Archive for the ‘MY LOVE AFFAIR WITH MATH AND TEACHING’ Category

MANY CHILDREN LEFT BEHIND, ONE TEACHER, TOO

October 26, 2011

 When I was 18, I guided four canoe trips into the wilderness of Ontario’s Algonquin Park.  I was in charge of several other teenagers for six days, navigating, choosing the campsite, cooking and first aid, 1-2 days travel from the nearest adult.  When the campers swam, I counted heads.  At night, anybody’s crying out made me wake up.  When a camper couldn’t carry his pack, I carried it and the canoe together–140 pounds–a half mile along the slippery shore of the Tim River.  I guided four times that summer, amazed I was given such responsibility at 18.

When I was 61, with far more real-world experience, I was not allowed to teach full-time with the nearest other adult a few yards away.  For 9 years, I was an active volunteer in math at two local high schools.  At least 20 times, I taught when a substitute did not know the material.  I wanted to be busier, and I saw a huge need.   I finally gave up, because I was not busy enough.

My father was a public school teacher, principal and superintendent.  I believe in public education; with liberty and the National Park Service, it is one of the three greatest gifts America has given the world.  If public education fails, and many hope it will, we will destroy the middle class and America.  There are 60 million school age children in this country.  I am open to other solutions, but for profit charter schools in the new pay as you go America do not seem to be workable.

Perhaps it is hopeless, and many of these children will be uneducated, given up on, not voting, and subject to the sharks and those who will take advantage of their money, and talk radio hosts, who will sway their opinions with outright lies.

In elementary school, I wrote a paper about Horace Mann.  His six principles are still valid today:

  1. The public should no longer remain ignorant.
  2. Such education should be paid for, controlled, and sustained by an interested public.
  3. This education will be best provided in schools that embrace children from a variety of backgrounds.
  4. That this education must be non-sectarian.
  5. That this education must be taught by the spirit, methods, and discipline of a free society.
  6. That education should be provided by well-trained, professional teachers.

I now have a substitute certificate.  But I wanted to create a statistics course at a high school that didn’t have one.  I have a Master’s in statistics; I taught it for many semesters as a graduate student at New Mexico State, Pima Community College and other venues.  For four years, I graded the free response portion of the Advanced Placement Statistics exam; only 3 of 400 graders were from Arizona, which in itself is a statement about how well we are doing.  I know how to develop a statistics syllabus, how to prepare lessons, teach and grade it.  I would have done it for free, too, because I could afford it, and I felt so strongly about the need.

Unfortunately, I was not allowed, because of No Child Left Behind (NCLB), even as children are being left behind in droves.  I encountered them every day I tutored.  I didn’t see the many others who needed help and didn’t get it, the ones who need to be pushed, or those who drop out.  With many teachers also dropping out physically or mentally, we have a major problem. With appallingly inadequate funding, many schools nationwide disenroll problem students, gaming the system to survive.  Our per capita funding is $2500 below the national average.  I told that to a friend, who said, “You can’t fix education with money.”  I replied, “You can’t fix defense with money, either.”  He was silent.

NCLB is like Clear Skies, Healthy Forests and Clean Coal:  all have titles that are purposely the opposite of their intent.  NCLB’s intent was to close schools and have more for profit charter schools,

As a neurologist, I saw many practicing in my field, who could do everything I did without my 9 years of post college training and military service.  But I couldn’t teach full-time unless I obtained 22 credit hours.  I went back to school when I was 50, ending up as a non-paid de facto volunteer in Tucson’s medical community.  I choose to volunteer in the schools, but I will not pay to get more education at age 60.  In emergencies, substitutes can be hired long term, making the rule inane.  Frankly, I consider the current situation an emergency. The America I served once used innovative approaches to solve problems.

Public education–an American invention– is in trouble.  We funded the poorly conceived Iraq war with an off budget emergency authorization.  Public education needs an emergency funding authorization, too, and thousands of volunteers.  Schools need to open their doors to volunteers, and both schools and public libraries need places for tutoring during lunch, evenings and weekends.  Parents need to be involved.  We need money, too, but with a concerted national effort, the money wouldn’t be an outrageous sum.  I don’t want to hear “children are our future,” until we start trying to achieve the six principles articulated by Horace Mann, whose credentials to lead reform were also questioned, 175 years ago.

THOUGHTS ABOUT A MATH CURRICULUM

February 13, 2010

(The following was written to the chairman of the math department at CDO High School in Tucson, where I have been a volunteer math tutor for 8 years.)

America has given three great gifts to humanity:  liberty, the national park system, and public education.  All three are under attack, but I will confine myself to the last.

I believe students must be able to do the following before entering high school:

  • Multiplication tables accurately and fast, mentally, with no calculator.
  • Addition and subtraction of all two digit numbers quickly–paper allowed.  No calculator.
  • Ability to add and subtract fractions, including those such as 5 1/8 -3 3/4. Paper allowed.  No calculator.
  • Ability to deal with negative numbers quickly and accurately.  It is inexcusable for students well into ninth grade continue to give “14” for 5 – 9.
  • Know the number of months in a year, days in a year, hours in a day, and minutes in an hour (yes, these are frequently not known.)
  • Know all basic English and metric measures:  these include ounces in a cup, cups in a pint, pints in a quart and quarts in a gallon.  They should know metric and English units of length, area, volume and temperature and be able to convert between them.  They should know sixteenths of an inch on a measuring tape.  They ought to know what a cubic foot of water weighs; maybe then we wouldn’t have drivers going through flowing washes every summer.   It wouldn’t hurt if they knew that 1:60,000 on a map is roughly an inch to a mile.
  • Know the decimal equivalent of all fractions with a denominator less than 10.  Be able to quickly and accurately convert mixed numbers to improper fractions and vice versa.      It is inexcusable for a softball player to require a calculator to figure out her batting average.
  • Be able to estimate square roots of numbers that aren’t perfect squares; many honors geometry students can’t estimate the square root of 5.
  • Know common factors of common numbers, which basically means understanding the multiplication tables.
  • Be able to estimate answers to common problems, including the proper sign and rough idea of the magnitude of the answer.  A quarter or the price scanners in Arizona are inaccurate; cashiers frequently give the wrong change.
  • Be able to work with large numbers using scientific/exponential notation.

American students need to be able to collect, format and interpret data at a basic level.  We have a data driven society:  today’s key issues cannot be addressed without understanding basic science and math.  We have a society where conspiracy theories are not debunked, vaccinations are believed to cause autism, and where a majority do not believe in evolution despite absolutely compelling evidence.   We have the highest percentage of non-believers in global climate change in the world.  More people believe in astrology than in astronomy; few know what a year represents, and fewer still can find ten well known countries on a map.  But my focus is on math.

It is impossible to use a calculator without number sense–some idea of what an answer ought to be.  Calculators give several decimal places when three to four would be sufficient (and better, in the case of scientific work, where the appropriate number of significant figures has to be used).  Inability to estimate an answer means that if a wrong button is pushed, the student will be unaware of it.  Too many graduate without basic skills; 80% of the students entering Pima Community College need remedial math training.  How many children in Pima County, Arizona, and America leave high school unable to do basic math?  We don’t know:  but from my experience in two affluent districts with high graduation rates, I would estimate the number well over 5000 annually.  These students are being left behind not just other Americans but the Singaporeans, Indians, South Koreans, Chinese and Japanese, all of whom recognize that in this century, as in all others, education is the way out of poverty.   They are strong competition, and if they succeed, this country won’t.

I believe if students had a solid background in basic math, they would be able to handle basic financial issues.  One reason we have the current financial crisis is that many cannot understand interest rates, doubling times for debt and money and budgeting skills.

We require students to memorize the alphabet and the numbers.  We practice writing and diction.  We similarly must require memorization, practice and homework in math to learn these basic skills.

I have nearly daily arguments with students about banning all electronic devices in the classroom, except calculators.  I have yet to be in any class in any school, and I am in school 5 days a week, where there isn’t surreptitious texting occurring.  Seniors probably can’t be controlled, but 9th graders should not be allowed to have electronic devices near them in the classroom.  How to do this will be difficult, but if we don’t stop the electromagnetic abuse, learning will continue to deteriorate.  I am well versed in studies of human error and know that multitasking hurts learning and that interruptions require significant time to reconfigure one’s ability to concentrate.  I think far too many are labelled as learning disabled, without seeing what happens with concentration, hard work and one-to-one tutoring.   I am bothered by grading participation, which to me has shown up in high school as loud, wrong guessing.  We need to bring back the concept of school nights, where there aren’t concerts or hanging out.  In these days of instant, easy electronic communication, parents ought to be fully aware of how their child is doing and should be acting upon it.

In passing, I mention the deterioration of dress, language and civility among today’s students.  I think that more formal dress correlates with better civility and a higher likelihood of learning.  I am no longer shocked by what students wear, and I’ve grown to tolerate coarse language, perhaps because of my Navy experience, where I’ve heard everything these students have said and a good deal more.  The lack of civility I have a more difficult time with.  I certainly wasn’t always polite as a student, but there were significant consequences for rudeness that when I violated, I still remember to this day.  The public displays of affection are nothing short of shocking.

Education requires society, not just schools and teachers.  Parents must inculcate their children with the proper values, dress, behavior and desire to learn.  My father was a high school principal, assistant superintendent of schools and superintendent in three cities.  He and my mother accepted nothing less than my best work.   Yes, education is underfunded, and we need to press for more.  But that does not excuse us from doing what we can with the people we have.  And we aren’t doing that.  We need society to be taxed to pay for the public schools.  We also need to stop promoting those who do not earn promotion.  We need volunteers, as I have been for the past 8 years.  And we need the schools to put them to use, in the classroom, library, before, during and after school and on weekends, which has not occurred.  Six schools have never returned my offer to volunteer.  This is unacceptable.  We need competent teachers, and we need competent substitutes, which I hope to be.  I decided to substitute for pay because I was not busy enough and was felt it unfair that I taught when a certified substitute got paid and did nothing.  We need appropriate tests, where to fail them would be so egregious that nobody, even in the Arizona legislature, would pass the student on to the next level.  (I would wonder how many legislators would pass such tests.)  Teachers promote students to high school who are incapable of doing elementary school–yes, elementary school–math.  Eventually, high schools and community colleges end up teaching remedial courses. This must stop.  Watering down AIMS because it upsets people is unacceptable.  Do we want educated students or don’t we?

America’s future at stake.  I am among the 7% of Americans who has served this country in uniform.  We need mandatory national service, some of which could be in the classroom, where we might find our next generation of teachers.  I refuse to let public education fail without a fight, for if it does, we will lose the middle class and this country.  I will continue to speak out on this subject; I will do all I can as a volunteer and as a substitute to teach students the skills that I have been blessed with, not just genetically, but through practice, hard work, and strict demands — timeless values —  made by my parents.

A DAY IN A TEACHER’S SHOES

December 3, 2009

After 7 years as a volunteer math tutor at a local high school, I was allowed to be an on-call volunteer math teacher, meaning I teach with a certified substitute present.  I address the occasional problem when a teacher is absent and a fully qualified math substitute is unavailable.  On my first day, I was given a lesson plan for algebraic inequalities and prepared one for geometry.  While I don’t find these subjects difficult, understanding a subject is far different from teaching it. 

I arrived at 7 a.m. with water bottle, lunch and objects needed to explain the material, for good teachers don’t parrot the textbook.  The official substitute took attendance, introduced me and I began teaching.  Fortunately, I had no problems with student behavior, because the teacher for whom I substituted is an exceedingly good disciplinarian, knowing when and how to act with words, inflection and body language.  My experience could easily have been worse. 

What’s it like to teach for a day?  I was on my feet nearly continuously for 7 hours.  I needed a bathroom break at 10:30, but preparing for the class before lunch took priority, and I nearly sprinted to the men’s room an hour later.  Other than a few swallows of water, I ate nothing until I finished at 2:20.  I left at 3:45 and wasn’t the last teacher leaving.  That evening, I relaxed, not having to grade homework or prepare the next day’s lesson. 

My parents were both hard-working teachers, and I frequently heard, “You can’t eat dedication.”  I’ve taught exactly one day and didn’t deal with problem students, parental e-mails, after school tutoring, worth $40/hr, but freely offered by many teachers or faculty meetings.  I’m 61 and want to teach math.  I can afford to; many of our best and brightest teachers, with whom I’ve had the honor and pleasure to be associated, struggle to pay their student loans.  Summers off?  Many teach summer school out of necessity. 

A properly educated populace won’t solve all our problems.  But it is a necessary condition if we ever hope to address them sensibly.  Arizona ranks last in per capita spending for what is arguably the highest yield and lowest risk investment of all – education.  Nationally, we invest far more in low yield/high risk unwinnable wars and impossible nation building.  Those whose high risk complex financial instruments devastated our economy receive annual bonuses greater than a teacher’s lifetime earnings.  Important, difficult jobs requiring significant training and long hours deserve appropriate compensation, which is how we attract and keep good people.  As a former neurologist, I was paid well for my training, work and hours.  Teachers are not paid commensurate with their extensive training, hours and immense responsibility preparing the next generation.  Teaching math, or any other subject, to 35 teenagers who’d rather be elsewhere is difficult:  doubters should try it – assuming they have the skills to do so.  Increased funding for teachers and education is one of the best investments Arizona and America can make.  Our future depends upon it. 

Michael Smith, retired physician and statistician, has been a grader for the AP Statistics examination.

MULTIPLICATION TRICKS

September 13, 2009

I timed it right, entering Dave Kukla’s calculus class to kill an hour between volunteering in a couple of college algebra classes.  Dave is one of the smartest mathematicians I know, a wonderful man, who mentored me through his approach to teaching, and has a way to make the students love his classes.  He has received many awards, and he deserves every one of them.

About once a year, I can “scoop” Dave on something, if I am having a good year.  This time I had a good year.  Dave was talking to the students about the tangent to a curve at y=x^2 at the point (0,1).  As y got closer to 1 x^2 got closer to 1, but more slowly, since it was less than 1 and being squared.  As he started with 0.9 for y and 0.81 for x, he kept going with 0.99 for y and 0.9801 for x.  He then wrote down y=0.99999999 and said he would need the calculator for x^2.

“You don’t need a calculator,” I said from the back.  The place went deathly silent.  Dave looked at me and smiled.

“You want to tell me what it is?”

“Sure.”  And I wrote down the answer 0.9999999800000001.  The calculators couldn’t do it, for it had too many decimal places. I sat down.

“You were the one who gave me the idea,” I said from the back.

Dave looked at me, a quizzical expression on his face.  “What did I do?”

You brought it up last year, and I found the pattern.

square 9 and you get 81

square 99 and you get 9801  Notice the pattern.  One less nine in front, 8 the same number of zeros as 9s  and a 1 at the end.

square 999 and you get 998001.

“You had eight “9s” up there, so I wrote down 7 of them, put an 8 in the middle, followed with 7 “0”s and a one. ”  Pretty cool, right?  Dave nodded.  That’s what I like about him.  He appreciates this sort of stuff.  And he got me to think about it.  He’s brilliant.  I’m just a run of the mill pattern recognizer.

To multiply anything by 9, multiply by 10 first and then subtract the number.  After all, a number x (10-1) is the number times 10 minus the number x 1, or itself.  This is the distributive law in action!

It should follow, then, that multiplying by 99 is easy.  Multiply by 100 then subtract the number, for you are multiplying by (100-1).  So, 83 x 99 would be 8217:  (83 x100) – (83 x 1)= 8300-83=8217.

Multiplying by 11 can be done without great difficulty.  To multiply by 11, multiply by (10+1), which is multiplying by 10, then adding the number (which is multiplying by 1).  So, 123 x 11=(123 x 10) + (123 x 1)=1230 + 123 = 1353.

Multiplying by 50 is the same as dividing by 2 and multiplying by 100, since 100/2 is 50.  That means that 98 x 50=(1/2) x 98, or 49 x 100=4900.  In the same fashion as above, we can multiply by 51 or 49.  The neat thing about multiplying by 51, is that for an even number less than 100, it is half that number with the number appended.  Say what?  64 x 51 is half of 64 (that would be 32) with the number (64) appended.  So 64 x51=3264.

Multiplying by 25 is dividing by 4 and multiplying by 100:

87 x25 is 21 3/4 x 100.  But that is 2175, since 3/4 x 100 =75.

Try left to right multiplication:  23 x 7

Write 23 as 20 +3.  So 7 x 23 is 7 x 20, or 140, plus 7 x3, or 21.  Add 140 and 21 to get 161.  In my head, 43 x6=”6 times 40 is 24o; 6 times 3 is/are 18, and 240 plus 18 is/are 258. ”

354 x 6= “6 times 300 is 1800; 6 times 50 is 300, and 1800 plus 300 is 2100.  6 times 4 is 24.  Answer is 2124.

498 x 8= 3200 plus 720 plus 64 or 3984.  BUT, isn’t that also 8 x (500-2) or 8 x 500 (4000) minus 8 x 2, or 16?  Same answer.

This one is really cool:  Square any number  x, ending in 5.  The answer is (x)*(x+1), with 25 appended.

So, 85 squared is 8*9 with 25 appended, or 7225.

295 squared is 29 *30, with 25 appended, or 87025.

Multiply two numbers ending or beginning with the same digit, like 1:

31 x 31=  30*30 (or 900) +(3+6) x 10 (or 60) +1=961.

96 x97= 90 *90 (or 8100) + (6+7) 90 (or 1170) +(6*7)=9312.

This one came to me in the car on my way to a clinic in Safford:

The product of two numbers whose sum is an even number constant is maximal when the numbers are the same.  For every unit away from that maximum number, the sum decreases by the square of that unit:

So, two numbers whose sum is 100 have a maximal product of 50*50 or 2500 (this can be proven with simple calculus).

51*49=2499; 52*48=2496; 61*30=2500-11 squared or 2379.

It works in reverse:  84 x 78 is 81 squared minus 9 or 6561-9=6552.