Units for Algebra 1

I had SBG and Learning Targets forced on me (and I don't mean that in a bad way) this past year.  The school I started at had been working to implement these for somewhere around 5 years (not exactly sure since I wasn't there).  And the teacher that left didn't leave anything for me.  So I fumbled around a lot at the beginning of the year trying to figure out what other teachers did, why they did that, what the benefits were, and how to apply it to my classroom.  What I ended up with are some poorly organized units with learning targets that are OK.  One of my goals this summer is to create some more solid units (with assessable learning targets) that are organized, teachable, and centered around the standards.

Our district's guideline for core math classes is Appendix A [pdf].  (I really don't want to get into an argument over whether this is a good choice or not.  I was not at the district when the choice was made, therefore I have little say.  Also I don't hate it.)  This means that the standards Appendix A suggests for Algebra 1 are the standards I have to teach, and I need to make sure my students demonstrate them to the level that Appendix A recommends.

I do not have to teach the Appendix A units, but it seems like a good place to start.  However Appendix A units are big, so I wanted to break them down into sections/modules/something smaller and more manageable.  What makes sense to me is to group certain standards together and create learning targets from there.  However I have kind of hit a wall trying to do this.  I've been working with Unit 1 and trying to break it into 2 or 3 sections/modules/whatever, but I am hesitant as to what is best.

The goal of Unit 1 is to continue students' work with linear equations and increase fluency/adaptability.  Students should already be able to solve linear equations and some simple systems.  Students analyze the process of solving and practice moving between forms of an equation (summarized from the Appendix A document).  To me, Appendix A is an opportunity for students to apply and play with linear equations and expressions.

Here are the recommended standards:

• N.Q.1 Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.
• N.Q.2 Define appropriate quantities for the purpose of descriptive modeling.
• N.Q.3 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities. 
• A.SSE.1 Interpret expressions that represent a quantity in terms of its context.
• A.CED.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.
• A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
• A.CED.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.
• A.CED.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm’s lawV = IR to highlight resistance R.
• A.REI.1 Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.
• A.REI.3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.

(Sorry for the list--please don't stop reading! I think you need this to tell if my grouping makes sense.  I'll try to not blabber much more.)

Group 1:  N.Q.1, 2, 3/A.CED.3
Mostly focused on precisely modeling and appropriately using units to assist with this.  (More "simple" modeling situations.)  Could get into expression vs. equation as that is an important distinction, but a lot of modeling situations could be solved either way.

Group 2:  A.CED.1, 2, 3, 4/A.SSE.1/A.REI.1, 3
Focused both on modeling and on the "pure" algebra.  Would get into more "complex" modeling situations where restraints on domain/range may come into play.

I have also thought of breaking a few from Group 2 to form
Group 3:  A.CED.4/A.REI.1, 3
This would allow a section focused on the algebra outside of modeling (not always, but a little bit more). Not sure if I want this or not...

Ok, so what do you think?  Two groups or three?  What are the pros and cons of each?  Would you do it a completely different way?  I am not an expert, so please tell me what you think.  I need something to help me continue moving forward in my planning for next year.  Thanks!

No Averaging!

I am here to sort through my issues with using SBG to determine an effective final grade.  I've already decided that I'm not going to allow any scores below a 2 (partial proficiency)...well maybe I'll extend it to allow some 1.5s, but definitely not 1s or 0.5s or 0s.  Students who score below this level (whatever I decide) will have to re-learn and re-assess.

The big challenge I'm thinking about though, is how to accurately give students a final grade.  I want it to be fair and make sense.  This past year I averaged scores students had earned (and I was not good about requiring re-learning/re-assessing), and I was very dissatisfied with the results.  I really don't feel like averaging everything is what I want to do.  (Consider this extreme example for why I dislike averaging.  For a student to pass Algebra s/he should meet these standards:  a, b, c, and d.  Say a student shows adv. prof on a and b and receives 4s, but then shows low learning on c and d and receives 1s.  This would average to a 2.5, which I believe in my grade book would be a C+.  But is this student really ready to move on to Geometry/Algebra 2 if they have failed to show proficiency for half of the Algebra standards?  This is the issue for me.)

Also, I want something that is more clear for students.  I could never predict how one retake would affect their grade, and neither could they.  My goal is to set up a rubric of sorts to determine their final grades (which we only give out 2/year--semester grades).  I'm not sure if this is something PowerSchool can do for me or if it is something I will have to do on my own.  I'm pretty sure Active Grade (http://activegrade.com/) can do it, but not sure I want to try to operate 2 grade books.  I may just have to hand calculate a few times a year.

Here are some ideas:

• To earn an A, 50% of the scores must be 3.5 or 4, and no more than 5% are 2.
• To earn a B, 50% of the scores must be 3 or above, and no more than 10% are 2.
• To earn a C, 25% of scores are 3 or above, and no more than 25% of scores are 2.
• To earn a D, 100% of scores are at or above a 2
• To earn an F, any scores below a 2.

This is just my thought for now.  Would love to hear feedback on what others are thinking.  My school does give more than just straight grades, so I would have to add more for A-, B+, B-, etc.  Which will be complicated...especially since there will be around 20 scores per semester (I think...haven't finished breaking down all standards into learning targets yet).

I'm done.  Now it's your turn.  Do you think I'm crazy or not?

Assessment

I have been reading Fair Isn't Always Equal  by Rick Wormeli and it has made me think a lot about assessment.  Obviously working through SBG makes me think a lot about assessment too, but through that and my conversations with math tweeps, I think I have enough thoughts that I need to process them.

Here are some random thoughts to start:

• Quizzes are a type of formative assessment; need to be used that way
• I want to pre-assess all my students for each learning target
• I want the pre-assessments and formative assessments to be a learning experience for students
• I don't want my students to feel like they are constantly being quizzed/tested
• I need more than just one data point to score a student on a learning target (thinking in terms of SBG)--this is where formative assessment comes in
• I want to assess retention of learning as well, not just once
• I want to allow reassessment to encourage (and sometimes require) students to relearn (probably a different post later...)
• If pre-assessments and formative assessments are used well few students should need to reassess

After a conversation with Garnet Hillman at #sbgchat, I decided that I want to report out achievement, growth, and behavior for each student.  I had already determined that I wanted to report out achievement and behavior, but I hadn't really considered growth.  (Not sure why, but I'm getting off task, so I won't reflect on that now).  However, growth is an important variable to measure, and with a pre-assessment, not that difficult.  So I added it to my lengthening list of goals for next year.  Achievement will be the only aspect that affects the letter grade students receive in the class (next blog post on constructing final grade), but reporting out behavior and growth are important as well.

Wormeli recommends that pre-assessments be as similar to the summative assessment as possible.  The goal is to pull out the basic ideas from the summative and give a short and sweet pre-asssessment.  I'm think that my pre-assessments will only be measurements of 0 (no proficiency), 1 (minimally proficient), 2 (near proficient), or 3 (proficient).  I will likely exclude advanced proficiency from pre-assessments mainly for the sake of simplifying and shortening the assessment.  (If you have comments on this feel free to leave them!)  

But like I said, I want pre-assessments to be more than just one-and-done.  I want to be able to use them for learning experiences.  One way I thought I could do this would be to help students prepare for the summative assessment.  Here's is how it would go down.  I would give groups of 2-3 the pre-assessment again (one copy).  Group 1 is instructed to work through the assessment at a 2 level.  Group 2 is instructed to work through the assessment at a 3 level.  (They wouldn't know what other groups had been assigned.)  Then group one and two switch papers and score the paper they were given.  Then either as double-groups or a class, students would discuss why they gave the score they gave and students would defend if they feel the "earned" a higher (or lower) score as they were initially instructed.

This past year in my implementation of SBG, my quizzes and tests were organized by Learning Targets, and students received a score on each LT.  I started giving quizzes on 1-2 LT to show students they could be successful if they would work on retention (low test scores after knowing students had learned at one point). I included both the quiz and the test score in the grade book.  I want to have only one score for each LT next year, based on multiple data points over time that demonstrate a student's learning.  (I'm not sure if I'll do one score per standard or LT yet, but that isn't a huge difference.)

With this change I know quizzes still have a use, but it is likely much more formative.  I would still like to score them myself so that students have feedback from me, but I'm wondering how else I could use them as a learning experience...I might have students use their IN to reflect after taking the assessment and then again after receiving my feedback.  I'd also like to have students do corrections on their quizzes.  Might be fun to have the class collect and organize common mistakes.  Could lead to good analysis...

Not sure what to do with students who then demonstrate proficiency...or how much instruction time would be needed between quiz and summative assessment (do I need to do more instruction?).  However, I think I'm getting ahead of myself.  I can't plan everything out ahead of time, because I need to meet my students where they are at.  So I guess I will have to play it by ear...however I don't think I'm going to less proficient quizzes "test" out of the summative.  I want to summative to ensure some sort of retention.

Even with a pre-assessment and a quiz as formative assessment, I want more formative assessment to gauge where students are and to record as data to support the learning I feel students have (that then get included in their grades).  So I was asking some tweeps for ideas and these are some of their comments:

• Scott Hills:  my favorite no (I've never heard of this before, so I'll have to look into it a little more), twitter, exit ticket, navigator, conversations
• Summer Sartain:  3-2-1s, exit tickets, clickers, muddiest point, journaling, wiki

Of these the ones I can see working in my classroom (not even evaluating the actual effectiveness of each) are: my favorite no/muddiest point/exit tickets, clickers, and conversations.  I'm not sure that my favorite no/muddiest point/exit tickets would give me enough information to actually score a student on a level of proficiency, which is a downside, because I want to know where students are at.  I think clickers can be limited and you have to ask really good questions to get good information from students...which I can't always do well...but I think I'm going to give it a try.  I will also work to incorporate more conferences with students where I can analyze their learning effectively.

Wow that was a lot and not sure it's any good, but those are my thoughts on formative assessment for now...wait I forgot portfolios!  Just a quick sentence or two.  I also want to try portfolios at some point (not sure every LT is necessary), but tried last year and wasn't very effective.  Perhaps a photo portfolio would be better...students could take pictures of their interactive notebooks and use it as well as other assessments to demonstrate their learning.  Ok.  I'm done.  Those are my current thoughts on assessments.  Perhaps as I continue my read I will have more thoughts, but I'm done for now!

Links to Things

As I become more popular on twittter, people have shown some interest in this project (as they very well should because it is amazing).  I promised them some links and such, so I will work to compile things we have done/seen.  I will organize it by the three main parts of our project.  Then if you are not an Algebra teacher, you don't have to worry about all the Algebra stuff.

Standards Based Grading

• Kid President
• The Collateral Damage of SBG
• Info from Nathan Wear
• Solon HS Info
• Understanding by Design

Common Core State Standards

• Understanding by Design
• Appendix A Pathways (This is where our units are coming from.)
• Kansas Flip Book
• Illustrative Mathematics
• New Hampshire Stuff
• Manchester Curriculum
• PrBL Curriculum Map
• Stacking Cups (and more)
• Standards for Mathematical Practice Rubric
• Implementing SMP and supplement

Tiered Instruction

• RtI Action Network
• Kid President
• AAIMS (Progress Monitoring)

Day 4 (For Real)

The first thing we discussed today was progress monitoring.  We had been emailed some probes to give to our students in May, so we were all eager to hear more about it.  The probes have come from AAIMS (not to be confused with aimsweb) and our AEA is going to receive training on how to score the probes and use them to progress monitor our Algebra students.  The exciting thing is that it will be at no cost to our district!  My principal was pretty much doing cartwheels when she found that out.

The rest of our morning was spent delving into standards based grading.  As a table we investigated SBG from Daniel Schneider's blog post on The Collateral Damage of SBG, which included following some of the links to other blogs.  Due to my intense twitter addiction I had already read this blog, but was excited to discuss it with my collegues.

We continued our SBG chat with a guest speaker.  Nathan Wear, principal of Solon High School, drove 4 hours each way to visit us SW Iowa teachers!  His school has been implementing SBG and so he shared their experiences and resources with us.  The principals got to spend a little bit more time with him, but for all of us it was very valuable.  It is obvious that he is passionate about SBG and good assessment, and that helped increase our passion, too.

In the afternoon we participated in a lesson originally from Dan Meyer (Stacking Cups) and student work provided by Andrew Stadel.  Us math teachers had a blast with it.  Aside from the huge environment issues with buying hundreds of styrofoam cups, it is for sure a Unit 1 go-to lesson.  Cannot wait to try with my students.  We discussed similar situations and tried to figure out what our RtI classes (mine will be called Algebra Topics) would look like to support this lesson.  That's the hardest thing for most of us to wrap our brains around what are we going to provide for these students to get them to be successful in a subject they have not been successful in previously.

That conversation about what our support/topics/RtI classes will look like wrapped up our day.  Overall I got the energy, motivation, and ideas to jump-start all of my summer projects.  I hope that I don't lose it!

The Practice Predicament (Part 2)

I followed through with my goal of differentiating for more in-class practice for students.  It was nothing too extreme.  One day the warm-up problems were based off of the previous day's lesson (for example factoring x^2 +bx+c).  Students who were able to show proficiency from the warm-up problems were given a worksheet immediately.  They could ignore the rest of my teaching and begin working or they could listen and work once the rest of the students got to work.  It was up to them.  Either way I would not be able to provide assistance as I was still working with the rest of the class.  (I let them listen to music while they worked, and they always get excited for that.)

With the students who had not shown proficiency from the warm-up problems went through them with me.  Different students had been able to factor different ones, so together we made it through all four and restated some hints along the way.  At this point I gave them a few more to do independently and put a star on their paper if they could now show proficiency.  Those students received an intermediate assignment (from the textbook in this case) and began working.  I moved students who still had not shown proficiency to the back of the room and worked on a beginning assignment with them.  This was a guided reteaching worksheet from the textbook.  Most of my energy for the remainder of the class period needed to be focused on these students.

Oh I forgot to say how I introduced this idea...I shared a "new plan" with students the day before that would include "less homework" which excited them (even though I rarely assigned more that 5 problems/night and none the whole month leading up to this...).  I clarified that this was an important part of our learning and students who were not willing to work in-class would be sent to the office and would work with me after school in the form of a detention.  I also clarified that my goal was to reach all students where they were.  Sometimes it might be embarrassing if you are beginning, but the goal is to get you to proficient, and then it wouldn't be embarrassing anymore.  I also reminded students that the same people are not always going to be advanced; I'm basing it off of what I see.

Random Thoughts:

• I am not capable of creating 3 assignments from scratch at 3 levels for each concept that students need to practice.  I wish I was, but I do not have enough time.  So you can see that I relied heavily on the textbook and its resources.  In this case it was a smooth alignment because it was a "traditional" Algebra 1 concept.  However, my textbook is not fully aligned to the Common Core, so there will likely be times when I will have to create 3 assignments (but maybe not from scratch) at 3 levels.
• I did not like making the lowest students get up and move to the back, because I think it was still a little bit embarrassing for them.  I still think it's better than making them come to the front of the room. Perhaps if we had been doing this all year it would be less embarrassing.  Perhaps there is something else I could try that would be less embarrassing...still thinking.
• I need to consider students with an IEP and how they would best fit into this.  They are able to leave the classroom to work with an associate, however I want them to have some understanding before that happens.  Is this fair to them?
• There were almost equal students in each group in some classes.  Others had more beginning than intermediate.  What are good amounts to have in each?  I would think that intermediate should be the largest group, what are ways that I can quickly help students who did not demonstrate proficiency at the beginning of class be able to demonstrate proficiency?  Maybe working through examples is not best?  
• Perhaps if I had been doing this throughout the year not as many students would be beginning.  Some students who really don't struggle with learning the Algebra, but struggle with the motivation to do the Algebra, were in the beginning group.  Maybe they thought they could get out of doing work???  But I think if they would see that the beginning group is sometimes more work they would show proficiency earlier.
• I posted all answer keys on my bulletin board and made students check their own work during this time.  I did not spend any class time going through answers the next day.  I'm trying to teach them to be responsible for their own learning in that way.
• It was a little bit frustrating that so much of my attention was focusing on the beginning students.  There were other students who had questions, and I felt like I wasn't able to do them justice because of needing to push those beginning students. Once again, perhaps if I had been doing this throughout the year not as many students would be beginning.  And then I could meet all students needs better.
• During this time I had almost no behavior issues.  There was one (out of 85 all day) student who showed intermediate level of proficiency and then didn't get much of the practice problems done.  I didn't notice soon enough because I was so focused on the beginning students.  That was about it.
• I also had one student who "showed" advance proficiency by copying (I think) and really thought he was ready for the advanced assignment.  He got almost none of it done and score extremely low on that target on assessments.  I likely should have "manned" up and just told him no.
• It ended up that usually students who were absent the day before were in the beginning group.  Not sure if this is OK or if I should address them in another way.

I did have a student directly thank me for doing this.  Other students offhandedly mentioned that it was better, but this student directly approached me after class and thanked me for doing this.  He said that it really meant a lot to him because he knew it took more work for me.  I count that as a win, at least for now.

Day 4: Cancelled

Our Day 4 was cancelled due to snow :(  I was extremely disappointed.
Also they didn't notify us in time for me to make it to school on time (I lived closer to the training than school).