Trying to find math inside everything else

Posts tagged ‘Education’

Satisfying Work

Earlier this year, Justin Aion wrote a post about how he tried to make his class boring on purpose by just giving silent independent work, to make them appreciate what he was normally doing, and how it backfired gloriously. At first, he wondered what he can do to break them of this preference for what they are used to and what is easy. About two months after that, we wrote about a similar situation, and wondered the following:

I’m beginning to wonder if my attempts to give them more engaging lessons and activities have burned them out.  I’m not giving up on the more involved activities.  I want them to be better at problem solving, but I think by trying to do it every day, I haven’t done a good job of meeting them where they are and helping to be where I want them to be.

As I read more of Reality Is Broken, though, I encountered an alternative explanation. In the book, Jane McGonigal wonders why so many people play games like World of Warcraft and other such MMORPGs where the gameplay is not, shall we say, the most thrilling. Many people find enjoyment in what other players call “grinding,” playing with the sole purpose of leveling up. In general, it’s a lot of work to level up in the game to get to what is considered the “good” part of the game, raiding in the end game.

But it’s work that people enjoy doing, and that’s because it is satisfying work. Dr. McGonigal defines satisfying work as work that has a clear goal and actionable next steps. She then goes on to say –

What if we have a clear goal, but we aren’t sure how to go about achieving it? Then it’s not work – it’s a problem. Now, there’s nothing wrong with having interesting problems to solve; it can be quite engaging. But it doesn’t necessarily lead to satisfaction. In the absence of actionable steps, our motivation to solve a problem might not be enough to make real progress. Well-designed work, on the other hand, leaves no doubt that progress will be made. There is a guarantee of productivity built in, and that’s what makes it so appealing.

Well, now, doesn’t that sound familiar? It kinda hit me in the gut when I read it. As math teachers, we are often preaching that we are trying to teach “problem solving” skills – but the thing is, people don’t like solving problems! It made make think of those poor grad students who are working towards their PhD – grad school burnout is a big issue, and one of the major contributing factors is that grad students are trying to solve problems, and so often feel like they are getting nowhere. Their work is inherently unsatisfying, which makes those that can finish a rare breed.

Our students, of course, are not all made of such stuff. But I’m not at all suggesting we drop our attempts at teaching problem solving and only give straight-forward work. Rather, I feel like we need to find a balance – for the past year, as I embraced a Problem-Based Curriculum, I may have pushed too far in the problem-solving direction, and found my students yearning for straight-forward worksheets, just as Justin did. But they also enjoyed tackling these problems, especially when they solved them, and I do think they had more independence and problem-solving skills by the end of the year.

So what should I do? Dr. McGonigal ends the chapter by noting that even high-powered CEOs take short breaks to play computer games like Solitaire or Bejeweled during the work day – it makes them less stressed and feel more productive, even if it doesn’t directly relate to what they are doing. (This reminds me of the recess debate in elementary school.) So even as I go forward with my problem-solving curriculum, I need to weave in more concrete work, and everyone will be more satisfied by it.

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games, schooling

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The Problem with Gamification in Education

(I suppose I shouldn’t say “the” problem, because there are many problems that I won’t be directly addressing, like extrinsic vs internal motivation.)

I’ve read a lot about gamification in the classroom, and while I’ve often thought about it and borrowed some elements from it, I’ve never gone whole hog. The motivation aspect is one of the reasons, but today, as I started reading Reality Is Broken: Why Games Make Us Better and How They Can Change the World, by Jane McGonigal, I realized there’s more to it.

In the first part of the book, Dr. McGonigal provides a definition of games. A game has four defining features: a goal, a set of defined rules, a feedback system, and voluntary participation. And if you think about gamification, you can easily pick out which of those elements is missing.

Because schooling is mandatory and, if you are taking a particular class, the gamification of that class is also mandatory, gamification of ed itself is not a game. If I gamify my chores by playing ChoreWars, I am choosing to take part in that game (even if the chores need to be done regardless). But if my teacher chooses to use a system of leveling up and roleplaying in my class, it is no longer a game; it is a requirement.

When I tried to think, then, about what in education would best fit these four requires, the first thing that came to mind is BIG, Shawn Cornally‘s school in Iowa. There students choose to participate in some project of their own devising, creating the goal and the voluntary participation. Then it is the school’s job to provide the feedback and the rules.

(An aside on the importance of rules – Dr. McGonigal quotes Bernard Suits who said, “Playing a game is the voluntary attempt to overcome unnecessary obstacles.” The rules are those unnecessary obstacles, and the excellent example given was golf. The goal of golf is to get the golf ball in the hole, but if we did that the most efficient way (walking up to the hole and dropping it in), we would get little enjoyment from it. But by implementing the rules of the game, we make the goal harder to achieve and thus much more fulfilling.)

So the big warning to those who want to gamify their classroom is this: if you require it, it’s not a game, no matter what game elements you include.

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games, schooling

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Being Out in the Classroom

Today I did the How I Met Your Mother hot/crazy scale lesson, which was strange this year. The past two years I did my statistics unit in October/November, so this lesson fell pretty early in the year. So I had a lot of fun because I was able to play with students’ expectations by using the androgynous names, the fact that the year is still new and they don’t know me as well and as less blatant about asking things, made for an overall enjoyable experience.

It’s funny because I don’t come out with intention every year, but it can sorta happen at times. I feel like this year my students still don’t really know across the board. And if they don’t know about me yet, they definitely don’t know about the other 3 gay male teachers. [4 out of  11 male faculty members seems like a lot. (The joke is that my old principal only hired either attractive young female teachers or male teachers that weren’t competition for the ladies’ attention.)] So I’m wondering if I played the game too well this year.

When I think about last year, there’s three moments that stood out. First was this lesson, which could plant suspicions but nothing confirmed. Then in December I did a lesson about the definition of a function. At one point, I ask for examples of functions that would map from the domain of people. Things like age and weight are examples, whereas race and hair color do not, since you can be more than one race or have more than one hair color. Then they say eye color, and I say it’s not, because someone could have two differently colored eyes. “In fact, my boyfriend has two differently colored eyes – one brown and one blue.” But if no one says eye color, it might not come up. And sometimes students jump in and mention that fact themselves. The third moment is when a student asked me who my Valentine was, with my response of “my boyfriend.”

For the current seniors and juniors, I feel like word spread quickly. The current sophomores a little more slowly, but by Feb 14 everyone knew. But this year it’s been somehow different. Partly it is due to the Common Core. I moved function definitions to the very beginning of the year, and so, I don’t know why, I said “I know somebody who has two different colored eyes” instead of specifying. Maybe I thought it was too early in the year? But then this lesson shifted later, so those two natural moments didn’t occur.

I mean, this didn’t stop individual students for talking about it. Most of my lunch gang knew because we’ve just had many more conversations and it came up. But my answer to “Do you have a wife/girlfriend?” Is always no, and the conversation often ends there. I won’t push it if they don’t, because we have math to do.

But because it wasn’t across the board acknowledged, somehow today was weirder. Maybe I’ll address it tomorrow.

This was longer than I thought – leave it to #MTBoS30 to make me ramble. I’m not sure what the thesis of this post was, other than “This can be surprisingly difficult to navigate, even if you aren’t trying to make it difficult or trying to navigate it at all.”

Category:

LGBTQ, reflection

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Rubrics for Standards

So my grading experiment has been going on for a month now, and so far I think it’s going well. But I was pretty stressed about getting it up and running, because a lot of the work was front-loaded. The thing I was particularly working to get done was my mega-rubric. I wanted to make a rubric that showed what exactly students needed to prove they understand to move up a level in a particular learning goal.

So here’s what I made (I call it the SPELS Book to go along with the students’ SPELS sheet):

I started by making the proficient categories, and for the first 8 (The Habits of Mind/Standards of Practice) it was pretty easy to scale them down to Novice, and then to add an additional high-level habit to become masters.

I was stuck, though, on the more Skill-Based Standards. I had all the things I wanted the students to show in each category, but how do I denote if they “sometimes” show me they can graph a linear equation? If I was doing quizzes all the time, like in the past, I could say something like “70% correct shows Apprentice levels.” But I wasn’t, and it seemed like a nightmare to keep track of across varying assignments.

So instead, my co-teacher had the idea that, if each topic had 4 sub-skills that I wanted them to know, we could rank them from easiest to hardest and just have that be the levels. So my system inadvertently became a binary SBG system, but still with the SBG and Level Up shell. Now if a student shows they understand a sub-skill, they level up. If they don’t, I write a comment on their assignment giving advice on what they should do in the future. What remains to be seen is how much they take me up on that advice. We’ll see.

Also, I’d LOVE any feedback you have on the rubric, and how I can improve it. Thanks!

Downloads

SPELS Book (pdf)

Updated Student Character Sheet (pdf)

Updated Student Character Sheet (pages)

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assessment

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Slow Rollout

This year has been weird so far. In the past the first week with actual students has never been a full week, usually just 1 or 2 days. So we’ll have some intro days, do intro stuff, and then head full steam into math class the next week. Last year September was so disjointed because of the Jewish holidays that we couldn’t even really get started.

This year, we started with a full week, and have 5 weeks straight of 5-day weeks before the first day off. So because we didn’t have weird intro days and odd days around holidays, I didn’t have a day introducing my class and systems, and instead went straight into math. I also have a lot more systems and routines now then I did in the past. So what I’ve wound up doing was introducing basically one new overarching idea or routine each class.

First class, Habits of Mind survey, then we did the Broken Calculator. (I’ve decided to loosely follow Geoff Krall’s PBL curriculum.) Next class, I introduce my new grading system (hope it works!) and then had to give them a stupid baseline assessment the city demanded. Next time, we set up our Interactive Notebooks, then did the Mullet Ratio. Today, I handed out the rubrics I’m going to be use to grade them (more on that next post), as well as introducing them to Estimation 180, and then we finished with day 2 of the Mullet Ratio. So every class has been a little routine, a little math. But I kind like it. We’ve been building up how the class works, layering it on. By the end of the month, we should be full steam ahead.

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schooling

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Habits of Mind, Standards of Practice

For the past three years, I’ve loosely organized my classroom around the Mathematical Habits of Mind which I first read about in grad school at Bard. I would give the students a survey to determine which habits are their strengths and which are their weaknesses, group them so each group have many strengths, and go from there. Last year I even used the habits as the names of some of my learning goals in my grading.

As I was planning for this year and the transition to the Common Core, I was thinking about how to assess and promote the Standards of Practice. And I realized that they are very similar to what I was already doing with the Habits of Mind. In fact, having a habit of mind would often lead to performing a certain practice! In that way, the SoP are actually the benchmarks by which I can determine if the habits of mind are being used.

Let me demonstrate:

Students should be pattern sniffers. This one is fairly straight-forward. SoP7 demands that students look for and make use of structure. What else is structure but patterns? Those patterns are the very fabric of what we explore when we do math, and discovering them is what leads to even greater conclusions.

Students should be experimenters. The article mentions that students should try large or small numbers, vary parameters, record results, etc. But now think about SoP1 – Make Sense of Problems and Persevere in Solving Them. How else do you do that except by experimenting? Especially if we are talking about a real problem and not just an exercise, mathematicians make things concrete and try out things to they can find patterns and make conjectures. It’s only after they have done that that they can move forward with solving a problem. And if they are stuck…they try something else! Experimenting is the best way to persevere.

Students should be describers. There are many ways mathematicians describe what they do, but one of the most is to Attend to Precision (as evidenced in things like the Peanut Butter & Jelly activity, depending on how you do it.) Students should practice saying what they mean in a way that is understandable to everyone listening. Precision is important for a good describer so that everyone listening or reading thinks the same thing. How else to properly share your mathematical thinking?

Students should be tinkerers. Okay, this one is my weakest connection, mostly because I did the other 7 first and these two were left. But maybe that’s mostly because I don’t think SoP5 is all that great. Being a tinkerer, however, is at the heart of mathematics itself. It is the question “What happens when I do this?” Using Tools Strategically is related in that it helps us lever that situation, helping us find out the answer so that we can move on to experimenting and conjecturing.

Students should be inventors. When we tinker and experiment, we discover interesting facts. But those facts remain nothing but interesting until the inventor comes up with a way to use them. Once a student notices a pattern about, saying, what happens whenever they multiply out two terms with the same base but different exponents, they can create a better, faster way of doing it. This is exactly what SoP8 asks.

Students should be visualizers. The article takes care to distinguish between visualizing things that are inherently visual (such as picturing your house) to visualizing a process by creating a visual analog that to process ideas and to clarify their meaning. This process is central to Modeling with Mathematics (SoP4). It is very difficult to model a process algebraically if you cannot see what is going on as variables change. To model, one must first visualize.

Students should be conjecturers. Students need to make conjectures not just from data but from a deeper understanding of the processes involved. SoP3 asks students to construct viable arguments (conjectures) and critique the reasoning of others. Notable, the habit of mind asks that students be able to critique their own reasoning, in order to push it further.

Students should be guessers. Of course, when we talk about guessing as math teachers, we really mean estimating. The difference between the two is a level of reasonableness. We always want to ask “What is too high? What is too low? Take a guess between.” Those guesses give use a great starting point for a problem. But how do you know what is too high? By Reasoning Abstractly and Quantitatively, SoP2. Building that number sense of a reasonable range strengthens our mathematical ability. We need to consider what units are involved and know what the numbers actually mean to do this.

What we do, or practice, as mathematicians is important, but what’s more important is how we go about things, and why. A common problem found in the math class is students not knowing where to begin. But if a student can develop these habits of mind, through practice, that should never be a problem.

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assessment, schooling, theory

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Twitter Math Camp ’13

Twitter Math Camp has come and gone, and once again it was truly amazing. The energy of all these other exuberant math teachers just recharges my batteries and gets me ready to go again. (Ironically I go on vacation in exactly one week, but I think this will be a productive week!)

I don’t feel that I learned as much at #TMC13 as I did at #TMC12, but that makes sense to me. Before last year I was only at the edge of the #MTBoS. I had only discovered Dan the summer before and was only following a handful of people by the time TMC12 came around. But after that, I dove in with full force, and absorbed so much great teaching. So this year, when TMC13 came about, I was more up to date and had less to learn.

What I did notice instead was that TMC13 was much more collaborative in nature. Last year, there was a focus on sharing things we knew, and exploring new math (the Exeter problems) together. That was still present this year, but so many sessions I went to focused on creating things together. I look forward to many of those projects coming to fruition (and have a lot of work to do on my half to make that happen).

It makes me wonder at the direction TMC will take next year. I have no idea, and that’s exciting.

 

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conference, TMC

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Math Games

Back in January I participated in a panel on Math Games over at the Global Math. I meant to write this follow-up post shortly after, but January was a hell of a month for me and it slipped to the wayside. See my talk here, at the 2:55 mark.

I sorta hit the same point over and over, using six different games as examples, but that’s because I truly believe it is the most important point in both designing math games as well as choosing which games to use in your classroom. If the math action required is separate from the game action performed, then it will seem forced and lead students to believe that math is useless.

Global Math - Math Games.003This can be fine if you want. Maybe you want to play a trivia game, where the knowledge action is separate from the game action. But if you pretend that they are the same, then you have problems.

This is the same essential argument as the one against psuedocontext. It may seem like you could say “It’s just a game,” but students see it as a shallow way to spice something up that can’t stand on its own. (I’m not saying review games and trivia games don’t have their place, but they can’t expand beyond their place.)

Below are the six examples I gave, with the breakdown of their game action and math action. I hope to use what I learned in this process to have us make a new, better math game in the summer, during Twitter Math Camp.

Example 1 – Math Man

A Pac-Man game where you can only eat a certain ghost, depending on the solution to an equation.

Global Math - Math Games.005

If we apply the metric above and think about what is the math action and what is the game action? Here, the math actions are simplifying expressions and adding/subtracting, but the game actions are navigating the maze and avoiding ghosts. If I’m a student playing this game, I want to play Pac-Man. The math here is preventing me from playing the game, not aiding me, which makes me resentful towards that math.

Verdict: Bad

Example 2: Ice Ice Maybe

Global Math - Math Games.008In this game, you help penguins cross a shark filled expanse by placing a platform for them to bounce over. Because of a time limit, you can’t calculate precisely where the platform needs to go, so you need to estimate. That skill is both the math action and the game action, so that alignment means that this game accomplishes its goal.

Verdict: Good

Example 3: Penguin Jump

Global Math - Math Games.011Here you pick a penguin, color them, and then race other people online jumping from iceberg to iceberg. The problem is that the math action is multiplying, which is not at all the same. The game gets worse, though, because AS the multiplying is preventing you from getting to the next iceberg, because maybe you are not good at it yet, you visibly see the other players pulling ahead, solidifying in your mind that you are bad at math, at exactly the point when you need the most support. A good math game should be easing you into the learning, not penalizing you when you are at your most vulnerable point, the beginning of your learning.

Verdict: Terrible

Example 4: FactortrisGlobal Math - Math Games.014

This is a game that seems like it has potential: given a number, factor that number into a rectangle (shout-out to Fawn Nguyen here in my talk), then drop the block you created by factoring to play Tetris.

Again, the math action is factoring whole numbers and creating visual representations, which are good actions. But the game action is dropping blocks into a space to fill up lines. As Megan called it, though, we have a carrot and stick layout here, and often in many games. Do the math, and you get to play a game afterwards. (Also, the Tetris part doesn’t really pan out, because all the blocks are rectangles, which is the most boring game of Tetris ever.)

Verdict: Bad

Example 5: DragonboxGlobal Math - Math Games.017

I’ve written about Dragonbox before, so I won’t write about it too much here. The goal of Dragonbox is to isolate the Dragon Box by removing extraneous monsters and cards. The math actions include combining inverses to zero-out or one-out, or to isolate variables. The game action is to combine day/night cards to swirl them out, or isolate the dragon box. The game action is in perfect alignment with the math action, which makes the game very engaging and very instructive.

Verdict: Good

Example 6: Totally RadicalPlaying the Root

The board game I created last year (and you can also make your own free following instructions here, or buy at the above link). In this game, the game actions were designed to match up with math actions. Simplifying a radical by moving a root outside the radical sign, as in the picture above, is done by playing the root card outside and removing the square from the inside (and keeping it as points).Global Math - Math Games.021 You also need to identify when a radical is fully simplified, which you do in game actions by slapping the board (because everything is better with slapping) and keeping the cards there as points.

Verdict: Good

Final Note

One of the real challenges of finding good math games, as a teacher, is curriculum. Most math teachers know of several good math games, like Set or Blokus. While these games are great and very mathematical, they’re not the math content that we usually need to teach in our classes. So the challenge falls on us to create our own games, but making good math games is hard. (Making bad ones is pretty easy.) On that note, if you know of some good math games (that meet the criteria mentioned in this post), drop a line in the comments!

 

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games

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Level Up! +1 to Exponents, +2 to Equations

Previously on The Roots of the Equation: You All Have “A”s, You All Have “0”s, and Grade Out of 10? This One Goes to 11.

I like games. All kinds of games: video, board, tabletop, role playing. And so I often think about how games and teaching align. One thing (good) games really do well is provide a sense of progress (especially role-playing games). You start off with not many skills, but as you advance you build them up, learn new things, and can conquer tougher tasks. By the time you reach the end of the game, those things that were hard from the beginning ain’t nothing to you now.

Games don’t usually score you on every little thing that you do. What they do is take a more holistic view and then, at some point, say that you’ve done enough to go up a level. And I say, why can’t I grade that way?

Many people have lamented that the best grading system would have no grades, just feedback that students respond to to improve their learning. But grades are required from external factors: school districts, colleges, parents, principals. But maybe there’s a way around that.

Last time, I said grades should just be a sum of the levels of the learning goals. So now I’m picturing students having a “character sheet” that looks something like this.

I maybe have created that name just so I could tell students to take out their SPELS sheet.

I maybe have created that name just so I could tell students to take out their SPELS sheet.

Student Character Sheet 2

The N/A/J/P/M are my current grading system, Novice –> Apprentice –> Journeyman –> Proficient –> Master

At the beginning of the year we can do a pre-assessment to determine their “starting stats and skills.” Then as the year moves in, we do our work in class. But none of that worked is graded in the usual sense. We would write feedback on the assignment, giving areas for improvement, but the only time a grade is mentioned is when a standard improves. Even then, we don’t focus on what they are (“You now have a 3 in Exponent Rules”), but rather in how they’ve grown (“You gained one level in Exponent Rules!”). The former just highlights that they are not the best they could be. The latter highlights their constant growth and improving.

(Then, at the end, based on what I said in the last post, their grade is literally how many boxes are shaded on the sheet. Have 75 boxes shaded? That’s a 75.)

In order to do this effectively, what we really need to have are rubrics for each standard. That way we know what counts as evidence of a certain level in a standard across all assignments, so it doesn’t matter which assignment provides the evidence. The upside to this is that you do not need to then have a rubric for each assignment! You only need your standards rubrics, because that is all you are using. (The collection of these rubrics, then, in the hands of the students, are a road map to success.)

I’m pretty excited by this idea, and can’t wait to try it next year. This is my idea from the last two posts taken to the next level, with a clear focus on growth, and not deficit. We can’t get rid of grading, and I’m not 100% convinced that we should. But we can definitely minimize the damage that it does and use it to actually promote students’ learning. All we need to do is focus on how we always get better.

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assessment

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You All Have “0”s

Last time, on The Roots of the Equation: You All Have “A”s.

To follow-up on my last post about grading, I wanted to talk about what I do in my class. What I do is applicable to all classrooms, whether they use SBG or not.

As I said last time, the promise of SBG is to promote a growth mindset with regards to grading: instead of being penalized by mistakes, you earn for proving you understand the standards and your grade rises. However, the responses I received belied that idea. When I asked what you would tell a student who asked their grade mid-marking period, most referred to something like a “snapshot” of their grade, simply averaging whatever they’ve done so far (whether it is standards in SBG, or test and projects and HW in more traditional grading).

If a student gets that snapshot every day, then it is quite clearly going to fluctuate and lead to some distress. Since my school uses on online gradebook, students can, in fact, check it. But I wanted my promise of rising grades to go through. So, I had to make it actually happen.

On the first day of class, I tell all my students they currently have a 0. Instead of 100 and dropping, every single thing they do in my class that is assessed will improve their grade. Even if they do terribly on an assignment say, getting a 50, that still improves their grade, because 50 is higher than 0.

That actual implementation of this, however, is hard. It means that, at the start of every marking period, I need to think ahead about what things I’m going to be assessing for the whole 6 weeks, and then enter those into the gradebook with a grade of 0. That way, everything will start at 0 and go up when actually completed. (Students can still see how they’ve done on things completed so far, and can determine their own “snapshot average” if they like, but this gives the view of the whole marking period.)

On the left, averages and assignments we have already completed. On the right, U grades mean “Unrated,” usually for assignments we have not done yet. The student who got an A- last marking period currently leads the pack with a 60.

But…thinking ahead 6 weeks about what I’m assessing…shouldn’t we be doing this anyway? Isn’t that just unit planning? My current Algebra course has 7 units, so it does work out to be almost one unit per marking period. And the process isn’t that inflexible: if I delete an assignment because I decided not to do it, or add something in, that’s a small fluctuation compared to the overall experience.

By the end of the marking period (as you see in my picture), everything will match up to the number it would have been had I gone top-down. But the way we get there is important. It is always better to grow.

ADDENDUM

After being questioned by Andrew Stadel and Chris Robinson on Twitter, I have some more explanations.

Andrew Stadel: I’d like to know more about this. Admin & parent understanding? Student response? Pros, cons, etc.

Me: Parents felt it was unclear at first, until I input marks that differentiated between “not done or graded yet” and “missing.” Then they were more on board. Students were confused by it at first, but liked it in the end. Admin supports it.

Pros include feeling like we are always improving and, a big one, it makes grading so much more enjoyable for me, because no one goes down.

Cons are that it’s hard to gauge sometimes (in terms of “snapshots”), especially when you get a big rush of grades at the end of the marking period.

Chris Robinson: James, can your “grades” go down per individual standard/learning target through the term?

Me: I’ve seen it go both ways in SBG. For me, they can’t go down in content standards, but can in practice ones. I do continuously assess but I feel like once someone has shown some understanding, they keep it, and they just need a refresher. (But I think I got that from Dan Meyer’s original “How Math Must Assess” post.)

Stadel: Thanks for explaining. What percent of students adjusted to & welcomed it? I like the premise of zero understanding and working towards mastery.

Me: Adjusted to, I would say over 95%. Welcomed, in the 80%. (Super rough estimates.)

Stadel: Do you have any materials/handouts explaining the philosophy to parents & students?

Me: I…really should.

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assessment

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