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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!

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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 majority.

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 days.

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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.

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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.

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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.

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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. But in Math, no way.

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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 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.

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SO HERE IS THE POINT:

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!

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So I repeat:

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 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.

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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.

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. 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 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, 50% of all high school students will flunk at at least one math math course, generally Algebra I. (See http://www.cde.ca.gov/nr/ne/yr13/yr13rel73att.asp)

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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 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.

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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 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.

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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.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 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.

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!!

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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 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!

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Thanks for listening,

Steve.