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Full Spectrum Communication
Analytical reasoning (primarily mathematics) is a part of all fully-developed languages around
the world. One of the primary reasons the United States falls short in mathematics education is
that, in the USA, mathematics is taught as a bunch of tricks and shortcuts to get a number. Much
of the more progressive parts of the world know that mathematics is comprised of logical
structure and patterns pulled together to relate complete thoughts, and that goes far beyond
simple numbers.
I want you to know that, when I agreed to do a talk today, it was my intention
to use power point so I could amuse you with cartoons and clever sayings.
After thinking about what I really wanted to do, I decided that, while power
point is a wonderful way to present certain kinds of learning, it only distracts
from what I consider to be a serious dialogue about what we really strive to do
with students: We want to focus on learning to think, improve thinking, and
focusing thinking on thoughts of today and the future. Thinking without
communication with other people may give personal satisfaction but those
thoughts are lost in the electromagnet ether. Please, accept my apology for not
focusing on entertainment.
It appears to me that there are increasing pressures on faculty and public
school teachers to not develop thinking in students. Just make certain that
they pass with acceptable grades and/or high scores on standardized exams.
I don’t know much about many things, but I have opinions about key
components of a basic education. Please, join me in an educational
adventure.
This is a quote from Feynman’s talk when he was reminiscing about why he wanted to go on an
adventure to Tuana Tuva:
“The way to have an adventure is to do things at a lower level. It is not to
ride on the freeway and stop at the Holiday Inn.”
Richard Phillips Feynman (a great thinker)
Today, I would like to give you some glimpses into the abstract world of
mathematics and some reasons why some people find mathematics to be
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more difficult than is necessary. I would like to talk about the meaning of
life, but I know a little more about mathematics, which is a much simpler
topic.
Communications people and mathematics people have a lot in common.
Any adventure in education is mostly about thinking, improving thinking,
and communicating those thoughts with other intelligent beings. When
we, as teachers, are working with our students to help them develop sound and
powerful thinking skills; we must establish a sound, dynamic foundation
(lower level) that will grow with-and-in their developing directions. This
means there has to be a goal. When I teach Intermediate Algebra, College
Algebra, and Trigonometry, my goal is to develop the student’s thinking toward
being able to handle calculus concepts. Any other goal is handicapping
students and limiting their future options. My most difficult challenge is to
help students “unlearn” crippling thought processes that students have
developed in their past lives. Any thought process that does not lead to a
“higher” thought process is crippling. I intend to lead you through some very
simple examples of damage caused by strange thinking (and strange teaching).
Feel free to ask questions or comment at any time. I am hoping that there will
be some interaction and insights from you.
Some simple examples:
1) Additive Inverse: My first big trauma with mathematics occurred in
second grade. I was really good with addition and had dutifully learned the
100 elementary addition facts. I had made the observation that 2 + 3 got the
same result as 3 + 2 and that reduced the amount of memorizing considerably.
Then came subtraction: I learned that 7 – 2 = 5, then I wondered about 2 – 7,
so I asked the teacher. She said, “Oh, silly, you can’t do that.” She then
proceeded to put two building blocks on her desk and said, “Now, take away
seven.” I stared at the blocks and said, “I guess that I can’t do that.” Years
later, I took out a checking account and suddenly realized that I could do that.
I had been lied to.
2) Reciprocal: This unary operator is defined as a property of real numbers as
follows:
∀ 𝒂, 𝒃 ∈ 𝑹 ∋ 𝒂 ≠ 𝟎 ∧ 𝒃 ≠ 𝟎; 𝒕𝒉𝒆 𝒓𝒆𝒄𝒊𝒑𝒓𝒐𝒄𝒂𝒍
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𝒐𝒇
𝒂
𝒃
𝒊𝒔 ∧ 𝒕𝒉𝒆 𝒑𝒓𝒐𝒅𝒖𝒄𝒕 𝒐𝒇 𝒂 𝒏𝒖𝒎𝒃𝒆𝒓 𝒂𝒏𝒅 𝒊𝒕′𝒔 𝒓𝒆𝒄𝒊𝒑𝒓𝒐𝒄𝒂𝒍 𝒊𝒔 "1".
𝒃
𝒂
This same concept is self-evident in the concept of the Multiplicative Inverse
1
1
Property: ∀ 𝑎 ∈ 𝑅 ∋ 𝑎 ≠ 0; ∃ 𝑎 ∈ ! 𝑅 ∋ 𝑎 (𝑎) = 1.
Reciprocal: ∀ 𝑎, 𝑏 ∈ 𝑅 ∋ 𝑎 ≠ 0 ∧ 𝑏 ≠ 0; 𝑡ℎ𝑒 𝑟𝑖𝑐𝑖𝑝𝑟𝑜𝑐𝑎𝑙 𝑜𝑓
𝑎
𝑏
𝑏
𝑖𝑠 𝑎.
Notice that I write in complete sentences that will not be misinterpreted.
There may be more than one way to write a concept but there is no
misinterpretation if the concept is written correctly. The is no
misinterpretation in mathematics.
When I was a child, I was extremely shy. I hated having to go to the chalkboard to do mathematics. We had to go up to the board, six at a time, and do a
mathematical problem. The first one done with the correct answer was
considered to be smart and would stay at the board with the next group. I
always made certain that I was never first done and would sometimes figure
out how to get the wrong answer without appearing to be too stupid. One time,
I was at the board and the teacher asked everyone to write the numeral “5” on
the board, so I did. Then she said for us to write the reciprocal of five on the
board. I stood there waiting for someone to write the answer. The teacher then
said, “Oh, come on Steve. You know, just turn it upside down. I said, “Okay,
and I wrote “
“. The teacher told me how to do it wrong and I just did what
I was told … “Was I traumatized?”
Student Evaluation:
"Sometimes I don't understand things because I am more of a memorizer than a deep thinker
so I wish he would approach both aspects more, rather than thinking about everything."
An evaluation from an anonymous student from Jerry Compton’s Chemistry class
"Tell the kid to memorize where the unemployment and welfare offices are, he will soon need
that information."
A comment from a friend of Jerry’s at Oakland
Part of any teacher’s job is to push students out of their comfort zone.
There is very little learning in the comfort zone.
Young, immature students retain (“learn”) repeated information, seen and
heard, … whether the information is correct or not. Therefore, it is
extremely important that teachers be more mindful of what they say and
write for students. It must always be presented with correctness and
uniformity. Teaching (thinking) should be simple but precisely true.
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Sometimes, the “simpler way” leads to incomplete and crippling
thoughts:
3) Absolute Value: This unary operator is defined on any well-ordered set.
The set of Real numbers is a well-ordered set. The set of Pure Complex
numbers is also a well-ordered set. For the set of Real numbers, this
operator has a formal definition as follows:
𝒙 𝒘𝒉𝒆𝒏 𝒙 ≥ 𝟎
∀ 𝒙 ∈ 𝑹; |𝒙| = {
.
−𝒙 𝒘𝒉𝒆𝒏 𝒙 < 𝟎
Absolute value is frequently taught in a very damaging way to young
students: Since |−7| = 7 ∧ |+7| = 7, students can get the correct answer by
simply throwing the sign away. That is voodoo, not mathematics.
When students get to the calculus level (or even before calculus), they must
correctly interpret:
∀ 𝒂, 𝒃 ∈ 𝑹 ∋ 𝒂 > 𝒃; |𝒃 − 𝒂| = ?
If a student throws the sign away, what happens? There is no mathematical
operator called “throwaway”.
Obviously, since 𝒂 > 𝒃, (𝒃 − 𝒂) < 𝟎. ∴ |𝒃 − 𝒂| = −(𝒃 − 𝒂) = 𝒂 − 𝒃.
Cancelitis: This is a nearly fatal disease of the mind where inflicted students
wander around aimlessly “canceling-out” anything that looks alike in an
above and below relationship.
Examples:
?
?
17
1
= 71 = 1 = 1 (Wrong.)
71
17
?
?
𝑥+1
1
= 𝑥+2 = 2 (Wrong.)
𝑥+2
𝑥+1
?
?
16
1
(Well, even a blind pig finds an acorn once in a while.) This is,
=
=
64
64
4
16
also, the sign of a very sick mind.
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4) When reducing or expanding fractions, the Fundamental Theorem of
Fractions is used: ∀ 𝑎, 𝑏, 𝑐 ∈ 𝑅 ∋ 𝑏 ≠ 0 ∧ 𝑐 ≠ 0;
𝑎
𝑏
𝑎𝑐
𝑎𝑐
𝑎
= 𝑏𝑐 ∧ 𝑏𝑐 = 𝑏.
The FTF is used along with the Multiplicative Inverse Theorem:
∀ 𝑎 ∈ 𝑅 ∋ 𝑎 ≠ 0; ∃
1
1
∈ ! 𝑅 ∋ 𝑎 ( ) = 1.
𝑎
𝑎
In order for us to help students cultivate abilities to communicate their
thoughts, an education must include strong basics for writing and
speaking skills. All living things are connected within the fabric of the
cosmos, however, our physical structure limits our conscious communication
(mostly) to speaking, writing, images, and body language. We humans are
limited to the dimensions that we “understand” and we think in pictures,
words, and patterns (an adventure at a lower level). However, we seldom
play with a full deck of cards.
Example: Our eyes see only a very small portion of the electromagnetic
spectrum. In other words, what we see with our eyes is only a very small part
of the “big” picture. Our thinking can compensate for our visual shortcomings
by using our minds to build technology that can feed us information about
other segments of the electromagnetic spectrum. Mathematics helps lead
thinking about dimensions within the fabric of the cosmos, for which, we can’t
even draw pictures (We will discuss this a little later). Yes, I do believe in
technology and I use it when it is an advantage.
Bad writing:
3𝑥 + 2 = 2 + 𝑥
Bad thinking.
−2 − 2
3𝑥 = 𝑥
3=1
−1 − 1 (This is the sign of a very sick mind.)
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Education in America falls short in three areas: 1) Students are taught
shortcuts whenever possible. This leads to not thinking through a complete
process “moving” from start to finish. 2) Students are not taught to think or
write in complete thoughts (complete sentences). Incomplete writing leads
to incomplete thoughts. 3) Students often cannot find the words to express
themselves because they have not “seen” the words and symbols that
they need used in a complete thought context. This is the reasoning
behind “The Great Truths of the Universe”.
Show bad writing:
Let’s elevate the thinking level.
1) Quadratic Theorem:
∀ 𝒂, 𝒃, 𝒄 ∈ 𝑹 ∋ 𝒂 ≠ 𝟎; 𝒊𝒇 𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄 = 𝟎, 𝒕𝒉𝒆𝒏 𝒙 =
−𝒃 ± √(𝒃)𝟐 − 𝟒𝒂𝒄
.
𝟐𝒂
2) Factoring Patterns:
In an attempt to “simplify” mathematics for students, teachers often tell
students how to do things with wrong thinking:
Many teachers teach FOIL to “help” students remember how to multiply two
binomial factors with like corresponding terms.
FOIL: First times first
Outer product
Inner product
Last times last
This cute, little acronym does not help students in any way when they need
to multiply two polynomial factors: The general way to multiply two polynomial
factors is to multiply each term in the first factor by each term in the second
factor and then combine like terms.
Example: Find the indicated product (the FOIL way). Show silly face:
(2𝑥 − 3)(5𝑥 + 4) = 10𝑥 2 + 8𝑥 − 15𝑥 − 12
= 10𝑥 2 − 7𝑥 − 12
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Example: Find the indicated product (the general way).
(2𝑥 − 3)(5𝑥 + 4) = 10𝑥 2 + 8𝑥 − 15𝑥 − 12
= 10𝑥 2 − 7𝑥 − 12
Notice that FOIL is never an advantage over the general method. However,
try to use FOIL on the following example:
Example: Find the indicated product (the general way). There is no FOIL way
for this example.
(2𝑥 − 3)(4𝑥 2 + 6𝑥 + 9) = 8𝑥 3 + 12𝑥 2 + 18𝑥 − 12𝑥 2 − 18𝑥 − 27
= 8𝑥 3 − 27
When students try FOIL in the above case, they get foiled.
Incomplete teaching leads to incomplete thinking and confusion: We
have seen absolute value defined as an operator.
1) Absolute Value Equations and Inequalities: Since the absolute value
operator is not a one-to-one function, there is no inverse operator.
An example of a unary operator that is one-to-one and has an inverse operator
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is “square root” (√(𝑥)) and the inverse operation is “squaring” (√(𝑥)) = (𝑥).
Another example is the logarithm and exponential operators.
∀ 𝑏, 𝑥 ∈ +𝑅 ∋ 𝑏 ≠ 1, ∃ 𝑐 ∈ 𝑅 ∋ log 𝑏 (𝑥) = 𝑐 ↔ 𝑏 𝑐 = 𝑥.
∴ log 𝑏 (𝑏 𝑥 ) = 𝑥 ∧ 𝑏 log𝑏 𝑥 = 𝑥 ∴ log 𝑏 (∎) ∧ 𝑏 (∎) 𝑜𝑝𝑒𝑟𝑎𝑡𝑜𝑟𝑠 𝑎𝑟𝑒 𝑖𝑛𝑣𝑒𝑟𝑠𝑒𝑠 𝑜𝑓 𝑒𝑎𝑐ℎ 𝑜𝑡ℎ𝑒𝑟.
There are four forms involving absolute value. I am going to show only one
form of an absolute value equation: Show “lumps” and “blobs”: Logic is
about patterns.
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|2𝑥 + 3| = 1 − 4𝑥
A logically-equivalent statement:
1 − 4𝑥 ≥ 0 ∧ (2𝑥 + 3 = 1 − 4𝑥 ∨ 2𝑥 + 3 = 4𝑥 − 1)
−4𝑥 ≥ −1 ∧ (6𝑥 = −2 ∨ −2𝑥 = −4)
1
𝑥 ≤ 4 ∧ (𝑥 =
1
−1
3
∨ 𝑥 = 2)
−1
S.S.=(−∞, 4]⋂ ({ 3 } ⋃{2})
1
−1
S.S.= (−∞, 4] ⋂ { 3 , 2}
−1
S.S.={ 3 }
Not teaching formal definitions of operators leads to miscommunication:
Here are some examples of precise communication:
1) Limit of a Function at a Point:
∀ 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝑦 = 𝑓(𝑥);
lim 𝑓(𝑥) = 𝐿 ⟺ ∀ ∈> 0, ∃ 𝛿 > 0 ∋ |𝑓(𝑥) − 𝐿| <∈, ∀ 𝑥 ∋ 0 < |𝑥 − 𝑎| < 𝛿.
𝑥→𝑎
2) The Derivative:
∀ 𝑐𝑜𝑛𝑡𝑖𝑛𝑢𝑜𝑢𝑠 𝑎𝑛𝑑 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡𝑖𝑎𝑏𝑙𝑒 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝑦 = 𝑓(𝑥);
𝐷𝑥 [𝑓(𝑥)] = 𝑦 ′ (𝑥) =
𝑑𝑦
𝑓(𝑥 + ∆𝑥) − 𝑓(𝑥)
= 𝑓 ′ (𝑥) = lim
𝑤ℎ𝑒𝑛 𝑡ℎ𝑖𝑠 𝑙𝑖𝑚𝑖𝑡 𝑒𝑥𝑖𝑠𝑡𝑠.
∆𝑥→0
𝑑𝑥
∆𝑥
3) The Definite Integral:
𝑛
𝑏
∀ 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛, 𝑦 = 𝑓(𝑥), 𝑜𝑛 [𝑎, 𝑏]; 𝑛→∞
lim ∑ 𝑓(𝑥𝑖 )∆𝑥 𝑤ℎ𝑒𝑟𝑒 𝑥𝑖 ∈ [𝑎, 𝑏] = ∫ 𝑓(𝑥) 𝑑𝑥
∆𝑥→0 𝑖=1
= 𝐹(𝑥)| 𝑎𝑏 = 𝐹(𝑏) − 𝐹(𝑎) 𝑤ℎ𝑒𝑟𝑒 𝐷𝑥 [𝐹(𝑥)] = 𝑓(𝑥).
𝑎
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Only discussing 1-D, 2-D, or 3-D leads to handicapping our students from
entering fields like electromagnetic field theory, semi-conducting circuits,
nanotechnology, and astrophysics. If you think I am talking about science
fiction, look at where the new professions are developing.
1) 1-D absolute value equations (picture of a domain of values)
2) 2-D limit of a function at a point (picture of a domain and range of values)
lim 𝑓(𝑥) = 𝐿 ⟺ ∀ ∈> 0, ∃ 𝛿 > 0 ∋ |𝑓(𝑥) − 𝐿| <∈, ∀ 𝑥 ∋ 0 < |𝑥 − 𝑎| < 𝛿.
𝑥→𝑎
Note: √(𝑥 − 𝑎)2 = |𝑥 − 𝑎|.
3) 3-D limit of a function at a point (picture of a domain and range of values)
lim
(𝑥,𝑦)→(𝑎,𝑏)
𝑓((𝑥, 𝑦)) = 𝐿 ⟺ ∀ ∈> 0, ∃ 𝛿 > 0 ∋ |𝑓((𝑥, 𝑦)) − 𝐿| <∈, ∀ (𝑥, 𝑦) ∋
0 < √(𝑥 − 𝑎)2 + (𝑦 − 𝑏)2 < 𝛿.
4) 4-D limit of a function at a point (picture of a domain)
lim
(𝑥,𝑦,𝑧)→(𝑎,𝑏,𝑐)
𝑓((𝑥, 𝑦, 𝑧)) = 𝐿 ⟺ ∀ ∈> 0, ∃ 𝛿 > 0 ∋ |𝑓((𝑥, 𝑦, 𝑧)) − 𝐿| <∈, ∀ (𝑥, 𝑦, 𝑧) ∋
0 < √(𝑥 − 𝑎)2 + (𝑦 − 𝑏)2 + (𝑧 − 𝑐)2 < 𝛿.
5) 5-D limit of a function at a point (stop drawing pictures)
lim
(𝑥,𝑦,𝑧,𝑟)→(𝑎,𝑏,𝑐,𝑑)
𝑓((𝑥, 𝑦, 𝑧, 𝑟)) = 𝐿 ⟺ ∀ ∈> 0, ∃ 𝛿 > 0 ∋ |𝑓((𝑥, 𝑦, 𝑧, 𝑟)) − 𝐿| <∈,
∀ (𝑥, 𝑦, 𝑧, 𝑟) ∋ 0 < √(𝑥 − 𝑎)2 + (𝑦 − 𝑏)2 + (𝑧 − 𝑐)2 + (𝑟 − 𝑑)2 < 𝛿.
The study of quantum mechanics, which is the basis of nanotechnology,
requires 8 dimensions. The unified field theory of superstrings requires a
minimum of 11 dimensions. The research on building an operating system
for a quantum computer is being done in as many as 18 dimensions. Some
mathematicians are talking about infinite dimensional analysis. Do you
think that American education is educating students for the future or fifty
years in the past?
Take a look at the “standardized” tests (controlled by large testing corporations)
that high school teachers in Michigan are being forced to use as their primary
goal for teaching and you will understand my concern about the “dumbing of
American education”. Standardization leads to mediocre thinking. Every
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American should watch the movie Idiocracy. The best way to lower
expectations is to make everyone (other than the extremely wealthy)
stupid. Another way to speed up this process is to make educational
institutions (like community colleges) become subsidiaries of monster
“education” corporations. Corporations exist for one reason and it isn’t
thinking.
Questions?