CGI with Secondary Students

So, I have seriously heard about CGI over and over again at the many conferences I have spoken at and attended, but since it is geared for elementary, I never really learned much about it until I decided to better myself by reading "Children's Mathematics: Cognitively Guided Instruction" last year.

I started modifying and using some of the ideas with my students last year, and this year I actually have a small group of 7th graders that work on just word problems with me once a week and I am relying heavily on the information I learned about CGI.

While I didn't get to put all the kids into that group that I would have liked (because scheduling is STUPID), I do have some kiddos who were really having a hard time making sense of what a word problem was even saying when they started, and after 6 sessions, they just blew me away this week. Another teacher asked me to share what I'm doing in a blog, so this is the result.

Please realize, I am not some kind of CGI expert - I haven't even seen it done other than the videos from the book, and teachers at my school that I've convinced to try it with me but I like what I read, and am doing my best to figure out how to apply the core ideals to my teaching with junior high kids.

Here's what I've done so far:
My focus turned pretty quickly toward writing equations because it is such a difficult concept for students to master, especially when there are multiple variables. I have been AMAZED at how much they have grown over our few sessions:

First Session:

• Gave the kids multi-step problems with low numbers that could be directly modeled
• This day was mostly for me to assess what they really know/are able to do right now
• Students were able to solve the problems, but multiple kids mentioned that it was hard for them to know what happened in what order, so I could tell they were still struggling with comprehension, and some of them weren't connecting mathematical symbols with the real life circumstances that create them.

Second Session:

• Presented one step addition and subtraction word problems with NO NUMBERS (blanks instead).
• For example: "Justin had ___ cookies, then Becca gave him ___ more. Now he has ___ cupcakes" 
• I asked students to discuss then prove what was happening between quantities (they had to identify what the blank was talking about, and prove why they thought it was that operation by citing evidence from the problem and giving examples).
• Students created numbers for the problems and wrote and solved equations representing the problems. Many students also drew pictures. Lots of sharing and explaining.
• At the end, students got to make up their own problems - they LOVED doing this and trying to stump each other by making them with extra information or complicated story lines. 
• I did do names of problem types with kids this day, but wound up deciding that wasn't the most helpful thing so I dropped that off after this session

Third Session: 

• Introduced multiplication and division problems. We talked about what types of things in real life would create multiplication and what types of things would create division. 
• We tried a similar approach with blanks instead of numbers and students reasoning about what operation made sense and why. 
• There was still a lot of confusion about how to tell what is getting divided and by what in problems that created a fractional answer, and also with rate problems. This makes sense because these problems are harder for students to model.
• Students were still choosing numbers and picking large numbers that were hard for them to draw, so I started switching to variables so we could focus on the operations instead of the numbers.
• This was actually a REALLY great challenge for them and totally made the discussions way better.
• Students also started creating cases to make their point and would plug in smaller numbers to try to justify reasoning within the story (it has always been hard for me to "teach" kids to break an argument into cases and they just did it themselves because they were working to make sense!). 
• This was the first day we had visitors and the kids kind of freaked out and shut down while they were there. So we had a heart to heart at the end about that.

Fourth Session: 

• Started with more multiplication and division with variables because once they shut down while the visitors were there, I wanted to give them a chance to get comfortable again. So this day was basically the same as session 3.
• Students are starting to write multiple equations with the inverse operations because they are understanding the relationships between quantities.
• Students mentioned wanting more challenge, so I asked if they'd like to try a multi-step problem next week - they are all in
• Discussions were WAY more confident today, so I know they are ready
• We had one visitor who was a teacher form a prior year so students were more comfortable

Fifth Session: 

• 3 visitors. We did an ice-breaker to make sure the kids were comfortable this time. 
• Multi-step problem (AND not the easiest to model directly):
• Calvin paints pictures and sells them at art shows
• He charges ___ for a large painting
• He charges ___ for a small painting
• Last month he sold ___ large paintings and ___ small paintings

• I showed only one sentence of the problem at a time, then asked students to give me facts they know so far
• When we would get to a sentence with a blank, I asked them what the blank was and made them prove it by citing evidence from the words in the problem
• I wrote what they said on the whiteboard then asked them what variable would make sense to represent that quantity (we talked about what the word quantity means)
• We had lots of good discussions about things like not using variables that look like numbers, not using the same variable for different quantities, and not using multiple variables for one quantity. We also talked about how writing multiplication as x is confusing, especially if x is a variable. 
• Then I mentioned that there was no question, and asked them if they could think of a way to turn this into a word problem. We found how much money Calvin got for all his large paintings last month, then they really wanted to find the full total he made last month, so they set to work working on making an equation to represent that. 
• I had lots of mini discussions with individual kids while they were working about what different parts of their equations meant and how they knew what operation to use. 
• The visitors split up and talked with different kids so ALL of my students were forced to explain A LOT today - which was such good practice for them!
• The visitors were really impressed with how much more articulate the kids were than just two sessions ago. The kids were also so much more confident and excited to share their thinking. 
• This was the first day where kids really shared their thinking with conviction, and you can see in one student's work where she plugged in numbers to help prove that her equation would represent the situation.
• They ATE UP this challenge - we talked about the fact that MANY visitors were planning to come the following week and I asked if they wanted to keep the difficulty the same - they said no way! Bring on the challenge!
• I have posted some pictures of their work below:

Sixth Session:

• MANY visitors scheduled to visit for "Pineapple Day" (10 teachers besides me total)
• Unfortunate absences - only 3 kids! They were nervous about all the teachers with so few of them but they said they were still up for the challenge (yay!)
• Here's the problem they worked on: 
• Brayden and Gavin were playing touch foot ball against Cole and Freddy.
• Touchdowns were worth ___ points
• Brayden and Gavin scored ___ touchdowns
• Cole and Freddy's team scored ___ touchdowns
• What questions could we ask to make this a word problem?
• I picked this problem on purpose to address the following things: Two names for one quantity (because of the two-person teams), multiple quantities about touchdowns in some way so students would really have to keep track of points versus touchdowns, and I planned to ask a compare problem at the end if a student didn't come up with it. 
• We had REALLY good discussions about what variables should represent things and how to tell the difference between to two teams, as well as whether the blanks represented points or touchdowns. 
• Each student came up with a different word problem, so we solved them all. The students decided they wanted to do them in the following order:
• How many total touchdowns were in the whole game
• How many points Cole and Freddy scored
• The total points for the whole game
• The first two problems didn't take too long, but all the students were writing multiple equations and using inverse operations. We talked about how to prove that they are really saying the same thing and all representing the situation in different ways and we had good conversations about the commutative property and practiced saying inverse operations instead of just saying "backwards" and "opposite"
• THEN... there were there LOTS of interesting student discussions (between each other and with me individually) to figure out the last one. 
• The COOLEST thing happened - every kid came up with a totally different equation. I was pretty excited because I knew when we came back together to summarize and connect we were gonna see a LOT of great connections. 
• I am KICKING myself because I didn't remember to take pictures at the end :( Here is what I remember of their solutions:
• One student had the sum of each individual team's touchdowns in parenthesis then multiplied by the points per touchdown to equal the total points
• Another student had the points per touchdown multiplied times each of the team's touchdowns first, then found the sum of the team's points for the total
• The third student had separate equations for each team's points where she defined new variables, then summed those new variables to find the total
• We had SUCH a good conversation - we talked about how to know the first and second equation were connected and a student even remembered the terminology "distributive property," students were able to see how each equation was connected to the others and explained in multiple ways, including by plugging in values to help explain their thinking, and at the VERY end, one of the students (who started the year out as my least confident and most confused) asked if we could also say that the total points divided by the points per touchdown equals the sum of each team's touchdowns. He mentioned inverse operations and plugged in numbers to try to convince others. This was SO cool - this kid who has been practically silent in my regular math class, and who has STRUGGLED to make sense of concepts in 7th grade is starting move things around in literal equations, and be able to explain it. And with an audience of a bunch of teachers. I was seriously blown away - it was so cool :)

My Takeaways So Far:

• We might spend a whole hour on one problem, but holy cow we address a TON of stuff in that one problem
• Students are making huge leaps with symbolic notation because they are FINALLY making sense of the problem, then using the context as a tool to help them understand the mathematics
• I always pick the problem(s) I do for the day with certain goals in mind, especially in terms of common misconceptions, vocabulary, and mathematical structure, and EVERY day they find a way to surpass that in some way by bringing up something interesting, noticing something new, or asking great questions
• These little people have turned from shy, unsure, sad-about-math kids into confident, eloquent, deep-thinking mathematicians, and it has ONLY been 6 hours!
• I have these kids before lunch on a minimum day, and they go home after lunch, so I am literally the last class of the day, and before lunch, and instead of being tired and complaining, EVERY day they beg to stay longer because they just want to solve one more. I have to physically kick them out of my room to get them to stop doing word problems. They are showing me that math doesn't HAVE to be painful, and it also doesn't have to be a "show" to be fun for kids - if kids understand and get to make choices about how they solve problems, they actually can have fun. 
• I have so much fun teaching this class - they teach me just as much as I teach them and I am so grateful I have had this opportunity this year.

CMC North Presentation: Fitting Number Sense into the Secondary Classroom

I just got home from Asilomar and the CMC North Conference.

I enjoyed my time but I am exhausted and have too many things that need doing, so I will cut to the chase:

I presented on strategies to support number sense and how to embed those things into what we are already doing in the classroom.

Overall, I felt like it went pretty well. I made a couple silly mistakes (inevitable if you know me), and the room got REALLY hot, but I ran into a few of my participants throughout the rest of the conference and many shared positive words about what they learned/thought they could use going forward, so that was cool :)

Anyway, the real purpose of this post - here are my slides:

Please feel free to check them out and use whatever you find useful! :)

Measuring What Matters Pt. 2

So, at the beginning of the summer, I was thinking a lot about how to measure the things from my math class that are the most important to the future success of my students. If you care to read about my thought process I blogged about it here.

Anyhoo... After lots of back and forth, I created what I am calling the "Success Behaviors Self Evaluation"

Here's how I plan to use it:

Students have an agenda (used school wide), a math journal, and file folder which houses their math work. Students will pull from these sources to create a curated portfolio of evidence that demonstrates their current personal strengths (and weaknesses) in the 5 areas on the Self Evaluation. They will grade themselves and justify the grade they have given themselves.

Students will have short meetings with me to discuss their grading and calibrate (I predict that many will be harder on themselves than I would be, and I will probably have a few who over estimate their demonstration of skill as well).

This will happen at least twice per grading period (so students can show growth over the grading period)

I am still required to give traditional letter grades, and this does not account for their entire grade either (the rest is all assessments).

I have NO IDEA how this is going to go and I am totally nervous about it. It might flop... or blow up in my face.... But I just can't shake the feeling that at least my thinking is heading in the right direction by attempting this :) - Wish me luck everyone!

Measuring What Really Matters

I have been thinking a lot lately about the relationships between testing and assessment. Heading into state testing this year, all classrooms were asked to do a goal setting activity where students look at their state test scores from last year as well as scores from the two mirror tests we did this year and shoot for a certain score. This was supposed to be posted visually somewhere. I was SO reluctant... The idea just bugged me. A LOT. It took me a while to nail down why. 

I mean... There are SO MANY reasons - the first being that I had just spent all year trying to convince my 8th graders that ANYone can do math, that ALL voices were valuable and had something to offer, and that right answers weren't the most important thing. Doing this activity this way would send the complete opposite message - in a bright, cheerily-condescending, and impossible-to-miss way. Just what I need - a huge neon reminder for my kids who decided that math wasn't right for them that they were, in fact, right when they made that decision. And, even though this is a good enough reason to make me squeamish, it was not actually the root of my frustration.

What really got me fired up was the fact that we were putting so much focus and intention on this stupid arbitrary number. A number that measures what a kid remembers about some questions they were asked on one day a year ago. Does this number matter in the grand scheme of things? Is it some kind of predictor for future success as an adult? Maybe a little. But is THIS measure really the one that I want to stake a claim to as most important? Is this activity that will take multiple instructional days, and be referenced in the future, actually make my students better people? Doubtful. 

So I started thinking about how to measure what matters - what REALLY matters - for my students' future success. I decided that those factors that most greatly increase potential success AND that I had any control over teaching would have to be the Standards for Mathematical Practice (SMPs). 

Now I do actually talk about, reference, and teach the SMPs, but measuring them is a different story! I currently use a modified standards-based-grading approach (based on this post from Kelly O'Shea), but I was imagining how my conversations would go with parents if I have "Make sense of problems and persevere in solving them" as a grade in my grade book... what constitutes an 8 versus a 10? Not quite so clear cut as my standard rubric!

Also, I have had to wrestle with the idea of essentially grading students on something that feels more like a behavior than a demonstration of knowledge. While it is unfortunately true that most grades wind up doing this anyway, I don't want to purposely head that direction. 

(A side comment here - I WISH that I could go to all feedback and forgo grading - it would make everything so much more awesome. Unfortunately I am currently required to submit letter grades, and while I am considering the possibilities of heading more in the feedback-only direction, my brain just isn't there yet!)

Anyhow, my current thoughts about measuring what matters: 

1. This is important so even though I don't know what I'm doing yet, I need to figure it out.
2. Current idea: I can make rubrics for each of the SMPs, and focus on what types of things I should see demonstrated at each level so students can see a progression of what actions to take to improve both themselves and their level of work. I would need to focus on observable aspects of work (like naming/labeling, multiple representations/solutions/solution paths, explaining why a solution works, etc). 
3. I am concerned about "leveling" the SMPs and making it seem like certain things are more important, or creating some weird value system that messes with kids' relationship with math (like most of the things we've done to kids in math class of the past have). My goal here is to support growth and encourage hope and improvement, not pigeonhole kids into levels. 
4. Another possibility: Just have SMP checklists? This would avoid levels but still give some guidance. Unsure if this is more or less helpful for student learning...
5. Explore ways to head towards feedback only (some thoughts):
1. Have students collect evidence of learning (portfolios?)
2. When it comes close to a grading period, meet w/ each student and have a discussion about what grade they feel they deserve - they must back it up with evidence. 
3. If I have SMP rubrics/guidelines of some kind, this could support students in reasoning out what grade they deserve.
4. Need to figure out how to get admin on board and make sure parents are informed and don't freak out if I do these things...
6. I need to find/follow/speak to more people who are teaching the way I wish I was teaching!
1. If you're reading this and are interested/intrigued/already doing this and wanting to share, PLEASE let me know! :)

A final thought - even though I don't have many readers and this whole process is more for my OWN thinking than anything else, I feel like I need to state publicly that I don't completely hate state testing. I do wish that the way we tested and what was tested and how we score and how we use that information actually mattered more for student learning and informing instruction, but I do get what we're doing and why. I just don't believe that basing my teaching entirely on those numbers is what's best for my students' lives or their relationship to mathematics.  

Happy Thoughts

So... the schedule on my campus is crazy, especially on Wednesdays. We have minimum days to accommodate staff meeting time, and at the end of last year, a group of the junior high teachers had this idea of running an "elective wheel" on Wednesdays. We are a small school, and since we all teach required subjects all day, there isn't much time for electives (we have one period where electives are possible, they are all at the same time, and various interventions occur at the same time as well, so many kids never get an elective at all). 

I was originally supposed to co-teach a performing arts elective, but as things started falling into place it was determined that I needed to teach intervention, and I partnered up with the 7th grade math teacher to do so. The nice thing about it is that we get full control over what we do and which kids we get and when. So we decided to run various themed intensive interventions with kids from 6-8th grade. 

While I am STILL trying to figure out how to keep up with the rest of 8th grade math in my district with one less day of instruction every week, I am actually incredibly grateful for these times with students. I get to do the things I know students really need, and I have the time to work on mindset, exploration, inquiry, and lots of number sense. 

This past Wednesday was particularly fun, and since I spend so much time thinking about the things that need fixing, I thought I would indulge myself and think about the things that were good!

So, here's a little run-down:

Group 1: Number Sense focused on Multiplication and Building Rational Numbers

My first group is mostly made up of kids that I also had during last trimester and they are the ones who struggle the most in their classes. We focused really heavily on number sense with multiplication last trimester (as well as mindset), and most of them actually requested to be in our class again! We had done work on whole number place value and operations before, but this trimester, we are working towards dealing with rational numbers. 

During our number talk the prior week, one of the students happened to question whether 0.9 and 0.90 were the same thing. We had a pretty big discussion and the class convinced itself that they are in fact the same value, even though the numbers are different. 

Because this idea of equivalence and place value came up, I wanted to start with that, so I made a list of equations and students had to discuss and determine whether they were true or false and why. Part of the way down the list, I had the equation "0.10 = 0.010" and boy, was that controversial! One group of kids was convinced that the two sides were equivalent and they put up quite the argument. The rest of the kids went through 4 or 5 different ways of looking at the numbers to try to prove the statement false. I had kids jumping out of their seats in excitement over what other students said. Students were at the board drawing, explaining, justifying, asking their peers questions. I WISH I had thought to take a picture of my whiteboard at the end - it was COVERED with math from students. And these are the kids who are supposed to be "bad at math" - what a bunch of hooey. At the end, one of the students looked at me wide-eyed and said, "That discussion was CRAZY!" I asked if he thought it was important and I got "Yeah!" right away. Then he asked if we could do number talks again next week. This is a student who was loudly complaining about being placed in this class at the first meeting (the prior session). 

Group 2: Place Value

Our second group is the only one that includes 6th graders (because of timing). My partner teacher and I decided to ask the 6th grade teacher what he would like us to focus on, and he requested place value. 

I had done some research on various routines and strategies for place value, but instead of trying to "teach" place value, I decided to do an exploration with "Exploding Dots" from James Tanton (you can check out the entire course here). At the first session, I showed this animation and we did notice and wonder (we did this twice, since it's hard to tell what's happening in the video right away). Then, students worked on answering their list of wonderings. This time, I showed the video again (there were a couple kids absent), and then had the students work on creating the code AND the boxes for 1-20 (in base 2). Our focus was on finding patterns. As students were working, I just wandered around and asked them what they noticed and if they found any patterns. 

6th graders are really cute and small by the way. 

Anyway, the 6th graders take the longest to warm up to me because I am TOTALLY new to them and I also seem really weird because I make them do things like watch weird videos and make them look for patterns instead of taking notes. But, as different kids started to see patterns, they totally lit up and wanted to tell me about it. Some of the more notable noticings:

 - As you count up consecutively, the last digit will always alternate 1, 0, 1, 0, etc. 

 - There is a pattern of circling back around to all the things that happened before every time a new place value happens

 - Any double is the same number, but with a zero at the end (basically, shifting place value).

We didn't have time to get to talking about what "1" dot is worth in each box, but that is where we will go next week. 

Group 3: Number Sense in Application

My last group is more focused on heading into grade-level work, but they struggle with application. They tend to sort-of get and do procedures, but they don't really have the background of how/why things work and they get confused. We are working on making sense of problems and connecting representations. 

I just stole Steve Leinwand's suggested plan for improving problems from the curriculum (from his Asilomar 2016 presentation) - sorry... I can't find a link, but here is a link to his twitter page and his blog. Here's the basic idea: Give a one piece of info from the story (math problem). Ask the kids what three things they can tell you? Allow them to make inferences (they think this is hilarious). Then give them the next piece. Ask what three things they know... etc. Then, ask them for the question. They solve the questions the class posed. 

This went pretty much as expected, except that the kids went WILD with wanting to make up stories for their word problems. One of my squirrely boys who complained about being put in my class was ALL excited to make up stories about the boring word-problem-people. He would get so involved, he would often have to solve most of the possible questions as part of his story. This rubbed off on my MOST reluctant student, who begged me to let him make a word problem story to share with the class (part of the plan for the next session). 

So yeah - it was a fun day - lots of questions, lots of smiles, and lots of math. Who knew that math intervention could be such a magnet for joy :)

Help!

I know it's a cryptic title, but hey, I decided to go for as true and concise a request as I could muster - it got you here, right?

So, now to explain myself...

I just got home from the CMC North Conference in Asilomar. It was wonderful to finally get a chance to go, and even though the experience was wonderful, the feeling I am left with is bittersweet. 

You see, I am in - 100%, drank the kool-aid on things like building up my classroom culture to support productive discourse, using rich tasks and questions I don't know the answer to so I can learn to listen TO my students instead of FOR an answer, and changing how I think and speak about learning to honor the "partial understandings" of all kids without pigeonholing them into mastery (or non-mastery) boxes. 

But I have a LONG way to go, folks. I am trying. I really do give my best effort to make decisions that are in my students' best interest every day. But more often than not, I see a huge gap between where I am and where I want to be. Most of the time, this motivates me to be reflective and continue the work to improve my practice, but there are times where I just want to throw in the towel (or at least throw it in someone's face... HARD). A lot of those times are when it seems like factors I cannot control get in the way of my goals and vision for my classroom. 

And here is where you come in. Below, I will detail some of the situations where I am currently feeling stuck and frustrated and my hope is that some of you may be able to offer some beginnings to possible solution paths :)

First... Just a little background about where I came from:

• This is my second year in a new district. I came from a district that started implementing common core 5 years ago where I was the department chair during that shift. Compared to many people's   experiences, we were pretty well supported in this work. Over my time there, we did a lot to develop as a team, and by the time I left, we met at least weekly to discuss goals/outcomes,  build assessments, co-plan lessons, and just discuss the craziness that is teaching math in middle   school. We struggled with a lot, and certainly didn't always agree but we had a lot of really good  collaboration. The biggest obstacle for our team was overcoming student apathy and "behaviors"

and where I am going:  

• My new district is still struggling with the implications of CCSS implementation and hasn't had as  positive an experience with the transition. Some parents do not support the idea of National  Standards, and there are a lot of conflicting ideas about what math education should look like. I'm at a little K-8 country school that has an awesome family feel to the campus, but I am the only one   teaching 8th grade math. Most district-wide curricular decisions are made by the large middle   school and because I am not there, I tend not to be included in many of those discussions. My biggest obstacles in this moment are below:

1. "Collaborative Competition" - I was in a meeting this last Wednesday where my principal rolled out our new (or perhaps old and just new to me) mission & vision statement which includes this signature idea. My understanding of how it's supposed to work is that the teacher strategically assigns teams, the kids come up with team names which are displayed in the room, and teams earn "points." The most common way this is done is by counting up the number of test questions each person got right on an assessment. Then the team goes over the most missed questions as a group so everyone understand the right answer. Then they set a goal for how many questions the team will get right on the next assessment. Then they earn team points for improvement. I think. There is some variation too - something about doing math tasks in teams then the teams earning points but I'm not totally sure how exactly that works at this point.
• What bugs me: I am REALLY trying to do things in my class to convince those kids who have struggled with math their whole lives that they have important mathematical ideas. I have heavily emphasized the idea that making mistakes, reasoning, justifying, explaining thinking, asking questions, and listening to others are all MUCH more important to learning than right answers the first time. I also want students to feel pride and accomplishment when they figure things out for themselves, and I hope that that will (over time) translate to some increased internal motivation for doing mathematics. I feel like the competition aspect of this idea is basically an external motivator that trains kids to try to get "points" for right answers. I worry that those struggling kids who might have spoken up to offer an opinion or different perspective to a task might now shut back up because they will just wait for the "smart kid" in their group to tell them how to do the procedure so they can get the right answer and not lose points. 
• What I've done: I voiced my concern to my principal. At first he thought I was afraid that other kids would bully the low kids, and I explained that this is not my fear at all, and tried again to give my concerns. He said it is my responsibility as the teacher to "coach up" the kids so this doesn't happen. I'm not really sure how to coach up a kid to feel confident sharing what (s)he thinks is probably not right when I'm giving points for being right. We kind of went in circles about this and that's where it stands.
• What I need: There are a few possibilities here:
• Perhaps, I need a paradigm shift. Maybe I AM being short-sighted here and competitions with team points are a great way to maintain my classroom norms. Someone please explain this to me so I can understand. Right now I just feel sad and scared about what it means for my timidly emerging mathematical thinkers...
• A way to show my point of view and back my concerns so it is more clear to my principal. Dan Meyer talked about "problematizing" in his keynote at Asilomar, and while I am not sure I am using it totally correctly, right now I have decided it means to find a way to make your problem visible and understandable to someone else so that they can empathize with your situation and clearly see your point of view. Suggestions so far have included bringing him to Asilomar and having him read Jo Boaler's book. I will tell you that neither will happen. Too many competing priorities on his desk. I was thinking maybe a one-page study/article that shows how answer-getting and external motivators like "points" can further shut down kids who already have a tenuous relationship with math, and another well-thought-out conversation. Your thoughts? Especially how to get my views to make sense. 
2. Assessment Calendar: Another important part of the meeting we had was talking about assessment calendars. We are supposed to give an assessment every 2 weeks and enter in a bunch of information (like what topics, standards, ELD objectives are addressed). After students take the assessment, we are supposed to list the percentage of students who "met/exceeded" standards, "nearly met" standards, or did not meet standards. Then we list the names of the kids who didn't meet the standards so we can identify what we are doing in terms of intervention. 
• What bugs me: Even before attending Megan Franke's keynote today, I felt like the data I was putting in this table was totally uninformative to my instruction. I test multiple skills and sometimes multiple standards on the same assessment. I could have a kid who "exceeds standards" but demonstrates a misunderstanding about a certain concept in her work, or a student who gets every answer wrong, but still demonstrates understanding of certain skills or concepts. This document tells me nothing about what students actually know. Even my grade book is more informative because I have it organized by the important ideas I want students to demonstrate understanding around (so for one test, I might put in 3 separate scores). This document is busywork. But more than that, it is a huge focus for the school. If this is how we talk and think about students at least every two weeks, will the language and culture around "assessment" be heading in the direction that is best for teaching and learning? The biggest struggle for me? the title of the document is "FORMATIVE assessment calendar" - meaning it is supposed to inform our teaching. How? I don't get it. 
• What I have done: Attempt to fill it out. Asked questions about what was expected. I was too scared to rock the boat on this as of yet. I am new and I have had a hard time figuring out how to share my ideas and opinions so that they are understood. I also wasn't sure how to put into words why this bothered me SO much until seeing Dr. Franke's keynote today.
• What I need:
• Ideas: I haven't figured out how to do it, but I was kind of thinking it would be helpful if I could modify this form so that it is...
• meaningful to my teaching and students' learning
• still meeting my site's requirements
• manageable in terms of implementation (meaning both in terms of time, and level of complication involved)
3. Isolation: Not gonna lie, one of the hardest things for me about my job now is that I can't just pop next door and see if the lesson flopped there too. I knew that my team was really important to me, especially after I had spent so much time working with them to build up our capacity to collaborate, but I have really realized HOW important that collaboration was for me. I spent most of last year questioning my capabilities as a teacher, and my only respite was going to conferences and some intermittent conversations with old colleagues to make sure I wasn't going crazy or harming my student's mathematical future. Now, I don't want to give the idea that I have no support - I work with some awesome, helpful, collaborative people... I just don't happen to teach the same thing as any of them. I can't bounce content ideas off of others quickly and easily, which means I tend not to. I know #MTBoS is out there... but again, while helpful and informative, fast and easy hasn't really been my experience.
4. Isolation 2.0: Another piece of this puzzle is that I am already kind of the "weird" math teacher on campus. I tend to see and do things differently than others, which also separates me in a way. Part of this problem is exacerbated by the fact that I cannot convince ANYONE to come with me to conferences! Most of the things I do are part of my practice because someone else presented them as an important way to build students' mathematical futures. If other people came with me, maybe they'd try this stuff too, so even though they teach other content/grade levels, perhaps we could at least have a conversation around the practices that support mathematical learning. Part of this is my fault - I haven't found a way to fully communicate the awesomeness of the experience (otherwise they would OBVIOUSLY jump on the bandwagon, right?). Part of this is the fault of the teachers - there is a surprising amount of what I can only call fear about venturing out past Sanger and going to "PD" that isn't forced. And by far, the largest part of this, is the institutional system surrounding these types of events. First of all, I have to pay for EVERYTHING myself, even when I am a speaker and am basically a walking advertisement for the district and site. They won't even cover my sub. So I have to use sick days when I go. It is possible to use sick days for conferences in our contract, but we have to fill out additional paperwork, get it approved, and it can be denied if our reason for attending isn't worthy enough. So most people who go to anything just put in as an illness. But you can't enter in an "illness" absence ahead of time. Which means you can't guarantee a sub you know, or plan fully for your class while you are gone to at least attempt something productive. And it just feels sneaky and integrity-deprived. Oh, and if you happen to try to get units because you want to attempt to afford survival, they have to be pre-approved by the district (which obviously none of mine are because I am taking sick days to go). Think about the messages being sent here - What teacher who has never attended this type of event, gets paid a teacher's salary, and already has fear about doing something like this would be willing to venture outside the bubble? I can tell you exactly how many: None. Because I am the only one I know of who ever goes!
• Why this bugs me: Well... I guess I should have said spoiler alert above because the cat's already out of the bag here. But I will try to calm my jets and say what I mean more succinctly: I am in a district that has an incredible reputation, fantastic people, and a track record for being responsive. How do the practices and systems detailed above even make sense? Is this the image and set of ideals we want to embed into the teachers here? I just don't see alignment between ideals and actions, which makes me think and hope that it is unintentional. Even so, I feel like it is a dangerous way to run things.
• What I have done: Ask questions. Lots. To lots of different people. Get surprised by the answers. Ask more questions. Get disappointed. Then frustrated. Then sad. I DID happen to have a conversation with important CMC board type people about this fact and we talked about possibilities for structuring scholarships for first time attendees. This will help... but it does not solve the underlying issues.
• What I need: Tenure and some guts so I can go have some tough conversations? I don't know. I don't know who I would talk to, and like I said before, I haven't had a great track record of explaining myself in a way that is understandable to the people who make these decisions. 

Some final thoughts: I would like to point out that I do enjoy my work, my students, and my colleagues. I know it's easy to paint individual people as antagonists in my story, but the truth is, my principal is a charismatic and supportive guy who genuinely does what he thinks is best for students every day. I don't know that he sees mathematics education in the same way I do, but I'm working towards finding better ways to illustrate my perspective. The teachers I work with want to collaborate and learn from others and do new and innovative things all the time, even though they do it without leaving home. And I have to believe that my district does want teachers to be supported, innovative, and connected to the outside world, and that the systems just need to be considered and updated. I am not trying to blame any of these people for the obstacles I face, but as I stare up at them, I am realizing that I need to find ways to convince those people to come down here with me and stare up from my perspective so that maybe they will see my struggles and offer some support for the climb. 

My CMC Presentation: Division of Fractions Through Problem Progressions

*WHEW* I am exhausted. But CMC South was great (as usual!) and while my brain and my body are sore, I am grateful for all the great new information I was introduced to.

Anyhow - I had a pretty decent response from my session on dividing fractions - mainly about the materials and people requesting them - so even though I posted them on the website for the conference, I decided I should post them here too so it's easier for people to find.

Now - I made a number of points about my intent and how *I* think these things should be used in the talk, and those things are not obvious by just downloading my stuff... So I will attempt to summarize my big points as well. If you don't give a crap what I have to say and you just wanna steal my stuff, well I will be an enabler and give you those links at the bottom. I just REALLY hope you don't run off copies then stand at the front of the class and go, "Ok, everyone, watch me do number 1 so you can copy that process for the rest..." ;-)

My goal:
Give teachers usable ready-to-go materials that actually help develop conceptual understanding of division, of fractions, and of division of fractions, and that lead sequentially towards noticing the pattern that division is just multiplying by the reciprocal.

Required Background Knowledge:
Many of my activities assume that students had the types of experiences expected in the 3-5 grade common core standards dealing with fractions, and in the way(s) that have been presented in the progressions document. This means that if students do not have those experiences (understanding fractions with various models, building fractions from unit fractions then using this understanding to make sense of multiplication of unit fractions times whole numbers, division represented in many ways with different models, thinking about partitive vs. quotative division, etc. - the list goes on), they will struggle with what the heck they are supposed to do. It doesn't mean not to do this stuff... It just means you may have to give them some of those experiences first in order for things to make sense to them. If you want some example slides of what I am talking about in terms of background knowledge, check out slides 5-9 in the full presentation. (Yeah - I know it would've been nice if I pulled those slides and linked them separately here, but seriously - I still have a whole period of tests to grade before tomorrow - I have faith that you can figure it out).

Some of My Philosophy for Activities:

1. When I introduce a new concept in math, I think it's important to attempt to include context - mainly because context aids in comprehension. For this reason, I often start with a "situation" or a word problem. If you want to be really cool, you could actually put the students themselves into the situation and have them do it for real (give string, have them measure and cut, etc). 
2. For some reason, students often do not use comprehension strategies in word problems. I see a lot of kiddos who just circle the numbers and start doing random operations with no regard to situation. We do "Notice and Wonder" a LOT with word problems. I have also started having students justify which operation they are using based on the context of the word problem (embedded in first "assignment"). 
3. Models are awesome for supporting understanding of concept, and the types of problems you should focus on when you are trying to support conceptual understanding should be transparent - simpler numbers, relatively easy to make work with the model. Students need enough time with these simpler versions and the models so that they can start to notice the patterns that lead to a more efficient strategy (the algorithm). Our goal is NOT that students solve every problem with the model! Our goal is to give students a tool that models how math works so that they have the foundation of understanding necessary to uncover the pathway to where we REALLY want them to wind up - the algorithm. 
4. Avoid "teaching" algorithms at all costs... BUT - carefully select problems that expose patterns, and embed activities where students practice noticing those patterns so that they can see the "rules" for themselves. There may still be those kids who stick to the longer and more unruly methods because it makes sense, but they usually need that anyway. And the kids who figure out the "short-cut" tend not to forget how it works as easily as if I had just given it all away. 
5. Embed everything possible. In many of the documents, students are expected to prove the answers to their division problems by writing the inverse problem (like if I said 8÷2=4, my proof by inverse would be that 4x2=8). This isn't "the standard" for dividing fractions per se, but it supports future algebraic thinking, it reinforces the idea that you can check yourself in math to see if you are making sense, and it can start to expose the idea of multiplying by the reciprocal in certain circumstances if students are paying attention. If you see other ways to embed more cool stuff - by all means, please do! 

What is IN this packet of stuff? Why?

1. "Cutting Things in Pieces" - I start with division with common denominators. While I have many reasons, two are the most important to me. First, they just make sense because it is VERY similar to working with whole numbers. We want students to connect fraction operations to operations with whole numbers and to understand that all of the same "properties" of numbers and operations still apply to fractions JUST like whole numbers. Often students think everything has to go out the window and they are learning some new alien thing that isn't a number anymore but is called a "fraction" - I want to show that that is not the case. Secondly, there are some interesting patterns and ideas that start out easy to see, but also come back in ways that make things easier in the future. Also - a note about the word problems and reasoning: It seems stupid to have this here because (duh) all the problems are division. BUT, if this is something that is happening regularly in the classroom, it won't feel so weird. I like to mix up the operations of practice and then the reasoning feels more appropriate, but if we don't ever practice it, they get frustrated when the operations are mixed. I am imagining that students have practiced all of these things before, and could dive in to this without the teacher "showing" them anything. If that is NOT the case... perhaps go back and build up some of that background knowledge stuff first. 
2. "Noticing Patterns: Dividing Fractions with Common Denominators" - This is just an extension of the previous problems. Basically some practice as well as gathering enough "data" to analyze for patterns. Often, students will see the "pattern" ahead of time :) - Notice, the first 4 problems on the first "assignment" are part of this data collection as well. Once data is collected, students look for a pattern - have them make conjectures and test them out (usually, this one is easy to see, so it doesn't take long). Then have them explain...
3. "Dan's Division Strategy" - This is an Illustrative Mathematics task that jumps into the rule that the students probably just devised. The information about how students might justify the rule is really nice, so you should check that out here. I am imagining that these first 6 pages would all be copied double sided (so 3 pages) and would be given in series.                                             *A brief note: Students will need more time/practice/models/versions that this. They need to be challenged a little bit with things that DON'T work out perfectly (like 7/3 ÷ 2/3) etc. I did NOT make an entire unit - just some key activities that I think can support conceptual understanding.
4. "How Many Servings" - This activity is kind of designed for students to struggle. That is pretty much the point. When I do things like this with my students. I tell them straight out that my goal is to see perseverance and creativity. I tell them that I am going to challenge them with something that I have never shown them how to do, but that they can use everything they do know as a tool to try to make sense of going forward. We do notice and wonder (embedded in the assignment). In classes where students tend to struggle more, we brainstorm a list of strategies. They each commit to trying one (I have them write it down). And I set a timer (start at 5 min, tell them I will check in). Usually what happens is that when I check in at 5 minutes, they complain that it wasn't long enough and I should give them more time ("Oh, so you WANT me to let you struggle to figure out that math problem longer than 5 minutes... I guess I can do that" *win*). Sometimes certain strategies are panning out better and I might have some students share ideas if kids are stuck - or they work in groups, etc. This is where I would do a "5 Practices" type discussion and focus on sequencing student work. If no one figured it out, I would STILL do this and discuss where our sticking points are and have students determine what it is they need in order to proceed. 
1. I show ONE POSSIBLE model in my powerpoint with animation and all that good stuff on slides 26-31. It is NOT the only way, and students who solve the problem may not have used this method. That is ok too! But I do think it is a valuable model for students to play with. 
2. This is a procedure that you may have to teach, but it is really much cooler if a kid comes up with this type of idea first :)
5. "Dice Practice!" - This is just a fun way to practice. For some reason, when I give kids dice to determine numbers in a problem they like it more. Totally shameless, to be honest, but I thought someone else might like it too, so if you do, have at it :)
6. "How many ___ and in...?" - Another IM task. Now, I wouldn't just do this as is. I placed it here in the order of things on purpose because it is a really nice set of problems for students to have to differentiate which model makes more sense to use. I would give them THAT as their goal ahead. We want them to think about the type of "tool" they select for the task. Have them practice doing that here, while also looking at patterns for dividing fractions :)                          *Side note: There is a blank page because it would be printed double sided and go with the following two pages which have spaces for students to draw a diagram and justify answers by doing the inverse operation. 
7. "Multiplicative Inverse and Identity Investigation" - Basically, this is just dividing 1 by a bunch of different fractions. BUT, there are a LOT of mathematical properties that are really visibly when doing that. I would tell students that they will be investigating the Multiplicative Inverse and the Multiplicative Identity, and to see if they can figure out what those are and what they mean. Again - at the end would be a nice place to have a discussion with students about what they notice, any patterns, etc. This leads REALLY nicely into the whole multiply by the reciprocal idea, and also the idea that WHEN you multiply a number times its reciprocal, the product is always 1. 
8. "Developing a Rule for Dividing Fractions" - So again, this is basically a series of problems that is designed to lead students to seeing the pattern of the algorithm by collecting data about what problems they started with and what answer they got. I PURPOSELY chose problems where the numerator in both fractions was 1 FIRST, so students might first come up with a conjecture that the denominator of the second fraction just goes on top. Then, when one numerator is no longer one, they still have that idea but realize you are then multiplying across after you flip things over. I am hoping to COMPLETELY AVOID the idea of "cross multiply" - and here is why: "Cross multiply" has no meaning in terms of mathematical properties - how do you know what thing goes on top after you multiply? You are just memorizing some procedure. If students make the connection that the MEANING of division is the inverse of multiplication (meaning - multiply by the multiplicative inverse which is the reciprocal), this actually has important mathematical implications for the future. It helps make sense of things like negative exponents (positive and negative are inverses, so it makes sense that if positive exponents mean repeated multiplication, that negative exponents should mean the inverse - repeated division), and is used throughout high school mathematics. I could go on... but I think you get the idea. 

OK! So, that's it. A crazy long post. Dunno if anyone will even read it. If you did - gold star for you ;)

Let me know what you like/hate about this stuff. If you use it, let me know how it goes! If you change things, let me know what and why and if it worked better for you that way. 

And if you have questions, let me know. I hope it helps someone out there!

Here are links for my PowerPoint presentation and the Student Activities Documents

*Please realize that I uploaded editable versions, which means formatting is probably not correct. If things look ridiculous, just shoot me a message and I will email you the documents directly.