## Archive for January, 2010

28 Jan

So, get this: I’m writing my own grading software.  This is largely because I really like computer programming and I want to try out some new technologies, but it’s also because I have found existing grading software severely lacking (especially now that I use this skill-centered grading that is not focused so much on deadlines).

For me:

1. Automatic report generation
2. Internet access for students

Thanks for any comments – I’m really pretty serious about making the ultimate grading solution here, and I’d love to incorporate your ideas if they meld well with mine.  I should note that my school does NOT have an EdLine-like service for all classes, and our teachers all use whatever grading system they can figure out.  If you don’t have this sort of burden / flexibility, this question is probably different for you!

### Try bringing the problem to physical reality

22 Jan

It takes a lot of class time to have students build models, but here’s what I’m almost ready to call a pro tip: people love building models.  I wanted to bring Algebra kids from an understanding of the typical calculus box (cut 4 squares out of the corners of a rectangular piece of…) to a function relating height and volume.  I spent 100 minutes of class doing this – a huge amount of time.  But you should have seen how proud kids were when they got their equation to hit all the points that represented the various boxes they made.  I literally gave them 6 or 7 sheets of paper of the same size, had them cut corners out of them, and record, plot, and draw diagrams of their results.

To them, they got to spend 100 minutes on a single, rewarding task.  To them I’m going slow.

To me, they’re spending 100 minutes practicing and learning the benefits of switching between multiple representations of a problem, and finally coming up with the ultimate representation, a function, about which we are just starting to learn <shocked gasp!/>.

Kate Nowak is writing about a trig problem.  She just wants to review, so she doesn’t have 100 minutes, but literally making a pyramid is such a good idea, you guys.  Kids can start from the design below, and in a series of careful questions, you eventually ask them at what angle they should cut the triangles (what angle should that little arc be?) to get a height of 10 cm.

To me this is the ultimate “be less helpful,” right?  You have, verbally, given them literally zero information about the problem except the question.  They have to go get a ruler from the ruler drawer (that is always accessible) after realizing they need a ruler.  They have to come up with a way to check if the pyramid is actually 10 cm high (kind of hard!).  And, to boot, they have an intrinsically rewarding experience at the end: a sweet little pyramid.  They should probably color a diagram RIGHT ON IT for awesome mathiness that they can be proud of, and that you can glue to the door and have a sweet spikey door of trigonometry.  Make sure to give different groups different goal sizes so that as a class you can confirm any patterns kids are seeing.  And so that your spikey door is cooler.

### You can’t sit next to me!

16 Jan

A colleague of mine, D, has a theory that makes a lot of sense: when a teacher helps a student work on something all the way to its completion, the student associates the final success with the presence of the teacher. I used to help my students all the way through a math problem and then be surprised when they couldn’t seem to do the same work when I wasn’t there. D recommended that I help a student only to the brink of success, by making sure that the student has all of the necessary tools for the situation, and then getting out of there before the actual achievement takes place. In part, to make sure I’m not just giving away the answer, but also so that the student can experience success when he’s alone!

I offer “office hours” out of my classroom after school.  It’s free-form, and students can come in and ask for any kinds of help, test out of a skill, or trade math jokes. I usually get a lot of work done while 3 or 4 kids work on their homework. This year I started a new rule: no students can sit near me. They thought I was joking at first; some students would even sit down right next to me as I was telling them they couldn’t, and I’d have to make them get up and move. It’s hard to keep enforcing the rule, but it makes it much easier to get away from the students before they make their breakthrough!

### Yikes

15 Jan

Grades after the first week, students by rows, skills by column.  Really disappointing, especially since I thought my lessons went well this week.

Do you ever give tests that pretty much everyone fails?

### Flash Presentations

14 Jan

Flash as in really fast! I used a period today to assign a group project. I gave the kids (in teams of 3 or 4) about 40 minutes to:

1. Pick a function from a list (e.g. u(x)=2*sqrt(10-x^2)+5)
2. Figure out how to graph it in geogebra (the “sqrt” command was not obvious to them)
3. Figure out how to get a table of values out of geogebra
4. Figure out interesting things about it (my prompt suggested asking questions like “Are any points especially interesting or important,” and “What does the function do or look like?”, but they were encouraged to ask their own questions too)
5. Graph the function “very well,”
6. Make a poster describing the interesting parts they learned
7. Give a 3-minute presentation to the class

The functions from which they could choose all had interesting features, like discontinuities, endpoints, etc.  I was astonished at how focused the teams stayed throughout the period; I think the low amount of time helped a lot.  Since the mathematical content was pretty simple (it would be, what, maybe 5 or 6 questions on a worksheet?) they could focus on finding interesting features, and I could encourage them during their work to find ways to convince the other students that these were interesting features.  The small time limit made it so that they HAD to find a quick way to get the graph from geogebra on to graph paper, which meant that they HAD to identify the important characteristics and sketch on top of that guideline.

Never before has it occurred to me to slam through presentations like this, but honestly the quality of the presentations did not suffer too much (as compared with a week-long assignment) and the interest in the content was much higher.  The kids had to remind themselves of deadlines every ten minutes or so (I put one student in each team in charge of the time) and I think they got a lot of practice prioritizing and working efficiently.  The biggest benefit, of course, is that our treatment of domain and range in the upcoming week is practically covered already, and the students already have a refreshed understanding of numbers that can’t be fed into or gotten out of functions.  And, they got practice zooming around in geogebra, making tables (one group figured out spreadsheet view!!), etc.

Fantastic!

### The best thing I could do with my summer

12 Jan

If you went to summer camp as a kid and felt it change your life, you already know what I’m talking about.  The community that forms when you face a challenge – a real, fundamental, spiritually intrinsic challenge – with a group of peers and let it change you into a group of close friends.  The skills you learn as a youngster in the outdoors: camping, cooking, cleaning, certainly, but more importantly good cheer in trying times, entertainment from friends and nature, self reflection, and self reliance.

During the school year I am a math teacher because I am passionate about youth, education, and, well, math.  During the summer I am the director of Shiloh Quaker Camp, where I, along with my staff of twenty five, create a community of loving, supportive, laughing kids.  I do this because I am passionate about teaching kids to respect themselves, each other, and the connections that form between people who live together.  We create the community at camp with intention and great care, and in many ways it’s a lot like teaching math.  (I’m beginning to suspect that any kind of teaching bears great resemblance to any other (who’d've thunk it?)).

I went to a camp like Shiloh when I was a kid, and I’ve worked at Shiloh since I was a teenager in high school myself.  It’s been a spiritual and social foundation for the entire framework I base my life upon, and I think what I learned at camp informs every interaction I have with my students in school.  Math awes me because it is a fundamental structure of the universe; math is an undeniable order that we presume stretches to the edge of the cosmos (and beyond?).  Camp awes me because of the growth I see in kids; independence and interdependence flourish together in kids as young as eight years when they’re faced with the right kinds of challenges by staff with the right training and a supportive focus.

I miss out on a lot of professional development opportunities in the summer, and I miss out on a lot of travel opportunities.  My summer break ends two days before it starts, and I’m planning classes desperately in the three days a week when the kids are off climbing or canoeing or hiking.  If you’re not a camp person, you might not understand the magnitude of the experience I’m describing, but as a camp person I can tell you that its the most rewarding experience I can imagine!

If you’re an amazing teacher (or at least working on becoming one) and want a taste of this camp experience, send me an email.  If you’re looking for a way to spend your summer that will recharge your batteries and give you new perspectives on your classes and your role as a teacher, send me an email.  Even if it’s not at Shiloh (which is hard to get a job at, if I may brag briefly), I can direct you towards some programs that need good people, are a blast, and will hone your teaching skills like no summer institute can!

Are you a camp person?  Leave a note with your camp experiences!

### How to create a skills list

10 Jan

My last post focused on three major mistakes I made in my first semester of skill-focused, mastery-based assessment: separating skills into chunks that were impractically small, choosing some skills that were too simple (almost trivial), and neglecting to plan for the end of the semester.  This post will focus on my process creating the skills list for semester two (for Algebra 2).  I’d love to hear your opinions or your own process – leave a comment or a link below!

The place to start is your curriculum map, whether that’s a list of topics, a set of state standards, a final exam, some chapters of a textbook, or whatever.  Find or create the document that describes what you hope to teach this semester.  The list I used last semester I got directly out of my textbook.  I knew what chapters I was going to teach, and I just ripped concepts out of the table of contents.  This process got me a list that I was moderately happy with; click here to download it.  Your list will almost certainly need to be different, since I was planning for 36 class periods.

For semester two, I set out in much the same way.  I went through the chapters in my text book I was planning on studying, and every time something popped up that seemed like it would be a good candidate for a skill, I wrote it down.  This gave me the following list of 36 skills:

 Evaluating functions Analyzing the domain and range of functions Modeling relationships with functions Modeling arithmetic sequences with functions Modeling geometric sequences with functions Distinguishing between arithmetic and geometric sequences Recognize exponential growth from situations, tables, graphs, or equations Understand multiple representations of exponential functions Represent exponential functions algebraically Using basic laws of exponents to simplify expressions Use exponential functions to solve problems involving growth or decay Find equations of exponential functions through two given points Identify graphs of quadratic, cubic, square root, absolute value, etc functions Transform a graph by stretching, shifting, or flipping it Write a general equation for a family of functions Use the “completing the square” technique Model physical situations with quadratic functions Write equations in graphing form Invert functions analytically and graphically Form compositions of functions Express the relationship between a function and its inverse Understand logarithms and transform their graphs Use properties of logarithms Use logarithms to solve exponential equations Count possibilities in situations that require a particular order Count possibilities in situations in which order does not matter Draw a tree diagram to represent and calculate probabilities Draw an area diagram to represent and calculate probabilities Calculate expected value Using the fundamental principle of counting Calculate conditional probabilities Find the value of arithmetic series of arbitrary length Find the value of geometric series of arbitrary length Find the value of geometric series with infinite length Writing a series with summation notation Using mathematical induction

The next step in the process is to look at each skill from the brainstorm and ask,

1. How will I test this skill?
2. Is this skill big enough to be its own skill?
3. Is this skill small enough to be a single skill?
4. Does this skill have multiple levels, so that intro level tests will be significantly different from master level skills?
5. If a student does not understand this skill at all, am I willing to flunk him? (My grades are set up that each student must get a minimum of 3/5 in every skill to pass the class.  If you’re using a simple average, you can ignore this question).

Consider the first skill, “Evaluating Functions.”

1. How will I test this skill?  What leaps to mind is showing a kid f(x)=3x+6, and asking for f(2).  Maybe f(f(2)) – or is that composition?  They also need to be able to evaluate functions from graphs and tables.  Maybe the question should be a three-parter?
2. Is this skill big enough to be its own skill?  Hmm… it’s pretty small, isn’t it?
3. Is this skill small enough to be a single skill? Yes, I am confident that it is.
4. Does this skill have different levels?  So, I could ask them to evaluate f(x)=3x+6, or f(x)=3x-2/x+(x+5)^-3, but those aren’t different levels of evaluating functions, those are different levels of order of operations or something.  I could ask them to find g(f(2)), but maybe that’s composition.
5. Is this skill a requirement of passing the course?  Absolutely.  I am not letting anyone who can’t evaluate a function out of Algebra 2.

So this first skill has some complications.  I really like testing f(g(2)) because it requires students to understand the input/output aspect of functions where I feel like a simple g(2) might let them slip by without it.  Since this skill is so essential, I’m leaving it in, though it might be a little bit small.  It may end up conflicting with the composition skill, but there might be flexibility in that skill.  ”Evaluating functions” seems like a solid requirement.

Let’s take another skill, “Modeling arithmetic sequences with functions.”

1. How will I test this skill?  I’m looking for students to be able to come up with functions that describe arithmetic relationships, like, “write a function that outputs the number of gloves that x people will need,” or something.  I could show a table of inputs 1, 2, 3, 4, and outputs 8, 11, 14, 17, and have them write this function.
2. Is this skill big enough to be on its own?  You know, I think it’s possible it could be combined with the skill before it, “modeling relationships with functions,” and the one after it, “modeling geometric sequences with functions.”  These three skills are so closely related, with the only difference being the arithmetic skills required.  I’m not trying to test those arithmetic skills – I hope the kids already have them – so I’m going to combine these three skills into just “modeling relationships with functions.”  I’ll answer the rest of these questions for the new skill.
3. Is this skill small enough to be on its own?  Clearly the title can involve arbitrarily complex functions and relationships, but I think the kinds of simple relationships we’ll study in class can all be combined under one roof.  This skill may be a little bit too big.  If I was the organized man I wish I were, I’d note somewhere that I should revisit this skill after we touch on it to see how I felt.
4. Are there different levels for this skill?  For intro level questions I can ask the students to model the relationship between Celsius and Fahrenheit, or some other linear relationship.  For master level questions I could ask a geometric volume question which includes an extra level of abstraction.
5. Is this skill absolutely required for every student that passes the class?  Yes, I think so.  Who wants to teach precalc to students who can’t create their own functions?

Now, I don’t have time to write an essay about each of these questions for each of these skills, and neither do you.  In this post and in my brain I’m deciding to move quicklky here.  I only have so many hours to get this done, and it’s not going to be perfect.  I hope that you can make sacrifices like this – it’s taken me a long time to accept the impossibility of perfection in a finite time frame.  I spent about 30 minutes considering this list, eventually deciding to cut 7 skills and reword several.  Click here for my final checklist.

I hope this post has showed you how easy it can be to come up with a list of representative skills to assess.  It’s an unglamorous process, and the hardest part is coming up with the rough list, but once you do that you can have an effective list in less than an hour.  I don’t recommend using my skills lists wholesale.  I am in the process of trying out several different textbooks and my order is wonky.  My school is exempt from most standardized tests and if you have specific objectives you need to hit you’ll need to take them into consideration, obviously.  That said, here are my skills lists for Algebra II and Calculus this year:

Algebra II, Semester 1

Algebra II, Semester 2

Calculus, Semester 1

Calculus, Semester 2

Also, Dan Meyer has posted his suggestions for Algebra 1, Geometry, and Precalculus (imagine my chagrin when I saw he covered exactly everything but what I needed!).  So, if you’re considering getting started with this system, you at least have a launching point for your class.  If you have comments about my lists or process, I’d love to hear about them!  I’m especially interested what you think of the questions I use on each skill.  What else needs to be asked?