Fidelity in Math

In Algebra 2, the topic today was an overview of functions. Some students were having difficulty understanding the "each input has exactly one output" condition, and the previous example (percent score --> letter grade) just wasn't cutting it.

The follow-up example was much better. The domain was a set of boys' names, the range was a set of girls names, and the mapping was "dating". And, of course, one of the boys had an arrow pointing to three different girls. We discussed why this was not a function, and one student said, "So to be a function, they have to be faithful!". Exactly! I took her up on this, and had them add to their notes: Functions are Faithful! This instantly made sense to them, and this language carried forth through the rest of the lesson. I then added another boy pointing to one of the girls that was already in the list, and asked if everyone was still faithful. They said no, and we clarified things; our new "taken-as-shared" idea was that only the boys (the inputs) have to be faithful for it to be a function. (I mentioned that if all the girls were faithful too, then it is called a one-to-one function, and we'd look at that later.)

It was really amazing - even when we did examples involving decontextualized numbers, they were still very comfortable using the analogy: i.e., that set of ordered pairs is not a function because the 4 is being unfaithful! It even made the vertical line test a breeze to teach.

It's always nice to find something new to add to the bag of tricks.

Happy Tofurkey Day

If you've read my blog in the past, you may have noticed I haven't been writing much lately. I've had some bad health problems over the past few months, which have made it difficult for me to get anything but the essentials done. It's finally under control, and I am feeling relatively human again. I am thankful for strong drugs and medical advances!

So, my lack of energy along with the typical November doldrums have made my freshman Numeracy classes less effective and positive than I'd like. I hope that this mini-break will give us all a chance to recover a bit, and come back to end the semester strong. Here are some of the major things I need to work on:

1) Multiplication Tables.
I decided not to focus on teaching the tables this year to the whole class, as it is a waste of time to the 1/2 to 2/3 of the kids that know them. And, I'm not sure how to do it really effectively for those that are in high school and still don't know them. ALEKS doesn't deal with multiplication tables.

I've given students who need them 12 x 12 tables to keep in their binders and look at as they do things like reducing fractions. I need to find a way that they can actually work on improving, and I think this will be different for each student. Some combination of flash cards, games, incentives, and quizzes will be needed. But how to work this seamlessly into the class? Hmm... Needs more thought..

2) Classroom Culture / Readiness
I've posted about this in the past, and my readiness checker was working very well. But, recently, class hasn't been getting started very efficiently. The students recognize this, and freely admit that, now that they are well into school and have made friends, there is a lot more temptation to hang out between classes and avoid getting into their classroom or seat in the room until the last possible second.

I don't sweat it if we miss out on the first minute or two of class, but more seriously and annoyingly, school supplies seem to be growing ever more sparse (kids don't have pencils or binder paper, and act shocked when I ask them to take these things out!). It's like they buy some new stuff in September, and when they're out, that's it for the year. Also, binders have gotten to be a mess (i.e. holy terror) again - and that's for the kids who still have binders.

I need to come up with an incentive system to get this all back on track - and a way that forces me to stay on top of it.

3) Problem Sets - Singapore Bar Models
I overshot greatly with my initial stabs at assigning students problem sets. My goal was to get them started on longer-term planning, while assessing their ability to use the bar model method to solve word problems. My initial idea was to assign ten problems on Monday, due the following Monday. I would grade and return them, and they would have till the following Monday to do revisions to increase their scores. Sounds reasonable? Well, it was still way too much for them to handle. The second time around, I made mandatory progress checks during the week, to help keep them on track. I got more turned in the second time around, but they were still not very good. And, there is a big problem with cheating. Unfortunately, students don't really understand all the time what cheating is, and it's hard to get them to see it. I want them to work together to help each other; many of them think that copying someone's answers who is "helping you" isn't cheating. This is going to require a lot more coaching. Here are my initial ideas for how to run problem set #3:
- Reduce the number of problems from 10 to 6.
- Continue with the progress checks, for regular homework credit.
- The day the assignment is due, give a one- or two-question quiz with selected problems from the assignment, but with numbers changed. This will help me see which kids actually understand the work, and which kids copied.

Don't worry, I'm not dead...

My Numeracy students are now about a month into their ALEKS experience. I started all students out on the third grade standards level (the lowest ALEKS goes), and on average, my students scored around 50% mastery on their initial diagnostics. At this point, some of the students have completed level 3 and are onto level 4, and many others are close to completing the level. There are the stragglers too, of course. I'll do more detailed stats later on. The goal I've set with students is that they should try to complete 3 entire levels by the end of the year (i.e. 3 years of growth in math ability). I was skeptical at first, but after seeing how the students interact with the program, I have much more hope. ALEKS is not a creative, fun, snazzy program. Essentially, students get a sample problem to try. If they don't know how to do it, they read an explanation and try again. When they get a certain type of question right 3 or 4 times in a row, without asking for help, the concept is added to their pie chart. Periodically, they are re-assessed by the program, and concepts they no longer know are pulled back out of their pie chart.

I have been impressed by how self-reliant the students are being. They are managing to read the explanations and figure out the problems on their own. Some students are really getting into it, and are bragging to each other about how much of their pie they have completed. They have also figured out that getting a problem wrong, or clicking on the "explain" button causes the program to require more correct problems to add the concept to the pie. For that reason, they are actually trying harder to get the problem right the first time. The immediate feedback has been very helpful for the students. My favorite moments are now at the end of class; sometimes, when I tell students they need to log off, a few will be like "oh wait, let me just get this one last problem so I can add it to my pie".

Right now, I am just assessing them on time spent on ALEKS - not on the actual amount of progress being made; it seems to be effective enough, and the whole point is to allow students to work at their own pace. We'll see if I need to modify that policy in the future.

On a different note, we have been working on bar modeling to solve word problems every class for 15-20 minutes. I assigned the first problem set as homework last week, and I graded them this weekend. They were quite bad. It's always a bad feeling when you realize your students are a lot farther behind than you thought. I've pushed ahead into more complicated problems, but I just realized that many students are still having trouble with the basics. That's ok.. we'll just cycle back to the beginning and have another go at it.

In Algebra 2, we've started in with the basic idea of logarithms, using the Big L notation I wrote about in an earlier post. I think it is working well. We have been focusing on the similarities between roots and logs: in a root, the index tells you the exponent, and you are looking for the base. In a log, the subscript tells you the base, and you are looking for the exponent. Last year, many students had trouble in power expressions determining when to use a log or a root; I think they will have a much better understanding of it this year.

Our Department Picture: Math Rox!

Are you ready for this?

I quickly mentioned the Readiness Checker idea a few posts ago, as a new idea I got from another teacher. I want to revisit it here, now that school has been under way for a few weeks. I have to say, it is simply excellent. In the past, I've done a variety of readiness checks, with various degrees of success. This is working better than any other method I've tried.

So here's how it works. To be ready, students must, *by the time the bell rings* do the following:

• Take out binder


• Take out homework (and Readiness Checker)


• Have a pencil sharp and ready


• Put backpack in back of the room


• Begin working on the Do Now

If all of those things are completed, the student earns a sticker or stamp on their Readiness Checker. When the checker is filled in (I currently have 9 spaces on mine, but will probably extend it to 12 for the next round), it turns into a "get out of homework" pass. This has the benefit of putting a nice positive incentive on being ready for class, no negative consequence, and it is not directly tied to points in the grade.

It has been working like a charm in my 9th grade Numeracy classes. Most of the time, I have at least 3/4 of the students earn a sticker (often more), which lets class start quietly, focused, and on track. Of course, there are off days (like Friday afternoons), but overall this has been a fantastic new innovation. If you teach students that have difficulty getting started, I highly recommend a system like this. A couple of our 12th grade English teachers are doing this too (I was surprised, but they say the students love it.) Best of all, there is no added management on your part - if the student loses their checker, give them a nice new blank one. (So far, though homework sometimes "gets lost", I haven't had a single student lose their Readiness Checker. What a surprise! :)

What we have to watch out for (a partial list)

San Jose, though a relatively safe city overall, does have a significant gang problem. Our students are generally not involved in gangs directly (though we do get the occasional hard core kid), but their communities are infused in a wash of red and blue, and gang symbols are everywhere. We work hard to keep this out of our school, so that all our kids can be and feel safe, and part of that means clamping down on the little behaviors that can flare up into big problems. So, aside from all of the normal things a teacher needs to watch out for, here are some others, any of which will get a kid put on a strict "gang contract" (which usually means that further behavior will end them up in a discipline committee meeting to discuss their behavior and their desire to remain at DCP).

• Red or blue markings on clothes or shoes


• Red or blue hair rubber bands, red or blue pens sticking out of pockets or used to hold up hair, red or blue nail polish and makeup


• Students writing in red pen (blue is too ubiquitous to try to prevent)


• Crossing out 3s or 4s; replacing "e"s with "3"s or writing "e"s backwards


• Using the numbers 3, 13, 4, or 14 inappropriately


• Showing problems with colors (i.e. a student given a blue whiteboard marker to write with who refuses and trades for a red)


• Certain tags like Sur, Norte, 408, ESSJ, Sharks


• Markings at the base of the thumb (3 or 4 dots)


• Roman numerals XIII or XIV, and clever ways to write them, such as dotting your "i"s with an "x", like in the word "live"


• Certain hand gestures


• The UFW eagle has also become a gang symbol. I had a couple of students building it out of unit cubes last week!

There are more, but that's all I can think of off the top of my head. It may seem trivial or ridiculous to watch out for these things, but it is amazing how serious they can be, and how they can lead directly to students getting physically intimidated or hurt.

Another discrete learning moment

In Numeracy, we have so far been working on two concepts: solving word problems with the bar model method and adding integers with and without manipulatives.

The bar model work has been quite interesting, and I'll post more on how it's going later. Adding integers has gone pretty well, as it is not that difficult of a topic for most students. The hard part, as always, is breaking students of their deeply ingrained habits of wanting the "rule" or the "shortcut" that will let them solve the problem faster. They can't seem to figure out that they have learned these rules again and again over the years, and that they haven't stuck yet. And, even though they may think they know the rule, they might not. Several times today I heard "a negative minus a negative is a positive". But I digress...

Today, I began integer subtraction in two of my classes; subtraction is, of course, much more difficult for students to master. In my first class, there was a lot of buy-in. First, I showed them how to do problems where the second number is smaller in magnitude than the first number (8 - 5, -6 - -4), which is easy to show with unit cubes and an integer mat. I like showing how the second example is no more difficult than the first when you understand what you are actually doing. Then, things got really interesting when we moved to problems like 6 - -4, 4 - 7, and -4 - -9. I showed them why and how we add zero pairs to be able to subtract. After I went through it once, a couple eyes lit up. After the next problem, a couple more. And after the third, a few more. I could actually witness students engaged in the act of finally learning a concept. This is one of the joys of teaching basic math to older students. One of my repeating students raised his hand and said, "I don't get it. Why is this so easy? Last year this made no sense, and now it's easy." I think I was able to convince him that the fact that he was paying close attention throughout the lesson was the answer to his question (I didn't teach him last year, but I know he almost never engaged in his class). I'm not sure if this meta-knowledge will stick, but if it does, I think he may now be set to finally learn some math and pass algebra. For sure, when he does lose focus in the future, I'll remind him about what he discovered today.

But with all successes come setbacks (I didn't say failure! I must be getting less cynical). In the next class, the lesson did not go over so well. A couple kids showed me the bright-eyed look of victory, but most were just playing with their cubes. I think I need to invest in unit cubes that do not lock like legos... Some of the students know the "rule", and though they don't know why it works, they wanted to keep using it and not try the blocks. I wouldn't mind it so much (for the few who really do know how to use the rule), except that it prevents students who don't know the rule yet from seeing the value in using the manipulatives. It's like creating a short-circuit. I have two more classes to go on this lesson, so we'll see how the others react. I am still getting a feel for the different character of my different periods, but certain patterns are already surfacing.

My life just got a bit easier

Every year, a handful of juniors end up having a free period due to various scheduling reasons. But, we don't let them sit idle. Instead, they become TAs. This may sound like a cruel thing to do, but they really seem to love the responsibility. For us, grading endless quizzes, for example, just isn't quite as entertaining as it used to be. But for them, it is a brand-spanking new rush of power! Ahh.. remember that first A you gave? That first F? Can you smell the red ink? Ok, I use DCP purple and orange... They even seem to like things like organizing files and cleaning lab equipment. Go figure.

I just got a TA assigned to me during my one Algebra 2 period. She was in my Algebra 2 honors class last year, and earned an A+. She'll be grading the daily homework quizzes, grading other quizzes from my Numeracy class while I lecture (ok, she can do her own homework if I don't have anything to grade that day...), and helping tutor students in the class during individual/pair/group work time. She starts tomorrow and I am psyched.

In the news... again...

Yesterday's article about DCP's gain in the API, reflecting the fact that the students decided to actually take the tests last spring.

Writing in math rocks my socks

I handed out and used the reflection journals in my Numeracy class for the first time today. Of course my first class was too fast for me and half the covers were tagged before I could even react; but I learn fast and outlawed tagging script for the rest of the classes. "Aw man, no tagging??" But they listened for the most part..

I know countless other teachers do the reflection journal, but it isn't seen as much in math. And I've never tried it before. The kids were a bit unsure what to write, and I was a bit unsure what to tell them. My prompt today was something like "write about what you learned today in class and what you feel that you still need help in". I also told them they could choose to follow my prompts, or write something else. General ideas: what are you understanding? What are you confused about? How did the class go? Is there anything you want or need to let me know? I did confirm for one girl that the writing should, indeed, be at least tangentially related to what is going on in class. She seemed to find this reasonable.

So far, I am totally into this. The last 5 minutes of class are silent, as kids process what they just learned, and think about what they still don't get. At the end of the day, I read through 3 classes worth (~60 students), in about 20 or 25 minutes, and responded to what they wrote. The immediate feedback was awesome. Most found the Bar Model method long and seemingly difficult, but they almost all conceded that it helped them to understand the problem better and make it easier. The kids who were totally confused let me know. One girl said she was proud of herself for having learned the new skill. Another told me that I talk too fast sometimes but that she thinks I'm going to be a good teacher anyway and is looking forward to the year. One boy told me his stomach hurt from lunch and that he needed to use the bathroom (he's in Numeracy for the second time - but I dig his sense of humor).

I'm going to try to commit to reading their journals every Friday at least.. I think that the more I write, the more they are likely to write to me.

I collected their math autobiographies today (only 3 or 4 kids didn't do them!) and I am looking forward to reading them later on. I'll probably post a few choice excerpts.