Confusion type: unexpected answer

When the answer to a problem shows up in an unexpected or unfamiliar format, the student very well may fail to realize that result is the answer to the problem.  

Confusion:
Are we there yet?
Where is the answer?
This cannot possibly be the answer, can it?
This type of confusion can occur at any level.
It most commonly shows up in Algebra, the first time students are asked to find equations of lines, circles and so on. Before that point, chances are they had only been asked to find numerical solutions to equations, or systems of equations, or to simplify algebraic expressions.
So, up to that point, for them an equation is understood as the problem, while the answer needs to be a number, or a simplified expression. Therefore, in their minds, an equation cannot be the answer to any problem.
This can happen again and again, in many forms, even in Differential Equation courses. For example, when the answer to some problem takes the form F(x, y) = c, meaning the variable ‘y’ is interpreted to be an implicitly defined function of ‘x.’ The same kind of confusion as mentioned above can take place if the student is strongly expecting the answer to be an explicitly defined function, in the traditional form y = f(x). At that point the solution given by the equation F(x, y) = c may seem like nothing, like no answer at all, just one more step in the process of finding the “real” answer. However, this impression is of course mistaken because in those particular cases F(x, y) = c is the real answer, since an explicit definition of the form y = f(x) for ‘y’ is nowhere to be found.

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.

Missing Pieces of Information

Some search for doors, sometimes some do not want to see them

Last week I showed a student how to solve two linear equations in two unknowns. He knew perfectly well how to solve one equation with one variable but did not know how to combine two separate equations into one.
Also last week another student made the remark: “I do not know how to start solving this problem. What does ‘isoceles’ mean?” As soon as I gave him the definition of an isoceles triangle he successfully proceeded to solve the problem.
Earlier today another student asked me: “What is a frequency histogram?” When I explained the concept to him, he found it very clear. He said: “Just that? Documenting the numbers in a graph? That is pretty simple!”
Most times students take the initiative, and they spontaneously ask the meaning of terms they are not familiar with. Sometimes however, some students are near some sort of saturation point, and they do not want to even think about the remote possibility that maybe there is a concept they do not know, or a technique they have not seen, and they need this new information to solve the problem at hand. In these rare occasions they keep trying to solve the problem with only the insufficient tools they already have in their problem-solving toolkit.
Writer Kenneth Grahame said “The strongest human instinct is to impart information, the second strongest is to resist it.” So, I choose my words carefully when telling them there is something extra they absolutely need to know first before having any chance of solving the problem. Many times I let them finish their attempts, and check the solution in the back of the book so they realize their approach was wrong without me telling them so before hand, because that could increase their resistance.
There are several problem-solving techniques or approaches that seem indeed artificial, weird, or mystifying the first time around. Once you see how they work, and you use them a couple times, they become perfectly natural, and then you wonder why you never thought of that before.
A perfect example of this I saw also last week with another student.
It was a probability problem involving three coins. For me it is quite amazing to watch time and again how students keep trying to solve these problems by reasoning only about the three separate coins, as if the relevant probability space had only three points. The strong insistence in this naive approach is only matched in its consistency by the strong surprise students show the first time you show them the full eight-point probability space by branching out the development of the experiment at each successive flip, and recording the eight different combination triples. It is really interesting. Somehow these once missing pieces of information act like doors to a whole new realm of math knowledge when they are presented and opened. Many times the student’s reaction reminds me of that feeling of “Wow! I never thought that was a door!” I get when watching some sci-fi movies.

Quadratic equations in rotated form

Some long, time-consuming problems

The last three weeks have been very busy for me. I have been tutoring all math subjects, from fractions to Statistics and multivariate calculus.
Looking back over these past weeks it all seems kind of blurred but one topic stands out from the rest because, by coincidence, I had two sessions on the same topic with two different students, both during last week.
The topic in question is the rotation of quadratic equations in the two-dimensional coordinate (x,y)-plane. It had been a long time since I last taught this subject. It does not come up very often in my tutoring sessions, so I noticed the coincidence when I had two different students independently reviewing with me these geometrical transformations in the same week.
Also, each student separately made the same comment after we worked out problems of this type about quadratic equations: “Wow! This is a lot of work!”
They are right, it is a lot of work. The general problem starts with a quadratic equation like, for example, 5x²+2xy+10y²-12x-22y+17=0,
with a non-zero coefficient in the “xy” term.
The goal of the exercise is to find a specific angle, let’s call it θ, so that the transformed (rotated) equation in the alternate variables x’ and y’ lacks the x’y’ term.
The variables x and y are connected to x’ and y’ by means of these two equations:
x = x’ cos θ – y’ sin θ
y = x’ sin θ + y’ cos θ
Solving these problems requires several steps. I list them here, hopefully without going into too much detail:
First, finding the value of tan(2θ), the tangent of the angle double of θ.
Second, finding the measure of the angle θ itself.
Third, finding the values for cos θ, sin θ, and their squares.
Fourth, plugging those trigonometric values into the formulas below to find the new coefficients for the transformed quadratic equation:
A’ = A cos² θ + B sin θ cos θ + C sin² θ
B’ = 0
C’ = A sin² θ – B sin θ cos θ + C cos² θ
D’ = D cos θ + E sin θ
E’ = E cos θ – D sin θ
F’ = F
where A, B, C, D, E, and F are the coefficients of the original equation.
So you can see each one of these problems involves a lot of algebraic and trigonometric calculations. These problems are long, time-consuming, and you have to pay very close attention to all details to ensure an accurate result.
Anyway, in the video below you can see a room-size metallic structure (some kind of architectural sculpture) where Richard Serra, the artist, incorporated two congruent ellipses, one at the base of the room, and the other formed by the upper edge of the wall. The two ellipses are identical in shape but they are rotated with respect to each other. This is a real, tangible example of the rotation of a conic section. It is relevant to this post because quadratic equations represent conic sections, like the ellipses we see in the video. It is a very interesting structure. Take a look:

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.

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: