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.

Finding better explanations

Another advantage for individual students taking math tutoring sessions

Recently, one of my students told me:

“I like your lecture style better than my professor’s. You have a much better way of explaining the subject. He just starts doing problems, and that’s it. Last time he was having a hard time explaining how he was using the absolute value to solve a problem. We were all confused about it, nobody was understanding what he was doing.”

Sometimes when you are studying math, and you see a topic for the first time, you struggle to understand it, and you work out examples until you find a way to get it. Then if you are a teacher, and you only have that one way of understanding the subject, you go out teaching it that way, and sometimes you confuse all of your students.

Some teachers care a lot about their students understanding their lectures but some other teachers do not care that much. Sometimes they think: “Well, if they don’t get it, though luck.” However, teachers who care about their students understanding the subject, they spend a lot of time thinking up alternative explanations, or better examples, or better ways to illustrate what’s happening.

I remember a few times (years back when I was a teacher) I felt kind of depressed, disappointed, or frustrated at the end of a lecture because I couldn’t find a way for my students to understand what I was trying to explain. Then, afterwards, I would spend hours, days, even weeks sometimes looking for better ways to explain a particular topic, and the next time I taught that course I was able to explain those topics much better.

I noticed when I started private tutoring, that really sped up the process for me, of finding better explanations, because sitting with students one-on-one, and taking the time to go in depth and in detail with them over their doubts and questions, many times I was able to discover exactly how my students in class were looking at specific problems.

That allowed me to discover faster the reasons why they were not understanding a subject, or why some of my explanations were not working. By tutoring individual students, I was able to find a lot faster a lot more alternative ways of explaining subjects when my students in a large class felt the need for those better explanations.

Tutoring individual students has helped me to focus on finding the best way for each student to understand a given subject, rather than focusing only on covering the whole subject fast in front of a big class.

Helping Students Find Their Own Motivation To Learn

What is in it for the student, from the student's own perspective?

It is important for students to understand teachers are helping them to figure out what they want to do in life, and are helping them achieve those goals. It is not enough for teachers to give students examples of what students do not want to do in life. Teachers want to inspire students to learn; to give them an appreciation for knowledge; to show them how to put value into knowledge, and how to extract value from it. For working adults is easier to see how, in this technological world of ours, meaningful numerical patterns come from everyday life. Numerical patterns coming at us directly from real life make us think about reality in terms of numbers. The more comfortable we are with numbers, and with handling them, the better we can express the ideas suggested to us by those numerical patterns. Sooner or later, in one form or another, we realize the math we currently know is somehow inadequate to analyze the data we want to understand. Even for students who have always been good at math, there may come a point where their homework problems baffle them. This may be because such students tend to enroll in AP classes at an early age. By the time they are high school seniors, they are already covering material some science majors only get to learn about in their college sophomore year. Therefore, it is important for students to create good study habits since early in life. Study is not only preparation for work but study in and of itself can be an awful lot of work. Given that study takes time, energy, and other resources, it is important for students to be able to associate it with experiences of achievement, and empowerment. Success means different things for different people; it even means different things for the same person at different points in their life. Often students question themselves: "Is this effort worthwhile?" Some sort of confirmation is needed about it. The discipline of doing homework with a good degree of concentration, regardless of whether or not we like a particular subject that much, pays off when, thanks to that consistent effort, we are able to see the things that interest us in a new light. For many, it may mean just getting past a particular requirement, thereby clearing their horizon from a bulky obstacle. Clearing out such requirements can give students an improved sense of self-esteem, and a renewed confidence in themselves. Another important factor may be taking our time to learn things thoroughly, to make the subject ours, to make sure we really understand it, because then we know what to do at any given point, instead of feeling like randomly throwing darts in the dark, and hoping to achieve some result by chance. There are many factors involved in learning. Each student builds their own learning strategy, according to what they determine is best for them. Teachers can only hope to influence in some measure such decision making process on the student's part. Students constantly make these decisions on their own, multiple times a day, choosing the way they study, selecting what gives them the best possible outcomes in their own world, according to everything they consider important - not necessarily what other people consider important for them. For some students it is more important to find ways of having fun while learning. Others predominantly focus on their long term goals (passing exams with a good enough score) without almost ever giving themselves the chance to consider their learning experience from any perspective other than their test results. In summary, it is important for students to find educators who can provide them not only with facts but also with motivation enough for grasping those facts, and applying them.

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.

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.

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.