Matrices!

Matrices! In Algebra! Whoa! As you may be able to tell, I was surprised and delighted to find we were learning about matrices in class today. Not baby matrix math either. We were adding, subtracting, and multiplying. Multiplying! Multiplying matrices! Matrices! We even covered the idea of a transformation matrix and used one to scale the points of a triangle. The few students I was able to sit down with today got to hear about how matrices are used a lot in computer graphics. A couple students asked about programming games for their calculators. When I said "you could download them, or you could make them yourself" they seemed amazed at the thought that a person could actually program a video game.

"How do you think video games are made? Of course people program them!" Incidentally, my college roommate, Matt*, recently left his game development job to start a game development company. Check out a demo of their new game here! I wish I could bring up youtube links in real life. Until then, I guess retroactively posting on my blog will have to do.

Despite how difficult I find matrix math to be (there are lots of additions, which I think is the hardest part of math), matrix multiplication is easy to algorithm-ize. Give students a recipe, and BAM, they get it. Initially I was surprised how after 2 examples, everyone I talked to was able to multiply 2x2 by 2x3 matrices and 2x3 by 3x3 matrices without error. Perhaps it was the straightforwardness of the work. Compare this to solving 2x+4=y for x, which can be done many ways and still confuses some students. Maybe there's a lot to be said for not having choices.

This partnership program has inspired me to write a math book. I imagine anyone that has had a similar experience has thought similarly. So, how do you write a math book that's fun, useful, and free?

Happy Halloween!

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When I got up this morning, I realized that I hadn't prepared a Halloween costume, and felt a little out-of-sorts.  Luckily, I was able to find my bear costume from two years ago and put that on before grabbing my camera and dashing out the door.  I was pretty excited to see all of the kids at Ypsi High dressed up in crazy costumes...

...but then I got there and realized maybe 10 people had Halloween spirit.  Nobody was wearing costumes!  Disappointed and a little embarrassed that I was the only person in the room with a fluffy tail, I proceeded to grade tests from the previous hours while our students took today's big test.  Unfortunately, this caused me to forget to get pictures of 3rd hour, but above you can catch a glimpse of 4th hour when they're being good.

Regarding the test, there weren't any big surprises.  Your usual set of questions about functions and your usual set of answers.  Grading is nice for a lot of reasons.  It's objective, so you can get fast at it and do it without much thought.  Mrs. Colwell always mentions it is a big help to her, so I like that.  Pretty much the only thing that isn't great about grading is when a student misses points that you know they could've gotten.  If only there were some way to help them improve... 

baby steps

Remember Bill Murray in What About Bob?

Today in class when I was helping students with a problem, I noticed everyone just wanted to rush to the answer.  They'd read the question.  They'd see the empty blank on the right side of the page.  Then they'd freak.  Dizzy spells.  Nausea.  Cold sweats.  Hot sweats.  Fever.  Blisters.  Difficulty breathing, difficulty swallowing, blurred vision, involuntary trembling, dead hands.   Numb lips.  Fingernail sensitivity.  Pelvic discomfort. 

Math anxiety isn't any different from Bob anxiety and solution for both is baby steps.  Instead of getting frustrated, panicked, or pelvis discomforted when trying to throw an answer in the blank, take a baby step.  What is the problem asking for?  Baby step.  What information does the problem give us?  Baby step.  What's the first step in getting from here to there?  Baby step.

Teaching someone to take baby steps is hard.  Even if you do, oftentimes  it will still only get a student halfway.  Still, I think this is a is a good model.  What's a teacher for, besides helping a student find the next baby step when they can't find it themselves?  

One class-wide baby step I've been happy about is the willingness to ask for help.  Of the students that need the most baby-stepping, some of the shy ones are now asking for help and some of the loud ones are quieting down and doing the same.  At the same time, some students are unfortunately stepping down a class- a few faces I'd grown accustomed to helping are no longer there.  

Not much I can do but take it one step at a time.

after a break

You probably didn't notice, but I didn't go in last week. Today I'm back, so you get a new blog post.

Grading quizzes today, the one salient observation that popped out of the student responses was that they still don't know what a Range is. What is the range of the function y=-7? Answer? -7. What did students put? Everything from 0, to all reals, to x is in [-7,7], to my favorite: -7>0. Maybe 5 out of the 40 or so tests I saw answered -7. Everyone probably already knows my attitude towards math vocabulary, but given that I had to double check what the domain and range were again today I couldn't help but mention this once more Let it be known, all students who got that wrong, I didn't feel good taking the red pen to your paper. I feel for ya.

In 4th hour, I had an opportunity for more one-on-one tutoring with students. One had turned in a blank quiz after about 5 seconds. We absconded to the far corner of the room where we went through some examples. After about 10 minutes, I revealed to him that our examples were really the quiz questions and that he had just gotten all of them right. "Sure is a shame you gave up so fast..." He thought this was pretty funny and I'm not sure if he spent the rest of class paying attention, but at least spent he wasn't causing trouble.

Not a whole lot else to report, so I'll shut up. Later!

A class divided

I just found this story about Jane Elliot's classroom discrimination experiment from 1968:

• A Class Divided
  

What an extraordinary documentary.  Watch it.  The whole thing.

Update:  I just finished watching it and read an interview of her from 2002.  It's sad to hear that her exercise is still as enlightening today as it was in 1968- but it's also inspiring to see that racism is a learned behavior and that you can "teach an old dog new tricks."  The comments from the children, both in the classroom and years later, and the comments from people from her later trials were poignant.  Watch it.  The whole thing.

Dividing both sides

Today was the first of two review days for Monday's test: students formed groups and worked in teams on the review sheet.  Tomorrow, the review sheet will be reviewed and the students will get a chance to check their answers.  I mentioned before that I like this format and today was no exception. 

In 3rd hour, one group of girls was flying through the review assignment.  This group also happened to contain the girl who was elected to homecoming court.  See kids?  Math skills make you popular =)

In 4th hour, I had two noteworthy interactions.  In the first, I had a chance to talk with a student who is failing the course.  Citing the fact that "no matter how hard I try, I'll still fail, so why even try?" she refused to work on the worksheet.  I wasn't able to work any miracles here, she didn't work on the worksheet all class, but at least we had a chance to talk about it.

The second interaction was with an international student that is clearly frustrated, but has a great attitude.  We only made it through the 3rd question on the review sheet (there are around 20, i think), but even so I think we had some great progress today.  As we worked through a problem, we discovered that he didn't "get" dividing.  If he'd reduced a problem to 2k/4=40, he'd stop.  We worked through a few more examples, and he seemed really pleased.  He even asked for more examples of this sort of problem, so he wouldn't forget!!  I scribbled a couple more examples out as the bell rang.  I apologize if the numbers work out to be really ugly- but I didn't have time to check!  How's that for pro-activeness, though?  

This same student also mentioned that nobody was at tutoring last week.  I don't know if he didn't wait around long enough, didn't go to the right room, or what happened, but I told him I'd mention this to the tutoring folks tonight at dinner.  I also gave him my email address in case he had other questions.  I don't know if he'll email at all, and I don't know if I've just opened up Pandora's box.  We'll see!

Hung-Hsi Wu

Last night I found the website of a professor at Berkeley who has been quite prolific in the area of Mathematics education.  I've only had a chance to read one of them (here), but I found it very timely and interesting.  I look forward to reading the rest.

I bring this up today, because Wu's comments on "order of operations" and other vocabulary like domain and range meshed perfectly with some of my observations thus far.  The first sentence:

One of the flaws of the school mathematics curriculum is that it wastes fruitless exercises in notation, definitions, and conventions, when it should be spending time on the mathematics of substance

Wu goes on to explain order of operations, the Please Excuse My Dear Aunt Sally mnemonic, and how all conventions like this are artificial.  My favorite example was

Evaluate 4+5*6/10

"Now, one never gets a computation of this type in real life, for several reasons.  In math the / sign (in the paper it's the divide sign with two dots, but i'm not going to the trouble of encoding it properly here) basically disappears after grade 7.  Once fractions are taught, you almost always see 6 * 1/10.  Moreover, if anyone wants you to compute that, he would certainly make sure that you do what he wants done, putting parentheses in the right spot.  The original problem is therefore a kind of Gotcha! parlor game designed to trap an unsuspecting person by phrasing it in terms of a set of unreasonably convoluted rules."

Or, at least I thought it was my favorite example until page 9, when I ran into "most beginning algebra texts still devote pages to teaching the concepts of the domain and range of relations which are not functions, and teachers are known to proudly drill their students on learning htese terms.  But these concepts are not used in any part of mathematics except for rare circumstances such as the discussion of correspondances in algebraic geometry.  There is not much to gain by spending valuable classroom time on this topic."  Our friends the domain and range return!  

The point of all this was mostly just to make a record of these math education papers.  Check back later to see if I've reviewed any more.  Ciao for now!

Experiment 1

Today was a strange day.  During the homework portion of 3rd hour, the class was silent.  Just dead silent.  Everyone was sitting, working, and quiet.  This unusual occurrence was creepy enough for me to remark "I don't want to ruin this, but why are you so quiet?" to the students sitting near me.  My hypothesis was that, like crickets, students communicate less as temperature drops.  As today was the coldest day of the year so far, it seemed plausible, but I was really shocked when one student said "cold!"  Weird. 

Anyway, on to serious stuff: the same student who remarked "cold!" had some questions about hyperbolas and inverse-square functions today (who doesn't?!) that I volunteered to help with.  After we talked about it for a bit and class proceeded, she threw her arms up "I get it now!"  When I told her that she now has a responsibility to help her friends get it, I don't think she thought I was being serious.  =)

During lunch, Mrs. Colwell and I discussed the plan for our experiment with 4th hour.  Here's what we decided on:

• let students self-select groups of 3-5 students
• students work in teams to solve problems on the board
• groups that finish early should disband and help other groups

Here is what happened:

• Students seemed excited to try something new
• Group formation was hectic (Mrs. Colwell predicted this)
• Some students didn't want to join any groups (Mrs. Colwell predicted this)
• Eventually the room settled down (maybe after 10 minutes) and groups were hard at work
• Mrs. Colwell and I circulated the room answering questions, and adopted lone students into the Mrs. Colwell, Eric, etc group.
• Only one group started throwing things at another group
• A couple groups finished early and helped out their peers!!

Here are some student responses to "what did you think of today's class?":

• "Thanks for doing this.  I actually learned today"
• "I don't know... maybe I like regular class more?"
• "This helped a lot, thanks."
• "It was loud at the beginning but I think it was good.  Are we going to do this more?  I liked it."
• "I learned a lot today."
• "This was good."
• "It was helpful!"

 Here is my analysis of their work:

• Group 1:  Drew good graphs of 1/x and 2/x^2.  Got domain and range for both
• Group 2: Good graphs, didn't finish
• Group 3: Good graphs, eventually
• Group 4: good graphs, figured out what happens to k/x when k<0>
• Group 5: Everything good except for the names of the functions
• Group 6: Good graphs, figured out how things change when k<0
• Group 7: One good graph, the only group to find the function names we were looking for (rectangular hyperbola and inverse square function)
• Group 8: Good graphs, couple of domain and range questions

Some commonalities: Pretty much everyone was able to draw good graphs and called 1/x an inverse relation when we were looking for "rectangular hyperbola."  I think most people are confused about asymptotes.  The answers that were most commonly wrong were those that dealt with vocabulary.  The answers that were the most commonly right were those that involved playing with a function.  Incidentally, I happened upon this article about playing in math and science today.  

One thing I really liked about this format was the fact that I got to work with more students one-on-one, but with others looking over our shoulder.  Working in this way, it's easier to identify the specific concepts a particular student is struggling with and devise a problem approach that helps address it- and those with similar questions get automatic help.  When one of the backseat drivers has a question, they chime in and it helps guide the discussion so everyone's questions get answered.  One large group that seemed very skeptical of this format at the beginning was able to take advantage of the over-the-shoulder learning.  I helped them draw a graph of 1/x and mentioned "boy, this looks different from the one you guys drew.  why's that?  How would it look if it was -1 on top?  What about if we square the denominator?  Draw those, and I'll be back."  When I came back and we talked some more about Domains, Ranges, and Asymptotes, this once-skeptical group seemed to really like this.  I should note that 10 minutes into the activity, their paper was blank and when I asked how they were doing they said "we give up."  At the end, the "Thanks for doing this.  I really learned today" comment came from this group. 

Two other students today also had problems creating graphs of 1/x.  Sitting down and talking with them, it became apparent that the concept of "plugging in a value" for x was completely foreign.  Well, this is a pretty crucial thing in algebra! While we didn't get to the concepts of asymptotes, these students did get some extra personal attention with the functions and ended up creating some decent graphs.  In situations like these, I don't care if we don't cover all the material: going back and building the foundation is more important.  I'm glad we got the chance to do a little of that today.  Also, I think the ability to find these gaps in foundation are good: I wouldn't be able to do this from looking at the quizzes, but it was easy once we started talking.  Noticing that dialogue was so helpful, I really pushed tutoring on a lot of students today, and they seemed excited to go.

While from the above two paragraphs you might get the impression that I think this was the best thing ever, there are a bunch of areas for improvement.  One problem was having the two kids with the worst attitudes in the same group.  They were off-task and loud, but fortunately they were the exception.  On another day (perhaps a warmer one) this whole class has loud and rowdy potential.  Choosing groups beforehand will help with this, and also help us pair over-achievers with those that need more attention.  Also, I acknowledge that not all students learn better in this format.  At least one girl mentioned "I think I like regular class better," so this format certainly can't supplant traditional lectures.  The thing I was most happy about was the fact that almost everyone seemed to be engaged with the material and that there was a good amount of positive feedback.

As Mrs. Colwell mentioned, the real metric of success will be on the test!  I hope they perform well on their upcoming quizzes!

PS:  Guess who got her books and backpack back, and guess who had them?  BOOYAH!

Grading, talking, and experimenting

For a change of pace today, instead of doing the warm-up at the beginning of class, I graded quizzes while Mrs. Colwell gave the lecture. I made sure the students knew I'd be happy to answer questions during class, but only a couple came over to chat. One conversation was about math, and in the other I had a trickier problem. Tricky problem first:

• student: "someone stole my backpack"
• me: "uhh... wow. that sucks. you should get it back."
• s: "but I don't know who took it! I had my friend watch it on Monday, but she left it there and when I came back it was gone. It had four books. I didn't even tell my mom. She would be so mad"
• me: "uhh... you have no leads as to whom might have it?"
• student: "well... I think I know. This boy on tuesday said 'where is your backpack?' and I was like 'how did you know it was missing' and he was with my friend who I left it with on Monday."
• me: "so it sounds like he has it."
• s: "but I don't want to accuse him of having it."
• me: "Actually, a couple of weeks ago, my bike was stolen. It made me really mad, because it is my primary mode of transportation. You know who took it? My friend that I work with. I felt really bad accusing him, but I just knew it was him. Still... I didn't say anything and I called the police first. Sure enough, it was him and the police made a trip out to my place for no reason. You should ask this boy if he has your bag."
• s: "but what if he doesn't? he will be mad"
• me: "ask him if he can help you find it."
• s: "oooooh. that may work."
  

Boy was I glad that that advice seemed to be sufficient. What do you tell someone who has lost everything they use on a day-to-day basis? I felt almost as helpless as she must. Needless to say, the math question was relatively simple compared to that.

Grading quizzes today, it was good to see a lot of high scores for my first class. Our genius friend mentioned before didn't perform as well as I think he can, but we did get a chance to talk about one of my favorite "math" games: Nim. In Nim, there are three piles of stones. On your turn, you can take any number of stones from one pile. Your opponent then does the same. Whomever takes the last stone wins. In a standard game, the piles begin with 3, 5, and 7 stones each.

The great thing about Nim is how easy it is to pick up and how quickly patterns start to emerge. For example, after you've played a few games, you realize that if, after you move, there are two piles of the same size, you can always win. Another pattern is that if there are piles of 1, 2, and 3 stones you can always win. Generalizing, there is a really neat trick you can use to figure out if you can force a win, but we'll save that for later. I showed a few students this game when they had a few spare moments and told them to play it a lot, hinting that "like tic-tac-toe, you should be able to figure out who is going to win." The students seemed really interested and excited, so we'll see if they've figured any tricks out later.

Mrs. Colwell really seemed to appreciate the grading, and I can understand. With so much homework (daily), so many quizzes (weekly or so), and so many tests (every few weeks) and so many students (most of these classes are around 30 students each), there's a lot of grading to do. Mrs. Colwell says sometimes she grades until 9 before she has a chance to do anything else. Talk about long hours!! Grading today was actually pretty fun. After stumbling over the first few, it became easier to explain the trick for a given problem to a student right on the test. Hopefully they are able to decipher my chicken-scratch handwriting and it's useful. It was also good to see that a number of students were able to get the really hard problems and were only getting tripped up by a silly reading mistake. I always thought that was the hardest part of math: translating sentences into symbols. Some students seem to have the same problem, while other continue to struggle with symbol manipulation.

During lunch today, Mrs. Colwell and I sat down and discussed our goals for the semester. Mine are pretty straightforward: be as helpful to Mrs. Colwell and the students as I can, while trying new things, without causing trouble. Mrs. Colwell's included:

• increasing student achievement using her teaching fellow
• maximizing utility of the extra teaching fellow body
• having some interesting presentation (~10 minutes) related to lecture and including engineering applications

While I agree that this program (the UM-YPSI partnership) should do a good job advertising engineering to potential engineers, my focus is less on selling engineering as a career choice as showing the beauty of math. This may seem antithetical, as I am an engineer and it is only natural to validate one's career choice by pushing other towards it, but I strongly believe that a stable math education is accessible to all (including those not engineering-inclined), should be enjoyable, and is fundamentally more important than knowing what it's good for.

In the area of testing out new things, tomorrow Mrs. Colwell has given me the go-ahead to try out a new format. In this first experiment, we're going to see if it's possible to keep everyone interested, teach a subject, and do all the normal stuff we do (collect homework, grade homework, pass back homework, give new homework assignment, etc) all at the same time. In tomorrow's experiment, the plan is to go something like this:

• Split the students into small groups (3-6 ppl/group)
• Put some problems on the board for students to work on in teams
• While doing all this, be collecting/grading/passing back homework (I really do like parallel processing. In case you don't believe me, check this out.)
• Collect a sheet of answers from each group, possibly for use as extra credit.

My hope in the future is to randomly select students from groups to answer questions, with the extra credit for the entire group depending on a correct answer (giving an incentive to make sure everyone understands the material), but tomorrow's experiment is just to probe the waters to see what we have to work with, and if anything sticks out as great or horrible. As I may have mentioned previously, I have some big hopes for group work with the following rationale:

• Bored over-achievers will have an opportunity to share their knowledge
• Attention-seeking students will have more people to ask questions of
• Questions common to all groups will pop up, allowing these to be addressed with the whole class
• Mrs. Colwell and I will have an opportunity to devote more personal attention to those that request it
• The format is superficially less quiz-like or test-like, while helping promote quiz taking strategies in a different way:
• What does this problem mean?
• How do other people solve it?
• What ways work best for me? etc.

One concern that Mrs. Colwell expressed is that time may be an issue: what if we're unable to cover the material we want to cover with these new activities? My goal is to not only not fall behind, but to eventually work up to covering more than one chapter in a day. I suspect some will find this unrealistic, but I am confident we at least will be able to keep the regular pace. Lofty goals are good ones. That, or the motivational posters that tile the Ypsi high walls are lying.

Spiderman and Fermi

Whoa- this semester is flying by.  I know it's only the 3rd week of class, but at this rate the semester will be done before the weekend is through.  First off, I want to acknowledge Carol Cramer's helpful comments, thanks for reading and thanks for the great suggestions.  Yesterday and today, Mrs. Colwell and I began talking about how we wish we could fit in some different activities, but how we're already pressed for time as it is.  I think we're both excited to try out some new ideas in the coming weeks, so stay tuned to see what they turn out to be and how they go.

When I came into class yesterday, I was a little surprised to find out that we were having a test, as I expected that to occur on Tuesday.  Unable to help work things out, I started thinking about possible material to present during future "warm ups."  Also, I offered to help grade exams or homework, and boy did Mrs. Colwell's eyes light up!  It was like a kid on Christmas morning.  During the warm-ups at the beginning of class, Mrs. Colwell usually grades homework while the students occupy themselves with some practice problems, or listen to me ramble on the days I show up.  Mrs. Colwell had expressed interest in hearing stories of mathematicians before, and Thursdays lesson was on variation (not the statistical one, but how a function varies wrt some part.  e.g. f(x)=3ax^2 varies linearly with a, but by the square of x).  I figured it would be a good time to pull out some cool Fermi problems.

One of the new postdocs in my lab (Aaron Santos) is about to publish a great book on Fermi calculations that I had the good fortune to proof-read.  Blatantly ripping off one of his problems for the benefit of the students, today I gave them a short history of Fermi before we dove into calculating how much Spiderman would have to eat to shoot all of that web.  For those of you interested in the answer, both classes came up with 56 lbs per day.  

Both classes seemed to enjoy this storytime/problem but the after-lunch class was way more engaged, as is their full-bellied wont.  Everyone was shouting suggestions for the length of a spiderweb-shot, the number of shots per day, estimates at shot widths, and that Batman was better than Spiderman.  It was a lot of fun, and I agree that Batman is better than Spiderman- he's just a justice-loving hard-working billionaire whereas Spiderman was randomly given superpowers by a radioactive spider.  Gotta root for the hard worker.  The thing I found interesting was that this sort of silly example was actually as engaging as it was.  I always felt that such pandering to presumed interests ran the risk of insulting the student's intelligence.

Anyway, two really cool things happened today.  In my first class, Scott came bounding up to me as class began with the answer to my brainteaser from last week.  He found that the next value in this sequence:  1, 11, 21, 1211, 111221... is 312211!  I feel like I had bet everyone $5 that they couldn't find the answer, so now I owe Scott $5.  Good thing I didn't remember in class, because I didn't even bring enough money for lunch, and had to borrow some from Mrs. Colwell (let the record show I owe her $4).  After Scott impressed me with this, he proceeded to solve a Rubik's cube to the amazement of the back row (i'm serious).  I thought "holy crap- this kid is amazing!  Guess I don't need to worry about him anymore," as he beamed confidence across the room.  After my warm-up and some lecture by Mrs. Colwell, I wandered the room looking to answer questions as the kids solved this problem:

• y(x) varies by x to the 6th power
• when x=2, y=100
• find y when x=3

When Scott raised his hand, I was expecting to see one of the homework problems, him having blown through the in-class assignment.  Instead, he was still struggling with the above problem.  We worked through it twice before the "ahhhhh... I get it now" moment.  Going back to a comment I made in a previous post, I think Scott is a great example of the students in this class: they're very very smart but don't always know how to use the tools they have.  Once we show them, they're able to rock!  This is partially why I've been emphasizing group learning as a new activity- the students will be able to help each other reach the "aha!" moment while reinforcing what they've learned.  Students of this caliber may thrive with such activities (I have no data to support this). 

The second cool thing that happened today was the after-lunch class's interest in their grades.  After getting their test scores back, Mrs. Colwell was rushed by a crowd desiring to know how they're doing.  For many of them, it wasn't very good news.  But they care!  And, just like the class before lunch, they have the raw ability.  I'm hoping that our group-learning testing in the weeks to come is well-received by the after-lunch crowd.  I think their extra energy may facilitate discussions, I think the discussions may take away some of the extra energy, and I think any deviations from the traditional lecture style will be welcome for these students.  We shall see.

Also, for the record, I did not get a perfect score on the test yesterday.   

Rowdiness

Today I noticed a large difference in the level of attentiveness between the two classes.  3rd hour was great.  We got through the quiz review, everyone seemed happy with how things were going, and there was time for a couple neat examples.  After lunch, 4th hour was a bunch of hooligans!  They were loud, the exact same material didn't seem to stick, and we were rushed at the end of class to fit everything in.  It was like night and day.

Hopefully the quiz goes well tomorrow and we can make sure everyone is on the same page next week!

Week 2!

Over the weekend I received a concerned email from Mrs. Colwell regarding my last post.  Most notably, she highlighted (correctly) the facts that students do need to know vocabulary to talk about math and that showing work is indeed very important.  After clarifying that I didn't mean to disparage these portions of class, but simply identified with the students we began this week anew.  

Today was the day after a quiz, so instead of giving a special talk I helped answer questions about mistakes on the quiz as well as homework problems and other examples.  There were a lot of non-passing grades on the quiz, but when I spoke with the students one-on-one we were able to clarify their mistakes.  One student said "You're my new best friend.  Always able to help when I have a question."  Most of the mistakes on the test boiled down to:

• Not knowing how to interpret the question
• Calculator error (or rather, an error from its operator)
• Not showing work

Talking through the problems helped elucidate most of the interpretation questions, and I was really surprised about how many students sounded excited to go to tutoring. 

On Thursday we're learning about sequences, so I think I'm going to give a small example at the beginning of class that covers both sequences and recursion (Friday's lecture) and proves that there are no vampires.  Hope they like it!

Showing work and memorizing vocab = BLECH

Today was day number 2.  Like yesterday, I began each class with a short lecture and spent the second half of class answering questions.  Mrs. Colwell mentioned she likes this system, as she gets to take care of beginning-of-class details while the students are getting work done.  Parallel processing at its finest!

The portion I introduced at the beginning was a practice quiz in preparation for the real thing on Monday.  This went pretty smoothly with 2 exceptions:  1)  I realized I don't really care what "domain" and "range" are.  2) I was caught in a lie.  When the students had questions about domain and range, I answered them as best I could without explicitly saying "I hate this too," but I did mention that "I'm in the 23rd grade, and even I don't always remember this."  The students jumped on this:  

students: "You've been in college for 11 years?!  How old are you?!" 

me:"umm... well I went to undergrad for 5 and I'm in my 4th year of grad school."

them: "that's 9"

me: "uh yeah, I worked for two years at Dow" (liar!!)

them: "how old are you?"

me: "26."

them:  "you don't look 26."  "wait, did you skip a grade?  how old were you when you started school..."

Pandemonium.

So... I learned that their mental math is pretty quick.  I'm thinking of applying a trick I had heard an economics professor had used.  On the first day of class you tell them that every class will have one lie.  Whoever finds the lie wins something.  It keeps them attentive, excited to win stuff, and if you mess up accidentally that gets to be the lie for the day.  Then, on the last day of class you tell no lies.  When they question this, you tell them that on the first day of class, you told them you would tell them one lie per day.  That statement, you now inform them, was that day's lie.

More importantly, I noticed that the students tuned out the subsequent discussion of domains, range, vertical line tests, etc.  These are facts about functions.  They're not really important in the real world, but they're the only things the students can be tested on.  The students also were mad about having to show their work (or rather, losing HW credit for not showing their work.)  This is something I hated too.  What matters is being able to figure the answer out, not how you did it.  Mrs. Colwell noted that this prevents the students from just copying answers from their friends.  All it does, though, is make them have to copy MORE answers from their friends, if they are copying.

A potential solution:  Don't have any homework.  Make the kids do the work in class, but let them ask questions of the teachers and of each other.  If we had a 10 minute quiz each day that emphasized just one important concept, they may very well be more engaged to learn than by having to memorize dumb facts.  Of course, this needs more thinking through, but I think the important thing is that the students learn how to use the math and how it empowers them.  Knowing what "domain" and "range" are is pointless.

A plan!

Next week, Mrs. Colwell gave me permission to tell stories about famous mathematicians, or stories about famous numbers or functions.  I think I'm going to show the students everything in the entire book.  I have trouble learning something if I don't why I need to learn it, and I suspect the same may be true for some of the students.  My plan is to show them the the only thing we're learning this semester are patterns and puzzles.  This is seriously all that's in the book:

• f(x) is how we denote a function
• pictures of f(x)
• matrices
• two equations, two unknowns
• f(x)=x^2
• f(x)=x^a  where a is an integer
• f(x)=x^a  where a is real
• f(x)=exp(x), f(x)=ln(x)
• sin, cos, tan
• polynomials
• conic sections
• series

Literally, each of those bullets is one chapter of 9-10 sections.  Why does polynomials have its own section away from x^2?  Why do they explain the names of function parts without showing cool functions?  Nobody cares that the thing in the parentheses is the "argument".  Just show me the patterns!  This book is basically a journey in finding x from f(x)=c up through f(x)=a*exp(x) + b*ln(x) + c*cos(x) + x^d.  Perhaps if they see what's coming and see that we're doing the same thing all the way through (learning about new expressions, what they look like, how they're used, and how to solve them) it may help some of them deal with the immediate pain of domain and range.  In short, the idea is to show them they can do everything in the book and that the rest of class is just filling in the details.

First day

Today was the first day of class.  I was simultaneously nervous and confident:  nervous because I had no idea what to expect from my first class, but confident that I would be able to help Mrs. Colwell had asked.  The assignment was seemingly simple "talk about how functions are used in engineering."  I thought it would be a breeze!  Just sit down, make sure everyone knew the notation, and then talk about some famous functions used in the real world.  Easy, right?

Wrong.  

As I took the projector for the first class, I didn't know what the kids knew or didn't know, so I just started talking in the hopes of feeling out unfamiliar waters.  Whereas I thought the discussion of nomenclature would just be a side note, a preface to real engineering functions, it turns out that this was actually the lecture for the day!  After I fumbled for a while, Mrs. Colwell took over and began the day's real lesson.  The kids copied down the answers to the previous day's homework, and then Mrs. Colwell explained Euler and Mapping notations for functions.  A couple of examples of these, and then BAM, it was homework time. 

During Mrs. Colwell's lecture, I tried my best to run around the room answering questions that the kids had as she went.  This seemed to work well- she had fewer interruptions to deal with and I was having fun being useful.  The kids asked good questions, were typically interested, seemed motivated, and had good attitudes.  Maybe it's just because it is still the first week, or maybe because they're not used to having a new person in the room answering questions, but I was pleasantly surprised with how well-behaved the class was.

With lunch between classes, I get a chance to sit down with Mrs. Colwell, which I think will be a great asset in the future.  We should be able to talk about how things did or didn't work, and then adjust for the second class.  Today, she asked "when were you going to talk about engineering?"  Before class, I thought that it would just "come up" as I talked about functions in general.  As this answer left my lips, I thought "boy do I sound unprepared." 

In the second hour, I was way more comfortable with being up in front of the class and set out to give my 20 minute shpeel about functions in engineering.  I sat down, introduced myself, and found myself talking about why they should care about math as a preface to "real engineering functions."  When I said math was beautiful, one student (who may be a bit of a disruption, we shall see) said "you mean math is sexy!?"  I went off on a tangent about the golden ratio, explaining how it appears in proportions for things that are aesthetically pleasing, like faces, buildings, and bodies.  "So yes, math is sexy."  One student asked if I would measure the ratios of her face before I realized I had probably run out of time and hadn't mentioned much in the way of engineering.

Again, Mrs. Colwell stepped in and delivered the lecture as I attempted to help out with questions as they arose.  On average, the attentiveness and attitude of these students was the same as the morning class, but their distribution is much more broad.  There were both more kids working hard and more kids causing disruptions.  

After class ended, Mrs. Colwell asked "so, can you talk about engineering tomorrow?"  I chuckled to myself a little bit- as impressed the students had been with "sexy math," none of them realized that their new teacher had failed at his homework assignment.  Now that I think about it more, it's a hard assignment.  How do you talk about math in engineering in a way that is both accessible to neophytes in the subject and sufficiently interesting?  Also, what is the goal of talking about functions used in engineering?  Is it to show them that they can be engineers?  Is the goal to show them real-world applications of the work they are doing?  What if they don't care about engineering?- is it possible that such a talk may turn them off from math?

The impression I got from the students today was that nobody hated math, but that none of them really loved it.  I think that rather than just showing them what math can be used for (like engineering), we should be emphasizing what they can do with it.  Show them that math is just playing with patterns and that playing is fun.  It's my hope that they can learn all the necessary test material at the same time, but we'll see how it goes.  Today I learned that they have the ability.  The challenge, of course, lies in taking the rote memorization of "dependent, domain, and argument" and turning it into something they can actually use and talk about.  

Teacher selection

Today we met with math and science teachers from Ypsi high school for Fellow-Faculty pairings.  I was paired with Doriane Colwell and I'll be teaching algebra and advanced algebra this year!  Sounds like we'll be getting into the classrooms soon, but I still don't think I have any idea what I'm getting into.  

first post

Hi.  This is the blog for Eric's experiences as he teaches (or, attempts to help teach) a high school class this year.  On Monday we meet with the teachers whose classrooms we can potentially serve in- and I'm pretty excited.  Right now I think I'm leaning towards math, but with all the idiocy about creationism in the news it might also be fun to teach biology.  Regardless of the subject, I hope I'll be helpful and a good example of a scientist.  Just look at what one 7th grade class thought of scientists before they met real researchers at Fermilab!  As a side note, if you look through all the pictures in that link, many students seem to have chosen a man with a striped shirt and goatee as their new canonical scientist.  I wonder who they met at Fermilab that fits this description.