Trying to find math inside everything else

Posts tagged ‘Education’

The Integral Struggle

I had an extra day to fill for one of my sections of calculus, thanks to the PSAT, so I set about thinking up a game I could play. I gotta tell you, as much as there is a paucity of good content-related math games out there, it’s extra so as you move up the years in high school. I can still find a good amount of algebra and geometry games online, but Algebra 2? Precalc? Calc? Fuhgeddaboutit. Well, I’m teaching both Pre-Calc and Calc this year, so I guess I’ll just have to make them myself. (I brainstormed two more during said PSAT, so those might happen soon enough.)

Starting layout for The Integral Struggle

We just covered function transformations and how they affected integrals, so this game hits on that topic. (Yeah, we’re doing integrals first.) Here’s how it works: there are 3 functions/graphs, each of which has a total area of 0 on the interval [–10,10]. One team is Team Positive, and the other is Team Negative. Teams take turns placing numbers from –9 to 9 that transform the function and therefore transform the area. Most importantly, they also place the numbers in the limits of integration, so they can just look at a specific part of the graph.

With three functions, it becomes a matter of best of 3 – if the final integral evaluates to a positive number, Team Positive gets a point, and vice versa. If it’s a tie at the end of the game, because one or more of the integrals evaluated to 0 or were undefined, then redo those specific integrals. (I discussed with the students whether it would be better to have it be the total value of all 3 integrals instead of best of 3, but I think that makes the vertical stretch too powerful.)

That’s it! Let me know what you think. Materials below.

the-integral-struggle-rulesDownload
the-integral-struggle-game-boardsDownload
the-integral-struggle-graphsDownload
the-integral-struggle-numbersDownload

Category:

Calculus, games, math

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The Other F Word

It’s been a few years since a student has called me a faggot.

—

Not that I hadn’t heard the word, of course. I do teach teenagers, after all, and it does come up. But no more than once or twice a year, because I come down hard on it. I’m pretty jovial in class, and even when I’m mad it’s a quiet mad, but that’s a time when the full-shout comes out. The student needs to leave the room and have a pretty serious talk about the power of words and hate speech. Usually it is done out of ignorance. Usually we can move past it.

—

Today I was walking in the hallway, having just gotten my lunch, when I heard the word, solitary, a single statement alone. “Faggot.”

—

It has happened before, though not, actually, that word, but rather “maricón.” That was in my first year teaching, with a student I would spar with quite frequently. When I mentioned it to my principal that year he was suitably enraged – meetings were had, parents called in, etc. And then we kept on.

—

There were three of us in the hall at the time, so maybe I was unclear. “Are you talking to me?” I asked.

“Who else would I be talking to?”

—

I told another LGBT colleague about it the next period. They were visibly upset by the news, a quiet shaking, but a deep anger. “That’s not the kind of environment I want to work in. Something needs to actually happen about this.” Referring to, of course, the habit of the school to either let slide an incident or go for a suspension, with little in between.

—

How did I react? Stunned silence, I suppose. That full-shout wasn’t anywhere near the surface. Why was that? Direction, intent, they matter, I suppose. When I hear it errantly in class, I am still the teacher. It is my role to teach them the error of their ways, to make clear the severity of their transgression. The anger there is a teaching tool, in its own way. It’s building on the relationship I have with the student, showing my emotion to forge a stronger connection that can avoid it in the future. Maybe the anger was gone because the relationship was broken.

—

When I went to the dean immediately after, the school aide immediately left to pull the student from class. But we all wondered why. The student is not in any of my classes – I have not taught them since they were a freshmen. What purpose did this serve? What’s the point?

—

As I spoke with my colleague, the next period, they appeared, sheepishly, at the door. “I’m sorry.” “I was just talking about some sneakers and it just came into my head to point it at you.” “I didn’t know you were going to take it personally.”

How else was I going to take it? I’m a person.

—

I wonder, also, if my reaction was different because I know this student so well. Did I know that it came from a place of teenage stupidity, not a place of hate? There are certainly other students where it would be much more hateful if they said it, that I know. But here, the emotion I felt the most was confusion.

My colleague felt a lot better after the apology. We both talked with the student about how hateful speech can be and how the choices we make with what we say matter. The benefits of restorative justice, I guess. No suspensions will be made, but I am okay with that, of course. We can’t suspend our way to peace. When there’s a breakdown, well, we just need to build up again.

 

Category:

LGBTQ

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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.

Category:

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)

Category:

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.

Category:

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.

Category:

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.

 

Category:

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!

 

Category:

games

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