Issue Number 248 December 31, 2018

This free Information Age Education Newsletter is edited by Dave Moursund and produced by Ken Loge. The newsletter is one component of the Information Age Education (IAE) and Advancement of Globally Appropriate Technology and Education (AGATE) publications.

All back issues of the newsletter and subscription information are available online. In addition, seven free books based on the newsletters are available: Joy of Learning; Validity and Credibility of Information; Education for Students’ Futures; 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.

Dave Moursund’s newly revised and updated book, The Fourth R (Second Edition), is now available in both English and Spanish (Moursund, 2018c). The unifying theme of the book is that the 4th R of Reasoning/Computational Thinking is fundamental to empowering today’s students and their teachers throughout the K-12 curriculum. The first edition was published in December, 2016, the second edition in August, 2018, and the Spanish translation of the second edition in September, 2018. The three books have now had a combined total of more than 29,500 page-views and downloads.

ICT Tools and the Future of Education
Part 2: Math Education

David Moursund
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 Reasoning/Computational Thinking is a new basic “R” to be added to the three basic “Rs” of Reading, ‘Riting, and “Rithmetic. 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:

  1. What is mathematics?
  2. 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:

  1. You have a clearly defined given initial situation.

  2. 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.)

  3. 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.

  4. 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).

  1. 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.

  2. 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.

  3. 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.

  4. 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.

  5. 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.

Reader Comments

We are using the Disqus commenting system to facilitate comments and discussions pertaining to this newsletter. To use Disqus, please click the Login link below and sign in. If you have questions about how to use Disqus, please refer to this help page.

Readers may also send comments via email directly to moursund@uoregon.edu.

About Information Age Education, Inc.

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.