Struggles and Frustration

Today was a hard day. 

First, I was hoping we could develop the definition of a tangent line in our shortened class period. It's a discussion I've done every year, and it can go very quickly. I give the girls these 6 pictures of tangent lines, carefully chosen so their previous definition of "touches a function at only one point" will not hold, and tell them that they are all pictures of tangent lines. Their job is to come up with a definition of what a tangent line is. 

Graph A

Graph B

                                                      

Graph C

Graph D

Graph E

Graph F

 The first problem is that none of them had even taken the time to look at these pictures ahead of time, like I asked them to. They came in cold to the discussion. Of course, I didn't realize it until I started moving around the room, noticing that there was nothing written in their notebooks. No wonder it was like pulling teeth to get them to list anything. 

The second problem is that when a student offered up "It touches at only one spot, it doesn't go through..." and trailed off, I jumped in with, "What about graph C? Or E and F?" Centering myself in the conversation seems to be the fastest way to kill it. 

I tried asking a question and just waiting. And waiting. And waiting. Then, I'd cave, and ask another question (or a variation on the last one). I still got blank stares. Thank God for the three girls who were willing to eventually speak up.

On the bright side, one student asked if I could give them a definition of a secant line, so they could work from there. Here's what I gave them: "A line that cuts through two points on a curve and whose slope equals the average rate of change of the curve between those two points."

Eventually, I told them to think about these bullet points we'd developed and come up with a statement about what makes a tangent line different from just a secant line:

• Touches the curve at at least one point
• Touches but doesn't go through (what about E & F?)
• Some tangent lines can also be secant lines (B, C and E, but not F)
• Slope?
• Terminology: point of tangency (See D)

I had to end the discussion a little early, because they have a take-home assessment on proving functions are continuous that is due tomorrow, and they asked for time to ask questions on it.

Enter frustration #2. On both sides.

They're somewhat upset that they don't have any examples just like these problems to refer to in their notes or practice problems. No, they don't. Instead, they have lots of examples and practices problems about different aspects of continuity that they can pull together to develop an understanding of the functions they have to prove are continuous. They get frustrated because I'm not pointing them to an exact template to follow. Instead, I'm suggesting that they think about what they know about what continuity and think about how it can be applied to these unusual functions. 

Who said take-home tests were supposed to be easy?

Here's what I want to say to them. I have the email written, but I'm not sure yet I'm going to send it.

Struggle is good. Embrace it. Work through it.
You have all the pieces you need. Now you have to put them together. Think about what you know about these functions. Think about what you know about continuity. Think about what you know about how properties and theorems work: conditions to be met, results you get when they are.

So the question of my day is, how do I teach them how to think and synthesize information? How do I get them to engage in the activities that will allow them to practice those skills instead of begging to be told "the answer" or "the method"?

"Temporary grades"

We just had conferences and mid-quarter report cards, and the Galileo Girls are now focused on their grades again. There's relief that reassessment scores for each topic replaces the old, but they're also recognizing the flexible nature of their grade.

In this class, I haven't been giving them weekly voluntary reassessment opportunities. I've been choosing what gets reassessed when. To balance this out a bit, I've provided a voluntary opportunity for reassessment: a 5-photo project on the course topic of their choice. (See below - I took AAPT's photo contest idea and ran with it.) They even have the choice if they're going to do it at all. One student came in see me about it after school and wrapped up her questions with, "And this is just temporary help for my grade, right?"

Uff-da.

Yes, love, it is temporary, in the same way that every assessment you've taken thus far is temporary, and every one will be, until the last one on that topic. But the point is that you're going to show me how you've improved your understanding of the topic or your ability to explain your understanding.

It's about growth and feedback and then more growth.

I have to do a better job communicating this to them. I have to help them change their outlook on what grades mean (at least in my classroom) and what those comments on assignments that are returned to them are for.


The project:

Honors Physics Photo Project.docx by klwarsin

My Year with Standards Based Grading (part 1)

I had big plans. I was ready and rarin' to go. I had a shiny new blog to talk about my shiny new grading system.

I made it to October. After that, blogging always got pushed off as other, more immediate, things came up. 

This is the last post I started:
"Several things are spinning in my head as I think about how this new grading system is working. I'm not thrilled with how I go about grading the quizzes, because of how long it takes. I struggle" 

I can't remember how I was going to finish that last sentence, but I think it sums up my year as it stands. I struggled. I chose to try a whole new grading system while I was still working on my masters degree, and I'm still not sure if it was a wise choice. Balancing my time between work, school and family is hard normally. Changing my classes into SBG classes was like adding a 4th ball before I'd really learned to juggle 3. On the other hand, I was unhappy with what I'd been doing. The philosophy of SBG clicked with me. I was happy and excited with my planned structure. There was an overall sense of change and growth in our upper school because we had created a think tank of faculty to determine how we needed to change our school and curriculum for the future so that we would be competitive and meeting our students' needs. It seemed like it was the right time to revamp my classes. I'm not sure that I could have waited 2 more years until I finished my degree to implement these changes. I wonder if I wouldn't have lost my drive and excitement in that time. 

Really, though, the question of "should" or "should not" is a moot point now. I did; it was hard, and now I'm making the time to reflect on it. 

I know I did one thing right. Whenever I thought of something I should write about, I scribbled a quick note on a slip of paper and set it aside on my desk. It's time to turn those prompts into real posts. 

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?

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!

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. 

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)