"Flipped Data" (PART 1 FOR Algebra 1 - Chapter 8a)

To see all data collected, please see the "FLIP Data" tab up top

I'm having a little issue with using this data to compare my current students to my 2010-2011 students because the classes were so different in terms of incoming ability level.  My students from this year were already so much lower than last year, that I don't know if I can expect to see improvement in comparison to the two years from test to test based on the flipped classroom.  But, I think I can look at improvement in the GAP that existed between the two classes.

The picture below shows comparison scores for the End of the First Semester for last year and this year.  Both courses were taught identically (no flipping, same materials, resources, tests, etc).  Right off the bat, you can see that my students this year have had more than twice the failure rate that I had last year (15.2% vs. 32.1%). I had more F's 1st semester in my three Algebra 1 classes than I think I have had in all my years teaching combined (or at least close to that!)  In addition, students receiving A's or B's at the First Semester was 9.7% lower this year than last year.

In addition to the overall semester grade comparison, I also pulled some test scores from the first semester of each year (both non-flipped) to compare the two classes and to see if the "samples" were similar...  All of these averages include the FINAL scores students received after being given the opportunity to retake the test.  In 2010-2011, the retake score was capped at 75% (school policy).  In 2011-2012, I was given permission to allow a full 100% retake of tests.  (So, the averages in 2010-2011 would have been even higher had students been able to receive a full 100% in the gradebook).

Chapter 1  2011-2012 were lower by -4.47%
Chapter 2a (did not have 2010-2011 raw data)
Chapter 2b  2011-2012 were lower by -9.77%
Chapter 3  2011-2012 were lower by -10.92%
Chapter 4  2011-2012 were lower by -8.51%
Chapter 5 & 6 tests were not comparable because the content was switched around between 2010-2011 and 2011-2012.


On average, my students this year scored 8.42% lower than my students last year on the first semester tests.

My First "Flipped" Chapter (experimental phase, no WSQ, just watching the video at home and coming to class to practice)
Chapter 7  2011-2012 were lower by -0.83%

So, with that, I will still be providing the comparison data for each test, since I have it, and we shall see where it goes.

My current students averaged 3.5% lower than last year.  At first glance, that looks bad (25 D's and F's vs. only 9 from last year on Chapter 8a).  However, when comparing the two samples of students and their prior achievement, this actually shows improvement by almost 5% from how they had been performing pre-flipped classroom.

Conversations and Observations - Every Single Student

Had a conversation with one of my Math Analysis students yesterday about the flipped class.  I guess one our other math teachers decided to have the students do a statistical analysis with some data this teacher got on the flipped classroom that "proved" it doesn't affect student achievement.  Several of my students are co-enrolled in both courses, so of course those that prefer the traditional classroom took it as an opportunity to loudly share their opinions in that class, while the others just kind of sat there quietly.  I don't really want to discuss that situation any further, but more so what came about because of it.

Anyways, this student who told me about it is one who loves the flipped class, so it was an interesting conversation to have.  We talked about one of the benefits of the flipped class in terms of use of class time.  I gave him this scenario:

"Imagine I was teaching you about the Trig Functions today.  I would have 40 of you in the 'audience', half of you listening, half of you spacing out, half of you getting it, half of you lost, half of you participating, half of you slouching back in your chair mindlessly copying notes.  Even if I was super exciting, doing crazy things up front, making amazing connections...that is still what I would see."

I then asked him to take a glance around the classroom right then.  Here is what we saw (I wish I had my video camera with me):

EVERY SINGLE STUDENT was working.  

EVERY SINGLE STUDENT was engaged.

EVERY SINGLE STUDENT was either solving a problem, leaning over helping someone else, talking about a question, or otherwise involved.

EVERY SINGLE STUDENT. 

I NEVER got that in my traditional classroom, even on my "best" lecture days.  Even when I thought I was the most engaging, innovative, and creative teacher, I still never had that.

No matter what complaints I hear, I know what I am looking for as an educator.  And, when I see more engagement, involvement, higher-order thinking, questioning, achievement, improvement, and feelings of competency than I have in my five years of teaching thus far, I know that I am doing something right.  

Or, THEY are doing something right.

THEY are the ones who are doing the work, leading the discussions, making the connections, and challenging themselves.

I am just there to guide, to provide the structure, to explain when necessary, to encourage when needed, to push when they are stuck, and to fully support their learning.

Their success is in their hands.  They have all the tools necessary to succeed.  They are responsible for their own learning.  It is up to them to do what they should with what they are provided.  That is very freeing to me, as a self-admitted overly stressed and overly involved teacher.  I can't do the learning for them; I can just provide them with everything they need to take charge of their learning and succeed.

A student-centered classroom.

I really like this change.  Although it is still a big work in progress and I am always learning new things, the Flipped Classroom approach has changed my teaching in amazing ways.  I can't wait to see what the rest of this semester holds!

Freedom...........will it work?

I'm taking a scary step...

I'm giving my Algebra 1 kids more freedom this week in my flipped class.  Just like I've been doing with the Math Analysis "waivers" (they can take a concept quiz to prove to me that they know it and they don't have to finish the required practice problems for that concept; they can move on to the next concept), I started doing that today in Algebra 1.

Things I'm excited about:

1. Let my higher kids move on and not feel bogged down with "busy work".  Might be able to do some more challenging stuff with them.

2. Motivation factor - if you do you work to actually "learn" in and get it, you might not have to do as much!  Work with a purpose, not to just "do the time"

Things I'm nervous about:

1. Can my 9th-10th graders self-evaluate enough to know when they are ready to take the quiz?  Out of the 25 or 30 kids today that took the Ch8b Concept 2 quiz, I did have 4-5 not pass it (a few weren't even close).  That means they have to fully finish the assignment.

2. Is the extra practice I assign necessary for them to make it concrete in their brain?  Is the repetition important, even if it does become a little "busy work-ish"?  Or will my students continue to remember how to do it once they've proven it to me on a quiz?

3. Will this become unmanageable since they are not as "self-directed" as my Math Analysis Honors students?  Will they be able to handle the freedom and the ability to take quizzes when they are ready and be honest about it?

We will see how this week goes and I'll blog about it in my reflection on Friday. :)

Also, this will be "survey results" week from my kids as they do their mid-semester evaluation/comments of the Flipped Class.  I'll be posting a bit each day, hopefully.

My Goals for the Flipped Classroom

I think it's about time I actually verbalized the goals that I hope to accomplish (for myself and for my students) by using the Flipped Classroom.  Then, in June, I will have a measuring stick to see if my goals were actually met or not.  This is list still a work in progress and I will be adding to it as needed.  I will be starting a new tab on the top of my blog titled "Data" that will track the progress of these goals.

For Students:

Using the Flipped Classroom ideology will increase student understanding of the material as shown by:

• An increase in the percentage of students receiving A's and B's on my Chapter/Unit Tests by an average of 5% over my 2010-2011 students (taught with a Traditional Classroom)
• A decrease in the percentage of students receiving F's on my Chapter/Unit Tests by an average of 5% less my 2010-2011 students (taught with a Traditional Classroom)
• An overall class average increase of 5% over my 2010-2011 students on each individual Chapter/Unit Test

Using the Flipped Classroom ideology will increase student motivation in math as shown by:

• A decrease the number of homework cards students receive by 20% for the entire semester. (1st semester - 219, 230, 206 = 655 HW cards in 3 Algebra 1 classes; 113, 156 = 269 HW cards in 2 Math Analysis classes)
• A consistent decrease in the number of homework cards students receive each week as we continue through the flipped classroom model.
  

For Myself:

Using the Flipped Classroom ideology will allow me as a teacher to:

• Interact with every student (ALL of them) on a daily basis in at least a short math-related conversation.
• Be able to more easily and readily assess student mastery of the content on a daily basis and provide the immediate support they need to succeed.

What are your thoughts on these goals?  Do you have similar goals for your students?  What are some goals I may be missing that I should consider setting?  All ideas and feedback welcome :)

What WSQing actually looks like...

A "WSQ" [wisk] is what we call 

"homework" in my Flipped Class.  

 *The purpose of this page is to have a place to continually update new strategies and ideas regarding using the WSQ in class.*

It is a way I have found to help my students TWRLS in math class in a way I've never seen before.

My previous posts on WSQs (read these if you don't know what a WSQ is!)

My Favorite WSQ (Jan 23, 2012)

Student's thoughts on the "WSQ" for homework (Part 1 of 2 - Algebra 1 9th & 10th graders) (Jan 26, 2012)

Student's thoughts on the "WSQ" for homework (Part 2 of 2 - Math Analysis 11th and 12th graders) (Jan 28, 2012)

Using the WSQ to deepen student understanding and academic conversations in my Flipped Classroom (Feb 21, 2012)

I don't believe a "flipped classroom" should become something where students receive direct instruction at home and then work on worksheets in class.  Class time must be made meaningful and purposeful, where students have the opportunity to (1) make meaning of the material and make connections to other content in an environment with the support of the teacher and other classmates, (2) understand the concepts at a deeper level through practice, answering and posing questions, or explaining problems/solutions to others, and (3)  receive one-on-one support and explanation from the teacher or other student "experts" when needed.  Using the WSQ has enabled me to provide this type of environment for my students in my Flipped Class in a way that is somewhat structured and holds students accountable for their work and learning.

Ways to "WSQ" in class:

*It is important to me that the following things happen in my Flipped Classroom every day:

1. Students come prepared with the video watched and WSQ completed.  If not, they must use a classroom computer to do it at the beginning of class.
2. Students are given opportunities to discuss their summaries and the key points of the lesson, practicing expressing math content in their own words and using math vocabulary in context.
3. Students are given opportunities to ask questions about the lesson and get them answered in detail during class.
4. Students are given opportunities to think critically about the lesson and pose questions to their classmates that will require deep thinking and making connections to other material.
5. Students are given opportunities to practice working out problems with the support of their classmates and myself to guide them when questions or problems arise.
6. Students are given opportunities to prove their mastery of concepts via concept quizzes that are taken when they feel they are ready.

The following is a list of ways to make sure #2,3,4 on the list above happen.  It is nice (and necessary) to provide variety for the students so the task of WSQing does not become monotonous.
 1. Whole Class

•  Pick a WSQ to put on the screen.  Read it as a class, discuss it as a class.  Ask questions about it and have students turn to their groups to answer, and then share out as a class.  "Score" it as a class.  Have students look at their own WSQs and give it a score as well.  Answer the question on the WSQ as a whole class (again, have small groups discuss it and then share out)
• Have all students get out their WSQs and SSS packets.  Together as a class, construct a summary of key points from the lesson, with each student giving one sentence or key point at a time.  Have a student up front writing the sentences either on the document camera or on the laptop for everyone to follow along.  At the end,  students add anything to their WSQ that is important and left out.  Students can ask their question to the whole class if they want to volunteer; otherwise students turn and ask their questions to their groups.

2. Small Group

•  Choose one, two, three, or all four students to read their summaries out loud with group members looking on.   Group members stop the reader, question the reader, and add to what the reader is saying as they read through their summary.  Then, reader goes over their question and the group discusses it before an answer is written down.
• Choose one student from each group to use as the base for a "perfect" summary.  The group members all look on to the one summary and break it down and tear it apart.  They cross things off, add sentences, clarify sentences, etc to make it a truly "perfect" summary of the lesson.  They are encouraged to look for places to include specific math vocabulary words in context.  Once the summary is perfect, group members look at all four questions and do the same thing - make the questions better by phrasing them more clearly, having math vocabulary in the question, and then making sure the answers are complete, detailed, and include proper explanations and vocabulary. After groups are given time to discuss, I come around and have an interview/interrogation with each group about the lesson, prompted and guided by the math vocabulary words they have written in their summaries.  See my post on the first time I did "Perfect Summary" in Math Analysis here.
• Instead of working in their groups of 4, have students switch summaries with their partners. Partners will individually read the summary and decide "if I didn't watch the video lesson last night, would this summary help me in starting the practice problems today?"   If the answer is no, the partner writes down anything that needs to be added or clarified to make the answer be "yes" and then explains to their partner what was missing.
• Students work in their small groups and go straight to their questions.  They answer the questions together with the help of their summaries, and then choose which question is "Their Best Question" to pose to the class.  This may be the question they feel is the "HOTtest", the toughest to answer, or one that they couldn't even answer themselves.  Questions are put on the board (put on sticky notes on the whiteboard, written down to project from the DocCam or Laptop, etc).  At some point in the period we discuss the questions - either as a whole class, or I assign each group to a different question to answer from the one they put on the board.

3. Individual

• Students read thru their summary individually and critique it.  They look for ways to revise it and make it better.  This would best be done near the end of a period after they have had a chance to discuss the concept and probably would have more to add to their original summary.
• Students read through their WSQ with me.  I provide the guidance, questioning, probing, and follow-up explanations that are needed to improve their summary and answer their question.

 *In any of these "methods", the questions can also be answered in a variety of ways.  The students themselves can write down the answers to their own questions once they find the answers, or the students can trade notebooks and "quiz" each other by making their groupmates write the answer to the question they posed.

Guidelines for questions:
1. Answers must  be written down.
2. Questions cannot have just a yes or no answer.  If so, the student must come up with a follow up question or an explanation beyond the yes/no.
3. Students are encouraged to make connections when answering their question and be detailed and descriptive.
4. Students should be able to discuss all questions posed in the group in detail, using correct math vocabulary and explaining the material clearly and concisely.

Presenting the Flipped Classroom (part 3 - answers to teacher questions)

This is the third in a series of posts about Presenting my Flipped Classroom at my school.  This post includes answers to questions that my staff asked after I presented it to them on Wednesday, February 8th, 2012.

To see my post about presenting to my teacher leadership team, click here.

To see my post about presenting to the whole staff, click here.

To see the Prezi I used in my presentations, click here.

*If you have any answers or resources for the questions that were asked below, please feel free to comment about them. I have answered the questions to the best of my ability with my experience with the flipped classroom. I would love to hear anyone else's perspectives and answers based on their observations.

• How do you know if students watched the video versus copying their friends' notes?  I would love to flip my classroom for grammar lessons, but I don't want students to copy each other's notes or summary just so they can avoid the homework cards.  

*Students who want to cheat will always find a way to cheat.  However, with the WSQ process I have constructed, and the way that class time is used where I can have conversations with each student/group on a daily basis, it is pretty easy to tell which kids actually did the lesson and which kids have no idea what is going on. Students who just copied notes will not be able to answer questions that require thought and connections, and will not be able to participate in the discussions to the extent that a student who did watch the video can.

*Also, if you want to, you can always do a couple of things - (1) have a quote, joke, or comment in the video that students have to know and explain the next day [yes, they can still cheat with that] (2) embed a google form at the end of the video that students have to submit to you verifying they watched it and answering a few short questions about the content.

•  How could I use this with the students with special needs for enrichment and extended learning (slower paced videos, etc.)  Also, I could see how this would be great for some students with special needs, but for others, I could see this being a disaster.  How does it account for different learning styles and disabilities? 

 *Students with special needs can benefit from the flipped classroom model because it provides an opportunity for them to learn the same depth of content as their peers but at their pace and in an environment conducive to their learning.  Videos can be watched at any pace the student needs, and because of the rewinding/pausing/re-watching capabilities a flipped classroom provides, students with special needs can take their time and go back over the material as much as they want.  Videos provide an audiovisual representation of the content for initial exposure, but it can easily be applied in ways that appeal to other learning styles outside of the video and during class time or tutoring sessions.

• What is the approximate number of students in your classes who are not doing the work of watching the videos at home and completing the summaries?  

*Algebra 1 - about 5 kids a day on average from each class (about10%).  Some days I have seen a few more, some days I have none.
*Math Analysis - maybe 1 or 2 a day?  Most days all students are prepared. (Math Analysis is also at the end of the day after lunch, so some of them watch it during lunch time) 

• Have you had comments or feedback from resistant parents?  

*I have not had any resistant parents contact me.  The ones that I have spoken with (mainly for their students getting homework cards for not doing the videos) can't believe how easy it is for their students to complete their homework now since they just have to watch a video and write a summary/question.  I have one parent who is starting to sit down with his son every night he's home and watch the video with him.

• So is the school policy changed to allow the use of student owned electronics in the classroom and on campus? 

*Students have not needed to use their own personal electronics in my class 
  during class time (they have during tutoring before/after school) since I have three 
(and more coming!) computers available for them to use if needed. If there are more 
than three students who need to watch the video, they just have to wait. However, I have 
received approval from the administration for students to use their electronic devices for 
the purpose of watching/re-watching the video lessons during class if needed. 

• I am looking forward to using "flipping", I have already spoken to some of your students who have been very positive about their experience. My only question is in using copyright images of which I use a lot of for reference.

*I do not know the legalities or answers to this question.  Maybe the specific textbook publisher where you get the images from would be able to clarify the legalities?  Anyone else have an answer to this?

• I like what I see, I just need more evidence for the CP kids. I wonder how this would compare for an Honors/AP class vs a CP class? 

*I am collecting data on both classes this semester and should have some solid evidence in either direction at the end of the semester in terms of how it affects test scores.  However, just by observation, Honors/AP students definitely pick up on the Flipped Classroom easier and do not have as many problems with motivation.  I can challenge them at a deeper level and they respond.  My CP Algebra 1 class requires a lot more training, guidance, and reminders, but I think they are coming along (now, at four weeks in).  I will be able to answer this question better at the end of the semester.

• I don't oppose doing this, but I would like to see some validated results prior to committing to the project. I would like to see a controlled test and comparisons of the outcomes.

*I am collecting data all semester comparing 2011-2012 students with 2010-2011 students.  During the first semester, in general, my current students generally matched last years' students in terms of performance (class averages, number of A's,B's, etc) for Math Analysis and were generally lower in performance overall for Algebra 1.  I am using test scores (overall averages, number of students in each grade category, etc) to make these comparisons.  See my first data post here.

More questions?  Please post them in the comments and I will do another Q/A post soon! 

Reflections on Week 4 (smooth sailing with a few rough waves to brave)

http://upload.wikimedia.org/wikipedia/commons/2/2a/Sailing-yachts.
Tuiga.Lulworth.Cambria.Cannes.2006-09-26.jpg

Week 4 of the Spring Semester is now over and I feel like this has been another great week in my Flipped Class.  I feel like everything is starting to be "smooth sailing", with a few rough waves along the way. 


All Reflections from This Year can be Found Here. 

*Each week, I spend some time personally reflecting on the week - what I did, what worked, what didn't, what I liked, what I didn't, etc.  I try to organize my reflections in a similar manner each week, since they do get pretty long: (1) Math Analysis; (2) Algebra; (3) Sharing and Collaboration; (4) Other Thoughts; (5) Running lists (Things I've heard this week that I love; Characteristics and qualities of my flipped classroom that I want to keep; Changes I've made this week that I like; Ideas I'm still contemplating and experimenting with).  I hope these reflections give you insight into my classroom and give you some ideas to try in your own flipped classroom.  I appreciate any comments, feedback, ideas, and follow-ups that you provide, so please comment and join in on the conversation! 

Math Analysis

It was a great week in Math Analysis.  I am very excited to see the test results for Unit N (they take it on Wednesday of next week, so I will have another data post soon comparing their scores to last year's non-flipped scores) and see how well they do! (I hope!!)  I feel like I am pushing my kids past the simple memorization and computation of the Unit Circle (that's what we are studying right now, and it can be very memorization-focused) and into actual understanding of why things are the way they are, why and how things connect with others, and what sort of patterns exist throughout the unit.  In my four years teaching Math Analysis, I've never really been able to have these higher-level discussions and conversations with my students, and my flipped class completely enables this to happen!  In addition, these conversations happen in small groups where participation can easily increase and student accountability is higher.

A big catalyst for helping this happen was the "My Perfect Summary" activity that I did on Wednesday.  I asked the students for feedback on Thursday and they said they didn't necessarily like the critiquing the summary part, but they really liked the questioning part.  Several students said it was "fun" and that it really made them think.  A few students said it was "easy" to answer the questions, but that it proved to them they knew what they were supposed to know.  Also, this activity helps my students see what "HOT" questions really are and how they can take a simple question and dive deeper into it.

So, how can I make "My Perfect Summary" activity more effective and fully useful for the students?  I think making the focus transition from analyzing the whole summary to including as many vocabulary words IN CONTEXT as possible (and giving them a minimum - like make sure you have 3, and then be ready for the interview/interrogation by me!...but letting them have as many as they want).  I'll try it that way next week and we'll see how it goes.

Student comments on "My Perfect Summary"

Student 1: The "My Perfect Summary" activity was very helpful and made me realize how hard it actually is to SAY what you mean about a topic.  It's easy to have important information written down because you can see it on the paper as you think, but out loud it is much harder and requires more concentration as seen in the "interview" part of the activity.  The summary part was especially helpful because it required listening to another student's work for things to add or fix.  This helped to remember some key information that I'll be sure not to exclude next time the concept comes up.

Student 2: I really enjoyed the activity that we did Wednesday, I like how you made us highlight the key math vocab, then you came over and interrogate us to see if we really knew the vocab we were using or we were just putting it in our summaries to sound smart or just because you use it. You made us think hard and it helped me because now i know the actually definition of a reference angle":) 

In other news, I do have a concern with my students watching the videos.  I post them on YouTube, SchoolTube, and then have them available offline in my classroom or via flash drive.  The majority of the students watch it on YouTube, and only a few overall watch it via flash drive because most (all but 2-3 total) have internet access at home.  I keep track of the views each video gets, and every night it should be averaging 80 views.  Most of the videos over the last couple weeks have been averaging 60 views.  Assuming the few that watch it offline, that still leaves room for a lot of students who aren't watching it but somehow coming to class with the notes, summary, and question done.  I talked with the students on Wednesday about my observations and how if they are choosing to find a way to get around the system, they aren't hurting anyone but themselves.  I also mentioned that I actually feel bad for them that they would feel the need to cheat and compromise their integrity on something as simple as a 10-15 minute video, when all they have to do is tell me and they can watch it in class.  I kept the conversation very short (about a minute), but I hope it hit home with the students.  In a flipped classroom, students are given more responsibility than most of them are used to, and they have to find a way to take charge of their learning.  A lot of them will continue to find the easy way out of actually doing the work, but I hope that by the end of this semester they have learned a lot about themselves and how to take that responsibility and greatly succeed because of it.  Friday update: I had 94 views for today's video on YouTube alone (7 on schooltube, and 5 in my classroom this morning), so that was MUCH better after our "talk". I hope it continues.

With that whole responsibility thing, I have found it very helpful to keep a visible "task list" on the standing whiteboard in my room to guide my students.  It lists out everything the students have to work on (in order by what I feel should be priorities).  Students constantly have things to do, and when they are done with one task, they move on to the next one.  It is a visible reminder to them of what needs to get done and what they can work on in case they don't feel like doing a certain task at that moment.  I am trying to give them as much ownership as possible, but still feel they need to be guided, and this allows them that freedom within my expectations and boundaries.

On a final note, on Thursday I did something new with Math Analysis.  Normally, the classwork assignment for Thursday would not be signed off until Friday, because if a few students didn't finish it for whatever reason, they would have to finish it at home in addition to the videos. However, on Thursday I told the students that they had to finish the classwork and get it signed off before 4pm that day or they would receive a "red line" (late credit) for the assignment.  I did this because I really felt a lot of students were not making good use of their time with the excuse "Oh, I'll just do it at home". (Then, of course, they complain that they have too much to do at home and I give too much homework!!).  They don't all know how to manage their time or realize that it is more beneficial to get it done now than to have to do it later.  Anyways, it went well and I don't think it is something I will do every day, because it did make several students feel very pressured and rushed, but every so often, I think it's good to throw it in there to keep them on their toes and focused.

Algebra 1

Algebra 1 was definitely an up and down week.  We started off pretty miserably on Tuesday after the long weekend.  I had the most number of students ever come to class not having watched the videos.  (11 in period 1, 7 in period 3, 9 in period 4; compared to usually 3-5 max each day in each class).  I had mixed feelings about this - one thought was "I'll have more than usual because the kids will be out of town" while the other was "I should have no kids missing because they had a whole extra day to watch it".

Of course, it had to be the day that every principal in our district (ES, IS, HS) came to our campus to do walk throughs of all the classrooms on our entire campus.  I was upset/nervous/anxious at the beginning of period 1, knowing that the administrators walking through would not see exactly what I wanted them to.  However, it ended up not being too bad.  I have three classroom computers now (the tech guy told me I should have two more within the week!), so we just rotated the kids and about 8-9 of them got the video watched in class.  I had three administrators walk through (one of them being the AP I work with a lot and he was able to explain to the other two what was going on).  They talked with the kids, they saw what they were writing, they listened to their conversations, etc.  I didn't talk with them personally, but I did overhear them whispering to each other, "They're writing in a math class!".  Our AP will be presenting results/data on Monday so we'll see if he mentions anything.

On Wednesday, I tried something new. It was the day before the test, and I decided to give them a little more freedom (with a "task list", of course) and see how they handled it.  I actually really liked it.  The kids had three things they could work on - review problems from previous chapters they would see on the test, review problems from the current chapter they would see on the test, or taking/retaking concept quizzes for the five concepts in this chapter.  I was up front most of the time and the kids were bringing me their quizzes and I was grading them automatically to give them feedback.  If they got something wrong, I either (1) tried to quickly identify their error if it was small, (2) circle where they made their mistake and tell them to go fix it and bring it back, or (3) (if they had no idea how to do it), told them to go grab a computer and re-watch the lesson.  The period flew by, and I feel it was really productive.  I will be grading those tests this weekend and will post the results with data soon.

Upsides - Students worked on what they needed to work on. Students were given independence and freedom.  Students received immediate feedback on what they were doing right or wrong.

Downsides - there were kids who didn't bring me a quiz all period.  I was not walking around to monitor so some kids I'm sure were off task. I gave them the guidelines/protocol for taking the quizzes (put everything away, don't talk to anyone), but didn't fully monitor it because I was up front giving feedback.

Student thoughts on the independence/freedom:

Student 1: I think it was a really great idea because we have our opportunities to re-take our quizzes at times when we don't have time or we can't just make it. But it will improve our quiz scores, so we'll improve much better on our chapter test.  :)

Today (Friday) we started our new chapter on Factoring.  Here are some goals I have for myself for Algebra 1 this chapter:

1. Re-explain the WSQV chart.  I guess my students are confused by what they are supposed to do.  I'm not sure why they are confused, but I've received requests to clarify it, so I will.  There's a column labeled classwork that says "Chapter 8a PQ2" and then they have a worksheet that is titled "Chapter 8a Practice Quizzes" (that's what PQ stands for) and on there is a heading labeled "Chapter 8a Concept 2".  We've done it all year.  Who knows.  There's another column labeled "Homework assignment (video)".  In the column is the title of the video they need to watch "Chapter 8a Concept 4".  If they don't have a video that night, it says "No video" followed by the actual assignment title. I did this on Friday in class and hopefully it is clear.  Some kids laughed when I told them I was going to explain the chart because they thought it was so easy to understand.  But, what will it hurt to explain it one more time? :)

2. Work with them as a class on their summaries so they know what I want and so they are more meaningful to them.  Their summaries are still pretty crappy overall and don't show me any deep thinking or connections.   I think some modeling is needed. I did this on Friday for Chapter 8b Concept 1 (Divisibility rules and GCF's).  We wrote a complete summary as a class and talked about what I am looking for.  I told them that right now I am getting "short and crappy" summaries, and I don't want my example (which was pretty long) to make them think that I want "long and crappy" summaries.  I gave them a visual of how a summary needs to include all of the important information that is given in the lesson and not just bits and pieces to fill up the minimum five sentences.  I am hoping to see improvement next week.

3. Give them question starters from the HOT question list so they can start asking better questions.  I don't think giving them the whole list would be helpful.  I need to pull out good question starters for them and give them a modified list.  I'll work on that this weekend and post it here.

Sharing and Collaboration

My "Teachers Using the Flipped Classroom" survey is up to 45 responses, but I am still looking for more before I close the survey on Sunday night to collect the data.  I will still accept responses after that in case people come across this post too late, but I am going to use that as the stopping point for looking at the information as a whole. 

This week will include two Data Posts.  One for Algebra 1 Chapter 8a (later this weekend) and one for Math Analysis Unit N (end of next week).  I'll link to those here once they are written.

Other Thoughts

Differentiation

One of the things I'm loving about the flipped classroom (and am looking forward to even more next year when I have all my units and videos finished ahead of time!) is the ability for students to work ahead.  This year, I only have one student really taking advantage of it fully.  To give you an idea, the class just finished learning the material for Unit N today.  This student took the Unit N test a week ago and is taking the Unit O test tomorrow. I think he finds pride in a couple of things: (1) knowing he is ahead of the class (2) being able to catch mistakes I may make before the rest of the class (3) being my "answer key" - now I hardly make answer keys because I just use his because he does it a couple weeks ahead of the class.

He is limited by the materials I have ready. I try to be a unit or two ahead of the class, but depending on time, it's not always that way.  Next year I would like that option to be open to more students.

Math Analysis Summer Packet and the Flipped Classroom

Several years ago, we would give our incoming Math Analysis students a summer packet to do to prepare for the class.  We stopped doing that the last three years (don't know why), but I'm really thinking about bringing it back.  Our first four units (about the first month of school) are really a lot of review.  I'm thinking that with this whole flipped classroom idea, students could get the packets and video links for the first few units and those could be completed over the summer (our school is big on summer homework - they have summer reading for English, summer stuff for history, etc.  I think they want to keep the kids busy so they don't get into the bad stuff they could in the area we live in).  Because of the videos, students would be able to get help on the material they don't remember from Algebra 2 and be given all the resources needed to start the year off great.  Then, when they come back to school, we could spend one week reviewing the material and testing them.  We could jump into the new stuff and that would give us a lot more time throughout the year to cover certain concepts in depth and to re-instate some of my favorite things we have had to cut out because of time (parametrics, polar equations, 3-D graphs, etc).  I'm talking with the other Math Analysis teachers about it and we'll see.  The only downside is I didn't start recording videos this year until Unit F, so I would have to get Units A-D recorded before summer break.  It's doable, but it will take planning...

THINGS I'VE HEARD THIS WEEK THAT I LOVE: 

(I had students fill out a "Second Semester Survey" to give me feedback on the Flipped Classroom so far this semester.  Here are some notable quotes.  Once I receive all 200 responses (it is due Tuesday) I will write a post about the results. 

Question: What have you learned about yourself as a student because of the Flipped Classroom?

I learned that it is helping me become more responsible as a student and later on as i grow older i will have learned to be responsible for myself

I learned that I am responsible for my own learning. I realized that if I want to succeed, it is all up to me.

I learned that I can work in a roomful of people talking about different things at one time.  I can easily ignore conversations that do not fit into what I am doing at the moment. 

I am responsible for my own learning and if i do not get the material, i can always go back and re-watch the videos. 

I've learned that maybe you're smarter than you think you are.

I have learned, that even as a busy student, with the flipped classroom I can manage my time wisely. I have learned to also become more productive. 

What I have learned is that it helps me more than a traditional classroom. It allows me to try to manage my time more as well as understand it

I have learned that when left to my own abilities, as I no doubt will be in college, I need to intentionally set time aside to watch videos and take responsibility for my learning. 

i think I've learned that if you are actually trying at something then you can do it. because in the beginning of flipped classroom I didn't try and I almost got a homework card but now I'm actually trying and I got a perfect WSQ chart.

I've learned how to be responsible and how willing i am to make a commitment.

what I learned was that I'm capable of learning things and become successful in my math class only if I stop being lazy and actually do my homework and learn what I'm suppose to be learning.

I learned to help others in the flip class room. It has help me a lot this flip class room.

I do much better in my class work, I get my work done

I learned that I am actually responsible about my education than I thought I was, since I never miss a video and always do my homework.

That I can be very lazy, and that I need to spend more time trying to make myself accountable for my learning. It teaches me that if I don't pay attention, i can often find myself not understanding the concept at all.

I learned that I learn better in a more convenient environment where there are less distractions.

As a student, using flipped classrooms, I notice that I actually do have a problem in focusing when a teacher is talking. My surroundings do affect and distract me. Learning at home and asking for help in school to the teacher or peers is better than sitting at home asking the wall.

CHARACTERISTICS AND QUALITIES OF MY FLIPPED CLASSROOM THAT I WANT TO KEEP (running list each week):

1. Students MUST ALWAYS have a written answer to the question part of their WSQ.  This can be written by them individually or answered with the help of a group member or myself.  The questions must require more than just a Yes or No answer.   This ensures that their question does get answered, and it forces students to practice using academic language in writing.

2. Picking a place in the classroom next to a group and sitting on my stool for a while, helping if needed, but listening in and guiding the group along.  (did not do that as much this week, but I like it). This includes sitting down to work with them if they have problems, but also sitting down and just questioning them, probing them, and getting them to think, speak, and make connections.

3. In the videos, always have at least an example or two that students need to work out on their own.  Two ways to do this - #1 - in the middle of the video, tell them to pause it and try it on their own.  Then they can follow along with me once they get stuck, and then pause me again and try from there. #2 - at the end of the video, assign 1-2 problems for the students to complete on their own before class.  Work out these problems in a "part 2" of the video for students to reference if they still get stuck. 

4. Every so often, make the classwork due at the end of class to keep students on their toes and to keep them on task during class.

CHANGES I'VE MADE THIS WEEK THAT I LIKE: 

1. "My Perfect Summary" - In Math analysis, I added a questioning/probing time with each individual group.  I liked that part better than the actual "make the summary perfect" part.  Watching the light bulbs come on in my students' brains were amazing!
2. More independence and freedom (but still very structure and guided) for my Algebra 1 kids.

IDEAS I'M STILL CONTEMPLATING & EXPERIMENTING WITH (running list each week with updates):

1. "Waiver" for assignments once students have shown mastery on a quiz- I think it went really well in Math Analysis this week and I think I will be keeping this.  One thing I want to change is that they have been "waived" from all future assignments for that concept (practice, videos, and WSQs).  I think they are just going to be waived from the practice, but they still need to watch the videos.  There is information presented in the videos that they need to hear, even if they don't need to do all the extra practice.

2. Coming up with a list of "key questions" myself for each concept to have handy to ask students, to have students discuss in groups, and to show students what "good, HOT questions" look like and sound like (modeling)... still haven't had time to really think through that for this week, but still want to do it.  It might turn into a summer task to kick off next year.

3. Letting the students pick their own groups and who they work with in class.  I think I want them in their set groups for the WSQ discussions, but then let them know they are free to work with whoever they want.

Thoughts, comments, ideas, your own experiences? Please share!!!
All Reflections from This Year can be Found Here. 

Today my flipped class was... amazing

Wow!  If every day were like today... all I can say is wow!

My Algebra 1 classes were great, and I'll talk about that in my weekly reflection on Friday or Saturday.  What makes me so excited was what I did for the first time in my Math Analysis class.

I've talked about "My Perfect Summary" before, but this is the first time I introduced it to my Math Analysis class.  Here is the premise.


1. Students pick one notebook/WSQ to use as their starting piece.
2. Students read it together and cross things off that don't make sense, add phrases or even complete sentences to the summary to make it "perfect".
3. Students make sure that they have at least 3 math vocabulary words used correctly in context and highlight those in the summary.
4. Students go around and answer each others' HOT questions from their individual notebooks and write down those answers as well.

Then comes the fun part...

I get to spend about 4-5 minutes with each group (I have 9 groups total in my class, 54 minute periods) basically grilling them about the content.  I use their summary as a guide, but mainly focus on their math vocabulary words.  I ask them everything about them and have them go deep into the content.  I ask follow-up questions and continue to probe deeper.  The funny thing is, students think they can just carelessly highlight random math vocabulary words, but with this, they actually have to KNOW them, and know them WELL.  It was AMAZING to see the students brains working and to see the material clicking and connecting together in ways that I don't think it had before!  I saw them go from not being able to explain a word or concept to being able to clearly and concisely explain and make connections within a matter of 5 minutes!

Example 1:  I saw the word "quadrant" in their summary.  Here is the list of questions that came from this (with student answers between every question that led to the next one)


What is a quadrant?
How are the four quadrants labeled?
How do the quadrants relate to trig functions?
Which trig functions are positive or negative in the third quadrant?
Why would tangent be positive in the third quadrant if both parts of the ordered pair are negative?
What about the fourth quadrant? Why would cosine be able to be positive?


Example 2: "trig function"


What is a trig function?
What trig functions relate to each other and how?
What are the ratios for the six trig functions?
How do the ratios correspond for the quadrants in which each trig function is positive or negative?
What trig functions can have values greater than 1?
What trig functions could possible have values of exactly 1? -1? 0? undefined?


Example 3: "reference angle"


Students' original definition included them showing me with their arms and pointing.  "That thing" and "closer to there" and "that gap here" is what they used.  Together, they were able to identify the parts (terminal side, closest x-axis) and the qualities (must be positive, must be acute) to define a reference angle concisely as:
"A positive, acute angle located between the terminal side of the angle and the closest x-axis.  This angle corresponds to three other angles around the unit circle by having the same ordered pair values (but different signs)"


...the list of follow-up questions could go on and on, but I limit myself to 5 minutes of questioning with each group...

While I am having these sessions with each group, the other students are working on practice, taking quizzes, and finishing their own discussions to prepare for my arrival.  I have a feeling next time we do this in class they will be more prepared.  Today I think they were a little shocked and how I kept asking them questions, but they liked it.

These are all questions I would have loved to do in a whole class discussion, and I may have done in the past.  But, as well all know, a whole class discussion like that engages maybe 10% of the learners and not everyone participates.  Doing this with a "My Perfect Summary" means I get to do that same questioning with 3-4 students and everyone is engaged, involved, thinking, and participating!

Talk about Thinking, Writing, Reading, Listening, and Speaking!!!


Now that is an amazing flipped class!

*Read a few student's thoughts on this activity in the Math Analysis section of my Week 4 reflection!

Here is what the WSQ's looked like today... sorry they are sideways!  Click on them to read them closer!

Using the WSQ to deepen student understanding and academic conversations in my Flipped Classroom

See the "Revisited" version of this post, published in August of 2016, by clicking here.

Tomorrow is my one month blogging anniversary, and I am celebrating over 2,700 views!!  When I first launched my blog on January 22nd, 2012, I was stepping bravely into somewhat unknown territory.  I was about to start fully flipping all five sections (2 Math Analysis, 3 Algebra 1) of math that I teach for the second semester of this school year.  My definition of fully flipping is that all lessons would be taught via video where students would have constant access to the material anytime, anywhere.  Most lessons were previewed with a video, while some lessons included videos that would review or supplement what was discussed in class.   I knew I wanted a place to reflect and make sense of all of the changes that would be coming, as well as to have a way to connect and share ideas with other educators using the Flipped Classroom. 

It has been an amazing month of personal and professional growth!  

If you have never blogged before, I highly suggest giving it a try - it has helped me immensely.  If you are a reader, but never join in on the conversation, I urge you to step out and share your views.  It's amazing how much we can learn from each other!

When thinking about what I wanted my Flipped Classroom to look like, feel like, sound like, etc., I had come up with the idea that I wanted my students doing more than passively watching the videos - I wanted them Thinking, Writing, Reading, Listening, and Speaking every day in my Flipped Classroom.  Thus, came the development of what I called the "WSQ" (pronounced 'wisk').

In the last month, I have adjusted, modified, and developed the WSQ beyond my original post, so I thought it would be good to give an update.

I truly believe the WSQ is a way to 
(1) deepen student understanding of the material (they have to be able to explain it in their own words, they have to ask detailed questions and answer them, they have to be able to hold a conversation with a group using math vocabulary in a way that makes sense.  I question them and probe them daily and if they don't make sense, I keep probing!)
 and 
(2) increase academic conversations (they are forced to talk about the material and not just blindly follow protocol and work problems out.  I walk around and am just asking, questioning, probing, and checking for understanding, and trying to make them do most of the talking.  Sometimes a student will ask me a question and I will jump into answer it right when their partner jumps in.  I stop and let the classmate answer because I want them doing the thinking and speaking as much as possible.)
in my math class.  

WATCH

• All students are required to watch the videos on a nightly basis and take notes in their SSS notes packets that correspond to what I am writing on in the videos. I check to see that students have written down the important information I talked about the video; highlighted key information, and worked out the few problems I instructed them to try on their own before class.  It is usually evident to me who actually watched the video and who just "watched" or didn't watch at all.
• Students usually have 4-5 videos a week.  The only time they generally bring home "regular" practice homework is on the night before a Unit Test.  I have chosen to keep it this way to maintain consistency.  At the beginning of the year when I was just testing the waters, we would flip one lesson but not the other and the students (and myself) got confused on if they were supposed to watch a video or do regular homework.  I like having consistency.  However, I still have the opportunity to teach or preview a lesson before students watch the video if I see fit.
• Videos range from 8-15 minutes long.  I try to keep them short and cut them into Parts if they extend past 10 minutes.
• Most videos include a "Part 1" - theory, instructions, vocabulary, introductory examples as well as a "Part 2" (and even sometimes "Part 3" or "Part 4" of additional examples for students to view). 
• If we spend more than one day on a concept, students will still watch a video each night.  The first night will be more introductory, and then after we work on the concept in class, they will watch a second video that is a little more advanced.
• Students may work ahead and watch videos ahead of schedule if needed or desired.  Students may watch videos in class if they choose, and work on practice at home.  This is not suggested, but is an option available.
• Students are encouraged to rewind and rewatch videos, and also to pause and try examples on their own before watching me work through it or checking their work compared to mine (see "The Power of FFW" - a great post by a colleague at flippingmath.  I always try to have an example or two that I specifically ask the students to pause and try on their own, or leave a few examples for them to try at the end of the video on their own before class (of which they can watch me work out in "Part 2" of the video).  I am starting to encourage students to take initiative and try problems on their own in the video before they watch me try them.
• See my video library on my YouTube channel. Videos are also available on SchoolTube under the same titles as the YouTube videos (you just have to search for them by name)

SUMMARY

• After watching the video, students write a summary of the important pieces of the video.
• Summary must be a minimum of 5 sentences long, but most are much longer.  My lower-level Algebra 1 students struggle with writing complete and detailed summaries.  Students are given sentence starters to help them in writing a minimum of five sentences.
• The purpose of having students write a summary is to try to put in their own words what they just saw and heard on the screen.  It is easy to copy notes and then say "oh yeah, I got this. That made sense."  It is a whole new level for them to watch the video and then have to condense the information into a comprehensible summary, using their own words.
• I am pushing students to be using math vocabulary in their summaries.  In certain instances, I make them highlight all of the math vocabulary that is used in the summary (minimum of three words per summary).  I may even have them define those vocabulary words in their own phrasing in the "Answer" portion of their WSQ in class.
• Students will complain about the summary, because it makes them think.  I like it :) 

QUESTION

• Students have to come up with a question about something from the video.  Sometimes this is a question about something they didn't quite get, but more often than not it is a question they have to come up with that (1) someone else might have or (2) is an important piece about the concept.
• Questions cannot be answered with a simple yes/no.  Students will be asked to re-write their question if this is the case.  See the "Algebra 1" section of my Week 3 Reflection for some details on this.
• All questions must have a written answer after them.  This is done in class in a variety of different ways.  Sometimes I have the students write the answers to their own questions and sometimes I have them trade notebooks and write the answer down in their partner's notebook.  Either way, students are given the opportunity to ask the question, discuss the question, and make sure to get the right answer.
• I really try to push my students (mainly Math Analysis at this point) to as HOTter questions (Higher Order Thinking) that move up Bloom's Taxonomy from basic recall and understanding to application, analysis, synthesis, and evaluation.  They are getting better at it, but I still often see those very basic recall or comprehension questions.  When I see those, I ask them follow-up questions and probe deeper, and then have them write the answers to all of those follow-up questions as well.  With probing, we can turn most of their "non-HOT" questions into "HOTter" ones.

IN CLASS TIME?
 I am still trying to come up with different "activities" to vary the in-class time.  My Algebra 1 class is definitely much more structured, while I am slowly letting my Math Analysis class have more freedom and take full control of their pacing and learning.  Here's what a sample day may look like:

Algebra 1:
1. Students come in and get out their WSQ's (kept in a spiral notebook), SSS packets, and WSQ sign-off charts. *Students who have not watched the video must tell me before the tardy bell rings and they start watching it on a classroom computer.  I usually only have about 4-5 kids who don't watch it, however I have had bad days where 10-11 don't watch it (I have 36-38 students per class).

2. We generally do a quick 1-2 minute review of the important pieces of the video, and if a student has a dire question they want to ask to the whole class, we go over it.

3. Students get in their WSQ groups and do one of the WSQing methods I am developing (see below).  All WSQing methods require that students get a written answer to their question.

4. While students are WSQing, I am walking around, listening in, commenting, and probing with deeper questions.  This process takes 10-15 minutes, depending on the concept and on the group's ability to stay on task.

5. Students get to work on the assigned practice problems for the remainder of the period (30-45 minutes).  I walk around and sign off on their WSQ charts.  I do not sign off on the summary if it is not complete (at least 5 sentences), and I do not sign off on the question if it is not answered.  At this time, students are allowed to work with people outside of their WSQ group and possibly move seats to work with a different group.

6. If students finish the assignment early, they may start watching the new video for that night.

7. For each concept, students must take a concept quiz.  This is a day or two after they have learned the concept.  Right now, the whole class takes the quiz on the same day at the same time, usually after the WSQ and before the practice problems.  I am hoping to modify this as my students get more used to the process (like I am currently doing in my Math Analysis classes).

Math Analysis:

1. Students come in and get out their WSQ's (kept in a spiral notebook), SSS packets, and WSQ sign-off charts. *Students who have not watched the video must tell me before the tardy bell rings and they start watching it on a classroom computer. I generally only have 1-2 students a day not prepared.

2. We generally do a quick 1-2 minute review of the important pieces of the video, and if a student has a dire question they want to ask to the whole class, we go over it.  Usually if a question is asked in Math Analysis, before giving the answer, I ask the groups to discuss the answer and then have one of them share out.  That way, they are still doing the thinking and I can just guide, support, and follow-up.

3. I go over the goals for the day, posted on a "Task List" on one of my standing whiteboards.  This includes everything the students need to try to get done, reminders about anything that needs to be turned in, etc.  Then I let them loose to work.  I try to give them more freedom than my Algebra 1's and they generally do very well with it.

4.  The WSQing process for Math Analysis is a little less structured.  My expectations for them are that they discuss their summaries in their groups and get their question answered somehow.  Some days I may give them more specific directions (today I want you to have all your math vocabulary highlighted and/or defined; today I want you to develop one "perfect" summary from your group before you get started on the problems, today I want you to pick your groups "Hottest" question and post it on the whiteboard, etc).  Much of this is done on a more small group basis rather than as a whole class.

5. Students get to work on their Task List for the day.  This includes practice problems, concept quizzes, and videos.  Some students may need to watch or re-watch a video.  Some students just need to practice a few problems and then they are ready for the quiz.  Some students need to spend the whole period practicing the problems.  I have started to let them have more and more freedom to self-monitor and self-assess.  While I have a few students work ahead, most students are still on pace with the daily expectations.
6. I walk around and listen, probe, question, and correct while signing off their WSQ charts for the day.  If I feel the need to go over something as a whole class because I am noticing common misconceptions, or if I want to do a quick check of understanding and have every group solve a problem and show me the answer (like on a group whiteboard or something), that can happen.  It all depends on what I am seeing from the students.


Ways to "WSQ" in class:
1. Whole Class

•  Pick a WSQ to put on the screen.  Read it as a class, discuss it as a class.  Ask questions about it and have students turn to their groups to answer, and then share out as a class.  "Score" it as a class.  Have students look at their own WSQs and give it a score as well.  Answer the question on the WSQ as a whole class (again, have small groups discuss it and then share out)

2. Small Group

•  Choose one, two, three, or all four students to read their summaries out loud with group members looking on.   Group members stop the reader, question the reader, and add to what the reader is saying as they read through their summary.  Then, reader goes over their question and the group discusses it before an answer is written down.
• Choose one student from each group to use as the base for a "perfect" summary.  The group members all look on to the one summary and break it down and tear it apart.  They cross things off, add sentences, clarify sentences, etc to make it a truly "perfect" summary of the lesson.  They are encouraged to look for places to include specific math vocabulary words in context.  Once the summary is perfect, group members look at all four questions and do the same thing - make the questions better by phrasing them more clearly, having math vocabulary in the question, and then making sure the answers are complete, detailed, and include proper explanations and vocabulary.

3. Individual

• Students read thru their summary individually and critique it.  They look for ways to revise it and make it better.  This would best be done near the end of a period after they have had a chance to discuss the concept and probably would have more to add to their original summary.
• Students read through their WSQ with me.  I provide the guidance, questioning, probing, and follow-up explanations that are needed to improve their summary and answer their question.

WSQ samples from today's class.  Here are the "perfect summaries" two of my Algebra 1 classes came up with.  Notice the things crossed out, added, and highlighted.  Students discuss and work together through this exercise.  Then, they answer each other's questions.  My Algebra 1's are still struggling with coming up with good questions, but they get better once I probe them. You can click on the image to see it better to actually read what they wrote.

This answer needs much more detail that we would discuss together in class... What would a 3x3 look like?  Would the same patterns we saw in  a 2x2 and 2x3 still exist?  What were those patterns?

Same probing questions as the previous WSQ... what would it look like?  What patterns would exist?

This question required a conversation about the patterns that exist when combining like terms and what we call them (diagonals), because you can see their first answer was not correct.

This was a great extension question to the box method - what to do when we are multiplying three sets of binomials together?  I was able to draw out and explain what would happen.

Again, another example of a Q/A that needed to be discussed and probed further in class - okay, so what do you think would be the same about it?

Answer: "Depends on the degrees"... okay, so what would it depend on exactly? Can you show me an example of something you could combine and something you couldn't combine?