Student to Student Discourse: NWMC Workshop Part 2

Today I am back with the second half(ish) of my presentation at the Northwest math conference. To get a feel for the flow or to get caught up:  Part 1 of my presentation can be found here.

After the discussion on the importance of classroom culture, we jumped right into trying out a few of the activities/strategies. The first was Which One Doesn’t Belong. I have used this activity has a whole class period learning opportunity, but for this presentation I wanted to highlight it was a way to jump start math talk and group students with peers who might think about things a bit differently than they do.

Increasing Student to Student Discourse (3)

I put up the previous picture and asked the attendees to move to the corner of the room representing their first impression of which picture was the odd one out. Once there, they were to discuss in their corner the reason they choose to see if they were all the same and then each corner had a chance to share out a reason (or a few) that their picture didn’t belong. Then the attendees regrouped into table groups that had at least one member from each corner represented. Teacher Note: Most of the time I’d do this activity with a more math-y example, especially one using similar math to the activity at hand. This way, the grouping by unlike thinking is even more powerful for having different views represented.  I chose a non-math example for the presentation since I had no idea of the background of attendees coming in and I wanted to showcase the breadth of examples on the WODB site.  This grouping strategy can be done relatively quickly (in place of a warm up, perhaps) and gets kids brains and bodies moving and in math discourse mode right away.  You can find more examples of WODB on this site.

After everyone was in their new groups, we jumped into math debates. The following slide gave a jumping off place with a few examples and tips.  We had a quick debate on the prompt “Which form of quadratic functions is most useful?” And then in their groups, they spent some time creating prompts or questions which could spark a debate in the classroom. I wanted to highlight a few key ideas:

1. Debates don’t have to be long, complicated or time consuming. We often have 2 minute warm up debates (like the quadratic prompt) as a way start talking and review important features.

2. Kids don’t have to be expected to stake a claim right away. Like the prompt at the bottom right of the slide below. I sometimes assign teams/sides and make them come up with reasons. As they become more comfortable, then I give them a prompt, let them pick their own side, listen to the ‘arguments’ and adjust sides if they were swayed.

3. Almost anything can be debate-able.

Increasing Student to Student Discourse (4)

Some of the best math debate resources, including the TeamA/TeamB prompt above came from Chris Luzinak and Matt Baker (@pispeak and blog & @stoodle and blog respectively on twitter). I’ve heard them speak live once and have done the twitter thing to find other cool debate resources from them both.  Teacher Note: Debates seem scarier to a lot of people than other forms of math talk, but we all have lots of practice arguing so its natural for a lot of students. I make sure I start with prompts that have no right side and often preface the first debates with the idea there is no right answer. Sometimes non-math prompts are good for setting up the structure you want to use (eg: What is the best superpower?). Also… talking about structure you can be rigid: “My claim is….my evidence is…” one at a time or for free flowing letting students jump in respectfully** as they wish depending on class culture and how much experience they have. I often start with more structure and relax it as the year goes on.  I don’t cold call as I have seen problem with the stress/trauma of that approach, but I haven’t had a problem with students unwilling to add their voice either directly or by discussing with a peer/group and having someone else share out.  

Next we moved into Stand and Talks which are very well explained (and originated?) by Sara Van Der Werf. Her post can be found here. It is basically a play on the turn and talk, but instead of sharing with the person next to you, students physically stand up and move somewhere else in the room to discuss with a peer. I had everyone do this (we ended up in small groups do to space constraints since they needed somewhere to write for the prompt).

Increasing Student to Student Discourse (5)

I then put up the following slide (again, after the movement). The structure is Conjectures and Counterexamples which I learned about from Dan Finkel (@mathforlove on twitter and blog) at a math teachers circle.  Basically, I put up a conjecture and show my examples and the goal is to come up with counterexamples to disprove. I combined this is Sara’s stand and talks as they work really well together. Teacher Note: One of Sara’s tip is to ask for a high number (I used 10) so that students are less likely to be “done” and disengage while others are still working.  This doesn’t mean they actually need 10 to continue on to the next part of the activity.

Because this S&T was leading into a bigger activity, I had them move from chatting to grouping at a table and drawing out the counterexamples.

Increasing Student to Student Discourse (6)

The next part of Conjectures and Counterexamples is lots of fun. Groups were tasked with trying to change the conjecture to make it true (unbreakable!) using the counterexamples as ideas for where the original conjecture ran into problems.

Increasing Student to Student Discourse (7)

Once the groups had some time to write a conjecture, they used a gallery walk to look at other groups new conjectures and tried to break them if they could. Most were re-broken, so groups had a chance to make their conjecture even better. The idea of drafts is important. Things don’t have to be perfect right away, and looking for counterexamples can help you find holes. (Some example work from my math teacher circle can be found at the bottom of the post here.)

All too quickly, time was running out, so I switched to the last slide to leave everyone with two important thoughts. 1. Our students are kids (even the big high school seniors). We have to remember that in everything we do. This means meany things: they like to have fun/play….but also importantly they still are developing their brains so we need to be okay with them making mistakes and Acting like KIDS!! Acting up doesn’t make them bad, it makes them human. I do dumb things/say things I don’t mean and I’ve had a lot more time to mature. 2. Our kids are Brave and Awesome. When we try new things, it can be scary. Asking a teacher to introduce new ideas/structures/activities is hard and we might screw up….BUT it is also hard for our kids. WE are asking THEM to take the leap of faith and try something new/scary/exciting and more often than not they step up to the challenge and do awesome things with our crazy ideas and they are usually super okay when our idea flops as long as you model how to take responsibility and come back with another go at it.

Increasing Student to Student Discourse (8)

And as a very last note (if anyone is still reading), one of the best ways to build discourse and culture in a classroom is through circles. I didn’t end up modeling this at the conference because we had too many people (an awesome thing!!) and not enough time, but I kept the slide in case anyone had an questions. We use them at my school to build community, as a piece of restorative justice, and I use them as a way to talk about math sometimes. (Here is a link to some information on circles and I’m happy to chat/answer questions via comments below).

Increasing Student to Student Discourse (9)



Student to Student Discourse: NWMC Workshop Part 1

Back in October I gave my first ever presentation/workshop at a big conference. It was actually my first time attending a big conference as well. I’m not sure what I expected but 1500+ people is busy! My topic was on increasing discourse between students. It was a workshop format which I took to mean as interactive as possible. The room was packed, so movement was difficult, but all the attendees were super gracious about the space constraints and were awesome participants.  I’m going to give a run down on how it went and access to resources I mentioned. (This post will be the first few slides).

Opener: Increasing Student to Student Discourse

I went around and shook hands and introduced myself to as many people as possible as they were coming in and getting settled. I chatted about their conference so far and what they were hoping to get out of the workshop. I introduced people to someone at their table as they came in when possible. Throughout the workshop I bounced between “teacher” role and “explainer of teacher moves” often. I’ll try to capture that here in regular typeface and bolded side notes.  ((Teaching Note: I try to greet my students at the door or as they get settled every day. I also try to ask group-like questions to get them talking to each other if they aren’t naturally inclined to do so. Non-math talk is still a positive to build relationships with the teacher and among peers. Also, if they have a chance to chat before class starts, they are more inclined to get to math talk sooner once we get into it).  Continue reading “Student to Student Discourse: NWMC Workshop Part 1”

Students Play with Dots: Math Circle Followup

I talked about my experience in a teacher math circle in a recent post and spent some of that time describing how we played with the initial prompt: How many squares are in a 4×4 dot pattern. What are the areas? You can read that post to get a run down on what I did and where the group went with it.  I wanted to present the same task to my students and see what they did. 

I had a plan for Friday morning which had students revisit some work from the day before and provide justifications to each other on the rules they had written for a series of pattern questions. I knew they all had had a rough go at it the day before and I wanted a bit more time to look into original responses and plan a better way to debrief and move forward so at the last minute I decided it was the perfect time to pitch the dot pattern prompt from the Math Circle. It was Friday. They could play and hopefully I’d inspired at least one to go home and think over the weekend. Here is what went down:

I projected the 4 by 4 dot pattern on the board and and asked students how many squares they could find and what the area of those squares were. They worked alone or in pairs with either graph paper on the table white boards.  Right away students started to dig in and after a bit of chatter, 14 became the it number to arrive at (with areas 1, 4, and 9 units squared). High fives all around, we were done!

Except… one student chimed in: “Mmmm… do diamonds count?”

Me: What do you mean? What is a diamond?

Std: This! (holds up some sticky notes at an angle)

All Stds: Shoot!

Me: ….?

Stds: Looks like we have more work to do.

They all quickly jumped back in to find more squares. I heard a few comments like ‘My brain hurts!” and “There is so much going on here!” but I wandered around a took note of different approaches. By this point, they’d self grouped into larger teams of about 3-4 students around a central paper or white board. (This seemed to fall in line with my experience. I played on my own until it looked more exciting to join up with others).

Part way through the group work, I asked by students to call out sizes, number of squares, and areas that they found to begin tracking on the class board. Then I asked students to come up and add in squares they had more trouble describing. As can be seen in the pictures, originally a group gave me a 4×4. I left it up until the group itself had trouble drawing it and then realized it was actually a 3×3.

Areas and Size of the diamond squared proved tricky for some students. I didn’t give away Pythagorean theorem (Thanks, Dan!) And instead encouraged them to visually determine the area. The 4 1/2 squares came out and they found an area of 2. I then asked them to tell me what that meant for a side length (√2) . While I was having that conversation with one group, another leaned over and said they’d used Pythagorean’s Theorem and asked if that was okay. I had them go up to the board and explain why it worked and then more groups began using it, especially for the 5 u² squares.

The students were excited. They’d done some great math, worked really well together. I didn’t have any cell phones or bathroom break requests. We could have stopped here. But, we end the fun. I clicked a button, and suddenly by 4 by 4 projected dot pattern was now a 5 by 5.

Me: “Hmm, maybe it was supposed to be a 5 by 5. Does that change your answer?”

Immediately students began debating  on how to count new squares.


I let them ride out the period in small groups exploring the larger pattern and them told them we’d revisit the question on Monday.

Monday: I had collected all the student work and made tables on the board with Size, #, Area for the 4×4 and 5×5 that had been found by students and left a blank 3×3 table and grid as well. Students came in and as they did, I asked them to draw up an example of one of the types of squares until the board was full and the 3×3 was completed as well.

I then asked the to ‘Notice and Wonder.’ We were really wrapped up in some great ideas, so I only have the aftermath board photo.  The two notices that seemed to peek the most interest are in the upper right hand corner of the photo (“Nice” was defined as teh non-tilted squares). But you can see lots more of the student thinking all over. A favorite of mine was the list of areas with the 13 * . I asked what the asterisk meant and a student added the ideas in the squiggly Areas Possible” section and said: “I predict that a 13 u² should happen, but we didn’t find it in the 5×5. I think we’d need a bigger grid, but that we’d still find it.” I told them a bit about Math Teacher Circle and how we all played with this task and found avenues to explore as well.


Those three ideas became the basis on my next move. We did have to get back into the regularly scheduled unit, so I told them they could choose one of those 3 questions (or anything else they noticed/wondered) and do some more exploring on their own and I’d give them some points on the MET. We have a Math Exploratory Tracker which means I ask students to do mathy things outside of the normal math class. Students keep track of how often they do mathy things. Brain teasers, puzzles, blocks, reading about math people….anything that sparks their interests. Points are just colored boxes. But they like to see their own colored box lines grow.  (The idea is similar to the reading log they keep for ELA class). I’ll update when I start getting their ideas in.

At first I wished I’d let them play more in class. We probably could have waited one more day to get back to our unit. But, on the other hand, we’d never reach “the end” since there isn’t really one, so I like the idea that they have something that they are invested in that still has room to explore. As students, they haven’t been asked to “explore” or “play” with math that often, so guiding them through the process a bit worked well to give them the tools to find something on their own and since we spent time and the questions were their own, hopefully they are more excited to continue to explore outside.

Teach 180 Round Up Part 2

Teach180 is a movement to document one picture (or maybe even a video or a thought) from each day of the school year. I am tweeting daily and then posting here every 2 weeks or so. Most provided without much comment, but happy to explain if anything catches your eye. Join use using the #teach180 on twitter!

The last two weeks have been busy and full of some great math learning:





Random count for Me:   4 Bridge to College, 2 Alg B, 2 Geo B, 2 Advisory

Teacher Math Circles: Take 1

Last night I joined my first ever math teacher circle! The awesome Math for Love founder (Dan Finkel) and the director of the Washington Experimental Mathematics Laboratory (Jayadev Athreya) teamed up to run circles for elementary and middle school math teachers in the Seattle area. I don’t fit into either group, but Dan let me crash the middle school group anyway.

My first big takeaway: I don’t do enough math. Between teaching, being a parent/spouse, finishing grad school (done as of last Jan!!!) and other random life things I let the practice of just playing and thinking with and about math slip. Even if I didn’t bring anything back to use in class (which I will) it is worth it just to spend 2 hours doing math again with other people who were also excited to just play with math. Continue reading “Teacher Math Circles: Take 1”

Teach180 Weekly Roundup

We have reached the end of the first week and a half. Its been a great way to make sure I am at least semi-active posting on twitter, not just lurking/finding great ideas but sharing out and connecting with other math teachers.

Also, I’ve noticed interesting things already, so I’m excited to see where the year long trends go. I get really busy and often forget to take pictures, so usually I only have one at post time. I’d love to actively document more, but it also brings up interesting things to watch/track.

Continue reading “Teach180 Weekly Roundup”