The Common Core Shift in Algebra I
I’ve spent this year trying to teach a genuine Common Core Algebra I curriculum to high school freshman (my first time doing either of those) and I keep trying to find a way to write about with my experiences, but it’s hard not to get lost in all of the moving pieces that’ve happened this year. As the year wind downs, I guess the biggest thing I feel is: the Common Core shift is real and I feel it and I have a lot of uncertainty about what my students should leave my class with or that I’m preparing them in the best way for what’s coming. This post is about me really have to readjust what I thought the content of an Algebra I curriculum was, and trying to figure out what it needs to be now.
I used to have a feeling for what students didn’t know when they walked into Algebra I – things I could assume I needed to teach from scratch and things I could assume they had seen before and had some familiarity with. I used to know what the non-negotiables were for when they left my class and moved to Geometry or, much down the road, Algebra II. I used to know the balance between procedural fluency and conceptual understanding; when to ground something in a purely mathematical process versus something necessitated by a real-world situation (example: are logarithms motivated by the need to take the inverse of an exponential? Or by the need to measure sound frequencies on a decibel scale?) I used to know where the overlap between Algebra, Geometry, and Algebra II was – where to draw limits in Algebra I that would get picked up in Algebra II. I’m much less confident now and I’m not sure what to fall back on. I feel things needing to be done differently but not knowing how or to what extent.
A lot of the uncertainty I’ve felt this year is also the product of some outside forces that I’ve been reacting to. Most of my district has adopted the Carnegie Curriculum, complete with textbooks, support materials, and once-a-week computer time with the the Cognitive Tutor program. Our middle schools are using this curriculum as well, which then ties directly to the curriculum we teach in high school. I teach in Arizona, which adopted the Common Core Standards and aligned itself with the PARCC assessment consortia a few years ago. In March of 2014, Arizona withdrew from PARCC. In November 2014, they adopted a new test dubbed AZMerit, which is similar to how Utah and Florida are implementing their state tests (colleague and fellow blogger Jason Dyer has a play-by-play of what those tests looks like). In February 2015 (this year), Arizona almost repealed the Common Core standards – instead, a group will review and revise them for next year. As a result: it’s hard to find solid instructional ground while also adjusting to a new curriculum, somewhat-new standards, and a new assessment.
Right now, thinking about next year, there is a tension with every Algebra I topic and how it:
- aligns to the Common Core standards or my best prediction of what the Arizona standards might turn into
- aligns to the AZMerit test, for which all I know is this blueprint.
- aligns to my textbook & the common curriculum my district is trying to use, especially since our feeder schools are using this curriculum as well
- aligns to skills or knowledge that my students need to know in order to be successful in their future classes or future careers.
So – reflecting on this year, starting to think about adustments for next year, and (I’m realizing now) as a place to process my thoughts, here are some things I’ll just call The Shifts and some thoughts on how to react to them
Shift #1: Self-Contained Units
I think the biggest thing I feel as a result of all of these moving pieces is the need for units that have a clearly defined ending and ‘wrap everything up’ quality. There were some topics where I used to be able to say “You need to know this so we can do ______ later”, but I’ve lost all intuition about when these statements are true anymore. Instead, my best units this last year were the ones that built towards a specific problem to solve or scenario to investigate or project that tied everything together – that was a natural culmination of the material we had been covering and didn’t rely on “trust me – you’ll need this later”. Trying to motivate material with “You need to know this for Geometry / the next unit / the state test” was a failure because, honestly, I no longer have any idea if what I’m teaching right now will truly be necessary as we move forward (I’m looking at you inequalities and solving absolute value equations).
Instead, thinking back to last year, I can remember a few units that had this nice self-contained feeling (start, middle, end). These were the ones I enjoyed teaching the most and the ones with the most in-depth questions, investigations, and sense of independence from the students. Thinking about next year, I want more of these – units with a natural progression towards some kind of self-contained question/scenario/project and with a clear beginning, middle, and end. I don’t want the motivation to purely be “they’ll need this later” because, really, everything is so fluid that I don’t know if they’ll really need it later. I want students to know something because they need it now, in this moment – otherwise it becomes just another random rule or procedure to memorize without any internal connection to it.
Shift #2: More Application, Less Procedural
One big shift I see in how my textbook presents new content is that it is almost always grounded in investigating a problem or scenario that is real and has consequences. Systems of Equations are introduced via break-even points, inequalities are introduced via ‘at least’ or ‘at most’ problems, exponentials are grounded in patterns or ‘doubling’/’tripling’ scenarios. I’m used to ‘word problems’ or ‘real-world scenarios’ being the last topic of a unit or even a unit all on its own – but in the age of Common Core, I see these showing up more and more as the entry point of problems and then again as the finishing point. I’ve become a big fan of this for lots of reasons: it has a lower floor for students to discuss a problem, it eliminates the ‘here’s what we’re learning next because (arbitrary reason)’ style of curriculum, it’s more relevant to what they’ll see outside of school, and its easier to engage students in something concrete instead of something abstract. I plan on trying to model every unit in this way – find something for my students to dig their teeth into before barging forward with the math content, then circling back and using the content to re-examine these problems/scenarios and see how useful all that math really was.
Shift #3: More Calculator
Holy crap – this thing is a game changer. Arizona’s old high-stakes test was no-calculator, so students were taught methods to answer questions by hand. But now? Calculators can be used on our exams, which means calculator fluency is a big deal in my class. And, frankly, I’m really glad calculators can play a bigger role in an Algebra I class – they’re a legitimate tool and, from a planning perspective, they let me ramp up the intensity for the types of problems we solve and scenarios we investigate. It used to be that I had to choose problems where the numbers ‘worked out nicely’ or the graphs ‘fit nicely in the standard window’. This was me creating artificial blocks for myself and my students that aren’t realistic and aren’t valid anymore. Now I have more freedom and more tools to show my students so they can solve problems that matter.
Or, at least, this is what it should be. This last year: it wasn’t. I hardly did any calculator-based lessons with my students. I skipped the sections in the textbook that explicitly used the calculator to answer problems. When I would try, the lessons would drag on and on as I tried to troubleshoot calculator problems and keep the class together. These lessons also would show up in the middle of a chapter on something else, so the switch to calculators usually seemed random and forced and took some time to get used to. I was never sure what the payoff was going to be and I was still so used to students needing to know how to do things by hand that I just defaulted to teaching familiar lessons that could be done ‘easily’ by hand.
Thinking about next year, my default ‘do it by hand, easy numbers’ mindset needs to shift dramatically. I need to spend time getting students familiar with calculators and seeing them as a valuable tool to solve problems. I want to plan an entire unit which is just on using the calculator, specifically the graphing functions (finding mins/max, finding roots, finding intersections, using the table, adjusting the window, etc). I want students to see the calculator as a valid option as a way to start investigating a scenario. I think this is a big deal, and I’m looking forward to explicitly planning lessons around using a calculator.
Shift #4: More Statistics
This may be a revelation that’s more for me than other teachers who’ve done Algebra I before, but I never thought of statistics as being a vital part of the course. Things like representating data, measures of center (mean, median, mode), and linear regressions were more like an afterthought or extension or digression rather than an integral part of the curriculum (or, at least, that’s always what it looked like to me). The Common Core seems to have shifted this a fair amount (with drastic shifts happening in Algebra II). The idea of ‘big data’ and analyzing (rather than simply representing) this data seems to have taken a much bigger role in a Common Core Algebra I class. And, seeing what’s to come in the Algebra II standards (standard deviations, z-scores, normal curve analysis), there’s a responsibility to prepare students for these topics in the next levels.
I didn’t do any of that this year. Even though statistics is 17% of our AZMerit test, my class didn’t do a whole lot with data and regressions and measures of central tendency – these standards fell to the wayside as I desperately tried to prioritize my time and guess what students would need moving forward. But, I don’t think I can let this happen next year – in the bigger picture of preparing students for their next classes (especially Algebra II) and the real world, there needs to be a place to prepare students to look at data and interpret it. I think the standards have made a statement that these items no longer lie solely in the realm of a statistics class and, if I’m to genuinely teach the Common Core, I can’t have these units be an afterthought that eventually falls to the bottom of my priorities.
Edit 6/20: This Tweet pointed me in the direction of these resources: Publications in Statistics Education, which looks to be a collection of resources and publications aimed precisely at this issue: the new influx of statistics required by the Common Core and the lack of resources I have to present these standards. So – that’s awesome, and the reputation of the person sending the tweet (she blogs at statteacher.blogspot.com) is enough for me to take the recommendation seriously.
Shift #5: Is Algebra I necessary for Geometry?
This question really challenged me as the year wound down and I thought about the impact this year would have on my students. This question really comes from the shift I’ve seen in Geometry: these standards are now grounded in proof or transformations or real-world applications. Gone are the days where Geometry was an excuse to solve an algebra problem. Fading are the days where a student could be unsuccessful in geometry purely because they had weak algebra skills. With the new emphasis on reason and explanation and coordinate planes, a lot of the algebraic foundation that is given in Algebra I may not be needed in a Geometry class. I don’t remember any student telling me that they had to solve an algebra problem on the Geometry portion of the AZMerit exam.
Personally, I like that Geometry wants to become more of its own discipline, free from the chains of algebraic problems being forced into a geometric context. But, it makes me wonder about the types of skills students need to have as they enter this class. Do I need to impart algebraic skills as students prepare to enter Geometry? Or do I need to impart algebraic strategies for their geometry experiences? Will they be more likely to graph an equation or solve for x, or more likely to analyze a scenario and find an entry-point into the problem? From what I’ve seen this year, it seems to be the latter. This, also, is even more of an argument for self-contained units in Algbera I.
This, again, makes prioritizing units and standards a challenge because it’s unclear when a particular algebraic skill may pop-up in Geometry next year. I used to try and pick ‘non-negotiable’ standards that students absolutely needed for their future, but I’ve lost most of my intuition about what these are as I’ve watched the Geometry curriculum change. It also makes me wonder, radically, if our course progressions need to stay in this same Algbera I – Geometry – Algebra II progression that they’ve been for as long as I’ve been in school. Before the AZMerit exam, this progression was mandated purely by the fact that our old high-stakes test included standards primarily from Algebra I and Geometry and was only given at the end of their sophomore year. But, with narrower end-of-course exams given at the end of each year, what’s to stop a school from switching up the order? It’s much easier for me to think about how Algebra I transitions to Algebra II rather than how Algebra I leads into Geometry. Without derailing this post into an argument for rearranging the course structure, I will say that I’ve started to think about ‘non-negotiable’ skills in terms of what they need for Algebra II instead of Geometry, even though they still take Geometry before Algebra II.
Shift #6: We Need Better Teacher-Given Assessments
This has been one of my biggest frustrations/regrets/source of anxiety this year – the gnawing feeling that I could never capture what we my students were learning and how they could communicate that learning in a pen-and-paper test. Writing tests that are purely procedural does a disservice to the complex scenarios we’ve discussed in class and what I’ve witnessed in my room as students talk to each other and present their ideas. Designing a test with large open-ended problems is tricky and takes practice to phrase the question exactly the way you’d like it so you can really parse if a student knows something or if they don’t. I haven’t figured it out yet, but almost every test I gave ended with the feeling that “these questions were not aligned with what we’ve been doing in class”.
A lot of this, especially early on, was (I think) my own unfamiliarity with how much less procedural Algebra I has become. Standards like solving equation and graphing lines and even solving systems have moved down to 8th grade, which means Algebra I is reserved for applying those skills to situations and interpreting the results. I found myself leading classes that focused on using algebra to analyze a scenario and spending most of our time discussing this analysis, but then giving a test with problems that were solely algebra and devoid of any context. Trying to find a way to write questions which are more than just application of skills has been tricky, and sometimes leads to open-ended questions where students aren’t sure what I’m asking or, in answering the question, they don’t demonstrate the skills I’d like to see as they solve the problem. Finding this balance has been tough.
Thinking about assessments next year, I was pointed towards this document from Achieve the Core on Publisher’s Criteria for High School Mathematics and was drawn to the pages that talk about rigor (although I probably could have found this same revelation from other common core documents). Rigor is discussed as being composed of equal parts: conceptual understanding, procedural skills & fluency, and applications. Thinking about my past assessments, I was usually too heavy on one of these areas while ignoring the others – maybe one test was entirely problems devoid of context, while another had mostly open-ended scenarios for students to analyze and solve. Looking into next year, I want to try and design each assessment so it has pieces of each of these: some questions emphasizing skills & fluency, some questions emphasizing conceptual understanding (maybe finding mistakes? or agree/disagree & explain why? or compare/contrast?), and some questions with an application focus (probably grounded in some sort of real-world scenario). I’m hoping this will give me some guidance so I can avoid those tests that lead to me thinking “this doesn’t match what we were doing in class”.