Foundational Standards

At our last Tiered Algebra day, we spent an entire morning choosing foundational standards for Algebra 1 (specifically looking through the Appendix A standards that were identified for Algebra 1).  We wanted to be make sure to state that we are not choosing Power Standards.  Common Core specifically clarifies that all standards are equally important.  In our core instruction, we need to be teaching all standards.  
However, we felt it might be helpful to identify some standards that were critical to the learning of other standards.  We wanted our foundational standards to be ones that:

• would help students learn other Algebra standards

• would be necessary for life

• might help bridge between 8th grade math and Algebra

An example is that if students have trouble solving linear equations, it is likely that they would have trouble solving systems of equations, quadratic equations, etc.  In that way, solving linear equations is a foundational standard.

After much discussion we came up with the following foundational standards/groups of standards:

EQUATIONS/INEQUALITIES
A.CED.1:  Create equations and inequalities in one variable and use them to solve problems.
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.CED.4:  Rearrange formulas to highlight quantities of interest, using the same reasoning as in solving equations.

SLOPE
F.IF.6: Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.
S.ID.7: Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.

POLYNOMIAL OPERATIONS
A.APR.1: Understand that polynomials for a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.

MEANING OF A GRAPH
F.IF.1: Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of fcorresponding to the input x. The graph of f is the graph of the equation y = f(x).

A.REI.10:  Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).

INTERPRET PARTS OF AN EXPRESSION

A.SSE.1:  Interpret expressions that represent a quantity in terms of its context.  Interpret parts of an expression, such as terms, factors, and coefficients.  Interpret complicated expressions by viewing one or more of their parts as a single entity.

EXPONENTS AND RADICALS

N.RN.2:  Rewrite expressions involving radicals and rational exponents using the properties of exponents.

USE UNITS

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.

It was a morning of long debates, so I would easily say that almost all of us would argue to change something about what we came up with...overall what to you think?  Is there something you would add or remove?

-Kathryn

#MTBoS30

8/30

Linear Functions

We jumped into linear functions this week.  Here are some of the things we did (and by some I mean pretty much my entire week of lessons).

What does it mean to be linear?
I really wanted to emphasize the various representations of linearity, so that students could see how they all work together.  So I started with this foldable to define linear.  I gave them the algebraic and graphic definitions, but we used those to come up with what it means to be linear in a table.

I did this by giving them each a graph of a linear function, with four points marked on it.  (Here are the graphs I used.)  Then I had them in groups of two create a table on the board for their graph.  I was hoping this would help students with the concept that a line is made up of points as we have been struggling with that idea.  Once we were done with that we made lots and lots of observations about the tables.  This took a LONG time.  I let them notice anything, so it took a while to get away from "they all have a 0/1/-2" type of statements into noticing the various patterns that were there.

But eventually we came up with "x and y both change by constant amounts".  I was very happy because in every class students recognized that they could choose any of the tables on the board as an example.  And students were willing to try to think of non-examples.  So each student got to make that part their own.  I know that is the purpose of things like this, but we don't get there often enough.  It was nice to get that far with this.

Here is the inside.  (And here is a link to the foldable.)

I created a card sort as a way to help them practice telling if something is linear.  I really wanted to get at the fact that linear is a word that describes a group of functions.  So something must be a function before it can be linear.  So we sorted between "Not a Function", "Function-Linear", and "Function-Nonlinear".  It was a lot harder for them than I thought it would be, but eventually we began to get the hang of it.  They still struggle with messing up whether VLT is for function or linear...but one step at a time.

As you can see, instead of gluing the cards in, we recorded the answers in our notebooks.  I hope that students will try to sort the cards on their own as a way to review.

Thoughts:

• My card sort is not perfect.  Two of my tables are non-functions for the same reason, that was a typo...I want to fix that.
• Also didn't really mean to have the equation x=2 in there because we didn't discuss what makes an equation a function...but it did lead to a decent conversation.  I think I would prefer to replace the card with a linear function that is not in slope-intercept form.
• One class got into a really good discussion about whether or not y=(1/3)x was linear or not.  Even when b=0 was thrown out as an idea, one student was still adamant that it needed to be written down for it to be in slope-intercept form.  I kept the conversation going for a while and then moved on without giving up who was correct.  While I was talking a student looked to the student sitting next to him and asked, "so is it linear or not?".  Aren't I mean?  The next day I eventually showed them the graph and they all agreed that it was linear.  I also explained that mathematicians tend toward laziness and prefer not to write something if it is unnecessary.
• I went hard-core with colors in my notes, but students didn't in theirs.  That is something I need to work on being more intentional about.

Finding slope and y-intercept

To begin our discussion of slope and y-intercept, I had students write "what I know..." and "what I want to know..." on the board.  Here is one example:

We used this to jump start our graphic organizer for notes.  Starting with what they already knew, but still getting it into the notes was a win-win.  They got to feel smart for knowing it and I got them all to put it into their notes anyway, especially since not all students knew it.  Here is what we came up with.

I don't think they've seen the slope formula, and I really wanted to share that with them for the Numeric part, but they thought smarter than me and were ready with ideas by the time we got their.  They remembered finding the change in y and x when deciding if the table was linear or not, and figured that would work!  It is really quite brilliant, and I feel lame because I didn't think of it first, but by the time I made it to the end of the day I had abandoned the slope formula.  I want to work back to it for tables that don't have a constant change in x, but still might be linear...we'll see if I can manage that :)

Then we practiced finding slope and y-intercept with a boring worksheet.  But it was quick and worked as a check to see where students were at.  I used my name cards to call students for answers at the end of class.

Thoughts:

• I really wish I could have just had them create their own graphic organizer for this, but I'm not sure I could have been clear enough for them to understand the expectations
• Once again I went hard-core with color and they did not (but look how cute it is!)
• The worksheet was boring, but didn't take too much time.  I think that's OK, because I just needed enough for them to get back into things.  I would have done things slightly differently if this was the first time they had been introduced to slope-intercept form.

Unit 5 LTs (draft)

Well, here I am again, asking your advice on my learning targets!  But the difference today is that this is the last time :)  Here you go!

UNIT 5:  QUADRATIC FUNCTIONS AND MODELING

Section 5.1:  Graphing and Interpreting Quadratic Functions

                Standards Addressed:  F.IF.4, F.IF.5, F.IF.6, F.IF.7, F.IF.8, F.BF.1

5.1A:  I can find the intercepts and extrema for a quadratic function (in any form).

5.1B:  I can graph a quadratic function using the intercepts and extrema.

5.1C:  I can interpret the meaning of intercepts and extrema of a function used to model a situation.

5.1D:  I can calculate and interpret the meaning of the slope between two points of a function used to model a situation.

Section 5.2:  Comparing Functions

                Standards Addressed:  N.RN.3, F.IF.9, F.BF.3, F.BF.4, F.LE.3

5.2A:  I can compare properties of two functions (in any form).

5.2B:  I can describe and identify translations that determine function families.

5.2C:  I can find inverses of linear functions (in any form).

Thoughts/Notes

• All but three of the standards in this unit were addressed previously in unit 2.  In this unit they are to be extended to quadratics.
• I kind of would like to throw 5.1D in with 5.1A and C, but I couldn't get the wording right and so then I questioned whether it actually fit or not.  Thoughts?
• I honestly feel like this was the easiest unit...maybe that's because I'm to the point where I'm not as worried about perfection as I was at the beginning...?  Maybe that means this unit is awful...?

Once again, I ask for your feedback.  I want to know what needs to happen to make these awesome for my students.  I appreciate you taking the time to read these!  Thanks, and once again remember...ALL DONE!

-Kathryn

Unit 4 LTs (draft)

I still have a million thoughts whirling around in my head about Unit 3 based off of comments and twitter conversations, but onto Unit 4 I must go!

UNIT 4:  EXPRESSIONS AND EQUATIONS

Section 4.1:  Polynomial Operations and Modeling

                Standards Addressed:  A.SSE.1, A.APR.1, A.CED.1, A.CED.2

4.1A:  I can add and subtract polynomials.

4.1B:  I can multiply polynomials.

4.1C:  I can create equations to model situations and use them to solve problems in the context of the situation.

Section 4.2:  Zeros of Quadratics Algebraically

                Standards Addressed:  A.SSE.2, A.SSE.3, A.CED.4, A.REI.4, A.REI.7

4.2A:  I can find zeros of quadratic functions by taking the square root.

4.2B:  I can find zeros of quadratic functions by factoring.

4.2C:  I can find zeros of quadratic functions by completing the square.

4.2D:  I can find zeros of quadratic functions by using the quadratic formula.

4.2E:  I can solve an equation/formula for a specified variable.

Thoughts/Notes:

• Do I really need 4.2A?  I'm looking at A.REI.4b, which says students solve by inspection, taking square roots, completing the square, the quadratic formula, and factoring.  Well, that's a lot, so I feel like it needs more than one LT, but not 5!  And to me if you solve by CTS you are taking a square root...would basic square root equations build up to CTS?  I would prefer 4 LTs to 5 in that section...thoughts?
• I'm a little confused about how I can fully meet A.SSE.3 and A.CED.2 which both have graphing components when most of the graphing standards aren't until Unit 5 (F.IF.7, F.BF.3, and many others).  We will likely do all our graphs by plotting points or using technology for Unit 4. 
• Last year when we found zeros of quadratic functions we did algebraically, numerically, and graphically all together.  I'm a little sad that they are split up here.  :(  I like drawing all the connections, but it will have to wait to Unit 5.

Please give feedback by commenting below or tweeting me (@kathrynfreed).  I really appreciate how much everyone has helped me so far with processing these.  Every time I think my brain is over-capacity with thoughts on standards and learning targets somebody will make a comment that gets me thinking again!

For those of you who have been walking through this entire journey with me, there is only one unit left!  Hang in there!

-Kathryn

Unit 3 LTs (draft)

I promised Unit 3 LTs by the end of the week, and I think I made it on time :)  There's nothing like cutting it close.  I had hoped to have units 4 and 5 done by this time as well, but we'll just have to see where I can get.

Here they are.  More thoughts at the bottom!

UNIT 3:  DESCRIPTIVE STATISTICS

Section 3.1:  One-Variable Statistics

                Standards Addressed:  S.ID.1, S.ID.2, S.ID.3

3.1A:  I can create plots (dot, box, histogram) to represent data.

3.1B:  I can find and interpret the mean, median, IQR, and standard deviation.

3.1C:  I can compare the shape, spread, and center of data sets using the mean, median, IQR, and standard deviation.

Section 3.2:  Two-Variable Categorical Data

                Standards Addressed:  S.ID.5

3.2A:  I can create a two-way frequency table for categorical data with two categories.

3.2B:  I can find and interpret relative frequencies (joint, marginal, and conditional) in the context of the data.

Section 3.3:  Two-Variable Quantitative Data

                Standards Addressed:  S.ID.6, S.ID.7, S.ID.8, S.ID.9

3.3A:  I can represent data with a scatter plot.

3.3B:  I can find a line of best fit and use it to solve problems in the context of the data.

3.3C:  I can interpret the meaning of the slope and the intercepts.

3.3D:  I can find and analyze the meaning of the residuals and the correlation coefficient.

3.3E:  I can make inferences about correlation and causation.

Thoughts/Notes:

• It makes sense to me to study 1-variable statistics separately, so I'm mostly OK with the organization of 3.1.  We did get a quite decent twitter conversation on whether or not students should be required to calculate standard deviation by hand.  Any further comments on that are welcome.  Thank you to those who already participated and gave me ideas.
• I had planned on doing all two-variable statistics as one section, but there is a lot there!  And it seems weird to me to have 1 standard on categorical data when the rest are on quantitative data.  Hard for me to make that fit well.  There is so much with the quantitative data though that I couldn't combine them.  I hate having 5 LTs in one section (it feels so big) but I feel like I still haven't done the standards justice.  Also I think Section 3.2 being so short will provide an opportunity for a different structure of class.  Maybe we will do a project assessment instead of a test

Please, please, please leave your thoughts in the comments or tweet me (@kathrynfreed) Like I have said with every other unit, I really truly want to make these the best goals I can for my students.  I appreciate all of your feedback whether praising my awesome-ness (not usually) or giving me advice for improvement (usually), I really like how you guys can always make me think more--even when I think my brain is dead!  Thanks for that!

-Kathryn

Unit 2 LTs (draft)

I have been working on my Unit 2 Standards.  Please remember that my big unit breakdowns are coming from Appendix A (this is because of my school's curriculum and the project I've been apart of).  The second unit from Appendix A is GIANT.  Over twenty standards.  That is difficult to manage, thankfully I had did some breaking-down of it this past year as I taught it, so I had a bit of a foundation.  I have changed a lot though, so without further ado...

UNIT 2:  LINEAR AND EXPONENTIAL RELATIONSHIPS

Standards Addressed Throughout:  F.BF.1, F.IF.4, F.IF.6, F.IF.9, F.LE.5

Section 2.1:  Functions in General

                Standards Addressed:  F.IF.1, F.IF.2, F.IF.5

2.1A:  I can find the domain and range of a relation (in any form).

2.1B:  I can determine and justify if a relation (in any form) is a function.

2.1C:  I can use function notation to describe, evaluate, and graph a function (in any form).

Section 2.2:  Linear Functions

                Standards Addressed:  A.REI.10, A.REI.12, F.IF.7

2.2A:  I can determine and justify if a function (in any form) is linear.

2.2B:  I can find the slope and y-intercept given a linear function (in any form).

2.2C:  I can graph a linear function (in any form).

2.2D:  I can define an explicit function to model a given situation.

2.2E:  I can interpret the meaning of the slope and y-intercept of a function used to model a situation.

Section 2.3:  Systems of Equations

                Standards Addressed:  A.REI.5, A.REI.6, A.REI.11

2.3A:  I can state whether or not given values for the variables represent a solution to a system of equations.

2.3B:  I can estimate a solution to a system of equations graphically.

2.3C:  I can identify a solution to a system of equations numerically.

2.3D:  I can solve a system of equations algebraically.

Section 2.4:  Exponential Functions

                Standards Addressed:  F.IF.9, F.BF.3, F.LE.1, F.LE.3

2.4A:  I can determine and justify if a function (in any form) is exponential.

2.4B:  I can find the base and y-intercept given an exponential function (in any form).

2.4C:  I can graph an exponential function (in any form).

2.4D:  I can define an explicit function to model a given situation.

2.4E:  I can interpret the meaning of the base and y-intercept of a function used to model a situation.

Section 2.5:  Sequences

                Standards Addressed:  F.IF.3, F.BF.2, F.LE.2

2.5A:  I can identify if a sequence is arithmetic, geometric, or neither.

2.5B:  I can define an arithmetic sequence (in any form) recursively and explicitly.

2.5C:  I can define a geometric sequence (in any form) recursively and explicitly.

2.5D:  I can explain why a sequence is a function.

Section 2.6:  Exponents and Radicals

                Standards Addressed:  N.RN.1, N.RN.2

2.6A:  I can find powers and roots.

2.6B:  I can translate between exponential and radical expressions.

2.6C:  I can simplify exponential and radical expressions using the properties of exponents.

A few of my thoughts:

• I want my students to understand that a function has multiple representations, and I want them to be comfortable moving from one representation to the next.  This is why I put "in any form" in so many of the learning targets.  I think it's obnoxious, so I think I will take it out...but for now I need it to remind myself of that crucial important concept.
• I think Section 2.6 actually fits better before Unit 4 (when quadratics are introduced).  For Unit 2 all I need is the definition of an exponent, negative exponent property, and the zero exponent property to go along with exponential functions.  I don't think it's worth doing all properties that far ahead of when they can be used in context.  So I will likely break slightly from the recommendations of Appendix A and place this with Unit 4.
• [Edit 7/28]  There is nothing specific in here about comparing linear and exponential functions, but we do it as we explore exponential functions, and also with sequences.  That's one of the reasons why I chose to not integrate sequences into the other sections.  We will discuss similarities and differences, and I will ask students to compare on assessment (we have "advanced" questions so I can ask an extension of a LT fairly).  Does that make it OK that I don't have it as a specific LT?  I just feel like there are SOO many...

PLEASE, PLEASE, PLEASE give me feedback.  Either by commenting here or by tweeting me (@kathrynfreed).  I want feedback from you no matter what you think.  Making decisions like this for my classroom is a little terrifying for me, so your feedback helps me know what I need to change to make this the best I can for my students.

-Kathryn

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!

Day 3

We had Day 3 of this project and we got to present right at the beginning of the day (Well after a Kid President video--he's hard to follow!).  We shared the key points of our action plan...almost like what I posted previously, but not the "why?" part.  I figured that part would be better to share verbally if the conversation went that way.

But I have to admit I was totally wrong in my assumption that people wanted to know why.  Most teachers are totally on board with the plan.  They want to see it happen, they just need help figuring out the logistics.  What was best about presenting was all the questions that came from this.  Everyone is trying to figure out how to make it work and so not all of the questions were addressed to us, some of them were addressed to others, too.  And that helped us realize some of the things we still have to figure out.  What great collaboration!

For the rest of the morning we split into groups:  Math teachers, Special Education teachers, and AEA staff. As Math teachers we continued our work to unpack standards in Unit 1 and find appropriate assessments.  We were working with 1-2 other teachers on a standard of our choice.  I worked with a teacher I had never talked to before, and we picked a standard she had worked on previously.  I felt as though it was much more productive this time.  She and I communicated well together, and I was able to be a little bit more flexible about the whole process.  I tried to avoid the things that had stressed me out too much previously.  We made some progress and we will continue to work on Unit 1 and maybe Unit 0 next time.  I hope that we can get the first part of the year planned prior to the beginning of the school year!

In the afternoon we discussed standards based grading in a little bit more detail.  This is one area that I feel like I have a good start to because of our school's PD, but there are lots of flaws that we need to work through.  I felt like the conversation we had around this were good, but I feel pressure to iron out the flaws prior to next school year.  I have to put my grading plan in my syllabus, and I want to have something fair figured out the communicate to students and parents.  This is a big challenge that I discussed with some of the AEA people.  They understand and also want to work to find a solution for us, however not all participants are going to fully jump into this shift next year, so it is going to be challenging to meet the needs of all the schools/teachers represented.

To conclude our day we got to be students in the lesson DVR Dilemma from Yummy Math.  The purpose was to get us to think through the practice standard that we are working on improving.  We have been challenged to work to improve our teaching strategies to better address this for our lesson plans in the upcoming week.  I will be working on teaching students to multiply polynomials...it's really hard for me to think about how to use this to help students "Make sense of problems and persevere in solving them."  But I've been thinking about it a lot after that challenge...maybe I will post again about what I decide to do and how it goes.  Going to be a big risk!

Day 2

Our second meeting (also known as "Day 2") was packed full of thought-intensive analysis of multiple things. We started by discussing "Understanding by Design" lesson planning, where focus begins with the desired result.  There are 3 stages of this planning method:  the Desired Results, the Assessment Evidence, and the Learning Plan.  This process can be challenging because only after a long time looking at the standards and choosing an assessment method can the instruction planning actually begin.  As teachers it is hard to wait so long to plan instruction!

After discussing this method, we broke into groups and began unpacking various standards from Unit 1 of Algebra 1 as defined by Appendix A of the CCSS with this template to guide us.  This is really challenging!  At first we were supposed to base it off of our professional understanding of the standard (which is not always clear), and discuss our understandings.  Then we could look to some other documents, but there are few resources available from the authors of the CCSS to help us unpack them.  We spent all morning doing this!  It felt very unproductive to me for two reasons.  First, because we were working in groups with people we didn't know very well, and so it was hard to respectfully critique each others' ideas.  And secondly, because we did not define "understanding" and "knowledge" prior to getting started.  All in all a challenging morning for me.

We were actually supposed to do one more thing before lunch.  Choose one of the practice standards to focus on for the rest of the year--yikes!  I remember hearing someone say that the first four are the most important, so we chose to go with #1:  Make sense of problems and persevere in solving them.  Then we had the opportunity to change the rubric for it.  The rubric we were given was not structured the way I like to have rubrics structured, so I redesigned it with the help of my coworkers present at the meeting.

After lunch we created a proposal for how our district should implement Tier II of the instruction.  We had already had quite a few conversations with our principal and other math teachers in our district, so I felt as though the proposal was just formalizing all of those conversations.  The goal/plan is for me to teach only Algebra next year.  Four periods of Algebra (tier I) and two periods of Strategies (tier II), while a special education teacher would teach the students receiving tier III instruction during their guided studies time.  My hope is that I can have the "strategies" courses at the beginning/end of the day, so that the students have had the same instruction from the Algebra class.

Once we completed that, we discussed the benefits of standards based grading.  My district is already in the process of developing standards based grading, so we have an idea of what it is, what it might look like, the benefits, and the drawbacks.  However I still found this conversation to be very rewarding.  I jumped into standards based grading this year without all the background work that teachers in my district have done, so it was good for me to step back and remember why they chose to move in this direction.

The last thing we did is to split off as Algebra teachers (special education teachers did something else) and look through 7th and 8th grade standards that we might have to teach during the transition to the common core.  These are the standards that are necessary to meet the Algebra standards, and may not get mastered in 7th and 8th grade.  I think it was important to acknowledge that it is a transition to the common core, because at times I feel pushed to move my students over to it too quickly.  It is important to get there, but we also have to remember that we are in a transition time.

So when I left Day 2, my brain was working in many different directions, but I felt that all were beneficial for the future of our students and their mathematical learning.  I sometimes just wish that it didn't take so much work, but in the end it really does.  In which case it is important to remember the purpose--student learning!

Mathematical Practice Standards

Completed one part of our “homework”--analyzing one of my Algebra lessons using the Practice Standards Rubric (here).  I had to see how my lesson fit all 8 of the practice standards:  

1. Make sense of problems and persevere in solving them
2. Reason abstractly and quantitatively
3. Construct viable arguments and critique the reasoning of others
4. Model with mathematics
5. Use appropriate tools strategically
6. Attend to precision
7. Look for and make use of structure
8. Look for and express regularity in repeated reasoning

It was challenging to think through all of it in the context of one lesson.  It made me feel incompetent because I felt low in many areas, but I kept reminding myself that it is impossible to address all of the standards in one lesson. 

Day 1

I attended day one of our Tiered Algebra 1 Action-Research Project.  We got an overview of the project including the components that will be involved:  Iowa Core Standards, Standards-Based Grading, and Tiered Instruction.

We spent a lot of time discussing the tiered instruction because that is the newest idea for most of us.  This is based off of RTI, or Response to Intervention, a method used to keep students performing at grade-level and provide assistance when they fall behind.  There is not much information published about implementing RTI at a high school level, especially for mathematics, so we took a look at the basic structure of the program.

We will provide three tiers of instruction.  The first is universal.  All students receive the first tier of instruction.  The second tier is for some students (10-20%) that for a lack of a better phrase are "chronic mathematics strugglers".  These students need to receive an additional 30 minutes of instruction everyday in small groups (7-10 students).  The third is for a few students (1-5%) who have IEP math goals, with additional mathematics instruction written into their IEPs.  These students will also receive an additional 30 minutes of instruction everyday in even smaller groups (1-3 students) with a special education instructor).

Oh to think of all the scheduling issues this creates!  Thankfully our math department had already been discussing the need for intervention and brainstorming ways to make it work.  All that brainstorming with this guidance for how best to help those students allowed us to see this not as an impossibility, but a challenge we would have to work around.