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This is the 4th IAE Newsletter
in a new series devoted to the educational issue of Credibility and Validity of Information.
Credibility and Validity of Information
Part 4: The Discipline of Mathematics
David
Moursund
Emeritus Professor of Education
University of Oregon
“Mathematics is the queen of sciences….
She often condescends to render service to astronomy and other natural
sciences, but in all relations she is entitled to the first rank.”
(Carl Friedrich Gauss; German mathematician, physicist, and prodigy;
1777-1855.)
From Previous Newsletters in the Credibility/Validity Series
Credibility focuses on a
belief that the person who made an allegation about a phenomenon is
believable and can indeed be trusted. It is common to talk about a
person and what the person writes and/or says as being credible and
believable.
Validity is an important component of research. The word tends to be used in two somewhat different ways:
Validity is the quality of being logically or factually sound.
Validity is the extent to which a concept, conclusion, or measurement
is well-founded. Results produced by valid research can be tested by
others repeating the research. Additional evidence of validity is
produced by research designed to find contractions to the results, and
that fail to find such contradictions.
In education, a research instrument or test is valid if it accurately measures what it is purported to measure.
In brief summary, one can think of credibility having a subjective base
and validity having an objective base. The performance of a gymnast or
dancer is determined by subjective methodologies, while mathematics and
the sciences use objective methodologies to support their claims.
This IAE Newsletter discusses
the validity of mathematics. You “know” with great certainty that 3 + 5
= 8 and that 3 x 4 = 12. These are examples of math “facts.” They are
math information that has a very high level of objective validity.
The math content in the discipline of mathematics gains its validity through objective methods that mathematicians call proofs,
and from the refereeing process used in vetting and publishing their
proofs. The objective validity of mathematics is rooted in:
Very carefully stated definitions, assumptions, and notation.
Developing, sharing, and building on math proofs that are openly
available and can be checked by others who have the needed math
knowledge and skills.
What Is Math?
See Moursund (2014a) to read some answers to the
question, “What is math?” A very short answer is that math is a
discipline in which people pose and attempt to solve problems that can
be stated in the language of mathematics. See Moursund (2014b) for a
discussion of the language of mathematics.
Math is a science, but in terms of credibility and validity it differs
from the other sciences. Here is a quote that helps explain this
situation:
“God created the natural numbers. All the rest is the work of man.”
(Leopold Kronecker; German mathematician and logician; 1823-1891.)
Leopold Kronecker’s statement posits that the natural numbers 1, 2, 3,
etc., are part of nature, but that all of the rest of math was created
by humans. We know that humans and many other animals have some innate
ability to count small numbers of objects. The mathematical sciences
that humans have created are much different from the natural and social
sciences, where researchers observe the real world, design experiments,
gather and analyze data, and draw conclusions.
Researchers in math produce results that are independent of our
physical world and the people on it. These results have a very high
level of validity in the world of mathematics. They constitute the
“gold standard” of validity.
Research projects looking for extraterrestrial intelligence on far-away
planets look for mathematical patterns of signals from space. Click here
to read about the Search for Extraterrestrial Intelligence (SETI). Some
of the SETI work is based on the assumption that math is somewhat the
same throughout the universe. That is, SETI researchers tend to assume
that any intelligence that creates a civilization that can broadcast
and receive electronic signals necessarily has begun with counting and
has developed math somewhat akin to our math.
Rote Memory Sometimes Fails Us
One way to learn math is via rote memory. Another way is through
understanding and being able to “figure it out.” Here is an example. If
I ask a representative sample of English-speaking adults in the U.S.
what 9 times 7 is, some will give me a response other than 63. So, even
though at one time in their lives they memorized that 9 x 7 = 63 and
accepted that math fact as having great credibility, their memory is
not perfect.
However, if I indicate to an adult that the answer they have provided
is incorrect, most can “figure out” a correct answer—perhaps by
counting by sevens or nines, perhaps by drawing an array of seven
objects by nine objects and counting them, and perhaps by other
methods.
This example illustrates a very important aspect of math. One can
memorize math facts, definitions, formulas, and other math information.
Unfortunately, our rote memory sometimes fails us. Indeed, this is so
common that in some math testing situations, students are provided a
list of formulas that might be relevant to the test questions. The test
is designed to move beyond use of rote memory. Some math test
situations allow students to use calculators.
In math—perhaps more so than in any other discipline—there are a
variety of methods we can use to check our math rote memory and also to
check the results we obtain when solving math problems. This checking
process might be done mentally, by use of pencil and paper, by use of a
calculator or computer, or by checking a credible source of math
information. These checking processes can be thought of as informal
proofs of correctness. Remember, however, that using a calculator in
producing or checking an answer does not necessarily mean the answer is
correct. The electronics of the calculator may be broken or the
calculator user may have made a keyboarding error!
Some Math and Non-math Examples
Here are some statements that involve numbers. Think about the
credibility and validity of each. Do some of the statements seem to you
to have more credibility and validity than others?
There is no largest positive integer.
Seven is a lucky number and 13 is an unlucky number.
There is no positive integer which, when multiplied by itself, gives an answer of 7.
A math problem either has exactly one correct solution, or it has no correct solution.
If the three sides of a triangle in a plane have lengths of 3, 4,
and 5 respectively, then the triangle is a right triangle (that is, one
of the angles in the triangle is 90 degrees).
The statements 1, 3, 4, and 5 are all “pure” math statements. Their
truth or falsehood is not a matter of opinion. Each can be
(mathematically) proven to be a correct or incorrect statement.
For example, consider the third statement. You might observe that 1 x 1
= 1, 2 x 2 = 4, and 3 x 3 = 9. As you consider subsequent integers such
as 4, 5, 6, and so on, the square of each is larger than the square of
the proceeding, and the squares are increasingly larger than 7. Most
people are convinced by this type of argument (this proof) that
statement 3 is correct. So, statement 3 has the highest of validity
that math can produce. That is, mathematicians consider the arguments just given as an objective math proof.
However, although the two assertions in statement 2 contain some
numbers, the statements are not pure math statements. They are
statements that some people believe and some people do not believe.
These statements cannot be proved or disproved by the use of
mathematical reasoning. Each has nothing to do with math other than the
fact that it happens to involve a positive integer in its statement. Do
you believe in lucky and unlucky numbers? Have you ever told someone
that a particular number is lucky and a different one is unlucky? If
you do this, you are sharing a personal (subjective) belief. Click here to read more about lucky numbers.
Statement 1 is a rather deep, abstract aspect of mathematics. In grade
school you probably were told that the positive integers go “on and on
forever.” You may have been introduced to the word infinity and/or the
symbol ∞. We have evidence that mathematicians thought about and explored various aspects of ∞ nearly 2,500 years ago.
Can your prove statement 1 in a manner that meets your personal
standards of “proof”? Can you explain your proof so that it is credible (understandable) to your peers or to students you teach?
Statement 5 is a more complex math challenge. In your mind, you may
relate it to the Pythagorean theorem that you encountered in a Plane
Geometry course you took a number of years ago. Quoting from The History of Mathematics (Allen, 2014):
Arguably the most famous theorem in all of mathematics, the Pythagorean
Theorem has an interesting history. Known to the Chinese and the
Babylonians more than a millennium before Pythagoras lived, it is a
"natural" result that has captivated mankind for 3000 years. More than
300 [different] proofs are known today.
So, the Pythagorean theorem is mathematically correct. However, it is a
statement about a triangle in a plane. It is not correct for triangles
drawn on the surface of a sphere or on the curved surface parts of a
cylinder such as a “tin” can. One must use great care in taking results
from math and applying them to problems that do not satisfy the
assumptions of the math results.
Statement 4 Is An Incorrect Math Belief
Consider statement 4 in the list given in the previous section. Do you
believe it is a correct statement? Can you prove that it is a correct
(or incorrect) statement?
Actually, it is believed to be true by many students, but it is
definitely incorrect. It is easy to disprove the assertion. To disprove
a math assertion, one only needs to find one counter example.
Consider the “exactly one correct” answer assertion. Think about
the math problem of finding a positive integer greater than 1 and less
than 10. Hmm. That’s easy enough. The integers 2, 3, … 9 are all
correct answers. This math problem has more than one correct answer.
So, we have proved that the assertion is incorrect. The next time you
hear a student say that in math the goal is to find the correct answer, I hope that you will make use the opportunity to correct the student’s misunderstanding.
Is it possible that a math problem has no solution? Consider the math
problem of finding a positive integer that is greater than 4 and less
than 5. Hmm. This problem has no solution. (Think about your thinking
as you work to convince yourself of this assertion.) So, through these
simple examples you are probably convinced that a math problem may have
no solution, one solution, or more than one solution.
Undecidable Math Problems
This short section touches briefly on a relatively modern and very deep aspect of mathematics. There are undecidable math problems. Quoting from Bjorn Poonen’s paper Undecidable Problems: A Sampler:
A single [mathematics] statement is
called undecidable if neither it nor its negation can be deduced using
the rules of logic from the set of axioms [and definitions] being used.
The goal of this survey article is to demonstrate that undecidable decision problems arise naturally in many branches of mathematics. [Bold added for emphasis.]
In summary, a math problem may have no solution, one solution, more than one solution, or be undecidable.
Spend a little time thinking about whether this situation applies to
some of the problems that people encounter in disciplines outside of
mathematics. For example, we have problems such as hunger,
homelessness, disease, bigotry, crime, sustainability, global warming,
and so on. These are not math problems, although people use math in
attempting to deal with such problems. We can make progress toward
solving such problems—but can any one of these problems be completely
solved in the sense that we solve math problems and provide proofs that
they are correctly solved?
Why Is Math One of the Basics of Education?
Reading, writing, and arithmetic (math) are considered the basics of education. Why is math in the list?
It is not because math is a human endeavor that has a long history, and
that some people find math to be beautiful and a lot of fun. Rather, it
is because math is so useful in our everyday lives. We use math to
measure quantity, distance, time, and so on. If a “real world” problem
can be represented as a math problem, then we may be able to build on
the thousands of years of progress in math to help solve the real world
problem.
Some real world problems are easily translated into math. Suppose that
I can save $15 per week by making my lunch at home rather than buying
it at a cafeteria. How much will I save in 12 weeks? In this question,
we use 15 to represent $15, and we use 12 to represent 12 weeks. We
solve the (pure) math calculation problem 15 x 12. We translate the
result (180) back into an answer of $180.
Think about the complexities involved in this simple “story” problem.
What is money, what is a dollar, what is a week, and what does it mean
to save money? The arithmetic calculation is the simple part, and we
can be quite comfortable in the math result that 15 x 12 = 180. Have we
actually solved the real world problem?
Perhaps with all of that extra (non-spent) money in my pocket I buy a
candy bar from a candy machine each day at work. Now I need to know how
many days I work in a week, and what candy bars cost. And perhaps the
machine sells candy bars of varying prices. Perhaps I should also be
considering other real world issues such as the fact that carrying my
lunch makes a significant change in my social life, and the daily candy
bars may cause me to gain weight. My point is, the real world differs
from the “pure” math world. Real world problems tend to be “messy.”
Math teaching makes extensive use of story problems (word problems)
that describe a problem that can be solved by use of mathematics.
Often the problems are over simplified, such as the original version of
the carrying lunch to work situation. Many students find that even the
overly simplified problems are quite difficult. This suggests that many
students find it difficult to do transfer of learning from “pure” math
into applications of math. The statements “I can’t do math,” and “I
hate math” usually come from students who have been taught rote memory
approaches to learning math and who have not had much success in the
transfer of learning from this rote memory math to applications in the
real world.
Final Remarks
The discipline of mathematics produces mathematical results
(proven theorems and solved math problems) that have a very high level
of validity. Such math can be very useful in representing and helping
to solve problems in other disciplines. However, just because known
(proven) math is used in helping to solve a problem in a discipline
outside of math does not ensure the correctness or validity of the
results.
Because math is so important in many non-math disciplines, students
studying such non-math disciplines face the dual learning challenge of
learning both their specific discipline and math. They face the
challenge of deciding on credibility and validity of the results in
their discipline of study and what role math plays in determining this
credibility/validity.
David Moursund earned his doctorate in mathematics from the
University of Wisconsin-Madison. He taught in the departments of
Mathematics, Computer Science, and Teacher Education at the University
of Oregon. A few highlights of his professional career include founding
the International Society for Technology in Education (ISTE), serving
as ISTE’s executive officer for 19 years, and establishing ISTE’s
flagship publication, Learning and
Leading with Technology (now named Entrsekt). He was a major professor
or co-major professor of 82 doctoral students. He has authored or
coauthored more than 60 academic books and hundreds of articles. He has
presented hundreds of professional talks and workshops.
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