Why math can be "hard" - an essay for parents and students

In my opinion, the number one reason students have trouble in math is:


And the number one reason for that is:


More than any other subject, math is sequential. Everything you learn in chapter 17 will be needed in chapter 18. If you get 80% on the chapter 17 test, then there is 20% of the material that you didn't know. This gives you a 20% handicap going into chapter 18. You will bog down in chapter 18. And with each chapter thereafter, students bog down more and more.

In history, you can miss the 17th century and still do well on the 18th century. Indeed, you can take a course in 18th century history without ever having taken 17th century history.

But in math, if you haven't done chapter 17, chapter 18 will be impossible. And if you've only partly done chapter 17, chapter 18 will be difficult, and 19 will be overwhelming.

In art, you could miss out on watercolor and yet do well in oil paints. There are probably plenty of successful artists who work in oil but can't do watercolor.

But there is not a single mathematician that can do Calculus but not do Algebra!!

In other words, Math, more than ANY other subject, is cumulative. You MUST learn prior material before moving on to later material. Almost every single problem you work at any particular level will require all the skills of all the previous levels.

What I see happening is that the school system doesn't give the majority of their students enough time to master any one skill, before being rushed onto the next one. They just barely learn a skill, they kinda-sorta know it, get C's or B's on the chapter test, and then are immediately moved on to the next chapter where they will need those skills.

In math, you can't kinda-sorta know a skill and then expect to do alright on the next skill. It can't be done!

Imagine a gymnast who has just learned to balance on a balance beam and can kinda sorta do it, but not well, and is then asked to do a cartwheel on the beam. No amount of practice of cartwheels is going to do it. She needs to go back and learn better balance first! She can't "kinda sorta" balance and then move on to cartwheels. She must MASTER balance before moving on. Otherwise she will find it all "hard".


And then we incorrectly blame it on the student. And they begin to feel inadequate. They think they are not as "smart" as they thought they were. They come home crying. Mom begins to worry. They think maybe they're just not "cut out" for a professional career. And we tell them they just need to "work harder".

*** NO! ***


A parable: Suppose I enroll in a push-ups course. On day 1, in the first 10 minutes of class, we all do 1 push-up. 10 minutes later we do it again: 1 push-up. In 10 more minutes we do another: 1 push up. We do this 5 or 6 times and then our hour is up and class is over. On day 2, we repeat with 2 push-ups. On day 3, with 3. And so on. On day 10, 10 push-ups. If you have trouble on day 10, your homework is: "do 10 push-ups!"

Suppose that it has been medically determined that almost all human bodies can handle that gradient: 1 more each day.

So even an overweight out-of-shape human like me can do this course. It doesn't take great athletic prowess. Any "average" person can do it. It's not even hard. All I have to do is show up every single day.

So let's say right after day 25 I miss a few days. Now the class is at 30 and I'm still at 25. I can't do 30! And in order to catch up, I'd have to do 2 more per day for 5 days in a row, which is almost beyond my ability! Might as well drop out. On day 50 when the final exam is "do 50 push-ups", I won't pass it. I will always be 5 days behind and I will always find each day's class "really hard".

Now, let's look at the converse of that. Let's say I haven't missed a day. On day 25 I do 25 push-ups. And it's easy! The next day the teacher announces "Today we are going to do 26 push-ups!" And the entire class is thinking, "piece of cake!" and "easy", because it is.

Because if you can do 25, then 26 is not hard. Even on a bad day with a bad coach and a bad textbook and a cold, if you can easily do 25 then you can manage to do 26. It's not about the coach or the textbook or the cold, it's about being able to do 25. Once there, at 25, you can reach for 26. And if not there, you can't.


Math is like that. If you can do every problem, understand every concept, up to page 25, then when you turn to page 26 you will find it EASY.

I say it's not about whether the teacher is good or bad, the book is good or bad, the student has a high IQ or not, the student is working hard or not. Of course all those things matter.


If we don't stop to get 100% of each chapter, we are setting ourselves up to slide down worse and worse in all subsequent chapters. Every missing skill becomes a liability forever.

Mastery of a chapter will take some students longer than others. Even the best students will occasionally have a cold or other distraction and need more time. It doesn't matter whether it takes a long time or a short time, as long as the material is mastered. If it takes one student a year to work up to 30 push-ups, and another student spends a week getting up to 30 push-ups, the two students then become equal: they have both attained the same level of expertise.

This from a football coach:

Time and again I see an Algebra I student who can't easily add fractions. Quick, ask your child, "What is 1/2 plus 1/3?" If that answer takes a student more than a few seconds, that student will bog down in Algebra I. Yet I have seen it time and again, they can't answer that simple problem quickly. On a test, many problems will have fractions in them, and they will go that much slower and then they are pressured and don't finish in time. I've often seen an Algebra student who looks at:

   3x + 2 = 14

and instantly sees that x=4. But then give them:

   3x + 2/3 = 14

and you may have to wait a minute. Or two. Or forever!! Yet it is almost the very same problem!! The difference is in the inclusion of a fraction, which they were supposed to have mastered several years before. Ironically, they actually have the skill of the current chapter: they know how to solve this kind of equation. They just can't handle fractions. Like I'm saying: "PRIOR SKILLS NOT THOROUGHLY LEARNED".

Early on in Algebra they learn to factor equations. They should be able to look at

    (x^2 + 5x + 6)

and factor it EASILY, in a few seconds. It should be almost obvious that the key numbers are 3 and 2. They add to 5 and multiply to 6. The answer is:


If that isn't obvious to an Algebra I student, he will die in Algebra II. A huge number of Algebra II problems will have a factorization step, or other step requiring some other prior skill, and they will then balk on every single problem.

Yet we take Algebra II students, who made B's and C's, and can factor, maybe, kinda, sorta, but not quickly or easily, and put them in a Calculus class! And they are overwhelmed. And we wonder why they find it so hard! After all, they were making "decent" grades in Algebra!

And we tell them they need to "work harder". That is like telling me to do 30 push-ups when I can only do 20. I just can't do it! I can try, I can "work hard", but it will take me 10 days to get from 20 to 30, and by then the class will be at 40!

I like to ask a student who has taken trig, what are the sines and cosines of 0 and 90 degrees. Many hesitate! Some can't answer it! They will say "well, I used to know it, just give me a minute to think about it!" If they can't state that answer almost instantly, they will die in Calculus! It assumes you are fluent with these concepts.

These are not difficult concepts. Except when the classes in trig move you through each chapter too quickly and a majority of the class doesn't really get enough time to fully master their skills. They barely learn what *pi* is all about and then are moved on. We often see a class get good grades on each chapter exam, but by final exam time, despite good grades throughout the course, they flunk the final. Because everything they "learned" was just barely learned.

They are never allowed to stop so they can really master something, to practice it enough to really "have" it. In particular, the basics are quickly introduced and then immediately passed over. They are continuously forced to move on.

Almost every single time I work with a student who is having trouble, I find that they understand the current chapter just fine: it's a PREVIOUS skill that is giving them difficulty. You can try to teach the current skill, but over and over again it won't sink in. Because there is a missing prior skill. Clear up the prior skill and the current problem suddenly becomes easy, "like duh!"


Even in a good school with a good teacher, here is what is happening: Say we have 20 or so students in a class. How long should a teacher spend on a chapter before moving on? If they spend too much time, the faster students get bored. If they spend too little time, the slower students get lost. So what is right?

That is generally answered by "a curve". Roughly speaking, they spend just enough time so that the middle 2/3rds of the class understand about 2/3rds of the material. Thus about 2/3rd of the students will get B's and C's. At the tail ends of the curve, a few will get A's, and a few will get F's. And then the teacher announces "Friday we will finish up on this chapter, and Monday we move on to the next!"


And 2/3rds of the class has only 2/3rds of the skills, with a 2/3rd certainty on them.

That method works in a history class. If you learned 2/3rds of the material about the Civil War, you'd know a lot more than most people. You'd get B's and C's on tests and you'd feel pretty good about yourself. History is not an "all or nothing" affair where if you can't name every battle of every war you are not a good student.

But if you learn only 2/3rds of your math skills, almost every problem at higher levels is going to require at least one skill you don't have! And it *is* all or nothing because you don't get credit for not solving a problem!

So this idea that 2/3rds of the class understanding 2/3rds of the materials is "good enough" to let them move on as a group just doesn't work for Math. It just sets them up to find the next chapter "hard", and the next after that "harder", until, like over 50% of all California students, they FLUNK.

Google "percent of California students that flunk math" and prepare to be shocked.
From the Sacramento Bee, Sep 27, 2017: https://www.sacbee.com/news/local/education/article175572031.html

"More than half of California students miss English standard. Even more fail at math."

"Just over half of California students failed to meet English standards based on spring 2017 standardized test results released Wednesday, a performance that remained essentially flat compared to the previous year. Students performed even worse on math tests, with nearly two-thirds falling short, according to the California Department of Education."


(And of course, it's not just California. It's the same all across the country.)


I have no solution to propose *within* the school system. To more advanced students, for example, in High School Calculus, I advise NOT doing it within the school system, but instead get a good textbook, or use Khan Academy or similar website, go through it on your own (and at your own pace!), and see me once a week to help keep you on track! In that way, whether you are slow or fast, or have a good week or a distracted week, you can master the material. And I will insist on mastery each week and not let you move on to the next section until you have it!

Here is what I teach to my students: The correct way to study (ANY subject) is this:

Start on page 1. When you get to the bottom of the page, stop and take a look at yourself in a mirror. Are you smiling? Are you thinking, "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! DO NOT TURN THE PAGE!

Re-read the material. Get a dictionary and look up some words. Think about it. Figure it out. Ask questions. Draw a diagram. 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 + cocky attitude back. You may have to go back a few pages and restart from there.

This may take only a minute. Maybe there was just one word you needed to look up and now it all 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 knowing attitude and 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.


Here is my preferred way to work as a tutor:

When you first come to me, I will turn to page 1 of your textbook and ask you if you understand it. I may ask you a question or two, the definition of some terms, 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 just a few seconds on some pages. 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 when we finally get caught up to the page where your class is, I believe that you will suddenly find it terribly EASY!!


Unfortunately, within the current school system, all of that is a "pipe dream".

Time is short and tutoring is expensive. 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,
Steven Swift.

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