Mathematics and Push-ups: both sequential subjects.
Suppose I enroll in a "Push-ups" class.
On the first day, we do 1 push-up. Then we wait 10 minutes and do 1
push-up again. Then 10 minutes later we do 1 push-up again. And
again. We do this 5 or 6 times during our 1 hour class.
On the second day, we do the same, but with 2 push-ups. On the third
day, 3. And so on. On the 50th day we do 50. The final exam is: Do
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 condition, 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!
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. 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 and 19th. 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 of 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
And here is an all-important sub-point: It's not about "IQ" or
"natural ability". It's about being fully prepared. High School
level Algebra is very doable for almost anyone of average
intelligence. And just as you don't need to be a natural "athlete"
to do 50 push-ups, if you just come to class every day and advance
by 1 push-up every day you will make it!
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 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. Because it does not
understand the principle I have just described. It thinks that it's
"OK" to miss or partially understand a chapter because you can just
"catch up later". In Mathematics, far more than any other subject,
this is fatal.
In my opinion, one of the most terrible statements a math 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. We put 20 or 30
students together in a single class and expect them all to move at
the same rate. But the class will move on, even if only a few are at
And so a significant percentage of the class will fall "a little"
behind. And then they will find the next chapter "a little" more
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 today, they
flunk out (see www.cde.ca.gov).
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 90% mastery still means 10% non-mastery. Would you fly
with a pilot who knows what 90% of his levers are for?
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 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 a few will get A, some percentage will
get B, C, D, and a few will get F.
Yet my 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
Post 22/7/2015: "Three out of four high schoolers failed
Algebra 1 final exams in Md. district". From the article: "74
percent of high school students in Montgomery failed June’s
Algebra 1 semester-end final... Last year, 82 percent of high
school students failed the exams."
Note that Montgomery is a wealthy suburb of Washington D.C. with a
very highly rated school system! Ha!
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. 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!
If not, you will continue to have a "hard" time and you will be
studying it again later anyway.
I say it is imperative to MASTER each topic before moving on to the
next. 90% on a chapter test is not good 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.
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". Remember that we are talking about young teenagers
who are just beginning to develop their own ideas about themselves
and the world. 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 minute. Maybe there was just one
sentence you needed to re-read and now it makes sense. If so,
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 you can.
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.
I don't want to yell at you ever! Instead, I want to take you back
to where you were last doing well and help you move forward from
there, and at a pace you can handle. I want to keep that smile on
your face at all times!
I find that when I take a student back to a place where she was
still doing well, and help her learn something new, and really learn
it, I see a big smile! The challenge then is to move forward one
smile at a time!
I find that almost everyone will smile when they learn something,
truly learn it. If we can go back to find something *simple* enough,
and learn *that*, we get a smile. And then if we move forward from
there, we can keep the smile. I once met with a 12-year old student
who was flunking Geometry. He was more than half-way through the
course. And he could not define for me the word "diameter"! I spent
20 minutes with him to clear it up. I googled it and found that Home
Depot sells pipes of various diameters and has pictures of them. I
found a website explaining how car tires have different diameters,
with pictures. I had him take a ruler and measure the diameter of
objects in the room. In 20 minutes, he truly understood "diameter".
And then I said "That's all we're going to do for today!". He had a
huge smile! I was probably the first tutor or teacher or parent who
didn't beat him over the head trying to get him to do a "proof",
something he had absolutely no idea of what was about! Instead, I
found something SIMPLE that he could actually LEARN. His mom told me
this was the first smile she'd seem from him in quite some time!
So we MUST back up to wherever you actually are, the point where you
were still smiling, 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 or chapters 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, within the current school system, all of the above is
a "pipe dream".
Time is short and tutoring is expensive. 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!