Paging Doctor Math: Cube Net Task and the Excitement of Substitute Teaching

I miss working with middle school students on an everyday basis. There, I said it. Don't get me wrong, I enjoy my new job as a middle level coordinator in higher education. I get to work closely with middle level preservice teachers and make an impact on their future teaching and the successes of all the students they will ever have in a classroom. I had a wonderful group of students for my first semester who I thought were genuinely curious to learn about what really goes on in a middle level classroom and how to best serve middle level students.

One of the preservice teachers asked me point-blank "if you liked teaching middle school students so much, why did you leave?" That was a great question, and of course the answer was to have a more widespread impact on middle level education and NOT because I was trying to get away from middle school students. I took that question as making sure I had a genuine connection to middle school students, which I appreciated. Just like middle school students, college students will be brutally honest...and that is why we get along well.

I still have a strong connection to my previous school district, and I am very grateful for that. I enjoyed working with the students and staff. I go back frequently for many reasons, with the most important being access to the classroom...making sure I don't lose touch with the reality that is middle school. I feel like this gives me a lot credibility with my preservice teachers, and I enjoy it as well. I also went back for a few open gyms over break...I am only 30 something, but I felt like I was near the end after that experience. I didn't know my feet could fit so many blisters.

Recently, I had a unique opportunity to actually be a substitute teacher in my old classroom. I didn't have class scheduled in the morning when I was needed to sub, so I jumped on the opportunity. The day I was there was the start of a new trimester, so the current teacher left it up to me what to plan for the day. As you can tell from my previous blog posts, I am into robust problem-solving tasks, so I thought I would go that route.

From my past experiences, my sixth grade students always seemed to lack in justification, collaboration, and problem-solving skills (not so much in math content knowledge). I don't know why that was (maybe the middle school transition or using a more traditional curriculum?), but when I gave them one of these tasks, you would have thought I gave them something in an unfamiliar language. For example, when I first get my 6th graders, I would ask them to justify an answer, and many would write, "I used a calculator" or "mental math." They would give up at the drop of a hat when something was difficult (which it always was). It generally took a month or so for most of the 6th graders to be comfortable with what justification and perseverance, but when they did, we were able to reach some truly amazing mathematical heights.

I decided to select a task dealing with nets and if they could be folded into cubes. The closest 6th grade standard to this would be 6.G.A.4, but that is probably a stretch. You would only focus on the solving problems with nets and ignoring the rest. I wasn't too worried about the standard anyway, because I was more concerned with the mathematical understanding and collaboration. This task is very accessible to many different grade levels and that helps to focus on the process standards without worrying about the content so much.

Here are the directions for the task (note: this task comes from @NCTMillum):

https://docs.google.com/document/d/15_KoeEJ9hkW3-fn9AhUm6Vgo7PYGeRe-3AQgGeWKUq4/edit?usp=sharing

Before we started, I put up a simplified version of my classroom expectations. My expectations are centered on students improving and not worry about their current standing compared to others...just getting better all the time. This is a strange concept for most students who I haven't had. I can keep making mistakes and correcting until I understand? You won't leave me behind? I still get full credit if I end up showing understanding even if it isn't on the first try? I had no idea another person with this philosophy existed until I ran into Rick Wormeli at AMLE.

They are always louder than a whisper, but if you are going for a 2 on a 1-10 scale with 10 being the loudest they could be, and they want a 10, I'll happily take a 4 so everyone can have a meaningful discussion and hear themselves think.

I generally let students work with others they were comfortable talking with so there aren't a lot of conversational barriers. I printed off two copies of this sheet off of the @NCTMillum website so students had plenty of room to write, draw, and explain.

Keep in mind that the order of the nets shifts around every time you start the online portion of the activity over, but it is easy to find which net the students are talking about. We did not use their 1:1 laptops for this activity even though the activity was online. I like to save that for the discussion at the end. There are 11 nets that make cubes and 13 nets where something is incorrect (not enough faces, too many tops, etc). I did not tell them how many did or did not.

This activity really takes 60+ minutes to do it right. I was there on an early out day with 40 minute classes, and by the time we did a few introductions and went over expectations/task objectives, we had about 30 minutes of student work time/discussion. I had the students put the sheets in a safe place in case I got an opportunity to come back and finish the task (I know what you are thinking...fat chance a 6th grader could hold onto a piece of paper for 2 months...it could happen :-)

Right away, I could tell that some of the students were not strong with spatial thinking. I try not to butt in too much, but listen to the ongoing conversations. One of the students who was confident in spatial thinking was trying to help a student by explaining that it made thinking about the cube easier if you picked out one of the faces to be the bottom before starting, and go from there. I thought this was sound advice, so I let this student explain to the class what she was thinking. I think this idea helped jump start a few individuals.

I enjoy using tasks from @jstevens009 and his site, www.wouldyourathermath.com, and I saw a lot of that argumentative spirit come out in these students when I am not sure they had ever thought about math this way before. Two boys were paired up, and they were not getting very far in making a decision on this net...

This seems pretty cut and dry to me, but from a 6th graders perspective who has been used to going with a gut feeling, I could see why he thought it was a cube. The conversation between these two boys was basically a yes it was, no it wasn't back and forth, which was going no where. I noticed this as I was walking around and stopped to talk with them. I reiterated the need for justifications, and told them to separate for a few minutes, come up with their best arguments, and come back together to make a decision. I gave no indication of who was correct and who was not. When they got back together, they had actually both came up with the same reasoning for it not being a cube, meaning that after taking the time to think and justify, one of the boys changed positions. Both boys actually pictured the top/middle square as being the bottom and stated that there could never be a back side and the top would overlap. This is a great example of the power of individual think time before talking with a group. It arms everyone with ideas instead of just accepting what someone else has or denying someone's suggestions just because a gut feeling told you so.

I had been in this classroom before helping out, and like any 6th grade classroom, there are some student who just need to be doing something. Listening to someone talk at them for 40 minutes doesn't work for them. I noticed those students really shining. They were free to talk and discuss, and they were getting to create the knowledge. These types of tasks are perfect for middle school students and their unique needs.

I wanted to keep letting them work, but we were running low on time. I put the computer version of this task on the screen and called on random students to select one of the nets and tell me yes it makes a cube, or no it does not and why. The students had plenty of time to work through justification with each other, so I did not feel bad at all calling on different individuals. They were prepared because I gave them plenty of time. Some of the answers were simplistic but to the point. "This cannot be a cube because it only has 5 sides (note: we did have a conversation about what a face is to clarify)." Some of the students thought it would be best to show a model of what they were doing, so one student actually grabbed some textbooks and started folding them up like each book was a face. After the student explained, I click on yes or know and the animation folded up. In one instance, a student was incorrect, but we had a discussion on what their misconception was based off of the animation. This is another value of the improvement mindset. It's more of an environment, really. Sometimes you feel bad if someone has a faulty explanation and you have to correct the student, but when the goal is improvement, students don't see it as being so bad. It gives them a chance to learn and it gives other students a chance to further explain misconceptions. I only knew these students for a short time, and to them, this concept of improvement was an epiphany.

Overall, this is a very accessible task to start addressing the concept of justification with your students. This would be one of the first tasks I would do to make that point. If your students are not used to justifying, you will get weak justifications at first, but you have to understand that your students probably don't know any better. It is your job to get them to that point. Trust me, it will get better. Here is a collection of student work from that day. Hopefully you will notice how quickly we were able to go from "this is a cube because it folds into a cube," to some of the justifications listed on the student work.

(Notes: not all of the student work is correct...but I hope to have more time in the future to discuss, and some of the incorrect work was discussed during our discussion portion of class. Some of the justifications have letters like B to represent bottom or back and so forth...you have to look closely. There was a wide variance in how far the student got, which was alright. I would allow for as much time as needed normally).