September 20, 2015

Intro to Radians

In the past I've introduced radians by having students measure angles of a circle graph with these weird protractors and compare that to arc length and degree measure until we finally tease out what these things are. It's never smooth and it doesn't result in any major insights. The minor insight comes from this lovely geogebra applet and the understanding comes from continued practice using radians. So I reorganized this year and I'm much happier with how things went. All the notes I took during this lesson are at the bottom. I'm too lazy to give them to you page by page so they go with the correct paragraph, I'm confident in your ability to match things up.

We'd just finished As the World Turns which uses both linear and angular speed. So we compare linear and angular distance for some turns a figure skater makes. They want to know the radius, I tell them to just use r. They find a nice pattern - add one revolution and you add 360 degrees or 2πr. This pattern is not nearly so convoluted. We realize this makes sense.

I tell a nice little story about the history of 360 and how it's nice but arbitrary and the radian measure is not! Then we watch the lovely geogebra applet. It's still not obvious to them that it will take ~6.28 radii to get around but there are some aha moments when someone shares that idea. Someone asks what if the circle was a different size, I love these children! I change the size of the circle and it looks the same, a few more aha moments. We add this to our notes.

Great, now we have a thing that doesn't seem totally crazy, we really just need to practice with it. But before we jump into graphing in standard position, why not solve a little puzzle? I start out facing East (note - that's the positive x-axis aka standard position) and I make a quarter turn to the left. If you start out facing East, what other moves could you make to end up in the same position? Hm, now I'm wondering if it would have been better to just tell them I end up facing North, then ask, how did that happen? Only benefit of my initial scenario is it sets counter-clockwise as the original direction. Either way, they come up with -3/4 turn pretty easily. We express both as radians. Then I ask for more, we eventually get around to making Ms. Cardone dizzy and see that lovely 2π pattern again. A student describes the pattern beautifully and I make a big deal of it and write it nice and big on its own slide. I kept using the word co-terminal enough times and it ended up in the nice generalization.

Now we all practice drawing a few angles in standard position - start to the East/positive x-axis and turn in the direction that the quadrants are numbered. This is where I find out who was fully engaged in that class discussion we concluded just moments ago and who was nodding and smiling without taking any notes (mental or on paper). Once I'm sure that everyone can draw an angle and recognizes a fraction of π as a fraction of a half turn, then we're ready for our pi(e) eating contest!

A while back I saw this post (possibly linked by someone who was doing this activity with radians?). Having them start in standard position and then add the angles together meant they were labeling angles around the unit circle without even knowing it! I gave them the first quadrant angles with some extra π/6's so no one won too fast. Having them make the angles with the radian protractors means they're building their own wedges a la Shireen and Meg, which we'll be able to cut out and use next class. They needed the practice adding fractions so most groups were still on their first round when the bell rang (five minutes early! Possibly because we were having an assembly during advisory block or maybe just because the bells were messed up on our first advisory day). That means this whole thing took about 75 minutes (I was half way through passing out quizzes when the bell went off unexpectedly so we didn't get to the final two slides below).

Next class we'll build the first quadrant of the unit circle together (similar to second photo here) and then they'll work in pairs to extend it.

My pi(e) eating contest boards and radians measures.

Team Teaching and a Clothesline

The Algebra 1 team this year is entirely made up of teachers who taught Algebra 1 last year. In a department with turnover like ours this is no small feat. We also had an opportunity to work together this summer to design our first unit. (It's so nice to have the basic outline of my lessons complete when I'm scrambling to keep up with all the other start of year paperwork!) These factors combined make us all a little more willing to try some new things this school year. One of those things is team teaching. We all have a support block for a subset of our students that meets on the opposite day of the full class. Guidance did an amazing job scheduling- all of the support blocks meet on red day! Since these classes are smaller we can merge two into one classroom and teach together when two classes meet at the same time.

One day we team taught was our introduction to types of numbers. We wanted to preview some of the ideas with our support kids before having the whole class complete a types of numbers diagram. So we just gave them the card sort (the diagram that would make an appearance the following day is also included here).

When we prepped over the summer we said we'd do a card sort. Everyone seemed clear on that (all four of us who were there over the summer had attended PD that included this structure together a few years back) so we didn't include specific instructions. The first block I did this with we were in my room and so used my slides. I instructed the students to work with their partner to group the cards however they wanted to and then come up with names for each of the categories. We circulated and then when students seemed done with that task the other teacher told one partner to stay at their desk to explain while the other partner circulated the room. When I do a gallery walk I have kids all walk around and the writing on the desk (in dry erase marker) is all the explanation they get, I liked this method!

I then had kids share one category they saw (why is one so hard for ninth graders?) until we'd exhausted all the options. Then I had kids hold up an example of each category and ask for clarification if they didn't know what the category meant. During this part of the lesson it was nice to have another teacher around to check the numbers because I made the font too small for this on the cards!

Next up was the clothesline. I read Andrew's post on clotheslines after our team planning time this summer so I threw the link into the unit map in case other people wanted to try it. Instead, I got to run it and the other teachers figured it out as we went. Again, it was great having other teachers around because they asked good questions and shared insights that didn't occur to me. If you study the card sort carefully you'll notice that there are a few equivalent values (such as 3 and 9/3) so we got to talk about how those go on the exact same spot on the number line (covering the previous one). We had some good conversations about fractions greater than and less than one. We ran out of time to dig into sqrt(1/5) being greater than 1/5. But we did have a great conversation about 0.7 vs. .75 - the leading zero only on one was purposeful and did bring out a misconception in some students, determining the distance between them and even which one was greater was also challenging. These were great conversations to be having in relative isolation before we ended up stuck on these concepts in the midst of a bigger problem. Of course I don't hold any grand ideations that I have cured my students of all misunderstandings related to fractions and decimals at this point, but it was a great start!

I have another support block in the afternoon. This time instead of team teaching we split the class and each ran a mini-lesson, then traded. Our first attempt at grouping kids didn't end well, but the idea was that smaller groups would be nice in the afternoon since we have quite a few who are bouncing off the walls by then. It also means that the other teacher (who is in his second year) only needs to prep for one mini-lesson and he gets to run it twice. We don't get the benefit of adding to each others lessons in real time, but we are discussing ahead of time what we want to make sure the other person focuses on with our students. It should be fun to see how things develop with these two methods of team teaching. I hypothesize that there will be some lessons that lend themselves more toward each. We may have lost our special education co-teachers during these blocks, but that doesn't mean we can't co-teach!

Notes on running this activity:

  • Color code your card sorts! The fan was on and apparently I didn't follow my own advice last year so when they flew onto the floor they got all mixed up. There was some frantic sorting while the next class started the do now.
  • The folded index cards were great for sliding along the clothesline but you do lose a little not being able to see the equivalent values by clipping them below each other. 
  • Write in marker, not in pen. Duh Tina.
  • The best spot we could find to hang the clothesline was toward one side of the room where there were pipes running up the wall on both sides. But the desks were too close to the line and kids were tempted to duck under the line to get to their spot. Possibly because I was standing behind the line. This resulted in cards flying off at least once per class. Move the desks away and model staying in front of the line (or at least enforce it - standing behind the line does make it significantly easier to point things out without blocking)

Notes on team teaching:

  • I learned the first day we team taught (before the one I just described) that meeting in my room makes the other teachers look toward me for supplies. Even when they had suggested the activity. The answer is now "no, we can't change my plan unless you're going to run it and have all the stuff" because running activities that aren't in my slides apparently causes me to stress. (This is not to say I don't deviate from my plans mid-class, but that I only deviate to things I know how to run and have all the stuff for; especially because running to the copier mid-class isn't an option when you teach solo).
  • If someone else is going to run things I need to be open to different ideas. 
  • We need to be comfortable enough with each other and with students seeing us discussing to take teacher time outs. This will take a bit of relationship building with one of my team members who I haven't had as many opportunities to work with.
  • This seems different from co-teaching because my co-teachers haven't been super confident with their math. There's also a bit of "my students" vs. "their students" which may have more to do with how many names I've learned so far, I'll be interested to see how that develops over time. I'm really excited for the opportunity to get to know more students (teaching double blocks seriously limits the number of kids I get to interact with).

September 4, 2015

Cultural Competency

We had our first two days of professional development this year and I'm impressed to say I can't complain. Sure there was some stuff I've seen repeatedly (the same required powerpoint on the difference between an accommodation and a modification along with repeated reminders that 504's are legally binding) but the refresher didn't hurt and the new teachers needed to hear it. A new idea we are working on this year is building our cultural competency. The teacher handbook includes a couple relevant definitions:

Cultural competency:
The ability to interact effectively with people of different cultures and socio-economic backgrounds.

Ensuring that students have equal and equitable opportunities to take full advantage of their education, generally requiring schools to provide additional services or remove any actual or potential barriers that might prevent some students from equitable participation in certain courses or academic programs.

I have to tell you this amazing news right now, immediately after that definition: Free breakfast and lunch for everyone! No activity fees for clubs or sports! It's so unbelievably wonderful! We have always given free and reduced lunch to anyone who qualifies. We have always waived fees for anyone who needed it. But to qualify, we needed to process forms. To know that someone has need, kids had to tell us. When everyone gets free lunch and no one pays fees there's no stigma; there's no barrier to access. When there's no excuse not to eat breakfast and lunch I can nag kids who are hungry and answer my question of "Did you eat breakfast/lunch?" in the negative without worrying that they didn't eat because there's no food at home and they somehow didn't qualify for free meals. Most of the schools in the district qualified for the food grant (the school had to have a certain percentage of students who are economically disadvantaged) and the school committee decided to waive the activity fees (despite the budget crunch) presumably because equity is that much of a priority this year. I'm so proud to work here right now!

Okay, end side bar, back to reporting in order:

"The professional development in this category will enhance the cultural competence of all high school staff, in order to increase staff and student relationships and improve the overall school culture. As a result, we hope to see a decrease in behavior problems."

We are working with the Massachusetts Institute of College and Career Readiness as a Gateway City to achieve this. During five professional development sessions we will work on the following goals:

  • Establish that racial, cultural and economic differences are real and that they make a difference in education outcomes.
  • Establish the need for a personal and professional journey toward greater awareness of how race, culture and economic difference impact educational outcomes.
  • Demonstrate that difficult topics can be discussed in an environment that is honest, safe and productive.
  • Understand what a "welcoming community" is and develop a vision of excellence for all students.
We started with a comparison of Salem 20 years ago (when our principal was a student at the high school) and now. The statistics have changed dramatically. In 1995 the school was mostly white and middle class (I didn't write down the percentages). Now we are:
  • 60% Economically Disadvantaged
  • 48% White
  • 39% Latino
  • 5% Black
  • ~30% Special Education
  • 70% High Needs (economically disadvantaged, special ed, English Language Learner or some combination of the three)
  • I don't have current ELL data, but I remember hearing something surprisingly small last year (maybe 13%?) because it only includes students who are currently enrolled in ELL classes. I think the number of students who don't speak English at home would be much more informative.
Then we took a multiple choice test (standing to vote for our answer) that boiled down to predicting if we write up students categorized by ethnicity below, at or above their population percentage. I was disappointed to be accurate in my choice that we write up Latino and Black students far more often than White students (proportionally speaking) but I was heartened to see that a decent percentage of the staff was aware that this might be the case.
  • 68% of write ups and ~80% of suspensions are Latino students (39% of our population)
  • 22% of write ups and ~20% of suspensions are White students (48% of our population)
  • 7% of write ups (and some non-zero number of suspensions) are Black students (5% of our population)
We discussed these statistics in our tables (which were strategically grouped to be a cross section of grade levels and departments) and then shared out some ideas. We also discussed a situation where a teacher made some broad assumptions (a student was frequently absent from school so the teacher was going to talk to the parents about the importance of education). As a staff we have a lot of learning to do (myself included!) but I think that most everyone is open to learning. It's really hard not to be when the person leading these discussions is so engaging and dynamic that it feels like I'm listening to spoken word poetry when she speaks!

  • I'm struggling with the feeling that I want to defend our data. But I'm not going to. Are there reasons that might skew the data? Sure. Is that any excuse for how incredibly biased we appear to be? Absolutely not.
  • In one of the ELL presentations the presenter mentioned that the language (e.g. Formerly Limited English Proficient) is not very positive. I know about person-first language but I might not have done such a good job today. I'm learning, we'll get there.

August 27, 2015

Math Practice Standard Portfolios

At our standards based grading implementation meeting the teacher working with us from the charter school asked about how we wanted to include the CCSSM Standards of Mathematical Practice. At the time we weren't sure but I've done some more thinking:

Last year I asked PreCalc kids to journal sometimes on which one they'd used that day. They referred to posters I have in the back of the room to choose one. I never asked Algebra 1 kids to do the same, but they're fully capable of it.

While I want to assess them, because they're important, I don't want to require every kid to use the same practice on a single assignment (because that's not the point). However, assessing students on whatever standard(s) they do use sounds like a logistical nightmare, especially since we aren't due to get an SBG friendly online grading system until second semester at the earliest.

Instead, what if I made the kids keep track? Basically, a portfolio! Kids would get the sheet below early on in the year and I'd tell them that they will need to find four good examples in their work of how they used the mathematical practices each quarter. That makes sixteen examples, two per standard. I could check their binders mid-quarter and near the end of quarter (providing enough time for revisions before report cards) and give them one SMP score based on the quality of their examples and explanations.

Reviewing my student friendly phrases, I feel like the modeling standard should include something about interpreting in context. Do the rest seem sufficiently clear?

I wanted to make this fit on a single sheet front and back but I also want them to write a decent explanation. Maybe we should do one example of each per semester? It would still be four per quarter but that way I could give them one sheet for each semester which means more room to write. And that way they can't leave all the seeing structure examples until fourth quarter when they may not have two good examples of it. A possible alternate handout:

Have you done this with your students? How did it work? I'm still not entirely clear on what the evidence should look like. How would you make the reflection meaningful? Can you edit the following invented student responses to make them better? I'm definitely going to need some quality examples make this a valuable venture.

"You can see I persevered in this problem because I tried three different equations before finding one that worked. I highlighted all three trials and boxed my final answer. My final equation is the one that makes sense because ____."

"In this problem I made a table and a graph. I found the pattern in the table which helped me to identify the slope of my line. I labeled the y-distances in green and the x-distances in blue on the table and the graph. In the original problem that slope means ____."

August 21, 2015

Algebra, the first month

Our first unit in Algebra 1 is focusing on the priority standards of Number Sense and Evaluating Functions. Four of the seven of us who will be teaching the course had the opportunity to work together this summer to map out the unit. That's five math teachers and two special ed teachers - our double block classes changed this year from co-taught every day to co-taught every other day and the math teacher with a smaller group (a subset of the co-taught Algebra 1 class so kids will have the same teacher both days) on the opposite day for a support class. Our goal with the support class is to do some pre-teaching and work on foundational skills that we'll need for the full class. In the map below that's the grayed out B Days. To recap- some kids will have class 90 minutes every other day, they'll do the white rows. Some kids will have class 90 minutes every day, they'll do all the rows.

It was fun to work with the team to find resources because we had plenty of time to share our ideas and search for new ones - they know about the MTBoS but none of them have jumped in (yet?). All the materials for the unit are in a dropbox folder. I'm sure we'll stray from this map by the end of the first week, but I'm so excited to have something available to refer to!

Unit 1 MAP

Mad Minute: It's not a race to do the whole sheet in a minute, it's a promise we'll only do this rote practice for two minutes, and it's as close as I ever get to pre-testing.

Exit Ticket: We don't know how we're going to have to grade the support class, we think it will be pass/fail. If so, the exit ticket will be a problem or two related to what we worked on that class and will be something in the online gradebook to appease admin :)

Flapper: An index card summary: definitions/generalizations go on the front of the card, an example goes on the back. They're taped to cardstock so they flap.

TIP chart: Three column vocab organizer. Term, Information, Picture.