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.

Day 2, Reflection, and Plan for Day 3

What I did for my second day with kiddos:

1. Started with the problem 1/2 + 1/3 (3 minutes of individual time) - I reminded them that making mistakes is ok, but not trying absolutely is not.
2. 3 minutes to partner share (during both independent and sharing I walked around and jotted down all the answers I saw)
3. I wrote all the answers on the board, then called the whole class's attention back to the front. I told them that I was noticing a number of different answers, and asked them what they thought about that. 
4. A few students basically made comments that they were unsure about their own answers and thought maybe there were mistakes. I happened to overhear one really great conversation during the partner sharing where one girl told her partner that she got 2/5 but that didn't make sense because that was too small - she mentioned cooking and how the cups work. I asked her to share her insight. The class agreed, so I asked if we could use this same logic to eliminate any other answers definitively. We were able to narrow it down some more :) - Someone also noticed that one of the answers was way too big (by similar logic), so we decided that couldn't be it either. 
5. A student asked to defend the answer 5/6 (this happened in every class - someone was just DYING to say how to do it! LOL). I had the student explain step by step, pausing and asking for clarification, asking for agreement, and calling on other students to restate reasoning. This took a long time, but there were some really nice moments where students started to realize how fractions work. Even my higher kids who totally knew exactly what to do in the procedure struggled to explain clearly why we need a common denominator and why we don't add denominators (the answer EVERY single one of them gave me the first time was basically, "because your answer isn't right that way"). 
6. I had prepared two more problems: 3/4 + 1/2 and 1 1/3 - 5/6. One of my classes wanted to do the "harder" problem, so they jumped right over the middle problem. They had done well in the explanations, and the idea of least common denominator had already come up through the conversation. The other classes did the middle problem. There was not enough time to finish a third problem today. 
7. Again, I had the students write what they learned today. I also told them that they could instead tell me what the most significant/helpful thing we did today was for them personally. 

Reflection:

     Many students said that this discussion was helpful for them, and my higher kids were some of the ones who said that they enjoyed the discussion (so they aren't bored out of their minds by talking about something that they basically already understand themselves). I have decided that it might be better to actually have them just discuss "How do you add and subtract fractions?" and come to consensus around that, and if they don't hit all of my key points that I think are the most important, I could just disagree and pose a problem that their "solution" doesn't fully address. This is my plan for my next class session on Monday (even though we have already talked about it).

First Day of School

Well... I've decided that my focus for the year will be to "Teach Through Structured Problem Solving." I want to focus on how I structure and facilitate discussion specifically, and I've been thinking about how to best document both my plan for my lessons, as well as my reflections on them.

Although I am sort of nervous about having my completely new-to-me and unpolished ideas available for any old person to see, I figure there aren't too many people looking anyway, and this is the easiest way for me to keep track without creating some new way of organizing things for myself. Besides... my lackluster attempts at keeping a blog going otherwise have obviously been unsuccessful to this point :)

So anyway - Today was the first day of school with students. I decided to intro the whole focus on discussion by focusing on the question, "Why are students required to learn math in school?"

Some background: I teach 8th Grade Math in little K-8 country school in the Central Valley of California. I get 80 minutes with my students, but we only have 4 days per week. Oh... and it was crazy today because there were no schedules/rosters yet... but that's another story!

ANYwhoo... Here's what I did:

1. Students set up paper with name, date, topic, and wrote the question
2. 3 minutes of independent time to write their thoughts and opinions (I walked around and read answers)
3. 3 minutes of partner share time to discuss ideas
4. Introduce discussion rules:
1. One Discussion (meaning no side conversations)
2. One Facilitator (I told them that students can be facilitators too, but during that time, I can't also be in that role)
3. One-hundred percent consensus (whatever our "solution" was, we had to all agree with that completely - INCLUDING me)
5. Asked a student to start conversation (one I had chosen because their answer was a "good starting point" - honestly, today I purposely chose kids who I know don't get a lot of "air time" speaking because they wouldn't normally volunteer)
6. Asked questions, "What do you think? Do you agree? Do you have something to change/add?" Basically on repeat
7. We kept track of the discussion on the whiteboard
8. All classes came up with something along the lines of "Math is in practically everything and it helps you in your future both in life and for your job" 
9. I told them that I agree so far, but I disagree that it's complete. I asked them to guess what I might think they should add.
10. It was REALLY funny that two different classes said that I want them to learn math because if it wasn't required in school I'd be out of a job! LOL. I told them that I wasn't that selfish - my reason was for their benefit more than mine! But it still cracked me up. 
11. My reason - It teaches thinking. Thinking that is different than what you are required to do in most other subjects, but that is necessary for really being successful in life, especially when presented with new problems. Basically - the application part of the "you use it in real life." 
12. They agreed - yay consensus!
13. Then I asked them if they wanted to have to write down all of the different points from the board (we bullet pointed ideas as we went and checked to see if everyone agreed - most classes had 4-6 sentences of bullet points). They said they would rather not, so I asked them if they could be more concise. 
14. Partner talk about how we could take all our main ideas and write it in only ONE sentence. 
15. Share out - then lots of editing until we reached consensus. 
16. Most classes came up with something along the lines of, "Math teaches us important ways of thinking that help us solve problems in our every day lives and builds a foundation for all the things to come in our future" 
17. I talked about WHY I am making them do all this discussion instead of just telling them what to do:
1. I want them to THINK
2. I want them to be able to COMMUNICATE their thinking so others can understand them
18. I had them answer the following "Connection" questions:
1. What do you hope to do/learn this year in math?
2. What did you learn today?

Reflection:

• Overall, lots of good discussion. My homeroom class is 20 students, but 5 of them are on IEPs with math goals that are not in the range of my standards. So far, all of them participated and were able to do ok... We'll see how that goes when we get into DOING math instead of just talking, but one of them was SO CUTE! He seriously wrote so much (and I just found out he really struggles with reading at all) and at the bottom where I asked what did you learn today, he wrote down almost all of my messages about making mistakes being ok, that trying is the most important, that everyone can learn, and he had little smiley faces all over with exclamation points. I was planning on giving papers back tomorrow to add to their notebooks, but I kinda want to keep that one because it just makes me smile! :)
• Students were really unsure about taking over as facilitator. I basically stayed in that role, but I'd like to find ways to get students more comfortable so they can take over for me sometimes! I'm sure part of it is just me not really knowing how to get out of their way, too, though. 
• Timing was actually really good! I was able to do all my administrative junk (even with the schedule fiasco), and still finish the discussion with time for students do finish their reflection portion at the end. If I don't have admin junk next year on the same day, I should consider adding something else in because this activity alone will NOT fill 80 minutes (maybe 50 minutes or so?)
• I need to get better about keeping track of who speaks, so I can make sure to ask for opinions specifically from those who don't tend to volunteer as much. I need to think about some structure that will help me do this without making my life more complicated... maybe initials on the board next to what each one says? Worried that might take up too much "real estate" on the board though...
• I like it so far. I'm feeling excited about the possibilities, but nervous about everything I'm doing being so new to me. I guess that makes me human! :)

Trying to be More Japanese

I had the wonderful opportunity to attend the CMC-Central Symposium in Bakersfield this weekend and *especially exciting* - I finally got to see Phil Daro speak IN PERSON! And... as always, his message(s) just rang so true.

There were so many great nuggets of information, challenges to how I think about teaching and learning, and possibilities for how I could improve my practice. I was thinking about what would be ONE thing I could really focus on and do right now, and because I want to document for myself now (and return to process later) I am sharing that with you :)

He talked a lot about practices and beliefs of high performing countries that differ from those that are common here. Many of those that seemed to really resonate with me came from Japan - here are the ones that I am focusing on right now:
 - All students start a lesson "all over the place" - in America we think students who do not grasp grade-level material immediately have a mathematical illness called "gaps" while in Japan, they focus on the future mathematical health of students by connecting lower grade level methods of approaching the problem to the grade level methods and exploring how/why it works.
 - Student work (approaches) is presented in a progression from easiest to understand to highest level mathematical approach. Solving a problem does not mean getting an answer, it means explaining in depth how they arrived at their conclusion(s) to prove that their answer makes sense in a way that is understandable to the students in the class (audience is not the teacher).
 - The teacher solicits information about the connections between different methods from students, asking students who used one method to relate to/explain/show connection with another.
 - Teacher summarizes progression of approaches, focusing on leading students to grade level (or beyond) approach.
 - THIS is the point where teacher might do "direct instruction" (because all students have necessary prior knowledge to connect to grade level approach to problem).

Apparently, much of how math is taught in Japan has been shaped by writer's workshop. I thought that was interesting too.

Anyhow, there was WAY more stuff packed into that talk, but gotta process!

So here's my plan for the next two days:

Monday: Focus is on gathering evidence of student understanding (this is technically review, but I am doing it because I know it is an area of need still and also pretty much the point of 8th grade).

• Give student Dominoes Pizza prompt (I went to the online ordering tool and put a two topping medium pizza and a 4 topping medium pizza, then took a screen shot). This idea is totally based on a Mathalicious lesson, by the way. You can find that here.
• Do Notice/Wonder
• Pick some good "wonderings" to answer (I imagine we will predict the price of some other medium pizzas with different amounts of toppings, and I will push them to work towards an equation - it would be CRAZY cool if a kid asks that as a wondering... we'll see). 
• SILENT independent work for 1:30 (2 minutes is still too hard! LOL)
• Continue working but discussion is permitted (I will float around room, ask questions, see different processes being used, start to pick who will present and in what order)
• Full explanation of process and why it makes sense is required. All parts of any equations must be identified and explained. (I will check in with students who I want to share to let them know so they are prepared and there won't be drama when I call them up)
• Selected students will present solutions. Audience will be looking for the best thing that student did to help them understand, and the one aspect that was the most confusing/unclear. Random select for feedback.
• As each new student presents, I will ask audience about connections from one solution strategy to the next.
• At end, I will summarize, focusing on how the different strategies are related to the equation, then ask students to add reflection on the summary to their own explanations

Tuesday: Focus is really on the explanations, giving and getting feedback, revision.

• Give students Me N Eds prompt (A sentence stating that Me N Eds charges $9.50 for a 1-topping medium pizza and $0.75 for each additional topping. The equation m = 0.75t + 9.50 is given, but variables are not defined). Students are asked to prove that the equation represents the situation. They must identify what every piece of the equation means/represents and give examples to back up their reasoning. 
• Time limit for completing draft explanation (I'm starting w/ 10 min, but willing to extend some - will have to see how things go)
• Break students into pairs - one student presents to the other. The watcher chooses best thing that helped understanding to praise and suggestion for one area that was vague or confusing. Then they switch roles. 
• Students have opportunity to update drafts using feedback from partners
• Selected presentations
• I will summarize and do a quick mini-lesson on extracting y=mx + b from situations, using student examples as evidence. 

I am nervous about many aspects, but also hopeful that I may have finally found a way to address concept development AND reasoning/communication at the same time. 

Oh... and by the way... we are having walk throughs during this too. Wish me luck! ;-)

Am I Crazy? Ok... Silly Question. Are You Crazy Enough to Join Me? :)

Ok - been thinking about all the great things I've discovered through the MTBoS and how it could impact my teaching for the future. 

I find that I get some cool lesson ideas as well as teaching strategies, class structures, and other independent pieces that are all awesome from the MTBoS, but I feel like what I have as a product for my students right now is rather disjointed and doesn't really flow in the way I wish it would. I am going to go on an analogy tangent here - bear with me... It's like we all have this huge 5000 piece puzzle to build (curriculum). No one actually has the picture on the box (example of pre-made perfected curriculum... because let's be honest, it doesn't exist - but besides the point for the analogy), but there is this basic instruction that says the goal is that our puzzle will be a representation of the Golden Gate Bridge (standards). We have to go find the puzzle pieces and, while it is possible to force all the pieces we find together, in reality, they are all to different puzzles. So the trick is to figure out which pieces match your idea of the "representation of the Golden Gate Bridge" and try to put them together so that it makes sense in the end. I feel like we are all frantically trying to put the pieces together into our own puzzles separately, although we do shout out to our neighbors when certain pieces seem to go, or seem to go together in a theme (MTBoS sharing). At the same time, it is such a huge amount of man-hours going into all of us separately trying to do essentially the same thing. 

So... Here's the idea: What if I could find a handful of teachers who match my content and would be willing to REALLY collaborate. Like not just make a lesson myself, teach it myself, then share afterward. I mean the long hours of hashing out what is the point of the curriculum, what are the pieces that need to be included, how to include them in a way that actually fits and then work together on making that happen. Like open source curriculum design - true "global math department." What I wish I had was a team of people that could work together to actually make units that incorporate all of the awesome stuff we are all trying to do but in a way that is more seamless, more thought out, and with a support system of other educators where we could try, discuss, tweak, etc. Does this just sound way too Pollyanna and ridiculous to you? 

Like... I'm not totally naieve. I know the input of work would be CRAZY. But seriously? It already is! If I had three people and we all put in the kind of hours I already know we all do, but were working towards a common goal, we could have something really worth all that time. I also know that we would probably disagree. And since most passionate educators are also driven and stubborn it would probably be a lot. But at the same time, wouldn't the resulting conversations be extremely fruitful in terms of pushing us all towards better pedagogy? 

I mean... it sounds cool to me. Am I crazy? If so... tell me and I'll keep playing the rat race game. But if not, go sign up on my interest form: http://bit.ly/8mathUnitDesign  and be a part of the insanity! I promise I won't really bug you too much until the end of the school year (since I don't even have time to sleep right now...). 

Anyway... That's where I am right now. Let me know what you think. And... I'm not kidding - if this is just a total ridiculous waste of time, please tell me. I don't have spare time to waste on something that is pointless. But, if you think I have a chance at making this work somehow, I'd love to try. Even if it isn't perfect, the possibility for awesomeness would be too good to pass up :)

Thanks!

Back to My "Roots"

Well... Somehow I made it through all of the blogging challenges (almost) on time, which means that in the last month I have blogged the same amount as I had over the past two years before that. So I guess that is a good thing, right? If nothing else, it has proven that even if my digital musings aren't stellar and I decide to post them anyway, no one is going to die! :)

This week, my poor chicken-scratch-covered remains of a pacing calendar had me covering simplifying radical expressions. Now, last Friday, I went to a fantastic PD by +Robert Kaplinsky where we talked a lot about HOW to bring problem-solving to life in the classroom and I was totally all about it! So I went home and hunted around for something that would relate and it was seriously like staring into a void. Now... I didn't search exhaustively because let's be real people...

At least, not in the middle of the school year.

But I am NOT KIDDING YOU - I was convinced that the #MTBoS would save me every time and I just didn't find something that I thought would build understanding of how and why we simplify imperfect square roots the way we do (then expand that to other radical expressions). 

So... I made something:

Here's a link to the entire document for those who are interested. I was having issues getting the word doc to show up correctly, so I had to convert to PDF, but if you want the word doc because you want to edit please just let me know and I can email it to you directly :)

Here's the rundown of what I was going for, how it went, and what I would do to make it better:

My Goals:

1. Build understanding that we can distribute roots over multiplication 
2. Extend understanding of equivalent expressions to include radical expressions
3. Use a specific progression of problems to demonstrate the pattern of "factoring out" perfect squares to simplify imperfect squares
4. Create "fertile ground" for students to develop the "rules" for simplifying roots without me explicitly teaching the procedure at the start. 

The Lesson Run Down:

1. Stood at the door and informed students that their warm up was the first two questions on the paper (we usually do a lot of our warm ups on whiteboards, so this was a change in procedure). I told them there was NO right or wrong answers, but they HAD to have their thinking written there. 
2. I asked what they noticed about the table of numbers. I was hoping they would see that they could multiply the numbers inside the radicals on the right to get the number inside the radical on the left. I was also hoping they would ask (or maybe even assume) that they were equal. *How it went: My first class got there and that was about it, but over the course of the day I had some really fantastic "noticings" - for example, I had students notice that the right side was always made up of a rational number times an irrational number but the left was all only irrational numbers (yay for some generalization!). I also had kids predicting what the next number(s) on the table would have been and passionately trying to convince peers who disagreed at first... it was glorious. And I didn't take a picture of our list. Because apparently I am forgetful and not quite as glorious as I wish. But you can imagine it, right?  Anyway...
3. I told them we were going to try to prove whether the two columns were equal. The students were set up in pairs today already, so I explained that each partner would use a calculator to find the value of the radial expression. We talked about order of operations and different calculators, and they set off. The DID complain when I told them that YES, in fact they did have to write that whole decimal number, but I also told them they only had to do their own column (they would compare with their partner's column after calculating). They were also to move on to the bottom two questions after comparing (so the whole front of the paper would be complete). *How it went: There were a couple of kids during the day who did weird things on their calculators and had to go back, but it was something like 3 out of 75, so not too bad there. Right away they saw that the two columns were equal, which was a blessing and a curse. That was kind of the point, but at the same time, many of them decided to stop there and turn off their thinking - like that was the pinnacle and they were done now. I had a hard time getting the early finishers to extend thinking at all and think about why I had chosen to break down the numbers on the right so that it was always a perfect square as a factor. I think perhaps I should add a question about this (i.e., "Why do you think Mrs. Aoki chose the numbers on the right the way she did?" or maybe, "Could you write another equivalent expression for the following?                            (imperfect square root) = (factored square root) = ____(fill-in-the-blank)___"   
4. I brought them back together again and took their noticing again as well as predictions for simplifying. I then had them flip over the paper and look at the example on the back. *How it went: At first, there was a lot of, "Wait, what the...?" but a lot of them started to see how it fit the pattern after a few seconds. Of course, there were also a lot who just looked confused. So we talked about how the example related to what we did on the front and it seemed like it wasn't too crazy of a leap. I do think I confused some of them because they wouldn't normally jump to 16 and 5 as their first choice of factors for 80. Most of them would have said 8 and 10. Perhaps I should have chosen a problem from the list that was a little more intuitive. Also, having the decimal approximation at the bottom was misleading. We covered approximating square roots before this lesson and I really wanted them to keep connecting that idea and practicing, but I decided it made more sense to actually do that at the top first (to the right of the original square root) and then go into simplifying. We discussed through a few and then I turned them loose again. Most got close to finishing before the end of class but didn't quite get there, so timing was pretty good. 

Overall: I was actually pretty happy with the result. I used to teach simplifying radical expressions in my Algebra I class four years ago, but having taught 7th grade Common Core for three years, it wasn't part of my duties any more. I feel like this little exploration really set them up for a stronger understanding so that when I went into more complex expressions (messier numbers and variables) students had the foundation and seemed to catch on a LOT faster than I remember in the past. 

Ok. If you are still with me - I seriously APPLAUD you because that was a long post. Thank you for your time and your eyes! If you want to really be awesome and go above and beyond, please share any ideas/improvements you might see so I can be better next year!