Sunday, July 10, 2016

Sophistication of Methods

So I've been thinking about what constitutes a sophisticated method and when. Until recently, I've always considered that solving equations "algebraically" using inverse operations is obviously the most sophisticated method, and it's the one that we should be training our students to do. I'm usually a teacher who wants students to develop different methods of reasoning, and to make sense of the mathematics (and I encourage that during our first unit, when we're working with linear relationships), but I've always considered the method of inverse operations for solving equations a necessity for higher levels of math because students will struggle to solve a quadratic or a system if they don't know how to use inverse operations.

I was recently pushed to reconsider this by a few people (teacher educators who I admire/respect) that perhaps this is not the ultimate way for students to BEGIN learning how to solve such abstract relationships, and there might be some stepping stones they need on the way to developing that abstract method. But then, I was pushed even further in my thinking about this.

I've recently been exploring the Contexts for Learning Math book about the California Frog Jump, and using it to explore equivalence and solving equations. And I'm now wondering whether this method might be more powerful than algebraic expressions, at helping students to understand the relationship between quantities.

I think one major problem with our math curriculum is that we force students who are 13 and less into formal reasoning about math before their minds are ready for it. Most people don't develop formal operations (a la Piaget's 4 stages) until later in life, yet we expect 12 year olds to reason formally about abstract algebra concepts...

Saturday, January 5, 2013

Teaching without a Curriculum

In the new common core standards, 8th graders are expected to Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.

In the sample equation that the state of New York put out, there are all of the above in one equation: distribution of fractions, combining like terms, variables on both sides, and fractions on both sides. The sample problem asked the students to identify which step was an invalid first step.

However, in ALL of the curriculum resources I've been searching, I cannot for the life of me find equations that are the correct level of complexity for my students to practice solving. I have had to personally make every equation for them to practice. I cannot even begin to explain how frustrating that is!

How am I expected to prepare my students appropriately for a state exam when there are no curriculum resources for me to use to teach? Instead of preparing my thoughts on how to teach students to solve these equations, I have to spend my time creating equations to use!

Tuesday, November 13, 2012

Looking at Student Work to Improve my Practice


On Tuesday mornings, we have some time where the teachers look at student work. We're in groups of four teachers, of a variety of content areas, typically teachers who have some students in common. I'm grouped with an ELA (English Language Arts), a Science and a computer teacher. Today was my first time to share my own student work.

We have to bring a task (in this case a Problem of the Week, or POW), the rubric we use, and three samples of student work, typically a low, medium and high sample. In POWs, I require students not only to solve these problems, but also to make some analysis of the situation and to explain their thinking and their solution.

At the beginning of the year, I dedicated about three class days to working on POW "stuff" - we worked on POWs, then we used rubrics/check-lists to Peer Evaluate, revised our work, and resubmitted as necessary. I then gave them feedback and from some students, requested a second round of revisions.

This was their third POW, and some of the students are still struggling with their explanation (I find some write NO explanation, but then there are varying degrees of "correct" in the explanations, depending on the level of detail).

The three students whose work I brought really showed a good range of student work, while controlling for the gender factor (all three students were female). The low said "I used the table to find my answers" but gave no details about how she created the table (since it wasn't given), or how the table related to the context/questions. The medium student explained in more details about how to do the problem, but there was no real reasoning about why her answers made sense  or why she selected her strategy - it was very procedural. "I subtracted, I divided, this gave me this answer" (paraphrasing her response).

The third student (the high level) gave such a great explanation, it was really great to read. Her explanation included answers to all of the questions on my checklist "What was the problem asking?" "How did you solve the problem?" "What steps did you take?" "What strategy did you use?" "Why did you choose this strategy?" "Why does your answer make sense/is reasonable?"

The other teachers in my group complimented me that my expectations were really clearly laid out for the students in the rubric, and the checklist was quite organized as well: it was clear that student C had internalized the structure of my checklist, since her explanation answered each question.

Then came the suggestions/critique:
1. They said it was clear that I had shared the rubric with the students, but how much had I actually modeled for them? They suggested sharing high level models with the students so they can see samples of what they are heading for. Since this was the first year of this curriculum, I didn't have any samples from last year, but this year, I'm going to scan some of my student explanations and erase the names so that next year, I can read some of their explanations in showing my new students how to write their POW explanations.

2. Although we had done a peer review, they had suggested that each student should "self-rate" on the rubric before I collect it to highlight to the students their own knowledge of how they had done on it. I really liked the idea of having students self-assess, especially to then counter-point it with my assessment, so they can check their own understanding of what's expected (in addition to checking their understanding of the problem).

3. The final suggestion from my colleagues was that for some students, the checklist and the rubric being on two pages might be a bit much, especially since there were lots of questions to read, and the rubric is a 5 x 5 grid with (with an addition row and column as headers) explaining a 5-pt rubric in 5 categories! So we discussed what to do to help students who were overwhelmed by it. And here's where I turn into a geek: the ELA teacher suggested giving the students "sentence starters." I wasn't sure how to do that (since I don't have any experience with it), but she pointed out that my checklist already has the questions I'm looking for them to answer in their explanations. For the students who are struggling to get started, she suggested using the sentence starters like "The question asked..." "I found my answer by..." "First I... Then I..." "I know my answer is right because..." "I know my answer is reasonable because..." "I chose my strategy because..." She suggested posting it on chart paper in the classroom, and also going over it for 15 minutes when I give out the next POW.

And this is where I feel like I chose the right profession. The level of excitement that I feel over the idea of sentence starters is ridiculous. I feel like I'm watching myself magically transform from the confused and overwhelmed first-year teacher struggling to stay afloat in the Bronx to an educator who is constantly learning new strategies to support all of the learners in his classroom. I feel very lucky that I work a job I love, and that learning new techniques genuinely excites me. I am mentally engaged with the task of thinking about other ways to present topics/skills to help struggling learners improve.

The next area I feel like I need to improve is how to challenge the gifted students in my class. I feel like I'm so used to teaching to the middle and to the bottom, I don't know how to "differentiate up." I'm guilty of the classic "here's some extra credit" or "read quietly until everyone is finished" or "begin working on your homework." I don't feel like any of these are sufficient, however, and I'm worried because I know that sometimes students who are very bright drop out of school because they get bored and frustrated. I'm looking for ideas next on how to differentiate up. What do I do for the student who has gotten 100% on pretty much everything we've done so far, has the POW explanation down pat (i.e. was the example student for the high), and has been using the equations to solve all of the problems algebraically already. How do I help her delve into this math at a deeper level?

Sunday, September 9, 2012

First Day of School

So this year, I tried something different on the first day. I had my students come in and sit down in their assigned seats (some of which changed, as the roster has already changed several times!). I had them fill out an information sheet, where I asked them for preferred names, technology access at home, plans for the future, favorite memory from last year, and any students who they couldn't work with or any that they worked well with. In addition, I gave them a section to let me know of anything else they wanted to tell me about: themselves, life, math, questions about me, etc. I had a couple students tell me important things that were going on in their lives outside of school, while others shared details about their passions.

Once we finished the info sheet, I told the students since this was math class, we were going to discuss some important numbers - numbers I found important, anyway! I then announced the first slide:
"Is the number 2 important because it is
a. the number of children Mr. G has.
b. the number of dogs Mr. G has.
c. the number of times Mr. G ate Chinese food last week."

The kids were instructed to circle the answer they thought was true on the handout in front of them (as they followed along on the slides). I then revealed the slide with the answer and two or three photos illustrating it. For example, on the answer to the above, it said, "Mr. G has 2 dogs: Margartia, who is a 14 year old Chihuahua and Stormy (short for Stormageddon, Dark Lord of All) who is a 4 month old Pembroke Welsh Corgi.

I had about 8 or 9 of these multiple choice questions, including facts like 5 summers spent at space camp as a camper, 3 nieces and nephews, 1 is my Softball Jersey number, 83 board games that I own, 42 episodes of Warehouse 13 that I watched this summer, and 118 hours of video games were played this summer by me (roughly).

I then gave them the chance to choose 3 numbers that were important to them and had them write about it. For homework, they chose one of those three numbers to create a poster, collage or diagram about. I learned about the number of siblings many of them had, how old they were when their grandma died, and/or how many countries they have visited.

I saved the rules and syllabus details for the second day, and let the students learn a bit more about ME on day 1. So far, it seems to have been a good idea, and I think I will use it again next year!

Wednesday, August 29, 2012

Notebooks vs. Binders for Middle Schoolers

I'm trying to decide whether to do 2-inch binders with four or five dividers (one section for Class problem/HW, one section for handouts/notes, one section for journal entries, one section for vocabulary, and one section for quizzes/tests) OR to do Marble notebooks with Cornell Cues again like I did last year.

Certain aspects of my notebooking system worked really well last year.

First off, I spent a lot of time at the start of the year teaching the students how I wanted them to take their notes, so I could ease up somewhat as the year went on. I would make the first HW assignment or two to draw the line on the first 25 - 50 pages of their notebook, so that I didn't have to deal with the students getting out a ruler as they prepared to do that at the start of class.

Students knew how to keep a table of contents for their notebook, and look up the information. It was arranged by AIM, Date, and Page number, corresponding with the page in their notebooks.

Students indexed new words they learned and the first date that it appeared in their notebook. I think it's important for them to have an index, but perhaps this year, we'll also do a glossary (easier in some ways if it's in a binder).

Students engaged with their notes both in class (as they copied from the board, they had to write down the questions that the notes were answering), as well as outside of class (creating more questions and doing summaries for the lessons). I think that the summaries in the beginning of the year would be more effective if they were of the 3-2-1 variety (3 facts you learned today, 2 questions you still have, and 1 big idea).

On the other hand, students lost a lot of the beginning of class time from setting up their notebooks - with the slower writers, it took as many as 5 minutes or more just to get their notebooks set up! Secondly, some students didn't actually engage with the notebooks, and then it pulled down their grade when they weren't prepared with the notes taken properly.

If I do binders, I think I would still try the two-column notes, but I might explore a bit more outside just having the left side for questions that the notes answer...

I am worried about binders because the three rings can get broken pretty easily (especially inside a middle schooler's backpack), and the paper can get ripped out easily (again, especially in their bags). I know how sturdy the composition notebooks are, and last year, most of my kids (even the disorganized ones) had their notebooks well into the year...

As of right now, I'm still undecided about which option will be better in my classroom: notebooks or binders. I don't have much time left to decide!

Start of the School Year

It's so hard for me to believe that year 3 is about to begin! In a week and a half, my new crop of 8th graders will be coming into my classroom for the first time this year.

I still have a lot to do before next week, in terms of getting both my classroom and my materials ready for the new students. I am determined to do some team-building activities in the opening of the school year, as well as routine teaching, introduction to me as a person (as well as a teacher), and procedure building.

I have certain decisions to make, still, but there are certain parts of my beginning of the year procedures that are already set.

For example:
1. I know that I prefer to give the students my already made rules, rather than "brainstorm" the rules with them. I do, however, believe that last year's lecture on the rules was ineffective, and I need to create a different method to teach them the rules this year. I saw one interactive idea, where it's got boxes with numbers and lines, so they can fill in the important information, and you can check to see that they are paying attention.

2. I know that I plan to spend at least 1 - 2 days on problem solving activities before getting started with the first unit. I don't know exactly how we'll implement the first unit - but I believe doing a couple of fun, math-related/problem solving activities are important, especially since this year's curriculum is much more heavily weighted on the problem-solving side.

3. I want to do two early quizzes: one about the rules, procedures, policies, and Mr. Golan and one about a topic that I teach them to be successful on very quickly, in the opening of the school year.

4. I will continue to use hand signals (non-verbal signs) for certain interruptions to a lesson: a raised R for rest room, a W at the throat for water, pinching the nose for a tissue, and holding a pencil up in the air for the sharpener. This will allow us to focus our attention on the math at hand, rather than these distractions.

5. I want to do some kind of activity where I get the students thinking about what math IS - beyond formulas, procedures, and calculations. I'm not sure how to get them to the realization that math is about recognizing patterns, about order (and chaos), and can be beautiful.

6. I know I want to go over the materials/supplies expectations with them. I also was thinking about doing either a classroom scavenger hunt and/OR a rules/expectations scavenger hunt. I also plan to do a textbook scavenger hunt to familiarize the kids with their new textbooks (which may be different from what they are used to).

7. I was thinking about doing a "Mr. G by the Numbers" ice-breaker, where I share with them 10 - 15 numbers that are important to me (or perhaps do it as a quiz where they have to match the numbers to the important facts). And then have them create their own and share out (and decorate it nicely to hang up?).

Monday, May 21, 2012

An effective classroom is like a well oiled machine

An effective classroom is like a well oiled machine. Every procedure and process runs smoothly and efficiently, from passing out papers to checking HW. Figuring out the routines and planning them (and practicing them with the students) is what makes teachers more effective. When it becomes second nature for the class to do the "usual stuff," the teacher can dedicate their energy to doing the important stuff.

I've started compiling a list of routines and procedures - some I have, some I need to develop, some I need to optimize.

1. Procedure for handing out worksheets. (now that I had rows, I can hand out 6 papers to the "row leader" and they take one and pass it down)

2. Collecting papers (pass to the left, then the last column passes all the piles to the front). This could be streamlined a bit. #1: Having a responsible student sit in the back, pass all the papers to them, and that person is responsible for placing the papers in my mailboxes in the back of the classroom. #2: I also need to make sure there's no way for students' work to get lost along the way or copied.

3. Entry Procedure - I've vacillated between entry slips/quizzes, Do Nows (in the notebooks), and I think the kids don't always know what to expect. Next year, I'll train them early that it will always be one of those three.

4. Notebook set up: my kids' notebooks ROCK. This is one area that I've excelled this year. They know how to set up the table of contents, they know the summaries, Cornell Cues, indexing their vocab words, etc.

5. A routine for bathroom and water visits: I taught my kids some sign language at the start of the year, and they know to ask to go to the bathroom or get a drink of water by signing. We have school-wide policies about bathroom & drinks: kids are only allowed 3 visits per week during classes for ALL subjects, must sign in and out of class, and only 2 kids per class can leave.

6. Start & End of class: several of my classes STILL don't come in quietly and get right to work... Next year, I think I need to practice that more at the beginning of the year, so they do it consistently all year long. I also didn't emphasize that I dismiss the class, not the bell, so unless I say "freeze" to my kids, they start trying to pack up when the bell rings if I haven't already dismissed them.

7. Passes - I need to have a form pass that I can just fill out to give the kids - both for being late (from my class, or if I pull them aside in the hallway) and for lunch tutoring/detention.

8. Gum chewing policy - I've been so lax about it this year, and the undersides of my tables/chairs show it! I just haven't wanted to fight that battle all year long...

9. Absence policy: I have definitely had a hard time keeping track of students who've been late/absent from class and missed work. I've basically said, "it's up to you to find me and see what you missed." The good students do so; the weak students don't - and then risk failure. I need a better method that sets up the absent students to be more likely to check in with me.

10. Class jobs. I know I can delegate some of the responsibility for taking care of the classroom to the kids - and it will help us all! But I cannot figure out what the right jobs are or what those students do... Maybe next year, we'll work on a brainstorming session in homeroom about the jobs that are necessary to make the class run efficiently and smoothly.

11. Talking procedures & routines - how we talk as a whole group, how we talk in small groups, how we talk in partnerships... I didn't really get a chance to teach the students how to do this this year - they kind of got thrown into it. And I would say that it shows - some of my classes are much more effective than others at this skill.

12. Lining up vs. entering immediately upon class starting. I've wavered with my students about whether I want them to enter immediately and get started when they arrive or whether I want them to make two silent lines outside of my classroom when they arrive. There are benefits to both, and I think ultimately, I'm going to continue to make it a case-by-case basis.

13. Tardiness enforcement - I have  not penalized kids for being late to my class yet - and I should... But I haven't figured out how yet or what would be appropriate.

14. A signal for their attention. I've used the count-down, "In 5, 4, 3, 2, 1, 0" and then shortening it to "3, 2, 1, and 0." It worked effectively in the beginning of the year, where I then told them to "let me see your best slant." (Sit up straight, Lean forward, ask and answer questions, nod/note key points, and track the speaker) It works less effectively now, because I'm only doing the countdown. I also think I want to do something similar to SLANT, but call it "Star" (sit like a STAR - sit up, track the speaker, ask and answer questions, respect the class). I also heard from a colleague that he says something like, "800, eyes up." and if he has to say it 3 times, they know they've got detention. I know another teacher who uses a singing bowl to get the kids' attention.

15. A minor meditation/relaxing technique before each exam. I heard from a colleague of mine that they do this with their students, and I know how anxious the kids get before a test. I think it would be good to do a visualization and/or meditation/relaxation technique to help prevent test anxiety.

16. How to label their papers. I changed the heading format in the middle of the year, because I saw that the rest of the math department had a more efficient one. I do like my system of "A#s" though - the students get a number based on their place in the alphabetical order of their class, so I can alphabetize without thinking.

17. What do I do now? I need better procedures and routines for what the students should do when they finish the assignments. I have a lending library, I tell the kids to work on their notebooks, but I don't have any clear activities defined for them - and I should.

18. Get the students into a routine habit of copying down their homework from the board at the start of class (which requires me to be more sure about which assignment I'm going to give at the start of class!).

19. Folders and organization of their work. I want to manage all of the papers I give the students much better next year. To that end, I think a folder with one of those three-ring clips might help organize some of the reference materials for them, I think that having a glue stick or tape will help with being able to affix handouts directly to their notebook during class when I instruct them to. I also think that I need to get better about handing out their homework sooner, so that they can put it away before they're out the door leaving.

20. Self-assessment: how do students learn to assess where they are and ask for additional support or additional challenge?

21. What to do for kids who forget their materials: pencils, paper, rulers, etc. I have had too many lost, broken, and stolen materials. I need a pencil sherrif or something!

Any other routines, procedures, or policies that help make your class smooth? Or that you think are needed to make your class run more smoothly?