerlessness. So although I'm excited about Aquaman and the Wonder Woman sequel, I would really like to see another Superman solo film make it onto the slate. And it would also be great if the Cyborg standalone would get the green light because I thought he was a really intriguing character in Justice League and I think his story could be used not only as a commentary about race in modern society but also about our increasing dependence on technology.


(Back to me) There's so much promising here. Use of real classroom discussions with connections to your own. The focus on equity. And the idea that in running it with teachers there's a measurable change in the number of kids participating in discussion, as well as the frequency and quality of discussions - that's a dream. I'd love a chance to work through this with teachers.

Find out more:

• Episode of Sam's podcast with Beth Herbel-Eisenmann and Michelle Cirillo. Also, that's Sam playing the piano.
• Curriculum available at Math Solutions

Funville Adventures

This has been a long time coming. Funville Adventures by A.O. (Sasha) Fradkin and A.B. Bishop is full of fun adventures.

Sasha is a Twitter acquaintance, an elementary math enrichment teacher with an amazing personal math journey, and I probably heard about the Kickstarter from there. I love to encourage these passion projects in general, but this book is especially delightful. (Sasha on Twitter & her blog.)

As a story, it may remind you in flavor of The Phantom Tollboth or Dragon Tales. Emmy and Leo are kids transported to an allegorical land, Funville. Kids in Funville each have a special ability. The story makes sense and is enjoyable without even knowing the math in a formal way, because the math is the idea behind the people they meet, but not how it's discussed.  These quirky characters are brought to life in quick vignettes and charming illustrations.

Part of the charm is that, since there are mathematical ideas behind the kids of Funville, the way they work and interact is surprising but logical.  Readers can predict what's going to happen or wonder what would happen.

“Yeah,” said Harvey as if it was the most natural thing in the world to have a power. “My power is to halve things in size.” 

“Halve?” inquired Emmy. “But he is no more than a tenth of what he was!” 

“That’s because ...

What do you think happened? Small mysteries like that. Big mysteries, like how will Leo get back to full size? Harvey has a brother Doug - will that relate? And the biggest: how will they get home?

Want to know more? There's a whole Funville Blog Tour, with lots of perspectives. As with a couple of those, you might find yourself wanting to make your own Funville characters. Even now, whom do you think Emmy and Leo might meet?

I have one here... Fan Funville Fiction!

Dylan's Dangling 

Emmy and Leo worked hard all week to get done with school work and chores, so they would have an afternoon free to visit their friends in Funville. They had developed the habit of tucking things into an old rucksack that would be interesting to see just how their friends' powers would work on them. This time the rucksack held a tiny Ant Man action figure, an elephant toy, a stretchable rubber snake, and an assortment of snacks.

Emmy was particularly interested in Fay's and Randy's powers and how they interacted with other powers. So she was always glad to see them at the other end of the slide down the Thief. But she and Leo were both surprised to see someone new in the playground.

He was sitting on one half of a see-saw, but was up in the air instead of down on the ground. Maybe Heather had been here? He had on a shirt that was way too long, but otherwise seemed to fit him well. Adding to his stretched out appearance was a very tall cylinder of curly black hair.

Leo ran over to him immediately. "I'm Leo!" he announced, and the boy answered him. "Oh, I know. I came down here to meet you two because I was so interested in the stories that everyone told about you. People without powers, but you're fun anyway? And Pencilvania? Wherever that is!"

"Pennsylvania," Emmy corrected, "but that's a good synonym! You probably know I'm Emmy, but who are you?"

"Dylan," Dylan answered, "and I'm stuck up here. I was playing with- "

"Heather?" Emmy interrupted.

"Exactly!" said Dylan. "But her mother was calling, and she hopped off, and didn't notice that she made that end so heavy, and ... long story short, here I am."

"How can we help?" asked Leo.

"Do you have anything small enough to stand up under the other end?" wondered Dylan.

"Sure!" said Leo, and rummaged in the rucksack for a perfect skipping stone he was hoping Cory would be willing to use his power on. He slid it under the down side of the see saw. "Like this?"

"Mmmmmm hmmmmm," said Dylan, who was already concentrating. Slowly the see saw seat lifted up, pushed by the stone, which was growing. But not getting like a bigger stone - more like a tree. Once the see saw got level, Dylan hopped off.

"So your power is growing things?" asked Emmy.

"Not exactly..."