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 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 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 push-ups!".

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 problem!".

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 chapter 19.

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 suspect that!"

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.



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 concept.

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 thing!

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!



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 not mastered!

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 complete mastery!

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

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 the exam.
(Washington 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 permanently disabling.

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, 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 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 boss.

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!


Thanks for listening,