End of Year Reflections: Algebra II

Continuing analysis of student's reflections... (see first post for a full description)

I'm not sure I want to share this publicly. To be honest, I didn't include some quotes because they were too depressing. This class was by far my most frustrating. A lot of that was me: it was my first year in a new school and I didn't understand what was covered in Algebra I, I'd never taught Algebra II before, I got irritated when students didn't remember how to do things that they really, really should know how to do by junior year. Some of it was the kids: most didn't start coming for extra help until the last quarter, they didn't work as a cohesive class, they didn't have high expectations of themselves. Plenty of the blame can go elsewhere: some went to 3 weeks of summer school over a year ago and that meant they 'got Algebra I', many weren't ready for Algebra I so they didn't retain as much, lots didn't want to take more math beyond Algebra II and society says that's fine, society also says math is hard and it's okay to say you're bad at it.  I learned a lot from this class, but I learned it all the hard way.  It helped that the other Algebra II teacher shared my frustrations, but I wish that we'd had time to work together.  Instead, after each chapter or two we made plans for adjustments next time we teach it, which neither of us will be able to enact this year since I'm not teaching Algebra II and she's taking a year off.  However, this post isn't about all that, it's about what the kids wrote in their reflections.

Most investigations in this class were of the "graph a lot of equations to find out what the different coefficients and constants do" variety. Shockingly, those didn't make it onto the list of favorite activities. We also acted out some of those when we studied parabolas (turning the squares on the floor into a giant coordinate grid), that activity didn't make it onto the list either. Other investigations included lots of numerical examples, followed by a generalization. Still not on the list! So, which ones did make the list? Just two:

The rolling markers lab, and the interest rates activity.

The rolling markers lab was pretty cool. We set up 'ramps' of different heights (folders propped on books), rolled markers down and measured the distance traveled. It was a nice review of scatter plots, best fit lines and the different equations used to describe lines. We had some fun building crazy ramps and trying to find markers which would roll in a straight line. It was from the first quarter and they remembered it at the end so clearly it made an impression.  I'll probably try this activity with my Learning Skills class this year (math for students with disabilities who can't access the traditional high school curriculum yet).

The interest rates project was bad. A couple other teachers wrote an outline while I was in a geometry meeting. I missed the discussion, I didn't make it precise before distributing it, and none of us were really thinking about exactly how low interest rates have fallen! Students went to the bank (or internet) to get rates on savings accounts and CD's. Pairs were given a certain amount of money to invest and then compare the outcomes of different scenarios. The scenarios were too vague and the final products reflected that. The idea was cool, the kids appreciated the value of it, but it was poorly executed on my part.

The specific topics they enjoyed or found challenging weren't particularly noteworthy, as in the other classes some students listed a topic as a favorite, while others listed it as hard. The one surprise came from a student who listed the same topic (factoring) as both hard and his favorite. That kid gets brownie points in my book.

After reading this you may be surprised to hear that the kids learned anything. They did, the class wasn't awful every day, but my lasting impression was of a group of rather uninspired kids. I didn't get them excited about math. When I asked them what they needed, all they could offer was a change of scenery might help. They did get more work done when we hung out in the library, but it was a very unsatisfying class in my mind. Still, they learned, and they can tell you about it in their own words:

• If I try my hardest I could get a good grade
• My intelligence is not enough, I need to work and study hard too
• I found out that I can do it
• I learn best by doing the work by myself
• I need to study more and make practice problems
• Study best with flashcards
• Staying after can help you make up a lot of points and help you understand
• When I really want something I can achieve it
• I learn best by doing projects
• I can learn to solve any problem with practice
• There are at least 2 ways to do anything in math

That was the last of the End of Year Reflections (until next year!).  Maybe by now I have some more posts written, but this daily posting thing will definitely be a rarity, so I hope you didn't get used to it ;).

Take aways:

Thank goodness you're not teaching Algebra II again!! (jk, but not really)

Do rolling markers in Learning Skills

Emphasize importance of making up work early and often.

End of Year Reflections: Fundamentals of Geometry

Continuing analysis of student's reflections... (see first post for a full description)

In addition to two college prep geometry courses, I also taught two fundamentals of geometry courses. Our levels are fundamentals (SPED and struggling students), college prep (aka CP, the 'regular' class) and honors. The fundamentals courses were co-taught and I got to use an amazing book: Merrill, from the 80's with only blue and black ink, filled with discoveries students should do before theorems were presented. I used many of the ideas in this book in my CP class, and I followed the ordering of each book in most cases so the two levels were taught in a slightly different order. Another thing to consider when planning for this year.

Most of the responses in the class that I had do the end of year reflection (I'm kicking myself now for forgetting in the other class) were similar to the ones in CP. The investigations were the same across the courses, but there are a few things to note.

Students in this class mentioned struggles with: algebra, formulas, measuring and square roots. Oftentimes placement is based on Algebra 1 scores, so those topics are more likely to be an issue in the fundamentals classes. Next year I might want to do some more explicit review of Algebra.

There was also one student who listed "working with others" as the hardest thing to learn to do. We discuss classroom norms a few times during the year, and I try to start the year with an activity emphasizing the importance of team work, but it's good to be reminded that it's still a struggle for many students. In truth, it's still a struggle for me! Some people I immediately click with and we're ready to share and learn together, but with others I find myself shutting down. I would like to have a better way of assessing how students are working together. The participation quiz (Sam wrote about it here) that I saw at PCMI might be a good way to do this, and I'd also like to see students discussing with each other.

In one class I gave students a daily checklist based off of our school's Habits of Mind rubric. It included points for being on time and prepared, working with others, asking questions and participating. Of course, this is the same class that I neglected to give the end of year reflection to. Maybe I can modify it from a daily list to a every 5 classes list and collect it when I collect journals... One of the best features turned out to be the "Is there anything else you'd like to share?" section where students vented frustrations, owned up to misbehavior and were generally their honest and insightful selves.

And, just because they're corny yet sincere, here's the quotes from the "What I learned about myself" section

• If I push myself I can accomplish many things
• I can do better work when I put my mind to it
• I can achieve anything if I work hard for it
• I can do things, all I need is a push
• If I pay attention, I can learn more
• I'm too smart for geometry
• Homework is important
• Geometry is easier than algebra
• I am a lot better with geometry than algebra
• I am good at math
• I do well when I study
• Studying is really important
• If I try, I can get good grades
• I need to be organized

Take aways:

More Algebra practice in Fundamentals

Compare the order of the levels to see if there are compelling reasons to switch either

Ways for teacher and students to report group dynamics? (rubric?)

End of Year Reflections: Geometry Investigations

Continuing analysis of student's reflections... (see first post for a full description)


Even as early as student teaching I quickly realized that working from the book day in and day out was boring. For me even more so than for the students. When life is running smoothly I do what I call investigations weekly. Block scheduling messed up my weekly routine, so last year they happened sometimes, but not at regular intervals. In geometry it's easy to do a quick investigation of "everyone draw a triangle, measure the angles, tell me a hypothesis." But those aren't the activities that kids remembered at the end of the year; they remembered the applications, the field trips to the parking lot and the ones that involved coloring. Honestly, those are my favorites as well. I get to interact with students in a different way when we're identifying trees (how do MA residents not know what a white birch looks like??) or rolling circles down the 100 foot track.

Here's what the Geometry CP students named as their favorite activities:

Expanding turtle

Measuring height with mirrors

Measuring height with shadows

Pi day (4)

Sierpinski's triangle

Comic dilation (7)

Tree activity (7)

Tesselations

BINGO

Diameter (pi day or tree?)

Outdoor activities

I'm impressed someone remembered the expanding turtle, since we did that on the very first day of school! I presented them with a turtle, they measured whatever they wanted, predicted growth based on the package's claims and stuck it into a bucket of water. Over the following weeks students would occasionally remember the turtle was growing and poke, measure and smell it (smells like cheerios for some strange reasons). It's a nice way to get them thinking the first day, but no big deal if a kid switches in 2nd day and didn't get to see the turtle before it entered the water. Co-teachers and friends have started picking up expanding animals for me so I now have a whole crew of creatures to experiment with.

The pi day tradition started in my last school, and I've carried it with me. We gather as many circular objects as we can (wheels, jars, balls, baskets...) then students measure diameter and circumference to calculate pi. For increased accuracy on circumference (and a lot more fun!) we count the number of times it can roll down a 100 foot track. Anyone who accurately calculates pi, gets a slice of pie. I even got the grocery store to donate $25 worth of pies and the principal found funds to cover the rest!

I was happiest with the 'tree activity.' It was nearing the end of the year and I was feeling fine about where we were in the curriculum. It was hot in my classroom (no windows!) so I was looking for something to get us outside. We were studying circles and I happened upon a chart relating the circumference of a tree to its age. Each type of tree in the chart had a growth factor, so all we had to do was identify some trees and measure their circumference. In my first vision of this project we would all go out with those tree ID guides that work like choose your adventure books (if it has needles jump to page 45), but I didn't know how to find those in time. So, I gave them the identifying features of 4 common trees (dogwood, white birch, red oak and red maple) to go out and find. But, to prove to me they had the correct tree they had to draw or describe why both the leaves and bark fit the description.

I anticipated that this would be a fun, but simple activity. Oh was I wrong! Kids started by running up to any plant, plucking a leaf and presenting it to me to identify. Dear children: don't harm the tree, read the description, look at the picture, think for yourselves! They got better, but even the ones following all my advice were missing something I assumed all would have- a basic idea of what these trees looked like. One group was looking at a tree- its leaves had jagged edges like a birch and the bark was light, but it wasn't until I pointed at the tree 5 feet away for comparison that they realized a white birch is really white! It still boggles my mind that kids can live in a city filled with parks and not know the first thing about the trees that fill them. I can't blame video games or TV, those existed when I was a kid too. Is no one interested in nature? I'm ashamed of how few plants I can identify, maybe this is something that we'll all work on together next year. In fact, this just may be one of the early activities on proof that I need.


Take aways:
Prove you correctly identified tree to intro proofs.

End of Year Reflections: Geometry Favorite vs. Hard

Continuing analysis of student's reflections... (see first post for a full description)

Geometry CP students' Favorite Topics:

Ch 1

Ch 5

Ch 10

Ch 12

Triangles

Triangles

quads

quads

quads

Solving Triangles

Similar triangles

Ratio/proportion

Proportion

Area

Area

Area/Volume

Trig (interesting)

Pythagorean thm

Pythagorean thm

Translation

Polygons

Polygons

SAS etc. Theorems

Geometry CP students' Hard Topics:

Nothing

Nothing (if I paid attention)

I don't know

I don't remember

A lot

Everything

Everything

Everything

Everything

Everything

Everything

Triangles

Finding lengths and angles

Ratios/proportions

Trig

Trig

Trig

Trig

Trig

Trig

Trig

Trig

Trig

Trig

Trig

Trig

Trig

Proofs

Proofs

Proofs

Circles

Area formulas

What's the most striking thing? The million Trigs!

That pattern isn't surprising to me at all. I haven't figured out yet what I will do next year (suggestions??), but Trigonometry was the only section where I had students asking "When will I ever need to know this?" We studied plenty of topics that were difficult (at best) for students to see the applications of, but they never stopped to ask that question because they were interested, involved, curious and they understood enough to be able to work toward the problem. In trig, that didn't happen. I saw confusion, frustration and kids giving up. It may have started when the first investigation we did gave data that was too far off to see real patterns (perhaps technology would be better than measuring by hand for this?) or perhaps when we started synthesizing too many ideas at once. When 'solving a right triangle' (finding all the side and angle measures given a few) we applied angle sum rule, pythagorean theorem, trig ratios and a lot of algebraic manipulation. Next year I'd like to do more problems throughout the year that synthesize topics so hopefully that won't be so overwhelming. Last, but certainly not least, trig was the first topic we did after MCAS (the state exam, required for graduation) and students feel like they should be done when they've finished that test (even though it happens mid-May and we didn't finish until June 29 this year). Overall, trigonometry got a bad deal last year. I'll try to do it more justice in the future.

Proofs are another challenge, and I believe that a lot of that is related to how they are presented. Student in my classes are accustomed to "defending their answers" and usually can do so well. However, when it comes to writing a proof they get caught up in format, and formal language. Precision is key and I certainly want my students to be able to write concise and carefully worded explanations, but I wish that they were more willing to just write something to start with. Does anyone have a method of draft proofs or easy entry formats?

Otherwise, I appreciate the overlap of favorite topics and hard topics. Different students had different preferences, and geometry has plenty of variety so most students get to experience a balance of topics they enjoy and others that they struggle with.

Take aways:

Do multi-step, synthesizing problems (before Trig)

Make proofs more 'low threshold'

End of Year Reflections: In Geometry I learned...

For the past few years, I've had students write reflections on the course and their year after they complete the final exam. The prompt is as follows:

• This year I learned…
• About myself
• About mathematics
• About studying/school/how I learn
• In this class…
• My favorite chapter
• The most interesting activity
• The hardest thing to learn

At the end of the year I'm spent, so I put these away until a day I'm ready to reflect. Today was that day. I'm getting geared up to prep for the arrival of students, take what I learned at PCMI and put it all together into a great new year. I pulled out the pile of reflections and they sparked fond memories of: my students, how much they grew over the year, and how willing they were to give meaningful feedback even after taking their final exam.

I compiled their responses for analysis of the year past, but also to remind myself on those tough days that even though it isn't obvious, they are learning and the ideas are sinking in. I'll divide this analysis into lots of posts so they don't get too long!

In Geometry College Prep I learned...

• I'm actually kind of good at math
• I'm good at geometry
• I'm not very good at geometry
• I'm not good at geometry, I'm a number person
• You can be very good at algebra but really suck at geometry or it can happen the other way around
• I am horrible at geometry, but if I just push extra hard I can do it
• I'm better at math than I thought
• I'm worse at math than I thought

• If you miss one class you can be lost forever!
• I need an explanation for everything
• I learn well from examples/book

• To work hard
• Studying is hard for math
• Studying really helps you become successful
• I don't do well in math unless I study
• If I study, I will do well
• Study sheets and flashcards help
• Doing the same problem over helps me understand

• You have to work to understand math
• I liked the experience
• I hated having to redo everything but it made me more conscious I should study hard before the first time
• I can figure out a lot of problems myself
• I am capable of doing all the work, I just have to focus
• I do my work well if I am alone
• I work better: not under pressure, by myself, when mad

• I learned who the real me really is
• I learned too much to put on paper
• I can achieve anything I want if I put my mind into it
• If I try I can do anything
• I never knew that we could learn so much about math in one year

It may have been the phrasing of the question, but I love that no one said 'I can't do math' or 'geometry is too hard' or 'I don't care about it.' And nearly every kid who said that they struggled then went on to say they should have worked harder, studied more or focused better. The fact that people learn by effort, not genetics or intrinsic understanding was a recurring topic this summer. I'm glad to see that most students internalized that message last year.

One comment that makes me really thrilled is:

I hated having to redo everything but it made me more conscious 

I should study hard before the first time

I worried that allowing students to correct their work would make them blow off studying the first time and just settle for whatever grade they could get with corrections. But, if going back over their assignments and having to re-work problems makes them wish they'd studied the first time around then I'm looking forward to continuing that option.

I shared these with a few friends before posting, and the one that jumps out at others is:

I work better: not under pressure, by myself, when mad

I won't pretend to understand exactly what this student meant. I can say that she was quite the talker, so the 'by myself' part makes sense. I wonder if the 'when mad' comment relates to the idea of 'struggle' that we all try to find a way to discuss. In the summer camp I taught at we talked about writing problems that would make students 'frustrated' or 'challenged.' We don't want angry kids, but the point is, that if a problem is too easy no one learns.  We need to give assignments that make students think and that they have to work to understand. I have no idea if that's what this student was trying to say, but it's interesting to think that students are having these same thoughts and are aware that they learn and grow through challenges.

Tina starts blogging again...

This summer I attended PCMI and had my eyes opened on a lot of levels.  One of those levels was the existence of this awesome community of math teachers online.  I mean, I knew that any group you could even think of has a presence on the internet (I even managed to find a new family for my degu in a matter of days this spring thanks to google), but it never occurred to me that so many rich conversations were happening both on twitter and on an overwhelming number of blogs.

So, I decided to join in with the fun on twitter, which led me to all of the blogs on my sidebar (I have a feeling I will be sad I've subscribed to so many once the posting picks up during the school year!).  Then the other day I was reading my students' end of year reflections and realized that I could share how adorable and insightful my kids are if I restarted this blog.  (If you look back into the archives you'll find this was once an environmentalist's blog, I'm still living that life, but don't really feel the need to write about it.  Using this blog rather than starting a new one seemed easiest, especially since its address is the same as my twitter handle- can you tell I've been using the same username since the internet was invented? Okay fine, since I was introduced to the internet.)

So, I'll start off with an analysis of end of year reflections, it's good for me to do and gives you some sense of how things were in my class last year.  Before we start that I suppose there are a few things you should know about me:

• I will be starting my 5th year of teaching this fall.
• I taught for 3 years in one school, then last year switched (and it was a really good move)
• I worked within a 90 minute alternating day schedule last year, the previous 3 years were 47 min daily classes (although I taught some double periods so 90 minutes wasn't new to me).
• I end every class with journaling (math learned, another question about key ideas) so the idea of writing in math for an end of year reflection was nothing new for my kids
• I've taught everything from pre-algebra to AP calc. I miss my calc kids but geometry is way fun
• I've done PCMI, PROMYS at BU and undergrad with licensure at MHC (omg abbreviations!)

This blog is really all for me to get my ideas sorted, but hopefully you can get something out of it too.  I'm going to try to end posts with 'take aways' so that I can go back and remember what I want to do when the going gets tough, feel free to jump to the end and then go back and really read if the take aways appeal to you!

Math Blog?

I'm playing with the idea of turning this into a math teacher blog (because I really need one more thing to do during the school year!) but I'm not sure if it's too open to spam. Hence the test post...

If I'm still in your google reader from ages ago, hi! I'm still doing everything I used to write about, and I actually talk about environmentalism regularly on my craft blog (TintedGreenCrafts.wordpress.com) which is on summer hiatus. You're also welcome to continue to follow along and learn something about teaching and math!