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Clarifying Expectations

We’re doing a mini-unit on Probability in my Math Lab class this week. Probability is a tricky thing because it’s hard to predict how much a student has seen in their previous math classes and it’s hard to predict how ‘intuitive’ some students will find it. Today was our first day, so I wanted to take it slow: I wanted to explore how to find the probability of an event by examining the probability space. This breaks down into two separate sub-topics: situations where you already know all possible combinations, and situations where you first have to generate all possible combinations.

I designed the lesson so that we started by performing an experiment – I had every student write down the following on an index card: their name, their age, and their favorite number between 1 and 10. I collected everything, then asked questions around “If I pick a card at random, what’s the probability that the person I pick is…. a boy; a girl; is under 21; wears glasses”. These are all things students can answer from observation. Then I asked “Whats the probability that the card has a favorite number of 5?”, which they can’t answer by observation alone: I have to actually go through the cards and say what everyone wrote down. As I do this, students make a frequency chart – using the chart, we answer the question, then talk about what it means for something to be ‘more probable’ vs ‘least probable’. The whole time, I’m emphasizing the context behind the numerator and denominator of a probability fraction – what we care about / total possibilities (sidenote: I still haven’t found a good catch-all for how to think about the numerator. I shuffle between “What we care about”, “the event happening”, “number of ways to win”, and a few other things depending on the context).

After a bunch of explicit full-group and small-group practice based around this experiment, I give them this problem:

Probability 1These answers looked familiar – they were fractions just like what we had been doing the last 10-15 minutes as a class. They followed the ‘What we care about/total possibilities’ model. Most worked through this with ease. If there was a mistake, most students forgot to also change the total number of students in problems 14, 15, and 16 (more concretely: before the new student, there are 11 students in the class, so the denominator for #s 1-13 should be 11. After the new student, there are 12 students in the class, so the denominator for #s 14-16 should be 12. Most students left it at 11, even though they correctly changed the numerator)

After we checked this problem, I had them flip their paper over, where they saw this problem:


The goal of this problem is to introduce a situation where students must first generate the probability space (ie: find all possible combinations), and only then can they start thinking about probability. I wanted this problem to be the catalyst that forced them to think about systematic ways of listing things. My plan was to model a few possibilities, give them some ambiguous clues as to how to think systematically, let them explore independently and privately acknowledge the students who seemed to have found a system for counting, and then have those students share their strategies with the class. I didn’t tell them how many total possibilities there were, but I did tell them there were more than 16 and less than 30.

Here’s where I had this strange reflective moment that inspired this post

As I found myself describing this second task – listing all possible combinations – I found myself also clarifying my expectations for the purpose of their work. I found myself saying “I’m not expecting you to know the answer right away or to be able to see this immediately. I expect you to start listing possibilities in whatever way you can, but then I hope you find yourself looking for patterns or trying to organize things so that you’re sure you don’t miss any and you’re sure you don’t repeat any”. By the time I was teaching this lesson for the third time today, this speech evolved into something like this: “This task isn’t like the questions on the other side of your sheet where we had practiced and I was expecting you to know the answers right away. This isn’t something we’ve done before, and I’m purposefully letting you explore a little bit before I offer some more guidance. I’m not expecting you to see the answer right away because I want you to give this a try and see if you can find patterns or some clever way to organize your work. But again – the goal is to try something so we can talk about it, not to get the definite right answer right away”

Explicit practice is something we do a lot of in class. Problems for the purpose of exploration and discussion are also something we do a lot of in class. This isn’t the first time I’ve layered both in the same lesson: problems for practice and clarification, then problems for investigation and discussion. BUT – this is the first time I can remember being aware of the difference as class was happening and this is the first time I can remember explicitly describing this difference in expectation to my students. And, as I think about it now, this is also the first time I’ve realized how much of an impact this could have on how my students approach this task. In the past, I’ve probably glided seamlessly from one type of problem to another – “Alright, you guys feel confident about those ones? Let’s try these” without even realizing that my purpose behind the two sets of problems may be fundamentally different.

Being aware of my own expectations and communicating them to students is something I’ve been doing to help with classroom management and class culture, but I think today’s the first time I was aware enough to realize that I also have expectations of how students will react to problems or questions (“They should be able to do this on their own”, “They should do fine up to this step, then we’ll have to regroup”, “We’ll use these to frame our class discussion”), that those expectations can change rapidly mid-lesson, and that these expectations are important for the mindset my students have as they approach these problems. I wonder how many lessons I’ve given where I’ve had this transition and never mentioned the shift in expectation to my students. I wonder if it made a difference in the past – if students struggled with these investigative questions purely because I didn’t make clear that I was no longer expecting an immediate answer. If they shut down quicker because they weren’t sure how to solve it – or, even worse, if I was more impatient because I transitioned from practice questions to investigation questions without even realizing it myself.

Anyway – in this particular lesson, students took risks and started to write down possibilities even if they weren’t sure that they were on the right track; they were receptive when I had a fellow student explain one of their methods for counting; and they participated in the ‘how…? why…?’ discussion that happened afterwards. I wonder if the same thing would have happened if I had just given them the task without clarifying that this was a different type of problem with a different expected outcome. I wonder if students would have begun the task thinking it was just more practice and more familiar, only to become frustrated and shut down more easily. I wonder how many times I’ve accidentally done this in the past too.

Geometry Problem Generator

Update 1/22: Added another Problem Generator link to the bottom of the post. Original post is below:

Short Version of this Post: Watch This Video of me using Geogebra & Javascript to make my life easier

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!)

Luckily for me, I created a Geogebra program that lets me generate as many of these as I want. It uses Javascript to randomly generate problems and put them on the screen, then I use Geogebra to make it look pretty and save it as a picture, which I can paste into a word document. With this, I can generate 20 problems in 20 minutes – pictures, algebra, and answers in all.

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.

Here is the link to the video where I demonstrate how to make your own problems

Here is the link to the Geogebra Javascript code that you will need

Here is a link to a more sophisticated program using the features I talk about in the video

Here is a link to a program that will generate Similarity problems

Something Small & Positive Reinforcement

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.

Student Surveys

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?

Read more…

Classroom Design: How I Use Wall Space

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:

2013-08-09 16.16.07

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:

2013-08-09 16.16.24

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 also want to get this Growth Mindset poster, which I stole from Bowman Dickson’s post on Teacher Beliefs in Poster Form.

Read more…

SBG: My Standards, Assessments, and Thoughts on Grading

Hi Everyone,

I present, for your planning pleasure, portions of my Geometry Standards and around 30 Geometry Assessments that I used last year:

My Standards

My Assessments

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:

I think there is an ongoing effort to make assessments better and I know mine certainly could be. But, that effort is happening here:

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: 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:

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: 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.

Daniel Schneider
aka: Mathy McMatherson

The Wall of Champions

Wall of Champions

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.

Bellwork as Practice & How I Grade It

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.

Assessments: The Collateral Damage of SBG

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…

Read more…

Global Math Department & Understanding by Design

Hey all,

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.

So, if you’re curious, come on down tomorrow at 9pm EST. Tina and Elizabeth are co-presenting. It’ll be fun.

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).


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