Exploring Math with the TI-89

Technology changes possibilities

The TI-89 is a graphing, programmable calculator with symbolic manipulation capabilities. I know that is a mouthful but we can break it down to three main features:

1) This device graphs not only functions of one variable; it also graphs functions of two variables, parametric curves, functions given in polar coordinates, and data from number tables.

2) Programmable means it has commands you can use for writing code that is directly executable on the device. You can save, edit, and execute programs in it. This gives you plenty of flexibility when it comes to calculations for complex problems, because you are not restricted to the built-in operations.

3) Symbolic manipulation means the device displays, and uses standard mathematical symbols like integral, derivative, roots, exponents, and fractions, in a way that is consistent with algebra rules. That means it is not only a number-crunching little machine, but it does many operations of algebra, and calculus, like factoring polynomials, solving equations, multiplying matrices, deriving functions, and finding definite integrals.

All these capabilities present opportunities for teaching math. For example, using this calculator it is very easy to show relationships between formulas of functions, and their graphs. Presenting these relationships in a book usually takes several pages. Students often have some trouble reading, believing, understanding, remembering, and applying the general rules for this level of algebra of functions. On the other hand, typing the examples from the book into the calculator, and seeing the graphs of the functions being displayed in real time in front of their eyes, gives them more confidence to “believe in the evidence.” Therefore they can be puzzled but they do not reject out of hand the result shown. On the contrary, given the easy-to-use editing capabilities of the function editor, they can experiment with the function formulas by changing their parameter values and seeing what happens with the graphic output. In my opinion this may make learning more interactive, fun, and effective.

Math as a Freedom Source

An ongoing dialog

Math gives you freedom because it takes freedom to learn it. Learning math makes you ask questions all the time, like: "What do we have here?" "What could that possibly mean?" "Have we seen something like that before?" "How can I get that?" "How can I figure the answer to that?" "Am I supposed to do something here?" "What would happen if I do this operation?" "Am I off the mark? How far off?" "How can I tell the difference?" "Is there a shortcut for this?" “Does it always work the same way?" and so on and so forth, ad infinitum.

Questions bring answers, and they also bring more questions. Being able to ask questions all the time means the society of ideas that is your mind tends to be pretty open, fluid, flexible, and dynamic. You find ways to free up your mind to ask more questions, and to keep searching for the answers.

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.