Reflections on Being a Math Tutor
After retiring from my 40 year electronics engineering career, the natural thing was to put my almost
50 years of teaching experience in the corporate world to work by becoming a one-on-one math tutor in
a large institution. For the next 12 years, I worked with children ranging from 3rd grade to collage
freshmen, with the vast majority being junior and senior high school math students. This article
discusses my feelings on the math being taught or not being taught and how it is being taught in our
public school systems.
Many of my students, most of whom are not college-bound, ask me same question. “Why do I have to
learn this stuff, when I will never use it?” I always explain that; “Math teaches you to think, logically
and orderly. Critical thinking is a skill that you can use in all aspects of life.”
The Role of a Tutor
To a great extent, students just want someone to do their homework for them and make their lives easy.
Naturally, there is a great tendency to give into time pressure and “help” a student by doing just that.
However, it is necessary to constantly push back against that tendency. The tutor’s job is to enable the
student to be self-sufficient and be able to do their own work. I constantly tell students that; “You can
not take me into an exam. You must be prepared to pass the exam by yourself.”
The tutor’s job is also to supplement the teacher’s job by explaining clearly material that may not have
been covered well in class or the student didn’t really sufficiently understand in class. The next section
discusses the difficulties in fulfilling this aspect of the tutoring role.
Textbooks versus Tablets
Partially due to the COVID pandemic and implementing cost-saving measures, the school system
where I tutor eliminated all textbooks several years ago and gave students tablets. Some teachers
briefly provided their students with hand-outs, from which the students could create mini-textbooks.
Most teachers never told students that it is now their responsibility to take notes. The school system did
provide on-line tutorials, but they were so poorly and slowly presented that most students refused to
use the tutorials. As a tutor, I could no longer determine what a student was being taught or how it was
being taught. By flying blind and having, at most, a homework assignment as a hint, the tutor’s job can
not be well executed. Overall, the substitution of tablets for textbooks was a major determent to the
learning process.
In retrospect, the school should have provided access to on-line .pdf copies of suitable text books,
which are available. Fortunately, both the institution and I were able to purchase sets of used textbooks,
placed on the used book market by the schools.
Also, as a tutor, I repeatedly asked for a guest log-in to the school’s system, so that I could see what my
students were working on.
Homework
After eliminating textbooks, the school system where I tutor eliminated all homework. With no
homework and no preparation for exams, there is no longer any need for tutors. We simply do not knowwhat, if anything, the students are learning. It appears that they are just being pushed through the
system to graduation, only to face the world unprepared. The marked the end of my tutoring.
SINGAPORE Math
Educators and most other fields, tend to be like lemmings, all racing off the same cliff. When my kids
were small, they taught “guesstimation,” so as to not clutter young minds with facts, especially the
multiplication tables. While they pretended that children could do calculus in second grade, it was a
total disaster, with parents and eventually teachers rebelling against it. Singapore math, in my opinion,
is another such trend.
In the lower grades, math is taught using techniques that might be good for visual learners, growing up
with video games. The core belief is that a symbol such as “5” is too abstract for a child to
comprehend. If you are talking about five stars, it should be expressed as “* * * * *.”
However, these math techniques may teach concepts, but are useless for numbers of any reasonable
size. As a result and as a math tutor, we have had to reteach basic, conventional math, starting in about
the 4th grade. This already puts students behind by starting their math education at such a later age.
Most students never recover and are still behind when graduating from high school. Colleges are
reporting that new freshman students require remedial math to learn what they should have learned in
high school. The same is true in other academic subjects, such as reading and English.
Personal Finance
The school system where I tutor requires all high school seniors take Personal Finance, instead of a
higher math class. This is good for students that are not college bound. The math is relatively easy, with
table-based loan calculations being he most difficult topic. A topic might be analyzing the cost of
taking loan and buying a car now, versus saving any buying the car later with cash. Another topic is
figuring out how much you have to spend on housing and what size apartment you can afford in your
community. Fiscal responsibility is a good thing that many adults have never learned or have learned
the hard way. Unfortunately, it detracts from the math education of college-bound students.
Alternate Number Systems
The remainder of this article discusses mathematics topics that are useful in the modern age of
computers and for people entering any form of computer science, including software and electronics
engineering, but are not taught in the school systems. Students today still learn only the same math that
I learned in the late 1950’s and early 1960’s, just a lot less of it. Therefore, they are less prepared in
math today for technical fields than I was before the age of computers.
Humans all know and use decimal numbers, only because we have ten fingers. While decimal may be
good for humans, it is not natural or useful for computers, which are binary devices. Therefore, this
section will discuss numbers systems that are useful for computers and form a good compromise for
both humans and computers. With the possible exception of binary, and octal, the other number systems
should at least be introduced to students, although only computer scientists and related fields need to be
comfortable using these systems.
Decimal Number System
Decimal uses a number system where each digit can have ten possible values, because we have 10
fingers. These values are listed as ‘0123456789.’ The next value after ‘9’ is ’10.’ Nothing new here.
In this section, we will use the decimal number ‘987654321’ and demonstrate it’s values in other
number systems.
Binary Number System
Internally, computers use binary bits, where each bit has only two values, 0 or 1. Therefore, the decimal
number above, ‘987654321’ would be represented as ‘0011101011011110100010110001.’ Reading from
right to left, this would be 1+16+32+128… While very computer friendly, binary is not human friendly
at all.
Hexadecimal Number System
As a compromise, hexadecimal uses a number system where each digit can have 16 possible values, as
opposed to only 10 values of the decimal system. These values are listed as ‘0123456789abcdef.’ The
next value after ‘f’ is ’10’. It is read as “one zero’ and is equivalent to the decimal value 16. Note that the
values represented by the letters ‘a’ through ‘f’ may be either upper or lower case.
The binary number above is partitioned into groups of four bits each, and each group is assigned its
corresponding hexadecimal value, as follows. The ‘0x’ prefix is simply a way of stating the the value is
hexadecimal format. The ‘0x’ is not part of the value and serves only as an indicator of “hexadecimal.”
‘0011 1010 1101 1110 0110 1000 1011 0001” is then expressed as ‘0x3ade68b1’
Note that hexadecimal numbers are more compact and have less digits than their counterparts in other
number systems. This makes them very human friendly, especially with dealing with very large values.
As an electronics engineer and mathematical cryptographer, I have probably done more math in
hexadecimal than I have in decimal, which may seem surprising. As a cryptographer, I typically
worked with numbers in the range of 900 hexadecimal digits or about 1,200 decimal digits in length.
A hexadecimal multiplication table is included as an appendix to this paper. The hexadecimal value of
‘100’ in the lower right of the table is equivalent to the decimal value 256.
Octal Number System
While no longer used in the era of modern computers, the octal number system is included here for
completeness.
As an early compromise, octal uses a number system where each digit can have only eight possible
values, as opposed to the 10 values of the decimal system. These values are listed as ‘01234567.’ The
next value after ‘7’ is ’10’. It is read as “one zero’ and is equivalent to the decimal value 8.
The binary number above is partitioned into groups of three bits each, and each group is assigned its
corresponding octal value, as follows. The ‘0o’ prefix is simply a way of stating the the value is in octal
format. The ‘0o’ is not part of the value and serves only as an indicator of “octal.”
‘111 010 110 111 100 110 100 010 110 001’ then expressed as ‘0o7267464261’
Boolean Algebra
George Boole was an English mathematician and logician who, in 1847, invented the algebra used in
all logic and digital electronic circuit design. High school algebra already introduces students to ‘set
theory,’ with concepts such as ‘union’ and ‘intersection.’ It would be a simple extension of this topic to
include logic theory with the same concepts, called ‘or’ and ‘and’ plus ‘exclusive-or.’ There are simple
graphical tools, such as ‘Karnaugh Maps’ for translating logic problems into Boolean equation.
As an electronics engineer, designing numerous digital logic systems, I have probably done as much
Boolean algebra in my career as conventional algebra and always always had a stack of blank
Karnaugh Maps on my desk.
