Axioms to Teach By

School is out - whoa, what is this "free time" thing you speak of?

2010-06-25T22:22:00.004-04:00

Hello all!

After such an overwhelmingly supportive reception from the online math teaching community, I completely dropped of the face of the internet. Sorry about that, guys! I didn't die or anything dramatic. I am just one of those unfortunate souls that hasn't worked out how to keep five minutes on the internet from turning into five hours on the internet, and cold turkey was the only strategy keeping me alive during the long months of the school year when there are always 25hrs of work to do today.

But SUMMER. Summer is when I actually have time to reflect on all those hours of work and decisions made ten minutes before class. I know everyone else stops blogging and twittering in the summer but, honestly - how do you people have the time in months when the temperature is below 78 degrees?

So a quick summary of the past year for those who might have been wondering what I've been up to:

1. My implementation of SBG and varied skill levels was a success, at least in that I stuck with it for all my classes, I felt good about it, and I got positive feedback from students. Sadly, I don't think I kept the kind of data that can demonstrate the improvement it had on my classes more quantitatively than that.
2. My classroom management improved, even though I don't really know why.
3. I used guided notes in my Algebra class and loved it so much I am now hoping to do it for Geometry as well.
4. I got engaged! And I'm getting married THIS AUGUST!!

So more posts to follow, at least for the summer anyway. Although I've been silent, I've been keeping up with everyone's blogs pretty well, and it seems like you all had a good year of trying new things - so here's to a well-deserved summer for all!

Good things happen when you need them

2009-10-20T15:16:00.003-04:00

Just a few uplifting notes from the day as a counterbalance to all the venting.

First, a former student gave me a box of sour patch kids as a present. I don't even know why!

Secondly, a current student, and not one who I thought was very fond of me, made my day. I actually stayed home yesterday with a(n admittedly weak) sore throat and when he saw me in the hall he said "Ms. Blank, you're back!" and gave me a big hug! I said "Wait, I'm still sick!" and he responded "I'll take your sickness." I love my students, I really do!

Oh, P.S. I passed my road test last week! I drove myself in to school this morning. That is definitely one area where I'm doing fine.

Pain shared is pain lessened

2009-10-20T13:26:00.005-04:00

Wow, I have been blessed with an abundance of commiseration and words of hope. I've been really touched by everyone's kinds words and advice. This sounds almost sadistic, but what helped the most was hearing that so many of you had been, or are still in, a similar frame of mind sometimes. Realizing that my difficulties are so common makes them seem less insurmountable.

This comment from Jake also really resonated with me:

Unlike the people you mention, the students would not have been enough to get me through. Yes, they're the reason I'm there and I love being around them and helping them discover real learning. But what really keeps me going is honing the craft, because that's what the challenge is for me. Teaching is HARD. Scratch that. GOOD teaching is HARD.

I didn't realize it, but it is the challenge that keeps me going. The long nights are evidence of how hard this is for me, but that's what gives me the potential to feel very proud and successful whenever I'm doing something right. And I have to say, I do feel like my effort is paying off in the classroom. It's just...a lot of effort. But, most of the time, I do feel that this is a worthy achievement to sacrifice a year of my social life for. I eschew the martyrdom that is sometimes associated with teaching, but I think it can be worthwhile to give up much of yourself in order to master difficult skill, in any arena.

To get back into a positive mindset, I'm scaling work back as far as possible this week. I'm not going to feel guilty for doing the bare minimum or planning lessons the night before instead of a week ahead of time.

I'm also going to take stock in the fact that what I consider the bare minimum this year is still better than almost any day last year! This is actually the crux of the matter: I'm being crushed by my workload because I'm trying to do too much. I really have full responsibility for the burn-out I've been feeling. Here's what I'm trying to make from scratch every week:

4 lesson plans for Elementary Algebra, Geometry, and Precalculus
4 guided notes for Elementary Algebra
4 homeworks for Elementary Algebra and Geometry
Quiz questions (3 level 1 questions, level 1 remediation questions, 3 level 2 questions, level 2 remediation questions, and a cool level 3 question) for Algebra, Geometry, and Precalclulus.

Also, I grade stuff sometimes. Absolutely no one asked me to do any of this, and the administration is only barely aware of it anyway. However, it has made all the difference in my instruction. As put off as I am by the endless work, I am in love with they way I am running my classes. I really must write a post about the success I've been enjoying with guided notes and quizzes.

So the true story is that even though I have so much work that I fantasize about quitting, I haven't even considered letting go of some of the work I've chosen for myself. Everything is coming together so well, and I'll be able to reuse it all next year. Sometimes, as in my post, I doubt that the investment of time I'm making now will be worth it - but most of the time I realize it will be, even if I'm not happy about it. The challenge is keeping my eye on the long term and not wallowing in the stress of getting through each day.

Thanks again for all the support. I am amazed and inspired by all of you who have gone through this, and stayed the course.

It only took 6 weeks to get to my first Big Crisis Moment of the year.

2009-10-18T02:25:00.003-04:00

I haven't been posting since the school year started, because I've been "busy", which for me is just code for "I've made a mess of my work schedule since day one and it's only gotten worse from there". I have chronic work/anxiety issues, and a lot of bad habits that have consistently prevented me from having any kind of guilt-free free time. I really wish I had friends to go to for support with these problems, but the Catch-22 is that I would need some free time in order to make friends in the first place. I sometimes consider seeking professional help, but my problems are just so...mundane, and I can't justify the expense. Then occasionally I think I should just blare my problems and doubts on twitter or the blogosphere, but honestly, you all just seem so well-balanced, that when I'm considering quitting because I'm just too damn lazy and undisciplined to do this job, I just don't feel comfortable asking for advice, especially since I haven't formed the kind of relationships that earn that kind of "shoulder to cry on" support. Everyone out there seems so full of love for the students and the job that it carries them through the long hours, but it hasn't been enough for me to break out of the vicious cycle of frantic work and procrastination I've been stuck in since first grade. I am really starting to think that I will never change - that I will always be stressed out and throwing things together at the last minute, and wasting all of my evenings and weekends procrastinating and hating myself for it rather than having a life.

Honestly: screw this. I am just not cut out for it. I need a job where when I'm not at work, I'M NOT AT WORK, or I may as well resign myself to a lifetime of remorseful all-nighters like this one. It's too bad too, because otherwise I really love teaching, and pretty much can't imagine doing anything else.

I know the beginning of the year is always chaos, but really...

2009-09-17T06:33:00.002-04:00

From an email from the Associate Head of School:

So, the confusing bottom line is that you will either know or not know in advance of a test [of the fire alarms] that may happen around 9:00 am or at some other time in the morning,

Developing Good Habits

2009-09-16T20:10:00.004-04:00

I never really developed good habits as a student. It was just never necessary. I got by perfectly well doing things late at night the day before they were due. That changed last year for me. Teaching just does not allow for that level of procrastination. When you have over 24hrs of work due tomorrow, you are in big trouble. However, I was so overwhelmed I never did get into the swing of things workloadwise; I just worked late when I had to and tried not to think about it the rest of the time. Needless to say, this is not my ideal work situation.

So one of my major goals for this year is to develop a really solid routine with work, so I can actually have time in the evening to relax, free from the guilt of mounds of grading and planning. I had some excellent success setting up good work habits this summer, and I'm putting them into practice again as I start the school year. Here's some guidelines I've learned to follow.

Track your progress: Can you imagine dieting without checking your weight? You need some sort of feedback on your efforts to stay motivated. I actually rate my success at sticking to my planned work routines by a very specific rubric, and keep all the data on an excel spreadsheet. It can seem obsessive, but whenever I blow an afternoon off, just looking back at all my progress convinces me to keep trying.

Start small: It's overwhelming to take on a whole new routine at once, which makes it too likely that you'll just give up. I learned this the hard way; my inclination is to totally redefine my lifestyle every few months. But by just trying to add one ridiculously simple change at a time, I've been able to make better lasting progress. Currently I'm just trying to get up early, go to sleep early, and plan Algebra in the morning.

Set specific goals: I used to say things like "From now on, I'm going to finish my work on time". This is too big, too vague, and doesn't provide satisfaction unless I'm totally on schedule. I've had much better luck committing to things like "I'm going to start working on my Precalculus plans by 9am on Saturday mornings."

I'm trying to work my way up to a completely regimented weekly routine, which for some reason I think would provide the greatest peace of mind, but I'd be absolutely thrilled if I could just manage to get my work done before going to bed on a regular basis. I'm curious how much structure other people try it inject in to their workdays. I know a lot of people who are just fine doing what they can when they find time, but for me, actually having a habit of doing work at a given time makes an enormous difference. It's so much simpler to start lesson planning during 8th period when I just know I'm going to do it because that's what I've done in the past 30 8th periods. Good habits can make the battle with your will into an automatic victory. Here's to starting the new year with as many as I can!

P.S. It was my first day of school today! It was chaos, but I loved seeing all my students again.

Back to School Jitters

2009-09-13T20:23:00.004-04:00

I know everyone else has been back for weeks by now, but I start school this Wednesday. I was very excited to get back to school earlier in the summer, when I was fleshing out all the shiny new policies and systems I plan to implement. Now that the first day is just around the corner, I instead find myself actively avoiding any thoughts of school out of anxiety. I have had such lofty ambitions, and have designed such complicated schemes, that now I feel destined to fall flat on my face and have it all fall apart. I haven't even started and already I'm sure I'm trying to do too much. Maybe it would be smarter (or saner) to just tear my plans up, teach from the textbook, assess traditionally, and leave work by 4pm. Or maybe, this is just preemptive laziness. Either way, I've invested too much in my ideas not to give them a shot, but I am desperately hoping that this job isn't really as overwhelming as I've made it out to be in my head.

Cool Problem Stockpiling

2009-09-08T21:18:00.002-04:00

I've been coming across lots of great problems and materials on the internet recently, but I haven't developed a system for storing my finds in such a way that I can find them again when I need them. I think the ideal online problem-keeper would allow me to save pages to different categories and add notes to pages. It would be wonderful if I could also store problems that weren't from webpages at all, so I could keep all my cool problems in the same place. Is there an easy way to do this? I'm curious what methods other teachers use. I've tried delicious, but I'm not sure what else is out there.

Multiplication Mailbag

2009-09-03T22:04:00.006-04:00

I got a lot of wonderful responses to my last post which have been percolating in my head over the past few days. Two images of multiplication seemed to emerge:

Multiplication is a change of unit.
Multiplication is a change of scale.

These are equivalent, but each provides a slightly different picture which I think would make a difference to my students.

The idea of multiplication as a change of unit is essentialy that when you say "4 times 6" is "four sixes", you are making "six" the new "one". It's the new unit. The difference between this and just saying multiplication is counting groups is subtle, but important. Substantially, introducing multiplication as counting groups (or repeated addition) reduces it to a mechanical trick one might use to simplify an otherwise lengthy sum or counting exercise. There is no need for the student to adjust his concept of unit. In fact, since the purpose of counting groups is generally just to get an answer in terms of the original unit, it renders such an adjustment inherently useless. What multiplication really allows though, is for us to leave the old unit behind and talk purely in terms of a more convenient one. Burt from Hawaii sent this ingenious lesson plan for drawing out this point.

In the Russian curriculum of Davydov, a follower of Lev Vygotsky, multiplication is taught as taking an indirect count using an intermediate unit size. In one lesson, students are asked to find how many tiny cups of water can be served from a large container. They start pouring the water out and counting how many tiny cups there are, transferring the water into another large container. But it is tedious and difficult to pour into such a tiny cup. The teacher asks them to think of a better solution than pouring into the tiny cup. The next day the teacher suggests using a large cup. They pour 8 tiny cups into the large cup, so they know that each large cupful is 8 tiny cups at once. They record 7 large cups, and he teaches them to write it as 8 x 7, and reads it as “8 taken 7 times”. They don’t compute the product right away. They do more exercises measuring different things with a larger unit, measuring how many smaller units are in the larger unit, and writing the result as 4 x 5, for example. The teacher wants them to understand the concept of using a larger unit size and counting in that unit size. Then later they compute the product using repeated addition. Repeated addition is a useful computational strategy, but the concept is changing the unit size and counting in a larger unit size.

If I taught elementary students, I would absolutely do this activity. Maybe I should even recommend it to the elementary teachers at my school. Unfortunately, my ninth-graders are are too cool to measure water cups. Instead, I think I may show them something like this image that David put together. 'A' is the new one. That's what multiplication does.

The other image I now have in my mind for multiplication was suggested to me by a good friend who called me just to talk about it. Thinking of multiplication as a change of scale lead him to picture the real number line getting stretched like a rubber band. Zero is the center of course; it doesn't move, and the distance you move is directly proportional to the distance you were from zero in the first place. This is similar to picturing multiplication as zooming in or out. I like this visualization because it's just as easy to see what happens to non-integers as integers and shows that multiplication is truly the same operation across the real numbers, whereas when I think of multiplication as changing the unit I find myself still stuck thinking in terms of discrete groups.

The actual approach I plan to take in my classroom is to let my students propose their own definitions at first. I am fairly confident they will sway towards repeated addition. I will absolutely go along with this at first, because I want the understanding of how inadequate this definition is to arise naturally, when we start talking about rational numbers. At that point, I will prod them to fix their definition, and share both of these suggestions.

Making the move to exponentiation still proves a challenge. It's a change of scale, but not a consistent change of scale. I can just barely form a mental picture of the number line as transformed by x^2. Numbers far from zero fly off to extremes, while numbers close to zero huddle closer. I can't imagine getting my students to share the same visualization before they are all ready comfortable with exponentiation; it's simply not very intuitive. I presume I will try the same tact: let the students define it inadequately, then put them in places where they need a better definition.

Further reading on the topic from Keith Devlin: It's Still Not Repeated Addition! Keith actually argues against the tactic of introducing inferior explanations first and revising them later, but in my defense, I am hardly forming my students first impressions of multiplication, and in order to properly shock them out of their complacency with repeated addition, I believe it would be most effective to have them tell me what's wrong with it, rather than simply lecture them about how they've been lied to all these years.

While thinking about this, I've restricted myself to the goal of leading my class to a better understanding. However, the problem hiding in the margins of my mind the entire time has been that this isn't really my issue. This is an elementary school math issue. After nine years of math education, my students should already have an excellent grasp of multiplication. However, in general, the teachers who have the greatest opportunity to explore the concept of multiplication are not part of this conversation.

What is multiplication?

2009-08-30T21:56:00.003-04:00

I start my review of integer operations by having students try to define them. It's incredibly difficult, because it seems like it should be so obvious. When I ask people to try to define multiplication, they tend to say it is repeated addition. Mathematically, this not really correct. It works alright for integers, but that's it. Likewise, understanding exponentiation as repeated multiplication falls apart when the exponent is not an integer. I do rely on these interpretations of multiplication and exponentiation when initially defining them, but I always make a big point of the fact that these interpretations don't work in all domains. However, I'm really struggling to come up with another way of defining them that works on all real numbers that is appropriate for ninth graders. The best I can come up with is to make multiplication analogous to finding the area of a rectangle. For exponentiation, I'm completely stumped: it's not like I can talk about the volume of a n-dimensional cube. Any suggestions?

Why My Students Fear Tests

2009-08-27T22:53:00.006-04:00

With more experience, the number of potentially relevant elements that the learner is able to recognize becomes overwhelming. At this point, since a sense of what is important in any particular situation is missing, performance becomes nerve-wracking and exhausting, and the student may well wonder how anyone ever masters the skill.
- Hubert L. Dreyfus

There is another way learning (and struggling) to drive has given me perspective on teaching: it's taught me exactly what this feels like. I don't think I ever quite got the same experience as a student. I think I'll be more sympathetic to my classes this year.

A Model for Assessment: the Road Test

2009-08-27T13:17:00.008-04:00

As a native New Yorker, I grew up without any incentive to learn to drive. No one in my high school could drive, and many of my friends were from families that didn't even own a car. Now, after a year of living in Albany and growing increasingly frustrated with its inadequate public transportation, I am finally learning to drive. And boy, do I suck at it.

I have lots of excuses. You see, I am a very focused person. When I'm concentrating, I hardly notice my surroundings. You can call my name and I won't even look up. So driving, which requires diffused attention, just isn't my brain's style. Furthermore, I'm slow to process what my eyes see. I can look in my blind spot and still not notice the car on my left. I'm just not much of a "visual thinker"; I'm too abstract. Probably the thing that really makes driving difficult for me is that I'm plain terrified of it. Everyone tells me to relax, but I think everyone else should get a little less complacent: driving is dangerous!

Of course, none of this makes one whit of a difference to the DMV. No matter how much I've improved, no matter how far I've come in overcoming my fears, no matter what kinds of mental handicaps I have, I am not going to get a driver's license until I demonstrate that I am a competent driver, the same as everyone else.

This is in stark contrast, of course, to the sorts of accommodations I've sometimes been asked to make for students who for various reasons "don't test well" or just "aren't math people". Now, I absolutely believe that there are people for whom the kind of thinking required in mathematics is a particular challenge, just as I believe driving is a particular challenge for me. But the difference is that some students and parents think this means that they should receive different assessments, or a different grading structure, so that their children have a "fair chance".

Implicit in these requests is the notion that competency in math, unlike driving, is not essential for the general populace. If people actually thought that the math test they were being asked to pass tested important, meaningful understanding that everyone should have, it would be obvious that making accommodations would do a disservice to the student as well as the public. Rather, most people seem to hold the opinion that math tests are meaningless hurdles and that effort, rather than competency, is what the grade should reflect.

Clearly, I disagree with this position: I do think there is a real danger in a totally innumerate population, and that mathematical understanding is something everyone should possess to some degree. However, that doesn't mean that my assessments automatically reflect that. If I want to toe the hard line and impress upon my students and their families that an A means excellence, not effort, that knowing what I'm teaching is worthwhile, even if it's hard for you and you're scared of it, then I have to do what I can to align my grading with those ideals.

Here again, the DMV provides a model. I failed my road test today. To a truly embarassing degree. (I cut off a guy in a tow truck and he got mad and started tailgating me.) However, at the end of the test, instead of receiving a big F or a judgmental sermon on how terrible I was, I got a lot of sound advice on where I needed to improve, and some tips that would make it easier. I even got a printed list of every stupid error I just made. Now initially, being faced with a long list of your specific failures hurts. The DMV doesn't sugar coat it when they say "dangerous violation". But this is targeted remediation of the most helpful sort. I know exactly what I should be working on as I continue to practice.

And there's the best part: I will continue to practice because I get to try again. It's not all over. In fact, I can just keep taking it as many times as I need to until it's safe to let me pass. The DMV doesn't care that I learn to drive by a specific date (heck, I'm many years past due by most standards) just that I learn to do it safely once I've decided to make it a goal.

My takeaway lesson from this is that in so far as any assessment is supposed to promote skill mastery, it should provide meaningful feedback, hold students to fixed standard of performance, and allow students to make multiple attempts.

All of this is probably preaching to the choir. What actually surprised me in thinking about this is the contrapositive. In so far as my assessments do not reflect these principles, they are NOT promoting skill mastery. Which is usually just because they are promoting something else. I let students retake quizzes but not tests, for grading purposes but also because deadlines are a part of life. I don't hold all students to the same time-limit for tests because even though I think speed is an aspect of mastery, I think it's expendable. I believe it is fine to make concessions like this, because there actually is more to assessment than the specific skills they test, but it was kind of unsettling to realize that there's a little bit of "meaningless hurdle" in my assessments, after all.

....................................................................................

This post is already too long, and the analogy is imperfect, but I still can't stop thinking about what a great assessment the road test is. In fact, there are two aspects of it that are so wonderful they are mostly beyond my ability to mimic and thus were left out of the previous analysis. The first is that what it actually assesses, more than skills, are habits. I am actually an ace at parallel parking by now because I practiced it over and over. But what got me on the road test was just dealing with intersections I had never seen before. Learning to drive is not just learning to repeat a set of motions. It's learning judgment, informed by an understanding of the rules and experience dealing with a variety of situations. On the road test, you don't get a comprehensive run through of every possible turn and intersection, but you get a reasonable sample that's a little different than you've seen before. Would that my math tests were the same.

The second aspect, which I can never match, is that you don't have to take the road test until you choose to. In fact, you don't have to take it at all. Generally, there is a great incentive for teenagers to learn to drive, but if you had forced me to take it at age 17, I'm not sure I could have survived the process. It is an immensely difficult task for me, and though every else seems to remember it as easy, I really think that's just the rosy tint of nostalgia. Having made the personal decision to drive, out of an awareness of the difference it will make in my life, is the only thing that is motivating me through the process. I often regret that my students do not have the same aid in their struggles with mathematics. They can't imagine how it could help them, and I can't let them just hang around until they figure that out. The best I can do is try to paint a picture for them, of what it would be like to speed down the freeway, with the wind in your hair, and get weirded out by the fact that your speedometer seems to report your speed over an instant rather than an interval.

New Year's Resolutions

2009-08-24T10:06:00.005-04:00

In response to Sam Shah's challenge, I'm publicly announcing three resolutions for the new school year:

1. I will set clear standards for adopting better work habits and celebrate my progress towards them. This is actually thirty small resolutions in one. I'll probably write more about it later, in case anyone else has problems with procrastination.

2. I will take just a few minutes to review lessons after I've taught them, and learn from my mistakes.

3. I will steal other teachers' ideas and submit my own for comments and criticism.

Phew! If just partially succeed on these, I'll be in good shape, indeed!

A Prezi Presentation

2009-08-22T02:14:00.004-04:00

Have you seen Prezi, yet? I read about here and I just love the way it allows you to organize information. I don't even have a projector in my classroom and I want to use this.

Anyway, I got so excited about it I went ahead and made a whole presentation on ways to combat the idea that math is a strict hierarchy of topics in our students and the general public. I was mostly just trying to see what I could do with the program, but I hope you enjoy it! Think about what you could present this way.

Math is Not Linear

You can just click the next arrow to move through the presentation, or click on text to zoom in on it, or click outside the text to zoom out and get a wider view. (After clicking around you can still continue with the presentation by clicking the arrow.)

BetterLesson

2009-08-18T10:49:00.005-04:00

After applying over a month ago, I just got confirmed for the private beta at BetterLesson (Dan Meyer's name was the secret password). I've started uploading my materials, which you can check out here. I would LOVE some feedback, as these are untried lessons by an extreme novice, so they could surely benefit from revision.

Regarding the BetterLesson site itself, I find uploading my materials a bit of a hassle, and viewing materials is not as convenient as it is with Scribd. In particular, I wish I didn't have to download the material to get a better look than I do in the preview window. However, the ability to search for relevant material has the ability to make up for this one-hundred fold. So far, the community still seems relatively small, and many of the subtopics I was interested in returned no results, but I am optimistic that in the near future I'll be able to look up exactly what I'm planning for tomorrow, and get some instant inspiration.

Even upon finding good material though, I can't imagine just using it as is. I would be far more comfortable adapting it to my own interests and my classes' needs. Does anyone ever really just print out another teacher's material and hand it to students?

Edit: Looks like the link to my curriculum doesn't work as intended. Well, that's another thing I'd like from BetterLesson. You can just search for Alison Blank, though. I'd love to build up a greater list of "colleagues" (the teacher's version of "facebook friends"!)

My Assessment Plan

2009-08-15T16:10:00.005-04:00

Dan Meyer may not have been the first to try out the skill-based assessment structure, but he certainly has popularized it. Like many, I am totally hooked on the idea and planning to put it into practice next year. I've made a few tweaks to adapt the system for my college-preparatory school.

Firstly, Dan's plan preempts the appearance of meaty, complex problems on quizzes. I understand he uses these in class, but one of my major frustrations last year was that for all my rhetoric about critical thinking, it was not part of my assessment. I think the only way my students will take problem solving seriously is if they receive a grade for it. I also had a big problem with some students getting bored and frustrated with my class because the quizzes and homework were not challenging enough for them. So in order to add interesting questions onto quizzes, without losing the confidence building and targeted remediation of Dan's system, I broke it into three levels.

The first time a skill appears on a quiz, only "Level 1" questions are given. If a student gets a 4 on this level, he or she has a B- for that skill, and will get "Level 2" questions on their next quiz, which are harder. Upon passing these questions with a 4, they have an A- for the skill, and get a "Level 3" question for homework. Level 3 questions require the student to take a concept one step further into abstraction, or combine it with other skills. A satisfactory answer on one of these earns the student an A+.

So what happens to the students who don't get a 4 on the first shot? Well, one problem I had with Dan's system was that students could make up skills by just coming in for extra help and doing some problems he made up on the spot. A great idea, but here's the thing - I am down right awful at coming up with problems on the spot. So instead, I'm preparing three versions of the Level 1 and Level 2 questions for each skill in advance. So if a student gets a 2 on Level 1a, on their next quiz they get Level 1b. (They can still come in to take these quizzes outside of class time.)

Since different students will progress through the levels at different rates, everyone's quiz is going to be just a little different. To handle this, I've written a program in LaTeX that uses students' previous quiz scores to generate the appropriate levels. An unfortunate side effect, however, is that I won't be able to actually give students their quizzes back, since some will have the questions that others haven't seen yet. I've resigned to just giving them a grade report, and telling them they can see their quiz outside of class if they want.

I'm also using LaTeX to automate the targeted remediation Dan's assessment is built for. I write up a few questions to remediate each skill, and after each quiz, each student gets a copy of the questions for the skills they have not passed yet. These questions are due before they can take another quiz.

I'm very excited to try this out, but who knows exactly how it will work. I've tried to stay true to the spirit of disaggregation, while rigging the grades so that a basic understanding earns a B-, but you need a really deep understanding to get an A+.

Please, let me know what you think! I still have time to change my plans before the year starts.

Algebra Projects

2009-08-13T09:39:00.002-04:00

I've been watching the WCYDWT bug spread around for a while, and while I have been blown away by some of the great lessons that have grown out of it, the prompts have never personally tugged at my heart strings. I'm just not an applied kind of girl; I want to discover universal truths, not "real life" ones. Naturally, I'm interested in anything that would get my students to sit up and participate, but I admit my personal biases have kept me on the search for slightly different prompts.

So where does a pure-at-heart teacher go for provocative questions? I'm sure there are other resources out there but the absolute BEST book I have found (for Algebra, at least) is Fostering Algebraic Thinking by Mark Driscoll. Here are a few questions I've taken out of my dog-eared copy:

What is 1 - 2 + 3 - 4 + 5 ... - 120? What is 1 - 2 + 3 - 4 + 5 ... n?
Can you cut a square into 2 other squares? 3 other squares? Try each number up to 17. Can you come up with a set of instructions for how to cut a square up into n squares?
Show that the sum of any two consecutive numbers is an odd number. What can you say about the sum of 3 (4, 5, n) consecutive numbers? What numbers can be written as the sum of consecutive numbers?
What numbers can be written as the difference of two different squares?

This is just the very tip of the iceberg. I spent three days on the last question last year and it was the best thing I did all year in that class. A student who constantly deprecated his ability ended up arguing vociferously in favor of his solution - and his main competition was a student with a D- average. I think the questions which best duplicate this experience tend to have at least three of these qualities:

The question can be approached by a brute force or intuitive method. Otherwise, many students will be stymied from the start.
The question can be generalized or extended. Otherwise, many students will get bored once they have "the answer".
The question can be solved multiple ways. This helps break students out of the search for the "right answer" and gets them defending why their solution is the best.
The question is simple to state and understand, and at best, compelling.

I think questions like this are very in the WCYDWT spirit. Though they may fail to inspire students to see math all around them, when handled correctly they combat "impatience with irresolution" and provide the kind of learning experiences students remember.

I've just discovered the questions used by the Phillips Exeter Academy, but if anyone knows any other resources for questions like this (not just for Algebra!) I'd love to hear about them.

Hopping on the Blogwagon

2009-08-13T09:22:00.000-04:00

Hey there, I'm another math teacher starting a blog.

The story so far:
I was a math major in college. I fell in love with set theory, and tried going to grad school to pursue that, but it just wasn't the right place for me and I dropped out after just a semester. So I tried teaching instead, and I immediately felt at home, but concerned about my total lack of experience. After a year of learning from my mistakes, I really want to take my teaching up a notch, so I'm calling on the power of the blogosphere to keep me on the right track.

I teach at a very small private school where I have absolute freedom to do what I wish in my classroom. The only end of year assessments are my own. My intent is to make full use of this latitude by straying far from the traditional textbook path in search of genuine learning experiences.