Tuesday, December 25, 2007

A Jewish-Italian Hannakristmas

I'm in Cleveland at my dad's house for the annual event... the homemade gnocchi and sauce were great, the kids got lots of noisy plastic crap to forget about by tomorrow, and the vegan chocolate-peanut butter cake from Mustard Seed cafe was tasty: I had two, ok three, slices.

After desert, board games, and being shot at with nerf guns (which have gotten scarily high-caliber), I was talking with my 10-year-old half-brother about his school. Specifically, I wanted to see how he is doing in math. He is (not surprisingly) unable to articulate exactly what he is doing in math, so I was asking him specific questions to see what his math is like. Last year, I was surprised to find out he knew square roots already - I explained about cube and higher roots, and he picked it up instantly.

I wanted to see what he knew about fractions as a 4th grader at a typical Cleveland-area public school. I asked him if 1/2 or 3/4 was bigger.. way too easy. I asked him if 2/3 or 3/4 was bigger. He got it right, and quickly, but couldn't really explain why. I then asked him if 3/7 or 3/8 was bigger, and he said 3/7 immediately. I asked him to explain how he knew, and he looked at me like I was stupid, saying, "a seventh is bigger than an eighth, so..". I asked if he had worked with mixed numbers, and he hadn't, so I asked him to figure out what 3 1/4 - 1 1/2 is. He couldn't do it in his head, so I told him to get paper and draw a picture. That's all I said - he drew fractions circles correctly, crossed off a whole, the fourth, and then another fourth from a whole, and came up with 1 3/4.

I've only ever taught at DCP, so I don't have much of a frame of reference for knowing if he is above average or not, but this is the kind of thinking that all students must have to be successful in high school math. This is the kind of numeracy ability I want my students to develop; this way, when they get to a new problem, instead of giving up, they can reason it through and at least make progress. I struggle daily to get them to pay attention, to care, to think, to not give up when a problem is hard, and their mathematical progress is painfully slow. In a couple weeks, when we start reviewing for finals, and half the kids don't even remember what an integer is, it will be painful. But I know that, by the end of the year, most of my students will have improved their math abilities in many ways. Never as much as I want, but it will have to do! I just gave our grade-level equivalency test before break, and the median score has improved by 1.1 grade levels (from 5.9 to 7.0) and the average by 1.65 grade levels (from 5.76 to 7.42) since the summer. If I can squeeze that kind of growth or better out of them during the second semester, most will be in pretty good shape for next year.

Thursday, December 20, 2007

A case study: freshmen's ability to listen...

(Background: cell phones are not permitted. If seen/heard, they are confiscated until parents pick them up.)

{Phone rings; I answer}

Me: C, they need you at the front desk with your cell phone, because your sister got hurt and they don't have your mom's cell phone number.

C: (Looking angry) But I don't even have a cell phone!

Me: They don't want to take your phone, they just need your mom's number.

C: But I don't have a phone!

Me: Just go...

(5 minutes later)

C: Mr. Greene, do you know why they wanted me? My sister hurt her leg, and they needed my mom's number!

Me: That's exactly what I told you.

C: You did?

Wednesday, December 19, 2007

More decimal and percent work

For the next lesson in Numeracy, I wanted to keep building students' ideas about what percents and decimals are, and how they relate to fractions. I've mentioned before that I am teaching the students to use bar modeling to solve word problems, but I haven't been posting the problems, and some examples would probably be nice. This week, I've started incorporating percents into the problems, which is, of course, throwing the students for a loop. But, they will get it eventually (and some already have), and I think that continually reinforcing the visual connection between percents and fractions is important. So here are the three problems I'm using this week:

Lesson 1

Diego and Dora both took a test in Algebra 1. Diego got 70% of the questions correct, which was 42 points. Dora did very well, and even got the bonus problem right, so she got a 105% on the test. How many points did Dora score on the test?

Lesson 2
By the end of tutorial, Mariana completed 45% of her homework. She spent 54 minutes working (the rest of the time, she was giggling with Gricelda). If she works at the same speed at home, how much longer will it take her to finish all of her homework?

Lesson 3
At the school dance, 70% of the students were girls, and the rest were boys. Ms. Vasquez wondered why there were so few boys there – she counted only 36 boys. How many students were at the dance in total?

We only do one problem like this per lesson (3 lessons per week - block schedule), because it really takes 15 - 20 minutes for the whole process (more when the students are unfocused) to play out. What's nice about these problems and this method is that it naturally connects percents to the work students have been doing for months drawing whole number bars (first) and then fraction bars. Right now, lots of students are still struggling, but I think it's more due to the proximity of vacation affecting their ability to care about math than a conceptual problem. We'll pick up with this after break as we review for finals, and I think it'll go better.

For this lesson, after finishing the problem solving portion, we did another class activity. I gave each group a set of 10 post-it notes with various decimals and percents, all between a pair of consecutive whole numbers. They had to stick their post-it notes to the board (where I had blue-taped up a long number line) drawing arrows with marker to indicate more precisely where the number should go. This only took about 5 minutes or so, and then I had everyone sit back down so we could evaluate how we did. I told them that they would earn 2 team points (whoopie!) for each number in the right place, and 1 bonus point (what can I say? Freshmen love their points!) if they could find a mistake in another team's positioning. We went through team by team, and I asked the class to point out any mistakes. The mistakes that were pointed out lead to additional discussions and modeling, until students seemed satisfied that everything was in the right place.

After doing this with my first period, I wasn't sure if the activity had been all that useful. I asked students if they found it useful (many did) and to share something that they had learned. This lead to some good questions and observations - the key one being that 2.45 is less than 2.5 (I still had the papers on the board from the previous lesson to refer back to) because of the value of each number; similarly, several realized that 2.45 is not more than 2.5 even though it has more numbers in it. Other students said that it helped them see better how a percent and a decimal could both be plotted together on the same line (yes!). So this made me feel better about the whole thing, and I didn't modify the lesson for the other periods.

Now that we've used manipulatives and done a couple of class activities, in tomorrow's lesson, students will be doing a worksheet I put together to get some solid independent practice time. They will start by drawing paper pieces (as in the last lesson) to represent a variety of decimals, fractions, and percents. Then, given a number in one form, they will have to write it in the other two forms. Finally, they have a bunch of problems where they must compare a pair of numbers - written in any form - to see which is larger.

Friday, December 14, 2007

The decimal point's job... where's the one?

This has been a long and stressful week, but it ended well yesterday. The biology classes were on a field trip, so my Algebra 2 class only had about 5 kids in it. I didn't bother trying to teach anything new - I just helped them with the homework, and then we spent the rest of the time looking at different problems they needed help with, and talking about stuff that was on their minds... some "remember when", since I taught most of them as Freshmen - they remember amazingly well things that I said over two years ago, as long as they are not math related, apparently... we talked through some of their fears about going to college... we reviewed fractions... I wish there were more times available to just sit and chat with students.

In my Numeracy classes, the goal of the day's lesson was to learn how the base-10 positional system works, with a focus on the decimal side of things. Specifically, I wanted them to be able to represent fractions and percents as decimals, and to understand how each decimal place relates to a specific base-10 fraction. This lesson went very poorly on Thursday with my first two classes, due to management issues (i.e. they were behaving terribly). Is there ever a point in time when you can get through a week without feeling like a first-year teacher again? Anyway, I didn't rewrite the lesson for my Friday classes, because I was confident that it was actually a good lesson, and worthwhile.. I just tightened it up, and made sure to stay on top of things behaviorally, and it really paid off. The discussions in both periods (and these are my two weaker classes) were quite rich and productive. On my board, I had taped up papers as in the graphic above, and told students that each smaller piece had been made by cutting the previous one up into 10 pieces. I asked how big each was, and they easily saw that each paper was 1/10 of the one to the left. I then asked which paper represented one whole. Some students thought it was the 10 big pieces, some thought it was the single full-size piece, some thought it was one of the smaller ones. They discussed it, and as I kept asking more students what they thought, in each class, several students started saying that any one of the pieces could be one whole. They convinced each other without much of my prompting - it was like magic! I then pulled out my magic Decimal Point cutout, and told them that the decimal point has one and only one job - to determine which shape is worth one whole. I taped the point up on the board (making the full sheet of paper the one whole for this lesson), and we then proceeded to label all the place values relative to one whole. All of a sudden, why the tenths are called "tenths" started to make sense for some students.

I showed what the fraction 71/100 would look like by taping up 7 strips and 1 tiny square in the right columns, and then translated this to the decimal 0.71. We did a couple more examples (I called students up to try some), and then they began asking about adding zeros at the end. Some thought it was ok, and others thought it would change the value. So I had them debate for a while if 0.6 and 0.60 were the same or not. Eventually, they were convinced that they were the same. So, to push them, I asked why 6 and 60 are not the same, which allowed me to show them that the 0's only change the value when they push a digit into a different place value. For each example, I also went back to the definition of percent (a fraction out of 100) to show that the tiny squares must therefore be considered 1% each. This made it easy to see why 0.36 would be the same as 36%, as they could see the 36 little squares on the board (comprised of 3 strips and 6 squares). It also made it easy to explain why 0.237 is the same as 23.7% - you have 23 little squares, and 7/10 of a little square.

I saw quite a few light bulb moments over the course of this lesson. I need to find a way to get my first two periods back on board now... We'll review this on Monday; I hope that, by the end of next week, students will all be able to compare and order decimals, and easily move back and forth between decimal, fraction, and percent representations. That would be quite an accomplishment, and a good way to go into break.

Friday, November 30, 2007

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.

Wednesday, November 21, 2007

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.

Sunday, October 21, 2007

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.

Monday, September 24, 2007

Monday, September 17, 2007

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! :)

Saturday, September 15, 2007

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.

Thursday, September 06, 2007

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.

Wednesday, September 05, 2007

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.

Sunday, September 02, 2007

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.

Thursday, August 30, 2007

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.

Tuesday, August 28, 2007


The first day came and went in a blur. We started the day with a special assembly, in which all the departments came up with a little skit to present themselves to the students. There was a lot of energy, and the kids had a good time. The math department came out as if we were doing an encore to a show - we brought costumes and real instruments, and rocked out to "Cult of Personality" (i.e. Cult of Math Ability). Then we introduced ourselves, including our new stage names. This year, I am "Bass 10". Hah!

I saw 2 Numeracy classes today (80 minute blocks), plus SSR and homeroom classes. Lots and lots of freshmen everywhere. They are always so different in the first week or so of classes: silent and afraid to stand out as their brains are processing all of the new social cues and are trying to make sense out of their new world. Or something. I promised I'd know all of their names by the end of the week, but that may be pushing it.

In Numeracy today, I asked them to start by brainstorming around MATH - what images does this loaded word bring to mind? As this was the first day, discussion was hard to draw out of them. Someone would finally mention fractions, and I'd say, "Is there anyone in here who doesn't really like fractions", and every single hand would suddenly shoot up. See, you guys do have things to say! Then, I handed out a math survey to try to measure their self-perception and self-confidence, among other things. We'll repeat this at the end of the year so I can see what changes have been made.

Then, I handed out the Math Autobiography assignment. It had some questions to get them thinking about their math experiences thus far, and they need to write a full 3/4 sheet autobiography for homework! I'm sure there will be some really interesting ones, and I'll post them once they come in.

We burned through class rules and expectations as fast as possible, and then I began the whole-class unit on integers. (Differentiation will start later on, once I figure out when our laptops will be coming in.) I introduced what integers are (and we talked about applications like money, position, time zones, temperature), and then I showed them how to use unit cubes and an integer mat to model integers. We also learned about zero pairs, and how to simplify integer mats by removing zero pairs. I feel like not rushing things is a good plan. Spending a few lessons really scaffolding integer addition, I think, will pay off in the long run. The kids did a good job with the manipulatives, but reading directions is going to be (as usual) a constant challenge.

I'll try to keep posting about what is happening in Numeracy - though maybe not full lesson plans. If anyone has questions about details, always feel free to leave a comment.

Monday, August 27, 2007

It begins!

Ok, I think I just laminated more things in the last three days than I have previously in my whole life. I've got up the class norms (1. Show respect to yourself, each other, and the class. 2. Always focus on learning math. 3. Try hard and take risks.) I've got up the 8 step process for solving problems with Singapore bar modeling. I've got up the signs for the Readiness Check (Binder and homework out; pencils sharpened; backpack at the back of the room; working on the Do Now) and I made the Readiness Checkers - an idea I stole not 5 hours ago. Each time the student completes all the readiness tasks by the bell, they receive a sticker on their grid of 9 spaces (3 weeks). When the grid is completed, it turns magically into a coveted get-out-of-homework pass. I've got up the giant sign with the answers to "Why should I learn math?" (see previous post) which took the help of two other teachers and 4 boys from the soccer team to color in. Oh yeah, and three kids from the robotics team to laminate it and cut it out. I've got the homework checkers laminated and up on the wall; each day, students receive a check mark or an X depending on whether they completed their homework or not. This system is a good way for them to continually be reminded of the effort they are (or are not) putting in. Plus, parents love to see them when they come visit. On parents' night, the homework checkers are the single biggest draw (as they are easy to understand and give an immediate sense of parental satisfaction or dismay). My blank journals are unpacked and stored on the shelves, labeled by period number. I came up with a new idea today - the "Days since the last referral" wall. Each class period has a sign up, with a referral stapled next to it on the bulletin board. The signs are laminated, so I can put a check mark each day that no one in the class gets a referral (or I can rip it down dramatically when someone does!). If they reach 10 days with no one getting a referral (that quite a lot for a low-skilled freshman class), there will be some sort of group reward. There are famous mathematicians on the walls smiling smugly over the room. I've got up the number line on one wall and the place value chart on another wall and a magic eye calendar on the back wall. I've got up a map of the US and another of the world. These are great to have up in general because students don't know where places are or how big they are relative to others. I always think back to my Algebra 1 student (who is now safely graduated) who, in a scientific notation lesson busted out with "Alaska? What's Alaska?". On the world map, I can show them where Singapore is. The manipulatives are shelved neatly and the copies for tomorrow's lesson are all made and laid out on the desk. I helped the other teachers remaining here at this ridiculous hour move stuff and get stuff prepared. And I just finished making my seating charts now that the class rosters have finally been sent out. You have to be a teacher to really understand how much goes into preparing for a new year...

I'm excited about my classes - both my single Algebra 2 class and all my little freshmen that will be in Numeracy. Going into my 7th year of teaching, I finally don't feel nervous because I know how most things are going to play out and I feel prepared. But last year I had a light schedule because I was working on another project for the school part time; this year, I have the 5 classes plus SSR and Homeroom, so I'm going to have to fight to get my teacher legs back. It's time to go home, get some sleep, wake up, pack lunch, and dive in to the deep end. I think I'll be holding by breath till Thanksgiving.

Thursday, August 23, 2007

Leer es poder!

In preparing for Numeracy this year, I've been reading up on the Singapore Math curriculum and philosophy. They really seem to know what they are doing. Everything has a logical, mathematical reason, and it all fits together neatly. I just read through two books which I highly recommend for anyone teaching primary level math to high schoolers. In just under 2 hours of reading, I've gotten some really good and practical ideas - both big and small picture.

Handbook for Primary Mathematics Teachers

8 Step Model Drawing

I read this next one a while ago, and have recommended it on this blog before, but it bears repeating. It is a fascinating comparison study of teachers in the US and China, and what kind of mathematical knowledge and ability is required in order to teach primary math.

Knowing and Teaching Elementary Mathematics

What books do you find useful/enlightening/interesting with regards to teaching math?

Sunday, August 19, 2007

Numeracy 07-08: Project DI

If you're a new reader of this blog, it's important to know a bit about my school, which serves students 9th graders whose average grade level in math is 5th grade, and we seek to have all students ready for 4-year college in 4 (or sometimes 5) years. All students take Algebra 1 when they first arrive, even if they have "passed" it in 8th grade - which many have. The majority of the students also take the Numeracy class, which I have written a bit about before, though I mainly wrote about Algebra 2 last year.

I wrote the current Numeracy curriculum 3 years ago, but the students have not gained as much as they could from the class due to the wide range of skills (and deficits) they bring with them from middle school. My plan this year is to take some of the best elements from the old curriculum, but to differentiate instruction. The units in the old curriculum went like so: Place value and addition/subtraction facts; multiplication and factors; division; fraction concepts; fraction addition and subtraction; fraction multiplication; fraction division. The first three units were not that useful for about half of the class, while they went too quickly for the other half. No one really got what they needed. So here is the plan for the new year (revised a bit from what I sketched out in an earlier post):

All students will spend the first half of the 80-minute class working on a mandatory curriculum. This will start with a unit on integer operations, and then will move on to fractions/decimals/percents. There will be heavy use of manipulatives to ground the students in concrete understanding of the concepts (which is what they lack the most), but I will also move them to algorithmic proficiency as quickly as possible. I'm going to try this year to focus more on representational fluency between fractions, decimals, and percents, instead of teaching them sequentially.

The second part of the class will consist of shorter units that target specific skills: multiplication/division facts; multi-digit operations; place value; rounding; multiplying and dividing by powers of 10, and so on. But here, students will take a quick diagnostic before each unit. Those who need the help will work with me during this portion of the lesson. Those who don't will now work with ALEKS; this software is totally individualized, so students can choose to work on whatever skills they need most help with - and are ready to learn. This will allow me to focus on the weaker students, and to provide them with a conceptual foundation for whatever the skill is, as ALEKS is really only good at providing practice with procedural fluency. I am also looking into the possibility of having students work on ALEKS as their homework, instead of doing worksheets. This will depend on the percent of students who have ready access to the internet, and if I can make the computers accessible to them during tutorial. But if this works, and I don't need to assign and check worksheets every day, that will be a huge time saver for us all.

In addition to this differentiation scheme, I plan to add in two other key components. First, I want to incorporate writing and reflecting into the daily activities. Our students even worse at explaining their work than they are at doing it! We decry their inability to explain and justify what they are doing, and to see how what they are learning connects with their other classes, the real world, and their future - and yet, we never really give space in the curriculum for them to improve at this. I read the book Writing to Learn Mathematics by Joan Countryman; it is a slim little volume, but it has a lot of good, practical suggestions. I'm going to start by having a daily 5-minute quick-write, where students respond to a prompt (or can write about something they learned or still have questions on), and then a longer journal entry every couple of weeks, where students are asked to explain mathematical concepts in more detail. I plan on reading these journals every weekend, and responding to as much as I can. I hope that this will help the students make more powerful connections, and help me understand better what they are really getting (and still needing) from the class.

Second, I plan on teaching students the bar-modeling method for solving word problems that is used in Singapore Math. If you look at some of the problems that 6th graders are expected to do in this curriculum, you'll see that many of our high-schoolers would have trouble doing them efficiently (or at all). I think the bar-modeling method is simple and powerful, and that it will be a tool my students can really use. I've purchased the series of primary math workbooks (and their series of challenge problems), and I plan on adapting these to fit my classroom. My plan is to spend a few days at the beginning of the year (before the differentiation kicks in) teaching this method with simple addition, subtraction, multiplication, and division problems. Then, as the year goes by, students will be assigned a "problem set" (in addition to their daily homework) that will be collected and graded every other week. This will give them a chance to practice the foundations of problem solving, as well as multiple chances to meet a longer-term deadline. I expect that many students will wait till the last minute for the first few assignments, and will learn how to better plan as the semester goes by.

So this is the general plan. I am interested in hearing any comments, questions, concerns, and suggestions as I embark on this new stage of my teaching practice. I don't pretend to have it all figured out - I just have a lot of ideas and a lot of hope that this will come together and help my students really, finally learn some good math.

Saturday, August 18, 2007

Why should I learn math? (Take 2)

I'm back, and beginning to plan for the new year. There are going to be lots of changes this year as we revise our program. I will be working primarily on Numeracy again, which will be totally overhauled as I plumb the depths of differentiated instruction. I'll also continue to work on Algebra 2; however, we have decided to stop having a separate honors class. I will be collaborating with another teacher to create a rigorous class that our target student can pass (if they put in the work), and that also provides academic opportunities for those who want to dig deeper and prepare for pre-calculus. I'll write about both of these challenges more in the upcoming days. For now, I am starting to think about setting up my Numeracy classroom and what sorts of things I want to get up on the walls. Most of the stuff you can buy is tacky and uninspiring to students. I had the idea to create a giant butcher-paper poster with the title "Because" (in huge letters) "you can..." (in smaller letters), and then a series of hand-lettered-by-sharpie, colored-in answers to the implied question. Here is the list I've come up with so far. Any suggestions for additions or changes?

1. Design video games
2. Defy negative stereotypes
3. Become a doctor or nurse
4. Avoid getting cheated
5. Know when politicians are lying
6. Make stronger arguments
7. Graduate from high school
8. Manage your money better
9. Become a forensic scientist
10. Design and program computers
11. Help your family and community
12. Get rich in the stock market
13. Solve challenging problems
14. Show the world how smart you are
15. Get a college degree
16. Become a better thinker
17. Increase your opportunities
18. Become a teacher
19. Understand statistics
20. Become a lawyer
21. Study criminal justice
22. Design bridges, cars, and buildings
23. Run your own business
24. Discover a cure for cancer
25. Help your children with their homework
26. Fly airplanes
27. Run for Congress
28. Start a new school
29. Study psychology
30. Make your family proud
31. Change the world
32. Believe in yourself

Thursday, July 12, 2007

DCP Student in the Mercury News

I am still on vacation and having a great time. I just checked my email and found out that a student of ours, who recently attended the Mosaic Journalism Camp at SJSU, had a piece published in the Merc about the difficulty undocumented students face due to their inability to get financial aid for college.

The readers' comments are predictably negative (i.e. calling for her and her "entire clan's" deportation and so forth). However, the chancellor of UC Berkeley agrees with Dulce :)

Here is her piece if you don't have an account at the Merc:

College door closed for talented undocumented students
By Dulce Martinez

Mayra, a 17-year-old who graduated recently from Downtown College Preparatory in San Jose with top grades, had hopes of going to a four-year university and becoming a lawyer. There is only one problem, which she can't fix.
She entered the United States illegally when she was 4 years old after her parents determined that if they stayed in Mexico they could all starve. As an undocumented immigrant, she's ineligible for government financial aid.
At her high school, Mayra was in Leadership, a program for students who help with campus activities. She was a member of the associated student body and MEChA, a Latino student group. She prepared and distributed sandwiches and water to day laborers in front of Orchard Supply Hardware while they waited for jobs.
Mayra thought that she was as American as anyone. She never thought that being an illegal immigrant was going to be a problem until she applied to a university. Then she found out that she qualified for almost no aid. That limited her college choices.
"I feel that it's not right that I worked so hard to improve my future and now I'm not sure what would become of my plans," said Mayra, a tall, brown-haired and brown-eyed girl from a poor Mexican village.
I know many teens who, like Mayra, had their dreams destroyed when the U.S. Senate turned down the latest immigration reform bill. They shattered the hopes not only of illegal immigrant students in San Jose but also, students living all over the country.

Another deserving undocumented immigrant student who worried about her future is Perla, a thin "jarocha," or native of the Mexican state of Veracruz.
Perla's parents smuggled her into America when she was 9. Also a 2007 Downtown College Prep graduate, she has all the qualities universities look for: She participated in student government, passed Advanced Placement classes and tutored struggling students. And yet when it came to applying to a university, she had all doors slammed in her face. "I feel betrayed by the country I call my home," Perla said.
When the Senate derailed the immigration bill, they punished innocent young people who had no input in their family's decision to enter this country illegally. Undocumented immigrant students in schools are not treated any differently than citizen students. They are always told that they can succeed, become lawyers, psychologists or teachers. But that's a lie if they are not able to attend a college because they don't qualify for government financial aid. I have met many students who have an unclear future waiting for them. Some decide that there is no point to working so hard and they start falling behind and start hanging out with gangs.
These undocumented immigrant students never rest because they are afraid that they might get deported at any second. It is not fair because they have worked really hard educating themselves, learning our country's history, speaking our language, contributing to their community, taking the hardest classes. They don't deserve to be treated so badly.
What happened to the American dream? For these students, the dream became a nightmare and now they have no choice but to remain here, stuck in the lowest ditch in our society because they can't go back to a country they no longer consider their own. All people are created equal, but did Congress think about that when it put the lives of thousands of undocumented students on hold?
Members of Congress should face the problem with their heads up high instead of running away and hiding. Is this what America wants, to destroy the hopes of so many bright and hard working young people who want to become productive citizens?

Thursday, June 28, 2007


I know it feels a little dead around here lately... I'm leaving the country for most of July and probably won't be posting. I'll be doing my best to not be thinking about math or school. But rest assured, once we hit the middle of August, I'll be back and posting regularly. There will be big changes in my Numeracy class that should be quite interesting to document, and I'm sure lots of other fun things to think about. See you then!

Friday, June 22, 2007

Common Denominator Division

I just finished helping facilitate a week-long workshop for most of the people teaching 5th - 9th grade math in two of our school districts. It was very interesting to be a part of, and to see what kinds of content and pedagogical knowledge teachers do (and don't) have. I'm not going to write much more about it, except I did want to post about common denominator division (CDD).

I brought this technique up to see if any teachers use it, and none had even heard of it. I've been teaching it to my students, as part of the scaffolding toward understanding the "multiply by the reciprocal" algorithm. Though not as efficient (typically), I think the CDD method can really help students see what is going on in fraction division, in terms of "the number of times the divisor fits into the dividend". It also fits nicely with multiplication, as "divide across" works just like "multiply across". The only problem is that "divide across" only comes out nicely when the denominator of the divisor is a factor of the denominator of the dividend, which is why (I assume) it is not the standard algorithm that is taught.

Once students know both methods, I give them a set of fraction division problems, and ask them to solve them with both methods. They then discuss which types of problems are better solved with which method, and why.

Here are some pictures that illustrate what is happening:

In this last one, you can see that the bottom shaded area fits into the top shaded area 1.5 times. This is the type of problem that I think really shows the benefit of CDD.

Here is a short proof that "divide across" works. Note: when the denominators are the same, dividing across yields 1 on the bottom, which I didn't write above.

Thursday, June 14, 2007

Last Day

Another year, come and gone. We had an end of the year assembly today that was super long - about 2.5 hours. There were lots of recognitions of student successes, good-byes to leaving teachers, the Numeracy Project played, I presented the winners of the Algebra 2 treasure hunt, and we saw the DCP mini-movie (once it gets published to the web, I'll link to it).

I gave back the finals to my Algebra 2 honors students - all of them passed the final but two, and everyone did end up passing the class. Here's how the course grades were distributed:

9|0 0 2 2 4
8|0 1 2 2 2 5 5 6 6 7
7|0 4 7 8 9

The student with the 106% is a math monster... She's gotten more than 100% on just about every single assessment I've given (she always gets the bonus question, and usually doesn't miss any regular credit). She is also one of the pair who found the treasure. When I gave out class awards, she won the "I'll know more math than Mr. Greene in a few years" award.

After the assembly, there was a pizza lunch (along with food tasting from our new food company for next year, Revolution Foods), music, dancing, soccer, yearbook signing, and so on. I did photo shoots with different students, informed kids that need to take summer school (I wanted to tell them first before they got a notice in the mail, and encourage them to look at it as a chance to improve their skills), and got a marriage proposal from the student I wrote about in my previous post! I declined, but told her I'd be happy to help her make a study plan next year instead. I got some notes of thanks, and the best student gift ever: some of my A2H boys called me over, clustered around me, and gave me a bird-cage shaped present wrapped in toilet paper and tape. They said it was in appreciation of all that I've done for them this year. It was a Transformer from the new movie coming out this summer (they remembered me talking about my love for Transformers all the way back to the functions unit, when we did transformations and translations). This year, like all years, has had its fair share of challenges and problems. But it was nice to end it like this - out in the sunshine, shaking hands and giving hugs, congratulating students, and wishing them a good summer.

Tomorrow, I'll finish cleaning out my classrooms and packing, and tomorrow evening is graduation. Then, the year will really be over. I can't believe that I've been here for 7 years now, but I don't know how I can ever tear myself away from these kids.

Next week, I'll be helping facilitate a math/pedagogy workshop for all the elementary and middle school teachers in a couple of local school districts. The education department at San Jose State got a large grant to put this together. I don't know how it will go, but it will be very interesting. I'll post what I can about it as it happens.

Tuesday, June 12, 2007

The Final

Tomorrow, my students are taking their Algebra 2 Honors final exam. It was hard, as always, to try to distill a year's worth of material into a test that is comprehensive yet fair, that can be completed in 2 hours.

I decided to give them 25 multiple choice questions that cover lots of the smaller topics, like dividing complex numbers, simplifying radicals, powers of i, associative/commutative properties, etc. These questions are worth 50 points, or 1/3 of the total. I expect students to finish them in 30 - 45 minutes.

The remaining time should be spent on the free-response, worth 100 points. To be as fair as possible, I gave students a sheet listing the content that would be tested in these 11 problems. Though it does not include all of the material we covered this year, I tried to pull out the topics we focused most heavily on.

  1. Place numbers in the correct locations on a Venn Diagram of the complex number system.
  2. Given 3 points on a parabola, find the function of the parabola in standard form. You must be able to write and solve a 3 x 3 system of equations to do this.
  3. Graph a piecewise function.
  4. Answer graphical analysis questions (given a graph, determine domain, range, values of x and f(x), find when f(x) <0, etc.)
  5. Solve a polynomial inequality with a number line model.
  6. Given a verbal situation, set up a model of an exponential function and then solve with logarithms. For example: the current value of my car is $12,000 and it is decreasing by 9% each year; how long will it take for my car to be worth only $9000?
  7. Solve a logarithmic equation using properties of logs, and eliminating extraneous solutions.
  8. Translate/transform a graph. For example, given the graph of f(x), draw the graph of y = 2f(x + 3) – 7.
  9. Simplify a rational function, indicating values excluded from the domain. Determine intercepts, holes, and asymptotes, and make a graph.
  10. Solve a quadratic equation (with the quadratic formula) that has imaginary solutions.
  11. Divide with polynomial (or synthetic) division.
Extra credit:
  1. Given a rational function, determine its inverse, and the domain and range of both.
  2. Prove the quadratic formula by completing the square.
This afternoon is our final review session, which I expect to be very well attended. Looking over that list of topics reminds me how proud I am of what these students have been able to accomplish this year - especially those students I was teaching fractions to last year.

Friday, June 08, 2007

Software assisted differentiation

Is there anyone out there with experience using software to help differentiate instruction for students in math?

A couple of programs I've looked into: Aleks and Agile Mind.

I am interested in seeing if I can use something like this to help support my Numeracy students. Of course the websites are filled with anecdotal success stories and even data to support their claims of success. Has anyone reading this tried to use something like this? I'd really like to hear your experiences if you have.

Brainslug asks a clarifying question in the comments, and the answer is long, so I'm posting it here...

Short answer:
"differentiating" is edu-babble for providing different students with different instruction and/or assessment, as opposed to teaching the same thing/same way to all students in the class. Software might help with this greatly in my Numeracy class.

Long answer:
We have all of our students who test below 7th grade take our Numeracy class concurrently with Algebra 1. The problem is that skills range from around 2nd to 7th grade levels. Some kids need to work on place value and subtraction, while others are ready to tackle fractions. We decided not to split the class up into two or more levels to avoid the pitfalls of tracking in a small school.

So, to help students most efficiently, they need to be provided with instruction where they are ready to learn. One solution can be to split kids into flexible groups within the class, where the different groups are working on different skills. However, aside from this being an exorbitant amount of planning time, our freshmen generally do not have the student skills needed to work independently for long periods of time, or the ability to learn from static worksheets without direct instruction and good coaching.

The software that I mentioned above assesses students' "knowledge space" and then only lets them work on the skills they are ready for. The software provides explanations, examples, feedback, and so on. It also allows you to easily provide individualized homework and assessments. In my ideal scenario, I'd set up the class as follows:

Each week (or so), a new skill in math is taught. At the beginning of the week, all students take a quick diagnostic. If they pass, they don't participate in the lesson: they use the time to instead work on the software, on whatever skills they are currently building - plus, maybe some other problem solving curriculum. If they don't pass, they spend half the class (which is an 80 minute block) working with me on the lesson as normal: direct instruction of conceptual and procedural understanding, manipulatives when appropriate, and pair/individual practice. They would then spend the second half of the class working on their individual objectives via the software.

If this works, it would allow class time to be much more efficient, as students would only ever be working on material that was needed, and at the appropriate level. Of course, this all hinges on the software being able to make good on its promises. I'm hoping that the software is both understandable enough and engaging enough that my students can actually learn from it. Computer based learning could be just like a glowing worksheet, or it could make good use of video, animation, interactive demonstrations, and so forth to really move students forward. So my question: has anyone tried this with students?

Thursday, June 07, 2007

The end is near...

Like many other people, I've felt the frustration that comes along with the run-up to finals. Students who know they don't have a chance at passing have checked out; lots of students are still on the fence, but you can't manage to get them all on the greener side. Teaching new material is fun and exciting; reviewing old material again and again is, well, not. Especially to those students who seem to think you are still teaching them new topics. But I know that this is all part of the game, and I am not too discouraged. Some students will fail, but they'll have another shot over the summer or next year. Ultimately, most will be fine. And, there have been lots of success stories this year too - those students who I wasn't sure had it in them yet, and yet they found it.

My algebra 2 honors class, though, has been and continues to be the best class I've ever taught. (I haven't told them that yet - when I do, I know I'm going to get lots of "awwwwwws!") They have been working hard this past week getting ready for the final, and I haven't even had to prepare anything for them. I decided to see what would happen if I just handed out blank copies of the old unit tests and midterms and let them work and ask questions. It's worked amazingly well, and I'm pretty sure that every single kid is going to not only pass the final, but actually pass the class. If every student passes the class, it may be a first in DCP history!

To celebrate their hard work, tomorrow I will stop class a bit early so I can thank each student, say a few words about them, and present them with an award that I whipped up in photoshop:

Thursday, May 31, 2007

Puzzle #8

The last puzzle! One team is almost done with it...


The treasure has been unearthed! Pop the corks! It was found today (6/1) at 3:45 pm by Enrique and Carolina behind the old calculus books in the book depository and much rejoicing ensued!

Puzzle #7

Puzzle #6

Puzzle #5

Thursday, May 24, 2007

Puzzle #4

Puzzle #3

Monday, May 21, 2007

Puzzle #2

Puzzle 2 in the Algebra 2 treasure hunt...

Wednesday, May 16, 2007

National Board Certification

We had a presentation about this after school today. Sounds like it could be really powerful professional development, but it takes a lot of time to complete. If you've gone through the certification process, I'd love to hear your impressions. What is the process like? Is it worth it? What did you gain from it?

Tuesday, May 08, 2007

Puzzle #1

As I've mentioned, my Algebra 2 honors students are currently engaged in a treasure hunt. The idea is that each puzzle will require them to review material from earlier units, and to also do a little bit of independent research to move forward. Here is the first puzzle - can you tell me who to talk to?

Thursday, May 03, 2007

The Big "L"

There are some students who, no matter what, can’t seem to comprehend what a logarithm (when treated like an operation) is doing. I see students that:

1) Cancel the log.

2) Multiply by log.

3) Ask where the 2 went when log2(8) is simplified to 3.

These mistakes indicate that “log” is being perceived as some sort of quantity to be manipulated, not as an operation. This may be due to the fact that “log” is the first time students are exposed to an operation that is represented as a word instead of as a symbol or other numerical notation. Texts apparently assume that this is a natural transition, not even worth mentioning, but it’s pretty clear that it is not as obvious as one might think.

To help students see what is going on, I’ve tried expressing other operations in a similar manner and drawing parallels. For example, take a look at roots and powers:

Logarithm does not have a symbol; our initial idea was to therefore rewrite exponentiation in terms of the “word operation" exp. We then explained that logarithms are the inverse of exponentiation, and that they undo each other, just like addition and subtraction, multiplication and division, and powers and roots.

This seems to have worked moderately well in terms of getting students to be able to evaluate and solve the log problems that they encounter on the STAR tests. However, I don’t think it’s really helped them to understand what a logarithm is, and their ability to apply the concept flexibly is quite limited.

I’m wondering now if going the other direction would have been better. Instead of rewriting exponentiation as a “word operation", we could have invented a symbolic representation for logarithms – say, a big L. (Not to be confused, of course, with the L formed by thumb and pointer finger, raised to the forehead!).

Inverse operations could then be modeled like this:

When I ask my students what “the third root of 8” means, they are pretty good about saying something like “what number to the third power gives you 8”.

When I ask them what “the log base 2 of 8” means, they rarely can say “2 to what power gives you 8”. I wonder if using a symbolic representation of logs will allow this meaning to be clearer. After all, when you think of a log in this way, it’s not really that much more confusing than a root.

I’d be interested in hearing any thoughts on this. Would a symbol for log be helpful? Confusing?

Mad Video Production Skills

I'm linking to this, not because he asked, but because it is pretty damn cool. Videos used to scaffold a linear functions unit. Check it out.

Friday, April 27, 2007

Something I Tried Today...

...that didn't work out as I'd hoped, but that's ok because you've got to experiment. One of my goals this year has been to increase the efficiency with which I use class time, especially for review (going over homework, preparing for assessments, etc.). Today's lesson was the last before the STAR tests, so I decided to use all of the time for a final full-on STAR question review. Of course, for any given concept, there are some students who know it and some who don't. Some students need direct modeling from me, some need to work with their notes, some need coaching from a peer, and some need god only knows what. So I am always trying to design new activities to increase the overall value of the class time to the students.

Here was the plan today: students had a quiz of 25 STAR type questions to answer, due by the end of class. On the tables, I placed labels with the numbers 1 - 25. Students were told that they should start anywhere in the quiz that they felt they needed help on, and go to that table number. Students were only allowed to talk with other students at the same table as them, working on the same problem. If everyone at a table was stuck, they were told to call me over to explain. As students finished each problem, they were supposed to move on to another problem at another table. What didn't work about this is that students began to clump up immediately and then just didn't bother moving around. Essentially, they just formed work groups and then stayed with them for the rest of class. They were working, so the lesson wasn't time wasted. But there was no increased pay-off either (I was really hoping that I was on to something). I have some ideas about modifying this for the future, so if I end up trying them, I'll write about it more.

Has anyone tried anything like this, or have any ideas about efficient reviewing techniques or activities?

Tuesday, April 24, 2007

Staying on top of blogging is like solving a greased Log

Or something. Ok, I haven't posted in a while about what's going on in class. So a summary:

1) The STAR Search(tm) treasure hunt is in effect. The idea of giving away puzzle pieces for each correct answer on the daily 10-minute 5-question warmup has worked really well. Most of the teams are nearly complete with their puzzles, and a few have worked it out already. It was the smiling face of another teacher (photoshopped to make him a bit harder to recognize) with the text "Find Me!" and "Make my noise of disgust." (He has a patented barf-noise he makes whenever PDA is observed in the halls). So a couple teams have done this, and are now working on puzzle #1. I'll post them all eventually, but this one has them reviewing concepts of the real number system from unit 1. There are a bunch of true false questions that they convert to 1s and 0s, and then must research binary to figure out what number is being represented, which will lead them to the next teacher based on a look-up list. An insight into DCP student critical thinking: one student asked me today what to do, because he got the 1s and 0s, but didn't understand how to get any of the numbers. I asked him to read me the title of the puzzle. "There are only 10 people in the world: those who know binary and those who don't." He looked at me. I asked him if there were any words in the title that he didn't understand. He said, "binary". I suggested that finding out what that means might be a good place to start.

2) In Algebra 2, aside from the treasure hunt, I am now desperately trying to get them to grasp logarithms before the STAR test next week. The proximity of the test has forced me to teach the unit totally out of order, and it bums me out. On the test, they only need to be able to evaluate basic logs, change the base, use the log properties, and spot incorrect lines in a log simplification problem. There is nothing about the log function and its domain, the inverse relationship with exponentials, solving real log or exponential equations, and so forth. The order I would have preferred would have been:

  1. introduce the concept of exponentials with a "trick" problem like a lottery or the grains of rice on a chessboard type thing
  2. develop an understanding of exponential functions - growth and decay, and maybe some translations and transformations
  3. present word problems (population growth, interest, depreciation, carbon dating, etc.) and model them with exponential functions
  4. use these models to help students realize that we don't have a tool for finding the x when we know the y, and why we need one - springboard to the logarithm operation
  5. develop a feel for how logs work, by estimating their value to being between a pair of consecutive integers; compare the log operation to the nth-root operation
  6. convert back and forth between log and exponential form, and solve basic log and exponential equations
  7. learn how to change the base of logs
  8. go back and use logs to answer the questions in the word problems that we previously couldn't
  9. derive and learn the properties of logs, drawing parallels with the properties of exponents
  10. use the properties to solve more complex log equations, including discussion of domain and extraneous solutions
  11. explore the graphs of log functions
  12. use that as a lead in to a brief unit on inverse functions
So instead, I jumped in at 5 and continued with 6, 7, and 9. Then, after the test next week, we'll go back to 1 and move forward.

3) I have a student teacher now, and he is taking over the instruction, as of today. It's really cool to work with someone like that, and help them learn how to become a teacher. But I didn't realize how hard it would be to change my own work habits (I can't just plan where and when I want to), and it's difficult to know how much to do versus how much to let him do, knowing that he's got to try things on his own, yet also wanting to make sure that my students learn the material. He's got a great natural patience and rapport with the students, and once he gets the teachable stuff down, he'll be on fire.

4) We're in Spirit Week now. Yesterday was boy-dress-like-girl-girl-dress-like-boy-day (umm... student council came up with the themes...) and today was dress-like-your-culture-day. Lots of sombreros and mariachis walking around, and apparently "hoochie" and "jeans and t-shirt" are cultures too. I abstained yesterday, but today I wore my bar-mitzvah talit and kipah, which always leads to the expected questions: "You're Jewish? Really? Do you speak Jewish? What's Jewish? etc." Today, a freshman took one look at me and said, "What race are you supposed to be?" Tomorrow is class color day (Go purple! Sophomores! Wise Fools!) and the Numeracy Project will be playing; I'll be doing my world-premiere version of "Slope Is Rise Over Run" (The Animals). That will lead into "y = mx + b" (borrowed from Semisonic), and then the ever popular standard "Sweet Home Alameda".

Thursday, April 05, 2007

0% APR!

My algebra 2 students today marketed their credit cards to the Freshmen during lunch. We set up tables outside of the cafeteria, and my students had copies of fliers with their advertisement on front and their disclosure on back (which I photocopied from a real solicitation, used white out on the card names, fees, and APRs, and then filled in with the information they chose). As the freshmen came by, my little salesmen and saleswomen began shouting out the virtues of their card, while slandering those of their competitors. I heard things like "No, don't go over there! They'll just raise the rate on you later. Come to us - we have 0% interest and no minimum monthly payments!"

Some Freshmen made the mistake of signing up for the first offer they saw, and then wanted to change later. I told them they couldn't - once their name is signed on a contract, they can't just say "oops, just kidding!". One girl was really mad ("There was a back side? I didn't even know! That's not fair!"), and I told her she should go talk to the company and try to work out a deal. I listened, impressed, as they worked it out: she was given the choice to switch to the company's other plan, or to keep her original card and then cancel it without charge after a month.

Overall, the Freshmen took the whole exercise very seriously. Many were concerned about getting the best deal, and came to me asking which they should pick. I told them that I didn't know - they had to ask the companies if they had any questions - and they did! My favorite sales pitch: "Hey - you were at my quince! Come sign up with us!".

If you're interested, here are some of the names of the cards my students offered:

Mexican Express
Paiza Express
The Green Card
Viso (Platinum)
Viso (Gold)
Smart Latino
Red Foxx
High Five

Sunday, March 25, 2007

Update and Upcoming

I haven't posted much recently for a few reasons:

1) We administered the CAHSEE (exit exam) last week, which ate up a bunch of class time. Hopefully, our students will do as well this year as in the past. Last year, we had 88% of sophomores pass the math section on their first try.

2) Perplex City. Ok, I have a problem. :)

3) The last 3 lessons in my rational functions unit are BORING. We just practice adding, subtracting, multiplying, and dividing, and solving rational equations. Good mathematics, but I don't have any clever ideas on how to teach it, so it's just me modeling the method and the students practicing. Nothing wrong with that per se, but nothing much to be said about it either. On Tuesday, we'll have a review lesson before the unit test which is on Thursday, and the students will spend most of the class working on Showdown cards created for this unit.

To make up for the recent lows, I have a couple of cool things coming up which I'll preview here and then write more about later (after they've been, you know, actually created).

1) The Financial Literacy project I wrote about earlier is now coming to fruition. I met last week with the College Readiness teacher and we hashed out the outline for the project. It will look something like this:

  • Freshmen will earn weekly income by performing their "job" - i.e. doing homework, being ready for class, etc. They can earn "lobobucks" in all their freshmen classes (assuming we can get all the teachers on board!). Each Friday, students will deposit their lobobucks with their college readiness teacher, and on Monday, they will receive an account statement.

  • They will also receive a weekly bill for expenses. For example: "rent" = their chair in class, "utilities" = worksheets and materials they are given. They must use their money to pay their bills. We are considering consequences - i.e., if you don't pay rent, you have to sit on the floor...

  • To make things more interesting, students will be required to sign up for a credit card. That is where my Algebra 2 class comes in. On Friday, we begin our exponentials and logs unit. I will teach them about interest rates and credit cards, and they will design their own cards and rate plans for the freshmen to sign up. They can use their credit cards to buy extras (though they come at a steep price!) such as free dress, bathroom passes, homework passes, listening to music during tutorial, and a double lunch period. There will be credit limits to prevent out of control spending too.

  • To complicate things further, when students get detentions, the interest rate on their credit card will increase!

  • Each Friday, my students will get a log of purchases made by the freshmen, and any payments that have been made. They will take into account any rate hikes, and will then generate a new balance and create a bill, which will be presented to the freshman on the following Monday along with their income and expenses.

  • The freshmen that are able to stay in budget (or maybe hit some sort of savings goal) will earn a big prize at the end of the unit (like a pizza party and trip to the imax)

  • My students, approaching things from the opposite angle, will be competing to see who can get their "clients" most in debt. What better way to understand how things really work?

2) The "STAR Search" Treasure Hunt. (Can you help me with a better name??)

To help energize my students and prepare them for the STAR test in May, I will create a treasure hunt for them, beginning with a puzzle. I ordered some blank, printable jigsaw puzzles from this site, and I will create a picture/clue that will launch students into the hunt. Each day, they will spend the first 15 minutes of class working on released STAR questions in teams. For each question they get right, they will earn a puzzle piece. By the time the test is here, they should have completed most of the puzzle. Once they do, and they figure out the clue (which leads to a teacher), that teacher will give the group their next puzzle, which will lead to the next, and so on. Each puzzle will require the students to review some Algebra 2 concept, and will also incorporate some sort of fun puzzle, and will lead to another staff member. Ultimately, there will be a prize for the group that gets there first.

The front side of each puzzle will have the same picture. The backs, however, will be different, and will be part of a puzzle that the class will need to solve together, with a class reward as the prize. I don't think any of my students read this, but, just in case they do, I won't post any more details here. After the hunt, I'll post up what we did. For now, just send me an email if you want to hear more, or if you have good ideas for puzzles and clues I can use.

Monday, March 12, 2007

Next Lesson: The Hidden Dangers of Simplifying

(aka Holey Functions, Batman!)

Quite a few students got the hidden message, and they did it faster than I would have expected. Mostly, it went like this:

Student: "Mr. Greene, I got the answer."
(I look at student's paper and see only a string of numbers.)
Me: "What does it say?"
Student: "What do you mean 'say'? You can't read numbers!"
(I look at student meaningfully. Single or double arched eyebrow, with a slight off-center forward head-tilt. Admit it - you're doing it right now!)
(Student looks at the puzzle again.) "Oh, wait!" (Student excitedly grabs pencil and gets back to work.)

Anyway, we took a quiz at the end of the class, and they did fine, so I hope we are ready to move into simplification in tomorrow's lesson.

For the warm up, students will review finding vertical asymptotes and end behavior functions, and they will do this when given a graph only, or when given a function. (Go Representational Fluency!)

Then, I will have them analyze f(x) = (x^2-x-6)/(x+2). (Sorry, I haven't spent the time to learn LaTex yet...) They will assume there is a vertical asymptote at x = -2, and won't they be surprised when they see the graph on the TI! This will lead in to the discussion of 0/0 and holes in functions, and so forth. We'll do a couple practice problems, where students need to simplify (clearly writing the domain of the simplified form of the function) and graph (clearly indicating any holes). If there is more time left, I have a few practice problems for them to do on their own.

Wish us luck!

Thursday, March 08, 2007

Rational Review

Today, I did a lecture on the end behavior of rational functions. We used polynomial division to rewrite the rational function, and then figured out what terms would approach 0, leaving us with a lovely end-behavior function.

Students are starting to get overwhelmed - though they know the differences between x-intercepts, y-intercepts, and vertical asymptotes, when called on to figure them all out for a problem, they tend to mix things up. So I need to stop and take a day for review. So, I present:

Another Perplex City inspired creation for tomorrow's lesson. Enjoy!

Can you decipher the message? Leave a comment.

Monday, March 05, 2007

Next Lesson: Rational Functions

The unit 5 test is over, vacation is over, and I'm ready to get back on track with posting.

In the last lesson, we started the Rational Functions unit (as I described in a previous post). As a warmup, I had students do some division work to explore what happens to a quotient as the divisor approaches zero. They did this visually (i.e. fitting smaller and smaller boxes into a fixed space) and numerically (filling in tables of values).

After they were clear on the effects of dividing by a number approaching 0, I gave them a graph with two linear functions on it, and asked them to work in teams to find the quotient function. They had to look at each value of x, estimate the y-values of the two lines, divide, and then plot a point for the quotient function. It doesn't sound like this would take too long, but I knew from experience that it would take at least a half hour (and it did!). But the division warmup did really help a lot, and my main goal was for them to really understand why a vertical asymptote occurs.

We then moved into some direct instruction where we reviewed the difference between 0/4 and 4/0, I introduced them to hyperbolas (the shape of the graph generated when you divide two linear functions... conic section definitions will have to wait), and we looked at vertical asymptotes and x-intercepts, and where they occur. Students have a lot of trouble with fractions (duh!) and this translates to confusion when trying to deal with rational functions. I hope that continued reminders about what happens when you divide by 0 will help them remember. Finally, I taught them the "as x approaches 2 from the left/right" type notation, with the minus/plus sign as superscript.

We did some example problems, and that was that. I came up with a good way of testing their understanding in the homework: I gave a graph of a hyperbola with two linear functions A and B, and asked them to determine which line was the numerator and which was the denominator.

In tomorrow's lesson, students will continue to practice these ideas, and I will introduce them to Rational Functions as a concept. We will solidify their understanding of x-intercepts, y-intercepts, and vertical asymptotes, and we will discuss the domain of rational functions. I will throw in some factoring, but nothing yet that simplifies (holes will be discussed a few lessons later on).

Wednesday, February 21, 2007

Cool Puzzles

Check out this puzzle, just one of many you can find at the new version of Perplex City. Come on, check it out (you'll thank me later!). Root for yochanan1 on the leaderboard!

"Pictured is an amazing geometric figure: a rectangle partitioned into ten different squares... each square a different size (all have whole number lengths). Knowing only that the side length of the small, white square is 3 units, can you determine what the side length of the yellow square must be?"