Measuring Success

The third year was… okay? Could have been better? Good? I think it was decent. I think.

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.

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Sharing: Polynomial Functions

We’re studying polynomial functions in my freshman Advanced Algebra course.  I’ve been struggling with determining what it is they really need to know about polynomials. I’m not convinced that long division of polynomials is an essential enduring skill (feel free to try to convince me that I’m wrong). We now have CAS and W|A to do this. I do want them to be able to successively divide by linear factors to reduce a polynomial function to a product of linear factors and irreducible quadratics. Long dividing a 7th degree by a 5th degree? Eh.

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!

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Sharing: Multiple Representations of Systems

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.

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NCTM 2010 Initial Reflections

I just returned from my first NCTM Conference in San Diego. I was fortunate to be able to attend with my friend and coworker Amy. Gotta say, being able to meet up with old “friends” I’d never met was a good thing too.

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”.

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He beat me to it

Earlier this week we worked on writing equations given the solutions to quadratics. I was surprised that a few were surprised that given solutions of 3\pm \sqrt{2} we can expand (x-(3+\sqrt{2}))(x-(3-\sqrt{2})) and get a “nice” equation. I reminded them that if we’re multiplying something with three terms by something with three terms we should get nine terms (before we simplify). After doing one more example but with complex roots, I sent them home with two new problems on which to practice. I was a bit worried as I had rushed this at the end of the period (my timing has been… off lately).

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

x=5 \pm \sqrt{3}

x -5=\pm \sqrt{3}

(x-5)^2=(\pm \sqrt{3})^2

x^2-10x+25=3

x^2-10x+22=0

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.

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Maximum return?

Today I issued a challenge to my 8th period class:  Which group can construct an open box with the largest volume?

I explained that the box was to be constructed out of an ordinary sheet of paper – 8.5×11 inches and that they would need to cut out a square to fold up the corners.

Those were the only instructions I gave. I then handed out paper, rulers, and scissors. The only question they had was if the winning group would get candy. I suggested bragging rights. They wanted candy.

Then I walked around and observed.

Some groups immediately began cutting out squares.

A few groups argued about how they should approach the problem.

Some groups started making tables — length, width, height and volume.

They worked for about 15 minutes on this (much longer than I thought it would take my honors students). One group who had been making tables came up with the formula y=(x)(11-2x)(8.5-2x). I asked them what x and y represented. They told me. I said “So you have an equation that represents volume and you’re trying to find the maximum volume. Huh.” and walked away.

Two minutes later I asked for each groups’s volume. One group reported 72 cubic inches. The group with the formula told them it was impossible. Then they explained why.

They got their candy and I handed out the homework. Max/min modeling problems. As students were reading it I overheard “Oh, I get it. We just did this with scissors.”

I’m hopefull that they will have less difficulty than classes have had in the past with modeling. We’ll see tomorrow.

Posted in Math, Problem Solving, teaching | 8 Comments

Error Analysis

My students seem to think that it is much easier for me to find their errors than it is for them to find them on their own.

Well of course it is. Anytime you can get someone else to do the work for you, it is easier. Also, I’ve had lots and lots of practice in trying to find the mistakes in other people’s work.

So in an effort to get them to be able to find their own mistakes, I created a systems of equations “find the error” worksheet.

Nothing fancy, but here’s the .doc file in case anyone is interested.

Posted in Math | Tagged , | 4 Comments