Fair Warning: This post took a month to write. It’s long. It’s involved. It’s also a meditation on my entire year trying to implement a Standards-Based Grading (SBG) system and what that even means. But first, an introduction.
Why It’s Important to Think About Assessment & SBG: My classroom is a game that my students play. I set the rules by how I allow them to succeed or fail in my class. If I’ve done it right, then the rules I set should motivate genuine learning and reflect that knowledge in the form of a ‘grade’. In my experiences as an observer in ‘good’ classrooms and ‘bad’ classrooms, the most reliable way to measure this is through independent performance on consistent evaluative assessments balanced with frequent feedback in the form of formative assessments. So, I need my tests and quizzes to be the focus of the ‘game’ that is my classroom, and I need them to behave in such a way that my students find them motivating while I make sure they are an accurate reflection of student performance. And I need all of this to be transparent – the better we understand the rules of the game, the better we are at playing and winning the game. This is all much harder than it sounds.
I’ve been thinking a lot about assessment because I ended my last year unsatisfied with my assessments. I never thought anything was ‘broken’ or a complete disaster, but I never felt like my assessment and grading systems were operating as efficiently as they could be. I found myself constantly retooling my assessments in an effort to find a magical balance between how and when I presented my assessments, how I graded them, and then what me and my students did with those grades.
In looking around for resources, I found Standards-Based Grading (hereafter: SBG). I read Dan Meyer (and here). I read Shawn Cornally. I read Jason Buell. I read Sam Shah. I read Frank Noschese. If you haven’t ready these, you should. Seriously. Like, take a break from this, go crazy in the world of SBG philosophy, then have a cup of coffee to let it all process, then come back and finish this post.
Reading all of these authors (and the many others I read but didn’t list) and reflecting on my own experiences in the classroom, I think everyone implements SBG slightly different. These differences can manifest in a lot of different ways – some people’s SBG system includes changes to homework and quizzes; some people make changes to their classroom structure & procedures; some people make changes to how they grade; some people make changes to how often they assess. I also think some differences have to do with external factors, such as whether they teach in a science classroom or a math classroom; that some are teaching middle school versus high school, some are teaching in classes with high-stakes testing pressures, and some are teaching advanced students (both in the sense of mathematical knowledge and in other student metrics such as notetaking and focus). The thing I found most interesting was the difference in length of some teachers lists of standards, as well as the level of cognitive demand for each standard. Some teachers have 100 highly-isolated standards, while some have 20-30 standards that involve synthesis and a high cognitive demand. This is what made me curious about assessments in the first place – if both of these teachers said they were implementing Standards-Based Grading, it was hard for me to believe they were assessing and grading the same way.
Despite all of the difference, there is one thing that every SBG teacher has in common: They separate their gradebook into separate standards. On the surface, this seems like a simple change that any teacher can make. However, I tried to trace the effects that this change had on my classroom and found it to be fundamental to the other monumental successes I’ve had this year. In other words, I imagined ‘What if the first change I made to my classroom was to separate my gradebook into standards – how would this affect other aspects of my classroom?” I claim that this ‘simple’ gradebook change causes so much collateral damage that it forces you to fundamentally shift several aspects of your classroom, leading to all of the homework and classroom and grading and reassessment policies that I’ve read about on other blogs. Reading what others have written about SBG, I think we’re all finding ways to deal with the collateral damage that SBG has created.
So, what follows is what I’ve pieced together from how I handled this change to my classroom – the things I realized I needed to adjust and why I needed to adjust them. I think of them like dominoes falling on one another, and it all starts with…
So here’s some publicity about something I’m doing soon: I’m co-presenting at the Global Math Department on Understanding by Design with Tina and Elizabeth. You can see the details and register here. We’ll see how it goes – if you’ve heard of Understanding by Design (UbD) before, you should consider stopping by.
Why This is Something I Can Talk About: One thing that is difficult to cover in any preservice program (in my opinion) is ‘You’re in charge of planning a unit from scratch – how do you do it?’. Most teaches I’d seen defaulted to the textbook, which I didn’t like. Last year was my first year of teaching, so this question was very fresh on my mind. The school I started working was just in the process of restructuring – some buzz words you could apply are ‘turnaround’ and ‘reconstituted’. As part of this process, my entire school implemented the Understanding by Design (UbD) framework for developing units, which pretty much solved this problem for me. Since then, whenever I encounter new content that I need to design units for (coughCOMMON COREcough), I always start with a UbD mindset and I find it very satisfying. Everyone in my school does UbD (in theory – its hard to do in some disciplines) and we’ve been doing it for the last 2 years, which is pretty cool.
In other news: I’m still writing posts on assessments, but writing about assessments is like fighting a hydra: my posts keep splitting and forking every which way until my one post has become five. Those will hit the web sometime soon (hopefully).
To Recap: I made it a goal of mine this year to make better assessments. Enough has happened that I’ve started to process it all, which is one of the things I use this blog for: taking a snapshot of where my thoughts are right now, and documenting the things I want to keep and the things I want to change. This endeavor led to these monsters here, here, and here.
Also – thanks to everyone who’s given such positive feedback about my post on the Three Types of Standards. In looking at my curriculum and deciding what to assess, I realized I started to see three distinct types of skills/concepts/problems that I want my students to be responsible for. The post linked at the beginning of this paragraph talks primarily about Procedural and Conceptual skills and how I identify and assess them.
What I left out of the last post is what I mean by Synthesis Skills. These are the ones I’m still trying to get a handle on, both with identification and with assessing.
Some Background: My Assessments and Grading: Each page of my test contains a single skill that’s been isolated from other aspects of my curriculum. I do this so students know exactly what they’re being assessed on and how they can show mastery. Some skills inevitably build on previous ones, so some are less ‘isolated’ than others (ie: Geometric Probability necessarily relies on calculating area correctly and I can’t avoid that). However, the goal is to probe if my students truly understand a particular concept or skill, which means I want questions that target this skill or concept as much as possible. My goal isn’t to assess a wider variety of skills with a single question – it’s to assess a single skill at a deeper level. This helps me target students weak points and emphasize where their strengths are. It makes it easier to target remediation. It also makes it easier for me to teach concepts, since I’m more aware of when a problem has a potential pitfall that is ultimately unrelated to the skill or concept that I’m trying to teach (ie: If I were going to teach ‘solving 2-step equations’, my first example wouldn’t have a fraction in it).
Outside Influence: This post by Dan Meyer has an excellent explanation of what I try to accomplish with my assessments in terms of collecting data and targeting remediation. I’m going to reference this post in the next paragraph, so if you don’t read it now, here’s the gist: Dan was assessing the skill “I can find the surface area of a cone” and he debated including a problem that would have required his students to first apply the Pythagorean Theorem before finding surface area. Dan argues, and I agree, that this does not serve the goal of targeted assessment and remediation – if a student gets this question wrong, is it because the don’t know how to find the surface area? Or because they don’t know the Pythagorean Theorem? Also, if I were teaching how to find the surface area of a cone: I wouldn’t start with a problem that required them to use the Pythagorean Theorem. I would start with a problem that simply had them calculate the surface area so they could get familiar with the concept and procedure.
The Problem: Even if I don’t start a unit with this problem, and even if I don’t include it on an assessment, I still want my students to solve a problem like that. That requires an extra step or two or three. That requires them to apply a few separate skills in order to solve a larger problem. My fear is that if I ignore problems like this, students begin to see mathematics as isolated chunks of knowledge and skills that aren’t necessarily inter-related. That the types of problems they can solve are limited in depth and complexity. That students begin to see entire units within my curriculum as disconnected and unrelated – or, worse, that they see my whole curriculum as disconnected and unrelated.
This isn’t what I want – I want to feel like my units and curriculum are always building towards something. Something that, every once in a while, unifies what we’ve been doing for the past few weeks/months/year. That require those extra steps and combining skills together. Sometimes this place is within the units themselves, and sometimes they’re entire units of their own – a culminating concept or project whose whole purpose is to synthesis several other skills to solve a specific problem.
I call these Synthesis Skills – skills that remind my students that my curriculum and my unit are interconnected and help me fight this fear that students will see everything as isolated. Sometimes they’re a specific type of problem that requires my students to tie together everything from a unit – in this case, it’s probably better to call these Synthesis Problems. The problems are unit-specific and serve to cement everything we’ve talked about in a particular unit. But also looking at my curriculum, I can see examples of entire Synthesis Units – a collection of concepts and skills where the success of this unit is entirely dependent on how well my students have understood everything that came before it. These are the units where, if my students never mastered some previous skill, it becomes deadly as we work through the concepts and skills that make up this unit.
Within a Unit: Synthesis Problems
When I go problem-hunting online or in textbooks, every once in a while I’ll see a problem where I think “Yah… if my students can do this, then I’m pretty sure they can do anything I could thrown at them. I want this to be the final problem of the unit”. They’re problems where each student may have solved it in a slightly different way. They’re problems with a small amount of information given, but whose execution requires several steps. They’re problems where when I think about an assessment, I think “I only need to give them one of these – if they can do this one, they’re in good shape”. They’re problems that I imagine would be free-response questions on an AP Exam. The statement of the skill is in its purest form – no caveats like “… using properties of triangles” or “… using the formulas for slope, distance, and midpoint”. The solution path is relatively open-ended compared to how questions on my Procedural and Conceptual pages. Here are some of mine:
Skill: I Can Find Missing Angles
Synthesized Skills: Basic angle relationships (vertical angles, linear pair), Angle Relationships with triangles (sum to 180, isosceles triangles), Angle Relationships with Parallel Lines.
Analysis: I’m a huge fan of problems like this – give them a mess of lines and a few pieces of information, then tell them to figure the rest out. They’ll need to use parallel line relationships, polygon relationships, triangle relationships, etc. They have to choose which one to use and when. They have to know when they can’t use a certain relationship – otherwise it’ll mess up the rest of their calculations. These problems can keep appearing throughout the year as we learn more and more about angle relationships (like the problems on the second page).
Skill: I Can Describe a Transformation in the Coordinate Plane
Synthesized Skills: Describing Translations, Describing Reflections, Describing Rotations
Analysis: This was a gooooood assessment from my 1st Semester Honors class. There are many possible solutions. They require the student to use all the transformations we’ve learned so far, but there’s little direction on how. If they can do this, then I’m pretty confident they can problem-solve any transformation-type problem I could throw at them.
Assessing These Skills: When I put a problem like this on an assessment, there are a lot of different places where a student can go wrong. They could make a lot of procedural mistakes, in which case I know we need to work on our foundation first. Or, if it’s a small procedural mistake in the beginning, then the comparison to a free-response question on an AP Exam is especially valid: I follow their mistake through the problem and see if they truly understand what’s happening at a conceptual level. Or maybe they use vocabulary incorrectly or apply procedures incorrectly, indicating a deeper conceptual misunderstanding. Or maybe they get stuck in the problem-solving aspect of the problem – the open-ended nature of the problem causes students to get stuck and lose confidence, leading to an incorrect answer. Grading and remediation becomes a lot less straightforward with these problems.
All of this is contrary to the stated goals of my assessments way back at the top of this post: to target specific skills and collect data about where my students are at. But they still serve my overall philosophy of assessment: the things I assess are how I tell my students “Hey! This stuff is important and you are responsible for it!”, and these are all problems that I feel are important and that they should be responsible for. In fact, these are the skills that have the highest importance because they collect together everything – procedural, conceptual, and problem-solving. They are the ‘big idea’ of the unit in it’s purest form.
I’ve struggled this year trying to unify this tension between targeted SBG assessments that feel like isolated ‘checklists’ of skills, and my desire to teach and assess these bigger, broader problems that I still want my student to be held accountable for. And I think I’ve realized that these ideas are fundamentally opposed – that I can’t do quick SBG type assessments if I want to also assess how well my students can solve these complex problems which take time and whose remediation is complex. Which means I either need to change how I do my pen-and-paper assessments (which is what I’ve been doing), or I need to find another way to assess these synthesis skills (also something I’ve been trying via projects).
My goal for next year is to have something like this in mind for all of my units – some sort of complex problem that we’re building towards – and then find a way to incorporate this into a project or an assessment (or both).
Synthesis Units Within the Curriculum
When I look at my curriculum, there are a few places where I find myself thinking “If I want to do this topic justice – to have my students really learn and appreciate it at a deep level and not just regurgitate for the test – then I either need to spend 2-3 weeks on it, or not mention it at all”. Since I teach Geometry, the place where I see this tension the most is with Centers of Triangles. I love the way Kate Nowak teaches them – hands on, incorporating coordinate geometry, incorporating the Pythagorean Theorem, incorporating area, and culminating with a scavenger hunt that has them construct these points – compass and straightedge and all. This is the right way to include Centers of Triangles in the curriculum – acknowledging that it will span several areas that we’ve already talked about and building up to a singular activity/task/assessment which has them apply all of their skills.
I haven’t found the time to fit this level of depth into my curriculum. If I were to teach Centers of Triangles, I would have to sandwich it between several other units that are already laid out. We wouldn’t be able to go to the depth that we need into order to really understand, appreciate, and apply what we were learning. I would be too pressed for time to teach anything more than memorize the centers, their properties, and regurgitate for a test. Which isn’t really teaching. It’s ‘covering’ the material, which I try to avoid whenever possible.
In my mind, ‘Centers of Triangles’ is a Synthesis Unit – the success of this unit is mostly dependent on how well students have understood the material that builds up to it (constructions, coordinate geometry, triangle vocabulary) and it connects all of these things together. This unit is a place to show off a truly interesting problem that we, as developing mathematicians, now have enough tools to solve. And not just solve – but appreciate and understand why our solution works and how it involves all of this machinery we’ve learned throughout the year. This unit revolves around a singular problem/question – which point is equidistant from the other three? – and the ‘new knowledge’ serves the goal of finding an answer to this problem. You could fit this problem at the end of an existing unit – one on constructions or triangles maybe – but it feels awkward because the problem requires non-trivial connections between several different units and concepts. This means the real benefits of this problem revolve around communication and problem-solving, not as an excuse to teach brand-new material that we’ll use later. In fact, anything ‘new’ we discover in this unit probably won’t be used again anytime soon – which is fine, since it’s supposed to be the celebration of ideas and strategies across the curriculum all culminating with a single problem.
Here are some Synthesis Units I’ve tried over the last few years:
Skill: I Can Determine the type of Quadrilateral formed by 4 points in the Coordinate Plane
Synthesized Skills: Calculating Slope, Distance, and Midpoint; Understanding that congruent segments have the same length; Understanding that parallel lines have the same slope; Understanding that parallel lines have the same slope; Understanding that two segments with the same midpoint bisect each other; Using deductive reasoning to apply properties of quadrilaterals; Explaining your answer
Analysis: For the last few years, I’ve used this as the culminating problem in my unit on quadrilaterals. After having it go a bit poorly this year, I’m starting to think that it’s something that belongs independent of my quadrilateral unit. When we discuss these problems, the real emphasis is on deductive reasoning and how to explain your answer to a third party. These are the real reasons to talk about these problems – the deductive process and explaining your reasoning – not the focus on a single correct answer. The journey is the most meaningful part, not the final result. If I can’t spend the time on this discussion, then these problems become oversimplified and an exercise in procedures without connections.
Reflective Moment: The ‘failure’ of the unit above was the catalyst for this post about assessment and curriculum and ‘tricking’ my students into practicing old skills under new contexts. If I’ve done a poor job at teaching quadrilaterals or coordinate geometry, then this unit becomes an exercise in re-learning these skills and I can’t have the high-level deductive, logical discussions that I want to have. If I’ve done a great job teaching quadrilaterals and coordinate geometry, then the conversations happen at a higher cognitive level and have this argumentative and questioning quality. I think this is why we sometimes avoid these Synthesis concepts and units (especially Proofs and Constructions in Geometry – I’m guilty of this too), because it makes it apparent when our classes haven’t been mastering the concepts that are supposed to build up to these units, which means the units become a shadow of their potential for investigation and synthesis.
A Synthesis Unit in Calculus: Sam Shah’s Optimization Unit.
Assessing These Units: These are the problems where I’ve never been sure how to assess them on a pen-and-paper test. The problem itself is complex, and the important pieces seem like something bigger than just a pen-and-paper test.
And I think the answer I’ve decided on is: I can’t. I need a project. I need a presentation. I need writing and reflection. I need something more than just pen and paper for a student to show that their mastery of these problems. And even more than that – these should be units where we’re doing work that we should be proud of, so I should give my students opportunities to be proud of their work. So, this is something I want to try next year – having little ‘Synthesis Units’ every quarter or so that highlight how our curriculum is interconnected and then give students a meaningful project to work on that emphasizes this.
Outside Influence: A while ago, the Common Core Tools blog released one possible way to sequence the Common Core standards into high school units. Within each high-school course, there are places for ‘Modeling Unis’ and ‘Projects’. These Synthesis Units are the types of problems/projects/ideas that I think of when I imagine what would fit into those spaces (although they may not be what the Common Core has in mind).
Closing Thoughts: I’ve been writing this post over the last few days, and now that I’m at the end, I went back through and wondered “Why was I writing this in the first place?”. So here I am trying to answer that question:
I think I’ve realized that my idea of a traditional SBG assessment (several skills per assessment, given very frequently, targeted remediation and data collection) does not play well with these Synthesis Skills that I still want to assess. And I’ve decided that’s okay, which is why I’m changing my assessments so that I can address these Synthesis skills but still hold students accountable for their Procedural and Conceptual knowledge. I also think its important to realize when a certain problem or concept or strategy is too big for a pen and paper test or even too big for an individual unit – that it’s worthwhile to dedicate a decent period of time to it and some sort of project or presentation in order to assess it properly. And so now, when I find units like this, instead of looking for the right ‘pen and paper’ assessment, I need to look for the right project/presentation assessment.
Final Related Blog Post: One day, I hope to have some sort of all-encompassing-project on the same scale as David Cox’s Farming Project.
I’m proud to be on the ground-floor of a possible ‘better assessments’ movement here on the math Blogotwittersphere. I’m excited to see people talking about their assessment process and admitting that our assessments could be better than what they could be – which is reassuring, because I’ve been thinking that for a long time. See this and this for some background of where my head is at. This post is a reflection on the things I choose to assess and how I choose to assess them.
One thing I’ve really begun to understand is how much assessment is guided by curriculum, and how choices about assessment can have amazing impacts on curriculum choices. I’m a believer that the things we assess and the way we assess is how we send a message to our students “HEY! This stuff is important! And you need to be able to do it if you want to be successful in this class!”
The organization of these individual skills and knowledge usually falls under the header of a ‘standard’. The way I assess, each page of my test is a separate standard that is graded independently from the other pages on the test. My underlying philosophy of these assessments is: It should be clear both to me and my students the standard that I am assessing. It should be clear both to me and to my students what the expectations of ‘mastery’ are for that standard. Assessments should make it clear both to me and to my students where their gaps in knowledge are, as well as their strengths in understanding. Assessments should promote student-directed remediation. Assessments should provide accurate data for a teacher about the level of understanding of his or her students. That’s a lot of pressure for an assessment.
This means it’s a big deal when we choose to assess something, and its a big deal when we choose not to assess something. I take this choice seriously, which means I need to examine the curriculum for each of my units and decide what it is that I want to assess and how I want to assess it. After doing this for a year, I’ve come to the following realization: Not all standards are created equally, which means not all standards should be assessed equally. When I look through my units and decide what I want to assess or how I want to assess it, I’ve started to group skills and concepts into three types of standards: Procedural Standards, Conceptual Standards, and Synthesis Standards.
Optional Reading: The choice of Procedural and Conceptual as the terms I chose comes from the article Adding it Up: Helping Children Learn Mathematics. This was an article I read in college when becoming a teacher – ‘Procedural Fluency’ and ‘Conceptual Understanding’ are two of the 5 strands of mathematical proficiency. As I was searching for words to describe what I was noticing in my curriculum, I reread the two sections on procedural fluency and conceptual understanding and felt it matched up pretty well to what I was observing in my standards.
As I looked through my curriculum for individual standards to assess, I realized there are certain skills that were simply foundational for everything we would do for the rest of the unit (or, in some cases, for the rest of the year). I needed to make sure my students understood these things at the level of ‘consistently computationally correct answers’. These are usually skills from previous courses or the foundation skill for a particular unit. My assessments for these skills are barebone: here are several problems you need to know, all at roughly the same level of difficulty, and you need to be able to do them consistently. I grade these harshly because they’re the foundation – there’s no question on here higher than the level of ‘identify’ or ‘apply a procedure correctly’. These are the skills that usually appear embedded in my Conceptual and Synthesis skills later on. They’re the skills where, once a student understands these, their success in the other skills suddenly skyrockets.
Outside Influence: Kelly O’Shea’s post about A and B objectives in Standards-Based Grading. I’ve taken her ideas about A objectives (what I’m calling Procedural) and B objectives (what I’m calling Conceptual) and added a third one: Synthesis. This post really helped cement a lot of the ideas I presented above, so I’m really grateful that I found it. If you don’t read her post now, I highly recommend returning to it and reading her bullets at the bottom of the post describing the benefit of separating objectives.
Procedural Skills that come from Algebra: Solving Algebra Equations, Integer Arithmetic (Assessment Below), Graphing Lines
Procedural Skills that come from Geometry: Applying the Pythagorean Theorem, Angle Identification (Assessment Below), Trigonometry Ratio Identification
For these skills, I don’t trust one solitary question to let a student demonstrate understanding – I make sure I have several so students can demonstrate consistency. I also grade these pages very harshly. On the integer test – if a student misses any more than 2 problems, they’ve failed that page. Most of my students aren’t used to this – they’re used to ‘slipping by’ on tests from other classes because its several skills collected together, so their little mistakes get lost in the mess of 100-pt test that they’re taking. This system doesn’t let them hide anymore – I’m making a statement with my assessment: consistency and computational correctness is important. The entire point of this page is to get these very specific problems correct. If you miss them on a different skill later in the year, I’ll cut you some slack – but for this procedural skill, I’ve purposefully created very little gray area: you either know it or you don’t.
This was one of the toughest things for me at the beginning of the year – explaining to students that they failed one page of the test because of several small mistakes. They’re not used to this, so they’re not happy about this, so it created some friction early in the year. But I’ve gotten better at this conversation as the year’s progressed and as my students have started to understand how high I’ve set my standards for these procedural skills. The conversation I’ve started having is: “Let’s say I asked you to spell your name 8 times. Should be easy, right? You know your name – no big deal. So you go to spell it and you give it to me, but I tell you that on the 5th line, you mispelled your name. Even though it’s just one time, what am I supposed to think? Spelling your name is something I would expect everyone to do no matter how many times – if you can’t, we need to have a serious talk, or you better do it 8 more times and prove to me that you really do know it. That’s what integers are for me: you need to be able to do it every single time. And if you can’t, either we need to talk, or you need to try again and prove to me that you can”.
Teaching Note: Procedural Skills almost assume that students will need to reassess. Multiple Times. And the work they do to reassess is not at the same level as the more conceptual skills in your course – some students with very low skills will need these basics from scratch, but many students will just need lots and lots of practice. I’ve solved this problem with my Wall of Remediation and by having assessment templates that I can use to create reassessments quickly and on the fly. But, in my experience, my high standards makes earning a 100% on these pages sooooooo satisfying.
These are the meat of my unit – the central conceptual understanding that I need my students to walk away with. These are skills that I imagine as scaffolded – there is a basic understanding, a strong understanding, and a mastery understanding. These are skills that usually have a problem-solving component or ‘explain’/'justify’/'analyze’/'sketch’ component embedded in them. They’re the ones where I really spend time trying to think about how to assess: “What’s the right question to ask so I that I can tell that they truly understand what they’re doing? How do I know they’re not mindlessly applying a procedure?”. When I think of a ‘bad’ test question I’ve written, it’s usually a question trying to assess one of these standards.
Outside Influence: I think this post by Jason Buell does a great job of emphasizing part of what I’m talking about: on a traditional test, how do you handle a student who nails all the trivial application of skills/vocabulary questions, but falls short on the application and synthesis questions? The resulting conversation about grading is worth reading too.
I’ve started creating Tiered Assessments for these skills, which I first read about at the It’s All Math blog but have since rediscovered a few other places. The basic idea is: You make a decision about what types of problems/prompts demonstrate ‘Mastery’ versus ‘Strong Understanding’ versus ‘Weak Understanding’ versus ‘No Understanding’. Or, for students who think purely in terms of grades, what an “A” student can do, what a “B” student can do, a “C” student, and a “D” student (and if you miss all of them, you’re an “F” student). I use numbers to communicate these ideas:
1 = Weak Understanding, 2 = Basic Understanding, 3 = Strong Understanding, 4 = Mastery with Small Mistakes, 5 = Mastery
This satisfies pretty much all of my goals for an assessment: it clearly communicates my expectations, it informs students about the remediation that they need, and it helps me collect data about my class. Creating one of these assessments involves deciding what my level 2, 3, and 5 problems look like.
I design my level 2 problems with the same philosophy as my Procedural standards: “These are problems everyone should be able to do consistently. Low-level Blooms. If you can’t get this right, we need to have a serious talk about these ideas”.
I design level 3 problems with the idea “What questions can I ask that requires you to make a choice about how to apply what you know? That may be multi-step or rely on some foundational procedural skill in addition to the current conceptual skill?” I usually use released items from the state assessment to gauge where these problems should be.
I design level 5 problems with the idea “Okay – prove to me that you really know what you’re doing. You’ll either have to apply this skill to a slightly new context, or decide how to apply it multiple times, or explain your thoughts in a way that proves to me that you know what you’re doing”. I’ve told my students: “my assessments are like an argument from you to me: it’s your job to convince me that you really understand what you’re doing. You can do this with your scratch work, with your explanations, or with your pictures – but whatever you do, it’s your job to be clear and correct so I believe you”. I think this especially applies with level 5 problems: I want to design a problem that really requires a student to do some leg-work to show me that they understand what they’re doing. For really conceptual problems, I want them to really explain what they know for me to judge. For procedural problems, I want there to be some sort of problem-solving or ‘habits of mind’ aspect to the problem that they’ll need to apply. When considering these problems, I usually look at Common Core resources, the Park Math curriculum, or any set of problems grounded in problem-solving strategies or habits of mind.
Here are some tiered assessments I’ve made that I’m proud of:
Analysis: I feel like the level 2 question gives me an immediate entrance into how the student thinks – the shapes are simple and the questions are simple. If a student misses this, we definitely need to have a talk, although I’ve debated giving them the areas as well rather than have them calculate it. The level 3 questions are straightforward if you know what you’re doing and build on a foundational skill (Calculating Area). But the level 5 question really gets to the heart of the student’s understanding – it requires explanation, analysis and reasoning, gets to the heart of how a student understands probability and how it relates to area.
Analysis: For the Level 5 Problem: I’ve written about Parallel Line Mazes before, but the gist is: a student has to ‘jump’ from angle to angle using the different parallel line relationships (Alternate Interior, Vertical, Corresponding, etc) meeting a certain set of criteria. This problem challenges the student to know more than just the name of the relationships, but how to apply those relationships in a novel situation and they must be comfortable with certain problem-solving strategies and perseverance.
See Also: Sam Shah’s Favorite Test Question is a Level 5 question – novel, gets to the heart of a student’s understanding, and requires explanation.
I can tell already – whenever I make an assessment I’m proud of, it’ll be when I’ve found the perfect Level 5 Question and the right transition from Level 2 to Level 3 to Level 5. I’m not there yet with all my assessments, but I think this is a good start. I feel extremely confident about the labels of ‘Master’ vs ‘Strong Understanding’ vs ‘Weak Understanding’ with the way this test is broken up, especially since I haven’t padded my test with extra questions just to hide what they do or don’t know.
Creating Better Assessments
Michael Fenton has written about his frustration with SBG assessments being purely application of skills. This is something I can absolutely relate to – when I first started implementing SBG and following the guides that I read online, I began feeling that the only type of assessment I could write was one that acted as a checklist of skills for my students to do. I struggled trying to find a way to keep that balance – of promoting problem-solving skills and ‘habits of mind’ while still holding students accountable for basic application of skills. This is the struggle that led to this blog post and my curiosity about assessments – I haven’t had very long to implement these types of assessments, but I feel pretty good about the direction this is going.
I’m still curious how other people write assessments. Michael Fenton is leading the charge and I highly recommend reading and responding to his post over at his blog. Tina C has written about her process and Lisa Henry is asking for feedback on a test question of her own. I think this endeavor is related to the question of “How do we create opportunities for our students to exceed our expectations”, and I’m excited to see these conversations continue and grow so that we’re all searching for these Level 5 Questions to give our students.
Some Parting Words from Sam Shah: (If there’s one thing I’m good at, it’s aggregating posts from the Blogotwittersphere with a similar theme, even if they’re from ages ago). Here’s Sam from when he gave a test that really asked students to express their thoughts:
“For me the obvious corollary is that: we need to start rethinking what our assessments ought to look like. If we want kids to truly understand concepts deeply, why don’t we actually make assessments that require students to demonstrate deep understanding of concepts?”
Some Foundational Ideas: Assessments are how I communicate to my students “These are the important mathematical ideas of my course – you are responsible for them”. When I tinker with my assessments, there is collateral damage to my curriculum (the order that I present mathematical ideas) and my lessons/activities (the depth with which we explore mathematical ideas). It all has to be aligned.
This post is building up to a realization I had earlier today. It comes from two key ideas I stole from Standards Based Grading:
1) I dissected my course into discrete concepts and skills that I could assess individually. When my students see my tests, each page is a separate skill and goes into the gradebook as a separate grade. This gives me a way to isolate particular skills (such as solving an algebra equation, or performing integer arithmetic) away from other mathematical ideas that build on these (finding missing angles with parallel lines, or finding the slope of a line given two points). This makes remediation easier, but it also makes it more explicit to my students which skills are ‘foundational’ and are needed to solve more complex problems
2) I assess certain skills multiple times. If my class still struggles with integer arithmetic (-2 + 5, etc), that skill appears on later assessments as its own page. Because each page is designated as a separate skill, students are aware of the fact that this is the explicit ‘algebra’ page. I also include this page when it’s the building-block for a skill we’ve been working on recently (for example: when I teach distance and midpoint in the coordinate plane, I also reassess on integer arithmetic because you need integers in order to do distance and midpoint calculations).
I’m realizing that these choices have fundamentally impacted some of my curriculum choices. Here’s what I mean:
Typical Situation from Last Year (Before SBG): I assess basic algebra skills at the beginning of the year, including integer operations (-3 + 4) and solving algebraic equations (2x – 14 = 26). For the purpose of this post, let’s say the skill of choice is solving algebra equations (two other foundational skill students usually take time to master at the beginning of the year are integer arithmetic and drawing geometric figures).
My test has several skills on it so the grade is more of a ‘summary’ than an itemized analysis – we lose information in a purely numerical grade. Because of this, many of my students get an ‘acceptable’ grade on my test (for some students, a 61% is acceptable), so they stuff it in their backpack and don’t think of it again – they passed, so it doesn’t bother them that they missed every single algebra question. However, I as the teacher can see that most of my class doesn’t know their algebra, even if each individual student doesn’t really care that they don’t know their algebra (remember: they still passed my test, so they’ve moved on to think about other things). I need to figure out a way to revisit algebra so my students realize that they need this skill for work we’re going to do later. Therefore, I adjust my curriculum so that algebra magically reappears a few weeks later in a different context, forcing my students to again confront the fact that they don’t understand this skill. So we spend a few more days on algebra ‘wrapped’ in a geometry concept, and then several problems like this appear on the test at the end of the unit. This gives me a chance to stealthily remediate and reassess their algebra skills without it seeming like we haven’t moved forward in the curriculum. One of the most unmotivating factors in curriculum is to linger on a topic for too long, which is why I need to create the illusion that this is actually a ‘new skill’ and we’re moving forward with our year.
So, we test again. Several of the problems on the test are these algebra problems ‘wrapped’ in a geometric context. After the test, more of my students understand algebra but still not as many as I would like. So I repeat this process. Before long, half of my curriculum has some sort of algebra component because I know that’s how long it will take for me to stealthily remediate and teach this skill.
More after the break below…
Just had dinner with an amazing teacher I work with – the topic of assessment came up – I want to write down some of these thoughts before I forget.
The Question: How do we create opportunities for students to exceed our expectations? Is that opportunity something that belongs on an assessment, or does it belong somewhere else?
Possible Answer: Students exceed expectations in the way they make connections and problem solve when presented with new content/meaningful problem/deep project. You are more likely to have students surprise and excite you when presenting new information rather than assessing old information. New contexts rather than old ones. These are the things we blog about – how amazing our students did on a project, or how a particular lesson elicited an ‘aha!’ moment that was more profound than we were expecting.
Does that mean my traditional pen-and-paper assessment can only reach the level of meeting my expectations rather than exceeding them? If I change the level of rigor that I want – in other words, if I raise the bar of for what meeting an expectation means – does that mean I have to change the type of assessment I give?
I’m gonna ask that question again, just because I like it, and so it’s the last thing in this blog post:
How do we create opportunities for students to exceed our expectations?
About a month ago, I did an interview with @mythagon as the inaugural episode of the Infinite Tangents podcast – a new project in the Blogotwittersphere meant to share stories and strategies about teaching, much in the same way Inside the Actor’s Studio shares stories and strategies about acting.
We talked for an hour or so about: it being my second year of teaching and what that’s all about; the process of becoming a teacher and getting ‘shattered’; the community of teaching and the blogotwittersphere; and some of the things I’ve tried to do in my classroom for the last few years. If any of those things sound interesting to you, you can listen to the podcast below:
(but before you do, apologies in advance for the number of times I say ‘like’ and ‘you know?’, as well as a few extra-loud laughs at the beginning of the podcast)