My views about ADHD

The ADHD label only makes sense within the framework of a massive, standardized educational system.

Sometimes, parents tell me that their child has ADHD, and they ask me if I can help them raise their child’s math scores. I tell them it would take me more than a few sessions to figure out whether, or to what extent I can help them, because this ADHD situation works out differently with each child.

In my opinion, ADHD is not a disability, nor a disorder, or a sickness, or anything like that, unless the lab work in a given case proved that it comes from a specific chemical imbalance in the brain. Otherwise, this is how I see it: children are autonomous individuals, with their own interests, and their own preferences. Very often the school system tries to "tame," or "domesticate" students into a uniform, one-size-fits-all set of academic variables, like pace of study, homework volume, expected format for answers, and so on.

There is a whole new body of research supporting the view that academic ability is very narrow a parameter for measuring intelligence. There are several other types of intelligence, each one very important on its own right, besides just having English and Math skills. Kids may be interested in other things, actually they usually are.

I am a firm believer each one of them should be allowed to pursue their math studies at their own pace, and -as much as possible- in the information processing way that works best for them; instead of expecting them to do all the time exactly the same as the whole class is doing. Mass education may save budget dollars but society pays for it in lost creativity, and lost opportunities for individual happiness, and fulfillment.

Schools typically have no resources to properly (meaning, individually) address these so called “ADHD” situations. And parents often do not have the time, or patience, to find out in detail, how exactly their children's learning process is evolving, what their true talents really are, how far their talents and interests go along with their school work, and how to help them make difficult decisions when it comes to a conflict between the child's true talents, and what schools expect of them. Often parents tell their children: "Just do the work," or something similar, ignoring the child’s emotional process. This is not because they are not interested in their children's development but mainly because most parents want to prepare their children for the dreadful world of jobs, deadlines, and bosses most parents live in as adults.

Personally, except in the case of unruly teenagers who are involved with drugs, or gangs, I usually side with the student against his or her school's expectations, and I wish more parents would do the same, taking the time to bust the "ADHD" label into pieces, and figure out what exactly is going on behind appearances in the particular case of their child. It would be great if more parents took the time, and interest, to find out what their children really want to do in life, and help them be themselves, instead of helping them, or sometimes even forcing them, to trim their natural talents and interests, just to become some facade of personality they may eventually get used to but are never going to be happy with.

In short, ADHD is a code term for "distraction." If a young student gets distracted during math class, it is obvious to me they are not finding math interesting enough. This does not mean it is the teacher’s fault, either. There may be other activities the so-called “distracted,” “unfocused” student strongly prefers to math. My question here is: Just how strongly? Can they overcome their lack of interest in math, and produce the grades their parents want? For me, at the beginning of my tutoring work with them, this is an open problem, and I cannot form any specific opinion until I have worked with them for some time. I tell parents that, if they want me to work with their ADHD child, they have to know that my focus will be finding out the following two things:

1) How much does the student dislike math? Is it just a mild form of lack of interest? Or is it an active, gut-level hatred of math? I have seen these extremes, so I know they are real, but they can be subtly hidden. It takes time to find out. Some children do not even know the answer themselves, since it generally is subconscious.

2) Is it possible to present math to this individual student in a way that will sufficiently raise his or her interest in math? To what extent is it possible to suggest math exercises that will make math easier for them?

It is entirely possible that a better way exists for these students in life, outside of school grades and diplomas. Maybe they can be great artists, or politicians, or sales people, or any number of professions that require special talents, special types of intelligence but not necessarily a college degree. I am aware many parents do not want to hear any suggestion like those above, as most would just have their children go through the whole school cycle getting A's and B's, and then take up a profession like lawyer, or doctor, or business manager, for the prospect of finding a good job. Unfortunately, life is never that simple.

So I never promise parents (especially those with “ADHD” children) that I am going to make their child's math grades go up because I do not know if I can. All I can promise is my best effort in finding out to what extent I can help them improve their grades, and to keep in constant, open, honest communication with the parents all along, always respecting the child’s individual set of preferences, abilities, values, and interests because I do not "fix kids," as some parents in the past have expected me to magically do.

Is it possible to visualize a 4th spatial dimension?

A classic question about higher dimensions

Last week I had a tutoring session with a college student I help with his Advanced Calculus class. We did mostly exercises on line integrals, integrating functions of two variables along paths in the XY plane. After that, near the end of the hour, my student asked me some questions on Linear Algebra. He wanted to clarify some details about the dimension of vector sub-spaces spanned by finite sets of vectors, related to whether the spanning set was linearly dependent or independent. We looked at a couple of examples in the XYZ three-dimensional space. That was enough to answer his questions on that particular topic but then he asked me something to the effect of: “How can we visualize a four-dimensional space?” This is a classic question many students ask themselves when taking classes like Linear Algebra, Advanced Calculus, and other, more advanced math classes. There are some videos about projections of four-dimensional objects onto the three-dimensional space. Some mathematicians do specialized work on the geometry of four-dimensional spaces but I remember relatively early on in my math studies I gave up on trying to visualize a four-dimensional space. I have no problem working out abstract math proofs and formulas about objects in many dimensions, even an infinite number of dimensions. When doing such exercises I construct visual images to help me solve these problems but said images are always two-dimensional, or at most three-dimensional, not multi-dimensional. They are just schematic representations of the concepts at hand. I never (not any more, not in a long time) try to actually imagine how a four-dimensional space would “really” look like if we could move around in it. So, when my student asked me this question I was kind of hesitant in my answer. I said: “Well, that thing of trying to visualize or imagine a four-dimensional space is complicated. We have no real experience of such a thing. Spaces of dimension higher than three are all in the dark. The space we live in is only three-dimensional. The space we believe we see, is just an illusion created by our brains but it is the best representation our brain can come up with, based on the visual information collected as light by our flat retina. It is one reconstruction, or a representation of our three-dimensional world, and our brain spends a lot of time, energy, and resources reproducing these three dimensions because that is the world in which we have to survive.” Then my student said: “So, do not think about four dimensions?” And I said: “Think about four dimensions, just do not try to ‘see’ them.” That was the end of it but somehow I was left with the feeling there was something not quite right with my answer. Why did my student interpret my answer as an admonition “not to think about four dimensions?” Did I make it sound too hard, too complicated? I ended up thinking: “Why not?” Maybe there is a way to build the specific kind of imagination that would allow us to represent, in a visually realistic way (whatever that means), the experience of how it would be like to see in four dimensions with our two-dimensional retinas. Just maybe. What do you think? If you are a computer programmer reading this, please contact me so we can see about the possibility of doing a project on these ideas.

Is there a connection between Mathematical Writing and Fractal Geometry?

When it comes to math formulas, and equations, is it possible to assign to them some sort of measure similar to a fractal dimension?

For the above question to even start to make sense, let’s make one main assumption: let’s say that plain text has dimension one, whereas pictures of faces, landscapes, and other objects, have dimension two. Here is the case for assigning dimension one to plain text. In English, and other European languages, verbal information is encoded in written form by means of the alphabet, writing down one character at a time, in a linear sequence. We create words by placing letter after letter in a given sequence. We create phrases, and sentences by placing word after word in a given sequence. In practice, text lines are broken according to the width of each page, and pages are filled with many lines of text. However, in the abstract model for this way of encoding information, we can consider each text document as a single, long, uninterrupted line of text. To read text, we only need the basic linear connection from each letter to the next one, and from each word to the next one. Any text document can be considered as a sequence of characters, however long it may be.
On the other hand, when we look at images in the real world, like homes, people, faces, mountains, trees, animals, and so on, we process this visual information in a very different way. We see color, shades of color, light, texture, and a multitude of details that can only make sense when we consider them embedded in the full three-dimensional space around us. However, our retina is pretty much a flat surface, and our brains have to imagine the three-dimensional world based on the two-dimensional information our flat retina collects from the incoming light. So, the raw material our brain uses to process visual information is nearly two-dimensional in nature. When looking at an image, if we consider a little part of it, there is no such thing as “the next pixel,” because that could be located above, or below, or to the right, or to the left, or in any diagonal direction. Often we can find linear patterns inside some images but the whole image is fully two-dimensional.
So, where does this basic assumption about dimensions leave the written representation of mathematical expressions?

In a recent math tutoring session, I was helping a student prepare for the SAT, and we came across a problem that involved the expression a^(1/2). However, we got confused for a couple minutes because there were no parenthesis around the fractional exponent, the exponent was in a font size as big as that of the variable, and the fraction bar was too close to the variable. It looked something like this:

So, at first we thought the book meant 1/(a²). We momentarily (and incorrectly) interpreted the expression as if it had looked like this instead:


We were trying to solve the problem doing the calculations with that interpretation, and we were getting nowhere near the answer, until we realized the book meant a^(1/2), not 1/(a²).

The expression should have looked more like this:

This simple example shows that, when reading mathematical expressions, we process the information in a way that seems like a hybrid of how we read text, and how we look at two-dimensional images. In reading math expressions, it is very important to take into account visual clues like the size of each symbol, and the relative position they hold to each other, their spatial arrangement in the page, and how close or far away they are from each other. This is essential because mathematical notation implicitly uses our instinctive understanding of two-dimensional images to convey the fine details of each expression’s precise, hierarchical structure. This also has to do with the familiar PEMDAS rules of evaluation, and is key to getting the problems right. Correctly applying the PEMDAS rules is relatively easy when a particular expression is all contained in a line of text. However, when we start dealing with sub-indexes, summation notation, roots, integrals, derivatives, rational functions, powers of powers, upper and lower limits, fractions of fractions (and especially with combos of all of the above); deciphering an expression's structure requires a visually detailed inspection of the two-dimensional arrangement of all the different symbols making up the expression.

As opposed to a line of text, the structure of a mathematical expression is not necessarily linear. Most often than not, the hierarchy branches out. Mathematical expressions include symbols for operations. Operations usually are functions of two arguments, or parameters. These are called "binary" operations, like addition, or multiplication. Often we work with "unary" operations, or functions of only one argument, like the square of a number, or its absolute value. Sometimes we work with operations that take more than two arguments. The basic fact is that functions have input arguments, and produce output values that can, in turn, be used as inputs by other functions. A mathematical expression has a hierarchical structure given by all the connections between input values, and the functions using them. The written representation of a math expression has to present all these connections unambiguously. The set of all these connections between symbols constitutes a hierarchy that we call a rooted tree. This term (bear with me) denotes an acyclic, connected, directed graph with a finite set of nodes, including one main node (the tree’s “root”). Upon this underlying structure, each node gets associated with a particular symbol representing either a constant, a variable, or an operation. Let’s look, for example, at the quadratic formula (the formula used to solve quadratic equations):

Below we show the rooted-tree that is the foundation for the hierarchical structure of the quadratic formula (not including the equal sign, just the right-hand side); along with the constants, variables, or operations that are associated to each node in the graph. Looking at the arrows, you can see each individual symbol is connected to the one directly “above it” in this hierarchical structure:

In the diagram above, I use the square shape to represent the application of the function "taking the square of b." Note we are still making an implicit assumption based on our visual processing of images. We are relying on the left-right distinction to implicitly give the correct ordering for the arguments of division, and subtraction, the two operations used here that are not commutative.

The rooted tree makes apparent the formula's underlying, hierarchical structure, it shows all its components, and their individual connections. We could philosophically argue that this structure is what the quadratic formula "really is," independently of the format we choose to represent it. My purpose here, in showing the rooted tree associated with the formula's structure, is to make the point that the linear simplicity of written text falls short when it comes to encoding complex mathematical expressions. True, with suitable conventions, and enough parenthesis, you can make almost any math expression fit into a line of text but that does not make its structural complexity go away one bit. For example, you can write the quadratic formula like this:

x = (-b [+/-] sqrt(b^2-4ac))/(2a)

It is all written in a line of text but the hierarchical, branching order of its operations is still the same. Many students (and, consequently their math instructors) deal all the time with the relative difficulty of correctly deciphering the hidden structure of mathematical formulas based on its written representation. This is a fundamental skill that heavily affects students' performance in math, and therefore, their grades, and their future career choices.

Not long ago I wrote a related post in this blog, titled "Math is not English."

People who are not "math-oriented" may find this hard to believe but actually, the mathematical syntax, symbols, notation and conventions currently in use (at least up to Calculus and Linear Algebra) are pretty much the easiest, clearest, simplest, most convenient way mathematicians have found (laboriously through the centuries) for writing and reading mathematical formulas. Believe me, the guessing and reasoning behind the formulas is hard enough. No mathematician is interested in making the notation artificially complicated, quite the contrary.

This finally leads me to the reason why I wrote this post in the first place. I recently attended an online get together of fellow Twitter math enthusiasts. The discussion centered on the large gap between text editors, and math equation editors; particularly with the purpose of publishing, storing, and searching mathematical expressions on the Internet. Compared to the wide availability of high-quality word processors, text editors, and text-based search engines, there seems to be a perceived scarcity of free, online tools for authoring and delivering math expressions online, as well as for searching math documents by their mathematical formulas, not by keywords. These topics immediately made me think of the fundamental structural difference between text and math I mention above because, as a math tutor, I have to help my students deal with this chasm practically every day.

Mathematicians would absolutely love a software package capable of identifying, and extracting the hidden, hierarchical structure of a math formula from the handwriting they could do on an electronic tablet with an electronic pen. My contention is that one of the main reasons this type of software does not yet exist, is because of the large extent to which the conventions of current mathematical notation rely on our unconscious, instinctive, biologically hard-wired, visual processing of images to convey mathematical meaning. As crazy as it sounds, and no matter how many of my students I know would disagree with this statement, we have come a long way in making math very easy to read and write on a piece of paper. However, we have done so by tapping into our biological processing of images, and this has inevitably put us at a disadvantage when it comes to entering that information into a digital format.

Anyway, the question in the title of this post: "Is there a connection between Mathematical Writing and Fractal Geometry?" is motivated by the non-linearity (branching out) of hierarchical math expressions, on one hand, and our hybrid way of reading them, on the other; as something between dimension one (plain text), and dimension two (full images).

"Made-up" operations

Sparing some test-takers the abstraction of modern algebra

Here is a specific type of problem that usually confuses many students who are preparing for standardized tests like the GMAT, GRE, and SAT:

Let the operation Δ be defined as aΔb = (a² - b)/(a+b) for all real numbers a, b such that a does not equal -b. If a = 15 and aΔb = 5, what is the value of b?

One source of confusion here is the symbol used to represent the operation (either Δ, or θ, or @, or other similar symbol). To the student, these symbols seem unusual, odd, strange, or weird. The main confusion source is the word “operation” itself, referring to the odd-looking symbol. This causes a particularly strong reaction in students who have been away from school a long time, not taking any math classes in the last several years. When they hear or read the word “operation” in connection with math, they automatically think of the four classic operations they are familiar with since elementary school: addition, subtraction, multiplication, and division. They know that weird-looking symbol is none of them.
When they ask me questions about this type of problem, often the conversation unfolds like this (using the example problem above):
~~~~~
Student: What the heck is that symbol Δ? That is not an operation, is it?
Tutor: No, you are right, it is not an operation. Nobody uses that in math. It is nothing like the quadratic formula, or something. No.
Student: So, why are they saying it is an operation?
Tutor: Oh, do not worry about it, it is nothing, they are just making it up. It is a made-up operation.
Student: But, why? Just to confuse me?
Tutor: You got that right. They want to see if you can plug in whatever values they give you, and go along with whatever expression comes out of that. For example: let’s say a=1 and b=2.
Then we have 1Δ2 = (1² – 2)/(1+2) = -1/3. Now, I bet you can do this other example: if a was 3 and b was 5, how much would 3Δ5 equal?
Student: So, is that it? I just have to plug in the numbers?
Tutor: Yes, that is right, the numbers, or the expressions the problem gives you.
Student: O.K., then: (15²-b)/(15 + b) = 5. Oh, well, now I have an equation, and I can solve for b.
Tutor: Perfect.
Student [after solving the equation]: Pfff! That is easy.
Tutor: Good, excellent!
Student: It was just plugging in the numbers, and solving the equation but they make it seem so complicated at the beginning with that weird symbol.
Tutor: Yes, I know. That is exactly what they do. So, just be prepared for those weird-looking, out-of-the-blue, made-up operations. Do not let them surprise you.
~~~~~
In abstract algebra, a binary operation on a given set is a function taking two input values from that set, and returning an output value in the same set. The set does not even have to be a set of numbers. So, if you want to get technical, the question of whether or not a formula like (a² - b)/(a+b) defines an operation, really has to do with the domain and codomain of the function.
In this particular example (a² - b)/(a+b) is not a binary operation on the set of real numbers, because the restriction that the denominator needs to be other than zero excludes the set {(x, -x)} from the function’s domain. You could call it a partially defined operation. Other formulas, like sqrt(ab), the geometric mean of two numbers, are operations only on the set of positive numbers, because the product ab needs to be positive for the square root to be defined.

However, I do not get into any of these abstract concepts with my students, unless they specifically ask, with curiosity, and with an open mind because, otherwise, it would be Greek to them, and it would be a waste of their time. In most cases regarding this particular type of confusion, test takers only want validation that they are not crazy, and that they did not totally miss a whole classic operation (like addition, subtraction, multiplication, and division) during elementary and middle school. So, I want to address their concern, and make sure they know I understand their question; the source of their surprise and confusion. I want to increase their confidence in themselves, that they can successfully solve the problem on their own. To do it, they do not need to know anything about abstract binary operations in algebraic structures. That is a topic CSET takers need to pay some detailed attention to but not GMAT, GRE, or SAT takers. There is no time for me to go into such topics with them. The typical student only wants to know how to solve the problems. They are quite comfortable with their familiar belief that the word “operation” must mean addition, multiplication, subtraction, or division. They are not paying me to make them go through all the mental gymnastics it would take them to overcome their resistance to expand their concept of “operation.” So I just give them what they are looking for, that is, the fastest way for them to be able to solve the problems, and to feel good about it.

Who is driving?

Transferring control of the tutoring session to the student

Along the lines of my previous post, about how I help students in our math tutoring sessions, here is another ingredient of my tutoring method: I transfer as much control as possible to the student, over the tutoring session. The keywords here are “as much as possible,” meaning, making sure the students still learn all they need to learn. I do this by asking questions like: “What do we have today?” “Do you have any specific questions?” “What topics would you like us to review?” “What topics is the next midterm going to cover?” “Would you like to see a shortcut for doing that faster?” “Does this explanation make sense?” “What problem do you want to do next?” and so on and so forth. This is a major difference between tutoring one-on-one, a single student at a time, versus teaching a large class. A teacher in the classroom has to cover a large amount of material under a tight schedule; while the tutor can focus exclusively on the specific issues the student is having difficulties with. In a large class every student has different questions, and different difficulties, so the teacher cannot allow the lesson to wander all over the place by following the interests, questions, and difficulties of every single student in the class. That is practically impossible in traditional education. However, in a one-on-one tutoring session the tutor can answer most of the student’s questions without getting sidetracked. Actually, answering all specific questions each particular student may have, is not only possible but indeed necessary for the tutoring session to be successful. That is the very essence of private tutoring, as opposed to teaching a class of many students. Very early on in my tutoring business I discovered the educational benefits of developing the tutoring session along the needs of the individual personality of each student. Students learn better when they are learning at their own pace; when they are encouraged to ask all questions they have about a particular topic; when the instructor checks with them if the explanations make sense to them; and when the instructor lets them choose the order in which to work out the problems. Whenever I notice a student is showing signs of being bored, uninterested, impatient, or irritated, I try to find a way to give the student more control over the tutoring session. The ideal is to have the tutoring session resemble a casual conversation as much as possible. This cannot be done in the same way with every student. Each student is different. However, there are two very broad categories in relation to this topic of controlling the flow of the tutoring session. On one hand we have the working adults who are preparing to take a standardized test, and who pay for the tutoring sessions out of their own pocket. On the other hand we have the children, and teenagers, whose parents made the decision for them to take tutoring sessions. In the latter case the parents are paying for the tutoring sessions, not the students themselves. There are exceptions to every rule but, in general, I find it easier transferring control of the tutoring session to the working adults who are paying for themselves, than to the children or teenagers whose parents are paying for them. Working adults who pay out of their own pocket are already motivated enough to learn. They made the decision to hire a tutor; and they took the trouble of finding one. They usually have a better idea of why they are taking the tutoring, and what they want from it. On the other hand, the children who come to the tutoring because their parents made that decision for them, they are in a different situation. Often they are still struggling to get over the fact that they have to learn math even when they do not like it. Letting children start talking about whatever they have in mind leads much more quickly outside of math than it does with working adults. It may not show when viewing the process from the outside but actually, transferring more control of the tutoring session to the student, takes a lot more attention, and effort from the tutor than it would take otherwise but it is much more effective as far as the student achieving educational results.

To like or dislike math is an individual choice

Helping students regardless of whether they like math

One basic way I help my students is by respecting their right to dislike math. I do not try to make them like math. I refrain from insinuating, or even thinking to myself, that they should like math, because I believe they are free to make that choice by themselves. I am no longer one of those teachers who are always telling students how wonderful, important, or beautiful math is, and that they should like it. I personally love math but I very well know it is one of the least popular subjects among students. As a math tutor, I see my students as human beings first, then as clients, then as students. I know they hire me to help them pass their exams with a good score, not to make them like math, and I totally respect that. Often my students start their first session telling me they are not good at math, or they have always had problems with math, or they do not understand math, or they do not like math, or they hate math. I always listen to them, I acknowledge what they say, and I tell them that it is O.K., meaning, I have no problem with them hating math; I do not feel offended that they do not like math; I do not think they should like math; and I am not going to judge them, or criticize them, or give them a hard time just because they do not like math. Most times I do not even have to say it. Just a nod of the head, and a brief comment like “Yeah, that’s alright” make them feel comfortable with me from the very beginning because they perceive my attitude is sincere. Somehow they realize that, plain and simple, I could not care less whether they like math or not. It is their choice. I am still going to help them to the best of my ability. I do not believe there is anything wrong with them just because they do not like math, or are having problems with math. Once this basic understanding is established, that I am not going to try to change them, they trust me, and they are much more open to what I have to say to help them. In this way I can help them much better.

Negative exponents

Using powers of two for explaining the concept of negative exponents

Negative numbers confuse many students. Usually students tend to struggle with almost everything related to negative numbers. From the very concept of using the (-) sign to refer to conventional spatial directions (left, down, back), to the different rules for adding, subtracting, or multiplying positive and negative numbers; there are plenty of instances where the (-) sign is overlooked, or misinterpreted, resulting in a wrong answer. Given that a high number of wrong answers produce a low score, it is only natural for many students to react defensively whenever a new concept involving negative numbers shows up in their radar. This is the case for negative exponents. Remember that the first definition of exponent (positive) most students are introduced to is: “the number of times you multiply a number times itself.” When one tries to apply this definition to the case of negative exponents it does not make sense because, how do you multiply a number times itself “minus once,” or “minus twice”? One possible answer is that, when it comes to negative exponents, you do not multiply but you divide instead. Even with that interpretation the rule still needs some adjustment because, if you start with a number a, and you divide it by itself once, you get 1 as a result, so you would think a^-1 was 1 but that would be wrong because 1 = a⁰, whereas a^-1 = 1/a.
In order to avoid this kind of confusion, I follow another approach: By asking very simple questions I help the student build a list of powers of 2 from 2 to 1024, each related to their corresponding positive exponent. Then we notice the patterns how the numbers change, and how the exponents change. Then we move backwards all the way to the beginning of the list, and beyond, continuing into fractions, on one column, and into negative exponents, on the other column. In this way it becomes crystal clear to the student how the exact same pattern correlates the negative exponents to the fractions, and even the general formula a^-n = 1/aⁿ becomes apparent. I have successfully used this method many, many times, and so far it has worked wonderfully, clearing away all confusion and anxiety students of all ages had around negative exponents. The key to this approach is being very thorough, and asking the right questions, in the right way, at the right time.

Another Testimonial

This one from a GMAT student:

"I did tell you that I ended up with 690 on the GMAT, right? It was a good score for my purposes. I am accepted at the PhD program at University of South Florida. :) Thanks for all your help."
MAIA F. - March 16, 2009

Ratio Word Problems in Standardized Tests

Look for all the numbers the problem does not show you

Standardized tests like GMAT, GRE, CBEST, and ASVAB include ratio word problems. These may be, for example, problems about mixing water with alcohol, or about the ratio of girls to boys in a classroom, or any other type of situation where it makes sense to talk about ratios. There is a consistent pattern that shows in nearly every ratio word problem found in standardized tests. They give you the basic proportion between two parts, and then they ask you a question about the total. Or they give you the ratio between the total and one of the parts, and then they ask you a question about the other part. To give a simple example, let’s consider this problem:
In a certain school, the ratio of girls to boys is 5 to 7. How many students are there in a classroom with 15 boys, if the same girls to boys ratio applies to that classroom?
Notice how the given ratio is that of girls to boys but then the question is about the total number of students in the classroom. That is typical of these ratio word problems, and it tends to confuse some students, especially at the beginning of their preparation period. If you are preparing for a standardized test, when you see this type of problem make sure you keep track of all the quantities involved, all the different parts as well as the total, not only the parts that come with the numbers in the given ratio. Look for the numbers the problem is not giving you. More often than not, the key to the solution is in those numbers that pertain to the situation but are not shown in the phrasing of the problem.

Testimonial

From a Calculus student:

"The only class keeping me from a 4.0 GPA has always been my math class. Math has always felt impossible. However, Mr. Casteneda's tutoring has changed all of this! I just took my first Calculus exam, and scored 105 out of 100, so over 100%! His tutoring has helped me to not only get over my Math anxiety, but helped me to master the subject."
COLETTE D. - March 10, 2009

A Number Divided By Another Number

One more instance of Math not being English

Lately I had a few students who had some trouble with fractions. Part of the problem was a very specific type of confusion at the time of setting up a division calculation. When asked “How much is 5 divided by 12?” for example, they would sometimes correctly calculate the result but most other times they would set up the division as 12 divided by 5. One of them asked me a few times if the result would be the same. When I tried to explain that division is not commutative, using some visual representations of fractions, he was not totally convinced. So I just pulled the calculator, I asked him to give me a couple numbers, and I did both divisions (let’s say, 17/4 and 4/17) with the calculator, showing him the results. We repeated the “experiment” with two other examples, and then he was convinced that division is not commutative. However, such discovery created some anxiety in him because he doubted he would choose the right order of calculation in any given problem. Actually, it took multiple repetitions on my part for him to finally learn how to set up the right calculation when a word problem involves the phrase “divided by.” At first he wanted to transliterate the written phrase (“105 divided by 15” for example) word by word, and number by number, in the exact same order into the division calculation by writing, from left to right, the 105 first, outside the division symbol, then the division symbol (the one that looks like a rotated “L”), and finally the 15, inside the division symbol. I would tell him that the order of the numbers in the phrase “105 divided by 15” is backwards to the order of the same numbers in the actual calculation but this only seemed to surprise him, and confuse him. Once more, I resorted to the calculator. I said: “O.K. just do the division.” When he asked: “In what order?” I said: “Do them both.” Once he had calculated both results by long-hand, I gave him the calculator, and I said, “Now do them both with the calculator.” As he punched the calculator keys, I directed his attention to what sequence in the calculator was giving him the same result as he had calculated long-hand. I said “Do you see when you enter into the calculator: ‘105, division symbol, 15, enter’ that gives you the same result as when you did ’15, division symbol, 105’ by long-hand?” When he saw this evidence he still said “It is confusing.” Then I said: “Yes, I know, but that is just the way it is, so you are going to have to remember it that way. In the calculator division the numbers go in the same order as they are in the phrase ‘105 divided by 15,’ whereas in the long-hand division, to get the same result, the numbers have to go backwards.” It took some repetition over a few tutoring sessions but he finally got it consistently right.

Ubiquitous Numbers

One and Zero are always everywhere

The more you tutor math, the more skilled you become in finding good, clear ways to explain all types of math concepts to students. However, some concepts are more elusive than others. The difficulty of grasping a concept depends not only on the concept itself but also on the student who is assimilating it. One person, for example, can easily understand polynomial multiplication, and struggle with percentages, while someone else can find percentages very easy but have trouble with polynomials. There are also some concepts or topics that seem to be hard for a significant majority of students, like word problems for instance.
Substitution is a very powerful problem-solving technique, and it is widely used in a variety of situations. Students who find substitution easy have a clear advantage over students who have trouble understanding it. Substitution comes up in many different ways, some more complex than others. Some students understand the more basic forms of substitution but have problems applying the same techniques when dealing with more complex expressions. In fact, a very consistent general trend is the host of negative reactions students tend to show in varying degrees when facing bigger, longer, more complicated expressions. The more complex the expression, the more likely that students will get confused, or feel overwhelmed by it.
There is a particular way of using substitution to which most students react with a strong resistance: It involves transforming a given expression into another one that is equivalent in value but looks more complicated. This is done with the ultimate goal of simplifying the expression but it starts out by complicating it a little more. It is like climbing up a hill to find a way down the mountain.
The most common way of using this technique is by introducing a representation of the numbers Zero or One into the given expression.
Zero and One are very special numbers. They implicitly are everywhere in any given algebraic expression, even when we do not see them written out.
Zero is called the additive identity because zero plus any number is that same number ( x + 0 = x ).
One is called the multiplicative identity because one times any number equals that same number ( x∙1 = x ).
These two properties make Zero and One algebraically omnipresent in an implicit way.
Further, we have the following properties:
A number subtracted from itself equals zero ( x – x = 0 ).
A number different from zero, when divided by itself gives us one as the result ( x/x = 1 ).
These last two properties give Zero and One an infinite number of representations (“disguises” if you will) to show up in a formula. So, not only are Zero and One ubiquitous, they can come in a dizzying multiplicity of seemingly different forms.
The above properties of Zero and One, and their consequences, make them extremely useful in solving equations, and in manipulating algebraic expressions in general.
However, as I mentioned before, many students present a strong resistance to the idea of making an expression more complicated to be able to reduce it later. This is partly because they do not see the point of multiplying a number times one, or adding zero to it; partly because doing so seems to increase the problem in size and complexity; partly because they feel we are working backwards into some uncalled-for calculation, and finally because they believe they would not know what particular form of Zero or One they are supposed to introduce into the expression if they were doing the problem on their own.
Sometimes, when working with a student on a given problem, it is very easy for me to see a path to the solution using these types of techniques but I have learned to make sure the student does not feel like I was expecting him or her to be able to solve the problem in the same way. I only use these techniques when the student is completely stuck in the problem, not making any progress at all. When they see one possible solution, it gives them some perspective on the different factors playing a role in the problem. After showing them one possible solution method, if they do not feel comfortable that they would be able to successfully use the same method on their own, we focus on finding an alternate method that works better for them.

Tutoring Vector Calculus

Rich Problems

I really like tutoring Vector Calculus because the problems are very rich. Very often you get to do a lot of stuff in a single problem, like: partial derivatives; determinants; dot-product; graphing three-dimensional shapes, parametrizing curves and/or surfaces; substitution; double or triple integrals; polar, cylindrical, or spherical coordinates; trigonometric substitution and/or integration by parts. Plus usually there is some flexibility as to how to go about setting up the problems; with quite a few choices from ordering the variables to writing down the equations, and selecting the integration techniques. It is a lot of fun.

Math is not English

The order of operations messes with our reading habits.

The mathematical order of operations seems to be a source of confusion for some students, sometimes even frustration. For example, when presented with the expression
3 + 4(x-1)
some students ask: “Why can’t we just start by adding 3 + 4, and then multiplying 7 times (x-1)?”
My short answer is: “Because math is not English.” Then I ask: “Is there any parenthesis around the 3 + 4 sum?” When they say “no” I continue: “Then the parenthesis that is there right after the 4 claims that 4 for itself, for multiplication purposes. It will not let the 4 run away with the 3, oh no sir, no way! The multiplication operation has title to that 4, and to that (x-1) as well, and it does not care about the 3 the slightest bit. The addition operation holds a lesser priority than multiplication does, so it has to wait for its turn.”
I explain the PEMDAS rules using action verbs commonly applied to human situations, thus making the math symbols play the role of active, independent characters with human-like behaviors. This type of explanation makes my students understand the mathematical structure of the expression at hand but still some seem puzzled, or surprised, or even bothered by the fact that the applicable sequence of operation does not necessarily follow the simple left-to-right order. So in those cases I proceed with the following explanation:
There is a crucial difference between the way we read math, and the way we read English. This is very important. We always read English from left to right. Such a simple, linear, unidirectional way does not do it for math. It does not work. Reading math from left to right only, is insufficient, and inadequate. In math we have to read formulas and expressions not just from left to right but from right to left; from the top down; from the bottom up; from the inside out; from the outside in; and even around in circles. In short, every which way, else we run the risk of missing essential information about the structure of the thing. Reading math is not reading. Reading math is much more similar to what the eyes of a helicopter pilot do when they are flying over a mountain terrain, looking for a spot to safely land the helicopter. You look at everything, everywhere.
Written language mimics spoken language, going along with the flow of the story. English is perfect for telling stories. Math describes structures. It has an altogether different goal, so it cannot work the same way English does. Mathematical expressions do not resemble stories nearly enough the way they resemble gizmos, appliances, devices, or cars, objects made out of parts. The parts are connected to each other in a very specific way. Each part has its own function, and its own place within the whole thing. So, really, reading math from left to right only, makes as much sense as trying to “read a car” from left to right only.

The Square of a Sum

An arithmetic and geometric approach to Algebra

A very common mistake test takers make when they have been out of school for a while is that they automatically try to expand the square of a sum as if it was the same as the plain sum of the squares of the individual terms. For example, sometimes some students, when presented with say, (x + 5)², they wrongly expand it as x² + 25, leaving out the middle term 10x.
They forget they have to use the foil method to multiply the given sum times itself. Since typographically the sum of the squares looks like something that could equal the square of the sum, they guess they can expand the square that way, as if it was a real math rule, and then they get the problem wrong.
I try to preemptively address the possibility of this mistake, because both sums of squares and squares of sums show up a lot in the math section of standardized tests. For a long time I have used a drawing, with two little squares and two congruent rectangles filling up a larger square, to show the geometric interpretation (in terms of area) for the algebraic formula (a + b)² = a² + 2ab + b². I emphasize the middle term, 2ab, in this expansion, and I point to the corresponding two rectangles in the drawing.
The above approach is useful but many students forget the formula when it comes to actually applying it. Most do not realize it is a universal pattern where to plug any other expressions in.
So, lately I have been experimenting with a slightly different approach: I present two or three numerical examples first, before drawing the squares and rectangles. I ask the student to choose two numbers, and then I guide them through a sequence of calculations that allows them to actually compute and see the numerical difference between the square of the sum, and the sum of the squares. Then I ask them to multiply the original numbers, then to double this product. By asking all the questions in the right order, I have gotten comments from them like: “Wow!” or “How weird!” or “That is so funny!” or “This is very interesting!” or “Does it always work like that?” or “How can that be?” or “What is going on?” or “I’m sure there is a pattern here!” Then I draw the squares and rectangles with all the numbers in their proper places. So, creating the perception of a “mystery” with the numbers, and then explaining it away with the picture seems to work well. It makes sense for the students. They usually say: “Oh, right! Now I get it.”

Tutoring for the CSET

Matrix Multiplication, Fields, and Other Abstract Concepts

Lately I have been tutoring a few CSET takers. The CSET tests for math knowledge equivalent to what is expected from a math major. Consequently, the CSET covers a lot of topics, and some of them are quite abstract. One of the common CSET preparation guides out there starts with one of its most abstract questions right at the very beginning. The question asks to identify, among five possible options, one valid argument showing that the set of all invertible 3-by-3 matrices is not a field. So, this particular question checks whether the student knows what a field is (as an algebraic structure), and also that matrix multiplication is not commutative.
Many CSET takers who are not math majors have the goal of teaching middle school math. They are usually surprised by the amount of math in the CSET. At least two of them have made this remark to me: “Wow! Maybe I do not want to teach math. This is quite a lot, and very complicated.”
Individual reactions to that first question (on the set of invertible square matrices not being a field) vary widely from student to student.
One of them told me: “Let’s just skip this one. I do not want to waste time on this. If I see a question like this one in the test, I am just going to take a guess and that is it.”
However, some other students have the sort of driving curiosity that do not allow them to let go so easily, because they want to know what the question talks about. So, another student kept asking me question after question, during a couple sessions, until she understood the algebraic concept of “field”. As we went through several examples and counter-examples of number types, sets, functions, operations, and properties, along with the abstract names and notation, she kept saying: “Wow! This is mind-blowing! I never thought they would expect me to know so much stuff.” But she kept asking questions all the way until she made sure she understood that first problem in her practice test.
I very much enjoy tutoring for the CSET, precisely because the wide variety of abstract topics it covers. Just like Linear Algebra, and Vector Calculus, the CSET reminds me of my college years.

Look for Solutions with Less Math and More Logic.

One instance where often “less is more.”

The following question is an excellent guideline for solving math word problems:
“How can I solve this problem by doing the least possible amount of math?”

Oftentimes there are several pathways from the setting of a problem to its final solution. Some routes are safer, while some are riskier, more error-prone. Some routes are faster, while some are time-consuming. Some routes are clearer, while some may be confusing.
Usually, the routes with more elementary operations (especially long division), and bigger numbers, tend to be lengthier, longer, and riskier, because adding, multiplying, and dividing big numbers or expressions requires a laser-focus attention. There are plenty of opportunities for doing silly mistakes during these calculations. Besides, it is easy to lose sight of the big picture when worrying about the accuracy of the calculations.
Factoring whole numbers and algebraic expressions is a good habit because it allows you to simplify some expressions before diving into the calculations, so you can operate with smaller numbers, gaining time, and accuracy.
Using logic is a very good habit, too. Many problems lend themselves to solutions that involve more reasoning, and less calculation. This is usually a good thing because these solutions tend to be clearer, and shorter.
Organizing all the information about a problem in a way that makes sense to you, is an excellent habit because this way you keep track of where you are and what you are doing all the time through the problem, and having all these references available makes it easy to retrace your steps, and identify any possible mistakes.
Go visual at any opportunity. Pictures, drawings, charts, graphs, and tables often are a huge help in writing down the right equations, or even in avoiding equations altogether sometimes.
There are many problems you can solve with a drawing and a little logic. Just because the problem is a math problem, that does not mean you need to write down an equation to solve it.
Focus on your possibilities, on what you can do. Organize the information in a logical way, using a drawing, or a table. Above all, try to spend the least possible amount of time and energy doing long, detailed, time-consuming calculations. Instead, simplify the expressions, and ask yourself logical questions about the problem.

Missing Pieces of Information

Some search for doors, sometimes some do not want to see them

Last week I showed a student how to solve two linear equations in two unknowns. He knew perfectly well how to solve one equation with one variable but did not know how to combine two separate equations into one.
Also last week another student made the remark: “I do not know how to start solving this problem. What does ‘isoceles’ mean?” As soon as I gave him the definition of an isoceles triangle he successfully proceeded to solve the problem.
Earlier today another student asked me: “What is a frequency histogram?” When I explained the concept to him, he found it very clear. He said: “Just that? Documenting the numbers in a graph? That is pretty simple!”
Most times students take the initiative, and they spontaneously ask the meaning of terms they are not familiar with. Sometimes however, some students are near some sort of saturation point, and they do not want to even think about the remote possibility that maybe there is a concept they do not know, or a technique they have not seen, and they need this new information to solve the problem at hand. In these rare occasions they keep trying to solve the problem with only the insufficient tools they already have in their problem-solving toolkit.
Writer Kenneth Grahame said “The strongest human instinct is to impart information, the second strongest is to resist it.” So, I choose my words carefully when telling them there is something extra they absolutely need to know first before having any chance of solving the problem. Many times I let them finish their attempts, and check the solution in the back of the book so they realize their approach was wrong without me telling them so before hand, because that could increase their resistance.
There are several problem-solving techniques or approaches that seem indeed artificial, weird, or mystifying the first time around. Once you see how they work, and you use them a couple times, they become perfectly natural, and then you wonder why you never thought of that before.
A perfect example of this I saw also last week with another student.
It was a probability problem involving three coins. For me it is quite amazing to watch time and again how students keep trying to solve these problems by reasoning only about the three separate coins, as if the relevant probability space had only three points. The strong insistence in this naive approach is only matched in its consistency by the strong surprise students show the first time you show them the full eight-point probability space by branching out the development of the experiment at each successive flip, and recording the eight different combination triples. It is really interesting. Somehow these once missing pieces of information act like doors to a whole new realm of math knowledge when they are presented and opened. Many times the student’s reaction reminds me of that feeling of “Wow! I never thought that was a door!” I get when watching some sci-fi movies.

Writing upside-down, and sideways

An indirect measure of tutoring experience

Last week, a student made this comment to me:
“Wow! You write not only upside-down but also sideways!”

Then I realized I have gradually acquired this ability over the years as a direct result of my continued math tutoring practice.

As a tutor, many times you have to correct a result, or an equation the student has just written. You are sitting across the table from them, and the correction may be a minor one. Reaching for their notebook across the table, grabbing it, turning it, putting it in front of you, writing what you want to write, and giving the notebook back to the student may not be extremely time consuming; however, the couple seconds it takes to move the notebook back and forth across the table may sometimes add up to something of a hassle if you find yourself having to make a lot of corrections and/or suggestions to keep the session moving forward.
In such cases, especially when the corrections are minor, once your hand is on the notebook at the other end of the table, it is much easier just to write whatever you have to write, right then and there, without ferrying the notebook to your side of the table.
For me the process started inadvertently, just by changing minus signs to pluses. From there it went on to changing y’s into x’s, inserting parenthesis, adding missing zeroes at the end of a number, and things like that. Still easy stuff but increasing in difficulty a little bit at a time.
The easiest digits to write upside-down are 0, 1, and 8. Before long, you can write all the digits upside-down. One day, all of a sudden you find yourself writing whole formulas upside-down. By this time, most likely you have started writing a few things also sideways, because often the student sits next to you but at a 90 degree angle.
I never had any independent practice writing upside-down or sideways on purpose. The only times I write sideways or upside-down are during my tutoring sessions. So I can say this ability, in my case, is a direct result, and therefore an indirect measure, of my tutoring experience.

Finding Problems That Motivate Students

Time units may help with multiplication practice

According to my experience each child is a unique learner. Generally speaking, students are more motivated to solve a particular problem when the problem relates to something they find real, meaningful, or important. However, different children usually assign different degrees of importance to the same thing. So it is always helpful to find a topic that holds a student’s interest, and that can be easily connected to math.
For example, let’s say you want to help a student who is struggling with multiplication tables. You can start a session by asking how many hours are there in a day, how many minutes in an hour, days in a week, and so on and so forth. Most third graders will know the answers off the top of their head. Then you proceed to ask how many minutes are there in a day, how many hours in a week, and so on and so forth. Children will realize these are multiplication problems but each child will react with a different degree of enthusiasm –or apathy– to find the answer.
Students who somehow have a strong connection to time units will diligently work out these multiplication problems all the way down to the number of seconds in a week. With a little bit of help they will continue working until they get all the numbers right, even if they struggle with the multiplications tables all along. They do this because they want to know the answers. In their mind, these problems are real, not just an empty drill.
Some other kids do not care at all how many seconds are there in a day, so for them this particular set of problems will not be very motivating. You will have to find a different set of problems for them.
Even when students understand the concept of multiplication, and the basic rules to multiply numbers, they may feel that some questions do not justify doing all the work necessary to find the answer. It depends on how real or important the questions seem to them, because that is what makes them want to find the answers.

Calculus is harder than Algebra

A few sources of confusion to be aware of.

Sometimes I see students who used to get good grades in their Algebra classes but who are now struggling with Calculus.
They tell me they are confused by Calculus; they do not know when to apply the Chain Rule; they get uncomfortable around dx; they have no clue how to start the problems, and so on and so forth.
What most amazes them is they know they had no problems with Algebra, they understood Algebra well; so the question in their mind is: “What is it about Calculus that makes it so difficult?”
One problem is their implicit, non-accurate expectations. Sometimes they think they are doing something wrong because they have not gotten to the solution yet, after filling in one page with equations. However, they may very well be on the right track. They are not doing anything wrong. They just never expected for the solution to take so much work, and time. They were expecting to arrive to the solution after three or four steps, like they used to do in their Algebra work. Instead, with Calculus problems they may need to go through ten or twelve similar steps.
Sometimes they ask:
“Another integral? You mean, this is not the result yet?”
Apparently they cannot believe it. I tell them:
“Look, we have to keep going. We are not there yet but we are getting close. O.K.? Think of this as a little marathon. You were used to run the 400 meters, now we are going for three miles. We just have to keep going.”
Another feature that makes Calculus harder than Algebra is the huge number of options when it comes to selecting routes to the solution, many of which may turn to be dead-ends. Calculus incorporates all the operations of Algebra, including exponents, roots, and logarithms; plus all trigonometric functions; and it makes heavy use of function composition. All these ingredients can be thrown into a problem involving limits, or derivatives, or integrals. So Calculus really pushes you to get your Algebra up to speed. And because the expressions get more complex, there are more forks in the road at almost every step of the way. This is confusing for many students.
Now, in my opinion, the major source of discomfort when doing the transition from Algebra to Calculus, is that students who like Algebra like also order, and neatness. This proves to be a disadvantage when it comes to acquiring an intuition for Calculus.
They tell me:
“I do not know what to do, where to start, when to apply what rule, none of that! In Algebra everything was more clear, more precise.”
I ask them:
“And you liked that, didn’t you? Having a set procedure to follow in an orderly manner, right?”
They say:
“Yes!”
Then I say something like:
“Well, I hate to break the news to you but when it comes to Calculus, you are out there in the wild, and everything is moving, all the time, even if it does not seem that way. You have to get used to it.”
Many times they ask:
“But, why? Why is it that way?”
My explanation goes more or less like this:
“Well, Calculus was invented to deal with problems of a very physical nature: motion, volume, pressure, speed, weight, and the like, but not only that. The specific aim of Calculus was to provide answers as to how those physical magnitudes behave when they are changing, either growing or going down, and being related to each other at the same time. Originally, everything in Calculus happens in time, nothing stays the same, everything is changing, moving, decreasing, growing, building up, fading away, speeding up, slowing down, just like in the real world. So when you think of a variable x or y or v in Calculus, it is not just a number that is there, with some fixed value, but a number that wants to go somewhere else, a number that has already started to change, even if only a little tiny bit”
Somehow this bizarre wording paints a picture that makes sense for them, in that it puts their confusion into the proper perspective, eliminating a big part of it.
One of my students, after hearing this type of explanation, said:
“Oh! So, dx is the sneakiness of change! That’s why I don’t like it! But now I know I have to deal with it. I cannot just pretend it’s not there.”
And I said:
“That is exactly right. That is exactly what it is.”
Then she said:
“And, we always have to apply the Chain Rule, we cannot get rid of it, because the variables are connected to each other.”
And I said:
“Yes, the Chain Rule is always there.”
It is a big help to be able to tell students at least some of the reasons why they are confused, because when they do not even know the source of their confusion, everything gets exponentially more confusing for them.

Dealing with the difficulty of memorizing products of digits higher than five

Advantages of using Mayan and Roman numerals

When it comes to memorizing the multiplication tables, each child has his or her own pace. Some children find it easier than others to memorize the lists of numerical facts that make up the multiplication tables. Others need longer practice periods, maybe with the help of flash cards. Rote memorization is not everybody’s best act. Some children find these dry memorization exercises burdensome.
Here I offer a suggestion to help third and fourth graders who are not into rote memorization, to gain a better grasp on multiplicative manipulation of the higher digits.
The main idea is using the distributive property of multiplication over addition to calculate the result of a product of two “big” digits by breaking one of the “big” factors into smaller ones, doing two smaller multiplications, and adding up the partial results at the end.
The essential keys for this approach to be successful are:
First of all, do not even mention the phrase “distributive property.” That’s a big no-no. Do not do it. Show by example only. At this stage children do not need to know there is such a thing as a distributive property. It would be a waste of time. Avoid the confusion. Just do it. Show them how it works but do not try to explain why. The best way to understand why it works is for them to see how it works, period. Do not say aloud any abstract names like “property,” much less “distributive.” Stick to the numbers.
Second, use a standard “breaking scheme.” The number five is an excellent stepping-stone in this process. It is very natural for a variety of reasons. Mainly, because five is half of ten, the base of our numerical system, plus we have five fingers in each hand, so it is very easy to break any higher digit as the sum of five plus a lower digit. Children accept this fact very easily. Furthermore, the multiplication table of five is one of the easiest to remember.
Third, –and this is very important– spend enough time (at least half-an-hour) in a preparation period showing them, or reviewing with them, how to write numbers using Roman numerals and Mayan numerals. Do this before getting into any multiplication practice. This specific type of preparation has a dual purpose. On the one hand, it lowers their anxiety level. You have to understand they are under pressure. Their parents are at least concerned, maybe even worried. That is why they hired a private tutor in the first place. The child knows he or she is not doing great in the class when it comes to memorizing the multiplication tables. Some children may be even beginning to have some dents in their self-esteem, thinking that perhaps there is something wrong with them, or that they are not good at math, or whatever. The main idea in their head at this time is “multiplication is hard.” So when you –the expert– come along and start working with them doing lists of Roman and Mayan numerals, they go “Oh! This is really not that hard. This is easy.” Some children actually have fun with these numerals. Some prefer Roman numerals, while some prefer Mayan ones. The main point is now they are relaxed, at ease, and working with something they understand much better than the monolithic multiplication tables. That is the first goal of this preparation. The second purpose is for them to realize, or remember, or reinforce the idea of just how natural is the use of the number five as a breaking point, or a stepping-stone. Both Roman and Mayan numerals make heavy use of the number five, and of multiples of five, in this fashion. They consistently apply the principle of expressing higher digits as sums of five plus a lower digit. After doing this work with Roman and Mayan numerals, children are so much more receptive to the idea of using the number five as a standard "break point."
So this is the ideal moment for you to start practicing multiplication with the higher digits in this additive fashion. Here I give just one example to illustrate the main idea:
7*8 = 7 * (5 + 3) = (7*5) + (7*3) = 35 + 21 = 56
Do not expect them to know what to do. Guide them with questions. You are supposed to pause and ask as you write:
“Eight is five plus what? Three? Is that correct? O.K.”
“Now, how much is seven times five? Yes, thirty-five, perfect! Thank you.”
“And, how much is seven times three? Yes, twenty one, very good!”
“So, now we just add those two numbers. Can you please add 35 + 21 ? Thanks.”
Just by listening to you asking them these questions and watching you as you write down this multiplication process step by step, they get it, they understand it, and they end up empowered by knowing they can get the right result by themselves even if they do not have memorized the result, even if it takes them a little while doing it in steps like above. They are now much closer to self-sufficiency when multiplying higher digits.

Lowest Common Multiple

A Method and an Analogy to Clarify this Topic

During the last few weeks I came up with a way to explain how to calculate the lowest common multiple (LCM) of two or more integers or two or more polynomials.
Many students get confused by this LCM topic. One reason is the simplicity of the fact that, for any two expressions, their product is a common multiple, so, “Why look any further?” many students ask themselves.
They know their teacher told them in general the product is not the lowest common multiple of two expressions, so they know they are going to get marked down if they give that answer, but many do not know how to find the LCM.
Recently I improved my success rate at explaining how to find the LCM when I started using a table format, as follows.
In the head row I write the two or three polynomials or integers for which we are looking their LCM.
On the left margin I make a list (going down) of all prime factors of the expressions involved, without any repetition. Common factors get listed just once regardless of how many expressions they appear in.
Then, having one row per factor, and one column per expression, we fill in the table by writing the exponent each factor appears raised to in each expression, carefully including all exponents (even those with value zero or one).
After all exponents are listed in the table we make another column at the far right, under the heading “Maximum.” There we write the biggest number out of each row.
The next step is to form the LCM as the product of all individual factors listed in the table (in the leftmost column), each raised to its maximum exponent, as listed in the rightmost column (under “Maximum”). This last product is the LCM we were looking for.
Of course this process can be done without the table, but the table makes it explicit, and it helps as a visual aid for the student to see everything that is going on, all at once. It also helps in making very clear that we do not add the exponents, nor do any other operation with them, we only identify and select the biggest one for each factor.
Most students are happy with this process; the table is good enough for them. It gives them a clear method to follow, and it takes away the guessing and the mystery they formerly faced when trying to calculate the LCM. One of them even said: “You just saved my life with that table! Now I know how to do it!”
However, there are always a few students who also want to know why the procedure works, not only how to do it.
For those who ask “Why?” after seeing the table, I have this explanation ready:
“We have to imagine we are watching a movie about spies and intelligence agents, O.K.? Each expression is like a security checkpoint, where our agent has to show the proper clearances to get pass that point. The checkpoints have different sets of requirements. Each requires verification of a certain level of authority for each security category they are checking at that point. The factors of the expressions are the security categories, like “radioactive material,” “fire arms,” “chemical hazards,” and so on. The exponents are the different levels of clearance agents may have in each category. So when determining the LCM we are looking for the bare minimum possible set of clearance levels we need to give an agent for him or her to be able to make it through all the checkpoints, without any extra, unnecessary authority. They don’t lose their credentials when they go through a checkpoint. They only need to show their badges, they do not give them up. That is why we do not need to add exponents; we only need to select the highest from all the expressions for that particular factor.”
I have found this explanation works very well with all students with whom I have used it so far. One of them said: “Oh! I see. The x² from 3x²y is already included in the x³ from 5x³(x+1) because the exponent 3 is higher than 2. We do not need x⁵ or x⁶. Just x³ will be enough.” And I said: “That is exactly how it works!”

Quadratic equations in rotated form

Some long, time-consuming problems

The last three weeks have been very busy for me. I have been tutoring all math subjects, from fractions to Statistics and multivariate calculus.
Looking back over these past weeks it all seems kind of blurred but one topic stands out from the rest because, by coincidence, I had two sessions on the same topic with two different students, both during last week.
The topic in question is the rotation of quadratic equations in the two-dimensional coordinate (x,y)-plane. It had been a long time since I last taught this subject. It does not come up very often in my tutoring sessions, so I noticed the coincidence when I had two different students independently reviewing with me these geometrical transformations in the same week.
Also, each student separately made the same comment after we worked out problems of this type about quadratic equations: “Wow! This is a lot of work!”
They are right, it is a lot of work. The general problem starts with a quadratic equation like, for example, 5x²+2xy+10y²-12x-22y+17=0,
with a non-zero coefficient in the “xy” term.
The goal of the exercise is to find a specific angle, let’s call it θ, so that the transformed (rotated) equation in the alternate variables x’ and y’ lacks the x’y’ term.
The variables x and y are connected to x’ and y’ by means of these two equations:
x = x’ cos θ – y’ sin θ
y = x’ sin θ + y’ cos θ
Solving these problems requires several steps. I list them here, hopefully without going into too much detail:
First, finding the value of tan(2θ), the tangent of the angle double of θ.
Second, finding the measure of the angle θ itself.
Third, finding the values for cos θ, sin θ, and their squares.
Fourth, plugging those trigonometric values into the formulas below to find the new coefficients for the transformed quadratic equation:
A’ = A cos² θ + B sin θ cos θ + C sin² θ
B’ = 0
C’ = A sin² θ – B sin θ cos θ + C cos² θ
D’ = D cos θ + E sin θ
E’ = E cos θ – D sin θ
F’ = F
where A, B, C, D, E, and F are the coefficients of the original equation.
So you can see each one of these problems involves a lot of algebraic and trigonometric calculations. These problems are long, time-consuming, and you have to pay very close attention to all details to ensure an accurate result.
Anyway, in the video below you can see a room-size metallic structure (some kind of architectural sculpture) where Richard Serra, the artist, incorporated two congruent ellipses, one at the base of the room, and the other formed by the upper edge of the wall. The two ellipses are identical in shape but they are rotated with respect to each other. This is a real, tangible example of the rotation of a conic section. It is relevant to this post because quadratic equations represent conic sections, like the ellipses we see in the video. It is a very interesting structure. Take a look:

Making an Icosahedron

Geometry is fun!

Yesterday I helped one student with his Geometry project.
He had to build a 3-D model of a regular solid, so I showed him how to draw a net of equilateral triangles. We used the triangular net to cut out a template for icosahedrons.
The icosahedron is one of the five Platonic solids (the other four are the tetrahedron, the cube, the octahedron, and the dodecahedron). The icosahedron has twenty triangular faces, thirty edges, and twelve vertices.
Helping my student with this 3-D geometry project reminded me of a course I took in college, where we covered in detail the algebraic structure of the symmetry groups of the five Platonic solids. In that class each one of us built a few models of each Platonic solid, highlighting some of their features, like the cubes formed inside the dodecahedron by the diagonals of its pentagonal faces, for example. For me, that part of the course was a lot of fun.

In www.mathsisfun.com you can find ready-to-print templates for making paper models of the five Platonic solids.

In isotropic.org you can find similar templates, plus additional ones for the 13 Archimedean semi-regular polyhedra.

The following video shows how to make an icosahedron:



This other video shows a MatLab animation of an icosahedron turning itself inside out repeatedly displaying multiple icosahedral net configurations:

Solving the Rubik's Cube Puzzle

Step-by-step solution in a couple of YouTube videos by Dan Brown.

I just signed-up to YouTube yesterday, and this post is mostly meant as practice for myself posting videos into my blog.
While exploring YouTube’s archives I found a few videos about solving the Rubik’s Cube puzzle. Rubik’s cube is one of my favorites puzzles because it is closely related to both Group Theory and Graph Theory, branches of modern math. Playing with Rubik’s cube also helps somehow develop one’s intuition about the Cartesian (x, y, z) coordinate system in 3-D space.
In the two videos below, Dan Brown incorporates a little algebraic notation to precisely describe a few sequences he uses in his general solution of the Rubik’s cube.
So far I have not used Rubik’s cube as a teaching aid in any of my tutoring sessions, so this post really does not necessarily have a lot to do with tutoring but I decided to include it anyway because the puzzle does have to do with math, and it is fun.
I hope you will enjoy the videos!

P.S. After loading these first videos I decided to search for other videos with content related to that of my previous posts, so I will be including some more videos in those older posts too.

Marathon GMAT Tutoring Session

Very few students go for a three-hour-long tutoring session

Recently I had a marathon (three hours in a row) GMAT tutoring session, with a student who had only a short time to prepare for the test. The coffee shop we were sitting in closed at 8:30 pm so we had to move to another coffee place nearby to finish the session.
We went over exponents, roots, ratios, percentages, percent increases, decimals, inequalities, data sufficiency questions, area and circumference of a circle, averages, powers of 2, prime numbers, factoring, the Pythagorean theorem, special triangles, Venn diagrams, area of a trapezoid, task completion team time, probability, counting unordered pairs, counting with the multiplication principle, rolling two dice, word problem key words, approaching word problems, setting up tables, picking numbers, and a few other topics.
This is only the second time in three years a student has requested a three-hour-long tutoring session. Maybe not by coincidence, the previous time it was also for the GMAT.
I have had plenty of two-hour sessions, and 90-minute sessions too but in the last three years only two of my students have gone for a solid three hours in a row.
I have no problem with a three-hour session; I can go for longer than that. It is usually my students who limit their sessions to one hour. More often than not, after one hour my students clearly indicate they can use a break from math.

Welcome to Number City!

Find your way around. Don’t get lost.

Sometimes students ask me: “How long is going to take for me to pass this test?”
To which I reply: “It all depends on how fast you get to the performance level you need for the score you want.”
The key phrase here is “performance level,” which the tests are supposed to measure.
Sometimes I have to be almost brutally honest by saying: “Look, realistically, as long as you keep hesitating for more than three seconds to come up with the result of a single-digit multiplication, there is no chance you are going to solve a whole problem in less than two minutes. You want to have all those little things down to less than a couple seconds, with no hesitation whatsoever. You have to let go of all those thoughts about not being good at math, or not liking math. If you really want to pass this test, you need to learn how to handle fractions, and all these other things you always hated and have never completely understood so far.”
There is an interesting metaphor I find useful to help students start distancing themselves from their math phobias. I say:
“Think of it this way: Imagine Math is a city you used to visit when you were a child, a city you never liked because you always got lost, or maybe even someone stole your money, or you always got sick when you were there, or something bad like that. I acknowledge it’s only natural for you to harbor bad feelings about that city. Now, because you want to pass this test, it is like now you have to move to that city and live there for a few months. Not only that but, to finally get out of it, you need to work three jobs while you are there, and you need to excel at all of them. You are going to deliver packages during the day, deliver pizzas at night, and drive a taxi cab on the weekends. Do you think you can allow yourself the luxury of being lost again? Are you going to stand there all confused for hours about how to cross the street, or about what avenue takes you downtown? To really do well in those three jobs you want to know all the landmarks, the big buildings, the highways, street names, bus routes, trolley stops, shopping malls, different neighborhoods, and the like, right? So, it’s just like that in math, too. Welcome to Number City. That is why I recommend you to memorize by heart the times tables, square numbers, primes, powers of two, odds, evens, integers, and things like that, so you can easily find your way around and move from place to place as fast as you can without getting lost again. Number sets like “squares” or “primes” are like avenues. Each individual number is like a franchise brand name, with multiple locations around the city. Algebraic operation rules are ways to get fast from place A to place B, like taking the subway or the highway or something like that. You want to set aside your old fears and phobias for a while, and apply yourself to the task of getting to know your way around this city. Then you will pass your test and you will be able to move out and move on with your life. That is what’s needed.”
I find the above analogy helps some students to kind of materialize their math fears and phobias into something external, and objective. They know what is like to familiarize oneself with a new city, so this is a task that looks familiar, doable, and makes sense for them. So they can stop the negative workout on their self-esteem, and focus instead on these concrete and essential memorization steps.

Tutoring Physics

Just fine as long as it is not advanced

Curiously enough, these days I find myself tutoring Physics sometimes.
Three students I previously tutored in Calculus, as well as one I am currently tutoring for the GRE, all of them have recently asked me to help them with Physics.
The first time it was a surprise for me because my student called me over the phone and all she said was she wanted to schedule an appointment. I just assumed she was taking another calculus class. When we started the session I looked at her book and said, “This is Physics”! Her only reply was a monosyllable: “Yeah,” along with a completely natural, matter-of-fact look on her face. I realized she expected me to know Physics so I went ahead and helped her with the problems she had to study for her exam, and we got all of them right! It was a very nice surprise for me. I thought it funny that it was not a surprise for her because all along she simply had assumed I knew Physics. Fortunately I did not disappoint her.
So far I have been able to successfully help these four students except for the last session with one of them, who is taking a Static Mechanics class at UCSD for his engineering major. Most college courses are packed full with topics to cover, and professors typically move very fast through the material. I had no problem helping this student with the first few topics, including up to finding momentums of forces in three dimensions but when we got to the chapter on force couples and reactions in systems at equilibrium I realized it is going to take me a while to figure it out.
The reason is at first any new subject seems esoteric to me until I find the meaning of its concepts, and the reasons for each step in its particular problem-solving processes.
Most likely I wont be able to go any further right now with this student but chances are I will soon find time to do some research, and I will be better prepared on this Static Mechanics Physics subject the next time around.

Integration by Parts in Calculus 2

It is really neat when students make their own discoveries.

This morning I had a Calculus 2 tutoring session about Integration by Parts.
My student was having some trouble with those integrals that are calculated by applying the integration-by-parts formula multiple times.
In this type of problem you get somewhat complicated expressions with parenthesis nested inside parenthesis, often with multiplying coefficients and/or negative signs in front of each parenthesis.
Students usually get confused when doing these integrals for the first time. The main reason is because they do not expect so much complexity. They believe the problems are going to be shorter than they actually turn out. So one key for getting these integrals right is to keep track of every step separately, identifying each new integral, and labeling it with a new variable of its own (using I, I1, I2, I3, ... works well). Then you keep working all the way down until you finally reach an integral you can actually solve, without any new “left-over” integrals. Then you re-trace your steps back one at a time, substituting each (ever longer) expression into the corresponding place for the previous partially solved integral, until you get to the original one.
We were working with the function (x^3)(e^x), where you start by deriving x^3, and integrating e^x.
My student understood everything but he wanted to make sure he would remember it later, so he started all over again. I suggested this time to start from the bottom up, so he integrated x(e^x), then (x^2)(e^x), and then (x^3)(e^x). I suggested for him to keep going so he integrated (x^4)(e^x).
At this point I said he could even memorize all the resulting formulas just for the test but it did not seem a good idea to him. Then I asked: “Well, maybe there is a pattern here. Do you see after factoring out e^x the last number (the constant term in the polynomial factor) is a factorial? And the polynomial always starts with the power of x we have in the original integral, does it not?”
I started looking for a way to factor out the polynomials. It turned out to be not always possible (the second-degree one already had complex roots) but all of a sudden my student exclaimed: “They are all derivatives!” So the pattern was not apparently multiplicative, but one more closely related to calculus, because it involves successive derivatives.
The best part here is my student saw it himself without my help. He realized what the pattern was before me.
That really made my day. I enjoy watching students making their own discoveries. This was not a textbook exercise, but a question I came up with after we went beyond the examples in the book, and my student was able to come up with the answer faster than me, showing he really understood the question. And now he has a very effective reference point to remember all those integrals if they show up in his exam! He won’t forget it because he discovered it on his own.

Multiplication Tables

A very important foundation for understanding math

Sometimes I see students who are really intelligent but who are having problems at school understanding new material. Some of these students only need to go over a few specific examples to grasp the concepts and move on to the next topic. So it is sad and almost unbelievable to discover that, in a few cases, the real obstacle standing on their way is that they do not know the multiplication tables! I remember when I was in second grade I hated learning the multiplication tables because the repetition process was so boring and it seemed meaningless to me at the time. However, in third grade I discovered the benefits of knowing by heart the multiplication tables. It allowed me to understand division. Understanding division allowed me to build a solid understanding of fractions.
In this time and age, many generations have grown up and gone through school using pocket calculators. A few people have made it all the way to college without ever learning how to multiply two numbers without using a calculator. The problem for them is, the more advanced the math courses they take, the more trouble they have at trying to figure out how formulas work by looking at specific examples. They cannot think their way through the examples because they don't know their multiplication tables; therefore their mastery of division, fractions and exponents is very limited and shaky.
To students who are preparing for the GMAT, GRE or CBEST, I always recommend to review, polish, extend and reinforce their knowledge of multiplication tables. The importance of this foundation cannot be stressed enough.

Absolute value expressions in 5th grade?

Too abstract topics too early

One of my students is in 5th grade. Not long ago his homework consisted of writing down the full-blown English names of twelve-digit numbers, like 535,176,402,988. He kept busy writing line after line of tens of millions, and hundreds of billions. After a while, the assignment seemed pretty boring to me but my student was interested in the task all the way through. I think a big reason for his motivation was that he was able to do it. The big numbers seemed challenging to him, but the task was doable because he completely understood the principles involved in the translation.
Last week his parents asked me to go with him over some questions he got wrong in a quiz. I was amazed to find in this quiz questions involving absolute value expressions!
I was like: Absolute value in 5th grade? What for?
I don’t know about you but it does not make any sense to me. I mean, the first time I knew absolute value existed, I was in 12th grade, at the end of high school. Now they are covering absolute value in elementary school? Give me a break!
It was kind of hard to explain his mistakes to him, in part because the absolute value concept is way more abstract than the concept of hundreds of millions, and in part because he did not want to accept he made a mistake. So he was ecstatic when I discovered that in one of the three problems he was marked down he actually had selected the right answer. He was right on that particular problem, not wrong.
Which kind of proves my point, in a way. The absolute value concept is too abstract not only for most 5th grade students, but apparently for some 5th grade teachers as well.

You Don't Have To Do Anything

Think about what you can do, not what you "have to" do

Very often students seem to freeze when they see some kind of problem. In this situations I ask: "What are you thinking? What is going through your mind right now? What thoughts, feelings or ideas do you have when you read this problem?" By asking this types of questions repeatedly, I have discovered that, in many cases, when students find a particular kind of problem (the exact type varies from student to student), they think they are supposed to follow some steps, some fixed routine they were taught at some point in the past by one of their teachers. The problem is now they don't remember what are those steps they think they have to follow, and more importantly, in most cases they never totally understood the reasons why those steps work.
They typically give me answers like:
"Well, I think I have to multiply these two numbers, but I am not sure..."
"Ah, I need to add these fractions, but I don't know how..."
"I forgot what formula I have to use for this problem..."
"I am supposed to set up an equation, right? But how?"
The key words these answers have in common are verbal forms like: "I have to," "I need to," "I am supposed to," etcetera. They really think there is something very specific they have to do, and they just don't know what that is.
When I spot this blockage I say: "You don't have to do anything!" They look at me and they go: "I don't?" They look quite surprised but relieved at the same time. Then I say:
"No, you don't have to do anything. You don't even have to solve the problem. I mean, you want to solve the problem because you want to pass your test, right? But you don't have to, you want to. Now, to solve the problem, you can do that by going whatever way it works. Nobody is going to be looking over your shoulder to see how you do it. It's a multiple choice question. Nobody cares how you do it. The only thing that counts is whether your final answer is right or wrong, right?"
After I make my point clear, they usually ask: "But, then, how can I solve it? What can I do?" And I say: "Exactly! Perfect! That is the right question. What can you do? Well, what do we have? Look at the problem, look at the numbers, look at those expressions. What can we do?" Then all of a sudden they go: "Oh! I can take the 50 out on both sides, then I can substitute this variable for that other formula!" or whatever the case may be, but they start working their way to the solution.
The words we choose to talk to ourselves make a big difference. When students think in terms of "I have to," "I need to," "I am supposed to," those words take them to a mental and emotional state where they were just going through the motions and mechanically repeated meaningless tasks they didn't understand and they didn't care about.
If instead they think about their possibilities, their options, about what they can do, then it's much more likely they will find the clarity, creativity and initiative that will lead them to finding the right solution by themselves.

Find and Replace

The substitution method

Most students understand the concept of substitution when the task is to plug in a plain numerical value for a variable in a formula.
A typical example would be to evaluate y = 3x^2 - 5x + 2
when x = -1
Things change dramatically when the task involves plugging in an algebraic expression to replace a variable in another expression, even when the expression we are plugging in is of small complexity.
For example, from y = 3x + 5 plugging in the value 3x + 5 instead of y into the equation 2x - y + 4 = x + 3y - 1.
In the past I often had trouble explaining the process to some students. In my experience, a significant fraction of students taking the tutoring have some difficulties mastering this process. They get the concept in theory, and they are able to apply it in simple examples, but as the replacing expression grows in complexity, they quickly get stuck.
Lately though, I have dramatically increased my success rate for teaching this concept by using the following analogy. I go:
"O.K., time out. I have a question. Are you familiar with the computer program MS Word, the word processor? Have you used it to type some letters?"
They look at me as if I was asking them whether they are from this planet, and they say "Yeah..." Then I continue:
"Have you seen that little binoculars button that says Search & Replace? You know, when you have just finished writing a letter, but you are not very happy with a particular word you used several times, and all of a sudden you think of a better word. Then you click on that Search and Replace button, right? Instead of reading over the whole letter, looking for the word you want to change, and manually typing the new word over and over."
Then their eyes light up and they say: "Oh, yeah! And it gives you the total count for how many replacements were made!"
At this point I say: "Exactly! That is what we are doing here, search and replace. This equation is the letter and this other equation is the Search & Replace button." They silently look at me with a face that yells, "What are you talking about?" Then I proceed:
"Look, this is the letter, O.K.? Our document, from here to here, this equation: 2x - y + 4 = x + 3y - 1 . That is the whole document. And we are the program. This other equation here: y = 3x + 5, that is the Search & Replace button that says:
'Search the document for the letter y and every time you find it, replace it for this other phrase: 3x + 5.' So we perform the instruction, right? We go over the document, symbol by symbol. we copy the 2, we copy the x, we copy the '-' sign, and then we find a 'y.' Well, instead of 'y,' we write this other thing, we write '3x + 5' right? And then we just keep copying the symbols from the original equation until we find another 'y' and we keep doing that search and replace thing until we reach this last '1' here at the end, the last symbol in the original equation."
They totally get it! For confirmation, I ask: "Does that make sense?" They usually say: "Yes, perfect sense! I mean, I get it. Is that all there is to it?" I look them in the eye and I say: "Yeah, that's it" Then they go: "Gosh, let me do the next one!" And they normally get it right in the first try or at most two tries with almost no exception. I am very happy I found this analogy.

Different Learning Styles

Everyone learns in his or her own individual way

Some students like to go fast. They quickly pick up a new problem-solving method as soon as they see that it works. They take note of the new method and they are ready to move on to the next problem. Some other students want to stop and ask several questions and examine the new method from different angles before giving any credibility to it. They are not satisfied with one or two examples or with only one type of explanation. They ask questions like: "What if x was negative instead of positive?" "What if the root is a cube root instead of a square root?" "Is it always going to work like that?" "How do I know when I have to use that formula?", and many other similar questions.
Each approach has advantages and disadvantages. In an exam, people who like solving problems fast are more at risk of making simple mistakes in the details of the calculations but they have a better chance to work with all the problems. People who like to pay attention to detail and to carefully think things through are more at risk of running out of time and having to guess in a hurry several problems at the end of the test but they tend to have a higher ratio of correct answers in the problems they solved first.

Probability and the Binomial Distribution

Some helpful key concepts

Probability problems describe a random experiment and some specific event its probability you are asked to calculate.
The probability of the event is a fraction, a positive fraction between 0 and 1. This fraction is a smaller number divided by a bigger number.
The big number at the bottom of the fraction is the total number of all possible outcomes of the experiment, everything that can possibly happen in the experiment.
The small number at the top of the fraction is the count of exactly how many ways the very specific event we are interested in can come to happen.
Usually two different counting techniques are needed, one for calculating the big count, the denominator; and another for finding the smaller count, the numerator.
It is a good idea for GMAT or GRE test takers to learn well a formula called “the probability distribution of a binomial random variable.”
I know the name is quite a mouthful but go by this example:
A 60% heads, 40% tails biased coin is tossed 20 times.
The total number of times the coin lands “heads” is represented by X, and is called “the random variable” in this experiment.
The “number of trials” is represented by n, and it equals 20 in the above example.
The “probability of success” at each individual trial is represented by p, and it equals 60% in this example.
The “probability of failure” at each individual trial is represented by q, and it equals 40% in this example.
There is an established formula for calculating the probability that the number X will equal each integer value from 0 to n (0 to 20 in this example). You want to memorize said formula and get enough practice applying it until you feel quite comfortable using it in various situations.

Repetition Is Key

Practice makes perfect

Most students need several exposures to a procedure before being able to perform it correctly. Successful performance of mathematical procedures has several ingredients: understanding the concept is fundamental but is not enough. Memory plays an important part as well, you have to put in the effort required to commit the formulas to memory.

Then there are the concrete examples that give the abstract principles a solid meaning. The most important part is application to complex examples, how to actually use the formulas, how to recognize you have an expression in front of you that is calling for that particular formula. More often than not, each of these performance components needs the student to be exposed multiple times to the subject matter in order for the skill to be correctly installed into the student's mind. In these regard our brain differs vastly from computers. A typical computer program gets completely installed in one installation session. The programs our brain uses to solve mathematical problems generally need multiple installation sessions to become functional. Sometimes there seems to be a big, tall skepticism barrier guarding the mind of the student against any mathematical learning. Sometimes, with some students, I end up under the impression that I have to show them how the formulas work and what the connections are from every conceivable angle before they finally make the decision to believe in the process, to see it for themselves, and to use the tools. In these cases I feel they make every mistake they can possibly make, at every chance they get. I do not attribute this behavior to lack of intelligence but to a strong skepticism, some sort of lack of faith, and a need for attention. That is another reason repetition is so important, not only because it is required for most students to understand a procedure well, but also because some students need the teacher to have multiple opportunities to figure out a way to finally explain the subject to that particular student well enough that finally he or she makes the decision to trust the evidence that has been presented to him or her and adopts the procedure.
Practice makes perfect, for the student as well as for the teacher.

How to Make the Most of Private Tutoring

Useful tips for students looking for tutoring

Find a math tutor who is right for you.

Plan ahead, work out a budget, and find a tutor within your price range.

Make sure you feel comfortable communicating with your tutor by e-mail, phone, and in person.

Make sure you feel comfortable about the tutoring schedule, and the meeting place you use for the tutoring sessions, whether it is your own house, a coffee shop, a public library, or the tutor’s office.

Schedule at least one tutoring session per week through your exam’s date. Two weekly sessions 3 or 4 days apart from each other is the ideal schedule. Three weekly sessions may be necessary in some cases. One-hour-long sessions work best in most cases, although some students prefer 90-minute sessions. Your tutor should be flexible and accommodating about this. He or she should work with you to find a good working schedule. Once the schedule is set, strictly adhere to it as best you can.

Make sure the tutor clearly explains the subject matter, and answers your questions in a way that is easy for you to understand.

Bring all your study materials to the tutoring sessions: prep book(s), notebook, pencils/pens, scratch paper, and the same calculator you will use in the test (if one is allowed).

Set enough time apart to work out practice problems on your own before each tutoring session.

While you are working out practice problems by yourself, identify all the problems that give you trouble. Mark them with sticky notes, color flags, highlighter, or in any other way that reminds you what kind of trouble you are having with each particular problem.
Mark the problems you did not even have a clue how to approach them; those where you got the wrong answer; those that took you too long; those that you got right but you are not really sure why; those where you read the explanation provided in the book but you did not understand such explanation. Mark all such problems.

The purpose of the preparation work described above –prior to the actual tutoring– is for you to enable the tutor to maximize results in your benefit. You want your questions answered. You don’t want to pay money just so someone sits next to you watching you solving a bunch of problems that you can do on your own without any trouble.

During each tutoring session make sure to ask all questions that come to your mind. Do not allow you to keep the slightest bit of confusion to yourself, share it with the tutor. Help your tutor understand what is confusing you. Tell him or her how you feel, and what thoughts and/or emotions go through your mind when you read each problem. It is very important your tutor understands your thought processes (whatever they may be) so he or she can help you by pointing you in the right direction. When I say “the right direction” I mean the right direction for you with that particular problem. There are many problem-solving techniques and strategies available, and each student is uniquely suited to use some better than others. That is why telling your tutor all your reactions to a problem is so important.

Make notes about everything your tutor says that you find useful, new, insightful, or otherwise helpful in any way. Any problem-solving methods, techniques, strategies, rules, formulas, tricks or shortcuts your tutor shows to you, and you feel they will help you solve the problems faster and/or with greater accuracy or reliability, make a note of them to review them later, and apply them to similar problems.

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

Percentages of Percentages

How to apply the percentage concept to itself

Example problem:
In a class, 40% of the students have blue eyes and 20% of those students with blue eyes have brown hair. What percentage of the whole class has blue eyes and brown hair?

This problem does not say how many students are there in the class. So, we can pick 100 to simplify the first step. If there are 100 students in the class, then 40 of them have blue eyes because 40% of 100 is 40.
Now, 20% of those 40 have brown hair. Because 20% of 40 is (20/100)(40) = (2/10)(40) = (1/5)(40) = 40/5 = 8, there are 8 students in the class with blue eyes and brown hair.
So the answer to this problem is 8%.

Another way to state this result is by saying that the 20% of 40% is 8% because 20% of 40 is 8.

A very important key to solve this kind of problem is to pay close attention to the words following "percentage of." The phrase after "of" completely determines the situation.

If the problem had said instead:
40% of the class has blue eyes, 20% of the class has brown hair, and all the students with brown hair have blue eyes. What percentage of the students with blue eyes has brown hair?
This would be a completely different problem.
Here the answer would be 50% because 20% of the class is exactly one half of 40% of the same class.
We could express this by saying that the 50% of 40% is 20%.

The difference between the two problems above is that 20% of the class is some number, but 20% of the students with blue eyes is another number. These numbers are different unless we knew all the students in the class had blue eyes. Usually though, the numbers are going to be quite different and the key to figure them out is to pay attention to the words that come after the word "of" in "percentage of."

By the way, when translating these phrases into equations, the word "of" is translated as multiplication, the word "is" is translated as the equal sign, and both the word "percentage" as well as the sign "%" are translated as "division by 100". For example, the phrases:

"The number B is x% of the number A"
"x% of A is B"
"What percentage of A is represented by B?"
"B is what percentage of A?"

are all equivalent and they can be translated into an equation as follows:

(x/100)(A) = B

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.

One Way To Get Familiar With More Square Numbers

Using the algebraic formula for the square of a sum

At the high school level, most students know all square numbers between one and one hundred: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100.
That is, the squares of all numbers from 1 to 10. Many students also know 121 is the square of 11 and 144 is the square of 12.
That is about how far most people go memorizing square numbers. Relatively few people know 196 off the top of their head when asked: "How much is 14 times 14?"

I find useful the following exercise for students who are learning topics such as the quadratic formula and how to factor trinomials:
I remind them of the formula for the square of a sum:
( a + b )^2 = a^2 + 2ab + b^2

Then I suggest to them applying it in the following way to figure out the squares of numbers between 10 and 20:

(14)^2 = (10 + 4)^2 = 10^2 + 2(10)(4) + 4^2 = 100 + 80 + 16 = 196

(17)^2 = (10 + 7)^2 = 10^2 + 2(10)(7) + 7^2 = 100 + 140 + 49 = 289

Working out a few of these concrete, numerical examples helps students to see a general pattern and become more familiar with both the formula for the square of a sum and, at the same time, with squares of numbers between 10 and 20. This feels like a natural expansion from their previous knowledge of squares of numbers between 1 and 10.

Have a Calculator Handy When Doing Long Division

Not to skip the work, but to check your results

Long division problems are among the most boring, detail oriented, time-consuming problems math has to offer.
I agree that 7th graders should know how to perform long division by hand, without a calculator. In my opinion, even 4th graders could be required to perform long division accurately.
However, some teachers go into overkill mode when assigning long division homework. They throw in too many digits into the calculations and too many problems into the assignment. I find this practice counterproductive when it comes to long division. There is a fine line between drill and overkill.
When students face such a heavy workload, such a long time doing these boring, exacting, fine detail, uninteresting drill problems, row after row, it does not take very long before many people start hating math with a passion, I tell you.
Once you have satisfied yourself that you understand the procedure, that you know how to do it, that you can actually do it and you can really do it well, what is the point of keeping at it beyond that? People are not machines.
Some students lack the attention span required to accurately divide a seven-digit number by a six-digit number. Much less doing ten of these calculations in a row. For them, this type of homework is an exercise in discipline and endurance, not in math or understanding. It is not their intelligence that is at play or in question, but their ability to submit to an arid, boring, meaningless routine.
At some point they start speeding up, they stop paying attention, and they start making mistakes.
If you find your child or yourself in this situation, I strongly recommend using a calculator. Not to skip the work altogether, but to check your result and make sure whether or not you made a mistake. Long division problems are exactly the kind of problems calculators are for.
If you are not 100% sure, beyond any doubt, that your division work is absolutely correct, then the calculator is almost the only way to find out if there are any mistakes there.
I say it is valid to use a calculator to check your long division answers.

Some Questions In Test Prep Books Make No Sense

But you can find the right answer anyway

Test prep books are very well written and edited. They generally present the right solution for almost every single problem included but every once in a while (seldom though) you can find a mistake. Sometimes the given answer for a problem may be wrong or, even more rare, some questions don't make sense from a strict, formal, rigorous, mathematical point of view.
The following question is a perfect example of this:

Select the number that is not a factor of 6/288.

All the five options given in the book are fractions (and different from zero).
Four of those fractions have a numerator that is a factor of 6, and have a denominator that is a factor of 288.
The other fraction is 2/11, where the denominator, 11, is not a factor of 288.
It turns out that is the answer the book indicates as the correct one. So thought by whoever designed that particular problem.

But strictly speaking, the question does not make any sense at all.
The set of rational numbers is a field, an algebraic structure where every non-zero element is a unit. In a field every element can be divided by any non-zero element.
That means, according to the technical definition, any non-zero fraction is a factor of any other fraction.
That is why the question does not make sense, because, in the domain of the rational numbers, the fraction 2/11 is a factor of 6/288.
How come? Simple: (2/11)(33/288) = 6/288.
There is a fraction, namely 33/288, that multiplied times 2/11 gives 6/288 as the result.
That makes 2/11 a factor of 6/288.
In the same way, given any non-zero fraction, we can always find another fraction to multiply it by and get 6/288 a result.
For that reason, it makes no sense at all to talk about factors of a fraction. Once we are dealing with fractions instead of restricting ourselves to whole numbers, every non-zero fraction becomes a factor of any other fraction.
So the term "factor" totally loses meaning in this context.

Fortunately in this case, it is still possible to figure out the answer the creator of the problem wants you to give.

Successive Changes Given As Percentages

Beware of addition in this case

A car increases its speed by 25%.
After that, it increases it again, this time by 20%.
By what percentage is the car's final speed greater than its original speed?

I solve this problem as follows:
(1.25)(1.2) = 1.5, so the answer is 50%.

Most students solve it the following way:
Assuming the initial speed is 100, the next speed is 125.
Now, 20% of 125 is (20/100)(125) = 2500/100 = 25
Therefore, 125 + 25 = 150, so the overall increase is 50%.

This method is of course correct, and equivalent to mine. It is just a little slower because it's more explicit.

What you must avoid when solving this problem, is the misguided notion that the result can be achieved simply by adding the two successive percentages.
In other words, 25% + 20% = 45% is the wrong method and it gives the wrong answer for this kind of problem.

When a change (increase or decrease) is measured as a percentage, there is an underlying assumption. We assume the percentage is relative to the initial and final states of that particular change, 100% meaning the amount being measured at the initial state, before the change.
So, if a second change occurs after the first one, it automatically modifies our frame of reference.
For example, let's say a magnitude (speed, volume, length, time or whatever) changes first from A to B, and then from B to C.
In this situation, 100% in the first change means exactly A, but 100% in the second change means exactly B.
The combined change from A to C expressed as a percentage is the number (100)(C-A)/A, and we put the word "percent" after that number.
Now, (C-A)/A = (C/A) - (A/A) = (C/A) - 1 = (C/B)(B/A) - 1, which is the basis for my multiplicative method.

Sessions on Pre-Calc and GRE prep

A good tutoring day

Yesterday I had two sessions. The first one was to prepare an entrance placement test where the student wants to qualify for a particular Calculus course, so he is being tested on Pre-calculus. The second session was with two students who are preparing for the GRE. Both tutoring sessions were early in the afternoon at the same Starbucks.
In the pre-calculus session we covered a wide variety of topics. Every problem in the study guide was about a different subject. For example, we saw inequalities with absolute value, exponents, logarithms, function evaluation, subtraction of algebraic fractions, factoring algebraic expression and so forth. The difficulty level was not hard, and the student understood all the explanations, even when he had not taken math in about ten years, since high school.
In the GRE prep session we focused on Geometry problems, working out of the ETS book. We reviewed basic properties and formulas for parallel lines, polygons, angles, triangles, special triangles, the Pythagorean formula, square roots, perimeter, area, circumference, volume, and surface area.
Both sessions had a nice flow regarding the problems. We did many different exercises and we did not get stuck at any single problem. Yesterday was a good tutoring day.

Sessions in Trigonometry, GMAT and Pre-Calc

A busy tutoring day

Last Saturday I had three students. The first session was about graphing sine and cosine functions, high school trig. The second one was a GMAT prep session, and the third one was about circles and parabolas, pre-calculus.
During the first session I was partially awake, not very alert, so I had to do the problems myself, kind of slowly, organizing the data in a table format, just to have it all in front of me and to be able to see what’s going on with the series of numbers. I personally like to solve the problems this way, but I don’t like having to do it like that during a tutoring session. The reason being the students understand the reasoning and the results but they sometimes end up with this expression on their face, like saying “How am I going to do that by myself when I am all alone?” This time it was pretty clear though (the hot chocolate helped me to wake up).
The GMAT prep session was the first one for the student I met last Saturday. It was one of those talks when they fully see what kind of test they face, and all the amount of work they will have to do, and for a moment they seriously consider quitting. This is a good sign. The GMAT is not a walk in the park by any means. The student is better advised to expect a heavy workload and to make some sacrifices in their schedule. We covered the inevitable “What for?” question. I am always straightforward about it; the test is just a hurdle for them to win admission over other applicants. Math is the cheapest filter applicable in a mass scale. Then we worked on a few problems covering sub-indexes, recursive formulas, Venn diagrams, and volume and area formulas. We scheduled a two-hour session for next week.
The third and last session last Saturday also lasted two hours. We had several problems where you are required to find the center and radius of a circle given three points on the circle. It was a little frustrating for my student to see how vastly different these problems can be in terms of difficulty level, computational detail, and time consumed. A set of points forming a right triangle, with horizontal and vertical legs of even length is like candy, but when the problem throws at you points with decimal coordinates, and fractional values for the slopes of the sides of the triangle, you can fill three pages and spend almost half and hour with just one problem. This can set the student in a state of panic, you know? Just realizing how long and tedious it can get sometimes may be alarming for many. Some books are like that, with problem sets that escalate quickly in the difficulty scale. At least my student ended up with the idea that it may be hard but is not impossible to work these problems out.
After the three sessions I was fried, and hungry. I went to eat a tasty bread bowl of soup at the Quizno’s Sub store in the Renaissance Town Center, off of Nobel Drive.

Moving over to Blogger

My previous blog host went off line. I think Blogger.com will offer a much longer lasting service. Anyway, I am re-posting here the backup copies of my old posts, so I apologize for the inconvenience if you already saw this material. I expect to catch up pretty soon.

Negative Zero

Yes, it is a real number.

Every once in a while, I find students who show surprise or disbelief when they first encounter the concept of -0 (negative zero).
Let's say they are solving some equation, and close to the end it reads like x = - (a - b), where a and b represent two numbers known to be equal by virtue of the conditions set at the beginning of the problem.
So, in this example, the next step would be to write x = - 0.
My observation here is that, some students in this situation freeze, turn their head towards me, with a strange look in their face, and go: "There is no negative zero, is there?"
Usually I reply: "Why not?"
And they go: "But ..., what is it?"
I say: "It's zero."
Then they say: "Oh! Really? Just that, zero? Are they the same?"
And I say: "Yes, they are the same thing."
And they go: "O.K."
They seem to suddenly realize that the concept makes sense and it's not really that big of a deal.
I mean, what else could it be? What else could negative zero be if it wasn't equal to zero? There is no other option.
Actually, being equal to its own negative, is a defining feature that uniquely identifies zero.
Zero is the only number equal to its own negative. If you find any number x for which x = - x holds true, then you know x must be zero.
However, the momentary puzzlement, surprise and disbelief some students show when confronting this concept for the first time, is quite natural.
Remember it took centuries for Western civilization to come in contact with the concept of zero, and to fully adopt it as part of the number family. At first it was not considered a "true" number, but only an artificial placeholder used in the representation of "true" numbers.
Not only zero had difficulties being accepted as a number, but also the number One went through a period in Greek history when it was considered more like a philosophical, psychological, or even a religious concept, not a plain mathematical entity.

Another Important Key to Problem-Solving

Write down ~everything~

Very often I see students struggling with confusion when they try to solve a word problem all in their head.
It is so common, it's amazing. First of all, nobody says you have to produce the answer by just looking at the problem.
This is what happens, they read the problem, they understand the first sentence, or the first few sentences, and they are already asking themselves: "How do I solve this?" "What do I do with these numbers?" "What formula do I apply?" "What operation am I supposed to perform?"
They are obsessed with the idea of taking action steps. This is the first obstacle.
As soon as they come up with an idea about what to do, they start doing it, they start performing the operations, all in their head.
Then they get a partial result, and they immediately jump with that result into the next operation, without writing down anything. It's just unbelievable!
When I see them doing this, I tell them: "You are using your mind as a calculator and as a piece of paper at the same time. Don't waste energy like that."
For the average person, the mind can be much more effective as a calculator than as a piece of paper. The short-term memory that stores numerical results from previous calculations is very volatile.
When you try to use your head to do the operations and to remember the results at the same time, you are headed for trouble and confusion.
Let's say you make a mistake. If you write down all the steps of your calculation, and don't erase anything, you are much more likely to catch your own mistake when you go back and check your steps.
If you don't write down anything, you won't even remember what operations you performed, let alone catch a mistake.
So, write down everything, not only the partial results from each operation, but the whole calculation.
Write down not only the calculations you perform, but the ideas that made you perform those calculations.
Write down everything, every single step, all of it, your ideas, your examples, the formulas you are going to use, everything.
You will be amazed how easy the process becomes when you create this habit of writing down everything as soon as you think of it, and not erasing anything.
Even with mistakes, in the end it works better just to mark them with a red circle, and rewrite the correct expression somewhere else in the page, instead of erasing them. Many times mistakes are useful for reference.
As a general rule, the more you write, the better. The more you write, the less stress you put on your mind and the easier the process becomes.

One Key to Problem-Solving

Not "How To?" but "What do we have here?"

I always emphasize this to my students. When first facing a math problem, especially a word problem, do not try to get the answer right away. This is an unrealistic expectation. The answer will come as the result of properly developing all the relevant, detailed information contained in the problem. Pay attention to the wording. Create a clear mental image of the situation the problem is describing. Sketch a graph, a table, or a drawing to represent the situation. Avoid jumping into pre-packaged, memorized formulas after having read the problem only once.

Moving the Decimal Point

When you have a fraction, meaning you are dividing one number by another, you can always move the decimal point in both numbers, provided you move it the same number of places in each number (both numerator, and denominator), and in the same direction.

Now, when you are multiplying two numbers, you can always move the decimal point in both of them, provided you move it the same number of places in each factor, and in the opposite direction.

These two manipulations can significantly simplify the given operation.

Chords in a Circle

The diameter is the chord of maximum length. If the circle has a chord of length L, then the diameter cannot be less than L. The circle can be as large as we want but it cannot be too small. It has to be large enough so the chord of length L fits inside.