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.