When Algebra Was Fun

I have heard the claim that most mathematicians either liked algebra or liked geometry in their early training. Part of my personal mathematical mythology is that I didn't much care for mathematics until I got to geometry.  (Partly, or perhaps largely, because of a really awesome geometry teacher.)  To this day, I have to admit that I mostly find (high school) algebra somewhat tedious, although obviously useful and necessary.  (What is sometimes called abstract algebra, starting with groups, rings, and fields, is obviously a totally different creature.)  I can appreciate some of the patterns and such in high school algebra, although of course at this point finding this type of algebra interesting is a little like finding the alphabet interesting: I'm much more interested in using it as a tool to do other things with.  But when I look back on my algebra classes in junior high and high school, I found those classes sort of boring, although not as bad as the arithmetic classes which preceded them.  I guess I just didn't find algebra that interesting.

Or did I?

I was recently talking about an "enrichment" program I participated in for one year in fifth grade, in which a small group of students from across the school district were gathered and bussed off to a special class one day a week.  On program days, we got to do all kinds of great things, like reading and discussing cool books, engaging in research projects, doing experiments, and working on a computer. It was actually pretty awesome. There were two real problems with the program, 'though.  The biggest problem was that the fantastic educational opportunities we got in this alternative class were really what everybody probably should have been doing all the time, instead of a special one-day-a-week pull-out activity for whosoever was judged to be the "best and brightest."  The second problem was that the program was a set of additional pull-out activities, because the students in the program had to make up all the work we missed in our regular classes.  (This, by the way, is why I only participated in fifth grade: I didn't do so well with keeping up with the other stuff, which was frankly mind-numbingly boring.)  So in the enrichment program, we'd research and report on ancient Egyptian burial practices, then come back and have to read a passage out of the social studies text book to fill in the blanks on a mimeographed worksheet.  Or we'd collect cell samples from our mouths and examine them under microscopes to learn about cells, and then come back to have to copy a diagram of the human nervous system out of the textbook.  (Interestingly enough, I remember that picture because I remember coming to the conclusion that we must be less sensitive in our forearms than our upper arms, because the diagram clearly showed more nerves in the upper arms.  This was not a misconception that I ever got to discuss in class.)  Or we'd go learn how to solve problems using algebra, only to come back to "Do the following 25 fraction addition problems."

Wait, what was the last one?  I'd forgotten about that! We actually learned some algebra in the program.  I don't remember all the details, but I think we had a worksheet, and I remember the idea of introducing a variable for an unknown quantity, setting up an equation to represent a problem, and how you could go about finding out what the x (or whatever) represented.  The problems were puzzles, and they were wonderful.  Some were quite difficult; I'm not sure we solved all of the problems.  I remember being fascinated by the very idea of working in some sense "backwards" to figure out an unknown quantity.  It was an exciting adventure for us to figure out, a marvelous mystery.  We were figuring stuff out, guided (loosely) by the teacher, who introduced just enough hints for us to make it through each new challenge. Each new idea and discovery was shared and traded with great relish.

I remember wanting to learn more about algebra and thinking it was wonderful.  Until of course I had some problems with finishing up the necessary arithmetic by hand, which led to various adults tut-tutting to me about how I obviously should have been doing more arithmetic drills. That was pretty much the end of my interest in algebra  since it was clear to me that expressing interest in algebra would lead to being punished with more arithmetic drills first.  So instead of picking up some of the arithmetic incidentally as I studied more interesting stuff, I just ground my way through the required math classes as best I could, hoping they would be over with soon, and forgot about algebra.

I finally took a regular algebra class in the eighth grade, but I'm not quite sure if I remembered how much I had liked it once.  But my eighth grade algebra class was a bit of a nightmare, taught by a man who was best known for yelling at the students and picking his nose.  (I suspect the latter would have been more tolerated and ignored were it not for the former.)  It was definitely not an adventure, and the problems were definitely not puzzles.  There were just a bunch of rules, and an algorithm of some sort for solving every sort of problem.  Every day was a new type of problem, and mostly an expectation to memorize an algorithm for solving it.  There was no "figuring" anything out, and the techniques were no longer mysteries to be discovered, but miseries to be endured.  Hell, I barely passed that class.

But as I think back on it, I realize that my interest was not completely crushed, even if I didn't realize it at the time.  I remember at one point during a summer vacation suddenly thinking about graphs, and wondering what feature in an equation made a graph "straight" versus "wavy."  I actually developed a hypothesis (by experimenting) that equations in x and y which didn't have any powers except for "1" were the only straight lines, and other powers gave bent curves.  (I have no idea whether I had already been told this before or not, but if so it hadn't stuck until I noticed it myself.)  And it's also clear that I must have had some interest left in math, because seriously, what high school student spends part of his summer vacation plotting multiple graphs by hand to test out a hypothesis about which graphs will be shaped which way?

So my personal mythology is wrong.  I did once love algebra  almost if not as much as I later loved geometry.  And I wonder:  What if my early interest in algebra had been allowed and encouraged, even if I was yet unsteady at arithmetic?  What if my first formal algebra teacher had been the same teacher who later taught my geometry class in high school, who encouraged my exploration and experimentation?  In retrospect, what I relished so much about the geometry class was that the problems were once again puzzles: No algorithms, no sequence of steps to memorize, just a statement starting "Prove that...," and it was up to us to figure out some way of getting from Point A to Point B.

In fact this spirit of investigation, of figuring things out, is at the heart of my favorite movement in mathematics education, known as Inquiry Based Learning, or IBL.  In IBL, students are set problems of some sort to solve, something to figure out.  The steps are small enough for the students to figure out on their own, and they are led along a path of discovery.  That's what happened back in the fifth-grade enrichment program:  We were introduced to the idea of using a variable, or of "doing the same thing to both sides of an equation", and asked to figure out how to solve the next problem using what we knew.  We figured the stuff out "on our own" (in actuality with plenty of guidance), and we were excited to be doing it.  When I got to the eighth grade class, what I got instead was "Day 23: How to Solve a Digit Problem.  Step 1: Let t be the tens digit in the unknown number...."

Now I have a bit of a dilemma:  I now remember what joy in algebra felt like, but can I bring that to my students?  In particular, I've recently been teaching a remedial algebra class.  It's required for many students who have poor math placement scores on entering the university, and it covers a great deal of material in fairly short order to make sure the students have all the needed algebraic skills for their next mathematics class.  Because of this, it is very algorithmic, using a very step-by-step, one-topic-at-a-time approach--the very approach I was bored to tears with. Can I bring any of the joy of algebra to my students?  I can imagine running an algebra class in the spirit of that first encounter I had, following an IBL approach, but I also think it would require more time than the one semester I would generally have.  (Now in high school, algebra is usually spread over two years, which I think would be ample time for a careful, and ultimately quite rigorous and thorough IBL algebra course.)

I'm sure that if our remedial students had a more inquiry oriented algebra class, they would be more likely to find some enjoyment in the mathematics (as I once did), and they would probably grasp some of the basics more fully.  I wonder what the longer term effects of such a remedial program would be.  Would the students with a stronger basic foundation in algebra and an interest in the material do fine in a later class without learning all the needed techniques, more or less filling-in material as they went?  Or would they end up struggling and failing to keep up because they did not know the assumed prerequisite?  Maybe I need to think about this question.

Cartoons for future educators

I'm a bit of a nut for cartoons in general, and love children's cartoons.  So when my partner was preparing to teach an intro to education style class for future teachers, he asked if I had any suggestions for cartoons about education related topics which might spark good conversations about issues like teaching, testing, bullying, diversity, and others.  I thought of a few off the bat, but then I spent a while going through the listings on Wikipedia for some of my favorite series, and used those to jog my memory so I could find some good suggestions.  He liked the list so much, he suggested I put it online, so here it is.

Below I have listed episodes from three of my favorite children's cartoon shows:  Arthur (PBS), Hey Arnold! (Nickelodeon), and Recess (Disney) which are meant to inspire discussions among future educators, organized by series.  I've also included a separate link and some comments at the end about some episodes of The Simpsons which might also be relevant.

I should note that the episode descriptions for the first three series listed below are copied verbatim from the Wikipedia episode list for the series (as of early July 2012); I provide links to these three episode lists.  After the description, I have provided a few comments of my own about why I included this on a list of cartoons for future teachers.  

Arthur  Children's cartoon show on PBS, based on the popular Arthur books by Marc Brown.  (Wikipedia: List of Arthur Episodes.  Click on the link for an individual season to see the episode descriptions, which I copied in italics below.)

• "Arthur and the True Francine"  Muffy and Francine were best friends since second grade, when Muffy was a new student. This episode is a flashback from that time. This particular memory is when Muffy and Francine decide to study together for an upcoming math test, but Muffy continually procrastinates, claiming that she knows her addition and subtraction. When the test day comes, Muffy cheats off of Francine's test and claims that she would never cheat, getting Francine into trouble which almost costs her her friendship with Francine.
  My comments:  This episode includes cheating, and a punishment meted out to the wrong student which causes strife among the students.  Teachers end up having to make tough decisions about cheating all the time.
• "Arthur, World's Greatest Gleeper"  Arthur and Buster sit at the only available table with the Tough Customers. Buster claims that Arthur is the world's greatest "gleeper" to stop the teasing, only to find out later that "gleep" means "steal". Arthur lets the lie grow into a school-wide rumor. 
  My comments:  Deals with peer pressure (especially with pressure to be considered "bad" rather than "good"), rumors, and how teachers might respond.
• "Sue Ellen Moves In"  Rumors are circulating about the new family that has moved in and Buster thinks that they may be art thieves, spies or aliens, even after he meets Sue Ellen. When Buster's mom invites Sue Ellen over for dinner, he finds out that Sue Ellen is not an alien, but just a kid who has lived in various places around the world.
  My comments:  A good diversity episode, featuring a new student who seems very strange to the rest of the class.
• "Bully for Binky"  Binky has a reputation for being a bully. He bullies Sue Ellen, who is still a new student ("Sue Ellen Moves In"), and she demands that he apologize. When he does not, she challenges him to a fight to settle it. He gets nervous when he finds out she knows Tae Kwon Do. He decides to beat her at music but loses, so he apologizes to her so they would not have to fight.
  My comments:  An episode on bullying.  The bully here (Binky) is interesting because he is in fact a little complicated, even 'though the show does not quite pull the old stereotype about bullies just being "misunderstood".  (Binky is actually a pretty interesting character throughout the series, since he's sometimes a bully, and sometimes a pretty decent kid, depending on the context.  We're all pretty multi-faceted, I guess.)
• "Arthur's Substitute Teacher Trouble"  Mr. Ratburn loses his voice and his sister, Miss Rodentia Ratburn, substitutes for him. She has them doing things such as reciting the one times table, reading words like "dog" and "cat" and assigning no homework. The class are excited at first, as they were sick and tired of Mr. Ratburn's unbelievably difficult classes, but they soon become bored. The class are glad when Mr. Ratburn returns the next day. 
  My comments:  What are reasonable expectations for students, and how will students react?  What's too hard?  Is it possible to be too easy?
• "Draw!"  When Francine offends Fern, Fern draws a comic humiliating Francine, which everyone finds amusing. Other comics about Francine are drawn. During the school carnival, those teasing Francine are convinced by Mrs. McGrady to dump green slime on Francine. As they prepare to do so, they notice how much they hurt Francine. Meanwhile, Miss Tingley tries to avoid Miss Sweetwater and Mr. Haney, both of which want her to take part in their acts. 
  My comments:  An interesting take on teasing and empathy.  Here, a bit of bullying goes both directions, and a quiet, shy student ends up leading a pile-on against a more outgoing student.  A teacher (in this case, the lunch lady) helps the students awaken their own empathy, and stop the teasing.
• "Sue Ellen and the Brainasaurous"  Sue Ellen and the Brain are assigned to work on a project together. The others say it will be easy as Brain does almost all of the work by himself. Sue Ellen wants to help, but the Brain refuses to let her. This turns into a struggle that nearly ruins their project.
  My comments:  How will students handle group work?  Most people have horror stories about groups they were in, and several of the standard problems are on display here.
• "Buster's Breathless"  When D.W. has a brush with poison ivy, Buster tells a story about how he once got asthma. When Buster learns he has asthma, his friends start treating him differently, thinking that Buster needed special attention. Buster educates his friends by doing a science project about asthma.
  My comments:  Nice diversity episode, about Buster trying to deal with the fact that his friends treat him differently after his diagnosis.
• "Prunella's Special Edition"  Prunella is excited to get her new monogrammed limited edition Henry Skreever book in the mail, but finds that it is in braille. When Prunella goes to the library to get the book with words, she meets and makes friends with a blind girl named Marina, who is looking for the same book in braille.
  My comments:  A diversity episode, featuring a new character with a physical disability.  (Also as a sidelight, a fun reference to popular children's literature with a book series which has more than a passing resemblance to Harry Potter.)
• "Arthur and Los Vecinos"   A new family moves in next to the Reads after their neighbor, Mr. Sipple, moves away. The Reads get to know the Molinas and discover how similar they are. 
  My comments:  A bit of cultural diversity is covered in this episode.
• "The Boy with His Head in the Clouds"  George has dyslexia and does not want anyone to think he is dumb. He takes Binky to be his mentor to teach him to be more hard-headed so no one will tease him, but realizes he was not meant for being tough. When he discovers his dyslexic problems, he tries to finish his reading project.
  My comments:  A diversity episode, involving an ongoing character with dyslexia and some of how he tries to deal with it.
• "Prunella Sees the Light"   Prunella is inviting Marina over for a Henry Skreever sleepover. However, she worries that Marina may not see the decoration in her room or may get injured because she is blind. Marina does not approve of the special treatment Prunella is giving her, and Prunella learns to treat Marina just like any other friend.
  My comments:  A bit of a follow up to "Prunella's Special Edition", focusing on how Prunella tries to decide how to act toward her blind friend.
• "April 9th"  [Part 1] It is April 9th, but the school day turns disastrous when a fire destroys the school. Arthur's Dad was in the fire but recovers, but then Arthur then has a nightmare involving the aquarium. Buster is upset that he was not there to experience it. He then meets Mr. Morris, the janitor, who was injured in the fire. Sue Ellen's journal is left behind in the school and is ruined, Muffy however buys a new one for her. The kids are sent to Mighty Mountain, but the fire alarm is pulled during a test.[Part 2] It turned out that Binky pulled the fire alarm because of his fear of the flames. Binky talks to Francine's dad, who was a volunteer firefighter. When Arthur fakes an illness, his dad finds out that Arthur is worried about him because he was trapped in Lakewood during the fire. Arthur's dad helps him that something similar happened when he was a kid and tells that it is his job to worry about Arthur. Sue Ellen and the others paint a mural at the wall of Lakewood. Soon life returns to Lakewood Elementary for the students.
  My comments:  A full length episode (containing two regular length parts) focusing on students dealing with a tragedy.  This is a fairly mild example, featuring a fire that results in few injuries but shuts the school down for a while, but it provides a lot to think and talk about regarding tragedies of any magnitude, which could strike any school.
• "Dear Adil"  Arthur writes letters to a Turkish boy who is looking for a pen pal after reading his Dad's letters from Japan. He isn't sure what life is like in Turkey, so to give him an idea, he looks at Buster's comics set in said country. However, these give him a stereotypical picture of Turkey, and Adil is befuddled when Arthur asks him about his camel, tent or the taste of lamb's eyes. Arthur finds in his paper, Adil's email address and they find out that they are alike in many ways.
  My comments:  A cultural diversity episode.
• "Brain's Shocking Secret"  Brain is afraid of the kids learning that he got held back in Kindergarten, and for what reason. 
  My comments:  This episode deals with holding students back for a grade and with psychological and social development.
• "Arthur's Number Nightmare"  Buster finds a piece of paper with names of his classmates and numbers beside them. Arthur, Buster, and Francine believe it's a class ranking system based on behavior, and Francine tries to find ways to have a higher rank.
  My comments:  The episode looks at the possible effects of ranking students.
• "When Carl Met George"  George meets a new friend named Carl, who has Asperger syndrome‚ a form of autism. George is unsure about how to act around his new friend, but Brain helps put autism in perspective for him so he can understand some of Carl's mannerisms.
  My comments:  A diversity episode featuring a new character with Asperger's syndrome. 
• "The Wheel Deal"  Brain is put in a wheelchair after a leg injury and copes with the encouragement of Lydia, a disabled girl who uses a wheelchair.
  My comments:  A diversity episode featuring a new character with a physical disability and a permanent character with a temporary physical disability.  
• "S.W.E.A.T."  The third grade students at Lakewood Elementary are stressing about their upcoming S.W.E.A.T. aptitude tests, including Sue Ellen, who panics that she doesn't have any #2 (HB) pencils, the Brain, who worries he will do badly after he accidentally skips a question on the practice test, and Arthur, who struggles to find a place to study quietly. 
  My comments:  Standardized testing and its effect on students and schools is the subject here.

Hey Arnold!  Cartoon series on Nickelodeon, created by Craig Bartlett.  (Wikipedia:  List of Hey Arnold! Episodes gives the episode descriptions I have copied below.) 

• "Tutoring Torvald"  Arnold must tutor a 13-year-old fourth grade bully in math.
  My comments: There's a bit here on bullies, on students being held back, on students struggling academically, and even peer tutoring.
• "New Teacher"  After their last teacher, Mrs. Slovak, retires, Arnold's class gets a new one, Mr. Simmons. They play the usual pranks for all new teachers, which he is familiar with. They don't bother him, but when Harold eats his lunch, he loses all hope of being able to teach them and quits. When Lieutenant Major ends up being their next one, they try to get Mr. Simmons back.
  My comments: The hazing of new teachers (and substitutes) is, of course, legendary, but I think the best part is the way the students initially reject Mr. Simmons' unusual and rather touchy-feely approach, but eventually come to grudgingly appreciate him.  Students resist change sometimes, even if they might think it's OK in the end.
• "Longest Monday"  The kids try to avoid getting thrown in dumpsters and trash cans as part of an annual ritual.
  My comments: A hazing ritual.  You have to wonder what's wrong with the adults.  The ending gives some idea about how these things get perpetuated.
• "Ms. Perfect"  Lila, a new student, comes to P.S. 118. Helga and the other girls play tricks on her in an attempt to drive her out of the social hierarchy.
  My comments: An exercise in group bullying and piling on focused on a new student, largely unnoticed by the adults.
• "Principal Simmons"  Mr. Simmons becomes principal after Principal Wartz has one too many outbursts.My comments: A great episode on the yin and yang of authority, from the overly domineering and explosive Wartz to the walking pushover who is Simmons.  Also features one of my favorite lines:  "No principal should ever call a small child a 'wicked minded animal'---even if it's true." 
• "A Day in the Life of a Classroom"  A film crew shoots a documentary of Mr. Simmons' class to be screened on TV, but he feels that they should follow a script, rather than be spontaneous.
  My comments: This is a really interesting take on being observed in a classroom, the kind of anxiety that might provoke in a teacher, and the temptation to "put on a show" vs. actually teaching.
• "Phoebe's Little Problem"  Phoebe is embarrassed after passing gas into a microphone and is afraid to come back to school, despite Arnold's, Helga's and others' efforts to console her.
  My comments: This episode on group teasing and embarrassment shows that sometimes it's really hard to make someone feel any better, but sometimes things do get better eventually.

Recess A cartoon series which aired on the Disney Channel, created by Paul Germain and Joe Ansolabehere.  (Wikipedia:  List of Recess Episodes gives the episode descriptions I have copied below.)

• "Kids in the Mist"  A researcher named Dr. Quilty wants to research by video about recess. When she first tries it, she fails, and T.J. and the gang decide to help her out. How will it turn out?
  My comments: This episode should be appealing to future teachers who will study educational research, observe students, and maybe even perform their own studies.  (It also slyly suggests that if you're interested in educating children, perhaps it would be a good idea to actually like them.)
• "Gus' Last Stand"  Gus stands up to Gelman the bully.
  My comments: Not a bad episode about a bully.  It includes some ineffective efforts by adults to deal with the problem.
• "Schoolworld"  The school gets a new technology system installed called the SAL 3000 which controls everything. This episode loosely parodies 2001: A Space Odyssey.
  My comments: A great episode to start a conversation about technology in education, and about rigid, top-down approaches to teaching.
• "The Dude"  A school legend, T.J.'s idol, comes back as a teacher.
  My comments: A great episode for future teachers, focusing on the tension between having to be the authority figure at the front of the classroom and still feeling like (and wanting to be) "one of the kids."  I imagine this hits home particularly hard for student teachers who are coming back to a school they once graduated from.
• "Spinelli's Masterpiece"   Spinelli lets off some steam by creating a chalk drawing and T.J. does everything he can to keep Miss Finster from erasing it.
  My comments: OK, this is the one episode in the list for which I have trouble explaining why I included.  It's a nice take on student creativity and different perspectives, 'though.
• "Bonky Fever"  Mikey has problems turning the big 1-0, and still has strange childlike obsessions with the dinosaur character Bonky.
  My comments: This one is all about psychological development of students, including the urge to regress to simpler times when things get stressful.

The Simpsons  OK, everyone knows what The Simpsons is.  The Simpsons Wiki has already put together a list of education related episodes under the Education page, and I've copied that list below.  I've added a brief description and a few comments to each episode.  (These episode synopses are my much-abbreviated versions.)

• "Bart the Genius"  After cheating on a test, Bart is placed in a school for the gifted, but he can't keep up.
  Comments:  Great scenes on testing near the beginning, plus some fun satire about progressive schools later.  (I periodically tell my class to "Discover your desks, people!" in the hopes that someone will recognize the quote.)  My only complaint is that I think Bart would have probably done better in the more progressive school.  I often wonder why the really good stuff is only offered to the high testing students.
• "Lisa's Substitute"  Lisa develops a crush on an extraordinary substitute teacher, who supports and encourages her love of learning.
  Comments:  Features an impossibly good substitute teacher contrasted with the rest of the school and Lisa's home life.
• "Homer Goes to College" After causing a disaster at the nuclear power plant, Homer is required to take a basic college class on nuclear physics.  He approaches the whole experience based on cheesy comedy movies about college.
  Comments:  While this one has fun with Homer's ideas about college and studying, since it's focused at the college level, there is less here for future public school teachers.
• "The PTA Disbands"  Bart helps start a teacher's strike, then tries to end it when his mother ends up subbing in his classroom.
  Comments:  The scene featuring Krabappel and Skinner debating over funding for the school in front of the town is priceless, and does a great job of capturing all public funding debates.  The larger issues of funding, unions, and conflict between administration and teachers are all relevant, and the scenes featuring other Springfield residents trying to sub for the teachers are hilarious.   The "solution" reached at the end is priceless.
• "Lisa Gets an 'A'"  Lisa discovers the joys of procrastination, and, in a panic, turns to cheating.  But when her high score gets the school additional funding, she feels the need to come clean.
  Comments:  This episode covers cheating, use of student scores to measure school performance, and funding of schools.  
• "The President Wore Pearls"  Loosely a parody of Evita, with Lisa playing the starring role running for (and winning) as school body president.
  Comments:  Although a fun episode which does briefly deal with issues of student power and with the cutting of classes like art and music, I'm not sure there's really much here that would generate discussion among future educators.  
• "The Monkey Suit"  Springfield outlaws the teaching of evolution, and a Scopes-like trial ensues when Lisa is arrested for trying to keep it alive.
  Comments:  Possibly a fun way to introduce the difficulty science teachers (and others) may have with some topics, including evolution.
• "Girls Just Want to Have Sums"  Springfield separates boys and girls math classes, but Lisa finds the "girls" math class unchallenging and sneaks into the boys class.
  Comments:  Single sex education is a big topic these days, and the way the school handled "girls" and "boys" math classes is indicative of common superficial approaches to closing the gender gap.  (And I refer you to this comic, which makes the point perfectly.) 
• "Little Girl in the Big Ten"  Lisa ends up being mistaken for, then impersonating, a college student.
  Comments:  There is some good material here about smart students and the problem of "fitting in" with their peers.
• "Bart Gets a 'Z'"  Bart conspires to get his teacher fired, then feels guilty.  A "hip" young teacher replaces her, making heavy use of technology and entrancing the students with his "coolness."
  Comments:  There is an opening montage showing Ms. Krabappel getting ready for school which is a priceless (and enormously human) commentary on teacher burnout.  The episode spoofs a lot of educational fads centered around overuse of technology, and the "too cool for school" teachers who try too hard to bedazzle their students.

Ponderings on Geometry

At a conference this summer I saw an interesting invited talk about teaching geometry. The speaker discussed some of the historical challenges to Euclidean geometry, and discussed how these were eventually overcome. The problem for schools being that while mathematicians like Hilbert managed to fill in all the gaps left by Euclid, the resulting geometry was very complex. It took a great deal of work to get from the axioms up to the interesting Euclidean results. In fact it took too long for a single year high school course to even get to the beginning of Euclidean geometry. As a result, the speaker claimed, most high schools eliminated axiomatic and proof-based geometry from the curriculum, replacing it with practical, computational geometry only. (He went on to detail a new approach, which allows a more rigorous approach to Euclid's geometry in a high school without starting from scratch, and using a book which is designed to be used in an inquiry based class. It looks fascinating, and I'm hoping to teach a class based on this book soon, but that's a digression.)

What caught me off guard was his statement that rigorous axiomatic approaches to geometry using proofs had been all but eliminated from most public high schools due to difficulty dealing with the challenges to Euclid. I had to wonder, was this actually true? I certainly had a proof based and axiomatic geometry in high school, and that was... ok, a little over twenty years ago. (Geez I feel old now.) And while my high school was mostly fine, it wasn't really an elite school. (When I taught an analysis class, I actually used several problems phrased in the form: "My high school calculus teacher said ___. Prove rigorously that he was wrong.") So had schools actually eliminated proofs from geometry since I had taken it? Or had my school been an odd stand out for not eliminating it?

Tonight I was about to start a geometry unit with a class. I asked the students for a show of hands of how many people remembered doing ruler and compass constructions. Less than a third--and maybe less than a quarter--raised their hands. Now I know from experience that some of the students may very well have seen constructions and just have forgotten everything five minutes later. But still, it seems that a number actually didn't do constructions. And I'm inclined to think that if they didn't do any constructions, they probably didn't do any proofs, although I could be wrong on this. (And of course some may have done constructions but not proofs.) So maybe the summer speaker was right, and axiomatic geometry is mostly gone.

But I question his reasoning about why the proofs are gone. He believed it to be because it proved too difficult to fill the gaps found in Euclid. I can't imagine boards of education, state legislatures and such largely care--or even know--about that. My first suspicion was that someone, somewhere decided it was "too hard", or that students just couldn't do it, and replaced it with other things.

But why too hard? Abstract reasoning and logic is hard, but given time and effort, most people can make progress. Ah, but there's the problem: Time. To do a good job with a class about proofs, it takes time spent working on problems, trying ideas, failing, and trying again. Learning to reason is a long and difficult process. And curriculum tends to fill up with all sorts of nonsense as everyone and their dog proposes new lists of things that "everybody" ought to know. Most of those things are lists of facts and formulas. I can almost hear the litany start for a geometry class: "Students must be able to give the formula for the area of a circle, a semicircle, a rectangle, a square, a triangle, a trapezoid, a parallelogram, a rhombus; Students must be able to find the perimeter of a circle, a rectangle, a square, a triangle, a trapezoid, a parallelogram, a rhombus..." (And it's worth noting that the only really interesting things in the list I just gave are probably the area of the circle and the rectangle, and perimeter of a circle. The rest should be easily derivable from some good geometric thinking. But the students will instead be given a list of formulas to memorize.)

We add and add to curriculum, and nothing is ever taken out. We add numerical approximations and work with computers. We add calculators, then graphing calculators, then geometry software. We add three dimensional shapes, and trigonometric functions, and whatever else anyone can think of because ... well, because someone else happened to remember it and thought it would show how rigorous we were being if the list of things for students to know was really long. (It should be noted that I am not opposed to any of these topics in high school math classes. But we must recognize that we cannot do every possible topic all at once.)

But a long list of topics is not rigor. That's just memorizing a bunch of stuff. Take any of those topics and spend some time with the students doing a long and careful analysis of some challenging problems, and you'd have a recipe for a great math class. The central questions will always be the following: What do you think is true? How do you know that? Why? Is this like anything we have done before? Can we generalize this result?

All of this of course misses the biggest elephant in the room as to why high school math has dropped most reasoning and replaced it with lists of tasks and formulas: It's very easy to write a statewide multiple-choice test which to see if students can choose the formula for the area of a circle. It's very difficult (or perhaps impossible) to write a state-wide multiple choice test which determines how skilled students are at reasoning and solving complex problems. But I'd much prefer a student who can work out how to find the area of a trapezoid based on what she already knows than one who has only memorized the formula. I regularly have students who have memorized a formula corresponding to a figure, but can't figure out if it's the formula for the area or the perimeter. (When I ask how to find the area of a circle, I can guarantee about half the class will respond "2 pi r.")

And I'd really love to have students who learned enough reasoning to write proofs in their geometry class, but apparently that's rare now. No wonder we have trouble teaching proofs later in college.

The semester so far

Today began the second week of a new semester.

I have two particularly small sections of one class, in part, I think, because I'm the only person teaching the second semester of the course who was not teaching the first semester in the fall. I love having small classes, so I'm not complaining. One of my students who didn't show up on the first day claimed she had switched to Professor D's section, but was still showing up in my roster. I double checked with Prof D, since today was the last day of add/drop. It turns out that Prof D had signed an override to allow her to enter his already overfull section. He now has 33 students. I went from 16 to 15. I'm laughing. I'm not sure Professor D was when he found out what had happened.

On the other hand, my Gen Ed class (aka, "So you think you can math?") is not small at all, and has stayed firmly at the enrollment limit of 40. Students have some incentive to pass the class this semester, since in the fall both the difficulty of the course and the prerequisites will increase. I'm being very straightforward (and maybe just a little easier than usual) to give them the best shot of getting through before the new course requirements kick in. We have just finished the second class (for a total of 2.5 hours of classroom time) doing nothing but unit conversions. Some students are completely lost.

I had a student show up in my office today wanting to do an independent study this semester. We discussed some possibilities, but I said we should talk to the chair to find out if it was possible such a study could be approved this late. The chair responded by putting her head in her hands and making a sound like a expiring mongoose. Since the student still needs to be in the chair's good graces, he wisely rescinded his request for an independent study.

But my joy this semester will be teaching the "Intro to Proofs" class for majors. I'm teaching the course using a technique commonly known as a modified "Moore method", which I've been interested in for some time. I went to a workshop for new practitioners two summers ago, and last semester I applied for and received a mentor so I could start trying it. It's the ultimate in student centered instruction, where most (or sometimes all) class time is spent with students presenting proofs to the class and the class dissecting the proofs as needed. I was prepared for all sorts of disasters to occur, but to my astonishment, my first two classes have gone extraordinarily well. My current major concern is to make sure all of the students stay involved in the class, rather than just a subset. But so far the discussions and presentations in class have been wonderful, and in fact beyond what I had hoped for. I still expect plenty of challenges ahead, but at least I feel like I've been well prepared for them, so I'm cautiously hopeful that we may have a really good semester.

Oh, and the unseasonably warm weather we have been enjoying is drawing to a close. Looks like we'll be down below freezing for a while now. Drat.

What students hear

I sometimes think about what students hear when we (their teachers) talk. I don't think undergraduate students (even good ones) manage to grasp the essentials of mathematics. I don't think they see what we are doing as logical reasoning based around some fundamental principles. They hear the specific words and explanations for a particular problem, but don't grasp that what we say comes from some coherent system. This is not necessarily because we don't say that this is so. It just seems that this concept doesn't completely register in the time we have them. It might later, or it might not.

When, for example, we want to find all solutions to x² = x, I might suggest dividing both sides by x, getting x = 1, which is one solution. I then note that we can only divide by x if we assume x is not zero, and in fact x = 0 is the other solution. We then have all solutions to the equation. To me, at this point in my life, I see division of both sides of an equation as a legal operation with very specific restrictions (namely, that we cannot divide by zero). I also recognize that since we can view both sides originally as multiplied by x, that x = 0 is certainly a solution. But I remember when things were not quite so coherent, and then the step back to note that we cannot divide by zero seemed like a trick just to justify a zero solution. It seemed that a lot of algebra (and some other mathematics) was made up of a bunch of special rules and exceptions, and it seemed like teachers had a never ending supply of these to pull out to justify whatever they said the answer was. It seemed a bit like playing pretend with a small child, where there is an exception to everything, to be made up on the spot: "Oh yeah? Well I had my invisible anti-force field magic belt on, so I could escape from your force field!"

How do people learn to see mathematics (or any field) as a unified whole? Perhaps part of it comes from time and experience. It may just take a certain amount of time working with the concepts before they become solidified and can be deftly manipulated. It may be similar to the feeling I can still remember in college, when I became sufficiently comfortable with algebraic manipulations that I could use them to do faster mental arithmetic, by disassembling and reassembling the numbers in convenient ways, making the numbers dance as needed. It's not that no one had ever suggested the idea before; it's popular to present these ideas to students, but until those concepts are internalized, it doesn't make mental arithmetic any easier. It just seems like a trick--and a somewhat painfully difficult one, at times.

Or maybe we need to give students more opportunity to see mathematics as a unified whole from the beginning. That would be one reason I've for years pushed for depth over breadth. Whenever a curriculum issue comes up, my first thought is usually, "What can we cut out?" Skimming quickly over dozens of topics and techniques encourages that sense of mathematics as a big collection of tricks, rather than something that has meaning and can be reasoned about. We need time to think about ideas, process them, wrestle with them, and make them our own. If students can understand a few concepts deeply, they'll have a better chance of figuring out something new on their own.

For the same reasons, I also feel drawn to inquiry based learning or Moore method teaching, where students are asked to figure things out with minimal guidance. We start with a few ideas--say, some definitions and a few axioms--and students are asked to build on that framework. Each new step has to be justified, and students need to figure out what works and what doesn't, without being told specifically what to do or what's right and wrong. They should rather be led by careful questioning to notice for themselves what works and what doesn't. If instead of just saying, "No, that's wrong," the teacher can simply present another problem and let the students figure out that their previous approach is flawed, they may internalize the structures much better. After all, structures you yourself have built are already essentially internalized. And it's hard to view a subject as filled with arbitrary tricks when you yourself have built up those "tricks" because they were needed.

In any case, what I would most like is for the students to come away with a sense of my subject (and others) as a unified whole, as something they can investigate and reason about, rather than just a collection of tricks and techniques. The techniques are useful, and the tricks are powerful, but without a strong foundation, it all falls apart.

Traveling Academics

Today my Alma mater hosted a conference for undergraduate research in mathematics and computer science. Since that school is a scant two and a half hour drive from our school, my colleague (and partner in crime in the department) spread word and attempted to find interested students to present at the conference. In the end, no students from our department managed to get a paper submitted in time, but three students were interested in going to the conference anyway. I was interested too, but with five of us going, that's a bit more than can be comfortably squeezed into one car for 2.5 hours, even if my car is fairly roomy. So my colleague and I debated whether or not to reserve a van through the university. We both pretty much agreed that the odds favored at least one of the students dropping out before the trip, but she decided (wisely, I think) to reserve a van anyway.

On Thursday, one of the students dropped out of the trip. So we canceled the van and planned to go in my car.

We planned on leaving about 6 am to allow enough time to arrive. One of the two students showed up at 6 am, and we finally gave up trying to find the other around 6:30 and hit the road. When we were about 15 miles outside of town my colleague got a call on her cell phone confirming that the student had indeed overslept, and in fact had just woken up. So in the end the trip with three students became a trip with one. So the van was definitely not needed, just as we suspected. It remains to be seen if it was canceled early enough that we will not be charged for it.

My colleague performed admirably when tasked with keeping me awake on the drive into Ohio, so we arrived without my getting my morning jolt of adrenalin by driving off the road or something. We even had enough time to get settled before the first talk started.

The talks were lovely and interesting. A few were in fields that went completely over my head. I have found in those cases that it is much easier to remain awake and alert looking if one stops trying to follow a talk that sounds like gobbledygook and instead just thinks about other things. Like, for instance, what I was going to write about tonight. This paragraph, for example.

But other talks were understandable and interesting. Even many of the CS talks I found fairly accessible, which was nice. The student who actually came was a CS student (but is also fairly mathy), and he seemed very charged up about a number of the topics, so I'm very glad he got a chance to come. He's actually about to graduate and plans to head off to grad school, so who knows what some of these ideas could inspire him to look into.

An invited speaker talked about art created via operations research, including such interesting projects as Obaminoes (which involved using 44 complete sets of dominoes to make a pretty darn good pictures of our 44th president). It was a neat talk, and the various art projects he'd worked on through operations research were pretty cool.

The student talks covered a wide variety of topics, including dissecting regular polygons into squares, simulating a robot, and solving sudoku and ken ken puzzles using some some algebraic geometry tools. All of the students did a great job. We need to get some students coming out and presenting at this thing.

Besides, our students might not have so far to travel next time. The hosts of the conference indicated they needed a host institution for next year, and my colleague and I talked to them about the possibility. It's a reasonably small conference and sounds like it might be manageable, so we are going to be looking into what resources we have and what support we might get if we wanted to try and host next year. It sounds like a lot of fun in addition to being enough work to make us truly frantic.

We briefly checked out the campus after the talks, grabbed some dinner, and headed home. On the drive, the three of us ended up in a light and fluffy discussion about education, history, social trends, biological engineering, the nature of knowledge, computability, the limits of human thought, and what constituted "writing down" or storing information. You know: the easy stuff. It was almost like being back in college again.

All in all a good day, although it was fairly tiring. And there goes Saturday. Tomorrow of course is Sunday, which means I have to get on the ball on getting ready for the next week, which thankfully only includes two days before Thanksgiving break.

Registration

It's registration time at the university again, so (some) of my advisees are coming to see me to get scheduled. Actually they pretty much have to come see me to get scheduled, since I'm the only one with their PIN that allows them to register. The school does that to force the students to actually get advising before they register for classes to help cut down on the students doing foolish things. This doesn't keep the students from doing foolish things, of course, but it might help.

Mostly I advise computer science majors, which I understand because we are a mixed department and most of our majors are CS. Most of the "math" majors are actually secondary education majors, and are advised in the education department. Although I still don't understand why I keep hearing about the few math majors we do have being advised by CS professors. But at least I get an opportunity to see the trajectory of the typical CS major. Or at least, the typical declared CS majors. They really don't have that many more majors than we do, they just get lots of people who think they want to be CS majors. I think some of the students think, "Hey, I love computer games; I'll major in computer science!" For many of those, things don't go so well. I often find myself advising a major who is repeating the introductory programming course and the introductory math class multiple times. Eventually, they usually give up and switch majors. Or fail out.

Some students are not so good at getting around to getting registration done. I had one who stopped by my office suddenly on Monday right before I was about to go to class and want to get his PIN so he could register. (Registration started some time earlier.) I spoke to the student briefly about the fact that he was failing his intro programming class and had withdrawn from his math class (see? I wasn't kidding). He indicated he wanted to change majors. So I took a minute to talk to another professor about a suitable major closer to what he wanted. Then he told me he actually wants to transfer to some other school, so he just wants to take some classes for spring and transfer credits. I asked where he was transferring. He didn't know. I told him I had to go to class, but I could talk to him tomorrow morning. He was worried all the classes would fill up and needed his PIN right away. I wasn't giving in on that, but offered to meet him after my classes finished that night at 8. He decided he could wait 'til tomorrow morning after all. But he didn't actually show up. I didn't see him again until Thursday. Then I could finally sit down and talk to him about what he was going to do, and make some semi-recommendations, which I know he isn't going to take. I suspect he's not going to end up transferring, changing majors, or doing anything else before next spring's registration, when I fully expect him to come rushing into my office towards the end of registration again, telling me he needs his PIN right away.

On the bright side, I do actually have a computer science major graduating this semester. That one, by the way, came to see me early in the registration period, always had a plan, passed his classes, and took what little advice I had the opportunity to give him. I wonder if there's any correlation.

Student projects again

My numerical class finished up the second (and final) week of student projects tonight, and I am still very pleased. So are the students. They told me they enjoyed it. I enjoyed it. I learned about a number of things that I hadn't before, including some tantalizing glimpses into topics such as the use of quaternions in computer graphics and how GPS works.

I had good students, and I think I structured the assignment well. Early in the semester, I asked everyone to pick (separately or in groups) a tentative topic one week as part of their homework. The next week, I asked everyone to find three sources about their topic. Next came a brief outline, and I started meeting with the students in their groups. We met about once a week, and I think it made a real difference. I got to find out what was going on and direct each group of students a little more each week, refining and refocusing their work. Most of the projects needed to be significantly reduced, but sometimes they needed redirected, too.

Over the past two weeks I also watched each group show me a practice run of their presentation and made some final suggestions for improvement. You know what the number one suggestion I had to give almost every group (including my very best students)? "Make sure you start by telling people what problem you are trying to solve." It was an odd sense of deja vu when I watched each new group during the practice presentations dive in and start explaining how to carry out, say, the Wronski-Schwarzchild Decomposition Algorithm,* without ever mentioning what the algorithm was supposed to accomplish. But this is why asking to see the presentations first is such a wise idea (for anyone who plans to do this); I got to let the students know that they ought to discuss such things. And to my delight, I found that the students by and large took my suggestions when they actually presented to the class.

It's been a good few weeks for this class, and we'll be off next week for Thanksgiving break, so it will be a while before I see them again. It was a good place to take a pause.

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*Yes, I just thoroughly made this algorithm up. And it was fun. Although I had a friend who came up with the idea that, should he ever develop some new mathematical operation which he got to name, he was going to call it the "Poopyface matrix," which I also like a lot. I think that also shows that he's funnier than I am.

Never mind

I mentioned recently that I was slightly panicked because I didn't see how I could finish everything in pre-calculus during the limited number of classes remaining. This week, I had to figure that out. I needed to put together the last assignment sheet that gets the class to the end of the semester.

So I sat down and did a major hatchet job over the weekend. Topics got pared down to almost bone. In the last chapter I cut all exposition and essentially covered two sections by just saying, "here's two equations, now use them." But I finally came up with a schedule that could (minimally) cover everything I needed to by the last day of classes, although I wasn't proud of what I was going to do.

To finish the schedule, I looked up our final exam time so I could put that on too. Then I... oh, look at that: We actually have two weeks after we get back from Thanksgiving break, not one. I can finish the chapter up in a reasonable way after all.

Never mind.

This is so hard, you should get college credit for it! Wait...

Our university has a math sequence for elementary education majors. The courses mostly take a very deep and comprehensive look at the underlying mathematics involved in about K-6 education. Since we have a number of mathematics education professors in the department, the courses are extraordinarily well designed. It meets in a room with large hexagonal tables where students can sit together in groups of six, and we have two large cabinets filled with all kinds of wonderful manipulatives which are used in many class activities. (A manipulative is any sort of physical object which can be manipulated to learn math. We have various types of colored chips, geometric shapes, and other cool toys for demonstrating mathematical concepts.) Many of the activities are actually similar to and based on activities which could be used to introduce concepts to elementary students, although of course the college students are expected to go a little deeper and are asked to do some things that we don't ask elementary students to do. (For example, we have the students in the course perform various operations in bases other than ten to emphasize the basics of a place value system. No one teaches base four or base twelve to elementary students anymore.)

It's actually a really fun class, and full of all sorts of wonderful discoveries waiting to be made. I personally find myself fascinated by the fact that in many cases, the way we explain a concept to our students parallels the abstract definitions which can be used to define that concept in advanced mathematics. So whereas in class we may use groups of red and yellow counters to define the integers, a mathematician might start tossing around scary sounding phrases like "sets of ordered pairs" and "equivalence classes", but ultimately mean pretty much the same thing. I personally found the demonstrations hugely enlightening the first time I did the class. It provided me with very concrete way to think about and explain concepts such as why a negative times a negative is a positive and why dividing by a fraction is done by multiplying by the reciprocal. The idea of the class of course is to provide our future elementary educators with similar insights.

Unfortunately, the class is always a struggle to one degree or another. One particular point which the students never seem to get (no matter how often they are told) is that this is not a class in elementary school mathematics. We obviously expect them to have already learned how to do things like add and subtract integers and fractions, how to multiply and divide multi-digit numbers, and the like. After all, they were supposed to have mastered these topics in grade school. (Except of course we know many of them actually can't do these things reliably, so the course also helps back up these concepts. But I digress.)

As a result, the students sometimes ignore instructions on how to complete an activity. For example, they are supposed to learn how to represent integers with sets of colored counters and then use the counters to add and subtract integers. (This is actually a really cool activity; I'll have to write about it sometime.) But since they know what 7+(-4) is, and following the directions to make representations of the numbers using the colored chips seems complicated, they instead just write down "7+(-4) = 3" and explain to me that "the model was too hard, so we just did it." Since they feel the class is about (or should be about) learning to add, subtract, multiply, and divide just like they did in grade school, there is no need to learn anything else about these topics as long as they know what the right answer is. They sometimes fail to understand that the colored counter model they have been asked to use is, in essence, the content of this course: we want them to learn to use a physical model which represents basic operations on integers, and to use that model to derive various known properties of addition and subtraction with integers. This issue is usually an uphill fight all semester with the students.

But this semester I'm getting even another argument from some students in one class. With almost every activity we do and with almost every mathematical model we describe and learn to use, the students complain to me that "this is too hard for any little kid to understand!" Which is completely irrelevant, since I'm not asking any little kids to do this work, I'm asking my class full of college students to do this work. I've told them I don't address the issue of how to teach their future students, but rather just teach them mathematics. I leave it to other people to teach them how to teach math. This doesn't sway the students.

I tell my students, "I'm not asking your students to to this."

The students respond, "Yes you are!" against all evidence to the contrary.

My students somehow feel that any topic which they consider to hard for a third grader should be too hard to ask a college student to do either. I suppose they want a refresher of third grade math without any of the "hard stuff." Remarkably, I seem to have little success with convincing the students that they are not, in fact, third graders. Do they really think that in a college math class they should learn nothing more than what grade school students are expected to learn?

But do you want to know what the worst part is? Most of the activities actually aren't beyond the grasp of moderately intelligent third graders. I consider it the dirty little secret of the course. Granted, it would take more time, but grade school students could certainly be taught rules for representing integers with colored chips. With practice, they could learn techniques for adding and subtracting with the colored counters and even explain how it works. Eventually they would find patterns in what happened when you add and subtract integers. The same is true for almost every other topic we discuss, from the most basic (addition of whole numbers), to the most advanced (division with fractions, perhaps). You couldn't do all of K-6 in a semester obviously, and children may not make as many connections as a college student ought to be able to, but they could do almost every activity we do in the college course, and learn a lot.

I don't even bother to argue the point with my students 'though, because whether grade school students could do what we do or not is entirely beside the point. My class isn't filled with grade school students. It's supposedly filled with college students. College students who want to be elementary teachers. The same teachers that will lay the next generations mathematical foundations. Which will, in another ten to fifteen years or so, become our next generation of college students sitting in my college classes. And that thought usually fills me with the urge to go lie down for a while.

Risk taking

So I heard passed on a complaint from some employers: That the current workforce is too risk-averse, that they only want to do what is "safe". Or I guess in the usual "business-ese", that their employees don't "think outside the box." This is, I suppose, seen to be a failure of educators. (That would include me.)

I'm inclined to call bullshit.

There's no question that US public education has an unspoken agenda to produce docile, unquestioning workers who will sit in cubicles all day doing mind-numbing tasks and avoid asking difficult questions at all costs. That's actually part of the history of what the public education system was for. But it's worth asking why this was ever a goal, and the answer is because that's what employers wanted.

I'm also inclined to think that people have been encouraged to take fewer risks because risk taking because they have so little overall security. Most people today worry about being laid off at every downturn of the economy. There is no long-term job security any more. And in a particularly screwed up twist, no one gets health care at an affordable cost without a really good job. Plus we have an otherwise generally eroding social safety net. So no one feels safe, and I think a lot of that lack of security can be laid at the feet of corporations that decided short term profits could be had by regularly laying off employees and trying to squeeze more out of the ones left. And now they're complaining that their employees aren't willing to take risks? Why would anyone take risks in such a precarious situation?

Plus I doubt they really want risk-takers. Risk takers might try some crazy scheme that no one ever thought of before, and that scheme may fail. Actually the crazy schemes probably fail more often than not. (How many start-up technology companies did not go on to become Google, Microsoft, or Apple?) I suspect what they mean is that they want employees to take risks doing things that turn out successful. But that's not risk-taking! I don't know. Maybe an employee that does try some wild new idea that fails spectacularly really does get a "Congratulations! You failed!" celebration a la Meet the Robinsons. Maybe innovations (including failures) are actually encouraged by some (or all) employers. But that just doesn't ring true. I think it's the businesses that are risk-avoiders, and the employees are picking up on that and following along.

And employees are easy to blame. What employee would disagree with his or her employer's assessment of the situation? That sounds like awfully risky behavior.

Projects

I gave my numerical analysis students this week off to work on their projects (which they start presenting next week). I told them I would be available during class time for them to ask questions. The difference between doing this with upper level and lower level students is that the upper level students will actually do it. (I was actually a little surprised at how many people I saw.)

I'm actually really pleased with my numerical students right now. Most of them have been working hard on interesting projects. Most of my work in meeting with the students up to this point has actually been in getting students to scale down their proposals to a manageable size. One group of students was originally starting with an ambitious project of figuring out how to guide a robot through an obstacle course using GPS guidance. I initially got that scaled down to just working with the GPS, then got them to massively reduce the number of factors they include in their GPS model, and finally we have settled on doing linear least squares fitting, which is simpler than the non-linear least squares fitting that GPS requires. I'm relieved, and so are they. I knew at the beginning of the semester that this was simply too big, and I think it's finally something they can finish in a reasonable time. (Actually, they are almost finished now.)

Almost all of the projects are progressing nicely and look really interesting. I have student showing how calculators evaluate functions like sines and cosines, another student solving linear systems using iterative techniques, one studying efficient matrix multiplication for graphics applications, a pair of students working on Bezier curves and their use in graphics, a group of three presenting on a numerical simulation of the Tacoma Narrows bridge collapse, one doing arbitrary precision arithmetic, and one doing on the fly polynomial interpolation to compensate for lag in networked computer games. All of the presentations look good. I just need to make sure they don't run too long.

Sometimes I really like teaching.

The grass is always greener

I subbed a class for someone who was out sick this morning. It was an algebra class and the second day the prof was out, so I was asked to try to actually teach something. Unfortunately, I never found out what he was doing in class, so I ended up having to ad lib something about graphs of transformations. (You know, a little algebra improv. "Now pretend you're a function translated to the left two units! OK, now you're getting stretched, s-t-r-e-t-c-h-e-d... AAAAANNNNND freeze! Reflect about the y-axis!" Well, something like that, anyway. Only less entertaining.)

After class I had several students asking me more about who I was and if I taught various courses. (And not in the "Ye gods, I must be sure of who you are so as to avoid scheduling you at all costs, you weirdo!" way, either. They seemed to like me.) This is always a little weird, because usually my own students aren't that crazy about me. They would probably like someone else who came in to sub for me and did some free-form math jazz better. I can just imagine my students all crowding around the other prof, asking if he ever taught the next course they needed to take.

This is why I don't take sick days.

Precalculus Despair

So this semester I'm teaching precalculus. Supposedly the students are already reasonably proficient in algebra.

Good parts of the semester include the fact that I found a book that I like. I really, really like it. It's beautifully focused. It feels like every time I start a new exercise, I think, "Yes--this is exactly what I wish my calculus students understood." And it's beautifully structured, spiraling through topics, adding layers of subtlety with each turn. I'm very happy about having such a good book in part because pretty much every semester I have been here so far, I've used books other people picked out or which were "typical" for a course at the university, and I've pretty much universally despised those books. I've stopped trusting anyone's recommendations.

I also like the way I have structured the semester, with lots of "mini" tests, which are cumulative, rather than two or three "big" tests that students cram for. It keeps the students up with the material, and it also keeps me apprised of where my students are. But this means I really do know how my students are doing, and I'm not feeling as happy about this at the moment. After the second quiz, I know there are many things that many of them cannot do. Many fairly simple things that many of them cannot do. Including things we have done repeatedly since the second day of class.

Part of the problem comes from previous deficits. Many of the students have trouble solving simple equations. Several need to be reminded repeatedly that there are real numbers between 2 and 3. Some are not sure what you might get if you were to square the square root of 5, or that -3 < -2, or whether a squared real number might be negative, or whether one might be allowed to take the square root of zero. (It's zero, by the way.) When students are struggling with these issues, it makes it difficult for them to learn about the domain and range of a function, and what the rate of change of a function on an interval might be, and how to sketch a piecewise defined function. How did these students end up in precalculus? Are they really expected to be able to complete calculus next semester?

So I have many failing students now. And tomorrow I must chide them to get the help they need if they wish to pass. There is still time, but the time to catch up is running out rapidly. And for so many of them, there is so much to catch up on.

Perspective

I had just finished a Calculus II class on antiderivatives using the natural logarithm, and as I was erasing the board (filled with indefinite integrals, u-substitutions, and things like "ln|sec(t)|"), one of my precalculus students came into class to ask me something. I shifted gears to answer his question, and realized that while I considered most of the stuff on the board pretty easy, to a precalculus student, it must look incredibly complicated, and perhaps like sheer gibberish. And of course it seems transparent to me; I've been doing calculus since 1988, so it's been over 20 years now. (Amazingly, things like this have stopped making me feel old.)

It's much like the conversation I had with another colleague once: We were talking about low-level, introductory courses at the university (Big State Tech U), and meant any of the various calculus sequences.* But we observed that for the general population, "calculus" is used as a metaphor for anything unbelievably advanced and difficult. (Sort of like "brain surgery" and "rocket science", although if you are a brain surgeons or rocket scientist, you mastered calculus long ago.) Most mathematicians (and a number of other scientists) see calculus at the starting point for our fields, while most of the general population sees it as the pinnacle of learning.

*Of course, that conversation was at Big State Tech U. Now I do teach at a school that teaches a wide variety of courses lower than calculus, including lots of algebra, a general education math course, courses for elementary teachers, and even remedial courses. Not that I don't still consider calculus the first real college level math class.

Contest Updates

This year I managed to organize our first ever Putnam team, and even got students together for practices. Unfortunately, the students dropped like flies. Two dropped after the first practice, and two more after the second. I ended with enough students to make a team, so we were going to be officially represented--but then one more student dropped at the last minute.

I enjoyed Putnam practice sessions, which I ran a bit like a Moore method class (with me in it). I didn't look up solutions to the problems; we just tackled what we could and tried to come up with ideas. It was interesting and moderately productive. (These are hard problems.) Unfortunately, two very strong math majors chose to participate because they are involved in too many other things. In the end, our Putnam team did not score any points, but our school does appear on the list of "Schools which took the Putnam exam", which I still think is cool. I also feel pretty good that I solved a few of the problems on the exam this year myself. (Did I mention these are hard problems?) Perhaps we will do better next year.

I also got together our department's "College Bowl" team, with significant help from another faculty member. We needed four players plus an alternate, and unfortunately we only had five people come to try out, but we did have some fairly good people. I had some schemes for this year, too. It basically tests how much useless trivia students have memorized, so I obtained for our team used copies of a great popular book on memory systems (to stuff pointless facts into their heads with) and a copy each of An Incomplete Education (full of pointless College-Bowl-style trivia to start stuffing). In the end, we did very respectably in our school; we almost made it to the finals.

I also had fun this year "consulting" with the programming contest team. The programming contests frequently involve some interesting mathematics, so I've been coming to practice sessions and helping them figure out how to tackle the mathy problems. The problems are often pretty cool. Between that and the Putnam, I enjoyed doing some occasional math last year. I also like working with the programming team because it lets the students see professors "bridging the gap" between math and computer science. For some reason, we seem to have a division between the math and computer science majors in the department, which is pretty weird, because there isn't much gap I've noticed among the professors.

So what new contests will the "I-don't-like-competition" guy find himself involved in? Funny you should ask; I actually thought about trying to get together a team for a mathematical modeling contest in the spring. Of course as with the Putnam, what I'm really interested in is getting together students to do some math, not really to compete. I didn't do it this year, but I'm looking into more information about mathematical modeling for next year.

Summer is over (Oh bother)

I did get to spend a little over a month all told with my other half, including about a week spent up here with me. (He hadn't been able to come up before.) That was really good on the one hand, but spending so much time together also made me remember I miss him a lot. I was pretty bummed when I had to leave.

It doesn't help that he had lots of adorable kittens living with him at the time, before they were going to go off to homes. One in particular decided I was the best bed EVER. I'd feel a couple of little paws grab my ankle and hear a "mew!", which I discovered translated to "Sit down so I can lay on you and go to sleep." That's hard to leave behind, too.

In terms of productivity, I went to a conference on Moore method teaching, tried to get back into some research, got (partially) ready for classes, and worked on some projects related to our library. (In particular, given the large temporary budget we had for books last year, I wanted to find ways to encourage students to go visit the library and take out some books. This ended with what I personally consider a rather nifty poster I made and hung in our hallway featuring cool new books we have. I plan to keep swapping out the featured books periodically.) I suppose I could probably count that I did clean the apartment pretty well, even though you can't really tell anymore...

But classes started last week, so I'm through my first week. Highlights from that first week:

• I discovered two students in one of my classes who had taken another class which duplicates it. On being informed that no one can receive credit for both, one was surprised (it turned out her advisor had specifically selected the class for her), and the other said she already knew that but thought she just had to take this one anyway.
• Get the impression our registration system is a little goofy? It is. One of my colleagues and I have conjectured that it does not actually enforce prerequisites at all. Her analysis class contains several students who have not taken one or the other (or both!) of two prerequisite classes. Some of these managed this by simply failing the prerequisite the previous semester, so that they registered before the system knew they failed, but others seem to have been able to slip in some other way.
• I helped with an introduction to the computer software Mathematica for a group of students on Friday. When we arrived at the lab, it turned out that the software was installed (as we had been promised), but that the license had expired. The first twenty minutes (in a fifty minute class) were spent talking the students through the registration procedure.
• One of my classes went more smoothly than it has before, I think because I successfully managed to cut a lot of stuff out of the class time and just leave the students to read and do it. I feel like I need to add a line to the Tao Te Ching: "I teach nothing, and nothing is left untaught." (This actually goes along really well with the Moore method conference I was at this summer, come to think of it.)
• I seem to have become the "contest" guy in the department. I'm trying to organize participation in one national mathematics contest, seem to have volunteered to take over a college bowl team, offered to help with running a small local math contest one professor is organizing, and have signed on to consult with a computer programming contest team. All this from the guy who basically doesn't like competition. One of my favorite authors is Alfie Kohn. Go figure.

It was a busy week, but I guess not too bad. I'm sort of glad it's a long weekend (even though I normally only teach one class Monday anyway). But it seems a shame this all has to start all over again now.

Regional meeting

I went to a regional mathematics meeting last weekend in Pittsburgh, which was fun. One of my colleagues got a van and brought a group of students down to the meeting. One even gave a student talk that evening. Unfortunately, the student government ran out of funds and couldn't pay for the students to stay overnight, but they still got to see some cool stuff. Including a trip with us for Indian food. I think that may have been a rather big leap for them; I've never seen people so happy to see rice and bread. I even got them to try gulab jamun, which is actually the most fantastic dessert ever created, but is described as cheese balls in honey, so I'm not surprised they were a little hesitant. (I told them afterwards, "But now you can go tell all your friends you ate cheese balls in honey.") I did hear that our lunch necessitated a side trip to Wendy's later when the group split up for a while.

I did stay the night, and I actually gave a short presentation on a little problem I started thinking about last summer. So that makes two presentations at meetings this year. I'm beginning to suspect that getting accepted to talk is not so difficult. However, I still have to do the publishing part, and that takes a little more work.

Fun story from the conference: On the first day, the organizers had a survey in which everyone put stickers on a poster to vote for various activities at regional meetings. When we finished, we wrote our name on the back of the sticker sheet and dropped it in a basket for a drawing the next day. When it came time for the drawing, someone stood up and told us, "The janitor threw out the basket this morning..."

Tutors

After Spring Break, I had a flurry of students coming by to ask me how to get a tutor, which they had suddenly decided they needed. I had not seen any of those students in my office hours up to that point.

I'm sure it's a direct result of students going home and hearing from parents that if they're not doing well in their math class, they should get a tutor or something. Unfortunately, they would have done better if they had just kept up and come in for occasional help when they needed it. This may be related to students who will do just about anything for extra credit but who skipped most of the credit that was offered all semester.

The Perversity of Self-Referential Teaching

When I taught my students version one of the Fundamental Theorem of Calculus (the one which tells you how to take derivatives of functions involving integrals), I said:

Isn't this easy? Don't you wish the whole test was going to be this? Well, I will go ahead and tell you that when I put one of these problems on the exam, half of you will forget and miss it.

I say this every time I teach the section, in the hopes that it might make the material stick. This time I went one step further and added:

That will be true despite the fact that I have just told you this.

I thought possibly that this statement might make at least a few more students decide to remember it. (It seemed to make an impression at least; they did laugh.)

Well I gave the exam this morning, and it turns out that in fact more than half the class forgot it. I think somewhere around 80% forgot it. Maybe next time I will try: "Everyone gets this problem every time, so don't bother to study this at all."