Learning Fractions at 40

The entrance of Evan's new middle school.

Learning Fractions at Forty

I am a forty-one-year-old man with undergraduate and post-graduate degrees and yet I cannot subtract or divide fractions.  Never could.  A standardized test from my fourth grade revealed higher than average aptitude for reading and language arts (though I tested poorly in punctuation–two correct, five incorrect), but lower-than-average math computational skills.  In every standardized test after that, 6th, 8th and 10th grade, I tested below average in math (and above average in everything else).  The results were embarrassing.  6th grade:  2/6 computing fractions; 4/12 computing percentages.  8th grade:  4/13 computing fractions; 0/7 signed numbers; 0/6 geometry.  10th grade: 100% multiplying fractions, but 0% dividing them and 75% adding and subtracting.  Huh?   In four years no-one could teach me how to add, subtract or divide fractions?   I still have the letter from the Lafayette (LA) Parish School Board that accompanied test results from 6th grade, May 20, 1981:

Dear Parent,

This is the first year that SRA (Science Research Associates) Achievement Series was given in the public school system.  This standardized test will make information available for comparing your child’s progress from year to year.  The Individual Skills Profile will provide information that will be used as an educational tool to help your child capitalize on his or her strengths and remedy any weaknesses.

I scored so well in Language Arts and Reading that I was placed in Advanced Literature Classes and ultimately entered a profession which leans heavily on reading and language arts, so one could presume teachers and schools helped me capitalize on my strengths.  Yet we never seemed to remedy my weakness in math.  The same Sophomore year that I scored a 0% on dividing fractions I skidded-by Algebra II with a final grade of 75 ‘D’.  I had failed Algebra I the year before forcing me to try again during summer school.   I never attempted Geometry, Trigonometry or Calculus in high school or college choosing, instead, lightweight math courses with equally lightweight names:  “Consumer Math” and “Math and Society”.   That is how public school helped me remedy my weaknesses, avoid heavy lifting.

Maybe my math failure was the school’s failure?

Or maybe it was a parenting failure?

Or maybe I just didn’t apply myself?  Let’s start there.

Applying Ourselves Toward Learning:  Spinning our own kaleidoscope.

We educate ourselves.  I just logged onto khanacademy.org  to learn how to divide fractions.  Easy.  You just flip the numerator with the denominator of the divisor and then multiply numerators and denominators.  Back in 10th grade I scored 100% on multiplication of fractions but 0% of division.  I’ve got two explanations; I either missed class on the day they discussed the inversion or I simply forgot the operation.  Either way it is my fault.  If I miss a class I have to track-down the lesson that I missed.  And what school or teacher can be held responsible for a student forgetting a lesson?  Students are expected to practice operations until they become second nature.  I didn’t practice enough.

But wait a minute; we forget things that don’t matter to us.  I used to know my 10th grade locker combination but have since forgotten it.  On the other hand, I recall that my wedding anniversary is September 18th.  Why should I care about fractions?  There is this great resource called Real World Math  that offers clever, modern day questions that reveal the usefulness of algebra (interpret a cell phone bill why don’t ya?) , statistics and yes, fractions.   We need fractions for cooking in the United States (and for using drill bits and wrenches).  Chefs and mechanics need fractions.

Should I learn fractions BEFORE I learn that I will need to cook or change my own oil?  Shouldn’t it be the other way around?  Former California school teacher Grace Llewellyn authored “The Teenage Liberation Handbook: How to Quit School and a Real Life and Education” in which she encourages teenagers to educate themselves by admiring the things that “still sparkle in your kaleidoscope.”   Imagine if I had  been encouraged to fully embrace and enjoy cooking as a entree to math!

Maybe my inability to compute fractions on that state test said more about the standardized test and schooling method than about my aptitude.

I Prefer Not to Take Your Test:  A New Day for Compulsory Education

Maybe the school system was incapable of remedying my math weakness.  Llewyln offers a list of ways schools prevent people from learning:

  1. Schools require passivity.
  2. Schools cram you too full too fast.
  3. Schoolpeople care more about appearances than about learning.
  4. School isn’t challenging enough if you are academically inclined.
  5. Schools present learning backwards, emphasizing answers instead of questions.
  6. Schools ask you to get stressed out attaining mediocrity in six or so subjects rather than be amazing at one or two you love.
  7. Schools are overly obsessed with and manipulative of the learning process.
  8. School won’t answer the door when real chances to learn come knocking.

What do all of these  exercises in missing the point do to us?  According to Jerry Farber in his  1969 book “The Student as Nigger”, they enslave us.   Farber offers a relentless deconstruction of public schooling:

You know how malleable we humans are.  So imagine what the effect must be upon our apt and impressionable minds of a twelve-year course in servility.  Twelve years of tarty bells and hall passes; of graded homework, graded tests, graded conduct; of report cards, GPA’s, honor’s lists; of dress codes, straight lines and silence.  What is it they’re teaching you?  Twelve years pitted against your classmates in a daily Roman circus.  The game is Doing What You’re Told.  The fear of losing the game is a great fear: it’s the fear of swats, of the principal’s office, and above all the fear of failing.  What if you fail and have to watch your friends move past you in glory?…Why does the medium of education affect us so deeply while it’s purported content [like fractions?]–the subject matter–so often slips our minds?…Under a coercive system it isn’t really the subject that matters; what matters is pleasing the authorities (p.19-20).

I bought this book second hand for 25 cents back in the days of my radical youth.  I’ve mellowed since then.  Things aren’t now as oppressive as I remembered them back then.  I was paddled with a ping pong paddle for throwing pine cones over a fence into a stream.  Even the Junior High bus driver hand a big wooden paddle swinging back and forth on his dashboard. But there are no more paddles and there is no more oppressive coercion, right?

I started reading John Taylor Gatto’s 2010 polemic “Weapons of Mass Instruction” where he recommends children resist standardized testing by writing “I prefer not to take your test.”   I read this in Barnes and Noble beside my eleven-year-old son and I ask him what would happen if he did that.  First he responds “Why would I do that?”  Hmm, why wouldn’t he want to spend a quarter or more of his time preparing for and taking a test that allows the school to ferret-out ineffective teachers (whom he may actually adore) and receive more federal money?  We leave that question alone and I ask him to simply imagine what would happen.  “I’d probably be sent to the principal.”  And?  “I’d probably be expelled.”  Anything else?  (I was hoping he would admit that they would ask him WHY he didn’t want to take their test, just as he had asked the question.)  He surprises me; “You and mom would probably lose your jobs because the principal would report it to his boss.”

John Taylor Gatto seems utterly disgusted by his experience teaching in public schools in spite of the fact that he won a teacher-of-the-year award the very year that he resigned in contempt.  He harps on public education’s Prussian origin faulting Massachusett’s own Horace Mann for importing compulsive schooling meant only to regulate wild behavior turning out compliant workers and complacent citizens.

Work in classrooms isn’t important work.  It fails to speak to real needs pressing on the young.  It doesn’t answer burning questions which day-to-day experiences force upon young minds.  The net effect of making work abstract–subject centered–external to individual longings, fears, experiences, and questions, is to render students of this enforced irrelevance listless and indifferent (p.64).

I have often heard from teachers that they would love to focus on the the burning questions of individual students.  That is what they signed-up to do!  But they are forced by bad student behavior into the roll of traffic cop.  Remember Natalie Munroe who was suspended from teaching duties when she Blogged about fantasy responses she would like to write to parents on kids’ report cards:

  • “Rude, beligerent [sic], argumentative f**k.”
  • “Am concerned that your kid is going to come in one day and open fire on the school. (Wish I was kidding.)”
  • “I called out sick a couple of days just to avoid your son.”

In the 2007-2008 school year 47% of schools (38,500) either suspended for over five days or expelled students for behavior problems, usually fighting.  The data from the National Center for Educational for Educational Statistics is overwhelming.  94% of High Schools and 65% of Middle Schools reported at least one violent crime within that one year period.  It would seem that administrators must necessarily place  higher premium on rules and order than on other goals related to learning subject matter.  At our Middle School orientation three principals spoke with great humor, at length,  about appropriate clothing for school–girls cannot wear flip-flops but boys CAN wear their larger-single-strapped sandals–and said very little about why they are studying what they are or why they are now in diverse classes.  The student handbook is 90% disciplinary issues.  Shouldn’t we expect public schools to train young people to be obedient?

But this seems to be a chicken/egg question:  Should we endorse a coercive, “Prussian” style of education BECAUSE kids tend to behave disrespectfully and aggressively or should we endorse an alternative to the present compulsory school system BECAUSE it leads to inconsiderate behavior?

Psychologist B.F. Skinner demonstrated the effects of operant conditioning on “Baby Albert”.  The year-old baby was introduced to numerous live animals, large and small, cuddly and not, including a rat.  The baby was curious and demonstrated no discomfort.  Then Skinner placed the rat near the child while an assistant banged on a pipe so loudly that the baby cried.  They did this five times.  Later, when Skinner brought the rat to the child (and didn’t bang the pipe) the baby cried and tried to escape.  The discomfort transferred to other animals as well.  Even the cute bunny.  So when I think back to my school years I remember the  clanging pipes–the bells, the bullies, the threats against chewing gum–and I scarcely remember curiosity about anything, only fear.

Is there any better design out there?  After reading John Taylor Gatto I needed some balance, something constructive.  The title “A Chance to make History:  What Works and What Doesn’t in Providing an Excellent Education for All”  by Wendy Copp (2011) fit the optimistic bill.

Copp is the founder of Teach for America which actively recruits, trains and places motivated teachers in at-risk settings for at least two years.  The book is full of program-promotion of that methodology for improving schools, but there were other great insights.  First, after recognizing our schools are indeed failing many students, especially inner-city students, Copp asserts that there are no “silver bullet” solutions.  We can’t simply throw more money at schools (not only does that not guarantee performance, it sometimes discourages it), or change to better curriculum or the best software, or blame and fire teachers.  on that last point, however, Copp bows to Jim Collin’s “Good to Great” management ethic which insists that before any real organizational improvement can take place, management has to get the right people on the “bus” and the wrong ones off.  Copp cites studies revealing that those who enter teaching had the lowest GPA’s and claims we need to reverse the school mindset that “We’ll take what we can get” which has led to uninspired teaching.  Here’s a quick example of what that kind of mediocrity looked like from a kid’s perspective:

I remember the stash of Payday bars Mrs. Payday [“Career Choices” teacher] kept in her desk.  She never gave us any.  She ate them while she took attendance and gave lectures from her chair.   Stand up.  Stay on two feet throughout class.  You can’t be a role model from your seat.  Be energetic. Be inclusive.  Be strong.  Be nuts if you have to.  Don’t be the teacher whose name I can’t remember.  Don’t be Mrs. Payday.  (“Be Honest: And Other Advice from Students Around the Country” p.178)

Copp’s main solutions seem to be refinement of the leadership pipeline (better mentoring programs, career ladders and performance pay) and improved learning loops (do a better job of collecting and analyzing data, establish a purposeful, workable theory, train toward that theory and then ruthlessly evaluate the theory and training based on results–data).  She also encourages poltical advocacy at all levels for improved education. But what I enjoyed most were examples she shared of cutting-edge schools like YES and KIPP and UnCommon Schools. I am most intrigued by the School-of-One model. (Maybe it has something to do with the great graphics!)

This graphic first reminds me of traditional schooling and traditional pedagogy–the teacher has the goods and everyone lines up in front of the spigot to get some.  When I tried to learn math this way I was lost in the crowd.  The sign over my head would have read “Needs to learn math in a practical context.  Try a cooking demonstration where he is forced to figure it our for himself.”  The second thing this graphic gave me was a sense of hope.  I imagine teachers already think of their students this way to some extent, but they cannot daily triage these different learning styles and needs.  The autistic, possibly abused kid who hates handwriting might get the most attention while the 10th grade reading level student doesn’t even really need the spigot since she brought her own.  My son is just wrapping-up his second week of school.  He is complaining about his math assignment “expanding decimals”, not because he doesn’t understand it but because it takes him three times as long to copy the question from the textbook onto his loose-leaf paper and THEN solve it (33.135=30+3+.1+.03+.005, I think).  Earlier that day Evan hears me drum my fingers on the steering wheel and tells me he is doing that in school more and more.  Why?  Where?  Mostly math.  He is bored.  I tell him many teachers will review what students should know in order to build on that foundation but I also tell him he should gently let his teacher know that he wants to be challenged.  That night he informs me that he told the teacher he needed a bit more challenge and she responded “Well, we move a little slower in the beginning.”  Do WE have to move at the same pace?  For me it was too fast, but for Evan it is too slow.

The School of One further illustrates the dilemma:

This is difficult task of bridging learning styles/needs with resources.  I am the kid who needs to cook in order to understand fractions.  On the right hand side there is a computer program that allows me to do that.  If Evan and I were in the same class he simply needs he printed worksheet to complete so he can move on.  Some other kid needs the textbook and a pretty good amount of coaching.  The traditional model only allows the teacher 45 minutes and limited real estate to teach, so she has to find one or two resources that work best for most.

The School of One relies on a software algorithm that takes input from the student (which takes place at home after completion of an assignment) and matches resources with that student’s unique needs and pace.  The resource schedule is then sent to the teacher and student.  That next day some of the kids will be working solo, some will be in small groups and some will work directly with the teacher/coach.  The teacher still offers lectures part of the time, but a greater percentage of his time and preparation would be individualized.

It appears likely that my son’s school will start using something called “Assistments”, a program developed by Worcester Polytechnic Institute right here in Worcester.  The program received millions of dollars in grant funding so that an idea hatched in 2003 might be GIVEN to public schools all over the country between 2011 and 2012.  It is intended as a WIKI which means that no corporation will own the content (HURRAY!) and no school would have to pay for service (HIP-HIP, HORRAY!).  As far as I can tell the program offers the same kind of real-time evaluation of student performance and comprehension of daily assignments that “School of One” offers.  Teachers create an assignment online through Assistments.org (using open-source lessons which appear at this point to revolve around standardized tests), students access that assignment, receive coaching (watch the video below to see how that works), and then the teacher can access the spreadsheet showing where a majority might have missed details.  That can be addressed the very next day BEFORE any other content is added.

I am impressed by the program and hope  it will one day include the kind of customized, next-day feedback School of One models.  As the video shows, the lesson itself is customized so that a kid who has no idea how to answer a problem moves through a series of steps that help to establish a foundation of understanding so that he will ultimately be able to solve the first problem, or one like it.   So if I get all of them right I can go on to more difficult stuff, right?  Then is there any point to attending class the next day while others figure out what they did wrong?

Shelley Blake-Plock listed among  21 Things That Will Be Obsolete in Education by 2020 this #12:

12. Centralized Institutions
School buildings are going to become ‘homebases’ of learning, not the institutions where all learning happens. Buildings will get smaller and greener, student and teacher schedules will change to allow less people on campus at any one time, and more teachers and students will be going out into their communities to engage in experiential learning.

These wonderful new innovations from School of One and WPI are harbingers of that shift from classroom-based learning to community-based learning and the end not only of the so-called “Prussian” style of education but also of compulsory education itself.  If so, who will “mandate” learning?  The Wiki-nature of the Internet will offer gobs of content to learn (see Khan Academy, for example), so we don’t need textbooks, but who will help us prioritize when to learn what.  Who will motivate me to WANT to learn fractions?

“Home-Schooling”: Somewhere between helicopter and hot-air balloon.

My dad was an engineer who loved math.  If anyone could convince me of the joy of computation, it would be him.  So it surprises me that I did not develop an appreciation for math or even the simple ability to remember how to deal with fractions.  Was I too embarrassed to admit that I wasn’t understanding what the teacher was offering?  By the time report cards came out I was already deep in the rough off of the fairway, hunting for that little white ball.  Did he just give up on me?  Or was he too busy to teach me his love for math?  I am curious about what happened to me in 4th, 6th and 8th grade because I still carry a level of guilt and a feeling of incompetence that has been remarkably constant through my life.  This exercise has been a way for me to make some sense of what happened to me.  But my larger concern is for my son and for other kids who may be like me.  They need schooling which is an improvement on what I experienced and on what is currently available.  And they need to WANT to learn.  How does that happen?

Parker Palmer, wrote:

“The world needs people with the patience and passion to make the pilgrimage [toward the true self] not only for their own sake but also as a social and political act.  The world still waits for truth that will set us free—my truth, your truth, our truth…Cultivating that truth, I believe, is the authentic vocation of every human being”  (“Let Your Life Speak” p. 36). 

I knew a teenager in Chicago who was bullied so badly that she stopped attending class and spent the equivalent time in Barnes and Noble Booksellers educating herself.  Her mother saw the results of her work and supported the truancy.   The girl went on to gain her GED, attend college early, graduate early and then complete Master’s then PhD degrees.   She certainly found the sparkle in her own kaleidoscope!  And I think her mother was instrumental in that alternative education.  I knew her mother.  Mom modeled curiosity, was aware of her own “kaleidoscope” of interests and was deeply rooted her own vocation. But most importantly, mom made the pilgrimage of discovery with her daughter, even when it became unconventional.

Parents like me have to figure how to accompany children on that journey of discovery walking neither to far ahead nor too far behind.

Natalie Munroe was suspended from her teaching position for foolishly publicizing her personal feelings about the kids in her classes AND their parents:

  • “Just as bad as his sibling. Don’t you know how to raise kids?”
  • “Nowhere near as good as her sibling. Are you sure they’re related?”
  • “Shy isn’t cute in 11th grade; it’s annoying. Must learn to advocate for himself instead of having Mommy do it.”

The teacher ended her ranting blog with this: “These comments, I think, would serve me well when filling out the cards. Only, I don’t think parents want to hear these truths. Thus, the old addage [sic] … if you don’t have anything nice to say … say ‘cooperative in class.’ ”

Earlier this September, Ron Clark wrote an article appearing on CNN.com entitled “What teachers really want to tell parents”.  He writes about an award winning principal he knows who is leaving public education:

I screamed, “You can’t leave us,” and she quite bluntly replied, “Look, if I get an offer to lead a school system of orphans, I will be all over it, but I just can’t deal with parents anymore; they are killing us.”

I sympathize with educators on this one because in my capacity as a pastor I have worked at educating youth and found parenting on a spectrum ranging from interference through hyper-involvement (helicopter) to non-cooperation through detachment (air balloon).   I worry that the parents who drop their kids in my class with no concern about the method or content will likewise drop the kids in front of a TV or other confusing, manipulative media with zero concerns while their job or some other “balloon” floats them away.  I almost think I prefer to work with parents who are over-involved, who send me long e-mails attempting to correct my interpretation of some detail or who find creative ways to speak to me through their child in a class settings.   I can work with interference in a way that I cannot work with apathy.

I am not an apathetic parent.  Neither am the kind of parent who disenfranchises my son by arguing his case for him to a teacher.  That is the least I can do.  But here’s what I can do for extra credit.  I can recognize those things that sparkle in my child’s eyes, something which requires daily attendance.  And I can also walk alongside the public school system with the same consistency and respect.

One Response “Learning Fractions at 40” →

  1. Cindy Deruyter

    September 13, 2011

    “walking neither too far ahead or too far behind…” which leaves letting them go it alone (which is the option most parents choose), or walking alongside. Be beside your children and you will see what they see; you will have steady discourse allowing each to interpret for the other using experience and understanding of the others path.

    Thank you for this piece.


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