Update 1/22: Added another Problem Generator link to the bottom of the post. Original post is below:
Full Version of this Post:
So here’s something I found annoying as a geometry teacher: it’s a pain in the butt to create my own geometry problems that incorporate solving an algebra problem. Some examples:
The difficulty in generating these problems is:
- Any algebra expression you create usually has to involve positive variables or a positive evaluated result (since it wouldn’t make sense for x = 7, but when you plug it in, the length of AB = -4)
- The algebra expressions depend on each other (for example: if you have two parallel lines and one of the angles is 47, then any algebra problem students need to solve must end up being equal to 47)
- There usually aren’t enough of these types of problems in textbooks to make them worth your time, but they’re really valuable for reinforcing algebra skills while also teaching geometry.
Right now I teach an intervention class for sophomores who are in geometry, so I would love it if I could find a ton of these problems because they let me talk about both geometry and algebra at the same time (which is great!)
So – if you’re interested in how I do this, I made a video and put it on Youtube. All you need is a copy of Geogebra and to be somewhat familiar with how Geogebra works. You don’t really need to know any programming, if you want to make more complicated problems, you’ll want to play around with what I have below.
I’ve been in something of a blogging rut lately, so I thought I’d try to find something small to share and hope it sparks more posts. So here we go:
I’ve been trying to find more ways to create positive reinforcement in my classroom. The response to the Wall of Champions in terms of attitude and motivation was more than I could have imagined, so I’ve been experimenting with other ways for students to receive positive feedback for their behaviors. This is also a manifestation of one of my core classroom beliefs: my students will care about the things that I care about. If I show them that certain types of behaviors are important, they will also think that these types of behaviors are important. So, I decided at the beginning of the year to create little ‘award’ cards to give to students when I see them doing something I like. Here they are:
These awards are meant to reward students who are doing things that I value in a student without the student necessarily realizing it or intentionally behaving this way. They’re not always the students who are incredibly eager to volunteer themselves in front of the whole class, which means its hard to find those moments when they open themselves up for positive reinforcement. I could find time to compliment them individually, but sometimes I forget or the moment has passed. These are the hard-working students who tend to fade into the background. And these awards are my quiet ways of saying “Hey – I notice you. And you’re doing a good job. Keep it up”.
Here’s how they work: I printed them on colored cardstock, cut them out, and carry a few in my back pocket throughout class. When I see a student doing something that fits in these categories, I find a moment to write their name on the award, and then quietly slip it in front of them. I try not to make a big deal about it and purposely ignore them when they ask “What’s this?” – they need to read it first. I don’t make a big deal about it, but my nonchalant attitude is sometimes more enticing and mysterious than the most elaborate performance I could create. Once a few of these start circulating, students pick up on it and start noticing it. The students who receive them feel validated that their hard work is noticed – that they’re doing something right – and the students who don’t receive them now know that these are the behaviors I’m looking for. And, hopefully, they’ll start imitating them.
A Fun Anecdote: These cards and this presentation is partly inspired by a story that a former colleague told me. Different organizations at my school sell candy bars in the hallways as a fundraising opportunity. My friend would always buy a Snickers in the morning, then put the candy bar on top of his smartboard. Throughout the day, he’d be on the lookout for a reason to give the Snickers away – something positive, worthwhile, and non-academic that one of his students would do. It was usually something different every day, in a different class period, to a different student – but he always gave it away.
One day, during a break in his lesson, one of his top students – without any prompting or explicit motivation – got up, changed seats so he was sitting next to a struggling student, and started helping him with the problem they were working on. After a few minutes, my friend grabbed the Snickers and handed it to the top student. Another top student saw this and asked the teacher “Wait, what just happened? Why did he get the Snickers?”. My friend responded, “I don’t know. He decided to help someone else out. It was a good thing to do”.
The next day, during a break in the lesson – without any prompting or explicit motivation - both students got up and went around and tried to help struggling students.
I’m doing something new this year: I’m teaching four sections of a math intervention course. I have all sophomores who’ve had a history of doing poorly in their past math classes. I’m loving it.
One thing I did on the first day with these students was give a survey. The purpose of this was for students to reflect on their past experiences and how their time in this class would work.
Their responses were honest and heartbreaking and sobering and important. Its amazing for me to see and interact with these students in my classroom, but then read about the experiences and self-perceptions that brought them here.
I’ve curated some choice quotes from their responses and I want to present them with a minimal amount of comment. I’m putting them here because they definitely made me reflect on some of my assumptions about this demographic of student, so maybe they’ll do the same to you.
Prompt 1: What is it that makes math classes hard? Is it the way that its taught? Did you have a bad class when you were really young? Is there something about Math itself that you just don’t like?
Over the summer, I changed schools and just finished my first two days in my new classroom with my new students. This means I had the chance to set up my classroom from scratch (again – this is actually the third time I’ve done it). One thing I’ve learned over the years is the value of using the physical space of the classroom to my advantage (see here and here). With that in mind, here are some of the things I’ve done with my physical classroom space this year.
1) Posters of Things I Say a Lot
I don’t know if there’s any scientific research backing this up, but I find there’s a huge difference between me saying something to a student 100 times, versus me writing that same thing on a poster and then pointing to it once or twice for them to reference. This is how my Rules of Math poster came into being:
I got tired of saying those phrases to students as encouragement, so I put them on a poster. Now I hear students saying it to each other. This year, I also added a How to Solve Problems poster and a Habits of Mind poster:
The ‘How to Solve Problems’ is heavily inspired by Polya’s problem-solving method. The Habits of Mind posters were made by stealing some of Bryan Meyer’s words and some phrases from the Park School of Mathematics. The posters themselves can be found here.
I present, for your planning pleasure, portions of my Geometry Standards and around 30 Geometry Assessments that I used last year:
There are comments at the end of each document detailing bits and pieces about how I made them and adjustments I would make if I had it to do all over again.
I’m not posting these because I’m especially proud or to brag or for feedback. In fact, I think most of it is pretty subpar. But, I’m posting them because someone emailed me asking what I did last year so they could have a place to jump-off from, so that’s what these are. I think they could be better. Maybe with these as a starting point, you won’t make the mistakes I made and your own standards and assessments become better. I hope they do, and that you post them, so someone else can jump off of those and we keep getting better and better.
I think there are better Standards documents out there on the web and a good place to find them is here: http://sbgbeginners.wikispaces.com/Skills+Lists
I think there is an ongoing effort to make assessments better and I know mine certainly could be. But, that effort is happening here: http://betterassessments.wordpress.com/
Update: This post was inspired by a teacher who emailed me asking about my assessments and standards and such. She also asked me about grading, which was a whole ‘nother long and complex email. I’ve copied it below in case you’d like to see even more into how I think about assessing and grading:
First, there’s the philosophy behind ‘grades’ and my desire for it to be more like feedback than like a grade. Most of that is well-documented on my blog (although if any of that is unclear, let me know and I’ll fill in the gaps). Then there’s my actual grading rubric – the 0-5.
Each page of an assessment is graded separately and entered into the gradebook separately. Each page receives a score of 1-5. The scores translate into the gradebook without any altering – a 1 on a test translates to 20%. A 4 translates to 80%. A 3 translates to 60%. This means, for a student to pass my class, they need mostly 3′s and 4′s on assessments, and a 2 represents a failing grade that necessitates remediation. I keep this in mind when I assign grades, and I’ll come back to this point later.
If a student left most of the assessment blank, I leave their score blank (not a 0, just blank) and tell them to come in and retake this. I think there’s something psychological about having a blank score vs a 0 score, and I find the blank score easier to motivate remediation with rather than the 0 score. Students are used to grades being final, so once any grade is given (even a 0), students tend to accept it. Blank scores, on the other hand, beg the question “Can I make that up?”. So, if I want a student to re-do something, I tend to leave it blank rather than give it a 0, even if the student already completed it but did a really poor job.
If a student gets 100% on a page, they get a 5. It has to be 100%. This is mostly for me so I don’t get too subjective with my grading and so I can be consistent. This is also why my 5′s are a big deal and why I started the Wall of Champions to help motivate students to get 5′s on my assessments.
Beyond that, a 4 is meant to represent “Understanding with 1 or 2 Small Mistakes”, a 3 is meant to represent “Strong Understanding, but inconsistent performance / one big glaring mistake that is straightforward to fix”, and a 2 is meant to represent “Little understanding – major mistakes, work does not convince me that you understand the material, we need to talk’. In my mind, 2 is failing, 3 is barely passing, and 4 is passing but not perfect. Here’s the handout I give to my students and I have posted in my classroom: https://app.box.com/s/36zaj5t1w6zmtjnsx6zo. Whenever I’m in doubt, I look at this to remind me. A few major influences for this rubric was Sam Shah’s rubric/explanation of his SBG system (there’s a link to this post somewhere on my blog), but also this grading rubric from a few teachers I know here in Tucson: http://edweb.tusd.k12.az.us/dmcdonald/documents/Rubric%20Math%20General.pdf
Update 2/21/14: In my next incarnation of how I describe what the different levels of understanding mean, I’m going to include some of the language from Evan Weinberg’s post of his own SBG Reflections. In particular, how he relates levels 1-3 around how independent a student is, as well as how he explicitly states “You won’t advance past here if you keep making this type of mistake”. I think his descriptions are spot-on and highly recommend reading his post.
How I assign 2′s, 3′s, and 4′s depends on what type of skill I’m grading and how specific their knowledge needs to be. For example, things like integer operations / linear equations / geometric definitions / coordinate geometry formulas (slope, distance, midpoint) / other foundational skills: I design the assessments to be very straightforward so that there is very little gray area in terms of the grade. This usually means those foundational skills are graded very harshly, but are also reassessed throughout the semester. This is me setting the bar high: everyone should be able to add and subtract signed numbers, and if you miss more than 2 questions on that assessment, you haven’t proven to me that you know it and you won’t earn higher than a 2. When I design these assessments, I want students to get a 5 on them, which is why some of my assessments look extremely straightfoward and simple – there’s no tricky or complex questions which means I can grade clearly and directly. It also makes it apparent when a student has a superficial understanding of a concept or skill, which makes it easier for me to remediate and fix.
For more conceptual skills – ones that are better measured with ‘explain’/'justify’/'sketch’ question – I usually think about the handout I give the students (linked above) and what that looks like for the specific skill I’m assessing. This is where separating the questions into “Level 2″, “Level 3″, and “Level 5″ questions helps make it easier for me to grade. If a student can answer the Level 2 questions correctly, they’ve earned at least a 2. If they can answer 2 and 3 correctly, they have at least a 3. If they make a mistake during the level 5, they earn a 4. This post was really influential in the way I think about these conceptual skills: http://itsallmath.wordpress.com/2012/08/23/tiered-assessment-for-geometry/. The rest is all subjective and based on the context of the assessment. In these situations, I think of their assessment as an argument to me – they’re saying “I know how to do this and here’s my proof!”. Which means if there are nonsensical statements, or a lack of work shown, or inconsistent mistakes (they get one question right but another question of the same type wrong), then I tend to mark down. If I’m debating between two grades and it takes me longer than 10 seconds to decide, I go with the lower one, since my internal debate must mean that they haven’t convinced me that they deserve the higher grade (if they did, my decision would be faster). The nice thing about SBG and offering reassessments is that if a student disagrees and talks to you about it, they can come in the next day and take another version of the test to prove they were right.
At the end of the day, the score on an assessment is both feedback and a grade. In the past, my final gradebook has looked like a reverse bellcurve – several scores below 40, several scores above 80, and a range of scores in between. When I was thinking about how I wanted my scores to translate into grades, I knew I wanted my grades to be more granular – I don’t really need the entire 0-100 range for student grades. I need extremely failing (20%), almost passing but still failing (45-55%), doing fine (65-75%), and exceeding (85-100%). This is why the scores translate exactly – a 1 is 20%, a 2 is 40%, a 3 is 60%, a 4 is 80% and a 5 is 100%. As a result, I found my gradebook looked like a true bell curve – a few scores in the low 20′s, most of them between 65-75, and a few A’s in each class. I found that it wasn’t until near the end of the semester that everyone’s grades leveled off where they should be. I found that giving assessments at the right time became extremely important – if my students aren’t ready, I don’t give the assessment. Having positive reinforcement for earning high scores is really important. Reassessing often is essential. Emphasizing a growth mindset is essential. Making it clear that I want students to ace my tests is important.
So…. there’s a lot of thoughts on grading. If something is unclear, definitely ask me about it and I’ll try to illuminate it.
aka: Mathy McMatherson
The Subtitle to this post: How I Get Students Excited About Acing My Assessments.
What You’re Looking At in the Picture: A bunch of post-it notes with my student’s names written on them (written by the students themselves)
Some Background: I use Standards Based Grading (SBG) in my classroom. This means whenever I give an assessment, students are graded on a holistic rubric from 1-5. A 1 means no understanding, a 3 means strong understanding and corresponds to a passing grade, and a 5 means mastery and corresponds to an A. In my system, 5′s always mean 100%. There is no argument or debate – it is black and white – to earn a 5, your paper must be perfect. As you might imagine, it can be difficult to get a 5 on some pages of my assessments. I set my standards high and my students know it. Which means when someone does get a 5, its a big deal. Like, a really big deal. I yell it out excitedly when I pass back tests – I can’t contain my excitement.
But, my excitement about 5′s wasn’t translating as well as I wanted to my students. Even though I was stoked and knew how big of a deal this was, most students just seemed to shrug and go ‘that’s pretty cool… I guess…’. And worse, some students were ‘satisfied’ with just earning a 3 or a 4, even though I knew they were capable of earning a 5. I wanted them to earn that 5, but they didn’t want it as much as I did.
And then I made the Wall of Champions. Here’s how it works: Whenever a student earns a 5 on a page, they get a post-it note on their test. With that post-it, they write their name on it and place it on the wall on the back of the room – a sign to the rest of my classes that this person earned their 5. In that picture above, each post-it represents a student who has earned 100% on one of the skills in my SBG system. Each color represents a different class, so students can see how each class compares with each other class. Throughout the year, I had them all mixed together – it was only at the end of the year that I separated them (like above) because I wanted to see which class had earned the most 5′s. And I threw that class a small party at the end of the semester to celebrate.
The philosophy behind this wall is simple – it’s a a simple, tangible, visible reward for students excelling in my class. And. Its. AWESOME! My students were getting pumped about having assessments returned – everyone wanted that post-it. And if they didn’t get it, they wanted to know why – so they would look at their test to figure out their mistakes and what they needed to do if they wanted that 5. Read that last sentence again – that’s a big deal. Then they’d say “I’m gonna retake this – I want that post-it!” (but first, they’d do some practice or come in for tutoring or whatever they needed so they’d understand the material). All students – even the ones who earned high marks - everyone wants that post it. Everyone wants to do better. This, coupled with the Wall of Remediation, led to an amazing ecosystem of motivation and action all on the part of my students. I just had to try and keep up with them
A lot of the results, from a social perspective, were fun to watch. Students started bragging to their friends – they’d come in before or after school to show their friends the post-it on the wall. They’d take a picture and upload it to Instagram. They’d see their friends on the wall and get competitive. They’d want to have the most post-its in their class. Students who weren’t reaching their potential would see their friends, who they considered ‘less-bright’, earning post-its and this would be the motivation for them to finally pick up their feet and start succeeding in my class. The amount of positive self-image that this created for my students was pretty awesome to watch.
Here’s my favorite part. For a lot of students, all they need is that spark – that initial feeling of success and progress before they’re ready to jump in head-first and take math seriously. Sometimes its bigger than math – after this spark, they start to take school seriously. But they still need that spark – as much as I say “You’re improving so much! Look how much better you’re doing!”, they still shrug their shoulders and don’t believe me. This is why I love SBG and my Wall of Champions – the Post-It is the spark. I’ve had so many students who, after earning that first post-it, become hungry for more. Their entire attitude about what they’re capable of completely changes – all because they’re able to put that post-it up on the board. They brag to their friends and parents and ask ‘What do I need to do to earn another one?’.
Full Disclosure: I stole this idea from another teacher. I was visiting a fellow math teacher in Phoenix (Hi Sarah!) when I saw this basic setup in another math classroom. This teacher doesn’t do any kind of SBG-related grading – he gives traditional tests at traditional times (ie: end of the unit). The teacher had two pieces of butcher paper on the wall with lots of post-its on them – one piece of butcher paper had the smaller post-its and the other one had the normal-sized post-its on it. The smaller post-its were for students who earned at least an 80% on his tests and the other regular post-its were for students who earned 100% on his tests. The 80% wall was reset every quarter, but the 100% board stayed up for the whole year. Same idea – a tangible, public incentive for students to be successful in his class. So, if you’re reading this and wondering if this can be done in a non-SBG classroom, the answer is yes because I stole it from a non-SBG classroom.
A Few Last Words: When I think of the usual motivation in schools, most students are satisfied to reach a certain threshold and stop. “I just need a D – as long as I’m passing, I’m good”. Traditional grades reinforce this. Very few students have the inherent incentive to earn high grades, and these are usually students who have this incentive before they walk into my classroom. I guess what gets me so excited about this is I’ve seen students change from the type of student who was satisfied with a D to the type of student who wants to achieve excellence. There’s probably lots at play here, but I think these post-its create a reward system based on mastery rather than an acceptable level of understanding, and I don’t think this happens too often in most schools (maybe honor roll?). The thing I’ve been the most impressed with are the students who earn passing grades but still want to earn that 5. This is the same as a student earning an 85% on a test, then asking to retake it because they know they can do better and they want that 100%. This is something I’ve never really experienced before and the only thing I can think to explain it is this Post-It reward system.
This post started as a comment on Lisa Henry’s post about Homework & SBG, but it was getting long so I decided to turn it in to a blog post. The essential question on Lisa’s post is: “what do you do about homework/practice problems when you are doing Standards Based Grading?”
Most of my daily practice comes in the form of bellwork and exit tickets. We have bellwork pretty much every single day – this is a very consistent part of my classroom. Bellwork is included as 5% of their grade in my class (very small). Here’s the typical routine:
The bell rings – I start a timer for 5-10 minutes. I take attendance, then I walk around and stamp students who have begun the bellwork. If a student hasn’t done anything more than write their name on the bellwork, no stamp. As I do this, I answer small questions but try not to get caught up tutoring individual students. When time is up, students take out a colored pen or pencil and we go over the bellwork as a class. This is mostly me calling on students to explain their answers (Cold Calling), unless there’s a specific point I want to clarify or a common mistake I’ve seen most students make. Students give themselves points when they have correct answers. If they don’t have a correct answer or they didn’t finish, they correct their bellwork with their colored pen or pencil, making notes about what mistakes they made. This last sentence is key. I collect the bellwork every day, but I never return it – I skim over it, then throw it away.
This is important because of how I grade bellwork: each bellwork is always worth 3 points. They get 1 point for the stamp – a motivation to not waste time and get started on time. They get another point for having every problem attempted, whether in pen or pencil. This means if a student doesn’t know how to do a problem, but follows along when we talk about it as a class and fills in their answer with a colored pen, they still get the point. However, if a student leaves a problem blank and doesn’t pay attention when we discuss it as a class, no point for this person. They get the last point for having corrected any mistakes in their colored pen or pencil. Notice that none of this has to do with whether or not a student answered a question correctly on their first try – what matters is that they tried to clarify their understanding as we discussed it as a class. I make a big deal about this at the beginning of the year – the purpose of the bellwork is to get practice, then reflect on that practice to determine what you need to work on, which is why you need to switch to a colored pen when you correct. Lots of corrections = come talk to me after school so we can see what’s going on.
A student who’s on-track should get 100% because they started on time and they know all the material. A student who is struggling should get 100% because they attempt every problem and realize where they get stuck, then correct their mistakes when we go over it as a class (which still earns them a point). A student who’s been absent for the last week can get 100% by asking a neighbor for help as I’m walking by with a stamp, then filling in the correct answers when we talk about the problems as a class. The only way not to get 100% is to not care about the practice in the first place.
The purpose of this system is to encourage self-reflection and ownership of where they stand in terms of understanding the course material. When we talk about the bellwork, students are focused and ask questions – they know that their points come from fixing their mistakes and making comments about their work. Eventually, this attitude stops being about ‘earning points’ and becomes a habit for them as students – to analyze how they did and what that means for them as students. In my mind, the whole purpose of practice is to encourage students to realize where they stand with the material – to help students become more self reflective and realize where their weaknesses are and what they can do to turn these into strengths. So, I grade bellwork in such a way that encourages these reflective habits, rather than rewards completeness or correctness.
So, the answer I’m pursuing to the ‘homework/practice in an SBG world’ problem is finding ways of grading that rewards reflection, correction, and self-regulation about how much practice they need before the assessment.
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 read 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.