Issue Number 248 December 31, 2018

This free I

All back issues of the newsletter and subscription information are available online. In addition, seven free books based on the newsletters are available:

Dave Moursund’s newly revised and updated book,

Tools and the
Future of Education

Part 2: Math Education

Part 2: Math Education

David
Moursund

Professor Emeritus, College of Education

University of Oregon

Professor Emeritus, College of Education

University of Oregon

“God created the natural numbers, all else
is the work of man.” (Leopold Kronecker, German mathematician who worked
on number theory, algebra, and logic; 1823-1891.)

“Mathematics is the queen of sciences and
number theory is the queen of mathematics. She often condescends to
render service to astronomy and other natural sciences, but in all
relations she is entitled to the first rank.” (Johann Carl Friedrich
Gauss; German mathematician, astronomer and physicist;1777-1855.

“A new technology [such as Information and
Communication Technology] does not add something, it changes
everything.” (Neil Postman; American author, educator, media theorist,
and cultural critic; 1931-2003.)

Introduction

Problem solving lies at the very heart of mathematics, and mathematics is useful in solving problems in many other disciplines (Moursund, 2016a, link). Because math is such a versatile aid to problem solving across the curriculum, it is considered to be one of the basics of education.

Similarly, computers are an aid to solving problems across the
curriculum. In my book,* The Fourth R*, I present the case that
**R**easoning/Computational Thinking is a new basic “**R**”
to be added to the three basic “**R**s” of **R**eading,
‘**R**iting, and “**R**ithmetic. Thus, this
new** Fourth R **use of both human and computer brains
in all curriculum areas is a new basic of education (Moursund, 2018b,
link).

This and the next* IAE Newsletter* will explore math education
and some roles of computers in math education. The goal is to help
improve these two important basics of a modern education.

As we work to improve math education, we must take into consideration answers to such fundamental questions as:

- What is mathematics?
- What are the goals of math education?

Quotation from Leopold
Kronecker

As quoted at the beginning of this newsletter, the famous mathematician Leopold Kronecker said, "God created the natural numbers, all else is the work of man." That is, mathematics is a tool developed by people. Some natural counting ability to count is wired into human brains and the brains of some other animals (Goldman, April, 2013, link.)

Long before we had written languages and schools, people learned to count and make other simple uses of math. The development of agriculture about 12,000 years ago and increased trading for goods and services led to the need for more math than simple counting. The development of written symbols for numbers and operations on the numbers facilitated the teaching of math in the earliest schools that were designed to teach reading and writing more than 5,000 years ago.

Nowadays, young children learn from their early childhood caregivers how to count and how to determine the number of objects in a small set. Above that level of math understanding, knowledge, and skill, our formal math education begins to click in. For example, we have various written numeral systems such as Roman numerals (I, II, III, IV, V, VI, VII, VIII, XI) and Hindu-Arabic numerals (1, 2, 3, 4, 5, 6, 7, 8, 9).

In our natural languages, we have specific words and symbols for
addition, subtraction, multiplication, division, and other operations
on numbers. We have fractions and decimals. We have subdisciplines of
math such as arithmetic, algebra, geometry, statistics, probability,
and calculus. We have various systems and vocabulary for measuring *distance*,
*area*, *time*, and *quantity*, such as the
metric system and the English system. As Kronecker noted, all of these
math-related concepts are the inventions of people.

What Is Mathematics?

Your education has included the study of mathematics. Based on your experiences and interests, you can provide a definition of mathematics. However, math is such a broad and deep field that it is difficult to give a short, comprehensive definition (Moursund, 2016b, link).

Math has a very long history and is a huge field of study and application. Quoting from History of Mathematics (Wikipedia, 2018b, link):

- The area of study known as the history of mathematics is primarily
an investigation into the origin of discoveries in mathematics and,
to a lesser extent, an investigation into the mathematical methods
and notation of the past. Before the modern age and the worldwide
spread of knowledge, written examples of new mathematical
developments have come to light only in a few locales. From 3000 BC
the Mesopotamian states of Sumer, Akkad and Assyria, together with
Ancient Egypt and Ebla began using
**arithmetic, algebra and geometry for purposes of taxation, commerce, trade and also in the field of astronomy and to formulate calendars and record time.**[Bold added for emphasis.]

The first schools were started in about 3,400 BC, shortly after the
development of reading and writing, However, mathematics is rooted in
the need for and use of measurement many tens of thousands of years
before the creation of these schools. For example, consider a small
group of hunter-gatherers who want to get back to their home cave
before it gets dark. They look at the position of the sun. They think
about their current location, the location of their cave, and about
the need to get back to their cave before dark. They may think about
the quantity of food they have gathered, and whether they can carry
all of it. *Time*, *distance*, *quantity*, and*
travel directions* are all important measurements.

Money is one of the things that we measure (count). Quoting from Who Invented Money? (Wonderopolis, n.d., link):

No one knows for sure who first invented such money, but historians believe metal objects were first used as money as early as 5,000 B.C. [This was well before the invention of reading and writing and the development of the first schools that taught reading, writing, and arithmetic.]

- Around 700 B.C., the Lydians became the first Western culture to make coins. Other countries and civilizations soon began to mint their own coins with specific values. Using coins with set values made it easier to compare values and trade money for goods and services.

In summary, the tool we call mathematics is thoroughly integrated into our natural languages and daily lives.

**We Each Have Our Own Opinions about Math Education**

As an adult, you know a lot about math and you use your knowledge every day. If I ask you what time it is, you likely will consult your wristwatch, your cellphone, or a more general-purpose computer, and give me an answer such as 1:35. Presumably, I know whether this is in the middle of the night or middle of the day, so I know whether it is AM or PM. A more precise answer could include the correct AM or PM, the day of the week, the month of the year, the year, and the calendar system being used.

Think carefully about the difference between reading a watch or clock
and understanding the meaning of the measurements they provide. (This
reading process is more difficult if an analog watch or clock is being
consulted.) Time is a very complex topic. Through years of experience
you have developed *time sense*. You can relate the symbols
1:35 to your knowledge about time and how you routinely make use of
this knowledge. If you are working a 9:00 to 5:00 job, you can quickly
figure out that it will be about 3½ hours until you get off work, but
just a little under a half-hour before you get your 2:00 rest break.

Through doing this thinking exercise, you have uncove<br
/><br /><br /><br /><br /><br
/><br /><br />d one of the major goals in math
education. This goal is to develop *number sense*. Consider an
analogy with knowing how to use a handheld calculator or a memorized
computational algorithm to do arithmetic. It takes little time to
learn to key numbers into a calculator and get an answer. It takes
much longer to memorize an algorithm and develop both speed and
accuracy in using the algorithm. It takes a very long time to
understand the meaning of what you are doing and to make use of the
answer. It also takes a long time to develop estimation skills (part
of number sense) that can help you to detect errors in keyboarding
and/or in reading the numbers you want to use in order to do
arithmetic.

Memorizing computational algorithms and/or using a calculator to do calculations contributes little to developing number sense. The inexpensive handheld calculator has been with us for more than 40 years. It is a useful tool, and many adults routinely use this tool. However, we have scant evidence that the current ways of using such calculators in elementary school makes a significant difference in the level of number sense these students are gaining.

George Polya’s Answer to
“What Is Mathematics?”

George Polya was a leading 20th century mathematician and math educator (Wikipedia, 2018a, link). Problem solving was one of his areas of study and writing. The following is quoted from a talk he gave to preservice and inservice teachers in the late 1960’s (O’Brien, n.d., link).

To understand mathematics means to be able
to do mathematics. And what does it mean doing mathematics? **In
the first place it means to be able to solve mathematical problems.**
For the higher aims about which I am now talking are some general
tactics of problems—to have the right attitude for problems and to be
able to attack all kinds of problems, not only very simple problems,
which can be solved with the skills of the primary school, but more
complicated problems of engineering, physics and so on, which will be
further developed in the high school. But the foundations should be
started in the primary school. And so I think an essential point in
the primary school is to introduce the children to the tactics of
problem solving.** Not to solve this or that kind of problem,
not to make just long divisions or some such thing, but to develop a
general attitude for the solution of problems.** [Bold added
for emphasis.]

The discipline of mathematics, as every other academic discipline, focuses on solving problems and accomplishing tasks (Moursund, 2016a, link).

Proofs

Mathematicians is a vertically structured discipline in which new results are built on previous results. Such a system of building on the previous work of self others falls apart if the previously done work is found to be incorrect. Thus, carefully chosen definitions and assumptions, and then results (theorems) based on the definitions and assumptions, form the foundation of mathematics. Quoting from the Wikipedia (Wikipedia. 2018c, link):

In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and generally accepted statements, such as axioms. A theorem is a logical consequence of the axioms. The proof of a mathematical theorem is a logical argument for the theorem statement given in accord with the rules of a deductive system. The proof of a theorem is often interpreted as justification of the truth of the theorem statement. In light of the requirement that theorems be proved, the concept of a theorem is fundamentally deductive, in contrast to the notion of a scientific law, which is experimental.

In brief summary, math is problem solving and proof. Math education instructional activities such as “show your work” and “explain the steps you have taken in solving a problem” are a common approach to teaching young students about proof. As students progress in their math studies, they learn more and more about proofs.

What Is a Problem?

Any discipline of study can be defined as an appropriate combination of the problems it works to solve, the tasks it works to accomplish, its accumulated achievements, and so on. I tend to view the world through problem-solving-colored glasses. Probably because of my many years of study in mathematics, computer science, and education, my colored glasses are tinted so that they emphasize these three disciplines of study.

Here is a formal definition of the term *problem*. You
(personally) have a problem if the following four conditions are
satisfied:

- You have a clearly defined given
*initial situation*. - You have a clearly defined
*goal*(a desired end situation). (Some writers talk about having multiple goals in a problem. However, such a multiple goal situation can be broken down into a number of single goal situations.) - You have a clearly defined set of
*resources*that may be applicable in helping you move from the given initial situation to the desired goal situation. These include your physical and cognitive capabilities, along with the knowledge and skills you have acquired throughout your life. In some problem-solving situation, there may be specified limitations on resources, such as rules, regulations, and guidelines for what you are allowed to do in attempting to solve a particular problem. This is certainly true in most school tests. - You have some
*ownership*—you are committed to using some of your own resources, such as your knowledge, skills, and energies, to achieve the desired final goal.

These four components of a well-defined problem are summarized by the
four words: *givens*, *goal*, *resources*, and
*ownership*. If one or more of these components are missing, we
call this a *problem situation*. An important aspect of
problem solving is realizing when one is dealing with a problem
situation and working to transform that into a well-defined problem.

People often get confused by the *resources *(part 3) of the
definition. Resources do not tell you how to solve a problem.
Resources merely tell you what you are allowed to do and/or use in
solving the problem. For example, you want to create a nationwide ad
campaign to increase the sales by at least 20% for a set of products
that your company produces. The campaign is to be completed in three
months, and not to exceed $80,000 in cost. Three months is a time
resource and $80,000 is a money resource. Your time and capabilities
are a people resource. You can use these resources in solving the
problem, but the resources do not tell you how to solve the problem.
Indeed, the problem might not be solvable. (Imagine an automobile
manufacturer trying to produce a 20% increase in sales in three
months, for $80,000!)

Problems do not exist in the abstract. They exist only when there is
*ownership*. The owner might be a person, a group of people
such as the students in a class, or it might be an organization, a
country, or the whole world. Global warming is an example of a
worldwide problem.

A person may have ownership “assigned” by his/her supervisor in a company. That is, the company, or the supervisor has ownership, and assigns it to an employee or group of employees.

The idea of ownership is particularly important in education. If a student creates or helps to create the problems to be solved, there is increased chance that the student will have ownership. Such ownership contributes to intrinsic motivation—a willingness to commit one's time and energies to solving the problem.

The type of ownership that comes from a student developing a problem that he/she really wants to solve is quite a bit different from the type of ownership that often occurs in school settings. When faced by a problem presented/assigned by the teacher or the textbook, a student may well translate this into, "My problem is to do the assignment and get a good grade. I don't have any interest in the problem presented by the teacher or the textbook." A skilled teacher will help students to develop projects that contain challenging problems, and the problems are ones that the students really care about.

Some Key Ideas for
Goal-setting in Math Education

Because there has been such a large amount of research in in the
field of mathematics over the years, there is a huge accumulation of
information about how to solve a wide variety of math problems. If a*
real world* problem can be represented mathematically, this may
be quite useful in solving the problem

Here are five ideas that can help us in setting goals for math
education (Moursund, 2018a, link).

*Problem solving*lies at the heart of mathematics and math education. Math is an important aid to representing and solving problems in many different disciplines. Schools have long included a focus on reading across the curriculum. This needs to be expanded to*mathing*and problem solving across the curriculum.- Each discipline of study makes progress by
*building on the previous work of others*. This is especially true in mathematics, because mathematicians develop and prove math theorems that endure over the ages. You might wonder, how many math theorems are there? This question is discussed on the*Ask a Mathematician/Ask a Physicist*website (The Physicist, 11/23/2012, link). The answer provided is that there are many millions and perhaps an infinite number. This suggests a serious question. How many theorems do students need to learn at various grade levels, and which ones? Many teachers and some standardized tests provide students with a “cheat sheet” list of often used math formulas. A sophisticated calculator or a computer can be a substitute for memorizing many theorem, formulas, and algorithms.*Just look it up on the Web*and have the computer carry out the necessary calculations and symbol manipulations. *Math fluency and sense making*is being able to read, write, speak, listen, think, and understand (make sense of) communications in the language of mathematics or that include some math. This is somewhat akin to developing fluency in a natural language.*Math maturity*is being able to make effective use of the math that one has studied. It is the ability to recognize, represent, clarify, and solve math-related problems using the math one has studied. Thus, a fifth grade student can have a high or low level of math maturity relative to math content that one expects a typical fifth grader to have learned.*A good math education*helps students to develop competence in 1 to 4 given above. Information and Communication Technology (ICT) provides us with steadily growing teaching, learning, and doing aids to achieving 1 to 4.

Many of the math-related problems that people encounter in various disciplines—particularly in the sciences and in all areas using statistical analysis or large databases—require an amount of computation that is beyond what can be done by hand. Just think of a company such as Amazon trying to deal with many millions of customers and millions of different products by using a paper-and-pencil system!

Final Remarks

It is easy to see why math is such a significant part of the PreK-12 school curriculum. While essentially all humans have the ability to learn and use a significant amount of mathematics, people vary widely in both innate ability and interest in mathematics.

For more than 5,000 years, schools have struggled to determine what math to teach and how to teach it both effectively and efficiently. The National Assessment of Educational Progress is a test in the United States that has been used since 1969, with graphs of results from 1971-2012 available online (NAEP, 2012, link). Data from more recent years are also available online (NAEP, 2018, link). Roughly speaking, 9-year-olds and 13-year-olds have made some progress in math scores over the years, but 17-year-olds have not. The United States tends to be in the middle of the pack in terms of international assessments.

Computers are a major change agent, both in math and in many other disciplines. The next newsletter in this series discusses some possible roles of computers in the teaching, learning, and doing (using) mathematics.

References and Resources

Goldman, B. (April, 2013). Scientists pinpoint brain's area for
numeral recognition. *Stanford Medical*. Retrieved 12/19/2018
from https://med.stanford.edu/news/all-news/2013/04/scientists-pinpoint-brains-area-for-numeral-recognition.html.

Moursund, D. (2018a). Improving math education. *IAE-pedia*.
Retrieved 12/20/2018 from http://iae-pedia.org/Improving_Math_Education.

Moursund, D. (2018b). *The fourth R (Second edition)*.
Eugene, OR: Information Age Education. Retrieved 12/17/2018 from http://iae-pedia.org/The_Fourth_R_(Second_Edition).
Download the Microsoft Word file from http://i-a-e.org/downloads/free-ebooks-by-dave-moursund/307-the-fourth-r-second-edition.html.
Download the PDF file from http://i-a-e.org/downloads/free-ebooks-by-dave-moursund/308-the-fourth-r-second-edition-1.html.
Download the Spanish edition from http://iae-pedia.org/La_Cuarta_R_(Segunda_Edici%C3%B3n.

Moursund, D. (9/30/2017). Impoving math and other education. *IAE-Newsletter*.
Retrieved 12/21/2018 from https://i-a-e.org/newsletters/IAE-Newsletter-2017-218.html.

Moursund, D. (2016a). Problem solving. *IAE-pedia*. Retrieved
12/20/2018 from http://iae-pedia.org/Problem_Solving.

Moursund, D. (2016b). What is mathematics. *IAE-pedia*.
Retrieved 12/17/2018 from http://iae-pedia.org/What_is_Mathematics.

NAEP (2018). 2017 report. *National Assessment of Educational
Progress*. Retrieved 12/20/2018 from https://nces.ed.gov/nationsreportcard/.

NAEP (2012). Trends in academic progress. *National Association
of Educational Progress*. Retrieved 12/19/2018 from https://nces.ed.gov/nationsreportcard/subject/publications/main2012/pdf/2013456.pdf.

O’Brien, T. (n.d.). George Polya. *California Mathematics Council*.
Retrieved 12/21/2018 from https://www.cmc-math.org/george-polya.

Pólya, G. (1957). *How to solve it: A new aspect of mathematical
method* (2nd ed.). Princeton, NJ: Princeton University.

The Physicist (11/23/2012). How many theorems are there? *Ask a
Mathematician/Ask a Physicist*. Retrieved 12/21/2018 from https://www.askamathematician.com/2012/11/q-how-many-theorems-are-there/.

Wikipedia (2018a). George Polya. Retrieved 12/19/2018 from https://en.wikipedia.org/wiki/George_P%C3%B3lya.

Wikipedia (2018b). History of mathematics. Retrieved 12/18/2018 from https://en.wikipedia.org/wiki/History_of_mathematics.

Wikipedia (218c). Theorem. Retrieved 12/25/2018 from https://en.wikipedia.org/wiki/Theorem.

Wonderopolis (n.d.). Who invented money? *National Center for
Families Learning*. Retrieved 12/20/2018 from https://wonderopolis.org/wonder/who-invented-money.

Author

**David Moursund** is an Emeritus
Professor of Education at the University of Oregon, and editor of
the *IAE Newsletter*. His professional career
includes founding the International Society for Technology in
Education (ISTE) in 1979, serving as ISTE’s executive officer for 19
years, and establishing ISTE’s flagship publication, *Learning
and Leading with Technology* (now published by
ISTE as *Empowered Learner*).He was the major
professor or co-major professor for 82 doctoral students. He has
presented hundreds of professional talks and workshops. He has
authored or coauthored more than 60 academic books and hundreds of
articles. Many of these books are available free online. See http://iaepedia.org/David_Moursund_Books .

In
2007, Moursund founded Information Age Education (IAE). IAE provides
free online educational materials via its *IAE-pedia*, *IAE
Newsletter*, *IAE Blog*, and IAE books.
See http://iaepedia.org/Main_Page#IAE_in_a_Nutshell .
Information Age Education is now fully integrated into the 501(c)(3)
non-profit corporation, Advancement of Globally Appropriate
Technology and Education (AGATE) that was established in 2016. David
Moursund is the Chief Executive Officer of AGATE.

Email: moursund@uoregon.edu.

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