Some Funny Comments and Questions.

From some of my students

Following below are a few questions and comments made by some of my students in the last year or so. They are all from different students, and they are unrelated to each other. The only thing they have in common is that I found them very funny at the time I heard them, maybe in part because of the student voices or facial expressions when they made them. Anyway, I quote them here in case you find them interesting:

“Are these all the primes there are? Or, is there any prime bigger than 13?”

“How can you stand so much math? My head hurts!”

“You are just fascinated with this stuff, aren’t you?”

“When I see logarithms I just freeze.”

“What is e to the negative x square? I don’t like ‘e’s! ‘e’s always confuse me.”

“Yes, a big number minus a small number we know how to do that, and a small number minus a big number is possible only if it’s money, or something like it.”

“Negative numbers are evil!!!”

“Wait! You just set those two values equal to each other! Why? Are you allowed to do that? How do you know you can do that?”

“Wow! All that math just to get a result equal to zero? That doesn’t make any sense!”

“I hope I will never see you again. I mean, don’t get me wrong; it’s nothing personal. You are a nice guy, and I am grateful for all your help but man, if I pass this test, I am never going to open another math book in my life again, ever!”

“I just feel like school is sucking my life away!”

“So, do you have a real job or you just tutor?”

Negative Numbers

Building up concepts a little bit at a time.

Beyond a certain age, most students can handle negative numbers. Some make mistakes sometimes, like forgetting writing the negative sign in front of the result, or subtracting the numbers when they should add them but in general they have the concept of negative numbers and their operations.
This is different for little children who have not yet been exposed to the subject.
Today I asked one of them:
“How much is 3 minus 5?”
He said:
“That’s impossible!”
I said:
“O.K., well, let’s see.. Have you ever borrowed money?”
I led him through the example of owing five dollars, having three in our possession, paying that amount and ending up owing only two.
Then I pointed out the fact that five minus three equals positive two, while three minus five equals negative two, and I continued:
“So, it is possible to subtract a big number from a small one, and the way we do it is we really just subtract the small number from the big number, and we write the result with a negative sign in front of it.”
Then he summarized his understanding as follows:
“Yes, a big number minus a small number we know how to do that, and a small number minus a big number is possible only if it’s money, or something like it.”
I thought that was really funny but the key point here is in his mind he moved the concept of “small minus big” from “impossible” to “possible only if it’s money.” So now he accepts the possibility of such an operation at least in some cases.
This illustrates another point, that learning most often than not is a gradual process, where we build up concepts a little bit at a time. Students require several exposures to negative numbers and to the rules governing operations with them, before they can feel comfortable handling such operations. These exposures better be gradual, clear, consistent, and such that the student gets a feeling of success about them. Otherwise confusion sets in, and with it the seed of long-term frustration.
I remember the following dialog with another student a few months back, when I asked her:
“So, when we multiply two negative numbers, what is the sign of the result?”
She said:
“Negative numbers are baaad!”
I asked:
“Really? How bad?”
She answered:
“Negative numbers are evil!!!”
I found that comment very funny, I smiled and I said:
“O.K., well, somehow we have to deal with the fact that your teacher for some obscure reason wants you to add and subtract and multiply those evil numbers so, how are we going to do that?”
Then she said:
“Well, maybe they are not always that bad after all.”
Usually it is not easy to discover (let alone clear them) the blockages installed in a student’s mind around a concept by virtue of unsuccessful teaching techniques.
The problem here is that every teaching technique is very effective with some students, while at the same time being totally useless with some others. Given the amount of material in the syllabus, and the limited time available, teachers in the classroom have to go with whatever technique proves useful for the majority of the class, and some students are left behind.

Factored Integers

A memorization exercise

Last week I added a new page to my tutoring web site. The new page’s title is “Factored Integers.” It is a reference page listing about 200 positive integers, completely factored out as products of smaller numbers, including their prime factorization.
The purpose of such a list is for some standardized test takers to do a memorization exercise. The idea is for the student to copy this list, and to write by hand a portion of it every day, anywhere from 20 to 50 numbers a day.
Just writing down the numbers and their factorizations has a cumulative effect in the student’s memory, as long as they do the exercise every day. Standardized test have many problems that can be solved much faster by factoring numbers out than by doing long multiplications and divisions.
Time is the most precious resource in a timed test, so the goal is for the student to have readily available, fresh in their memory, these factorizations, instead of wasting time thinking about what could be a possible factorization, or even worse, going down the path of long multiplications and divisions, because these operations become very time consuming, and prone to errors when the numbers involved are large.
So the best way to solve these problems is by factoring all numbers as much as possible, and simplifying all expressions as much as possible by canceling out any common factors that can be canceled out before getting into any multiplication.
So the value of the memorization exercise resides in the increased awareness of factors the student develops a little bit each day by writing and re-writing the list of factorizations.
The goal is for the student to start thinking about a number’s factors as soon as they see the number in the problem; naturally, automatically, by default, without even thinking about it. See a number, boom! Factor it. The less time you spend on this process at test time, the better. So the time spent at home writing and re-writing the list will pay off on test day.

Recommended Calculus Books

Updating my "Recommended Books" web page

Last week I reorganized my “Recommended Books” web page, now including seven calculus titles. Three of these books I have come across through my students: Stewart’s Calculus; Larson/Hostetler/Edwards Calculus: Early Transcendental Functions; and Frank Blume’s Applied Calculus for Scientists and Engineers.
Stewart’s Calculus book is popular among college professors; many students have it as their textbook. Of course it is a very good book but personally I prefer the way the topics are organized in the Larson/Hostetler/Edwards textbook.
Then there are a few books I was familiar with since I was in college myself. Spivak’s Calculus is an excellent book with superb explanations of the concepts and the proofs. This is a classic book for mathematicians. The vector calculus books: “Vector Calculus” by Marsden & Tromba; and “Div, Grad, Curl and All That” by Schey are also very good, with very clear explanations and examples.
Among these calculus books I listed a book that is not a calculus book per se: “e: The Story of a Number” by Eli Maor. Here there are many illuminating stories about the number e, the base of the natural logarithms. The number e is customarily defined as a limit in calculus and it is used extensively in exponential functions and hyperbolic functions, as well as in many types of differential equations. Many students are often mystified by the number e and its strange looking definition, because e is a transcendental number, as pi is but e does not have an obvious geometrical point of reference like the circle to understand its meaning. Eli Maor in its book does an outstanding job of illuminating many of the relationships the number e has with the real world and with other parts of math.

The Tower of Hanoi classic puzzle

The latest addition to my site, a page with an interactive puzzle.

Last Friday I added a new page to my site.
The new page is about the Towers of Hanoi puzzle.
Solving this puzzle is a fun way to develop stamina for keeping your mental focus sharp for longer and longer, when you solve it with more and more discs each time.
The new page is interactive, featuring a DHTML javascript that renders a simulation of the puzzle. The script lets you choose from three to eight discs. It also has a “solve” button, allowing you to see a step-by-step solution with the minimum possible number of moves. When you play, the script keeps a count of the numbers of moves you make.
(The javascript code author is Glenn G. Vergara. The script is featured on www.dynamicdrive.com, )
The object of the game is to achieve the minimum possible number of moves in transferring one stack of discs from one tower to another, observing the two rules of the game, which are very simple.
Please visit my new page and have fun with this puzzle. The script is very good in that you move the discs by clicking on them and dragging them with the mouse, instead of pressing more buttons.
After the javascript, I included some basic information on how this puzzle relates to math. Then there is the classic legend of the Tower of Brahma, followed by a cosmological speculation on the total time-span of the universe as predicted by math calculations partly based on assuming the legend was true, and partly based on current scientific beliefs about the universe’s age.
At the end of the page there are links to other web sites featuring information on this brilliant puzzle, and some links to Amazon.com, where you can buy a wooden model of the Tower of Hanoi puzzle, or the book "The Liar Paradox and the Towers of Hanoi: The Ten Greatest Math Puzzles of All Time" by Marcel Danesi.
Please visit my new page and let me know your opinion, because I appreciate any feedback you can give me. Thank you.

The Psychological Order of Operations

The risks of seeking safety in the avoidance of difficult tasks

Some students, when evaluating an expression like 8 / 3 - 1, they go:
8 / 3 - 1 = 8 / 2 = 4
When I stop them by asking, "What is going on there?" they sometimes look at me like, "What? I am just doing the operations here." Then I ask, "What about PEMDAS? Isn't there some sort of order we are supposed to follow for the calculations? Are there any parenthesis around the 2 and the 1 in the original expression?" Then they ask, "Oh! You mean, do I have to do the division first?" and I say, "I think that's the rule, going from left to right and doing the multiplications and divisions first, then the additions and subtractions later, isn't it?" then they say, "Oh, yes, that's right!" and proceed to correct the mistake.
For a long time I used to leave it at that but I was always intrigued by that behavior. Why do they do that? They already know the rule, as they have seen it before. They even apply it correctly most of the time. Yet, sometimes they make the same kind of mistake again.
After seeing this happening over and over with several of the many students I have worked with during my tutoring years, I have noticed a pattern associated with this behavior. It is likely to occur more often when the student is under pressure, when they feel they don't have enough time to complete their assignment or to finish studying for the exam. The likelihood of this type of mistake increases also when the students are working with material that is new for them, new formulas, new procedures, or new concepts.
I call this type of mistake "the Psychological Order of Operations," as opposed to "the Logical Order of Operations," which is basically just following the established rules.
The reason why some students make this mistake is because they tend to back away from operations they have not mastered completely, operations they don't fully understand, or that they do not know exactly how to perform. Instead they choose to execute first the operations they are more familiar with and they feel more confident about. Somehow they feel safer that way. For these students, this false safety impulse is strong enough they subconsciously choose to ignore the risk of breaking the rules when changing the prescribed order of operations. Somewhere inside their mind, there is a monologue going on like this: "Rats, a division! That is not an exact division; it's going to be a fraction. How do I deal with that? I don't know. I hate fractions! Oh but look! There is a subtraction over there! I know how to subtract those numbers, that's easy, I can do that and the result will be all right. Let's do that one first."
How do I know this is what they are thinking? Well, when the idea first occurred to me, I started asking them "Why did you do that?" They say, "Do what?" and I explain, "Why did you choose to do the subtraction first, before the division?" They blankly stare at me and say: "I don't know, I forgot PEMDAS." And I insist, "Yes, but why?" To this they usually give me a look like saying, "Geez! Give me a break, will you? It was just a mistake, and I already fixed it!" Then I ask, "In general, for you, what's easier, divisions or subtractions?" and they say, "Subtractions!" Then I ask: "Now, when dealing with divisions, do you prefer the ones where the result is an integer or the ones where the result is a fraction?" They say, "The ones with an integer result." My next question is: "When you have to do two tasks, one of them you are not sure how to go about it and the other you know exactly what to do, which one do you do first?" They say, "I do first the one I know how to do." Finally I ask: "Do you think that could be what just happened here?" pointing to the part of their work where the mistake occurred. They look at the paper, they show an expression of surprise in their face, and they look at me as if thinking, "I'm busted!"
After all this, I tell them: Look, just be aware you sometimes have this impulse to do first what you already know, but this impulse increases your chances of making mistakes when you change the order of operations. Sometimes we can change the order, but not always, you want to be careful about it. Besides, when you find an operation you are not sure how to perform, what you really want to do is to learn more about that operation. Postponing it is not going to make it go away, you are going to have to face it at some point, right? Just pay attention and try to catch yourself when you are making this kind of decision.
They usually like this advice.

Buy the right math prep book

Pay attention and read the book’s preface, not only the title.

One of my students was going to take a standardized test specific to some job he wanted to apply for. The test includes a math section, and this is the only part of the test where this student felt he needed tutoring. My student has a friend who already took the same test, successfully. This friend told my student that, if he prepared well for the GRE, he would sure pass this other test, because it is an easier test than the GRE.
So my student went to the bookstore and bought a book he saw there titled “Cracking the GRE* Math Subject Test.” He did this a few weeks before we started our tutoring sessions. He brought this book to our second session for us to work on its problems.
When we started going through the book I was amazed to see double integrals, differential equations, matrices, power series, three-dimensional surfaces, polar coordinates, and almost any calculus topic covered in college, not to mention Abelian groups and topological spaces.
We found only three problems that could actually be in the general GRE test.
I suggested for my student to go back to the bookstore and get the normal Kaplan GRE prep book.
When my student realized he had no use for the book he had bought, he gave it to me.
I was puzzled by this “GRE” prep book because it obviously has nothing to do with the general GRE test. So when I got home I started reading the preface. I found a paragraph with the heading “What is the GRE Math Subject test?” explaining that this test is taken by students who are applying for admission to study math at the graduate level. Now it all made sense!
So, the moral of the story is: be careful when buying a math test prep book. Before paying for it you want to make sure it’s going to help you, and that you are not buying something that has nothing to do with the actual test you are preparing for.

Reasoning, Creativity and Mental Flexibility

The mental equivalent of a boot camp obstacle course

Sometimes after solving a relatively hard problem, GMAT students ask: "Can you give me a few problems similar to this one? Can you make them up? Or, is there a book with several problems just like this one? I want more practice with this kind of problem."
Seemingly, there is nothing wrong with wanting more practice but when GMAT or GRE students make this request they are missing an important point. That exact kind of problem is not going to be on the test. They may encounter a somewhat similar type of problem, but it is going to have its own twist, in some way it is going to be different, tweaked. The companies that produce tests like the GMAT and the GRE work very effectively to make sure that, when you take the real test, you will need to invent a solution on the spot by yourself, for every single problem. Each problem is going to be unique, completely new, something you would have never seen before, no matter how many preparation books and courses you have been through.
Traditional instruction during normal school courses has students drilling through stacks of problems of the same kind, applying the same formulas over and over. And, generally speaking, students can expect to find in their finals and midterms the same kind of problems they practiced with while doing homework.
The GMAT and the GRE do not cater to this drill-based learning model. Granted, you still need to know how to perform the operations, you need to know which properties to use and how to apply the formulas. But that alone is not enough. You need an extra skill that traditional, drill-based homework cannot give you.
Allow me to make this metaphor. Mathematics is a certain kind of mental activity, a mental exercise. Tests like the GMAT and the GRE are designed to measure a specific type of mental performance but, what type exactly? Well, let's compare drill homework (solving a ton of problems of the same type using the same procedure and the same formulas over and over) to working out by lifting weights at the machines in the gym. If you train only at the machines, that can give you muscle mass and strength. But that alone will not prepare you for successfully negotiating a Navy Seal obstacle course. The GMAT and the GRE are the mental equivalent of a boot camp obstacle course, while drill homework is the equivalent of the machines at the gym.
Real preparation for the GMAT and the GRE has to address your mental flexibility, that is, your creativity and your ability to discover patterns you have not seen before. You want to learn how to solve problems by reasoning, not by memorizing routine procedures. Once you understand this, you will be asking for new problems, for problems of a different kind, not for the same kind of problems.

Signs of Numbers vs. Signs of Variables

The inner value vs. the outer look.

A variable does not have a sign of its own. Numbers have signs of their own. Variables represent numbers. Variables may come with different signs in front of them in any particular expression. Still the number represented by the variable has a sign of its own. That is why sometimes I tell to some of my students: "Variables have an inner sign," which we cannot determine from outside just by looking at the sign in front of that variable in a particular expression or formula. I am not talking about astrological signs here but about the tendency some students show to assume a number represented by x in an expression has to be negative just because in that expression there is a negative sign in front of x. Don't go only by the sign in front of the variable to determine whether the number represented by that variable is positive or negative. In most cases we start working on the problem without knowing the value of the variables. At the beginning we don't know whether x is going to end up being positive or negative. Until you find the exact numerical value of that variable, we don't know if it's positive or negative. It can turn out either way, regardless of the sign that is showing "on the outside," in the expressions the variable appears in. For example:

If 7 + x = 4, then x = -3 (a negative number), even though the sign in front of x in the original equation is "+"

If 7 - x = 2, then x = 5 (a positive number), even though the sign in front of x in the original equation is "-"

These are very simple examples but you would be surprised to see how often people make assumptions or jump to conclusions about the mathematical properties of the numbers represented by variables, based only on lexicographic features of the algebraic expressions the variables appear in. In other words, very often students think they know things about the numerical value of a variable based only on how the variable itself looks like when surrounded by all the other symbols in an expression. People do this even in the case of very complex expressions. In general the more complex an expression is, the more reasoning time and effort it will take to infer the exact values of the variables involved, or just to determine some of their properties.

Imagine your friend is telling you a story where at some point they mention their doctor's mother. You wouldn't assume their doctor is a female person just because the noun "mother" is female, and the only time at which your friend mentioned their doctor was in connection with that doctor's mother. Would you?

Getting To Your Destination

You can be the best driver in the world and still you can be driving for hours on end without ever reaching your destination.

Some students have an interesting reaction when they first overcome some difficulties. For example, some people are used to making mistakes at solving equations, because they don't know the rules or they don't know how to apply them. Meaning, they cancel identical terms out of a fraction without realizing it is an illegal move because those terms are just terms, not factors of the whole numerator and denominator. Or they somehow mess up other algebraic steps of solving an equation. They know they are making mistakes, they just don't know what those mistakes are. They are used to getting the wrong answer because they know they make mistakes then cannot even identify. Then they come to get some tutoring and they start finding out what their mistakes are. They learn the correct way to apply the operations to algebraic expressions. At first they seem unsure. Little by little they gain confidence. And then, when they finally stop making the mistakes they had been making before, something interesting happens. All of a sudden they seem confused. They seem surprised to see they are still not finding the solution to the equation, even now when they are not making mistakes any more! They seem to think: "Where is the solution?" They used to think the reason why they were not getting the right solution before was only because they were doing all those mistakes. So, why are they not finding the solution now? They are not making mistakes any more, so where is the solution? Now they seem to keep on going, correctly applying operation after operation to the equation, without getting anywhere near to a solution. Why is that? Well, this is the metaphor I use in this situation. When you drive a car, you have to learn the correct way to use all the controls, the brakes, the steering wheel, the accelerator and everything. If you don't use them correctly, you will crash the car, or you will break the transmission or you will overheat the engine or cause some other catastrophe like that. Once you learn how to correctly use all the controls you can drive safely. Still, that is not enough for you to get to your destination. It is a necessary condition, but by itself is not enough. You can be the best driver in the world but if you keep aimlessly driving in circles all around town when you have to go from Los Angeles to Las Vegas, you are never going to make it. You need to pay attention not only to the car's controls, but also to the road signals. You may even need a map. That is similar to what happens when you are solving equations. Applying each operation correctly and following the rules well at every step is the equivalent of driving safely and operating the car correctly. Now, actually being able to reduce the equation to its simplest form and finding a solution is the equivalent of getting to your destination. You can be correctly applying operations for hours on end, producing more and more equivalent equations like pop-corn in a microwave, but in order to get the final numerical solution, you have to choose what operations to apply and when to apply them. You have to arrange your algebraic steps so that the equation actually gets simpler and shorter with almost every step. That is another skill you have to master but you couldn't see the need for it while you were crashing the car all the time. It's like when you are climbing a mountain and after getting to the top of a hill, you discover another hill in front of you, which you couldn't see before because its view was obstructed by the previous hill you just conquered.

Solving Equations For a Particular Variable

A very basic principle

An equation has one equal sign.

The equal sign divides the equation into left hand side and right hand side.

The two sides may look totally different from each other as expressions but the equal sign says their numerical value has to be the same.

The fundamental principle of equations says that, when two expressions have the same numerical value, if we apply one operation to both expressions, the resulting expressions after the operation is performed will also be equal in value. They will be equal not to the original expressions, but to each other.

So, if A, B and C are three algebraic expressions, and we have the equation A = B, then all of the following will also be valid equations:

A + C = B + C

A - C = B - C

(A)(C) = (B)(C)

A/C = B/C [provided C is not zero]

A^2 = B^2

Square root of A = Square root of B

This fundamental principle is used over and over to solve equations for specific variables, one step at a time.

For example, in solving for x the equation (3x + 1)/2 = 5y - 4, we can do it like this:

1) Multiply both sides by 2 and we get

3x + 1 = 2(5y - 4)

2) Subtract 1 from both sides and we get

3x = 2(5y - 4) - 1

3) Divide both sides by 3 and we get

x = ( 2(5y - 4) - 1)/3

Now the equation has been solved for x in a series of steps, where each step consists of applying one and the same operation to BOTH sides of the equation.

The fact that the resulting expression for x can be simplified to

x = (10y -9)/3

is not relevant here. I am only illustrating the process we use to isolate x one step at a time by applying the same operation to both sides of the equation.

The following YouTube video from InterAlgebra12 shows several more examples:

Word Problems Are Not That Much of a Mystery

Most of it is just common sense

Many times, my students are surprised by how simple some problems seem when I explain them. They go "Is that all there is to it? Can it be that simple? You didn't use any formula!"
A big part of the difficulty students often have with word problems is they think there is or there should be a special type of formula suited for each particular problem. But, for many word problems out there, that is not the case.
If your first reaction to a word problem is trying to remember a formula that would solve that problem, chances are you are never going to remember such a formula, because you have never seen it, because it is not there in any book, and no teacher teaches it as "the" formula for this problem.
There may well be a formula for that particular problem, but the formula is nowhere to be found on record. Because nobody has bothered to figure it out or to pass it down, because even if they did, the formula would be applicable only to that particular word problem and to no other problem. It would be a very limited, almost useless formula.
So, the first thing you have to do with word problems, is to forget about formulas altogether and just read the problem, over and over and over again, as many times as you need to understand what the problem is talking about, what situation it is describing.
You want to really understand the situation, the process described in the problem. You want to understand it as clearly as you see sunlight. You want to be able to express it in your own words, you want to be able to imagine it, you want to be able to tell a story about it, you want to be able to draw a complete picture of it.
Once you do that, the solution presents itself to you naturally, the numbers practically work themselves out. When you really know what is going on, you know what to do, you know what operations to perform, they make sense.
So, again, it's not how to mechanically make the problem fit into a canned formula, but how to make your very own mind wrap itself around the problem completely, with total abandon, accuracy and precision

The General Quadratic Equation Formula

Benefits of showing where the formula comes from

I often show my student how the formula for solving a general quadratic equation is derived. This, understandably, seems quite complex to them. The benefits of showing them the development of the formula in full detail are:

1) After that, memorizing the formula seems an easier task in comparison.
2) Plugging in the correct values in the formula and evaluating it to a numerical result seems now way easier than having to figure out the solutions for each particular equation by completing the square. The fact that the formula is available saves them the work of completing the square in each particular case.
3) They now have seen how the formula is developed. Even if they do not understand the process 100%, even if they forget the process within five minutes, the formula itself is no longer a mystery. They know there is an algebraic derivation of it, and they have seen it, at least once. They feel now much more confident in using the formula.

So, because of the previous reasons, it is totally worth it to go through the process of showing them how the formula is developed. It is time well invested.

Self-Talk Is a Performance Factor

A very important factor to achieve good results

Many students, when they first start working with me, show the following behavior, it's really common: they are working on a problem and all of a sudden they start saying things like: "Well, I don't know what to do, I have never been good at math," "I really suck at math," "I always get these problems wrong," "I do not understand percentages," "When it comes to algebra, I just don't get it," "Oh, boy! I hate these problems. I don't like math at all," "I am not smart," etcetera.
So, very often I have to explain to them the huge impact and importance of daily self-talk. I look them in the eye and I say:
"You mean you never were good at math before, but now you are, and you will."
Most times they look surprised and a little confused when they hear that. Then I go on telling them about how the subconscious mind works, in a totally different way than the conscious mind. I tell them:
"Look, in the long run, nothing is more important than what you say to yourself. Because your subconscious mind literally believes everything you say. It does not judge, it does not analyze, it does not argue. It just stores the information you put in there and later it retrieves it like that, unprocessed."
If you go on repeating things like you are not good at math, then that is what your subconscious mind stores and believes. Later it will make you act from that belief in ways that will produce results consistent with that belief, and the results will of course validate and reinforce the original belief.
If you want to pass your test, stop saying that and start saying: "I'm good at math," "I like math," "I can solve these problems." Just say it, even if it sounds fake at the beginning. Your conscious mind may say "That's a lie," but it doesn't matter. Your subconscious mind won't say anything like that, it will just take the new affirmation and store it. Then, later it will incorporate it in your subconscious decision making process.
By repetition, you can make the new, positive affirmation, outweigh the old, negative ones.
So, starting today, change what you say to yourself about math, about you math skills and your performance. You don't even have to say it aloud. Only by thinking it, it is having that effect in your subconscious mind. Watch carefully what you say to yourself about anything you care about. This process has a huge impact. It makes a real difference.