Information Age Education
   Issue Number 186
May, 2016   

This free Information Age Education Newsletter is edited 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, six free books based on the newsletters are available: 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.


Joy of Learning and Using Math

David Moursund
Emeritus Professor of Education
University of Oregon

“God created the natural numbers. All the rest is the work of man.” (Leopold Kronecker; German mathematician and logician; 1823-1891.)

“If I have seen further it is by standing on the shoulders of giants.” (Isaac Newton; English mathematician and physicist, in letter to Robert Hooke, February 5, 1675; 1642 -1727.)

Many people enjoy learning about the history of science and technology. I enjoy learning about the steady progress humankind has made in these disciplines.

The first quote given above emphasizes that the natural numbers 1, 2, 3, etc., are all around us—we grow up with them and they are part of our natural languages. The second quote indicates that humans have accumulated a tremendous amount of knowledge (in math, the sciences, and other areas) going back even before reading and writing were developed more than 5,000 years ago. We routinely use and build on this accumulated knowledge.

As two simple math examples, negative numbers were invented about 2,600 years ago and the number zero was invented a little more than 2,300 years ago (Rogers, February, 2011; Matson, 8/21/2009).

The point is that math is a vertically structured discipline, and each student is faced by thousands of years of progress that math researchers have contributed. We expect grade school children to learn about negative numbers and the number zero—things that the leading mathematical minds of the world did not know 2,600 years ago!

What Is Mathematics?

As noted above, math is a very large, steadily growing, and vertically structured discipline of study. Here are some of the reasons it is an important discipline of study:

  1. Math includes an oral and written language that can be considered to be part of natural languages and also part of the languages of many other disciplines of study (Moursund, 2016a).

  2. Knowledge of math and skill in using it empowers people in their everyday lives. Such empowerment brings pleasure and joy. Put another way, the inability to do mental math and make mental math estimations certainly takes away some of the joy in life.

  3. Math can be used to help represent and solve problems in math and in a very wide range of other disciplines. For example, math is an indispensable component of all sciences.
For these and other reasons, mathematics is part of the basics of a modern education. Our schools require precollege students to study math year after year. Many academic programs at the college level require still more study of math.

Here are some statements that help to capture the breadth of the discipline of math. Many people find joy in using math in one or more of the areas listed below:
  • Math is an art form. Mathematicians talk about the beauty of certain math results and proofs.

  • Math provides a way to study and create patterns, and many people find such patterns have beauty or other special meanings.

  • Math is an important component of the study of music and musical instruments.

  • Math includes the study of logic and the analysis of logic-based arguments.

  • Math provides a basis for understanding the probability and statistics that are a routine part of our lives.

  • Math provides an environment in which a student can have the joy of discovery and can struggle with cognitively challenging problems.

  • Math aids in the creation of and use of measurement systems. Quoting Lord Kelvin, "If you can not measure it, you can not improve it."

  • Math is an essential component of computer modeling and simulation approaches to problem solving.

  • Math is important in computer animation, such as in videos and video games.
Some of the glory and mystery of math is presented in the NOVA video, The Great Math Mystery (NOVA, 4/15/2015).

Teaching and Learning Mathematics

Here are some questions faced by our precollege educational system:
  1. What math should all students be required to study?

  2. How should we assess the math learning of students, and what standards should we set?

  3. What aids to teaching, learning, and doing (using) math should we make available to teachers and students? Remember, adults use such tools if they find them useful and know how to use them.

  4. How can we make math education more intrinsically motivating and fun for students?
These questions are intertwined, but the next four subsections provide my thoughts on them as separate questions.

Math Requirements

At the current time, the “average” adult in the U.S. performs at about the eighth grade level in math, and does poorly relative to adults in other countries (Anderson 3/15/2016; Kornell, 11/27/2012). About 90 percent of U.S. adults have completed high school or a GED (U.S. Census, n.d.). This type of data suggests that the results of three or more years of required math coursework beyond the eighth grade may be cementing earlier math knowledge and skills, but may have little long-term effect in moving average students above that level.

Here is a different way to think about this situation. For an average person, routine life in the U.S. does not require the use of math beyond the eighth grade level. Those people having a need for a greater level of math knowledge in their vocation, avocations, and everyday life learn, use, and retain the math that they find useful.

Our math education system faces an uphill battle with many students after about the seventh or eighth grade. The math being taught simply does not seem relevant to a great many students. Many begin to claim that they hate math and cannot do math. Math classes are not a joyful part of their days!

Math Assessment

To a large extent, the content of precollege math focuses on learning to solve the types of math problems for which a student’s answer is either right or wrong. Tests are designed to determine whether students can produce a right answer under the constraints of the testing situation.

However, math is far more than producing right answers in paper-and-pencil or computer-administered test situations. Math, itself, is an authentic component of our everyday lives. Most math assessment is not authentic relative to the lives of the students being assessed (Concordia Online Education, 1/9/2013).

This topic reminds me of when I spent quite a bit of time observing teachers who were teaching elementary school math. Once I singled out a couple of third graders and asked them what time it was. They looked at the clock on the wall and told me the correct time. I then inquired about how long it would be until school ended for the day. They were not able to answer that question. They could read a clock, but they lacked an understanding of time needed to answer my question. They are missing out on some of the joy of being able to tell time and use this knowledge to plan for the future.

So it is with much of student understanding of math. By rote memory, most students reach a level whereby they can pass the tests. But, they do not reach a level at which they can understand the math in a manner that allows them to apply it to novel and challenging situations. They do not readily meet the transfer of learning challenge—moving their knowledge of math from the classroom to outside the classroom.

Some students experience joy in scoring well on math tests. However, we know that math learned through rote memory, or that passing math tests by being “taught to the test” produces little long-term retention. I believe this test-passing joy is superficial relative to the joy of understanding what one has learned and using it when the need arises in one’s everyday life.

Aids to Learning and Doing (Using) Mathematics

Beginning with the invention of the counting board and then the abacus about 2,500 years ago, humans have found it helpful to have mechanical aids to calculation (The Abacus, n.d.). This use began at a time when very few people received formal schooling, so merchants and others needed aids to do arithmetic. The abacus is still in use in some parts of the world, and bead frames are commonly used in elementary school math instruction.

A wide range of by-hand and computerized math manipulatives are now commonly used in elementary school (Moursund, 2016b). Skilled teachers can add considerable joy for their math students—as well as improved learning—through the use of such manipulatives.

Let me share another story with you. More than 30 years ago, I was teaching a computers and math course. Essentially all of the students in the course were high school math teachers. A few hours before one of the class meetings I received a copy of Wolfram Mathematica for my Apple computer (Wolfram, n.d.). This is software designed to solve math problems. I installed the software, found my old freshman calculus book, and took my computer and book to class. I then “amazed” the students (and myself) by keyboarding in problems from the calculus book, and having the Mathematica program solve them. It even did well in the “starred” (extra difficult) problems at the end of the chapter.

In essence, for many years computer users have had free access to such computer software that can solve all of the types of computational problems that students are taught in precollege math and up through the first two years of typical college math courses. What this means is that, if we wanted to, we could transform our precollege math education system so that it placed a great deal more emphasis on the non-calculation aspects of math—such as understanding and using math to represent and solve problems that one encounters in everyday life, at work, and at play.

This does not mean that learning and understanding math does not require considerable rote learning and practice. A person who is unable to mentally, rapidly, and accurately do simple addition, subtract, multiplication, and division is significantly handicapped in our world. Also, the language of mathematics has a large vocabulary and special notation. One must know quite a bit of the vocabulary of math to be able to communicate using the language of math.

We now have computer systems with sufficient intelligence to individualize and speed up this type of rote learning instruction to the needs of each student. We have substantial research evidence on the value of this type of instructional use of computers.

We also have a growing collection of carefully crafted educational games that are designed to be both fun and educationally sound (Moursund, 1/20/2016).

Making Math Education More Intrinsically Interesting

It is my contention that students find math education more joyful when they gain knowledge and skills that empower them to solve problems and accomplish tasks that are meaningful to their everyday lives both in and outside of school. In school, this means that the math they are learning or have learned should routinely be used and found useful in the other disciplines they are studying. Outside of school, this means that students should find that the math knowledge and skills they are gaining are useful aids to improving their quality of life (Moursund, 2/15/2016).

As noted earlier in this newsletter, there can be much more to math education than just “covering” the required textbook or syllabus. The “I hate math” and “I can’t do math” outcomes that occur for many students are certainly undesirable. It is my impression that this type of situation occurs much less frequently in other required parts of the school curriculum. That is certainly suggestive that we can do better. My 4/25/2016 Google search of the expression “I hate math” syndrome produced over 45,000 hits. Many contain suggestions for improving this situation.

A Book on Math Tutoring

My friend and colleague Bob Albrecht and I have written Becoming a Better Math Tutor (Moursund & Albrecht, 11/27/2011). The focus is on engaging students in ways that they find to be fun. This free book contains numerous examples and success stories.

References and Resources

Anderson, J. (3/15/2016). Americans are spectacularly bad at answering even the most basic math questions. Quartz. Retrieved 5/22/2016 from http://qz.com/638845/americans-are-spectacularly-bad-at-answering-even-the-most-basic-math-questions/.

Concordia Online Education (1/9/2013). Authentic assessment methods for mathematics. Retrieved 4/25/2016 from http://education.cu-portland.edu/blog/reference-material/authentic-assessment-methods-for-mathematics/.

Kornell, N. (11/27/2012). US math achievement: How bad is it? Psychology Today. Retrieved 5/22/2016 from https://www.psychologytoday.com/blog/everybody-is-stupid-except-you/201211/us-math-achievement-how-bad-is-it.

Matson, J. (8/21/2009). The origin of zero. Scientific American. Retrieved 5/1/2016 from http://www.scientificamerican.com/article/history-of-zero/.

Moursund, D. (2016a). Communicating in the language of mathematics. IAE-pedia. Retrieved 4/25/2016 from http://iae-pedia.org/Communicating_in_the_Language_of_Mathematics.

Moursund, D. (2016b). Math methods for preservice elementary teachers. IAE-pedia. Retrieved 4/25/2016 from http://iae-pedia.org/Math_Methods_for_Preservice_Elementary_Teachers. See the specific section on math manipulatives at http://iae-pedia.org/Math_Methods_for_Preservice_Elementary_Teachers
#Math_Manipulatives
.

Moursund, D. (2/15/2016). Improving worldwide quality of life. IAE Blog. Retrieved 4/25/2016 from http://i-a-e.org/iae-blog/entry/improving-worldwide-quality-of-life.html.

Moursund, D. (1/20/2016). Learning problem-solving strategies through the use of games: A guide for teachers and parents. Eugene, OR: Information Age Education. PDF file: http://i-a-e.org/downloads/free-ebooks-by-dave-moursund/278-learning-problem-solving-strategies-through-the-use-of-games-a-guide-for-teachers-and-parents/file.html. Microsoft Word file: http://i-a-e.org/downloads/free-ebooks-by-dave-moursund/279-learning-problem-solving-strategies-through-the-use-of-games-a-guide-for-teachers-and-parents-1/file.html.

Moursund, D., & Albrecht, R. (11/27/2010). Becoming a better math tutor. Eugene, OR: Information Age Education. PDF file: http://i-a-e.org/downloads/free-ebooks-by-bob-albrecht/221-becoming-a-better-math-tutor-3/file.html. Microsoft word file: http://i-a-e.org/downloads/free-ebooks-by-bob-albrecht/220-becoming-a-better-math-tutor-2/file.html.

NOVA (4/15/2015). The great math mystery. (Video, 53:10.) Retrieved 4/25/2016 from http://www.pbs.org/video/2365464997/.

Rogers, L. (February, 2011). The history of negative numbers. NRICH. Retrieved 5/1/2016 from https://nrich.maths.org/5961.

The Abacus (n.d.). Brief history. Retrieved 4/25/2016 from http://www.ee.ryerson.ca/~elf/abacus/history.html.

U.S. Census (n.d.). Educational attainment in the United States: 2015. Retrieved 4/24/2016 from https://www.census.gov/hhes/socdemo/education/.

Wolfram (n.d.). Wolfram Mathematica. Retrieved 4/25/2016 from https://www.wolfram.com/mathematica/. A somewhat limited but very good version of this software is available for free online use at https://www.wolframalpha.com/.

Author

David Moursund is an Emeritus Professor of Education at the University of Oregon, and coeditor 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. 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 books. See http://iaepedia.org/Main_Page#IAE_in_a_Nutshell.

Email: 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.