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America.
A Simple (But Very Challenging)
Math Education Question
David
Moursund
Professor Emeritus, College of Education
University of Oregon
“Mathematics is the gate and key of the
sciences.... Neglect of mathematics works injury to all knowledge,
since one who is ignorant of it cannot know the other sciences or the
things of this world.” (Roger Bacon; English philosopher and
naturalist; 1214-1294.) [For more math education quotations see
Moursund, 2017b.]
Recently I received the following email message from my long-time friend Bob Albrecht:
Ahoy Dave,
What if you were a 1st-grade teacher and you could have only one dice game.
What dice game would you choose?
Bob
Bob Albrecht and I go back a very long way together. In 1979, he played
a pivotal role in my establishing the International Council of
Computers in Education—which eventually became the International
Society for Technology in Education (ISTE). In more recent years, we
wrote two books together and collaborated on a number of other writing
projects (Moursund & Albrecht, 11/27/2011; 9/2/2011). For more
information about Bob see Moursund (2017c) and the tail end of this IAE Newsletter.
Dice Games
Games involving the use of dice go back a very long way. Perhaps
the first dice were sheep knuckle bones that had four flat sides.
Evidence of use of such dice goes back at least 7,000 years.
Eventually, someone got the idea of cutting off the rounded ends of the
knuckle bone, making a six-faced die (Carr, n.d.)
Dice are a fun and very useful math manipulative. My recent Internet
search indicated that one can purchase dice in bulk for about eight
cents apiece. They were 10 for a dollar at my local Dollar store. At
that price, it is inexpensive to provide a collection of dice for
children at home and at school to use for play and for learning.
My Internet search of dice games for first graders produced about 3.2
million results. Needless to say, there are a huge number of dice games
suitable for use at the first-grade level. Bob Albrecht has contributed
considerably to the field of dice games in education. See two of his
dice game books at (Albrecht, 2/21/2017; 4/19/2016).
Responding to Bob Albrecht’s Question
My quick, off the top of my head response to Bob Albrecht’s question is
that I do not have a favorite dice game for use by first graders.
However, I can suggest things to think about as you and others ponder
the question.
For example, I assumed Bob was thinking about roles of dice in helping
first grade students learn about math. Of course, there are other
possibilities. One can study the history of dice. Dice are made of many
different types of materials, come in a variety of colors, and some
dice are works of art. So, one can study the materials used to make
dice, and dice as art objects. Dice also can be used as building
blocks, to make various two and three-dimensional objects. Dice come in
many geometrical shapes, including the five Platonic polyhedra
(tetrahedron, hexahedron, octahedron, dodecahedron, and icosahedron).
However, let’s stick to math. My first question to myself was, “What
are the goals of math education in the first grade?” Immediately I saw
that I was in over my head. I have never taught first grade. Of course,
I was a first grader at one time, and I helped to raise four children
who all went to public schools starting with kindergarten.
Also, I helped to create a home environment for my children that
included math in general, and lots of board games, card games, and
other types of games. Games can play an important role in children
learning to interact with each other, cooperate and compete with each
other, learn some rules, learn to follow the rules, and learn to deal
with winning and losing. There is also the task and responsibility of
getting the game out of its container, setting the game up, putting the
game pieces back in its container, and storing the container in a place
where it can later be found.
These social skills are very important for children to learn—indeed,
they are a critical part of a good education. Thus, one measure of a
good dice game to use in the first grade is the extent to which it
contributes to these very important social aspects of education at that
grade level.
Goals of Math Education
Math is both a broad and deep discipline of study in its own
right, and is also an aid to representing and solving problems in other
disciplines. It is part of our everyday lives. So, one goal in math
education is for students to gain the math knowledge and skills needed
for life in our society.
Consider very young children who are learning to communicate orally in
their native language(s). (Here, I am assuming the children do not have
severe physical problems that impair speech and hearing.) Many children
grow up bilingual and some grow up trilingual. Humans have a great
capacity to learn languages. It is a huge learning achievement to move
from hearing and making sounds to understanding oral communication. It
is a huge achievement to develop such skills to a level where one can
carry on a “meaningful” communication, and use this communication to
help represent and solve problems and as aid to further learning.
Math is a very important part of every natural language. One goal of
math education is to learn to communicate in the language of
mathematics at a level commensurate with the level that one is learning
oral communication in the other parts of one’s native language(s)
(Moursund, 2017a).
In math, we call this developing number sense. Quoting from the Wikipedia (n.d.):
In mathematics education, number sense
can refer to "an intuitive understanding of numbers, their magnitude,
relationships, and how they are affected by operations". Other
definitions of number sense emphasize an ability to work outside of the
traditionally taught algorithms, e.g., "a well-organized conceptual
framework of number information that enables a person to understand
numbers and number relationships and to solve mathematical problems
that are not bound by traditional algorithms".
…
There are also some differences in how number sense is defined in math
cognition. For example, Gersten and Chard say number sense "refers to a
child's fluidity and flexibility with numbers, the sense of what
numbers mean and an ability to perform mental mathematics and to look
at the world and make comparisons."
Dice Games in Education
Dice games have these three characteristics:
Many people find them to be fun (intrinsically motivating).
They involve working with numbers that are in a coded form (dots on dies), oral form, and perhaps in a written form.
A player is actively involved, using numbers, perhaps doing mental arithmetic, making decisions, and so on.
Of course, there is much more. A student who is repeatedly rolling a
pair of dice will likely detect that some totals come up more
frequently than others. Aha! This student is at the start of learning
about probability, one of the important branches of mathematics. This
student may begin to make a written record of the outcomes when a pair
of dice are rolled many times. Another Aha! This is a type of research.
It involves collecting experiential data and then trying to draw
conclusions by analysis of the data. Similar research can be carried
out using eight-face or other polyhedral dice.
If we add in the social benefits of playing games with other people,
you can see that we have put together a strong argument for using dice
games in education.
However, I want to make a few more observations.
When a group of students are playing a game together, they are
communicating with each other and helping each other. Each student is
both a learner and a teacher. Although they are all playing the same
game, the overall game-playing process creates an environment of
individualization. For example, students can play with widely varying
levels of previous experience, performance, and understanding of what
they are doing. This is not a “one size fits all” type of education.
Teachers who want to make more use of games in their teaching
need to have clear learning objectives in mind. For example, you might
want students to learn to keep written records of their moves or
decisions. You might want students to observe that when rolling a pair
of dice, some totals come up more frequently than others. You might
want students to improve their skills at glancing at two six-faced dice
and giving their total from rote memory rather than laboriously
counting the pips on the two dice faces. Student learning can be a
combination of discovery-based learning, learning from each other, and
learning that you specifically direct.
This type of education typically does not include written, oral,
or other type of performance tests. A game winner does not get an “A”
while those who do not win get lower letter grades.
Many games can be played in a solitary manner. For example, I
like to play various forms of solitaire card games and computer games.
Typically, such solitaire games lack the social education learning
experiences that are such a valuable part of playing a game in a group.
More About Bob Albrecht
Throughout his professional career, Bob Albrecht has been a
pioneer in the field of computers in education and a prolific author.
Quoting from Jon Cappetta’s interview of Bob Albrecht (7/9/2015):
Jon Cappetta: What do you think sparked your interest in computers?
Bob Albrecht: Well, let’s see; it began in 1955. After going to college
for quite a few years, I finally quit halfway through a master’s degree
and went to work at Minneapolis Honeywell Aeronautical Division in
Minneapolis.… At first, I worked on analog computers there—REAC analog
computers.…Then upstairs [from the room I worked in] they got an IBM650
computer. One day after I had been there for three or four months my
boss called me in and said that he would like me to go upstairs and
learn how to use that computer. Once I learned how to use it he then
wanted me to spread the word down where we were. So that was my
introduction to computers, an IBM650 drum computer. The memory was a
spinning drum [coated with material like used on magnetic tape]. We
used punched cards for input and such. So that was my introduction to
computing.
…
Jon: Can you further elaborate on your experience of teaching and computing in the early 1960s?
Bob: So, in 1962 I began to teach high school students. Some of whom
are well known now such as Randy Levine and Bob Kahn, both of whom were
in the first group of students that I taught Fortran to in the Control
Data office. I talked the University of Colorado Denver Center into
going for a National Science Foundation grant. Control Data then
provided a 160A, which would run Fortran paper tape, punch a paper tape
on a flexiwriter and feed it in. So, we ran that. My students were the
teachers; so, they taught students and teachers in the evening classes
under this NSF grant.… We picked up the 160A, moved it into George
Washington High school, and for an entire day my students ran
demonstrations for different classes that were brought in.
For more about Bob Albrecht, and especially about his interest in using games in education, see his IAE-pedia entry (Moursund, 2017c).
Final Remarks
Games are not the answer to
improving education. However, appropriately used, games can make a huge
contribution to improving both informal and formal education. Here is
my mangled version of a famous quotation from Shakespeare:
All the world’s a game,
And all the men and women active players:
They have their exits and their entrances;
And all people in their time play many parts.
(Dave Moursund—Adapted from Shakespeare.)
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.
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.
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 Executuve Officer of AGATE.
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