Why is substitution so confusing for so many people?

The short answer is, because it is not a mathematical operation. Not in the sense addition, subtraction, multiplication and division are.

For most people, mathematics is about crunching numbers. When they are doing math homework, or preparing for an exam, they expect to be adding, subtracting, multiplying, dividing,  deriving, or integrating, or calculating square roots, or standard deviations, averages, percentages, or otherwise combining two or more numbers into a final numerical answer.

However, when we are following the step-by-step solution to an algebraic equation, oftentimes substitution is one key step in the sequence, and substitution is an editing operation performed on a line of text. It is a special kind of cut-and-paste, find-and-replace action that has found a proper place as a drop-down menu item inside word processing computer software.

If you use a computer-like device to type letters, notes, or messages, you are probably familiar with that menu item, following the sequence of buttons 

Edit > Find > Find and Replace > Replace All

that you can use to change one word or phrase into another all across the selected part of your document. The computer automatically does that for you, saving you the time, effort, and risk of errors you would take if you did it yourself visually and manually, searching line by line, phrase by phrase, and word by word to make your desired replacement in all places.

That is what substitution is about, replacing one expression by another expression, under the assumption that they represent the same numerical value.

What stumps many people when they stumble upon a substitution step while following the solution to an algebraic equation, is their own expectation that every step had to be “mathematical,” meaning: “number-crunching-y” in one way or another. But substitution is editing, it is cutting and pasting, it is not multiplying, it is finding and replacing, it is not adding.

So, in many people’s minds, substitution is not math. They just don’t see the substitution step because they are not expecting to see it. I mean, they see it but they don’t recognize it as math because it is not math in the same sense addition and multiplication are.

They get so confused by the fact they are seeing something they were not expecting to see, that the surprise does not allow them to see whatever else is going on at that step.

It is as if, in some part of their mind they are going like:

“Math, math, math, not-math, .. wait, what?”

It is O.K. There are some seemingly non-mathematical procedures that are part of math, too, especially logic. You just have to learn to expect them to show up every once in a while, so they won’t take you by surprise.

Solving systems of equations by substitution

A topic that is hard to explain because it is so simple

The substitution method oftentimes works as a powerful technique for solving systems of equations. This method is widely taught in middle and high schools, as part of the Algebra curriculum, along with the other standard methods for solving systems of linear equations in two variables: the elimination method, the graphing method, and the method of determinants (also known as Kramer's rule). Solving systems of equations by substitution is a very interesting process, especially when we consider that the fundamental basis of its concrete execution is not really any algebraic operation at all but a typographical one. Substitution means textual substitution. It is a typographical "find and replace" operation, whereby we combine two strings of characters into a new one, by means of "copy," "cut," and "paste" manipulations. It is a common experience for math teachers noticing many of their students get confused when learning the substitution method. I believe a big part of such confusion in the student's mind comes from the unexpected, unexplained, fundamental difference in nature between algebraic, arithmetic, numerical operations, on one hand, and such a typographical, textual, character-and-string oriented operation like substitution, on the other hand. Most teachers explain the substitution method by doing some examples on the board, and hoping that students will somehow "get it." Indeed, some students do get it. After watching the teacher doing a few examples, something clicks and, that is it, they now know it. They have gotten it. Moreover, usually they not only get it but they love it when they realize how it works. Unfortunately though, these students I refer to in the last few sentences, typically make up only between ten and twenty percent of the class. They are the intellectual high achievers of the class, many of whom will go on to careers in science, engineering, medicine, or money management. The other eighty to ninety percent of the class typically did not get it. They are confused, they do not know what is going on, they have no clue what the teacher did or is talking about. For them this is no happy experience. Actually, it can be really aggravating if the teacher is particularly enthusiastic about substitution but lacks the ability to infect the whole class with his or her enthusiasm. There are some particular examples of systems of equations that, when solved by substitution, seem to yield a spectacularly elegant and short solution. When students have not yet understood the substitution method, watching one of these spectacular solutions makes them feel like the teacher is practicing some mysterious magic trick in front of them. This only adds to their discomfort, and their distaste for math in general, since it is only natural to fear and/or reject what we do not understand. As a math tutor, I have the luxury of working with one student at a time, so I can focus my attention on delivering the particular information my student needs, in the way he or she wants to approach each problem. In the case of substitution, I make sure they understand how to do it, by doing the first example myself so slowly, so carefully, so explicitly, so spelled out, so mechanically, that I make my students feel for sure they can do it faster than me. When I explain solving by substitution I do not try to look smart. Instead, I become a machine, and I consciously take all the magic away from the process, so my student can clearly see how simple it is. The delivery here needs to be accurately tailored to each individual student. It is much harder to do this in front of a whole class, because teachers have to maintain their authority; and making the explanation so explicit that the last student in the class understands it, would probably lower the teachers' own status in the eyes of several other students. In part, substitution is difficult to teach and understand because it is so simple. Compounding the problem, we have to remember all the accumulated deficiencies students are still struggling with, and dragging behind since their first years in elementary school. When solving a system of equations by substitution, the actual substitution is only one step in the process. Even when it is done correctly, students still need to work their way through all the algebraic and arithmetic operations needed to solve the given problem. They very well may do the substitution correctly, only to mess up the problem two steps down the road because they do not know how to add/subtract negative numbers, or they do not know how to divide fractions, or they are still adding with their fingers.

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.

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: