Comparing rectangles by their area and by their diagonal

The other day I was in someone’s office waiting for some paperwork to be completed.
It was a small room, about 12 feet by 12 feet.
The back wall was orange, and its lower part was horizontally covered by a desk.
In the middle of the orange wall there was a picture of a green car. I thought that was an interesting color combination.
Looking at the picture on the wall, I wondered how bigger the visible part of the wall (including the picture) was, in area, in relation to the area of the picture.
The horizontal length of the wall was about two-and-a-half times the length of the picture, while the hight of the visible part of the wall was about three times the height of the picture.
Putting

b = length of the picture
h = height of the picture
a = area of the picture
B = length of the wall
H = height of the visible part of the wall
A = area of the visible part of the wall (including the picture)

we have, approximately

B = 5b/2, and H = 3h

therefore

A = BH = (5b/2)(3h) = 15bh/2 = 15a/2

So the proportion between those two areas is 15/2, 0r 7.5

Then I wondered what the proportion would be between the diagonals of those same two rectangles. As it turns out, when the two rectangles (the wall and the picture) do not have the same shape, the proportion between their diagonals depends not only on how many times the sides of the wall are on relation to the sides of the picture, but on the precise shape of the reference rectangle, as well.
If we use

d = measure of the picture’s diagonal
D = measure of the diagonal of the visible part of the wall

Then we have

D = sqrt(B² + H²)
 = sqrt((5b/2)² + (3h)²)
 = sqrt(25b²/4 + 9h²)

while

d = sqrt(b² + h²)

The coefficients of b² and h² inside the square root in the expression for D are not the same, so we cannot simplify that expression completely just in terms of d.

If we introduce a new constant, k, to represent how many times the height of the picture equals its length, then we can write

b = kh

In that case, the diagonals of the two rectangles can be expressed as

d = sqrt((kh)² + h²)  =
 = sqrt(k²•h² + h²) = 
 = h•sqrt(k² + 1)

D = sqrt(25(kh)²/4+ 9h²) = 
 = sqrt(25k²h²/4 + 9h²) = 
 = h•sqrt(25k²/4 + 9)

This shows the proportion between D and d being dependent on k, the proportion between h and b, because

D/d = sqrt(25k²/4 + 9) / sqrt(k² + 1) 

and this is a function of k that is not constant.

A very cool property of binomial coefficients

Last night I was reading a post in this other blog, by Ben Vitale. His post is about some special types of binomial coefficients.

In case you don’t remember, given two non-negative integers m and r, their binomial coefficient  C(m,r) (also known as “m chose r”) counts the number of subsets of size r out of a set of size m, and it is calculated by the formula:

C(m,r) = m!/(r!•(m-r)!)

In particular, when r=2, we get

C(m,2) = m!/(2!•(m-2)!) = m(m-1)/2

This is an expression for counting the number of all possible unordered pairs formed by two elements of a set of size m.
Somewhere near the end of his post, Ben presents the following formula:

C(kn,2) = k•C(n,2) + (1/2)•(k-1)k•n²

On seeing this formula, I noticed the second term on the right-hand side,

(1/2)•(k-1)k•n²

can be rewritten using the binomial coefficient C(k,2), producing the expression

C(kn,2) = k•C(n,2) + C(k,2)•n²

This formula holds true for any two positive integers n and k, each greater than 1.
In particular, by using the trick of swapping n with k in that expression, we get the following one:

C(nk,2) = n•C(k,2) + C(n,2)•k²

Now, multiplication being commutative, we know kn = nk.

Therefore, C(kn,2) = C(nk,2)

And we can connect their two equivalent expressions, making this formula:

 k•C(n,2) + (n²)•C(k,2) = (k²)•C(n,2) + n•C(k,2)

I don’t know about you but I believe this is just amazing!
Let’s illustrate this property with a couple examples:

a) First, we choose two numbers, let’s say n=5 and k=7

b) Then we calculate their squares: 5² = 25, and 7² = 49

c) Now we calculate their binomial coefficients for 2:

C(5,2) = 5•4/2 = 20/2 = 10
C(7,2) = 7•6/2 = 42/2 = 21

d) Now we plug the numbers into the formula, and verify it:

7•C(5,2) + 25•C(7,2) = 49•C(5,2) + 5•C(7,2)

(7)(10) + (25)(21) = (49)(10) + (5)(21)

70 + 525 = 490 + 105

595 = 595

Let’s check some other numbers, for example n=8 and k=14

Their squares are 8² = 64, and 14² = 196

Their binomial coefficients with 2 are:

C(8,2) = 8•7/2 = 28
C(14,2) = 14•13/2 = 91

and we verify:

14•C(8,2) + 64•C(14,2) = 196•C(8,2) + 8•C(14,2)

(14)(28) + (64)(91) = (196)(28) + (8)(91)

392 + 5824 = 5488 + 728

6216 = 6216

The reason I like this property so much is because both expressions

k•C(n,2) + (n²)•C(k,2)

and

(k²)•C(n,2) + n•C(k,2)

are linear combinations of the same two binomial coefficients. These linear combinations have different coefficients:

k and n² in one case, and k² and n, in the other.

In principle, right off the bat, I would not expect them to turn out equal but they are, every single time! What is more, the only change in the coefficients is that you move the square (the exponent 2) from k to n, and viceversa.
I find all this to be pretty cool. I wanted to share it with you, and many thanks to Ben Vitale for his blog post.

Powers of 2

From my early days as an undergrad math student, I knew that powers of 2, the numbers of the form 2ⁿ, have many properties, and are very important in many areas of mathematics.
However, it was only after I became a math tutor that I realized how common questions involving powers of 2 are in standardized tests.

Before becoming a math tutor I had held positions at colleges and high schools as an associate professor, adjunct professor, math teacher, and teaching assistant. Up to that point I had memorized only the first six natural powers of 2, from 2¹ to 2⁶, these are:
2, 4, 8, 16, 32, and 64, the powers of 2 that are two-digit numbers.

Later, when I became a math tutor, and I realized how often standardized tests present students with questions involving powers of 2, then I memorized larger powers of 2, up to 2¹², or:
128, 256, 512, 1024, 2048, and 4096.

It is really a big advantage at test time, for a student to be able to recognize a power of 2 when they see such a number in a problem, because then they can write that number in its exponential format, and use the algebraic rules of exponents to work out the problem’s math easier and faster than dealing with the number in its decimal form, using only arithmetic operations, because the longer route is more time-consuming, and more attention-intensive, therefore riskier and less efficient.

So, if you are preparing for a standardized test, from the CBEST or the ASVAB to the GMAT or the CSET, know your powers of 2. Memorize them well from 2¹=2 to at least 2¹⁰=1024, and you’ll increase your chances of scoring a few extra points in the exam.

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.

Linear Algebra - matrix dimensions

A nxm matrix is a rectangular array of numbers in n rows and m columns.
For example:
A 2x3 matrix has two rows and three columns.
A 1x3 matrix is a 3-coordinate vector in horizontal format, just one row with three numbers in it.
A 3x1 matrix is a 3-coordinate vector in vertical format, commonly known as a “column vector.” It is a matrix with a single column and three numbers in it.

For us to be able to add or subtract two matrices, they have to have the exact same dimensions. For example, you can add or subtract a 4x3 matrix with another 4x3 matrix but not with a 4x4, 3x3, 3x4, 4x2, 5x3 or any other type of matrix.

Matrix multiplication has its own, particular requirements on the dimensions of the matrices being multiplied. You can multiply any nxm matrix with any mxk matrix, and the result will be a nxk matrix.
For example:
A 3x2 matrix times a 2x4 matrix will result in a 3x4 matrix.
A 3x2 matrix cannot be multiplied with another 3x2 matrix.
A 3x2 matrix times a 2x3 matrix will result in a 3x3 matrix.
A 3x3 matrix times a 3x3 matrix is again a 3x3 matrix.
A 3x3 matrix times a 3x1 matrix will result in another 3x1 matrix.
A 2x7 matrix times a 7x4 matrix will result in a 2x4 matrix.

Matrix multiplication involves a lot more arithmetic operations than matrix addition, because just to get the number for the entry in the i-th row and j-th column of the product, we have to calculate the dot-product of the i-th row of the first matrix times the j-th column of the second matrix.

All this can be pretty confusing for students when they first read it in a book but after being shown how it’s done in a couple of examples, it becomes clear and then with more practice it becomes easy but it always is a lot of work, and you have to pay close attention to all the little calculations involved because the large number of them makes it easy to make silly mistakes.

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.

Guest Post: Tutoring at a Community College

By: Kathryn Lewallen

I just received the following letter from a reader, who graciously accepted my invitation to publish it here as the first guest post in this blog:

Greetings, Juan~

I am a math tutor at our local community college and want to thank you for your blog and website, which I found with a Google search.  I am currently tutoring six students, all of whom are women over forty, back in school to improve their lives.  It is such a rewarding job, and I learn so much every session about what works and what doesn't with each student. The college pays me for the tutoring sessions, which makes it possible for students to be able to take advantage of tutoring without personal expense. We meet in a room on campus.

I have students in varying levels of Algebra and one in Analytic Trigonometry this semester.  The Trig teacher teaches with, and requires, the TI-89 in the three levels of math that he teaches.  If a student can't afford one, he has TI-89s that he will let them borrow for the semester.  I was able to find one online and am thrilled with its capabilities and wish more teachers realized the value of training math students with calculators, as well as teaching the calculations the "old-fashioned way."

Working one-on-one is a key to good education, and I wish there were a way to make it possible in public education.  Seeing students gain confidence and grasp understanding of concepts for themselves is addicting and I love the challenge of finding ways to explain those concepts in ways that "click" with a student. To hear a student go from walking in the door saying, "I hate math," to "that was fun, can I do some more?" is like music to my ears.

Your experience and insights are so very helpful and I appreciate your sharing them online. Thank you so much!

Kathryn Lewallen
Math tutor, Gila Community College
Payson, AZ

You are very welcome, Kathryn. Your letter shows your enthusiasm for math tutoring, and for using technology in education. It also shows the positive difference you are helping create in your students lives. Keep up the good work! Thank you for letting me share in this blog your tutoring experience, and thank you very much for your positive feedback!

Math is the art of stating the obvious

Simple questions go a long way towards understanding

Recently I had the following dialogue with a sixth-grader during a tutoring session:

The little girl says:
I have a question I wanted to ask you. We were working on this problem earlier today in class, and I am confused, I don’t know how to do it. It’s 159 – 26.6 because, what do you do? This number 159 has no decimal point. Is it O.K. if you..? Can you put a decimal point, and write an imaginary zero here, like 159.0 even though the zero is not really there?

(I thought it was really interesting she referred to that zero after the decimal point calling it an “imaginary zero.” She even drew it with a series of little dashes, instead of a solid line)

I say:
Yes, that is correct, you can certainly do that. 

She continues:
See, I know how to do 159 – 26 because you just do it, and it is 133 but with 159.0 – 26.6 it’s not going to be the same, or is it?

I say:
No, you are right, it’s not going to be the same. Why not?

She hesitates, and says:
Well, you cannot do 159 – 26 = 133, and then for 159.0 – 26.6 just put the .6 after the result, and say that it is 133.6 because that is not right, or is it?

(She was looking more and more confused as she moved closer to facing her idea that she did not know how to do that operation)

I say:
Exactly, that’s not right. 

She says:
But then, how much is it?

I say:
Well, let’s see, you got 159 – 26 = 133, right? 

She says:
Yes.

I say:
And, what is 159.0 – 26.6?

She says:
I don’t know!

(While saying this she gave me a look like “I just asked you the exact same question!”)

I say:
It’s going to be some kind of number, isn’t it?

She says:
Yeah..

(The look she gave me in that moment I can only translate as “What are you, an idiot? Of course it’s going to be a number. What else could it be?” She is very polite, and she would never say out loud any of the sort but I could just see it in her eyes)

I continue:
How are 26 and 26.6?

She ventures an answer, not sure about what I am asking:
Are they positive numbers?

I say:
Yes, they are positive, definitely, and, which one is bigger, 26 or 26.6?

She says:
26.6 is bigger than 26

(Then the look on her eyes seemed like “I cannot believe you are asking me such simple questions! Is there a point to this? Do you think I don’t know that?”)

I continue:
Now, which one is bigger between 26.6 and 27?

She very patiently responds:
27 is bigger.

I say:
O.K., can you please do me a favor, and do these two subtractions? Just write them all out, please. 

While saying the above I write 159 – 26 and 159 – 27

She quickly writes down both answers: 133, and 132.

I say:
Thank you. Now, since 26.6 is greater than 26, and smaller than 27, would it make sense to say that 159.0 – 26.6 it’s going to be some number in between 132 and 133?

She says:
Yes.

(Then it was apparent she was deeply focused, thinking about that order relationship between the numbers, and the subtraction operation)

I say:
So, it’s going to be 132 point something, right? But point what?

(At that very moment something clicked in her mind)

She quickly scribbles the right numbers on the paper and says:
Wait! Wait! I know how to do this! Yes! It’s going to be 132.4. I got it!

I say:
Excellent! That’s perfect!

There are a few points I want to emphasize in the above dialogue:

1) In my math tutoring, the most relevant factor is not the math itself but how the student is feeling about the math. I did not even ask her how the teacher explained the problem in class. The most relevant fact right then was her confusion, and her confusion was coming from a lack of confidence of some sort. For whatever reason, the teacher’s explanation had not been enough for her to acquire a firm trust in the procedure described by the teacher, or in her own ability to follow all the steps. It was mostly a matter of her stopping herself, holding herself back.

2) It is very likely that, at some point during class, when the teacher first started explaining the problem, she originally had the idea of “just adding the .6” from 26.6, after the 133 from 159 – 26, to get 133.6 (the wrong result). As her initial idea turned out to be wrong, she might have felt discouraged, frustrated, and confused. These feelings may have led her to reject the validity of the procedure proposed by the teacher (“the right way” to do it), and to forget it, blocking it from her memory. It is worth noting that coming up with plenty of wrong ideas is extremely common in math, for everybody, even for professional mathematicians. There is nothing wrong with wrong ideas in math; you just have to keep working until you find an idea that works.

3) When she acknowledged that the possible solution she was thinking about was wrong, and I confirmed that it was wrong, she directly asked me for the solution, and I purposefully avoided giving her the answer. This conscious avoidance on the tutor’s part is absolutely crucial for the student’s self-esteem, and for her future success. This is very important. The best I can hope for when working with my students is to instill in them a strong conviction that they can figure out math problems on their own, by themselves, even when they don’t know what to do.

4) I not only repeated the question back to her but I phrased it in such a way that allowed me, when she said: “I don’t know,” to produce a bafflingly obvious, non-answer answer. That was when I said: “It’s going to be some kind of number, isn’t it?” Getting to such level of obviousness has a few key advantages. To start with, she immediately felt smarter than me. This effect is an invaluable asset in math tutoring, making the student feel smarter than the tutor. Deployed at the right time during the discovery process, this effect can create wonders of understanding.

5) Another advantage of pointing out the obvious is that, when you start seriously asking very simple questions with very obvious answers, all assumptions start flying out the window. It may be very confusing but it is also liberating. It unclogs the mind, and it clears up your vision, allowing you to see what’s there hidden in plain sight, and you were not able to see before because of your assumptions, or because of your mood, or emotional state. Resorting to the obvious allows you to take a long series of tiny, little baby steps towards the solution.

6) Last but not least, another great benefit of stating the obvious is the fact that you move the student from negatively focusing on himself, or herself (“I cannot do this problem, I’m going to fail the class, I am not good at math.”), to focusing on the particulars of the problem at hand, meaning, focusing on the numbers as objects you can manipulate, instead of letting them ruin your life. Focusing on the obvious sets you free from the paralyzing grip of your own emotions, and lets you realize numbers are truly inert in relation to you; you are the one who is alive.

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.”

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“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?”

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“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?”

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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?"

Exploring Math with the TI-89

Technology changes possibilities

The TI-89 is a graphing, programmable calculator with symbolic manipulation capabilities. I know that is a mouthful but we can break it down to three main features:

1) This device graphs not only functions of one variable; it also graphs functions of two variables, parametric curves, functions given in polar coordinates, and data from number tables.

2) Programmable means it has commands you can use for writing code that is directly executable on the device. You can save, edit, and execute programs in it. This gives you plenty of flexibility when it comes to calculations for complex problems, because you are not restricted to the built-in operations.

3) Symbolic manipulation means the device displays, and uses standard mathematical symbols like integral, derivative, roots, exponents, and fractions, in a way that is consistent with algebra rules. That means it is not only a number-crunching little machine, but it does many operations of algebra, and calculus, like factoring polynomials, solving equations, multiplying matrices, deriving functions, and finding definite integrals.

All these capabilities present opportunities for teaching math. For example, using this calculator it is very easy to show relationships between formulas of functions, and their graphs. Presenting these relationships in a book usually takes several pages. Students often have some trouble reading, believing, understanding, remembering, and applying the general rules for this level of algebra of functions. On the other hand, typing the examples from the book into the calculator, and seeing the graphs of the functions being displayed in real time in front of their eyes, gives them more confidence to “believe in the evidence.” Therefore they can be puzzled but they do not reject out of hand the result shown. On the contrary, given the easy-to-use editing capabilities of the function editor, they can experiment with the function formulas by changing their parameter values and seeing what happens with the graphic output. In my opinion this may make learning more interactive, fun, and effective.

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.