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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:
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).
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.
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:
What math should all students be required to study?
How should we assess the math learning of students, and what standards should we set?
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.
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.
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.
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