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.

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.

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 and 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 and 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 and 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.