Monday, October 1, 2012

Going for the "A"

Why am I so annoyed that they're all trying for an "Advanced" the first time they assess a skill? It's a good goal to have...  I suspect the goal is not about the advanced level of mastery, but the requirements for getting in "A" in the course, which is some of what bothers me.

I think the other part of the problem is how long they take to do the assessments. They're writing novels instead of a concise explanation. They're trying the shot-gun approach to getting an advanced. You know, "if I just keep writing, maybe I'll mention all of the important things I need to mention for an advanced," instead of "Here are my reasons, and here's how I know they're right."

The length of time concerns me, because (1) it takes away from instruction/exploration time and (2) I have the belief that if you don't know it quickly, you don't have the understanding you should.

For example, I gave a 4 problem quiz on piecewise graphs today, where they either had to sketch the graph of a given function, or figure out the function from a given graph. I gave them 40 minutes to do the quiz. 10 minutes per problem. Is it really so unreasonable to expect them to be able to do these problems in that time?

Thursday, September 13, 2012

First Quiz!

I gave my first quiz today! This is super exciting because I have something to put in my grade book. We get to see my grading policy in action. :)

Actually, to be honest, I'm more excited about the quiz itself and the girls' work. SBG has forced me to re-evaluate the assessments I was giving out, and  revise them. Now I ask for more explanation. It's not enough to write the correct answer, or show some limited algebra. I'm also trying to find more creative ways to assess their understanding. For example, "What did this student do wrong, what should she have done and why?" is something I haven't used much. I should.  Their answers are very informative; I can pretty quickly see what they do know, and where the errors in their thinking are.

Now, to the grading!

Friday, September 7, 2012

Meet the Teacher's Grading Policy

Introducing the parents to my grading policy last night at Meet the Teacher Night went really well! Of course they had questions, of course they had concerns. Actually, the concerns were not quite the ones I expected. They were troubled that I was introducing this uncertainty in the senior year. No one really knows how this will work out. For seniors who depend on the first term grades to help cement their acceptance into the colleges they want, not knowing how the grading system works is really scary. (They know what the policy says - they don't know how it works. Experience is needed.) So the parental concerns were not that I was grading differently, but that grading difficulty is really scaring/stressing their daughters out in an already stressful year. 

The highlight of MtTN was when one mother pulled me aside after it was all over to talk about her concerns that her daughter hadn't really mastered the material in previous classes. Her response to my new grading system was along the lines of "I think this is wonderful! Why isn't everyone doing this? Why haven't they been using this in other math classes? It would have helped my daughter!"

Gotta admit - that was one response I wasn't expecting. At all. 

Tuesday, September 4, 2012

This is Math

"The important thing is not to stop questioning. Curiosity has its own reason for existing. One cannot help but be in awe when one contemplates the mysteries of eternity, of life, of the marvelous structure of reality." --Albert Einstein

"Each problem that I solved became a rule which served afterwards to solve other problems." --Rene Descartes

Somewhere, in and between these two quotes, is what mathematics means to me. It's not the techniques and the procedural ways to solve problems. It's not even knowing theorem after theorem. It's in the exploration I do myself, and the discoveries I make.

In my last two years of high school, I decided I was going to be a math teacher. As far as I was aware, I was good at math. I mean, really good. My mom loves to remind me how my HS math teacher, Mrs. C, told her how I would act as a translator for other students in the class, re-explaining during work time what Mrs. C had meant. I had almost no trouble getting it the first time. Math was recreating what the teacher had shown us on a variety of problems. Sometimes there was a cool little trick you needed. I liked those. They made me feel smart when I figured out what the trick was.

My first semester of college, I got a wake-up call. Sure, most of my math classes were recreating what the teacher was showing us, but one class stood out. Everything was open ended. We'd get a question (the first one was about ants and how their tunnels connected - hooray for graph theory!), and then we'd play around, trying to find an answer of some sort. David, the instructor, had some rules about the format of our notebooks and everyone having to share something in class each day, but he didn't seem to do much. (Now that I'm on the other end of the classroom, I know what a lie that was!) We met in study groups (required), shared with each other, got excited for each other's brilliant ideas, and hoped some rule we came up with would be named after us. (David started that, too.)

One of my group mates spent hours drawing and looking at graph after graph, counting vertices, faces and edges, until he came up with an equation relating them all. Our study group was so proud of his result - it was HUGE. I was envious of his dedication and discovery. (To this day, I think of it as the Wilmot Equation, not Euler's Formula.) That was math. That was what it meant to do math.

I promise I'm not really straying from my quotes. Yes, problems were given to us, but they were only the starting point. David encouraged us to extend and expand the problems, go down a variety of different paths, and ask more questions. We became curious, and more curious, as we found more answers. And every problem built upon the one before. The definitions, rules and explanations from one problem guided us and pushed us on the next one. Sometimes something forgotten would pop back up. It was like finding those little tricks, but so much bigger, so much more meaningful. There's nothing like it to make you feel like a genius and an idiot all at once.

That, my friends, is real math. Not school math. REAL MATH. Mathematicians-turning-coffee-into-theorems math.

How do we get it into our schools?

Personally, I stole the research log from David. (Though he helped a bit - so is it just borrowing? I'm not giving it back!) I pose open questions to my students, and let them try stuff, play around, suggest ideas and shoot them down. I'm nowhere near as accomplished at "doing nothing" as David was, but I'm working on it.  I make up questions related to calculus, and ask them to play with them. I've gotten some beautiful definitions of tangent lines and explanations of how curves can have slope (without me introducing the topic first!), and I've gotten frustrated, stuck in the mud spinning the tires students. (It varies, depending on how willing they are to throw out how they've defined Math for all of these years...) I'm learning, slowly and painfully, how to scaffold the questions without making them obviously leading. I've got a loooong way to go on that. I need more calculus teachers in my circle of friends, so I can pick their brains.

Here's the notebook guidelines I give my students:

Research Log Guidelines
(Based on the syllabus of David Olson)

The Big Reveal!

I've had a couple of posts now where I reference my new grading policy, but I haven't really said much about the details. I wanted to wait until I had all of the major pieces done. But I think if I wait much longer, I'll never get it posted.

To start, here's the explanation of how it all works. This is actually a subsection of my syllabus and procedures document. It isn't specifically mentioned here, because homework has its own section in the syllabus, but I do not grade homework. (Though I foresee a language shift in my future. Some assessments could be worked on at home, I think... so better to say I don't grade practice problems)

SBG Policy

A lot of these ideas aren't really mine. They come from the great minds of people like Sam Shah, Shawn Cornally, Frank Noschese, Joel Ochiltree, Dan Bowdoin... Shawn was kind enough to exchange emails with me, answering a bunch of questions I had. Many thanks, gentlemen!!

One piece that wasn't included in the policy above was how, at the end of the term, I was going to convert all the standards marks into a letter grade. (As the only person using SBG at my school this year, I still have to put a letter grade and a percentage on the final report card. But we may be moving to a school-wide grading policy - and this one has generated a lot of interest among the other faculty...) I was hoping I'd get more buy-in if the students had input on what was a fair conversion. The conversation quickly revealed a lot of their doubts and fears about this new policy. In the end, I came up with a compromise between my initial impulse (which was very strict) and their I-need-to-get-an-A-so-let's-make-this-easy ideas. (To be fair, I'm not really sure that's what they were thinking, but it's what it felt like!)

This is what it looks like:

Grade Conversion

I'm not really thrilled with the last bit about using my discretion to assign in-between grades, but it was the easiest way to account for the in-between percentages and the wide variety of combinations of marks. There's going to be around 60 skills for the term (broken into units of anywhere from 8 - 12 standards), which means to get an 80% (B-), they need to be about 90% proficient.

I'm going to use the table above to convert their standards marks for the term into a percentage, which goes into the 85% category. On the midterm and the final, I'll grade based on the standards that appear on each assessment, and convert their marks using the chart above into fixed percentages for the 7.5% categories.

So there it is. I'd love hear any thoughts you have. Especially if you've been here and done (something like) this before.

Thursday night is Meet the Teacher Night. I hope the parents are open to this!

Wednesday, August 29, 2012

Discouraged and Scared

Maybe it was a mistake to ask the students to help me figure out the conversion from standards based grading to a percentage and letter grade. 

Yesterday, when I introduced the grading policy, there was a thoughtfulness to their reaction. One or two said they liked it. The rest seemed open to it. 

Today, as we discussed what sorts of marks they thought a person should have to earn an A, an A-, etc., they had more and more "what if" scenarios, which led to more and more complicated acceptable conversion rules. I'm not really surprised by that, but I am pretty discouraged by how readily they accepted the thought that you aren't going to be able to be proficient in every skill in a math class. That there's at least always one idea that you can't get no matter how hard you work. But as long as you work pretty hard at it, your teacher should still give you an A, because you know everything else.

There was a bit of an outcry when I admitted that my original inclination was that if you have a beginning score on any skill, you can't get any higher than a C. 

(Here's the criteria for "beginning" again:

  • Demonstrating little to no understanding of the concept.
  • Using inappropriate strategies to solve problems
  • Using inappropriate or very little appropriate mathematical terminology or notation.
  • There is no mathematical reasoning
  • Mathematical claims or statements are not justified.

I'm willing to allow for a "beginning" mark up to a B-. I just don't see how I can say you've earned a B or an A if you there's a concept that you have essentially NO understanding of.)

The atmosphere of the room just felt like it was getting darker and darker. A few times my responses to the "what if" questions actually reassured the girl asking, but it wasn't often. 

It was discouraging to watch them push so much against something I've worked so hard on because they're still so focused on the end grade. I realize I opened this can of worms by starting a conversation specifically about the end grade, but I wanted them to have input on it, because I know they have strong ideas about "fair and right". I just didn't realize they were so different from mine! (At least when it comes to grades...)

I do understand where they're coming from. They don't really have any idea what a "developing" response would look like. Or an "advanced" response. In fact, they asked if I could provide some examples of student work that would fall into the different categories.  They're scared that it's going to be next to impossible to earn an "advanced" mark. They're scared that there will be so much to have memorized at all times. They want to know that this grading system will give them an A- if the traditional grading system would have given them an A-. 

I'm hoping once we get into the routine they'll be reassured - that experience with the process will make more sense of it all. 

Tuesday, August 28, 2012

"How it works..."

I'm taking part in a new blogger initiation. Some really great people (<--list of names at bottom) are putting it on for 140+ people. Are they not awesome?!

Anyway, each week for a month they send us newbies some prompts we can use to write a post. Here's what I picked for this week:

5) Here’s a comic. Respond.:

"How It Works"

How could I resist this?!* I have a classroom of girls who don't suck at math.

How could I resist asking THEM to respond?!

Here're some highlights:

"Just because one girl does something wrong doesn't make all girls bad at math. Some girls are better at math than men. This is inaccurate because boys think they are better at everything and that makes boys stupid." 

"If the teacher is on the left and a student is on the right, this is unfair to both children, whether the kid was a male or female. :p"

"One person could be bad at something, but that doesn't mean that everyone like them is bad at it. Boys often think that they are better at EVERYTHING, but in reality, they're not."


"I'm so mad. That was so sexist we are all wonderful at math and we are females. Look at us, we are all in calculus class. This is unacceptable. >:0"

"This comic shows how people generalize about all women or girls instead of treating each one as an individual, in this case about their abilities in math. When the boy was wrong, it was just about him, not men in general.
Ignorant people!"

And my favorite response...

*My response: It's a centuries-old attitude... 

"She proved to the world that even a woman can accomplish something worthwhile in the most rigorous and abstract of the sciences and for that reason would well have deserved an honorary degree." - Carl Friedrich Gauss, about Sophie Germain [Emphasis mine]

Someday, we'll grow out of it. Norms change, but not overnight.

Also? This.

Tuesday, August 21, 2012

Experimentation, Part 2

I have another experiment with my calculus courses this fall. I've never been happy with continuity following direct substitution. It feels too much like being asked if your feet will get wet after you just put on wellies. The only functions you can (immediately) use direct substitution on are continuous functions! 

I want my limits and continuity unit to grow out of a review unit on functions and graphs. I want my students to look again at polynomials, rational functions, holes and asymptotes, and start putting together some observations about when there's a hole (or an asymptote) and when there isn't. I want them to talk about when we might "assume" a function to have a certain value, and when it would be wrong to have that "assumption". (We all know what happens when you assume... well, at least part of the time.)

At some point we'll switch our language from "assumption" to "limit". We'll make tables and look at graphs, and compare the actual function values to the limit values. And then we start categorizing, conjecturing and discussing, and viola! Continuity!

And from continuity? From that comes "Mrs. Hamilton, if we know this function is continuous (see? Look at the graph!), can't we just say that the limit of f(x) as x approaches a is whatever f(a) is?"

I can dream, right?

Now I just have to figure out how to work/weave in the properties of limits and continuity and more formal proof work.

Monday, August 20, 2012


I’m switching to standards based grading.

I've grown increasingly uncomfortable with my traditional grading system. I can't quickly tell by looking at my grade book what a particular student is struggling with. I mean, I can narrow it down to 3 or 4 possible topics, but that's it. I can't even tell if it was really a problem with the calculus ideas, or if it was due to algebra errors! I give lots of feedback on the what I return to students, but there isn't a good way to concisely incorporate that into my grade book. 

Then there's getting students to actually read the feedback I give them, and do something with it. Tests, quizzes, etc. get shoved in a folder, a locker, a laptop bag, who-knows-where and that was that. It took asking them to make corrections (entirely redoing each problem) for half points back on correct work for them to start reading what I wrote and asking me what I meant. And we all know why they did it - for the half-points they could earn. In the end, for my students (and most parents), it was about the final grade, NOT the fascinating, mysterious, totally awesome math (or physics) we'd been studying. 

While I was getting frustrated, I started working on a personal learning network, collecting blogs in an aggregator - especially blogs by other calculus and physics teachers. And the more I read about SBG, the more it felt like something I should do.

So now I'm meeting with my director later this week to go through the formal write-up of my policy (posted when it's done - it's mostly bullet points, at the moment.). She sounded excited about it. I'm looking forward to explaining my reasoning, and getting her advice on any possible improvements and phrasing. And if she knows exactly what my policy says, she can help correct any misunderstandings parents or students might come to her with. 

Next Monday, I'll introduce the policy to the girls. I like the idea of starting with a discussion of what grades mean and what they're for. I also want to get their opinions on what an A in the course would mean in terms of the values and number of scores earned. (I'm planning on a conjunctive grading scale - so something like "To get an 80% (B-), the lowest score you can have is 2, and you have to have at least a 3 on at least 75% of the standards.") I have a pretty good idea of what I want the final grading scale to look like, but they're more likely to buy in if their ideas are incorporated. And who knows - they might actually be tougher than I would be. That's what happened when we asked for student input on the academic integrity policy!

Parents will get to see the policy (sans the part about converting to a letter grade for the report card) when their daughters bring it home next week. The week after, we'll get a chance to sit down at meet the teacher night and discuss it. I think the biggest resistance to the change will be because it's a change. 

That's the roll-out plan... now to get the details written out coherently so I have something to roll out!

Thursday, August 16, 2012

Introductions are in order

Hooray! My first post to this blog! (And as I type that, I have to think, "Holy crap, what am I doing?! I don't need one more thing to do this year!" But, really, this thing? Yes, I need this.)

Allow me to introduce myself. My name is Katrina Hamilton, and I am a nerd/geek. (I'm working on the social ineptitude thing...) I love Star Trek, Star Wars, Firefly, Tolkien, Asimov, Bradbury, McCaffery, Lackey, etc. I even have my ham radio license (N8XUG), though I don't really use it any more. College-wise, I went to Michigan Tech for my BS in math, a minor in German, a teaching minor in physics and my teaching certificate. Now I'm working on my MAED through Michigan State - math & science and technology concentrations. Nerd cred established, hey?

I currently teach at the Academy of the Sacred Heart in Bloomfield Hills, MI. Specifically, in the all-girl upper school. I probably have one of the cushiest teaching jobs ever. I teach the classes I really want (physics and calculus), my classes are made up of anywhere from 3-15 juniors or seniors (all girls!), we have a 1-to-1 tablet PC program and fantastic tech support, supportive parents (usually), and a high number of motivated students. Don't get me wrong, though. I still whine and complain - I just feel guilty about it, because I know I could be grading 150 papers instead of just 36.

Also, I use parentheses. A lot.