"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.