Information Age Education
   Issue Number 151
December, 2014   

This free Information Age Education Newsletter is written by Dave Moursund and Bob Sylwester, and produced by Ken Loge. The newsletter is one component of the Information Age Education (IAE) publications.

All back issues of the newsletter and subscription information are available online. In addition, four free books based on the newsletters are available: Understanding and Mastering Complexity; Consciousness and Morality: Recent Research Developments; Creating an Appropriate 21st Century Education; and Common Core State Standards for Education in America.

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:
  1. 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.

  2. 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:
  1. Very carefully stated definitions, assumptions, and notation.

  2. 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?
  1. There is no largest positive integer.

  2. Seven is a lucky number and 13 is an unlucky number.

  3. There is no positive integer which, when multiplied by itself, gives an answer of 7.

  4. A math problem either has exactly one correct solution, or it has no correct solution.

  5. 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.

References

Allen, G.D. (2014). The history of mathematics. Retrieved 12/7/2014 from http://www.math.tamu.edu/~dallen/masters/hist_frame.htm.

Moursund, D. (2014a). What is mathematics? IAE-pedia. Retrieved 12/7/2014 from http://iae-pedia.org/What_is_Mathematics.

Moursund, D. (2014b). Communication in the language of mathematics. IAE-pedia. Retrieved 12/7/2014 from http://iae-pedia.org/Communicating_in_the_Language_of_Mathematics.

Author

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.

In 2007, he founded Information Age Education (IAE), a non-profit company dedicated to improving teaching and learning by people of all ages throughout the world. See http://iae-pedia.org/Main_Page#IAE_in_a_Nutshell. Contact information: moursund@uoregon.edu.

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Information Age Education is a non-profit organization dedicated to improving education for learners of all ages throughout the world. Current IAE activities and free materials include the IAE-pedia at http://iae-pedia.org, a Website containing free books and articles at http://i-a-e.org/, a Blog at http://i-a-e.org/iae-blog.html, and the free newsletter you are now reading. See all back issues of the Blog at http://iae-pedia.org/IAE_Blog and all back issues of the Newsletter at http://i-a-e.org/iae-newsletter.html.