Pre-Calc is the only prep that isn’t new. I taught it last year and have a pretty good foundation.  Skill lists for each unit are good to go.  Unit Exams are in decent shape (AP style: Free Response and Multiple Choice, calculator and no calculator sections of each).  Last week our course alike team met and went through all the modified AP Calculus problems we have written thus far.  We decided which we’ll use and where. We each left with a couple of new questions to modify and will meet again in July to do more work with these.

What I still need to do for pre-calc:  Rewrite all of our individual skill assessments.

Advanced Algebra: This is somewhat of a new course.  Previously, our honors students began with Advanced Algebra as freshmen, then took Geometry, Pre-Calc, and AP Calc.  Last year we switched to Geometry as freshman.  The honors team met and reworked our plan for Advanced Algebra and Pre-Calc, to make it more of a two-year course.  We decided which topics would be taught in which course. So we have a lot of work to do on this “new” course. I’m excited that we will now be able to embed Geometry concepts into our problems.

What we have done:  decided on the order of the units, written skill lists for all of the first semester units. Began writing individual skill assessments.

What we still need to do: Write the final and unit exams for first semester. Write all of the individual skill assessments. Write modified-AP style problems. Make our supplemental materials. I wish we had enough time to begin working on second semester, but we’ll make time do that during the year.

Algebra I… for lack of a better name.  This will be our course for our ‘average’ level freshmen. This will be the first year we are not using IMP in this course (although much of what we have planned includes many of the activities from IMP1).  We’ll do some algebra, some geometry, and some prob/stats, using ACT’s College Readiness Standards as our framework.

What is already done:  breaking the course into units and rough outlining of the skills in each unit. We have more than a few workshop days scheduled in July during which we’ll get quite a lot of the work done for this “new” course. We already have a lot of materials written (we won’t be issuing textbooks).

What needs to be done: finalizing the skills lists… and then everything else. Organizing the materials we’ve already written (deciding what we’re keeping/modifying/trashing), writing the rest of our materials, writing unit/final exams. I am very very fortunate to have an incredibly strong team on this course.  We all worked together last year and do an incredible job of bouncing ideas off one another and dividing up tasks.

Writing this all down makes (part of) me regret not doing much work during the month of June. But only part of me. I really needed these past three weeks of … actual summer. Relaxing was very much needed. I haven’t had a stretch of time with so little that had to be done in a very very long time.  However, it is time to get back at it.

The second part of the activity (not shown) asks them complete three statements:

1. If the degree of the numerator is greater than the degree of the denominator, then…
2. If the degree of the numerator is equal to the degree of the denominator, then…
3. If the degree of the numerator is less than the degree of the denominator, then …

I thought it worked rather well to help them understand why the three different cases occur. We were able to have great conversations about rates of growth. “Weird stuff happens in the middle, but in the long run, the bottom gets bigger faster, so the function goes to zero.”.  I like this approach much better than presenting the “rules” for the three cases and just asking them to use them.

The students conjectures were not a precisely worded as I would have liked. I think that when I use this next year, I’ll still include the conjectures in Part II. However I’m thinking of changing/adding another part that we’ll complete as a whole class where we formalize the wording together.

Any suggestions for improvement would be greatly appreciated!

That is kind of what happened with this blog.  I’d think about it, but somehow never got around to making the first move to write.  It has been a crazy year for me (but aren’t they all?)  My mom passed away on the first day of school and for the first month of school I was just trying to … survive. Writing was not high on my list of priorities. Then as I came out of my fog, I meant to write. But I didn’t.

Then someone reached out. I recently received an email asking why I had stopped blogging. This person didn’t blame. Didn’t accuse. She simply reached out. And somehow, I was ready to hear what she had to say and ready to write here.

Despite not posting here, I have been writing this year – for National Board Certification. It was a  stressful experience for me (but mostly because I procrastinated and didn’t begin writing as early as I should have). It was also an incredible learning experience.  I started “planning” for it last summer. The portfolio requires 4 entries — with very specific requirements and questions to address. These questions were in the back of my mind just about every day in every class.

I did manage to complete the portfolio and am scheduled to take the test on May 20th. I’ll find out late next fall if I “passed” or not. Either way, I am very glad I did it. I chose to try to become a National Board Certified Teacher as I wanted to focus upon exclusively upon my teaching this year. I finished my masters last summer and wasn’t ready to begin the next one, so National Board’s seemed like a good fit.

It was a good fit. I was forced to video tape my classes and critically examine my practice. “Entry 2: Instructional Analysis: Whole-Class Mathematical Discourse” and “Entry 3: Instructional Analysis: Small-Group Mathematical Collaborations” require one to submit two 15 minute video clips of a whole-class discussion and small group discussions respectively. The video were painful to watch. Even more frightening was asking someone else to watch them in order to edit my writing for each entry.

Despite the fear factor, taping myself and my students and then writing about it was a great learning opportunity. It forced me to examine the way I respond to student statements and questions. To examine the way I  foster peer-to-peer communication about mathematics. To examine how I respond to misconceptions. To examine every thing each student says or writes for evidence of mathematical understanding. To examine how my feedback helps to develop and refine each student’s understanding.

I wish I could say that I had begun taping early in the school year as I had planned. But I didn’t. I did however think about each class as if I had. I guess I was taping in my mind. And reflecting upon each class as if I were writing about it for my portfolio. I can’t say that I drastically changed any of my practices for this process. I did however become much, much more aware of each comment I made and how that comment helped (or hindered) student understanding.

The effect this process has had on me this year really hit home for me recently.  A week or so ago, one of the students in my class shared an alternate way of answering a problem involving ratios. In response to his method, another student said, “That was nice evidence of mathematical thinking. Thank you for sharing it with us.”

I wonder how the way(s) in which we each learn influences the way we each structure our classes.

I also wonder if I incorporate enough different modes into my own classes. While I try to structure each class so that there is individual, small group, and whole class work/discussion time, I’m not sure there is enough time given to individual work. Then again, I want them discussing their work with each other. I’m sure there is a good balance. I’m also sure I haven’t found it yet. Something to watch closely next year.

There are approximately 90,000 members of NCTM.

Winter Kiss by mubblegum via Flicker CC-A-NC-ND 2.0

I have 87 people listed under my math teacher list on Twitter.

I have about 50 blogs listed in my “Math Teacher” folder of Google Reader.

While I value those I converse with on Twitter and what I learn from what y’all write, I have lately come to realize that my view of what “everybody” knows/has read is highly distorted.

By what measure do I judge how successful my year was? Student evaluations?  Grades on final exams? My students growth on standardized tests? Some other metric that exists only in my mind?

Student evaluations were mostly positive.

Mostly:

Those who disagree are eating away at me. Do I consider this a successful year- that most of my students felt comfortable asking questions in class? Is getting everyone to feel comfortable a reasonable goal? I don’t know. (The results were very similar for all of my five classes)

My final exam grades were… what I expected them to be. Most kids did well, a few did exceptionally well, and fewer didn’t do as well. While there was a surprise or three in the exceptional category, there were no students who didn’t do well that I didn’t expect. So, is this success? Knowing who knows what before giving a final? If I were a better teacher, wouldn’t I have caught and helped them all to do well?

Now for those standardized test results: My district uses the EPAS system; incoming 8th graders take the EXPLORE, frehsmen take the PLAN, sophmores take a released version of an ACT, and juniors take the PSAE (part of which is the ACT). We use this to measure individual student growth, growth by students in a particular course, and individual class/teacher growth. In the course I have taught for three years, on average my students’ growth has increased ever year. This is good. Did they grow “enough”? I have no idea.

As for my own metric… I don’t know. I think it was a decent year. I think that as a whole they learned to communicate their thinking, they learned to keep working at a problem even if the answer didn’t come right away, and they learned new skills. Oh, and we laughed a lot too.

Some of the free-response survey comments that help me to believe it was a decent year.

• I liked that you didn’t just give us the answers and made us figure things out.
• I didn’t like all the word problems; it’d be more beneficial if we did more worksheets.
• I enjoyed working in groups because when our knowledge was pooled together, we could learn off of each other well.
• You understand better when you teach someone else.
• You are my favorite high school math teacher.

Okay, that last one was made by a freshmen, so I’m the favorite out of a sample of one. But it isn’t a bad note on which to end the year.

So, I had to decide, what else do I want them to know? I have been talking to the pre-calc and calculus teachers to find out with what they struggle when they get to their courses. These are important conversations to have. We still have some work to do, but we’re getting there. In light of these conversations, I made this:

Among other things, I’m asking them to: identify the zeros, y-intercept, vertical scale factor and write the equation — determine if the function is decreasing faster between A & B or between H & I — determine the equation of the secant through C & G and through D & F — then for the big preview to calc piece: draw a tangent at E and determine which of those two secant’s slope better approximates the slope of the tangent. (link to .pdf or .docx).

Any suggestions for other questions I should be asking are welcome!

Filed under: Math Tagged: Math, polynomial, slope ]]> http://continuities.wordpress.com/2010/04/28/sharing-polynomial-functions/feed/ 7 Jackie http://continuities.wordpress.com/2010/04/25/sharing-multiple-representations-of-systems/ http://continuities.wordpress.com/2010/04/25/sharing-multiple-representations-of-systems/#comments Sun, 25 Apr 2010 18:30:24 +0000 Jackie http://continuities.wordpress.com/?p=776 Continue reading →]]> One of the realizations that I made at NCTM was that the idea of multiple representations is still new to some people and that I need to start sharing more.¹

This is one of my “worksheets”² from last fall that I made on systems of equations for my freshmen Honors Accelerated Advanced Algebra class (links: .docx or .pdf). The first problem was difficult for some of them, they didn’t like being asked to estimate the solution. Which was my whole point. It doesn’t fall on a “nice” intersection, but I wanted them to use that estimation to determine if their answer to part c was reasonable.

Even though they had all taken Algebra I in 8th grade, most of them are not used to working from multiple representations. One of my goals is for them to become comfortable in working from tables and graphs: equations they were used to (not that we’re going to stop working with equations).

The third problem asked them to solve the system from a table.

They approached parts a and c in a variety of ways . Some of them sketched a rough graph (didn’t occur to me, but okay, that works). Some of them took a more analytic approach and looked for the interval where f(x) became greater than g(x). A few got stuck. They didn’t know what to do, so they started with d and e and then worked backwards. Once others had shared their methods (yay document camera!), I heard a few  Ooh, I hadn’t thought of that type comments. Which means they were listening to each other. They weren’t used to doing that at the beginning of the year either.

I’m happy to say that as the year has gone on, they have gotten much better at working with multiple representations and at explaining and justfying their work. And in realizing that I am not the mathematical authority in the classroom.

¹ I think I forget that the people whose blogs I read are probably not a representative sample of math teachers.
²“Worksheet” has such a bad connotation. Just because something is on a piece of paper, does that mean it has no value? I don’t really think of this as an “activity”. It is a worksheet. One that asks them to work from multiple representations, see the connections between them, and to explain and justify. I am okay with that.

The most immediately valuable aspect of the conference for me was the conversations I had with Amy as we debriefed each night. Having someone to talk and reflect with has become a necessary part of any learning experience for me. I also think our attending together will be helpful as we return to school and are able to remind one another of what we liked and planned to do with what we learned. She’s smart. She keeps me on my toes. I’m lucky to work with and learn from her. Everyone should have an Amy in their department.

As for conferencing, I learned I need to have a better system for taking notes. By the time we left on Saturday, many of the sessions were jumbled up in my mind. As I began trying to type up the specific session reflections, I realized I had haphazard notes on my laptop, comments scrawled on session handouts, phrases on my iPhone, and random comments I’d made on Twitter that I somehow need to try to piece together into a coherent semblance of thought. I need a system. As always, suggestions are welcome.

When I was attending sessions solo or standing in line at Starbucks (I did a lot of that), I would start chatting with the random people around me. I had some interesting conversations. Overall impressions were that the elementary ed people were happy with the sessions they attended, while the secondary ed folks seemed to have a more hit or miss experience. I met a lot of very nice high school teachers who were frustrated with some of the more lofty-theory type sessions:  “How am I supposed to change this system, policy, structure, …, whatever as ‘just’ a classroom teacher.” or “I want something I can take back to my classes on Monday.”

Too many of the sessions I went to spent at least the first twenty of the sixty minutes setting up the need for whatever point they were trying to make. Yes, I know that we are… doing poorly on the TIMSS/getting some new common standards soon/teaching computation with little conceptual understanding/teaching in a flat world/training the students for jobs that don’t yet exist/ … /whatever crisis you want to use to make your point.  But get to the point. Sooner. PLEASE. I sat in one session for twenty minutes without ever figuring out what the presenter meant by her title.  I wasn’t sure she was every going to get there, so I left. Maybe I need to learn to be a bit more patient, but I’m not sure that will actually happen, so if you’re presenting, GET TO THE POINT.

Another issue was that too many of the presenters ran out of handouts. But don’t worry, if you email them next week, they’ll send them to you as a .pdf. (I find it ironic that those who spoke of the flattening of the world were some of the same who’d email you the handouts. Although it could be worse – they all could have had this option). There were a few that were able to provide a website with links. Crazy idea: maybe next year we can have actual links to session materials on the NCTM conference site.

Oh, and WiFi would be nice.

Overall, I’m glad I went. I did learn. I’ll be writing more detailed descriptions soon. After I get caught up on planning and the return to the real world. And after I make some sense of my “notes”.

The next day, as they were comparing results and I was walking around observing, I noticed one student had this:

So I asked him to put it up on the board and explain his steps. There was a collective “oooh” voiced as they watched him work.

I asked him how he thought of that. He said, “Isn’t that what you showed us yesterday?” After the class stopped laughing, he continued “Well, I didn’t actually write anything down that you did. So I just, uh, thought about it when I got home.”

However, the classes amazement quickly wore off, grumblings of “Why didn’t she show us this yesterday?” were heard. I replied, “Yesterday you wouldn’t have appreciated it. And I planned on showing you today, but he beat me to it.”

Have I mentioned that I really enjoy having an honors section?

Honestly, I just saw this method in my grad school class this semester. More than a few of the other math teachers in class had never seen it either, so I thought it was worth sharing.