Monday, September 27, 2010

The Pinnacle of Journalism?

4,100 Massachusetts Students Prove ‘Small Is Better’ Rule Wrong

I think this title sucks, but what about the message?

Claim: Brockton High School has moved from being unsuccessful to successful in spite of its size, “proving” that size doesn’t matter (tee hee).  This contradicts “certain education circles” that believe small schools are a useful reform tool (to be transparent, I fall into this category…I personally find it much easier to build a positive learning community in a place where people know each other).

After reading, I think the message it also pretty lousy.  In a two page article, the following is the evidence given for how the school was previously unsuccessful: “[a decade ago] only a quarter of the students passed statewide exams. One in three dropped out.”

Pretty damning.  Evidence of improvement? This is it.  All of it. “In 2001 testing, more students passed the state tests after failing the year before than at any other school in Massachusetts. The gains continued. This year and last, Brockton outperformed 90 percent of Massachusetts high schools.”

Don’t get me wrong.  This is an improvement worth commending.  Regardless of where you stand on the importance or measure of state assessments, it’s no small feat to make this kind of improvement. But the skeptic in me didn’t linger long in this success.  What about the dropout rates?   Here’s an interesting article from the local newspaper that appears to contradict the “one in three dropped out” fact and claim that, in fact, the drop out rate was lower in 1997-1998 than last year (about 3.5%).  I guess it is possible that both of these statements are “correct” since drop out rates are notorious for statistical meddling. 

Most of the remainder of the article talked about how these reforms were put in place which, again with the transparency thing, some of which made me cringe.  The sole example of a math lesson: “Bob Perkins, the math department chairman, used a writing lesson last week in his Introduction to Algebra II class. He wrote ‘3 + 72 - 6 x 3 – 11’ on the board, then asked students to solve the problem in their workbooks and to explain their reasoning, step by step, in simple sentences.  ‘I did the exponents first and squared the 7,’ wrote Sharon Peterson, a junior. ‘I multiplied 6 x 3. I added 3 + 49, and combined 18 and 11, because they were both negatives. I ended up with 52-29. The final answer was 23.’  Some students had more trouble, and the lesson seemed to drag a bit.”

Really?  Drag a bit?  Please shoot me now.  Oh, keeping with the theme of transparency, I should say that I think this ‘lesson’ is worse than useless from a mathematical or a writing perspective.  Alas.

“It had become dogma that smaller was better, but there was no evidence,” said Mr. Driscoll, who since 2007 has headed the National Assessment Governing Board, which oversees federal testing.

Nice logic buddy.  Test scores now > test scores ten years ago imply large school = good.  Ergo, small school ≠ better.

Ack.  Apology for all the snark.

Thursday, September 23, 2010

For an apple and I'll let you paint this fence

So this'll be fun.  If you were to write a 10-minute-ish individual assessment on pattern sniffing for...ah, you pick the age (or tell me it doesn't matter)...what would it look like?

Here's my most recent definition of pattern sniffing:

  • On the lookout for patterns
    "Ok.  We’ve been working on this staircase problem and it seems that you can’t write powers of two as a sum of consecutive whole numbers."
  • Looking for and creating shortcuts
    "It would be nice if there were a faster way to find the greatest common factor of 2 numbers other than listing all the factors. Think we can find a way?”
Challenge question: How would you grade this little quiz of yours?

EDIT:  I should be clear.  The end goal is for students to be able to apply these habits in unfamiliar problems without a big neon sign saying *Use Pattern Sniffing*.  This is in no way an assessment of that end goal, but the first step in what I suspect will be a relatively long road.  

Wednesday, September 22, 2010

Confessions of a Math Teacher

I believe that my job as a math teacher is not to make things as easy as possible for my students.  I do not give them the shortest explanation with the most straightforward formula and ask them to solve different versions of the same exercise for homework and tests.  I do not tell kids how to do problems from the next section in the textbook so that their 1-29 odd homework will be as painless as possible.  I will not become impatient and compromise a deep conceptual understanding and let kids get away with just knowing the procedure.

Or at least I try really hard not to do these things.

I always let kids explore an idea and try to build an understanding themselves before I tell them anything.  I will give homework problems that are unlike any problem we've talked about in class.  I do think "play with it...look for patterns...make observations" is a legitimate homework assignment.  I do ask students to work on messy problems, problems that I don't know the answer to, and open problems.  I have let kids leave the classroom with a misconception that I know about and did not address.

Or at least I try really hard to do these things.

I see the need to teach math as a verb.  I teach the activity of doing math as much as I teach the subject of math.  This means teaching kids to enjoy, embrace, and feel confident working with hard problems that they won't solve in thirty seconds, five minutes, one sitting, one week, or maybe ever.  This means caring about beliefs, perceptions, and attitudes as much as I care about fractions.  I want kids to enjoy the process of thinking, playing, trying, failing, and lots of other -ing words.

But this post isn't my call for you to see the wisdom of my ways and throw out your lesson plan for tomorrow.  No, it is a plea for help.  Tonight I don't need help with fractions.  I need help with teaching the activity.

This student has been on my radar after an email from the father about frustration at home and "disliking math for the first time."  A conversation was had, but I now know that the attitude equivalent of a deep conceptual understanding has not been made.  The one positive note is that the student is talking to me about this instead of repressing these feelings that will develop into yet another adult who hates math.

Via email: "  I just want you to know that even though you said [all] the [homework] problems aren't mean't [sic] to be finished sometimes when you hand in unfinished work it can make you feel stupid and like you failed. I thought that that was something you should know."

So any ideas on how to teach "the activity of not finishing problems in five minutes" to a super smart sixth grader who still participates actively in class but is clearly not enjoying math on the home/homework end?  If it helps, here and here are the homework assignments that sparked this frustration.

It also might be helpful to know that my policy is that students should spend no more than 30 minutes on homework.  If they spend any more than 30 minutes it should be because they'd rather be doing this than playing, reading, running around outside, talking with friends, or whatever else they're allowed to do.  For these two assignments, the first 2 problems gave me all the information I needed to know about their understanding of the content at hand.  Aside from the students who tried to compute 2 to the 57th power, everyone made it past #2 but only a handful finished the sheet(s).

Friday, September 17, 2010

Beliefs and Attitudes about Mathematics

What beliefs and attitudes about mathematics do you see in your students, in society, in the media, and elsewhere? Try and think of both positive and negative beliefs and attitudes. These can be beliefs that you agree or disagree with. I'll start with a few, but please add your own in the comments.
  • If you're good at math, math problems can be solved in a relatively short amount of time.
  • People do not solve math problems for fun; they do it for school, for their job, or to balance their checkbook.
  • Every math problem has been solved by someone.
  • Math is about numbers.
  • Math is a language to describe the world.
  • If you are good at math, you are smart.
  • If you can do computations accurately and quickly, you are good at math.
  • People who are good at math are eccentric and/or not socially adept.
  • Boys are good at math.
  • Asians are good at math.
  • There can only be one correct answer.
  • If I don't know how to solve a homework problem, I must be doing something wrong.
  • Math topics/classes are sequential; I need to understand A before I can learn B.
  • It is socially acceptable to say you're bad at math.
  • Math is more analytical than creative.
  • Using a procedure correctly to get the right answer is more important than understanding why the procedure works
  • With current technology, arithmetic is not important.
  • To be an engineer, you need to be computationally strong.
  • Math is a gatekeeper.
  • The value of math is in its connection to real world applications.
  • Math teachers sleep under their desk at school. 
Ok, I guess that was more than "a few".  Hopefully I left some for everyone else.

Sunday, September 12, 2010

Fun with Data Fitting

Holy page hit batman.    I will readily admit that I am a curmudgeony luddite when it comes to twitter.  The format perpetuates what I see as a disturbing trend towards shortened attention spans and disinterest in depth or subtlty. …or even knowing how to spell subtlety.  To be honest, though, my feelings may also stem from the question “What has twitter ever done for me?” 

Well, this has been a big week for me in the educyberbloggerinterwebmathsphere.  Firstly, I was really excited to meet three—count ‘em three—bloggers that I follow at an evening professional development get together where I may or may not have actually made the comment “One-half is not equal to two-fourths.”  It’s a long story that may or may not deserve a separate post, but somewhat related to Simpson’s Paradox. I had already put a face to Dan because I was blown away by his presentation last winter at CMC-North and, frankly, he’s sorta’ famous in that “famous within a specific subset of a specific field” sort of way. Nonetheless, it was great shaking his hand (ok, the handshake was actually just ok, but subsequent conversations were great).  I had not had a chance, though, to put a face to Jason or Sophgermain (although M. LeBlanc might be more appropriate).   

It appears that I also owe Dan and, begrudgingly, twitter a big thanks because my recent post about mathematical habits of mind was “picked up” via twitter by The O’Reilly Radar (and I thought they just wrote nerdy computer science books) and (which I had never heard of, but it appears that other people have).  Anyway, the table below says it all.

Page hits
Friday, August 20th
Friday August 27th
Friday, September 3rd
Friday, September 10th

It’s growing exponentially!!  And by exponentially, I mean fast.  And by fast, I mean something completely different from exponentially.

So don’t fall for that malapropism.  With a little data fitting, you can instead see that these numbers fit the function:

f(t) = 1063t3 -6368t2 + 11670t - 6361 (with a little rounding)

where t is the number of weeks after August 13th and f(t) is the number of page hits.

What does this mean? The table below says it all.

Page hits
Friday, August 20th
Friday August 27th
Friday, September 3rd
Friday, September 10th
Friday, September 17th
Friday, September 24th
December 31st, 2010
December 31st, 2014

Since this is greater than the United Nation’s projection of the world population at the end of 2014, this is unequivocal proof of either:
1.     Extraterrestrial intelligent life in the universe that will be reading my blog
2.     A tea party landslide in the 2010 midterm elections leading to a universal ban on contraception
3.     Bad math

I’ll let you decide.

Friday, September 3, 2010

Habits of Mind

This is still a work in progress (and feedback would be greatly appreciated), but I've decided to explicitly teach (and assess...more on that later) 4 "categories" of mathematics this year.
  1. Skills (I know how to...)
  2. Concepts (I understand and can explain why...)
  3. Connections (I see and can explain the relationship between...)
  4. Mathematical Habits of Mind (I can use and appreciate the process of...)
I've decided not to use the term "problem solving" because I believe this term is often misused to include be limited to solving problems and because the motivation for problem solving skills seems to be to solely help you get an answer.  While I believe that they can be very helpful in finding answers, I see mathematical habits of mind as also being mathematical in and of themselves.  So...while searching for patterns may help you solve a problem it is also DOING mathematics.

Here's the current version of the mathematical habits of mind I think are important.  I hope to explore (in varying depths) every one of these and have already shared the list with my 6th graders.

This is definitely a work in progress and some of these are based on work by Cuoco, Driscoll, Schoenfeld, and others.

Habits of mind
1.    Pattern Sniff
A.     On the lookout for patterns
“Ok.  We’ve been working on this staircase problem and it seems that you can’t write perfect squares powers of two as a sum of consecutive whole numbers.”
B.     On the lookout for Looking for and creating shortcuts
“It would be nice if there were a faster way to do 57x34 than adding 57 to itself 34 times. Think we can find a way?”
2.    Experiment, Guess and Conjecture
A.     Can begin to work on a problem independently
“I’m not sure how to solve this problem, but I’m confident I can make some progress.”
B.     Estimates
“Without doing any calculations, I’m guessing that it will take him 30 seconds to walk up the down escalator.”
C.     Conjectures
“Based on my work, I think the following is true.”
D.    Healthy skepticism of experimental results
“Boy, it sure seems like this 4, 2, 1 thing always repeats but we don’t have a proof yet.”
E.     Determines lower and upper bounds
“I know it will take the people at least 10 minutes to cross the bridge because the 10 minute soldier has to cross the bridge.  I also found a solution that takes 19 minutes so I know the final answer is somewhere between 10 and 19 minutes.”
F.     Looks at small or large cases to find and test conjectures
“I made a table of the first 5 cases and I think I see a pattern.  I’m going to see if this pattern holds for the 100th case.”
G.     Is thoughtful and purposeful about which case(s) to explore

H.    Keeps all but one variable fixed
“So I’m exploring the equation y=mx+b and I’m wondering how the graph changes as m and b change.  For now, I’m going to set m to 1 and just look at how the graph changes when I change b.”
I.      Varies parameters in regular and useful ways
(Even/odd example)
J.      Works backwards (guesses at a solution and see if it makes sense)
3.    Organize and Simplify
A.     Records results in a useful way
“I’m going to make a table.”
B.     Process, solutions and answers are detailed and easy to follow
C.     Looks at information about the problem or solution in different ways
D.    Determine whether the problem can be broken up into simpler pieces
“I think I can solve this problem by solving these other 2 simpler problems.”
E.     Considers the form of data (deciding when, for example, 1+2 is more helpful than 3)
“I’m going to leave my fraction as 6/36 because the 6 represents the number of ways you can roll a 7 with 2 standard dice and the 36 represents the total number of rolls.”
F.     Uses parity and other methods to simplify and classify cases
“Next time we play 21 Nim I’m going to keep track of whether the running sum is a multiple of 3, one more than a multiple of 3, or 2 more than a multiple of 3.”
4.    Describe
A.     Verbal/visual articulation of thoughts, results, conjectures, arguments, process, proofs, questions, opinions
B.     Written articulation of thoughts, results, conjectures, arguments, process, proofs, questions, opinions
C.     Can explain both how and why
“The algorithm for dividing fractions is simple.  Now I just need to work on making sense why this works.” 
D.    Creates precise problems
E.     Invents notation and language when helpful
“For the sugar weighing problem, I don’t want to have to write out every solution in words so I’m going to let the symbol 3w~3s stand for the act of putting the 3 pound weight on one side of the balance scale, measuring out 3 pounds of sugar on the other side of the scale, and then setting aside the sugar.”
F.     Ensures that this invented notation and language is precise
“I need to be careful that I am differentiating between sugar that I am measuring and sugar I am using as a weight.”
5.     Tinker and Invent
      A.   Creates variations
B.     Looks at simpler examples when necessary (change variables to numbers, change values, reduce or increase the number of conditions, etc)
C.     Looks at more complicated examples when necessary
D.    Creates extensions and generalizations
E.     Creates algorithms for doing things
F.     Looks at statements that are generally false to see when they are true
G.     Creates and alters rules of a game
H.    Creates axioms for a mathematical structure
I.      Invents new mathematical systems that are innovative, but not arbitrary
6.    Visualize
A.     Uses pictures to describe and solve problems
B.     Uses manipulatives to describe and solve problems
C.     Reasons about shapes
“I see how this structure is made.”
D.    Visualizes data
E.     Looks for symmetry
F.     Visualizes relationships (using tools such as Venn diagrams and graphs)
G.     Vizualizes processes (using tools such as graphic organizers)
H.    Visualizes changes
I.      Visualizes calculations (such as doing arithmetic mentally)
7.    Strategize, Reason and Prove
A.     Moves from data driven conjectures to theory based conjectures
B.     Tests conjectures using thoughtful cases
C.     Proves conjectures using reasoning
E.    Looks for mistakes or holes in proofs
F.  Uses indirect reasoning or a counter-example (Park School)
E.  Uses inductive proof
8.    Connect
A.     Articulates how different skills and concepts are related
B.     Applies old skills and concepts to new material
C.     Describes problems and solutions using multiple representations
D.    Finds and exploits similarities between problems (invariants, isomorphisms)
9.    Listen and Collaborate
A.     Respectful to others when they are talking
B.     Asks for clarification when necessary
C.     Challenges others in a respectful way when there is disagreement
D.    Participates
E.     Ensures that everyone else has the chance to participate
F.     Willing to ask questions when needed
G.     Willing to help others when needed
H.    Shares work in an equitable way
I.      Gives others the opportunity to have “aha” moments
10. Contextualize, Reflect and Persevere
A.     Determines givens
B.     Eliminates unimportant information
C.     Makes and articulates reasonable assumptions
D.    Determines if answer is reasonable by looking at units, magnitudes, shape, limiting cases, etc.
E.     Determines if there are additional or easier explanations
F.     Continuously reflects on process
G.     Works on one problem for greater and greater lengths of time
H.    Spends more and more time stuck without giving up

Thursday, September 2, 2010

My first day

It's taken me a little while to find the time to post this, but I wanted to post my first day experience.  Why?  Because it's great to be back!

The context
* 6-12 all girls independent school
* 1:1 laptop school (starting this year)
* 16 students in each class
* 4 desks of 4 students each
* 4 classes of 6th grade
* 50 minute period 3 days a week, one 75 minute period per week (staggered by class)
* most all students have (so far as I can tell at this point) relatively strong, but varying math background
* most students have very strong reading/writing skills
* every student has a (so far) positive feelings towards school

Cushy, I know.  I very much understand and appreciate the challenges that I don't have to face (large transient classes, different home languages, severe learning differences, unstable families, etc).  That being said, I am confident that this lesson would work in any classroom because 2 years ago I did a similar first day lesson in a very very different school; it was probably the best lesson of that year.

The lesson
We were on a special first day schedule so I saw each class for 60 minutes.

Introductions: names and something you enjoy outside of school (5 minutes)

The Desk Problem: Look at the people you are now sitting with.  In a short while, I am going to give you 5 minutes to get up, talk, strategize, and rearrange yourselves so that everyone is sitting with as many new people as possible.  I'll then ask each person individually how many new people they are sitting with.  If you're sitting with 2 new people, you get 2 points.  3 new people, 3 points.  I'll then add up all the points and that's your class score.  Let's see if you can beat my other classes.

This was inspired by a boat problem that I don't remember the details of involving fisherman sharing boats with different people every day of the week.

I tend to assume kids are going to figure things out faster than they actually do, but every one of my groups surprised me on this one and within 30 seconds figured out that one person from each table needed to go to 1 of the 4 tables.  After congratulating everyone on their success and talking a little about different methods of getting the 48 points, we brainstormed ways to "tweak" the problem in order to make it more challenging.  Some of the ideas:
  1. Winning class is the class to get 48 points as quickly as possible.
  2. Changing the number of people, tables and/or chairs at each table.
  3. Conducting multiple rounds where you only score points for people you've never sat with.
  4. Forcing every person to change tables
I haven't worked on it at all, but I think the problem of finding a maximum score as a function of # of people, # of tables, and # of chairs at each table would be an interesting problem to work on.

Breaking the code: I then gave a few short directions for quality group work (everyone's contributing, everyone's voice being heard, and not making any decisions without everyone understanding and being on board, no questions for me unless it was a question everyone had) and handed this out with no further directions.  They worked on this for the remainder of the period.

Some notable observations:
  • no table asked "What are we supposed to do?"
  • no table complained "We haven't learned how to do this."
  • every table eventually asked "Do we have to decode the whole thing?" which led to some variation of the following conversation:
    Me: When do you think it would be fair to say you're done?
    Them: When we figure out the code for every letter?
    Me: Sounds good to me.
  • Every student was engaged until the end of class
  • Not every table did a great job of following the rules of group work
At the end of class, I handed out the homework, which kids quickly figured out was the decrypted version of what they'd been working on.

Great first day!  See you tomorrow.