Words to Ponder and Chew Over

I recently came across a chart that contained these comparisons, while I was searching various high school websites. (hmmm, the formatting is wonky, but ...)
Comparing Solution Building with Problem Solving
Solution Building vs. Problem Solving

1. "How did you do that?" vs. "Why did you do that?"

2. Focus on the future without the problem vs. Emphasis on past with the problem

3. Solution talk vs. Problem talk

4. Attention on what is working vs. Attention on what is wrong

5. Student is capable. vs. Student is flawed.

6. Teacher skilled at "not knowing." vs. Teacher is "all knowing."

7. If it works, do more of it. vs. Just keep using what you think should work until it, hopefully, does.

8. Change is inevitable. vs. People cannot change.

This resonated with me and it's been popping in and out of my mind for the last few days. If I scan down the list, this past year I could count 5 of the 8 where I was more focused on what was wrong at my school than just rotating my thinking and concentrating on what I could change or fix or be a part of making better.

Ooh, 7 makes me wince because in some of my classes, that's what I did sometimes. Off the top of my head: if students were having continual problems understanding, I would tell myself that I was ALWAYS available after school and they had every opportunity to come to tutoring. Yet every year, there are kids who for a variety of reasons don't come to tutoring and STILL fail to understand various topics. Maybe I need to enhance my toolbox of skills so that more kids understand more topics in class or have different avenues of seeking help other than coming to my tutoring.

Anyway, more stuff to reflect on.

Learned & Reaffirmed Things

After a slew of math workshops all crammed together in one summer, I can generate a list of things I've either learned or relearned from my experiences:

1. Your math text is NOT your curriculum. Not for what you should teach, not for how you should teach, not for why you should teach something.

2. Every workshop seems to have the same set of characters (other than perfect people like me ... cough cough):
- the know-it-all that has to shout out all the answers quickly just to show they know things.
- the on-the-sly-constant texter
- the nary-a-peeper
- the challenged learner that asks TONS of questions
- the I-know-this-&-am-too-cool-to-REALLY-process-what-you're-saying person who will probably be unpleasantly surprised when they have to actually apply this current knowledge next year.

3. I'm set in my ways. Before my 1st out of town workshop, I was stressing, "oh no, I won't have my favorite tea, my vegetarian food, my own bed, ..." waa waa waa. I *mostly* had a change of attitude and used it as an opportunity to try new things ...... mostly.

4. I love my new school-provided laptop. When I was getting homesick, I could stream our local NPR station, and I could Skype my husband, and I could send e-mail and impatiently wait for replies to feel connected to friends.

5. "yelp.com" is a great new asset in my traveling life. I could just type in a city and "thai restaurant" or "breakfast" and get tons of opinions to scroll through to find places to go.

6. "mathforum.org" has a section called "math tools" and you can enter the subject you teach and the topic you're interested in, and it returns a list of resources for you to browse through to use: applets, calculator tasks, worksheets.

Even Better Calculator Trick

We explored a "different ways of payment" problem and learned a new calculator trick to boot. Suppose plan A gives you $20 the 1st day and increases your salary by $1 per day. Suppose plan B gives you $0.01 the 1st day and doubles your salary every day.

Calculator:

{1, 20, 0.01} (to represent 1st day, plan A, plan B)

ENTER

{ANS(1)+1,ANS(2)+1,ANS(3)*2}

ENTER, ENTER, ENTER, ..... to keep seeing current day, plan A, plan B updates.

Nice. And sandwiched between 2 nice pictures of a recent trip.

Ooh Cool Calculator Skill

Today at my math workshop, I learned some new-to-me calculator skills that I believe I'll use this year.

Let's say you want to compare 2 functions at various values. They don't need to be, but for ease sake here I'll assume the x values are integers, and the y values linear, say some geometric patterns where x represents the pattern number, and y represents the number of sides. In the main window type:

{1, 3, 7} and ENTER
This would mean: 1=1st pattern, 3=#sides for one shape in 1st pattern, and 7=#sides for 2nd shape in 1st pattern.

Then TYPE/GET
ANS + {1, 2, 6} and ENTER
This would mean you're adding 1 to the 1st number (1) in list, adding 2 to the 2nd number (3) in list, and adding 6 to the 3rd number (7) in list.

Now you can keep hitting ENTER, and you have a nice way of visualizing what pattern number you're at without keeping track of how many times you pushed ENTER, and what the 2 other functions are, so this list would look like:

{1,3,7}
{2,5,13}
{3,7,19}
{4,9,25}, etc.

Second skill: you know how sometimes you're populating L1 and L2 where L1 is just integers and L2 is some function of the integers? Well, in the past, I would just go to L1 and physically type in 1,2,3,4,5,... then type in the equation at the header of L2. Well. In your main window you can easily populate L1 by:

seq(x,x,1,100,1)--> L1

I guess it would only be quicker this way for large amounts of integers. "seq" is found under LIST>OPS.
The first value is your expression/function,
the 2nd value is the variable,
the 3rd value is where you want to start,
the 4th value is where you want to end, and
the 5th value is what you want to increment the input by.

Now the ironic thing would be if I "learned" this in the past, and just don't remember it because I haven't used it.

preAP math ideas

I'm at a week-long workshop for "experienced" preAP math teachers (for nonTexans, preAP is like an honors course or the higher level math students). After the 1st day, I already have some good ideas on what to do differently next year. In my school district, we have to get re-professionally-developed in preAP techniques every 5 years. At first I was thinking, hmph, I do so much other professional development on my own, this is not needed. Oh contraire. Even though I've taught calculus and precalculus, it is apparently still needed and useful.

Today our instructor did various things using calculus and statistics AP questions as the basis. But. The way he did it was very useful. We worked through a calculus problem. Then we brainstormed on earlier class skills it addresses. He showed concrete ideas on how to modify the problems to address alg, geo, alg2, precal.

But what I really came away with were the LOAD of things I could do with statistics and why it's useful. For example, at the end of the day, he had a sheet with a cm ruler on it, and we all measured our hand span from outer index finger to outer pinkie finger. We got that list in L1 on the calculator. Here are some of the things we did with it besides calculate mean, range, Q1, Q3,...:

1. What if the ruler on the page had a mistake, and we were 2 cm off, so we had to subtract 2 cm from each data point. Without using your calculator, discuss what that would do to the: mean, range, median, ...

2. What if instead of cm, we wanted inches. How would you change data? What would this now do to mean, range, ...

3. What if instead of the largest span we had, that person was replaced with (name some local basketball player who's tall with BIG hand span) ... and we put in a large reasonable number. What would this now do to ....

4. What if for our data of spans that ranged from 16.5 cm to 21.5 cm, someone got a mean of 15 cm. Discuss why this is reasonable or not reasonable.

There were a ton more good ideas. I liked it because it made us think and it would make the kids think and it was in context they could understand and it had them manipulating formulas and thinking about meanings (throughout the day).

Real Life Math (as opposed to the fake kind)

We're going on multiple longer-than-a-week trips this summer, lucky us, and our outside Texas plants are begging for dribbles of water. We decided to put a timer on a drip irrigation system, and then we wouldn't have to worry about coming back to an Adam's Family set of plants.

I was hooking up the 1 gal/hour drippers to each pot, and doing the math. I usually water them every two days with a huge dollop from a pitcher. Hmmm, how many dollops in a gallon? How many minutes every other day to equal a dollop? I guess I'll see if I did my math right when I come back to either a lush plantation or a brown oasis.

Student Teacher Interactions

My 2-week training is over, and now all that's left to do is practice, practice, practice and map out what I'll teach when. It was put on by these folks, and everything was so professional and well-thought-out. I also know from other teachers who teach the curriculum that you are well-supported throughout the year.

For 2 weeks I was in a class with 16 other varied-ability people - lots of time for reflection on how teachers respond to students and how different students handle their learning. Two people had their hands in the air basically the whole time - asking for tons of help and being a little gun-shy of exploring on their own. I'm wondering if there was something non-demeaning the teachers could have done to make them more self-sufficient. Maybe something along the lines of, "I'm confident that you can figure out the answer. Try 3 things first to see what happens and then I'll help you." or "Here's a hint, explore it for 3 minutes and then ask me." Instead, every time they went over to help them on the program, the teachers would take the mouse in their hands and solve the problem. To me that just kept the people helpless.

Another 2 students had already had a lot of exposure to the program, so something that would take me all night of homework to figure out, they finished during class. They weren't rude or bragging about it, but it was clear that was what was happening. But. By the 3rd day of our 2-week workshop, the teachers would frequently make comments such as, "I bet R. has it finished and has improved on it.", or "I bet C. has already figured out how to do that.", etc. As a student, that got annoying to hear. I'm thinking it wasn't helpful to either R. or C. because maybe they felt singled out, and the other students (me included) would feel that much slower. Then I started wondering if I did that in my class. I'd better stop it if I did/do.

Okay, one workshop down, 4 to go. Mwa ha ha ha.

My New Love Affair

Dear Webcam,

I love you. Today we started a "reverse engineering" project (don't ask), and you came through for me by easily allowing instant digital photos needed for our engineering notebook. Contrary to what your lens tells you, this is NOT a gun. It's a tape dispenser that we are to improve in some way. Thus, the inspecting and analyzing and taking apart and sketching and improving and drawing. Because of you, my technical toolbag has grown.

I also like your friend, Skype. We are concurrently working on a virtual project with other students in Ohio, Colorado, and South Carolina. Fancy Schmancy new skills.

I had to tell my husband about us. He now suggested I should reverse engineer the vacuum cleaner that keeps breaking because of SOMEONE'S long hair that keeps tripping up the inner workings. Good luck on that account.

Love,

Me (click click strike a pose)

Core Set of Knowledge

I'm currently at an engineering teaching workshop (ooh, math teacher teaching engineering) with many other teachers from all sorts of disciplines - math, english, science, computer languages, ... and during lunch or breaks or in the course of our training I have had various conversations with many people.

During one conversation, I was talking to a middle-school math teacher who was mentioning that she was helping out her science-teaching friend with one piece of their homework that involved plotting points on the coordinate plane. The other teacher was having trouble with the negative y-axis. During another conversation, I was helping my table partner figure out how to put one dimension from the paper (a radius length) into the computer which demanded a diameter length. I thought she was having trouble reading where the paper number was located on the busy drawing, so I pointed out the radius number, and waited for her to convert it to the diameter length. She blankly stared at me. We finally got her to the point of doubling the length, but I was astounded that she didn't know this basic D=2R fact.

This got me to thinking of what I assumed was basic adult math knowledge regardless of your job. I got on my high horse and was scared of what I was seeing. Then I flipped it around and wondered what science teachers would assume I should know, or what english teachers assumed I should know, etc. Maybe I'm one of those people who cause others to inwardly raise their eyebrows and say, "you don't know THAT?!?!"

I went to dinner with another participant who's a biology teacher and asked her what she thought was an example of what I should know as core science facts. She thought a while and said: what are the functions of different body parts (liver, kidney, pancreas) ... as one of her questions. Eek, eyebrow raising.

Sooooo, hmmm, I assume some math adults should know/retain-from-schooling:
basic circle, square, rectangle facts
plotting points

... so I started that list and then thought: why should they know that (for what purpose)? why did I think they should know that (because I thought those were "easy" to remember and basic)? Maybe it's more of a case of "I'm surprised they don't remember those facts .... like I'd be surprised if they didn't remember how to add, subtract, etc.".

Engineering Education Workshop

On Saturday I went to a free workshop about teaching engineering. It was pretty cool. We got to experiment on how to create "penguin houses" out of various materials for ice-cube-penguins to keep them from melting in 20 minutes of a heat-lamp-induced stay (conduction, convection, radiation) while keeping under a $200 budget. We got to build boats out of aluminum foil to see how many would fit before the boat sank (bouyancy). We got to create lunar landers out of cardboard and straws and index cards and such to land from shoulder height without jarring the astronauts (marshmallows) out of their shuttle (cup). ... and other activities.

It was a fun day and showed me what I DON'T know about engineering, and what I want to explore more about. By the way, our group's penguin house cost $2000 and only preserved 6g out of 10g of ice (whereas others cost less than $200 and preserved anywhere from 8g to 9.1g). Go us. But the presenter did NOT have to make a face when she read our results out loud.

I liked the concepts. Most seemed to assume you were teaching science or had the freedom to use up lots of time to do these things. So now, as a math teacher, I have to see how I can break these lessons up into smaller chunks to wedge into a packed curriculum. I do think the activities were worthwhile, now I have to see how to incorporate them into algebra 1 and geometry.

I did see one geometry connection. When we were making our lunar lander, our accordian shaped index-card "legs" kept flaying out. Then I recalled that triangle shapes are sturdy, so we fashioned some extra supports that used this geometry fact ... of course it didn't prevent our marshmallow astronauts from bouncing boisterously out of the cups to their sure death on the moon.

I did like their continued stressing of the fact that your 1st attempt was not it. As an engineer, you learn from your mistakes and go back to the drawing board to rethink and recreate.

I also liked the talk related to NASA and "Design Squad". There are videos of the DS show you can stream and discuss. It's broken down into chapters (she said), so you can show snippets. NASA has a ton of free resources for teachers. Also, there are video profiles (Pro Files) of engineers (on DS site) that show cool things they do and non-stereotyped people who work as engineers.

Whew! Stuff to think about.

Center of Mass Fun

This is an activity I did with my calculus students post AP exam. We had talked about center of mass, and we calculated it with integrals in a previous class. Then I saw this problem in my new favorite book, so I decided to run with it. I first had the students try to balance one ruler on the desk top with the condition that it should go out as "far as it can".

Then I gave them 2 rulers and said they should line up and for the future tasks, if a ruler is on top of another ruler, it should be farther out than the one below it. So they were to play around and get the whole system to be out as far as it could be.

You can get the total out 9". There's a pattern and a reason, and I'll be mean and not give it away yet because half the fun is figuring it out. Then they got 3 rulers with the same rules.

Do you see the coolness ... almost off the table (you can get 11" with 3 rulers). Then 4 rulers.

Woot! 12.5". And just for fun, 5 rulers:

13.7" off the table edge.

Stating the (not so) Obvious

You know how you go around all the time interacting with a set of people, and you have nice opinions of them in some sense but you never voice these things because in your mind you're thinking, "oh. THEY know that about themselves." Well, it turns out that either they don't, or they're unsure, or they just like to hear other people confirm what they *may* suspect or hope is true.

For example, I have a lovely friend in tap dancing. She's in her late 50's and funny and pretty and vibrant and wise. She's also a teacher (at a different school in a different subject), and she always has some common sense take on school occurrences. I'm forever thinking these things about her. Well, today she was telling me about meeting a gentleman for the first time and how unsure she is about her "package". Holy cow! I had to tell her that she was gorgeous, and he would be thanking his lucky stars when he laid eyes on her. She seemed totally taken aback and did not seem to think this about herself.

Now I'm wondering what other friends I need to pounce on and tell them all the good things I constantly think about them.

Alternate Finals

All year in calculus I told the kids that if they signed up for and took the AP Exam, and if the proctor mentioned that they worked hard and didn't put their heads down or doodle or such, then those students would have an "easier" final exam in my class. Everyone this year took the exam, and I debated what final to give them.

I had a discussion with an AP English teacher whom I respect, and she mentioned that she gives the kids (in the same situation) a college-style text (poem?) to read, and they have to analyze it in some such way. That got me to thinking that even though I want to teach textbook reading skills to my kids, I never seem to get it together to manage such a thing effectively. Then, VOILA, idea. I scanned some books and found a section on probability using calculus. It "looked" like heavy reading from the perspective of a high school kid, but once I plowed through it, I saw that it was very friendly and gave examples and such for the problems I had them do for their final.

I prefaced the final with, "you will be reading such things and learning on your own in college, so this is your chance to practice it. Read through the section, and look carefully at the examples, and do the 6 problems I've circled". They (for the most part) worked hard, and I got a wide range of scores back, and it was enlightening for me.

At the same time, I'm thinking of various precalculus students I have. They scored in the high 90's all semester, and I know that the final would just be an exercise of spewing back stuff to me. So for those kids, I copied some sections of a precalculus book on a topic or 2 that we didn't cover, and did the same thing with them. I did not get to cover induction proofs or binomial theorem this year, and those were my topics of choice. In my mind, I'm thinking of just averaging their semester grades and giving them that for their final exam grade, but I also wanted to challenge them a bit. It was extremely fascinating to see how various students handled it. Some just dove right in and tripped a bit, but with a wee bit of help did fine. Some quietly plugged away without asking any questions, but when I went over to see what they were doing, they were lost. Some were completely flustered at not having a good handle on things. They were so out of their comfort level and had to keep being reassured.

This makes me think that I want to give such a test to the whole class some time periodically, to give them an incentive to truly read a text for understanding. Of course the problem is time, time, time. We'll see.

Wrapping Up the Year and School

Technically I have 4 school days left with children. I teach 5 of 6 classes that have seniors in them, and since seniors are having some finals early, I have NO FINALS ON THE LAST DAY. Woo Hoo. That will give me a huge chunk of time to pack up all my stuff and do "leaving chores".

I have to say that in my 12 years of teaching, this fall counted as my worst ever. This was not kid-related but adult-related, so in January I decided to look for a new school to teach at for next year. I found one, and based on several e-mails and encounters with my soon-to-be coworkers, I think it'll be a great place to work.

That doesn't mean it's not sad leaving a place I've worked at for 6 years, leaving students I've grown to love and would have been teaching calculus to next year, leaving many other teachers that I respect and enjoy being around, leaving a comfort zone of routines that I know, leaving a place where students know me.

And then there's the constant battle of thoughts in my head:
- you're jumping ship when you should have stayed and fought for what's right
- you can't work for people you don't respect
- you're moving to a functional place
- you're deserting the kids
- you'll have better mental health next year and more to share with the kids
- you'll never see these teachers again
- change is good
- change is scary
- will it be weird
- will it be better

Crabby Pants Cookie

Raowr! It's really quite shocking that students would rather text with their friends or chit chat or stare of into space lately than learn super cool math (redundant, obviously). I'm shocked, I tell you. I do have some kids showing snippets of interest and other kids showing tons of interest, and I'm most likely at the end of my patience with this particular class because of year-long behavior of a handful of students.

Cases in point:

One student flitters in and out on attendance, and has missed some classes lately and all of a sudden (1 week before the end of his school year ... seniors take finals early) wants to make up a grade from 6 months ago, and can you please show me what I missed and can you please make copies of it for me and thank you so much.

One student after constantly coming in all year and starting every other class with, "oh miss, I was going to skip today, but I decided to come," has been absent the last 3 classes (one was valid, the other 2 shady). I gave her a zero for the quiz she missed on her shady day. Her current average is now a 25%. She suddenly shows up and has an interest in class and wants to please know what she can do to make up the grade. I question her about the validity of her absence. Oh, I went home sick. Cough cough. Oh, I'll get my mom to write a note. Oh please reteach me EVERYTHING I missed oh and thank you very much.

Two other girls start "maam-ing" me when they know they're getting on my nerves from disruptive behavior. "Yes, maam. No maam."

Deep breaths. They're just kids. They're still pushing buttons and learning how to act. It's spring. Repeat.

On positive notes: various students come up to talk to me after class about what we did and wanted to talk through further thoughts on the matter. Many students mentioned they'll be sad I'm not teaching at this school next year. Handfuls of students stop by periodically and chat about life and such. A student who got pregnant with twins her senior year 3-4 years ago and still managed to graduate now has cute little girls and comes to visit periodically and is going to school to become a nurse. A student I had last year who was so edgy and ADHD and rudely violent in the hallways last year has turned out to be one of my favorite students this year. She's pleasant (still edgy) and interesting and humorous.

A cool fact a student shared with me in calculus today while we talked about breaking the sound barrier: The 1st man-made object to break the sound barrier was ........ the whip. Cool.

Post AP Exam

After the AP Calculus exam every year (of the 4 I've taught it), I've done different things. The first year I did a volumes of cross section project with foam and hot glue and a ton of grief. That was the year I had a screaming match with a student outside of class. The 2nd year I took a break from that and did various other snippets of math topics with mini quizzes. No screaming matches. The 3rd year I revamped the volumes project and tossed in a volumes of revolution project with foam and hot glue and stricter guidelines. That year I had a child sit in class with his pants unzipped and then tell me later that HE WILL DECIDE what is socially acceptable.

This year I'm teaching snippets of advanced calculus to my 2 classes with "easy" quizzes at the end of each class that should be ace-able if they simply pay attention. So far I have one class being good about it, and I've even had one student finally perk up and get out of his morose I-suck-at-math state and pay attention to be able to pass the quiz. The other class was great for one day (when the loud students were gone for other AP exams).

There are students in that 2nd class though that are enjoying things. They come up to me after class to discuss the topics some more.

So far we've "covered" (just given them a taste of) double integrals used for calculating volumes of weirder shapes, surface area calculations, and Gabriel's Horn Paradox. I think I also want to do Fourier Series with my BC class and centers of mass and .... who knows what else. I have a great resource ... the Smith calculus book. It's the THICK blue one, and it has amazing problems and historical snippets and ideas. Anyhow, busy busy busy trying to learn things right before I teach them.

I was also intrigued by the Hubble Telescope news of late, so I assigned my precalculus class an assignment of bringing back 3 facts in their own words about anything to do with "Hubble". I told them that I wanted to learn about it, too, since I didn't know much about it, and I would also do the homework. I told them that we couldn't just exist in our own little bubble of everyday existence. We had to be informed about the world. Then a student said, "we'll bring Hubble into our bubble."

I told them to explore the "what,when,where,who,why, & math" of the situation. Anyway, I went on a particular website, and WOW, the pictures it sent back from space are breath-taking. Can't wait to see what they find out.

Surprising Gap of Knowledge

My teacher friend was having her advisory class address envelopes to their parents the other week to send out invitations to an acadamy picnic. She quickly realized that more than 75% of the class did not know how to do this and was shocked. After she relayed the story to me, I asked my freshman class and my mixed class of juniors and seniors if they knew how to address envelopes. I basically had the same result. We went through a quick lesson. In my freshman algebra class I wondered out loud when I had learned it and why it was useful (because they think they can just text and e-mail their way through life). I told them that it was fun to get letters in the mail, and when I was a kid, I had pen pals in different states. They then wanted to get pen pals with "littler kids" in other math classes. Hmmmmm, I think it's too late in the year, but maybe it's an idea for next year.

NCTM goodies

I went to NCTM in Washington D.C. last week and learned and bought some useful things.

One talk I went to was about how to incorporate web tools into your class. They mentioned a great website, "Everything 2.0", that lists techy websites the blog author finds. There's a phenomenally easy-to-use graphing site that you can use and then save as a document or such to put on your worksheets. You can find this by scanning the list on the left of the page and clicking on calculator 2.0.

There was also a session about "Visual Thinking Activities" that had some great ideas. Two ideas were:

"talking graph". He gives them a graph (say of a line). They have to verbalize some (any) information regarding the graph; they have to make a table; they have to symbolize (equation) the graph.

He gives a picture of the xy plane and plots the point (1,1) without any scale on the axes. Then he randomly puts another point somewhere and asks the students to estimate the point. He spends time with each answer and does not stomp on any estimate. Just through discussion, the student may either stand by their answer if it's reasonable, or self correct if necessary. The point (1,1) may or not result from identical scales on the x and y axis, so that was cool. He does the same idea with the (1,1) but has it on a line and asks them to estimate the equation of the line.

I also went to a useful talk about "jump starting" your class - basically activities related to your topic of the day that take about 5-8 minutes or so. They gave a link that lists all their ideas in a word document. I liked the culling through foreign math textbooks and presenting a page covering your same topic. They suggested going online to search or finding YouTube links to show to class.

There was also a funny guy presenting various math related humor and activities. One example: "Algebra - an intense study of the last 3 letters of the alphabet."

Finally, I bought some books:
"Managing Your Classroom with Heart" good ideas from a high school teacher about relating to students
"The Inspired Teacher" discussing ways "unaware" and "aware" teachers handle various situations that inevitably come up in the teaching day/year.
"Geometry Teacher's Activities Kit" because I have their Algebra book, and it has some good resources to copy immediately and use, and because in my NEW SCHOOL next year, I'll be teaching geometry.
"Math Games: ..." more of thinking activities for the students.

Inappropriate Religious Jokes

Whew! TAKS is over, and today we resumed our regularly scheduled program of classes. I taught the basics of sequences and series in my precalculus preAP class 1st period. I had lunch. On my way back to my classroom after lunch, I ran into a student who walked and talked with me down the hall to my classroom. She was gabbing at me once we got into my class, and then all of a sudden she half shouts, "Jesus!". I quickly turned around to see where she was looking, half expecting to see a mouse or rat or bat or some such thing.

But, no, she literally meant, "Jesus". There was a 1.5 inch doll / flashlight in the form of Jesus on the floor. She proceeded to put sticky notes on him and hang him on my overhead. The notes said, "Math Jesus" and (being a calculus student) "integral of f(x) dx".

Oh my. In the remainder of the day, I had to make reference to the Jesus on my overhead:

Jesus wants you to stop chatting and do your work.
Jesus says to study this weekend for your AP Calculus exam.

Boo Hiss.

School Schedule Craziness

Whew! It's TAKS week in TEXAS high schools. Four days of altered schedules and proctoring not-your-kids and not being able to teach "real" classes for most sections because otherwise the other classes would be off whack. AND to top it all off, thank goodness, we got the go ahead to hold some AP prep sessions during TAKS, so our seniors wouldn't miss a whole week of learning right before the AP exams. AND the schedule was such that I still have my 2 calculus classes all week. AND I'm holding an AP Blitz after school every day from 4:30 to 6pm to "rah rah rah" and tutor my calculus kiddies. Whew! (a "whew" sandwich).

I also see one of my 3 precalculus preAP classes this week, so I can't teach new stuff, but I don't want to have them just sit there, so I did a fun topic. I also gave them a quiz they were sure to get 100% on at the end of class, so that the "good kids" who actually showed up to class would be rewarded for doing the right thing and not skipping class.

I taught a topic I love that can be done in any amount of time. I had about an hour, so I covered: counting numbers in different bases. Then converting between base 10 and other bases (back and forth). Then adding in different bases. Then subtracting in different bases. THEN multiplying in different bases. I liked their wide-eye understanding of what it means to "carry" the 1 now in base 10 and how to "carry" in other bases. I also love what happened with one student. She has been struggling ALL year. She is never completely comfortable with a concept. She feels stupid. She's failing, and yet she shows up to class every day and pays attention and does not quit.

Anyway. This topic started out much the same for her. She sat there all frustrated, but she kept asking questions. Then. BLING. She got it, and she was whizzing through all the things we did. Then she was helping others. Then she truly got 100% on the end-of-period quiz. She walked out a happy camper.

Anyway. I present it by talking about how we can think of numbers in base ten as filling up bins from right to left. Each "chip" or "1" in a bin means I have "1" of that type of number. Once the bin reaches 9, and I try to add one more chip, then I am overflowing in that bin, and I have to scoop all 10 chips in my hand and put a "1" or chip in the next bin over to the left. The bins are:

... 10^4, 10^3,10^2,10^1,10^0 (and so on to the left)

So in (say) base 4, the capacity of each bin is 3 (one less than the base number),

... 4^4, 4^3, 4^2, 4^1, 4^0.

For example, 231 in base 4 means you have 2 16's, 3 4's, and 1 1's, so your base 10 number that means the same thing is 2x16 + 3x4 + 1x1, or 45.

I also peaked their curiosity about what decimals would mean in different bases.

Two Funny Incidents

Some of my precalculus students were staying after school to (gasp) study for a test coming up in a couple of days (new behavior). Anyway, they were working through logarithm and natural log problems, and one of the students muttered, "what are these used for (good for) anyway?". Then another students quickly pipes up with, "shhhhh, she'll assign us a project!".

This made me laugh. Throughout the year, I've given them various small "projects" to research things like:
* how do the conic sections show up in real life (what is it about their properties that lends them to the use).
* who uses imaginary numbers
* how is trigonometry used and find one fact you find interesting about it.
* what are fractals and what's one use of them.
etc.

These are usually small posters (8.5 x 11) or just handed in on paper. They have to cite their sources and have a good presentation. This way, I make them do the work, and sometimes, I have nice posters to hang around my room.

The other funny incident happened while they were reacquainting themselves with log and ln and "e" and their calculators. After a bit, I asked them to look on their calculators and notice the positioning as a pair of ln & e ... and log & 10^x ... and linked it to the positioning of x^2 & sqrt(x), and cos x & arccos x. I made the connection that they are all inverse operations of each other, and that's why they're placed that way. Then a student comes up with another example of inverse operations: "oh like the positioning of the ON & OFF keys".

Reteaching

Since it's the end of the 6 weeks, shockingly a hoard of students are now just realizing they better get their act in gear and turn in late work or do "retests" to bump up their smelly grades. Because of that, I ping-pong back and forth after school from one kid to another reteaching or helping and such.

Yesterday, two kids wanted their memory refreshed on old topics so they could understand their homework or take a test. I was just about to sit down and talk with each of them, when I thought to show them the textbook in one case and some copied notes in another case. I made one kid look in the index for his topic (solving systems of inequalities), and I showed him how to look through the examples and try to understand it. I told him to ask me when and if he had questions. With the other child, I gave him the same instructions with the notes (on graphing polynomials and finding their zeros). I told him to look at his old test and find similar problems on the notes and to ask me questions when he had them.

Whew. That worked out great. They both diligently poured over the texts and worked things out for themselves. I guess this is one skill they need to practice/learn: how to learn for themselves without thinking someone else is the ONLY source of knowledge.

High School Kids

Two funny incidents happened within the last week. In one class I was giving a polynomial test, and allotted an hour. Some kids were done early and put their heads down to nap. This is the ONLY time I allow this. At the same time I'm back at my computer taking roll.

We hear a key in the lock and see the door open, and the principal comes in. He frequently does this visiting of classroom thing. I instantly feel guilty being back at the computer. Then my brain kicks into gear, and I quickly glance over at the 2 kids I knew were sleeping. Their heads were up, and they were studiously pouring over their tests. I laughed on the inside. Boy do they know how to play the game. As a class after he left, and the test was over, they mentioned it and laughed about it, and I had to thank them for not getting us in trouble.


In my second class, I have a periodic visit from another man who basically oversees a math program at various schools. I like to talk with him, and we have many conversations about the state of affairs in education. I consider him a friend. Anyway, he left after a brief visit to discuss something during one of my precalculus classes, and a few girls started giggling. "Ooooh, he likes you. Is he your boyfriend? He's cute." etc. Oh my.

Well, yesterday he stopped by again to give me some information. It happened to be during that same class while they were practicing some problems. Though happy to see him, on the inside I was groaning because I knew the kids were on full alert. A third of me is listening to him, a third is keeping my ears open for my kids to see what they're doing, and a third is hoping he'd leave quickly, so we wouldn't have an "incident". RIGHT after he left, the kids stopped pretending they were working and started teasing me again about him being my boyfriend and gently mimicking things we were saying but in a sappy sweet tone of voice. Oh my goodness. I had to laughingly admonish them and say thanks because every time I see him now, I simultaneously see my class saying, "ooooooh".

I'm going to miss these kids next year.

Students With Issues

My algebra 1 preAP class started out horribly 2 days ago. It's the first period of the day, and as I'm herding kids in, I notice one girl with big sunglasses on. I asked her to please take them off. She ignored me. I waited to see what she would do before asking again about 30 seconds later. Again with the nonanswer. The bell has rung now. I don't want to make a big issue of it, but I can't ignore it and have any credibility with my students. She's a touchy, edgy kid I've gently butted heads with before. I stand right in front of her and ask her again. She refuses. I ask her to step outside and that I will talk with her in a bit. I start my kids on checking their homework, and go outside.

No she does not want to take them off. No she won't tell me why.
"I don't want to."
"Not the issue," I reply.
"I don't want people to look at me"
I can sort of see through the glasses and nothing seems to be wrong. This goes on for a few seconds. I indicate that now she has taken it to the next level. She's directly disobeying me, and if she continues to keep them on, I'll have to send her to the AP. I tell her I'll give her 3 minutes to decide and go back inside.

All the time while I'm teaching my other 20-some kids my mind is racing:
-ackh. I hate calling the office
-what's the office number?
-what's the procedure?
-how long will it take to fill out the paperwork and what a waste of instructional time.
-take OFF your glasses for the love of pete and just come inside
-heyyyyy, her bag and her beloved cell phone are still in the room.
-maybe I'll just let her sit out there all period.
-ackh I hate this.

She never decided to come in, and I never sent her to the AP. I like to deal with my own discipline problems whenever possible unless they're skipping or doing harmful things. The 1.5 hour class ended. I felt bad. I called her mom and left a message. Her mom called me later because the girl had texted her. Apparently, they'd had a HUGE fight that morning about my class and her mom wanted her to get tutoring and the girl is all frustrated because she's not used to not getting concepts immediately, and she's struggling this year. So I'm guessing the girl had red eyes from crying and didn't want to attract notice. Can you have just TOLD me that somehow! I'm so glad I didn't send her to the office.

Then later I sent home an e-mail about a calculus student who I haven't seen in a bit. His attendance has been spotty, and I'm worried about his grades and AP exam outcome. My class is the only class he has on campus one day, and it's at the end of the day. My cynical mind is thinking he's just finding it hard to make it back from the community college.

Hmph to me and my cynicism. Apparently, he's had this soap opera existence for the last little while, his mom wrote me. People and animals have died lately, and she's been gone a lot, and he's been sick tons, and and and. Today I saw him, and on the positive side, he's very smart, so he can quickly catch up.

A 3rd student has had horrible attendance in another 1st period class of the day. Starting around Thanksgiving she's been gone more than she's been here. Another teacher gave me the heads up that she was going through some personal issues. On one of the days I saw the girl (who I like), I asked if she was okay and did she have someone to talk to. She did. But then again with the hit and miss attendance. Today I saw the other teacher again and said I was worried about the girl and did she know if she was okay. Okay, TMI time. The teacher told me what was going on. Ahhhhhhh. But still. Come to class, don't fail. You're too smart. And now I have this other knowledge running around in my head.

Then there's the kids I know of who have moved out or have been kicked out of home. There's the super poor ones that have to ride the bus 2 hours each way to come to school. There are the ones that go on crying jags during advisory.

I'm sending out big hugs to all of them for just continuing to plug away and *mostly* take care of business.

The "Higher" Polynomials

Here is a lesson I love because of graphing calculators. My goals for my students were to be able to look at a graph of a polynomial and know its degree and find an equation given various points on the polynomial. I also wanted them to be able to look at an equation and be able to give a quick sketch of the graph with the intercepts and shape correct.

At the start of class before I mentioned anything about shapes of non parabola polynomials, I handed out graphing calculators and we all got the same window and turned the grid on. In Y=, I had them type in y=(x-2)(x+1)(x-1) (or something similar previously checked out to make sure it fits in the window). I told them not to graph it but to think: what are the x-int, y-int, and make a conjecture about the shape. THEN they could graph and see if they were right. We then had a discussion to link the intercepts with the equation.

Then they typed in y=2(x-2)(x+1)(x-1). I asked them to think about how this might affect the graph and intercepts and shape and THEN graph and confirm their reasoning. We did one more, and then I discussed the shape and degree.

Then same process with y=(x+2)(x-1)^2. They had to think and do the same analysis as above. We then did y=(x-1)(x+2)^2 and such.

Then the fun part: I typed in an equation and just showed them the graph, and they had to match it. We did that a few times. I made sure to change the constant in front sometimes and to make it "upside down" and to have various powers. Then I asked them to find an equation for a 5th degree polynomial with 2 "bounces" off the x-axis going in the same direction (both below the axis or both above). Then for the ones that finished that early: opposite bounce direction.

Then we took notes, and I believe they had a good sense of polynomial graphs.

First Time Teaching Concepts

On my broken record mode (you know, those things that music used to come out of and were bigger than 12" diameter?): I'm loving teaching algebra 1 this year since it's so fun to show them all the cool stuff they'll see over and over again - FOILing, lines, exponents, quadratics...


But it's also daily unsettling because I don't know what they'll find difficult or what will need extra practice or how the best way to explain something will be. Today I told them we were going to do "big kid math" (they were going to start the FOIL process without me saying FOIL just yet). To build up to it I needed to teach them like terms and adding like terms of expressions such as (5x^2)(y^3) or (3xy^7), etc.


I racked my brain (okay I just googled "wracked"/"racked" ....) for a way to present this since I was tired of worksheets and notes. I decided to fold up a white paper to have 20 squares (4x5) and opened it up and put various monomials in each making sure that there were sets of 2 to 3 like terms usually. Then I did it with another piece of paper making sure to put some sort of bubbly border on each square (on one paper only). Then I copied (from 1 to 2) the papers onto various pretty colored papers. The kids then chose their color and cut out the 20 pieces (yay, less work for me) and they had the 2 sets of 20 monomials.


I chose a side (border or no border) and randomly picked a card for the overhead, say (7x^2)(y^6), and told them to find it and cull it from the herd. Then they were to find like terms in their cards (without explaining what like terms were). I wanted to see what they knew. I saw various mistakes as I walked around the room, and it was interesting. So I went up front again and said, "this is like making chocolate milk. You have 2 parts milk and 6 parts chocolate (refering to the exponents). You want to find similar "drinks", and those are like terms". That seemed to work with everyone, and we practiced some more.


Then I had them find a set of like terms, say the cards were 3xy^2, -5xy^2, and 8xy^2. I asked them to add them up to see what they'd get. Instead of 6xy^2, a lot of students got 6xy^6. I was expecting the mistake of (6x^3)(y^6), but I guess the "silent 1 power makes it stay 1!".

Anyhow then we built up the skills for that: 3x + 5x, 10x^2 + 8x^2, etc. They still had the silent 1 problem: 3x+5x = 8x, but 10x^2 + 8x^2 = 18x^4. Then what seemed to work for most (all?) kids was the EXCELLENT box again. Think of the variable parts as boxes filled with x or x^2 or such, and how many boxes do you have and when you add them what happens.

All in all, I think it went well. I also want to share an excellent resource for algebra worksheets. It's been a godsend for extra practice problems all neatly laid out.

The Joys of Spring Break

Happiness is catching up on sleep to start out your break ... and also having lunch with friends and reading books and doing crafts and baking. I hope all teachers everywhere have a rejuvenating break.

I happened across a bookstore yesterday (and by happened I mean I zeroed in on it and ran inside), and found a book another teacher had recommended to me, "The Courage to Teach", by Parker Palmer. I've started to read it, and I see what all her fuss was about.

I'm also resting easier because I know my plans for next year. Daily, still, I'm frustrated by what's going on at our school, but now it only solidifies my resolve not to be in this situation in the future. In January I sent out my resume, and now it looks like I've secured a great job for next year. It's in the same district, which is a plus, but it's at a more sane-seeming school. All the people I interviewed with seemed like great people to work with, and they had intelligent things to say about teaching math. We clicked.

I'm also excited because I'm going to NCTM in D.C. this year. This will allow me to get great ideas for next year. It looks like I'll be teaching algebra 1 and geometry to begin with. I haven't taught geometry for a few years (after 7 years of teaching it), so I'm eager for fresh ideas. The school I'm starting at is just building up their high school component by adding a grade each year. That's also exciting to be in on the beginning of new traditions.

I also found out last December that I passed my National Board Certification, so, Yay. All in all I'm thinking, that if it hadn't been such a rough year, I wouldn't have looked elsewhere for a job, and maybe I wouldn't have happened onto a better situation (if that's what it turns out to be).

Whew!

Hectic times, low on sleep, high on things to think about. I like that even after 12 years of teaching I still have epiphanies. This last unit in AB calculus was derivatives and integrals involving e^x. In the past I've always grouped the derivatives and integrals on one day and was always amazed that my set of kids just confused everything and didn't know what to do and when to "go forward" and when to "go backward". I also used to just give them a brief intro to the basics and then start tossing challenging problems at them that I thought made them think, but apparently just confused them more. I also used to cut and paste from various sources and cobble together a presentation.

This year, I separated them into one day each. I also slowly and carefully picked my examples to build skills that gradually became harder with about 3 problems per skill .... thinking that I could semi walk them through the first one of each set and then they could practice on the next 2 before we built up difficulty. Also, I grouped the skills together, instead of mixing them up all higgeldy piggeldy. I also kept stressing: "The derivative of e to the box, is e to the box times the derivative of box". This seems to work better for my kids than "e to the u" which is just one more alphabet letter. I physically draw "e" with a blank box in the exponent.

This seems to have been MUCH more successful than in the past. Of course now I say to myself, "self .... DUH! What were you thinking??"

The Running Theme of the Week

A coworker e-mailed all of us a copy of this article on "Grading by entitlement", and it seemed to coincide with several occurrences this week on the same theme.

As I was proctoring the TAKS exam, the students finished in time to sit and chat, and I was casually eavesdropping. One conversation went, "oh did you have so-and-so for a math teacher? Well, on a test if you just call her over and say you don't understand a question, she'll basically work it all for you. That's how I passed the class."

As I was helping tutor my precalculus kids on Thursday for their conics exam on Friday, a student was there for help for the first time in this unit. ... A student who hasn't done any of the conics homework ... A student who was absent for one of the days and missed the treatment of ellipses. ... A student who has pulled this for all 4 of the 6 weeks and always thinks he can go for the "hail Mary pass" at the last week to "pass". ... A student who did this last 6 weeks and ended up with a 68% (not passing in Texas). So. During the tutoring (in which he basically wanted me to do all the work), he grumbled that I wouldn't even give him 2 extra points last 6 weeks even though he tried REALLY hard and came in every day for the last 2 weeks and passed the exams and he should be rewarded for effort. AAAAARRRRRGGGHHH. Don't even get me started on how I was berating him on Thursday and how he still didn't get it and how he still doesn't think he needs to do the hard work to understand the concepts.

As my algebra 1 students were taking a test on solving systems of inequalities, one student early in the test put his head down. I walked over to him and gently but firmly told him not to do that if he wasn't finished. He proceeded to work a little but then his head was down again. I let him be. Another weak student ... same behavior. Towards the end when I wanted to pick up all the tests and teach a new topic, a 3rd student was still working hard, and so I put him outside the class to finish. The other 2 head-downers I went to pick up their tests asking, "are you done?" Their response was, "I don't know what to do (on the test)." I believe they were wanting me to come and guide them through the process. I said, "okay, then you're done. Turn it in." I picked up the tests. Then they saw I was putting the 3rd student in the hall to finish. The 1st head-downer then piped up with, "I want to continue." I said, "but you said you didn't know what you're doing." He said, "I want to try." Okay, so I put them in the hall. At the end, the I-don't-know-what-I'm-doing kid turns in a completely finished test (whereas before 2 out of the 6 questions were finished). Hmph.

Oh my. Okay, but I want to end with a more uplifting article another teacher passed around on learning called, "Try and Fail".

Teaching Conics

I'm loving this unit more and more each year. It's a chance for the students to practice their ever-waning algebra skills (hello completing the square, I love you). It's also a chance to do some hands-on stuff. AND it's a chance to see some cool applications.

One year (not this one, because ... well, just because) I saw an application that you could build a pool table in the shape of an ellipse, and then if a ball was at one focus point and you hit it, then in an ideal world, it would pass through the other focus point. I then just had to try it. I had the kids construct an ellipse (lots of string and a large piece of paper. Then I had my patient/loving husband carve this out of some wood and hollow out the inside to be the "pool table" in the shape of the ellipse. I brought it to class and we recreated where the foci were and we tested it out. It worked most of the time and was cool.

This year for ellipses, I just had them create ellipses on paper with a partner and a loop of string and 2 sharp pencils held down for the foci. Then they took their notes on this creation. Every student had 2 ellipses, one on each side of the paper, one with vertical foci, and one with horizontal. It worked well.

I'm having them do a project of searching for practical uses of each conic (circles, parabolas, ellipses, and hyperbolas). Let's see what they wow me with.

The Worst Part of Turning in Grades

Grades were due at 2pm on Monday. The previous Thursday night I was driving out of town to go to a math conference and would be out Friday. I warned the kids and told them that the LATEST they could turn in grades would be 4:16 on Thursday afternoon. I made a joke of it to hammer it home.

"Do not run after my car waving your homework at 4:30"
"Do not slip your work under my door after 4:16 and expect me to get it"
"Do not secretly slip it in my mailbox"
"Do not come to my house, please, this weekend to turn things in."

Mostly it went okay. But then there's always the special cases that I make allowances for without telling anyone else. One student's dad had recently died and she was having a rough time of it. She was also out the end of last week for FFA. She talked to me before Thursday, and I said that she could turn in late work AND work that was way past acceptable to turn in.

One student is struggling socially and familially and scholastically and has been in tears and has a hard time keeping it together. I went to school Monday and found some test corrections on my desk from him. I accepted them. He still didn't pass, but this brought his grade up.

Then there's the students that beg and plead and such after the fact and after they have slacked off ALL 6 weeks (which is 7 weeks this time, but who's counting).

Student #1. Monday morning he shows up. "I see my grades a 50%. Is there anything I can do?". Hmmmm, well, you turned in no work all 6 weeks, you barely did your late work. You didn't take advantage of tutoring or retests or test corrections. "No, there's nothing you can do. The time has passed." ... "Please, please, please, PLEASE. I'll do anything. I'll do quadruple work, I'll turn it in before 2pm (due time), I'll, I'll, I'll." .... "NO!". I mean, quite honestly, it's nothing to me if he turned in more late work. I had time to put it in. But I made the decision that that would not be beneficial to him. What would he have learned in that case. "Oh, I can always slack off and then be super polite and hang dog faced and teachers will let it slide at the end." Still it put a sour taste in my mouth and I felt horrible for the day, but ultimately I know it was the right decision.

Student #2 Basically the same story as student #1, except he came in AFTER school AFTER grades were turned in. "IS there ANYthing I can do? I need to stay eligible for band.". ... NO. This one had the extra added effect of super politeness, "yes ma'am, no ma'am, thank you ma'am", and the dipped head of sorrow. He wouldn't leave. He kept waiting around looking all glum waiting for me to change my mind. No, no, no. It may seem nicer to give in now, but it's not useful to you in the long run.

Argh. Hard decisions. I feel like "mean teacher", but I have to remember that I'm trying to do what's ultimately best for each kid.

Teaching Neatness

I just graded my algebra 1 tests over Solving Systems by Substitution. Oh my. The kids generally knew what they were doing, but their sloppiness got in their way. Some couldn't read their own handwriting and dropped negatives or made 3's into 13's and such. Some wavered all over the place and then misread their work that way.

Today we had a lesson on neatness and how part of your job as a "mathematician" is not just to get the answer but to communicate to others how the problem is done, so they can just follow along by reading your work and you don't have to be there to interpret the "doctor handwriting" as I call it. I implored (ordered) them to write each step and keep it all lined up going down the page and not all higgeldy-piggeldy every which way.

We practiced. We practiced some more. We refreshed our memory on fractions. We refreshed our memory on the fact that "3x/4" means the same thing as "3/4 x". We refreshed our distributing skills of "5 - 3(x - 2)" types of situations.

We discussed how to check our work (plug (x,y) back into BOTH equations).

"But why do I have to check both?"
"It could be right in one but your previous mistake makes it wrong in the other. Check both!"
"But I'm not going to be a mathematician. I'm going to be a doctor."
"Well, after surgery you don't just want to check ... 'did I leave the scalpel in the body? No? Good, sew him up' and meanwhile, you didn't check that you left the saw in the body."

Differentiating

Yesterday, for some strange reason, I actually had more time than my usual 2 seconds to prepare for classes, and I was giving a test in precalculus, and I have a handful of very smart kids that are polite and bored in class because it's going too slow for them.

This all adds up to a differentiated test on vectors and polars. The bulk of the class just had the standard test that asked them to:
find the angle between 2 vectors,
convert a polar point to a rectangular point and visa versa,
plot r = 3 sin theta + 2, etc.

Without saying anything to them, I handed the super-smart kids a different version of the test and kept an eye on them throughout the period to see their reaction. They did fine. It took them the whole allotted time (whereas usually they're done in less than 1/2 the time of the other students). They're aware of this fact, so periodically, one of them would look up as another student handed in his test. I wonder what was going through their minds.

Their questions were more of the variety of (and maybe I could have made them harder, but ...) :

give me 2 vectors that are perpendicular to each other and neither lies on the axes or has equal components,

plot 4 points on the polar plane that when connected form a rectangle, none are on the axes. then give me their coordinates, each in 2 ways.

vector u is <6,> find me a vector in the same direction that is 1 unit long (and we did NOT cover this in our short time with vectors). I did give them a hint by making them answer a "similar triangle" type question right before this one.

Anyway, I'm glad I could "unbore" them briefly, and hopefully I can do this again.

Polar Graphs

I'm so excited. I taught the graphing r= 4 sin(theta), r = 2 cos(theta) + 3 and such today, and I think THIS method will stick more successfully than what I've done in the past.

Two years ago at the NCTM Atlanta Conference, a teacher from North Dakota shared her strategy, and it made so much sense, and this year I adapted it and tried it.

I made up a packet where I have 12 such graphs mapped out on a rectangular coordinate system. I don't even label which ones they are. Right next to these graphs are blank polar coordinate systems. The tick marks (or angle marks) on each are divided the same (into pi/6). This ND teacher stressed to make the connection between "y = f(x)" and "r = f(theta)" and link x to theta and y to r and to keep mentioning it. Then you transfer each point from (x,y) to (r,theta) accordingly, and voila! You have your graph.

On the front page I had 3 similar ones, and after they/we graphed all three, then we refreshed our memory on what the equations were. Then we discussed what the connection was between "amplitude BIGGER/smaller than vertical shift" was, etc.

We got through 4 in class, and they have the rest for homework. I'm thinking it will work, because even I can now remember what the graphs should look like by doing such an analysis (whereas before, I had to refresh my memory each year).

Scaffolding & Storing Teaching Materials

As I was searching for "teaching polar coordinates" materials from an NCTM workshop I attended a while ago, I came across a packet that described scaffolding. I liked their 4 tier process they layed out:

1. I teach, you watch
2. I teach, you help
3. You do, I help
4. You do, I watch

I got me thinking about how I've been teaching lately, and about how different uses of these 4 steps work for different populations of students. I think lately my philosophy is more "2", then practice based on "4" with help if I see they need it. I teach preAP and AP classes that are supposed to be for students willing to try things on their own, but realistically has a whole range of abilities. I internally balk at running through all 4 steps because it makes me think of them learning math by seeing, "oh, this is THE method/way of doing things. I will memorize this technique and parrot it back when tested".

I want them to think for themselves. But then another part of me says, "well, they have to learn the basic skills first, and THEN you can throw in some harder thinking problems". And then a 3rd part of me is inundated with comments of "hard homework" and "you didn't teach us how to do THOSE types of problems". I, apparently, need to have more time to think through each lesson and map out my strategy of presenting concepts then mixing the types of problems effectively .... maybe with some warning about various problems and hints (?) and admonitions to actually put forth some effort on the more challenging ones.

Anyhow. Then that got me to thinking about how I store my teaching materials, and how my plan has evolved since I started teaching. I use those large plastic tubs (with tons of hanging files to store papers). I have 4 tubs currently: one for precalculus, 2 for calculus, and 1 I just started for algebra 1.

My hanging file folders used to be: "chapter 1", "chapter 2", .... because I shortsightedly thought I'd ALWAYS be teaching out of the same book and the same school and the same topic. Then I think I moved to large groups of topics in each hanging folder. I'd have to paw through all the papers each year when the time came to teach concepts. Then, for some reason, I moved to "first 6 weeks", "2nd 6 weeks", ... (what was I thinking). Maybe I was a masochist or liked to take tons of time to sift through the whole pile every time I had to teach something.

I finally wised up (at least it's working MUCH better for me) and added manila folders inside each large hanging folder, and the manila folders have concept titles: "vectors", "triangle area", "graphing lines". I've also used large sticky notes attached to lessons to make my reflections about how a topic went after I taught it the current time and possible suggestions for the future teaching of it. MUCH more convenient and time-saving for me.

Fresh Starts

One thing I love about teaching high school is that you're forced to start every day fresh no matter what happened yesterday with the kids. Last semester I had a rash of cheating and some student crying and bad behavior and the usual drama of day-to-day high school life. Teachers usually don't have the luxury of banning the students from our vision/life/class, so we have to make the best of the situation.

This is good because it forces me to see beyond black and white to the gray: "cheating student = bad person" vs. "cheating student = bad decision and potentially a person that has other great qualities".

Today was another such day. I have a student in BC Calculus whom I had (ooh pompous-sounding "whom") in precalculus last year. He's a football player, smart as a whip, lazy as a cuss, funny as all get out. He struggled 2nd 6 weeks because of various life things and football things and laziness things, and didn't pass, but has since made up some grades and keeps coming to class and plugging away. Anyway, this morning (he's my aid in another class ... so that I can force him to spend 1.5 hours on his calculus homework) he looked to be in a foul mood, and he was texting and I was "put away your cell phone" in a grumpy voice. We didn't talk much the rest of the period as I was teaching vectors in precalculus.

Later on in calculus he comes in and asks how my day had been so far, and I mentioned that it was not good at all, and he commiserated with how crappy his day was and recalled this morning and how he almost lost it with me because of the "cell phone incident", but he thought better of it. Anyway, we had a good "grown up" discussion about horrible days and various other things.

These are the fun parts of teaching.

Mish Mash

As someone else has probably said before, when your mind is consumed with how bad things are at work (crazy decisions from "above", bullying of new teachers called "mentoring", being tested ad infinitum, ...), you have less to give your students (reflecting about the lesson, making sure in a positive way every kid is doing okay, developing new ways to teach concepts). This is the position I find myself in.

It's a never-ending loop in my mind about what I'd really like to say to so and so, what should be done in regards to being a successful math department, how you could make kids succeed. Copy room conversations with other math teachers revolve around the nightmare of this year and what/where they/we will do/be next year. Already 2 teachers have quit midyear. More are planning to leave at the end of the year. I'm sad and frustrated.

On a positive note, I've read some quotes lately that resonate with me, that I find myself referring to repeatedly:

When dealing with yourself, use your head.
When dealing with others, use your heart.

People first, paper second.

Great people talk about ideas. Average people talk about events. Small people talk about others.

And from my phenomenal teacher friend from the northeast when dealing with bad situations: Here is where you grow a little.

Accumulation Functions (Calculus)

I'm so excited. I think I found a more effective way (for me) to explain a certain part of Accumulation Functions in calculus. These are functions defined by

f(x) = integral (from some number to x) of r(t) dt

where r(t) is a graph. The graph can be (say) from -4 to 8 and the lower bound of f(x) could be 1, so: f(x) = integral (from 1 to x) r(t) dt.

Anyway, if r(t) is ABOVE the x-axis to the right of 1, then f(x) is accumulating "things" and getting larger, and if r(t) is BELOW the x-axis to the right of 1, then f(x) is losing "things" and getting smaller.

Well, everything is "reversed" in this example if you pick an x value to the left of 1. Say, f(-2) = integral (from 1 to -2) r(t) dt. Then if the graph is ABOVE the x-axis, between -2 and 1, then this defined f(x) is getting smaller.

This always confused the kids, and I hadn't a effective way to explain it. This year I tried: Suppose you took a movie of how f(x) is changing from start to finish on the SHOWN graph (regardless of the lower bound of your integral), so in this case, the movie would run from -4 to 8.

Now if f(x) = integral (from 1 to x) r(t) dt, you start this movie at "1" and show it either forward (for x>1) or backward (for x<1) and you see what is happening to f(x). This seemed to make sense to them, since ABOVE the graph r(t) means you're accumulating, and so if you show the movie "backwards", then you're doing the opposite.

Anyway, it looks kind of confusing written out here, but it was a small joy of my day to see their looks of comprehension.

I asked for it...

It was good to be back with the students today. I ignored the administrative silliness and just concentrated on the kids. My new mantra this semester will be asking myself, "what's best for my students at this time?"

For example, today what was best for my students was to not go to a department meeting that was slated for 30 minutes, but that I heard lasted (not surprisingly as it always does) 1 hour. It was also a meeting that sounded not purposeful. It was also a meeting scheduled after we'd proposedly spent 3 hours in a math meeting the day before during our "work day". Anyway. Today it was best for my kids (and my sanity) to skip the meeting during my one planning period and to concentrate on thinking through the best way to teach the next 2 lessons of this day.

One funny thing happened in class. I was going over calculus problems with my kids, and I was modeling the stream-of-thought way I work through the problem and how I ask myself questions about what is needed or what is to be done at each step.

I said to my students, "you could do this. Just pretend you have a little Ms. Cookie on your shoulders asking you these questions while you're working through a problem." And I put my thumb and forefinger together to mimic a small me and put my fingers near my shoulder. Then out of the corner of my eye I see one funny kid swat his shoulder to get rid of the nuisance.

Happy New Year

I have no idea how reasonable an idea this is, but what the heck. I was looking through my "teaching closet" at home and realized I had all these teaching books that are just sitting there gathering dust. I culled through the ones I wanted to keep and found many I'd be willing to share with someone who may get more use out of them.

I'll mail any of them to you on a first come first serve basis on the following condition. You will send a check to me made out to the charity of your choice for whatever amount you think is reasonable, and I'll pass the check along when I get it and mail you the book. If you're interested, please send me e-mail at math_mambo@yahoo.com indicating which book(s) you want. We can continue the conversation there. I also have favorite charities in town I'm willing to suggest (food banks and "safe places" and such).

Here's the list in "pulled out of the closet" order:

"The Passionate Teacher", Robert L. Fried
"Ordinary Children, Extraordinary Teachers", Marva Collins
"Among SchoolChildren", Tracy Kidder
"Small Victories", Samuel G. Freedman
"Escalante", Jay Mathews
"Why I Teach", Esther Wright
"36 Children", Herbert Kohl
"The Classroom Crucible", Edward Pauly
"Mentors, Masters and Mrs. MacGregor", Jane Bluestein
"Hot Tips for Teachers", Harrison and Spuler
"Teaching and the Art of Successful Classroom Management", Harvey Kraut
"Nothing's Impossible", Lorraine Monroe
"Growing Minds", Herbert Kohl
"In Code", Sarah Flannery
"Ed School Follies", Rita Kramer
"Positive Discipline: A Teacher's A-Z Guide", Nelsen, Duffy, Escobar, Ortolano, Owen-Sohocki
"Improving Schools from Within", Roland Barth
"Positive Discipline in the Classroom", Nelsen, Lott, Glenn
"Teaching Matters", Whitaker & Whitaker
"The Teacher's Almanac", Patricia Woodward

Holy Cow! What a book hog.

Winter of Discontent

Phew, a few days into the holidays, and I think I've finally caught up on sleep. It's lovely to nap when you want and to sleep past 5am. My mind keeps returning to how down and sour and stressed out I've been for this past semester. Most of it comes from policies "from up above" that are not sitting well with me.

For example, we have a huge tardy issue at school. They've tried various things with no great success, and "they" have to get the numbers lower. What do they do now? Let's ring the 6 minute warning bell as usual, but then let's start school and ring the "late" bell 1 to 1.5 minutes AFTER it should ring. Voila! Number of tardies has magically decreased (for now). And in response to teachers' indignation and concern about this bad precedent? "Teachers should trust the administration".

Another example, not-too-newish teacher is at her wits end with disruptive students and goes to administration for some extra suggestions on what to do. Response? Let's put the teacher on a "growth plan", which I think is akin to warning the teacher about possible teacher consequences, ultimately.

Another (biggest) example, let's make all the math teachers implement a disruptive TAKS remediation/pre-emptive teach-to-the-test strategy where students practice the 1st part of class and the last part of class and get taught the regular class material in between. Let's not test this process out for bugs or to see if the timing is appropriate. Let's not be open to any suggestions.

Arghhhh. In the past I've strongly believed in the public school system since it seemed most democratic (sorry I know that sounds judgmental about private and charter schools). But lately I can see the draw to teach at non public schools. I'm wondering if policies are implemented more sanely there and administration and teachers are not as badgered by the NCLB mandates.

Anyway, bla bla bla. Grumperina must go out now and spread her cheer on the human race during this last-minute-shopping time of year.

Finals Week

Tuesday we'll be into the 2nd day of finals. That's one of the (many) things to love about Texas. We consistently get 2 weeks off for Christmas break. Thursday is our last day with the kids, and Friday is just a minimal "check grade print out" sort of morning.

This year, I did something different for my precalculus final. In the past, at this school, it's been all multiple choice. Mostly because that's what the tradition had been and also because we are in such a rush to finish grades and turn them in. For example, our last test ends on Thursday at 1:10, and ALL grades are due by 3pm that day. Whew. Also, the finals are worth 25% of the semester grade, for however THAT fits into the equation.

Anyway, it never sat right with me, this multiple choice test, and this year, I made it a "fill in the box with your answer and show your work on a separate sheet" sort of test. This way, the kids are not overtly guessing at answers, and I can see more of what they know (or don't know as the case may be). I guess a time comsuming part of tests in general is hunting through all their work to find their answer, and even if they box it on regular tests, the boxes are all over the place. On my final, I placed the boxes, and there were only answers to dig through, and the grading went very fast. I finished up one set (33 tests) while I was monitoring another final ... and by monitoring I mean I was walking around and looking up every 10 seconds and answering questions and such.

Anyway, I'm REALLY looking forward to this break. It's been a stressful semester, and I've had to work with people I don't respect and whose focus is the ever-dreaded TAKS test at the expense of lots of other important issues. It's so bad, I'm wavering about coming back next year. Stress, stress, stress. Come ONNNNNNNNNNN holidays.

New Things On the Radar

It's been a busy year, and I don't know if I've mentioned yet (to everyone I know and strangers on the street) that I have 4 preps this year. So maybe it doesn't seem too surprising that unlike the past, I haven't found the time to make new seating charts and change the students' seats every 6-8 weeks or so. Finally, they were getting SO chatty and comfortable with each other, something needed to be done.

I made little slips of paper that I handed to them as they walked into class, and I made them choose their seats. These slips basically said:

You are (mostly) in charge of your seating destiny.
Pick a NEW seat by following these rules:

1. Sit in a new section of the classroom.
2. You may NOT sit by anyone you’ve sat by before (front/back/side).
3. Find a seat that will allow you to learn effectively.
4. Be flexible and willing to move (a wee bit) if doing so allows someone else to satisfy these conditions.

Good Luck, and Go to it.

I was nicely surprised, and only had to move a few seats after they were all settled. Some kids moved themselves after class started and they noticed their chosen seat wasn't effective. Anyway. Whew. That chore done for now.

The second thing I think I'll start trying has to do with implicit differentiation in calculus. ONE of the issues the kids have is incorrectly slapping down "dy/dx" in the wrong places and wrong times or forgetting it all together.

I noticed one student was always successful, and when I looked at her work more carefully, I noticed she did the following. When taking the derivative of an "x" terms she adjoined "dx/dx", and with the "y" terms, "dy/dx". Then she later canceled out the "dx/dx" term since it equals 1 and it's always multiplied. This way, she puts everything in the right place, and she doesn't forget that she has to use this "extra" appendage. I think this is the way I'll start teaching implicit differentiation now.

Spread the Warm Fuzzies

Last year a teacher started something at our school that I thought was so cool, I had to do it to. I've done it a few times, and this last time I got a great response from one of my students.

I've done it in advisory (we meet once a week for 30 minutes), but I've also used it when I've had an extra 5-10 minutes in math class. I ask my students to write a thank you note (on paper I provide, with markers I provide) to a teacher in school which I then put in the appropriate mailboxes. I mention that it shouldn't just be, "thank you". It should be genuine and indicate something they appreciate about the teacher, and it doesn't have to be "saga long".

Anyway. This last time I did it in advisory, a couple of students were balking. I tried to persuade them with, "it really feels good when you get an unexpected positive note or compliment or such from someone. Think about how you're making someone feel." One student finally, grudgingly wrote a note.

The next time in class, he talked to me about it and said (with a grin on his face) that that teacher had come up to him and said she was having a horrible day, and then she got his note and it cheered her up immensely.

Laws of Sine & Cosine

We've recently worked through those laws in precalculus. This is my 5th year teaching them at this school, and I think I've finally settled on a way to present the "ambiguous ... A.S.S. case" for the Law of Sines.

This I've done before and this year: I present a sheet that has 3 situations drawn in which I have a partially made ASS case but ask them to use their rulers to complete the triangle with the last "S". For example, for triangle ABC. I've drawn a long line for the "base" of the triangle representing side b. I've measured and drawn in angle A of 31 degrees and side c of 5 cm. B is at the top of the triangle, and they're to draw segment BC of length 3 cm to complete the triangle. I eventually get them to see that there are 2 triangles they could draw. Then we see how this plays out without drawing and why there are 2 triangles mathematically.

I also do this for the cases where there is one and zero triangles.

In the past, I used to make a big case of how you could tell there was a 2nd triangle by looking at the given information and if the 2nd "S" in ASS info given is longer or shorter than the 1st "S" then you make some decisions. Hmmm, made sense to me, and that's how I still do it myself, but the students weren't always successful.

This year, I just said: after you solve for your 1st angle, try for the 2nd option of that same angle and "see if it works" (you can add up to 180). That seemed to work for more students.

We also just did Law of Cosines. I had everyone create their own scalene, non right triangles and measure all sides and angles. Then we plugged into the formula to see why it may work (instead of just proving it to them or just showing them the formula). I always hesitate with this measuring thing because the numbers never work out EXACTLY. But I figure, it's a good opportunity to discuss human error and measurement tools and degrees of accuracy and such. I make a game out of it and make my own triangle and ask if they can beat my "closeness". Depending on the class, I was off by anywhere from 0.3 to 1.5 units when the 2 sides should have been equal. I blamed my aging eyes (cough cough).

Cheating

Ackh! It's a few weeks until the students have to start thinking about and studying for (cough cough) finals. For the past couple of class periods in precalculus I haven't really gone over new material since they were getting ready for a test on graphing all the trigonometric functions. Instead of not assigning homework, I decided to assign OLD topics and grade on accuracy instead of just completion like I normally do.

The first assignment was on function notation, and the current one is on function composition. I gave them the stern teacher face of "don't show up and say you don't know how to do it. Look at your old notes and look at the book examples and look at the web, but LOOK and recall and do something for yourself." Gee. My face is expressive.

Anyway. I was giving the graphing test on Friday, and the homework was due, but I didn't collect it until after the test. Some students finished early and were doing various things. Two friends started to look suspicious. I saw one pass back a paper to the other. Then I saw the other surreptitiously copying her paper. Crap. I walked over and quietly said, "do not do that! That is a zero for both of you. Do your own work.". I was so mad. First for her doing it, and equally for thinking that I'm so oblivious, that I can't notice what's going on.

I debated talking to both after class (I didn't). I had the whole speech prepared in my head about how once you lose someone's trust, it's very hard to gain it back. But I settled for the unhappy glances their way and the ignoring of them and a total change of my demeanor towards them after the test was over, and we were going over other concepts. I'd had a hard enough week, and I didn't want to deal with more stress.

November Slump

Sheesh. I forget this happens every year. I get super depressed and down on various kid behavior and teaching and the stress and all the whole bundle every year at about this time. I think I should make a cross-stitch sampler to remind myself that this will pass, and things do get better SOON.

It's been especially bad this year. We are mandated from above to have meetings up the wazoo that suck the life out of our planning periods. Then we are dictated to teach to the "exit exam" in a specified way and eat into our regular curriculum to do this. Then .... then .... then.

I think I have enough seniority to say "phhhphflt" and do what I think is best for the students: a type of don't-ask-don't-tell policy. My rationalization is that if questioned, I'll have a valid reason for why I think what I'm doing is the best for my kids.

On a positive note. My students are mostly great. Highlights:

One shared with me her way of remembering the sine and cosine graphs. She's Hispanic, and "sin" in Spanish means "without", so that's the one that is zero at zero. And "con" in Spanish means "with", so that's the one that has a value at zero.

A day or so ago we were doing the ambiguous case of "Law of Sines", and for the "no triangle case". I wanted to show them that in the A.S.S. case when they're first solving for the angle, the ratio of sides is 1._____. So I asked what happened on their calculators when they tried to solve sin x = 1._____. They said, "ERROR". Then I said, okay, close your eyes and picture the graph of y = sin x. My goal was to get them to see that the largest it could be was 1. So I asked, after they were thinking for a while on what the graph looks like, "what do you see?". One kid answered, "ERROR".

Anyway. Some of us got together tonight as a math department and went out for a drink after work and destressed and laughed and such. That will go a long way to making the rest of the 6 weeks livable.

The Power of Off-The-Cuff Words

Just recently, my memory was refreshed as to how seriously students take what we say. Things I just sort of blurt out because I think they're the right thing to say seem to make a difference.

For example, I have this incredibly smart and hard-working student in BC Calculus this year. Last year I had her for precalculus preAP. Back then she was talking about her math choices for the next year, and I mentioned that she should definitely take BC calculus as opposed to AB because she had such a great work ethic and cared about really understanding topics and such. I think I also said something to the effect that it would be a waste of her time to take AB which would not challenge her as much. I did truly believe this, but I didn't know how much weight this would carry. Well, several times this year as she's bemoaning the fact of how hard it is (in a good-natured I'll-still-muddle-through sort of way), she kept mentioning the fact that I made her take BC calculus. Hmph.

I have another student in AB Calculus. I also had him in precalculus preAP last year. He's an interesting, intense, strange, slow-working, self-stressing type of person. I really did not think he could manage the pace of calculus with how much he frets over EVERYTHING. He asked last year if he should take calculus, and I hate to discourage students/people, because, really, what do I know, maybe they will surprise themselves and me and if not, then the experience will be a learning one one way or another. So here he is in AB calculus this year. He is struggling and stressing and such all along. He came to me after school one day and said he wanted to drop out of the class. I said that well, the decision was his, but I think it would be a shame if he did because then if anything hard came up in the future then his first idea would be just to quit because it was too hard. I also said that he was smart enough to handle it, and personally I would stick it out. (inside, I think he can do it, and he just has to approach it in a different way, but I DEFINITELY know that if he dropped, he'd be WAY less stressed than he is now). Well, anyway, he decided to stay, and later wrote me a note thanking me for believing in him and in his ability.

I guess the point of all this is to err on the side of pushing the kids to do their best and to do things they don't think they could do or think possible. Maybe it will open up opportunities for them and make them think of themselves in ways they didn't in the past.

Foreign Exchange Students

At my school we seem to get a lot of foreign exchange students. Every year I've had at least one; always from Germany; and last year I had 3. Phew! This year I again have 2 girls from Germany, and lo and behold one boy from China.

Well, a while into the school year, I saw that he wasn't interacting with the other students, and they weren't talking to them. I don't think it was a rudeness thing on either part. My sense is, that the girls (and boy) from Germany in the past and present were so cute and approachable and maybe "looked more like them", that they naturally talked to them and became friends. That's my guess, and now that I put it down in words, it seems kind of wrong somehow (the situation). Why hasn't anyone struck up a friendship with this boy? Maybe I'm just misreading the situation, and it's only in my class that this phenomenon occurs.

Just yesterday I asked the boy from China (He's cute: when he first arrived, he said, "My American name is Eric" ... I finally asked him his real name, and he mentioned it, and I've been practicing it, so now I use it when I talk with him). Anyway, I asked him to talk to the class about how school differs in China from our school. Sheesh. That was a cultural wake-up call for my kids.

He said they start school at 7am, and have 4 classes until 12pm. Then they have a 2 hour lunch break, and at 2pm until 7pm, they have 4 more classes. Then they have 2 hours of studying. They have a month off in the spring for a sports festival. Every year (?) they take off one week for each of the following activities (?) working in the factories, farms, and army. The sports they play are tennis, running, ping-pong, and badmington (for some reason, that got a titter from my students). I guess I was surprised there was no soccer.

Anyway, hopefully, he's getting a chance to interact with students and such and not having a lonely existence of a day.

9th graders

I'm teaching one section of algebra 1 preAP this year, and I feel like I take extra care of them ... because they're "young uns", and they're learning the foundation for everything else that comes, and they're just so darned cute most of the times. With that said, there are still some "young uns" issues.

There's one whiney child that grates on my last nerve, and I have to work extra hard to not show it. Everything we do, her response is (insert whine), "I don't get it. I don't get anything." She needs to be hand-held through every step. It got to the point where she snapped back at me in class one day because I told her to get to work when she was chatting (because she JUST didn't get it and was waiting for me to do things as opposed to getting help from her more capable group mates).

I was super frustrated last week, so I sent e-mail to her other teachers to explain the situation and to see if it was just me and my class or if these issues came up elsewhere. Two other teachers responded and mimicked what I said about her neediness. They mentioned that she either has a capable friend sitting by her to help or the teacher literally DOES break things down into baby steps. This made me feel better and gave me an idea of copying extra "reteach" sheets for each topic so that she'll have extra practice to look at while the rest of the class is "zooming" along.

She was better in class this last time, but had to leave, so I didn't get to test out my extra-sheet idea yet.

I'm also dealing with a large portion of the class (4-6 out of 28) not turning in ANY homework. I think they're still used to middle school where you magically pass no matter what. I've made some calls home, so hopefully that will produce some late work this coming week.