Frankensongs and Frankenfunctions: Using Mashups to Teach Piecewise-Defined Functions

After a riveting session about brain science at the summer program I attended (where I met Sam!), I wanted to read a little bit more about epistemology. I chose a few books that the presenter suggested: I just finished reading “Made to Stick” by Chip and Dan Heath (about why some ideas stick in our mind better than others and how to turn your ideas into some of those better ones) and am about halfway through “Brain Rules” by John Medina (twelve basic rules about how the brain works). Both were fascinating and will absolutely influence my teaching.

One thing that I have really latched onto is the idea of working with students’ previous knowledge about everything and anything in order to guide and improve learning (both books kind of harp on this). Take this example from Made to Stick where they define a Pomelo (an example which the lecturer also talked about at the summer program):

A pomelo is the largest citrus fruit. The rind is very thick, but soft and easy to peel away. The resulting fruit has a light yellow to coral pink flesh and can vary from juicy to slightly dry and from seductively spicy-sweet to tangy and tart.

If you already know what a pomelo is, that should make sense, and if you don’t, you can still get a pretty good picture of what’s going on. But compare that definition to this one:

A pomelo is basically an oversized grapefruit with a very thick and soft rind.

Both define a pomelo, but the second one uses the crazy ideas in your head to build new knowledge, making a much more descriptive and much stickier idea – not only is it easier to learn what a pomelo is, you will remember it much better. AAAND, the big bonus, it’s more efficient! [Here is a picture of a pomelo, by the way, if you need one. They’re kind of gross, but I am still partial to pomelos – in Arabic, pomelo is “bomaly,” and at first the guards at school couldn’t understand my weird sounding name (“Booooowman”), so they chose to hear the closest familiar thing, and started calling me “bomaly.” The origin of one of my many nicknames.]

Using Schemas in Math Education

I was thinking back to my year to see if I used anything like this in to teach math. I thought of one example, which I wanted to share, and then decided to put out a call for others. Can you think of a specific instance where you used anything from students’ prior knowledge to effectively and efficiently make a mathematical concept stick?

Piecewise-Defined Functions and Music Mashups

When reviewing at the beginning of the year in my Calculus class, I found that a lot of students were surprisingly stymied by the idea of piecewise-defined functions, which kind of blew my mind (this was in my first two weeks teaching math, and I was not expecting this to be a tricky concept for seniors in high school). It dawned on me that piecewise functions (which I call “Frankenfunctions”) are a lot of like Music Mashups, like this awesome mashup of the Top 25 songs from 2009 by DJ Earworm:

I played the song for the class and before connecting it to math, we broke down what we were hearing – like, actually had a brief conversation about what a mashup is (basically, one song constructed from segments of many others, though we went into more detail). Then we talked about the piecewise functions with this context:

  • A piecewise-defined function is one function made up of pieces of many others.
  • Each segment on a piecewise function is just a little part of a much bigger function.
  • The segments are broken down into intervals based on the x-axis (or time axis).
  • In piecewise functions, only one “song” can be playing at a time for it to be a function.
  • Piecewise functions can capture more interesting situations where the relationships between the variables in play changes.
I still got some of the craziest graphs I have ever seen on the following quiz, but the metaphor gave me a way to talk through their mistakes with them and hopefully gave them a way to connect something that is easily comprehensible to the slightly more abstract idea here. Now, this maybe isn’t the best example because it feels a bit cheap and may not get at deep understanding of some of the “whys,” so I will repeat my call again…  Can you think of a specific instance where you used anything from students’ prior knowledge to effectively and efficiently make a mathematical concept stick?

from @bowmanimal 

How I Grade Tests to Mine Learning Data [quickly]

For my first year using Standards Based Grading, I was an SBG-hybrid teacher. The standards that I used made up about 30-40% of students’ overall grades (the category weights changed over the course of the year) and I still included the traditional categories of Tests, Quizzes, Homework etc. This is for two reasons: I was really hesitant to change everything all in one year and I also felt compelled to fit with our departmental grading policy. Next year will probably be the same, almost entirely because of the latter pressure. I got into a little bit of hot water because I didn’t really explain what was doing very clearly at the beginning of the year – anyone else have the same problem?

But traditional “summative” assessments can, of course, still provide data you can use to guide your teaching and student learning. When I first started grading tests I would try to eyeball which problem students were getting wrong and then try to remember at the end of grading what skills or concepts they were struggling with. I felt like I definitely would pick out the major ones, but also felt like I was missing a lot. So I brainstormed a way to solve this problem and began grading all of my tests with Excel spreadsheets. Now I see something like this when I grade a test:

That might be a bit hard to see, but basically it’s a breakdown of what percentage of my students got each individual part of each problem correct on a test from this past spring (the actual spreadsheet goes a few more columns over to have the overall score too). I found that this gave me two main benefits:

  • Surprisingly faster grading (even with compiling all the data) with less totaling points mistakes
  • Extraordinary amount of data about specific parts of test problems that I could use to guide learning and to revise assessments from year to year

–> Example of a completed test grading spreadsheet

So how does this work?

1. The Setup
(this takes me about 10 minutes now, though took longer at first)

  • First, I start by placing the breakdown of the points for each question in the second row. This all depends on how you grade tests, but I generally have 6-7 questions on a test that are all broken down into a bunch of individual points for various items like “splitting up the area into a few parts,” “setting up the integral,” “simplifying the expression” and “correct answer.” This forces me to decide beforehand what is important in each problem and how I’m going to grade it. I put little notes above each point for me to remember what each point is for (and again, these force me to award points for specific things rather than a 6/10 for an “almost got it” answer).Then, using a summation (this is important), sum up all of the points into a total for the question and place this under the question number.

  • Then sum up all of the question totals to make the total number of points on the test. 
  • Last, using that row you created, fill down as many rows as you need for as many students as you have (plus one row that you can keep at the top to remind you of what each question is worth and the totals). If you use the fill function, the equations that you created in the previous step will stick with you.

    Now you’re ready for the actual grading process.

2. The Actual Grading

  • Pull out a student test and enter that student’s name in the first column next to their row of points. You can either grade page-by-page/question-by-question or student-by-student, but if you do page-by-page (which I prefer) just keep the tests in the same order. Then, when a student misses something, just enter a 0 in the space for that point. So if Bart correctly identified that the y values are needed in a Riemann Sum as the height of the rectangles, but used the wrong x’s to calculate them, you can leave the 1 in the y column but change the column about the x’s to 0. Notice that it automatically totals how many points he earned both for the question and for the whole test.
    Continue doing this until you have graded all the tests. This is the part that I find makes everything faster. The spreadsheet automatically totals everything so you can concentrate on making helpful remarks on the test instead of totaling points.

3. Reflection
(the powerful part)

  • Okay, so all of that is nice, but wouldn’t be all that worth it considering hand adding works fine (math teachers = good at mental math). But here’s the powerful part – now with one click of a button, you can see how the whole class did on specific parts of specific problems. Just average the responses for a specific question by averaging the column.
     ………
  • Then fill the equation all the way from the left to the right covering all the individual parts on each question, the question totals themselves and the test total itself. You automatically have averages for everything now.
     .
  • So check this out: The students did overall mediocre-ly (I love making up words) on both questions, but now we can see that they totally understand specific parts on both questions and totally bombed others. I often color code it with the “Conditional Formatting” tool to make this even more visual (only works if everything is out of 1 or you scale everything to be a percent of the total points offered):
     .
  • Now when you go to review, remediate and revisit, you can ignore the green items and focus much more on the reds and oranges. You could even try to judge if the part that NOBODY got right was even a fair question in the first place and use this datum to analyze your assessment.
  • You can also use a lot of other Excel features to quickly or do a lot of things like order the students to see grade distribution, curve your test in creative ways if you do that (or see what a bunch of different curves would do), hide the individual points columns to leave the question totals so you can switch between a macro and micro view, and if you input the students’ names in the same order as your gradebook, you can just copy paste right into your gradebook.

I also tend to do a lot of color coding and separate questions by colored bars. This is unnecessary, but makes it easier for me to look at (along with freezing the first column so I can always see the students’ names) Here is my –> example of a completed test grading spreadsheet (same as above).

The Best Part?

One of my favorite things to do though is to compare my data from the Standards to the data from the test. Comparing my formative assessments and my summative assessments. If the Standards are telling me that 92% of the class is rocking the Quotient Rule, but the test problem indicates that only 45% of people can solve a test problem involving the Quotient Rule, what does that mean? Do I need to alter my standards assessments? Was I lulled into a false sense of security by the high marks so I didn’t bother doing any review, or didn’t bother effectively integrating this concept into later material? Did the question I asked on the test line up with the type of thing I had been asking previously, and should it have? Had I been assessing algorithms previously instead of understanding? Lots of grrrrreat questions it raises for me every time.

The Second Best Part

I save all of these files for the next year. This (theoretically) allows me to focus my curriculum revisions on things that weren’t particularly sticky the first time around, and gives me concrete data to compare different approaches used in different classes (if I use some of the same exam questions).

Anyone else do something like this?

from @bowmanimal 

Make it Better: Memory Modeling

“A monk weighing 170 lbs begins a fast to protest a war. His weight after t days is given by W = 170e^(-0.008t). When the war ends 20 days later, how much does the monk weigh? At what rate is the monk losing weight after 20 days (before any food is consumed)?” <– That’s an actual problem from our Calculus book, which I find very amusing. Though it doesn’t really fit Dan Meyer’s definition of psuedocontext, I just get a kick out my mental picture of a monk sitting in a dark room taking a break from protesting the war to scribble away on a notepad trying to make predictions with an exponential model… There are so many word problems that force “real-life” situations into the convenient framework of whatever math topic is being presented in that section. I guess these are supposed to demonstrate to students how useful and relevant math is, but I think we all know that students just find them to be tricky and unyielding disguises to math that they generally know how to do.

There was one word problem that fit an exponential decay model to someone forgetting information, so I decided that instead of just doing the word problem, we would test the model by recreating the experiment. The day after we had a midterm exam, instead of handing back their corrected test, I put them in groups and gave them the following list of 50 three-letter syllables that I generated with a random number generator:

SOQ XAC DOB NEB BAR JYS ZYW GEK TUD ZEM GAK KUR BEN XOQ DUX BYR NIT WAP ZIJ HOG HIQ DUW CUD SAM BIM LIH JEV VEZ QEM GUL ZIQ SEQ JYV GUT XYM XAX BIQ DOJ ROM ZIV QEW JEH CYS ZEM FOM KEG DUC GYK WYQ POD

I gave them 15 minutes to memorize as many as they could and then tested them by having them write down all that they remembered. Then, I handed out the midterms and we started going over them. About 5 minutes later, I had them write down as many of the syllables as they could again. Then, we went over a few problems on the midterm… then another memory test…. then more midterm… then another memory test. They had absolutely no idea why we were doing this, so each time they groaned and complained. And they groaned even more when I opened class the next day with another trial. And then again two days after that… And then a last time a week and a half later. All without studying the list after the original 15 minutes.

Finally, I revealed the purpose of the whole experiment. We collected data and used GeoGebra to fit various models to their data. There were four different mathematical models to choose from that I found from various psychological studies (which I had loaded into a GeoGebra file with sliders so that they could move the various models around to fit their data). Each student picked the one that they thought fit their data best (a function to calculate how many words they would remember over time), took the derivative of that to calculate their “forgetting function” (a function that tells them how fast they are forgetting words at any given time), and then used both to calculate how many words they will remember in a few weeks and how fast they will be forgetting them at the point.

We graphed all of their functions on the same axes (y-axis = number of words remembered, x-axis = time in hours) to analyze which model was best and analyze how their memories compared to their classmates. The results are below. The different colors correspond to the model that each student chose.

        CLASS 1 –

        CLASS 2 –

Now, the clean final result of that graph hides how messy the model fitting part was. Though some students’ data fit well, some didn’t, at all, which was actually really nice. They really struggled trying to fit the model and hopefully realized that a lot of these models that we are dealing with in cooked textbook problems aren’t as powerful as they purport to be. If I could do it again, I would have them use more mathematically sound ways of fitting the models than just eyeballing it (I hadn’t really considered this and realize now that, though it would be an investment in time, it would make the whole thing much better).

But besides doing some authentic math that was individually tailored to each student, my favorite part of the experiment was the followup meta-cognitive discussion. We ended up having a really great conversation on how best to memorize these random things, which then led to a great discussion about how to learn and study best (especially how you should go about studying math). We talked about how some people put the words in context by using a story, some people made patterns by grouping similar items together, and the ones that didn’t do very well talked about how they just tried to memorize these random unconnected things by rote memorization. Many also noticed that throughout the closely connected trials on the first day, their number memorized actually went up, so we talked about how assessment can actually help you learn something too (in addition, of course, to regular practice).

Make it Better.

I have one simple question this time: the thing that I really didn’t like about this experiment was that it was entirely teacher centered. They were in the dark about what was going on (for experimental purposes) until the day that we collected data, fit models and did some quick calculations. How can I make this more student-centered and add elements of inquiry? I have a few ideas, but I wanted to see what other people thought.

Files:

  1. Word document with list of random words
  2. Excel spreadsheet for collecting data
  3. GeoGebra file with various forgetting models, ready to drop data in
from @bowmanimal 

Make it Better: Drawing with GeoGebra

Hello! Though Sam may refer to me as Kiki, don’t be fooled. My name is Bowman and I’m an American dude teaching MATH at a 9-12 co-ed boarding school in Amman, Jordan. I teach mostly Jordanian kids, though we teach an American-style curriculum in English, with sort of international school type outlook. For the past two years I have taught Physics, then last year I picked up Calculus, and next year I’m dropping the Physics to pick up AP Calculus AB. All of my friends can’t really understand why I’m so pumped about this because they think I’m the only person in the world that gets giddy about Calculus. False.

I love the math blogging community and am excited to be delving into it. Though I already have an “I-don’t-live-in-America” type blog about my time in Jordan, I have relied heavily on edublogs to develop as a teacher and I’m looking forward to repaying my debt. And probably like you, Sam’s blog is my fave, so I’m honored that he would give me some airtime. If you ever get a chance to meet him in person, consider yourself lucky. Since the thing that first drew me here was the wealth of practical lesson planning ideas in his Virtual Filing Cabinet (which I check pretty consistently before I plan a unit), I thought I’d begin repaying my debt by sharing some of the creative ideas I have used to present specific material this past year. Acknowledging my youth and paucity of teaching experience, I’m going to title these posts “Make It Better” to indicate that while I think these ideas have a lot of potential, I’m looking for ways to improve them. Enough introduction… on to the math!

Drawing with GeoGebra

For anyone who has not discovered the magic of GeoGebra yet, download it right now and then spend some time this summer playing around with it. It has a nice mix of geometric and algebraic capabilities, with fancy looking sliders and animations to help students visualize or experiment with mathematical concepts. These can be used in front of the class or on individual students’ computers. I ended up using it so much throughout the year in student directed learning that when we did end-of-the-year individualized projects, a majority of my students pulled out GeoGebra on their own to graph something or fit a model to some data. Sweet.

One of my favorite GeoGebra exploits this year was a “drawing” project, where students converted an actual picture into a mathematical picture by fitting functions around the outlines and then using integrals to shade in the area between. For example, they could a picture of a guitar and turn it into a sexy mathematical image, like this:

 >>>   
Basic steps (more detailed procedure below):

  1. Upload a picture into GeoGebra and scale the axes to the right scale.
  2. Place points around all the outlines making sure to hit critical points
  3. Fit functions to the outlines.
  4. Use integrals to shade in the areas between the outlines. The basic syntax looks like this…
    which means the integral between the [top function f, and the bottom function g, from x=1, to x=4].
So I conceptualized this project as a low-key but conceptually rich thing to do during the craziness of APs, and as something that the kids who were going to miss many days of class could do on their own. But it turned out to have some other really cool benefits too. Here are some things I really liked about it…
  • The hardest part about integrals is setting them up and that’s all students practiced in this project. They did absolutely no calculation. (In the age of Wolfram Alpha and TI’s could that be sooomewhat a thing of the past?)
  • The visual nature of the project gave immediate feedback to wrong inputs. If a student chose the wrong endpoints or the wrong functions, the wrong integral that they typed in would show up. They could see what was wrong about it to hopefully figure out how to correct their input, and correct their misconception. The tinkering aspect was maybe my favorite part because they often don’t understand that just by trying something to solve a math problem,  it can point you in the right direction to solve the problem even if it’s “wrong.”
  • It unearthed deep misconceptions about integration. Some students were conceptualizing integration diagonally, some would choose endpoints at completely wrong spots and some couldn’t conceptualize what areas they were trying to “color in” in the first place. I had lots of great conversations to address misconceptions that were at the same time a bit scary because we had already been integrating for a few weeks by that point.
  • Everybody’s problem was different. Each student was forced to visualize what he or she needed to do and had to attack a rather large problem by breaking it up into much smaller pieces.
  • The whole thing was kinda fun. Sometimes I pretend I’m above this, but each student chose a picture they were interested in and then we hung them up in the classroom at the end. And you can color the integrals whatever color you want. Pretty!
Here are some things that I didn’t like about it…
  • Some students totally copped out and chose really easy picture. I wasn’t very clear with my expectations (well, I actually didn’t know what to expect) and as a result a few students chose dumb things. One student did a watermelon… uncut…. like, a whole green watermelon. Or other students didn’t know what would be a “good” picture and chose something that ended up being way too hard, or uninteresting.
  • The function fitting part was a bit ridiculous. You can go really high with the degrees of the polynomials for the function fitting so people would just put points along a really curvy surface and then pick a 73rd degree polynomial that nicely fit the whole thing. I don’t know if this is actually bad, but it felt weird to me.
  • The problem was slightly meaningless. I had them add up the integrals at the end to find the total area, but this was a bit meaningless unless they had chose a flat object.

I can picture something like this being done for lower levels too. The concepts that come to mind immediately to me are piecewise functions and transformations. Instead of having them fit functions to points, you could give them a basic set of functions and force them to manipulate them with various transformations to fit outlines (and then ignore the coloring in part). There are easy ways to limit the domains of functions in GeoGebra. Below are GeoGebra instructions for the various steps and waaaay below are some more examples of student work.

But first, the whole point of this post…
Make it Better.
What do you think? What would you do differently? Do you think this holds educational value even though the problem is contrived? How could I make this more meaningful (i.e. make the result, the integral, not just the picture, actually hold value itself)? Should I give them a set of predetermined images for them to choose from to avoid “bad” choices? How else could I avoid “bad” choices? With what other material could something like this work? Would you use this in your classroom?

from @bowmanimal 

The procedural instructions for the various tasks:

Examples of student work (I had to include the watermelon because I mentioned it):

Introducing Bowman

Here’s Kiki!

I mean Bowman. Bowman (@bowmanimal) is a math teacher I met at the Klingenstein program I attended earlier this summer. He has only been teaching two years, but he’s The Real Deal. He has this passion and curiosity for mathematics that infects everything he does in the classroom. He has a personality of a superstar — an eternal, nice-guy, optimist. He’s focused on student learning, jumping in the SBG waters in his second year. He is multi-talented, teaching physics (not next year tho!), and is able to sing (among other things) the hokey pokey in Arabic. To put it another and more crass way, he’s one of those teachers that when you meet them you immediately get insanely jealous because you want to be them. But you aren’t. (Shut up, guys, I know you all get that feeling. David Cox, you know I dream about becoming you. But shucking the wife and kids, and spider infestation. Wait, you think that’s weird that my only dream in this world is for us to play Freaky Friday? Naah.)

I asked Bowman if he’d like to “guest blog” here for the summer, and he said yes.

You lucky ducks.

YOU LUCKY DUCKS.

So now that I’ve sung his praises, I’ll let him take over and introduce himself. I don’t know when he’s going to start, or precisely what he’s going to blog on, but hells bells, it’s going to be fresh.

Indeed, one of these strange creatures may or may not be Bowman. One of these strange creatures may or may not be me. One of these strange creatures may or may not be Chief Justice Roberts.

Participation Quizzes

I am going to be doing a lot more intentional group work next year with my classes. I’m definitely envisioning this for my Algebra II classes, and if I can come up with some good materials, for my Calculus classes too.

Today at PCMI, I was introduced to a way to do groupwork well. I am dismayed that I haven’t seen this before, because in some ways, it’s so obvious that I don’t know why it hasn’t made the rounds into my brain. I need to type it out here to codify it in my brain.

It’s called a “Participation Quiz.”

What I’m going to do is describe the video we watched of a teacher implementing it.
The teacher has students sitting in groups of 4. She introduces a worksheet she created to help students multiply binomials, but with some positive and some negative constants — because she saw that it was tough for her kids to deal with negative numbers when multiplying binomials. She had everyone’s attention on her, at the front of the room, and she says “today will be a participation quiz.” She then lays out her classroom norms for groupwork, some of which included:

  1. Everyone in the group must participate equally. There isn’t a leader, or the same person leading the show. The voices are shared.
  2. Students should not work too quickly. If they work simply to finish the sheet, without any other consideration, they aren’t doing it right.
  3. No one moves on until everyone understands. This isn’t about everyone having the same thing written down — but everyone has to know why.
  4. Students should think out loud. Students should check in with each other. Students should ask questions of one another.

She then let’s them get started.

As the groups work, she is both circulating, and sometimes at her laptop. When she is at her laptop, she is taking notes on each group — and displaying her notes on her smartboard live. Initially, her smartboard has group names (“purple group” “red group” …) written on there. It also has some specific actions which can be copied/pasted under each group, if they occur. Examples are:

negative actions: too quiet, talking outside group, off task, texting, different problems [students in the group are on different problems, not on the same problem]

good phrases: “I don’t get how you…”, “What did you get for…?”, “Can you also do it this way…?” “How did you…?” “Are we ready to move on…”

good actions: quick start [group started right away], reading directions out loud, same problem [everyone in the group is on the same problem], pointing and explaining, WHY???, BECAUSE!!!, calling group members out, all heads in, checking calculations/work, thinking out loud, equal participation.

Notice that these are specific things the teacher is listening for and looking for. They are actions — body language, speaking, interactions, etc.

The teacher watches and listens as she walks around or is at her computer. If she noticed any of the actions/phrases/comments, she typed them in her computer under the group name. It appears automatically on the SmartBoard for all to see.
At one point, one of the groups wasn’t working together. The teacher sat down and re-explained what the participation quiz to them, and even said “I’d rather you all work together and be stuck on one problem the entire class. This is about working together and coming to a shared understanding.” She then started getting them talking to each other, and then left.

The teacher also didn’t only copy and paste from the pre-written list on the SmartBoard, but also transcribed specific phrases/actions: everyone trying combos,  oh right, you’re multiplying” “would it be -21?” “so you mult… and…” “I got… that’s because…” “what do you think about that?”

At the end, her SmartBoard was full — a bit messy, but full. She did not shy away from writing the negative comments too. One group had “off task” written 3 times! What’s nice is that the teacher had a mathematical learning goal, but the lens through which she viewed the class (and the lens through which she had students view the class) was about classroom participation/engagement/teamwork. The two aren’t divorced.

She recapped the mathematical goal, but then she talked briefly about what she observed. She asked them questions about her SmartBoard. Under one group, her note said “I don’t know what to do after this?” and then she asked the class if that was a good or bad interaction. Most of the class said “bad” but through discussion she got them to realize it was good! That by saying that, someone is going to help that student, and the student may soon understand something. Through this process, she started clarifying the group norms for teamwork.

Fin.

There are so many amazing things about this. For me, this sort of activity, done a lot at the beginning of the year, is a concrete way to provide meaningful feedback for kids when talking about something as vague and “in the air” as participation. It builds the expectations for the rest of the year. It generates good conversations about what good groupwork is, and why. It provides the teacher a tool to get students to talk mathematically, and provide feedback. (Carol, one of the PCMI organizers, told me she will sometimes told me that sometimes she will do this and tell her kids she will only be looking for students justifying mathematics and those are the only notes she puts on the board.)

I don’t know if the teacher in the video actually assigned grades to each group. I think that’s something we’re going to be discussing in our groups tomorrow. But at the very least, it’s a really powerful way to spend 50 minutes on a mathematical goal while you are inculcating your class with a more “hidden curriculum” goal too.

I also think that a class, collectively generating groupwork norms (and the teacher adding missing but important ideas) could be a powerful exercise before engaging in this activity the first time. And using those norms as the lens to which to watch and critique students.

Virtual Conference on Core Values: The Heart of my Classroom

The conference is here.

The question of what’s at the rapidly beating heart of your classroom is a tough one. Let me rephrase that: for me, it’s tough, because it is totally evolving. Also whatever is at the heart of your classroom is your hidden curriculum — something that isn’t content, but just as important (if not more so) for kids to take away. So it’s pretty hard to get a handle on. It’s values.

Beginnings

In my first three years, I would have said the heart of my teaching revolved around three words:

clear

consistent

fair

Yes. Those three words drove me. The thing about having a core philosophy is that: everything revolves around it.  Every assignment. Every interaction. Every expectation. And although there are hard decisions that have to be made, when I struggled through them, I found I eventually turned back to my core beliefs, and I saw the light. Do I let that kid, that sweet sweet kid, take a re-test? Do I really need to create a super involved rubric with benchmarks, or can I just outline the project? If everyone in the class bombs an assessment, what do I do? [1] When holding core beliefs, every choice has to be intentional. Because these are what you value, and you need to enact those values. If you can only “say” your values, but you can’t “see” your values… then you’ve failed. [2] [3]

This philosophy has helped me out a lot with classroom management. It has helped me gain the respect of at least a good number of students. But I have started to see that philosophy as a baseline, now, of what I am doing. I believe in more.

Current Status

In the past year, the heart of my classroom has expanded to include more than clear, consistent, and fair. Thanks to the philosophical reorientation that Standards Based Grading has given me, it now includes metacognition and proactivity. [4]

I want my kids to be aware of what they know and what they don’t know. I want them to aware of the process of learning, and strategies to help them along the way. And I want them to be able to act on that knowledge. This is my hidden curriculum.

In Calculus, I used Standards Based Grading, which is all about kids getting a handle on their own learning. It forces them to understand what they know, and what they don’t know, and really articulate it! [5]

Dismantling the course into individual skills allowed me to have a specific breakdown of what the student knows and what the student doesn’t know. A student might have mastered how to apply the product rule, but struggle with explaining in words where the formal definition of the derivative comes from.  With SBG, I know this. In a school newspaper article written about my calculus class, one student was quoted: “The fact that the material is broken down into very specific skills as opposed to chapters or sections means you can focus on what you don’t know and figure out what you need to improve.”  More than me knowing where my students’ strengths and weaknesses are, my students themselves can recognize them.

I talk about metacognition, but that’s only half the battle. Who cares? Kids knowing about their learning habits, that’s great. But it doesn’t help them unless they believe they can grow from it. This is something I’ve been thinking a lot about since reading Carol Dweck and her notions of growth mindset. If a student — especially my students who tend to come to class never really appreciating math — thinks they suck at math, that they aren’t a “math person,” they’ve already stabbed themselves in the eye, shot themselves in the foot, whatever. There is blood everywhere, and it sucks. My kids come in with a fixed mindset. To get them engaged, to act on the “metacognitive” work, to see that doing well in math isn’t a matter of being “born with it,” I need them to see themselves as people who can change through hard work. Because really, if they don’t believe that, they won’t be doing hard work. They’ll simply continue to try to get by in math.

The thing is, we’re human beings. We suck. It’s hard to alter our own perceptions of ourselves. It’s also hard to say “we suck” and then decide to move on from there to say “let’s do things to suck less”!

This year I’ve been trying to do some good work in getting kids to be proactive, and to build their confidence. It involves a lot of individual communication with students. It involves me showing them that I care. It involves me avoiding ever comparing a student to another. It involves me demonstrating passion which occasionally translates into passion in them. It involves me talking explicitly about how math is a process, a journey, and how anyone can do it. It involves me not falling into the trap of thinking of certain kids as “smart.”

Standards Based Grading has helped me get kids to be proactive. My favorite example of this is a student reflection I’ve blogged about before:

1. I like the way that even though I was falling rapidly into a hole, and it felt almost impossible to get out, once you talked to me I became proactive and tried my best to do better. I like to continue meeting with you. I also like to continue to participate in class and asking questions. I think asking questions in class was the biggest way for me to better understand the topics.

2. I wish I would have started from the first day of school in this attack math mentality. I was acting very passive and like ‘oh I don’t get it now, but I will later,’ which honestly was the worst thing I could have done. I also wasn’t used to the class setting and the grading system. But once you emailed me and I met with you and I know that this is a class that I have to be in it 100%, and that your method is one that helps us actually learn, it was just beneficial. I needed that scare and wake up class because I was in serious denial. I became more on top of things. However, I had to dig myself out of a huge hole that I put myself in, but eventually the rhythm has become one that I used to. And I’m almost in a weird way glad that I learned the hard way because now I truly understand Math.

But that’s just one example. For as many kids as I might have helped, I know the struggle of SBG was enough to turn some of my kids off to math. I couldn’t get them to act. I don’t think it was laziness on their part, but despair. They hadn’t fully embraced the growth mindset and realized they could do it. I failed to be able to counter this.
I value a growth mindset, and I try to promote it through my actions. That is the current central core of my classroom. I’m still working on it, but here’s where I stand now.

[1] Of course I don’t mean “clear, consistent, and fair” to mean everyone gets the same treatment. Context matters, and what’s fair is not always “treat everyone the same way.”

[2] See Sizer and Sizer’s The Students are Watching (my review here)

[3] A grand experiment would be to have someone watch a video of your class, and try to suss out your values, and where they are expressed through your actions and words.

[4] SBG also has helped me remember the point of teaching: student learning. And now I have a razor sharp focus on that goal.

[5] In Algebra II, I deal with metacognition also, but not as well. I do this by talking to my kids explicitly about categorizing what they know and what they don’t know.

I tried to make homework more meaningful, by creating a full feedback loop. If a student got something wrong, they were asked to re-do the work and correct it. Otherwise they would have practiced the skill incorrectly, or illuminated the concept poorly, and never fixed it. (The “ill-leave-it-to-learn-before-the-test” syndrome.) I did this using binder checks (and redux), which had the added benefit of keeping (most) students organized.