Mathematics and Push-ups: both sequential subjects.
Suppose I enroll in a "Push-ups" class.
On the first day, we do 1 push-up. On the second day, 2 push-ups. On
the third day, 3 push-ups. And so on. On the 50th day we do 50. The
final exam is: Do 50 push-ups!
Let's say that it has been medically determined that almost 100% of
all healthy students can handle this course. Unless you are sick or
have some sort of medical problem, you can easily do this.
So almost all students find it EASY, and almost all pass with an A+.
All you have to do is to show up every day!
Now suppose that one year we experiment with a more advanced version
of the class that does 2 push-ups the first day, 4 the next, 6 the
next, etc, and finishes with 50 push-ups in just 25 days instead of
50. And let's say that it has been medically determined that most
students CANNOT do this, it is just too steep a gradient for the
So we try this course and almost everybody fails, except a few who
were very athletic.
Now note: I am completely making up these numbers. I have no idea
how many push-ups most people can do. I'm just trying to present an
idea, a story. There is truth in the idea, I believe, no matter what
the actual numbers.
The idea is, most all students can increase by 1 per day, and few
can increase by 2 per day.
So, lets say that we stick to the version of the course that almost
everyone can do: a 1 push-up increase per day to 50 push-ups in 50
So on day 25 the teacher announces: "Today class, let's turn to
Chapter 25 in our textbook", which of course reads: "Do 25
Now everyone who is there has just done 24 push-ups the day before.
They look at this new assignment and think: "Piece of cake!" and "No
And it *is* that easy, because if you can do 24 push-ups, then 25
push-ups is pretty easy. And once you can do 25, 26 is pretty easy.
And so on.
In other words, if you just show up every day, and practice one more
push up than the previous day, you will find it easy and you will
continue through the course with ease.
Now, suppose you miss a week. A family emergency. A bad cold.
Whatever. You miss a week.
You are at 25, but the class has moved on to 30. You come to class
and cannot do 30. And in order to catch up, you'd have to do 2 more
each day for 5 days in a row. Which is very difficult. Our
experiments have found that very few people can increase by 2 per
day. And you would have to do that for 5 days in a row! So unless
you put in a herculean effort to catch up, you fall behind.
And the more behind you get the harder it is to catch up. Even if
you hire a coach, you'd still have to do 2 or more extra per day to
catch up, and that is hard if not impossible.
The End. Forget it. Might as well drop out now. You will be behind
all the way to the end and then not pass the "Do 50" final exam.
Start all over again next year.
Math, more than any other subject, is SEQUENTIAL. Each new topic
depends on understanding the previous ones. If you miss chapter 17,
you will have trouble in chapter 18, and then be totally lost by
Perhaps in a History course, you could miss the entire 17th century,
yet still do alright in the 18th. But in Math, no way.
Here is what I always find when I tutor a student. Say she is having
trouble with some particular thing, some sort an equation. She says
the teacher just didn't explain it well, and the book is no help.
Sorry, I don't buy this.
No, the real problem is that there is some PREVIOUS thing that she
didn't really master. If she had mastered every concept up to now,
then no matter how bad the book or the teacher, she would have
figured it out.
Just as, if you can do 25 push-ups easily, then even on a bad day
with a bad coach and a cold, you can manage 26!
If she understands A, B, C, and D, then one need merely mention "E"
and the student will say "Aha! Of course! I was just beginning to
So when a student says she is having trouble with "E", I know the
problem is with D or C. The problem is always PRIOR.
SO HERE IS THE POINT:
A student who comes to a class completely prepared, that is,
has learned everything up to the current point, will then find
the next point *easy*.
And a student who is having trouble has missed a previous
When a student has mastered 25 push-ups, he will find 26 easy! And
if he is having trouble with 26, I suspect he really didn't do 25.
When a student has all the concepts mastered up to a certain point
in math, then the next concept needs only to be briefly stated and
the student should say: "Oh, yea, this new concept totally makes
sense!" Even if given only a half-decent teacher and a half-decent
book, the student should see no problem in understanding the next
So I repeat: *** THE PROBLEM IS ALWAYS: EARLIER MATERIAL NOT
And I always back up to the previous point and clear THAT up. Not
the current thing she is having trouble with, but the previous
things. Then suddenly, the current thing is understood, without me
even having to "teach" it.
Even the best students experience the problem. I have often worked
with an Algebra II student who is doing just fine, making B's in the
course. They can do all the problems, but not without some thought
or difficulty. They say that math is "hard", but are (deservedly)
proud because they are working "hard"! And every time, in working
with them on a "hard" problem, I discover that they have not
MASTERED techniques from back in Algebra I. The only reason a
technique in Algebra II is "hard" is because of a prior technique
I say this: An "A" student is not necessarily "smarter" than a "B"
student. Nor is an "A" student one who is "working harder". No, an
"A" student is merely one who has learned everything up the the
current point. A "B" student is one who has learned most of it but
not all. I propose that if you take the B student and catch him up,
fill him in on all the little pieces he missed, he will suddenly
become an A student, and not because he works any harder or got any
smarter. He just has a stronger foundation.
Unfortunately, the way our school system works, the moment you fall
behind, you are going to get left behind. The system is *ruining*
students because of this. Not just in math, but in all subjects, and
for the rest of their lives. Even the best of students can fall
behind occasionally, even for valid reasons: a family emergency, a
brief cold, whatever. But once you fall behind, "the system" does
not help you. Indeed, it tramples right over you and moves on to the
next chapter whether you are caught up or not. In Mathematics, far
more than any other subject, this is fatal.
Completely related to this is the fact that we put 20 or 30 students
together in a single class and expect them all to move at the same
In my opinion, one of the most terrible statements a teacher can
make is, "OK, class, Friday we'll finish up on this chapter, and
Monday we will move on to the next chapter!". Now, I don't blame the
teacher. This is just the way the system works. But the class will
move on, even if only 70 or 80 percent are at complete mastery!
And so 20 or 30 percent of the class (and I'm being VERY
optimistic!) will fall behind. And so that 20 or 30 percent will
find the next chapter very difficult, and the next one after that
even worse. A higher and higher fraction of the class will fall more
and more behind, until, like over half of all California high school
students (according to http://www.cde.ca.gov/nr/ne/yr13/yr13rel73att.asp
), they flunk out.
For example, suppose a school spends a certain number of weeks on
learning concept X. Now the time period is up and some students are
only at a 90% or 80% level. That is considered "good", that's A or B
work. But 80% mastery still means 20% non-mastery. Would you fly
with a pilot who got 80% on his pilot's test?
So the class moves on to the next chapter, whether the entire class
has reached 100% mastery or not. And indeed, the course is timed on
purpose to where when some fraction are able to make C or above,
enough time has been given. If they waited for ALL students to have
full mastery, the course would take too long and bore the faster
ones. If they only go long enough for just a few to master it, they
will loose the entire bottom of the class. So they set the
curriculum timing "just right" so that some percentage will A, some
percentage will get B, C, D, and a few will get F.
Yet my entire thesis is, if you don't let EVERYONE master to A
level, you simply set them up to get a lower and lower grade with
each successive chapter. You end up with a huge mass of students who
are not necessarily any less bright, but who have not really learned
the material and will forget everything they "learned" immediately
after the exam.
I tell you this: whatever, even slight, concepts you miss now, will
come back to bite you when you get to the next level of math. The
curriculum doesn't have "extraneous" chapters thrown in just for
"fun". EVERY chapter in Algebra I will cover concepts needed to pass
Algebra II. And every chapter in Algebra II will cover concepts
needed in Calculus. If you miss something in Algebra I, I guarantee
you that when you get to Algebra II you will go slow and have to
study what you missed *again*! The solution is: study it well the
first time! Or you will be studying it again later.
I say it is imperative to MASTER each topic before moving on to the
next. 80% on the chapter test is not enough. Otherwise, the student
slows down, and with each more thing after that not fully mastered,
he slows down and down and down more and more and more, until he
becomes overwhelmed. In California in 2014, that overwhelm comes at
Algebra I. The statistics are clear: 50% will flunk it! (See http://www.cde.ca.gov/nr/ne/yr13/yr13rel73att.asp)
And then the student begins to feel inadequate. He thinks there is
something "wrong" with him. It's not just that he falls behind, but
also that he begins to feel he is not smart, cannot make it, is
somehow "bad". Not knowing algebra is not fatal. But loosing one's
own sense of personal worth, self-esteem, self-confidence, can be
I try to tell my students: IT IS NOT YOUR FAULT!
It is the system's fault! BECAUSE THEY INSIST ON GOING ON TO THE
NEXT CHAPTER WHETHER *YOU* ARE READY OR NOT. And as time progresses,
a larger and larger fraction of the class is NOT ready. And then it
is all over.
Here is what I teach to my students: The correct way to study
(ANY subject) is this: When you get to the bottom of a page, stop and take
a look at yourself. Are you smiling? Do you have the thought: "I
understand this!"? Could you explain what you just learned to
someone else, without looking at the book? If so, pat yourself on the back, then turn the page
and move on. If not so, DO NOT TURN THE PAGE! I REPEAT: DO NOT
TURN THE PAGE. DO NOT TURN THE PAGE!
Re-read the material. Get a dictionary and look up some words.
Figure it out. Ask questions. Hire a tutor. Google it. Watch a
video. Ask your mom. Whatever. But don't go on to the next page
until you get that smile + attitude back.
This may take only a few seconds. If so, great! But this
may also take a significant amount of time. And the school system
doesn't give you much extra time. You probably have a busy enough
schedule as it is. I'm sorry about that. You just gotta do what
If you follow this method, you will: (a) find each
page quite easy, because you were fully prepared for it, and (b) you
will go through your entire book with a smile on your face! People
will think YOU are one of those "naturals" who finds math easy!
As an adult, in "real life", you will be able to do this, to learn
things "at your own rate". But in school, you will be hurried along,
ready or not. I'm truly sorry about that.
When you first come to me for tutoring, here is what I will do:
I will turn to page 1 of your textbook and ask you if
you understand it. I may ask you a question or two, or to do a
problem or two.
And if you show me that you do fully get it, we will continue on
to page 2 and do the same thing.
And so on, page by page, until we catch up to where your class has
reached in the book.
This may not take long. We may spend less than 10
seconds on a page. You may understand chapters 1 through 6 really
well, and in just a couple of minutes we can fly past them. But WE
WILL CHECK! We will ENSURE that nothing was missed.
And of course, if something *was* missed, we will stop and clear it
up COMPLETELY! And if you have too many problems in the very first
chapters, we may have to go back to LAST YEAR'S BOOK!
But when we finally get caught up to the page where your class is, I
believe that you will suddenly find it EASY!!
If you can't do 40 push-ups, no amount of tender loving care, or
gentle coaching, or angry coaching with yelling, is going to get you
to do 40 push-ups. I can yell at you all day long, but if you can
only do 30 push-ups you are not going to suddenly do 40. The only
way is to get from 30 to 40 is to have you do 31, and then 32, and
then 33, and so on until you are back to 40!
So we MUST back up to 30, or wherever you actually are, and then
steadily come forward from there. There is no other way. I can try
to coach (prod) you into advancing by two push-ups per day or
whatever accelerated rate you can handle, but even so, we must still
start at where you really are and then work our way back up.
Unfortunately: Tutoring is expensive and time is short. Sometimes we
have to compromise. So sure, I can just go over the current homework
with you. Sure, I can help you "cram" for a test occurring in just
two days. After all, it's your dime, you are paying me, you are the
There is a compromise which I have used successfully: we meet twice
a week; on one day we go over the current homework -- a temporary
"fix"; and on the other day we review from the beginning of the
book, until we eventually catch up -- a permanent solution!