Geometry and Optical Illusions

Optical illusions show the need for proofs in Geometry

Yesterday I had two Geometry tutoring sessions. The first one in the day was only my second session with that student, while the second one was my first session with the other student [ a curious coincidence: (1,2) and (2,1); that is a symmetric pair in that relationship, with x = ordinal number of the session in the day; and y = ordinal number of the session with that particular student but I am digressing ].

One of the students showed progress in the sense that she was approaching the problems with more confidence than one week ago. Her mom told me she had also improved her grade in the last Geometry quiz she took. I attribute her improved performance to her increased confidence, which in turn I see as the result of our first tutoring session, because the confidence students have in their own understanding, and capabilities, usually increases when they benefit from the immediate feedback provided by the tutor subtly guiding them through their practice problems.

In the other session, the student asked some very good questions about proofs.
One question was to the effect of: “How do I know what properties, or theorems to use in a proof? Because at some point there are just too many of them, how do I know which ones to use in a proof?”
This is a very good question. It points to the core problem of looking for a path that connects the given statements with the desired conclusion. Many geometry students are confused by the uncertainty of the guessing involved in the process of finding a proof.
Elementary Arithmetic, an Algebra, are different from Geometry, and from Calculus, in that (among other things) most of their material can be presented as a set of prescribed, step-by-step procedures that specify what rules to apply, and in what order. They lend themselves to an algorithmic presentation more easily than Geometry or Calculus do.

Another good question this student had was about the very need for proofs, especially when the problem in question references a figure, drawing, or diagram. He said: “Why do I have to prove that those triangles are congruent, when you can just look at the picture and see they are congruent?”
Many geometry students ask this question. My short answer for it is: “What if the triangles in the picture are not really congruent because the picture is a tiny little bit off but we cannot notice the difference without a magnifying glass?”
We also talked about optical illusions, and how our sense of sight is charged with a heavy propensity to be “fooled,” “deceived,” or “misguided” by a whole variety of optical illusions, and so we can never trust pictures one hundred percent, not because of the picture but because the way our brain processes visual information.

Later in the afternoon, in between two tutoring sessions, I dropped by the bookstore looking for a CSET prep book, and I was gladly surprised when, by chance, I found a bargain priced book, precisely on the subject of optical illusions. It is this book right here, by Inga Menkhoff. It has plenty of quality pictures, with the corresponding explanations. I have always liked optical illusions.

Random Comments From Students

A math tutoring day with nine sessions

After nine math tutor sessions in the same day I barely remember where I was during the day yesterday but I remember some random comments my students made during the tutoring sessions. The show must go on, so I have no time to go into detail about any thoughts or opinions I may have about the following topics. I just present here some of the comments I remember, in case you find them interesting.
Some paragraphs below are single comments made by students. Two paragraphs are short pieces of dialogue, as indicated.

=======================

“I already understand everything we have seen so far but my mom wanted me to come take this tutoring session anyway.”

=======================

“This problem is actually pretty easy! That’s weird. I was expecting it to be this huge big thing but it’s just that simple!”

=======================

“Volume is everything that goes inside, it is the whole space!”

=======================

Student: “Well, they say these lines bisect these angles, then this half angle should be 33 as well.”
Me: “Exactly! Very good. Now, could you do the same with the two other half angles? Can we write x and y for those two other half angles, as well?”
Student: “No! You cannot do that because you don’t know how much x is.”

=======================

Student: “I’m sure I’m never going to use this stuff in my life, ever! This is so boring!”
Me: “What would you rather be doing?”
Student: “Anything!”
Me: “For example?”
Student: “Writing a speech.”
Me: “About what?”
Student: “About how math is silly, and stupid, and totally useless. They just make us waste our time struggling with these things.”
Me: “Imagine a world where nobody had to take math classes, only people who really wanted it.”
Student: “That would be awesome! But then you would be out of a job, and a whole bunch of other people would be out of a job as well. They just want the system to keep going the way it is because they want to protect their jobs. I don’t want to disrupt anything. I wouldn’t take away people’s jobs.”

=======================

“Oh, decimals! Decimals always confuse me. I always get confused when I see decimals.”

=======================

“These are my notes, sorry. I don’t know what was going on over here. He was going so fast, oh my God! I was just writing down everything as fast as I could but I didn’t understand anything. Well, I understood a little bit but not really.”

=======================

“Do we add these, or do we multiply them? This is getting so confusing! It is so much! And he said this is just skimming the surface of it, can you imagine?”

=======================

“At the beginning I was actually enjoying this class, and I was interested but now I don’t care anymore, I am not interested. I just want this to be over, I just want to get a passing grade in this class. The final is coming up, and I’m getting so stressed out!”

=======================

“Why did they invent all of this? What for? Is it used anywhere? Does it have a purpose?”

=======================

“Oh! So, that’s what remainders are! That is very good to know because there are a whole lot of problems that use those things.”

=======================

“That is so cool! I never knew that! Is this kind of math used in the real world?”

=======================

“What did I do now?”

=======================

“Oh, no! My bad.”

=======================

“First we move this bad boy over here to the other side. Then x equals 6, then x equals 3, then x equals 9, and this is what we add. Now we square the equation but there is no square root of 7, so that’s it. Am I good so far?”

=======================

The disconnect between expectations

Individual student emotions are often neglected during math class

Students have their own expectations about how much they should learn, and how fast. Usually these expectations differ from what their teacher expects them to do. Generally, math syllabuses are designed with an “ideal student” in mind. The material is covered and organized in a way that suites the teachers, and the small percentage of “straight A” students. Whoever can keep up with that pace, gets the A, others get lower grades.

In my one-on-one tutoring sessions, I have the opportunity to closely observe student reactions to their math workload, and to the pace at which their teacher is covering the material. Often I find myself addressing and somehow mitigating the emotional stress some students are under because of the huge chasm between their teacher expectations and their own expectations about their performance, and their learning pace. Sometimes, while going through their homework students make comments like: “Boy! They really want to drill this stuff into your head, don’t they?” Often all these students want is to get it done and over with as soon as possible, and forget about it.

Recently one student told me about the freezing fear she experienced in elementary school when one of her teachers timed the class on multiplication exercises. She felt terrified every single time they had such timed drills. All she learned from those experiences was that math is awful, something to avoid; that she was bad at math, and that she was going to struggle with math for the rest of her school life, which she did.

One more tutoring day

A new writing style (for me, in this blog)

This is my first post here in four months. It has been a long time without writing anything. Thankfully, I have been very busy lately. I have decided that, in order to keep this blog active, I need to change my writing style to something lighter, more alike social media status updates, if I want to find the time to keep posting entries in this blog at all. So, here it is, something that could fit somewhere between five and ten tweets, just a short report on another day of tutoring math, and loving it.

Yesterday I had six math tutor sessions, on Advanced Algebra, Discrete Math, Algebra 1 (two sessions), Calculus 1, and Calculus 2.

I like working out the multiplication table of specific finite groups, small ones. As the table builds up, I feel I am getting to know the group a little better, and more and more comfortable using the basic relationships between the group generators.

I usually enjoy tracing runs of algorithms to find out what they do, although sometimes with some simple examples I go: “Come on! Can’t you just see that value is not on the list? Just look at it, why run anything?” Then I marvel at how much we humans have come to rely on machines working for us, and we just keep doing it more and more, to the point where many times I wonder if it is not us who are working for them.

Learning styles are very varied. Some students want to race through the material as fast as possible, while others want to take their time to make sure they don’t miss any single detail, so they feel good about their skills at test time.

The more I tutor in Calculus, the more I like it. When I was a college student myself, many years ago, I did not appreciate Calculus as much as I do now. I noticed I started liking Calculus more when I started teaching Calculus classes. Since then, every time I have taught or tutored in Calculus, my fascination with its ideas deepens.

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.

Associative and distributive properties of multiplication

Illustrating a not uncommon confusion

Recently, working with a student who was preparing for the Algebra portion of a test, we came across this problem about simplifying the radical expression 3*sqrt(27).

He factored 27 = 9*3, and he wrote

3*sqrt(27) = 3*sqrt(9*3) = 3*(sqrt(9)*sqrt(3)) = 3*(3*sqrt(3))

Then he paused, hesitated, stopped, and he silently looked at me, telling me with his look that he didn’t know what to do next, pointing his pencil to the parenthesis, and the multiplication inside the parenthesis.

I said: “Probably the problem here is not asking for decimals, maybe we can do this multiplication first”, and I wrote

9*sqrt(3)

Then he asked: “But, don’t you have to also multiply the first 3 times the sqrt(3), as well?”

At that point I saw his confusion. It was about the associative property of multiplication, and the distributive property of multiplication over addition.

He was thinking that because

a(b+c) = ab + ac

maybe then

a(bc) would have to equal ab*ac, instead of just (ab)c.

So I did a quick example with numbers. First I wrote:

2(3+5) = 6 + 10 = 16

and then

2*(3*5) = 2*15 = 30, to establish a firm reference.

Then, making explicit how the associative property works in this particular case, I wrote:

(2*3)*5 = 6*5 = 30

Finally, showing the mistake of trying to distribute multiplication over itself, I wrote:

(2*3)*(2*5) = 6*10 = 60 which is different from 30.

Looking at this example the student agreed that the first procedure (using the associative property of multiplication) gives the right answer, while the second (trying to distribute the multiplication symbol outside the parenthesis over the multiplication inside) doesn’t work. After having the formats clearly illustrated with whole numbers, I wrote them with variables, for him to see the difference in the abstract level.

In this particular instance, the confusion arose in the first place because of the square root symbol in the factor sqrt(3).

When a triple product a(bc) involves only whole numbers, students don’t have a problem with that. They just multiply the two numbers inside the parenthesis, and then multiply that product by the number outside the parenthesis. However, for many students, the presence of a radical makes the problem a little bit too abstract. They don’t have a ready answer for 3*sqrt(3), so they think: “I don’t know how to do that multiplication, so I don’t know how to do the problem,” or “That multiplication cannot be done without a calculator, so I have to apply some rule but I don’t know which one.” Once in the abstract realm of properties, formulas, and identities, their chances of getting lost increase dramatically.

This type of confusion is actually quite interesting. Students often fail to make a connection between variables and numbers. They seem to forget that variables stand for numbers. However, here the student was applying the connection in one way, from the abstract rule to the concrete example. The problem was the rule he was applying was made up. It is not a rule at all. It only seemed a rule to him because of its typographical similarity with the distributive property of multiplication over addition.

First, the presence of a radical disconnected him from the concrete, familiar realm of whole numbers. Then he went searching his memory banks for an algebra rule to apply. The first rule he remembered was the distributive property. At that point, an oversimplification drive took over, suggesting the idea that the typographically corresponding expression should be a rule as well. So he was considering adopting such a made-up rule as a real number property, without checking its validity with numerical examples before applying it but just going by the typographical similarity with the distributive property, effectively making him want to distribute multiplication over itself.

Why are there so many numbers?

Where are they?

In my math tutor practice I constantly answer questions. I usually get a lot of questions from my students. All kinds of math related questions. Some of them are very common, meaning, I get them all the time. For example, Calculus and Pre-Calculus students very often ask: “What is the domain?” “How do I find the domain of a function?” The vast majority of standardized test takers ask questions to the effect of “Why do I have to re-learn all this math stuff?” “When am I ever going to use it?” Sometimes I get questions that make me laugh, like: “How can you stand so much math? My head hurts!” and “Do you also have a real job? Or, is this all you do?”

Recently a student asked me a couple questions I found just fascinating. We were going over some algebra rules. I started giving her some numerical examples to illustrate one of the rules. All of a sudden my student asked: “Why are there so many numbers? Where are they?” These are great questions! They get to the root of the concept of number. Just think about these questions for a moment. “Where are they?” Where are the numbers? It is almost like asking: “Where do numbers come from?” or even “How can I believe numbers really exist? Can I see them?” It is clear numbers are not physical objects but we use them to count physical objects all the time. You believe in the existence of something concrete, like cars, because you can easily see them (often in big numbers), but what about numbers themselves? Do we ever get to see a number? What we usually call numbers, like street addresses, or numbers in license plates, or ID cards, or page numbers in a book, all those are not actually numbers but numerals, the symbols we use to represent numbers. Numbers are in our mind. They are concepts, ideas, thoughts, more than things. My answer to these questions was along the following lines:

Numbers are everywhere. We do not see the numbers but we put numbers on the things we see. Numbers show up as soon as you are able to tell differences and similarities. Think about counting the chairs in this coffee shop, for example. When you count the chairs you do not count the tables, or the bookcases, only the chairs. So you count them because they are equal, they are all chairs. However, you do not keep pointing your finger at the same chair while going 1, 2, 3,.. You count that chair and immediately you go on to the next chair for the next number. So you count them with different numbers because they are all different chairs. What makes counting possible is our ability to identify a set of objects that are all equal, in a sense, yet different, in another sense. So we can “see” numbers when we look at the stars, at the grains of sand in a beach, or the cars in a highway, and so on. As long as our mind sees the world in terms of “equal” and “different,” numbers will be there, everywhere.

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.

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.

When it comes to math homework, slower is often better

Take your time to use the correct set of problem-solving questions

Students often approach math homework problems with the same level of interest and excitement many adults show when preparing their tax returns. They want to get them done, and over with, as soon as possible so they can do something else. Many times, this is the only interest students have in their math problems, just getting them done, and moving on to having some fun with any activity other than math. Ironically, though, this goal turns out to be counterproductive, since you can do something well and fast only when you already know how to do it. More often than not, however, homework is part of the learning process. Typically, when students do homework, they are still learning, still practicing. They have not yet mastered all aspects of the topic at hand. Very often homework problems present students with finer points they have not yet considered. We could say when students do homework they only partly know what they are doing, kind of. So their goal of finishing as soon as possible and go play, only causes them to get the problems wrong, because they use the wrong question: "What am I supposed to do here?" instead of the questions that can successfully lead them to correct solutions: "What do we have here?" "What does this mean?" "What is the question?" and "What can I do?"