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The first issue of this newsletter was published in August, 2008.
Bob Albrecht’s Free Book: Dice Probabilities & Statistics 01
David
Moursund
Professor Emeritus, College of Education
University of Oregon
“Play is the work of the child.”
Friedrich Froebel, Maria Montessori, Jean Piaget, and others.
“When tools become toys, then work becomes play.” (Bernard Louis
"Bernie" DeKoven; American game designer, author, lecturer and fun
theorist; 1941- .)
As a child, I learned to play many different kinds of games. I liked
board games, card games, checkers, chess, marbles, Ping-Pong, and
others. I grew up in a safe and friendly neighborhood, and played a
variety of neighborhood games such as kick-the-can, hide-and-seek and
outdoor sports like football, basketball, and softball.
During a recent vacation trip to California, I had the opportunity to
play Monopoly with my
granddaughter (age 12) and others. This brought back fond memories of
playing Monopoly and other
games that involved rolling dice, acquiring money, buying and selling,
and making decisions. I was surprised that I remembered that a hotel on
Baltic produced a rent of $250 and a hotel on Mediterranean a rent of
$450.
But, I was also somewhat surprised at how rusty I have become in
glancing at a pair of dice, “seeing” the total, and then “jumping” my
piece forward that many spaces without counting them out. My automaticity at such tasks had gone
down through lack of use.
At the very earliest grade levels, students can learn to recognize at a
glance the number of objects in a small collection. This is called subitizing. For example, a student
can learn to recognize the patterns of the pips on dice. In much the
same way that a person sees the numeral 5 and mentally says five, a
student can recognize the pattern of five pips on a die and mentally
say five.
Students are also taught to do simple additions by counting on. For example, suppose I
roll a pair of dice and the outcome is a six and a three. I subitize the six, and then count on
(mentally or out loud, saying seven, eight, nine) to get the total.
This is more efficient than subitizing the three and counting on 4, 5,
6, etc. Eventually, I develop skill in subitizing the total of a pair
of dice, and in quickly mentally solving other simple counting tasks.
Subitizing is also useful in making rapid estimates of the number of
objects in a larger group. See the short video at https://www.youtube.com/watch?v=C-O_5rk1ydo.
Games such as Monopoly
provide an excellent environment for gaining skill in automaticity, or recognizing the
number of objects in a small group and counting on (start with the
largest group you can subitize, and count on from there for other
groups in the overall collection of objects).
I also noticed that my automatic recall of the probabilities of rolling
a seven or a double was better preserved. Probability is an important
component of math, and games such as Monopoly provide an excellent
environment for learning about probability.
This game playing caused me to reflect on the important role that board
games played in my childhood education. I can mentally compare the
roles these face-to-face games played for me versus the roles that
current electronic games play in the lives of today’s children. Wow,
what a difference!
My colleague Bob Albrecht has been writing about uses of non-electronic
games in education for many years (Albrecht, 2017a). He is author or
co-author of a number of free books about gaming—specially as it
applies to math education (Albrecht, 2017b).
The next two sections of this IAE
Newsletter are copied from Albrecht’s recent 150-page book, Dice Probabilities & Statistics 01
(Albrecht,10/30/2017). This book is written for teachers and parents of
elementary school children. It is written with novices in mind, and
contains both detailed instructions and many examples. As you read the
following excerpts from Albrecht’s book, notice the emphasis on
mathematical vocabulary and dice notation. We want students to learn to
read, write, and speak the language of mathematics. It is important to
stress correct vocabulary when interacting with children.
Dice Probabilities & Statistics 01 Chapter 1: About Dice
Dice are many-faced polyhedra.
See https://en.wikipedia.org/wiki/Polyhedron).
If we could magically become elementary-school teachers just starting
out in our first classroom, our initial purchases of tools and toys for
learning and teaching math would be dice and base-10 blocks.
A regular tetrahedron has four faces. The faces are equilateral
triangles. The four faces enjoy the same shape and size – they are
congruent. Dice notation: A tetrahedral die with faces numbered 1 to 4
in an interesting way is a D4.
More about that down yonder.
A regular hexahedron (cube) has six faces. The faces are squares. Yep,
the four faces are congruent – they have the same shape and size. Dice
notation: A hexahedral (cubical) die with faces labeled 1 to 6 by pips
(dots) or numerals is a D6
(Wikipedia, 2017).
A regular octahedron has eight faces. The faces are equilateral
triangles. The eight faces have the same shape and size – they are
congruent. Dice notation: An octahedral die with faces labeled 1 to 8
is a D8.
A regular dodecahedron has 12 faces. The faces are regular pentagons.
The 12 faces are congruent – same shape, same size. Dice notation: A
dodecahedral die with faces labeled 1 to 12 is a D12.
A regular icosahedron has 20 faces. The faces are congruent equilateral
triangles. Dice notation: An icosahedral die with faces labeled 1 to 20
is a D20.
A digit die
(DD) is a die with 10 faces numbered 0 through 9. It is not a Platonic
solid. Roll a DD: Possible outcomes are the decimal digits 0, 1, 2, 3,
4, 5, 6, 7, 8, and 9. Use two digit-dice to roll percentages from 0 to
99. One digit die is the tens digit, the other is the ones digit of the
percentage. We like a silver-colored DD for the tens digit and a
copper-colored DD for the ones digit.
We think that D6s (hexahedrons, cubes) are the most familiar dice.
True? We don’t know. What do you think? Do you have D6s in your
classroom or home? A D6 has six faces labeled 1 to 6 with pips (dots)
or numerals.
D6s are used in board games such as Backgammon, Monopoly, Risk, and Yahtzee.
D6s (and other dice) are used in role-playing games such as Dungeons & Dragons, RuneQuest, and Tunnels & Trolls. Bob once
wrote a book for teachers about these games. [Adventurers Handbook: A Guide to
Role-Playing Games (1984)]
D6s are used in math games such as Roll, Pick, and Add, Number
Quest Dice Games, Number Race 1 to 12, Place-Value Games, many more. We
searched the Internet for ‘math dice games’ and got thousands of hits.
D6s are widely used: Your turn – please add your knowledge and
ideas.
Dungeons & Dragons (D&D)
was published in 1974. Soon millions of kids were playing D&D
in their neighborhoods, at role-playing game conventions, and
elsewhere. Serendipity! Millions of kids playing a game that requires
high-level thinking, understanding dice algebra, problem solving,
cooperating with other players to survive and thrive in the Dungeon
Master’s (game master’s) world. A D&D
game might go on for hours – D&D
players have long attention spans. Yeah!
D&D used many-faced dice:
D4s, D6s, D8s, DDs Digital Dice (also
represented by D10), D12s, D20s, and other dice we won’t describe in
this eBook. The creators of D&D,
Dave Arneson and Gary Gygax,
invented an elegant system of dice
notation and dice algebra.
Nostalgia: As editor of People’s
Computer Company, 1972-1977, Bob published much information
about role-playing games. He and Dave Arneson (co-creator of D&D) became friends. Dave ran a
D&D game with a Japanese
setting once a week at a school where Bob was running after-school
classes.
Dice Notation
We think D6s are the most-likely dice to be used in elhi (PreK-12)
classrooms. D6 Dice notation is displayed in Table 02.
We pilfered the bucket–of-dice image from our favorite source of math
manipulatives:
Nasco Math https://www.enasco.com/math/.
Click on ‘math manipulatives’. then click on ‘dice’.
More D6s:
3D6 are three 6-faced dice.
4D6 are four 6-faced dice.
5D6s are five 6-faced dice.
Et cetera, et cetera.
A D6 is a regular polyhedron called a regular hexahedronor cube.
A regular hexahedron (cube) has six faces.
Each face is a square.
The faces are congruent
to one another. Oops. What say? Congruent faces have the same shape and
the same size.
The faces are numbered 1 to 6 with numerals or with
pips (dots).
While
browsing elementary-school textbooks, we encountered the phrase ‘number
cube’. We conjectured that some textbook writers do not like the word
‘dice’, or perhaps they wish to add ‘number cube’ to the dice lexicon.
We found number cubes at Nasco. Aha! Number cubes have numerals 1 to 6
on their six faces instead of pips (dots). The term ‘number cube’ makes
us think of 1, 8, 27, 64, 125, and so on – instead of dice – so we
shall think of these dice as ‘numeral dice’.
• Nasco Math https://www.enasco.com/math/
NEXT: Digit dice (DDs)
Digit dice (DDs) are our favorite dice. Roll a DD: Possible outcomes
are 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. The decimal digits!
1st-grade students are learning how to add 1-digit numbers. Does that
Include 0? We include 0 in our list of 1-digit numbers. [Comment from
David Moursund: Some people make the mistake of equating the digit 0
with the word nothing. This
can be a challenge for young learners.]
1-digit numbers: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 [also known as
decimal digits]
(End of chapter
1)
Chapter 2: 1D6 Probabilities and Statistics
(excerpt)
If the possible outcomes of a 1D6 roll are equally probable (have the same probability of occurrence), then the die is a fair die.
Roll 1D6: the probability of occurrence of a possible outcome (1, 2, 3,
4, 5, or 6) is 1/6. In this eBook, 1D6 is always a fair die.
We rolled a 1D6 casino die
100 times and recorded the outcomes in Table 04 down yonder. Casino
dice are designed to be fair dice: same probability of occurrence (1/6)
for each possible outcome (1, 2, 3, 4, 5, 6). Find casino dice at
Amazon.
Go to Amazon and search for casino dice. You will find casino dice in several colors. For dice experiments, we like to use casino dice.
How many 1s did we roll? How many 2s? How many 3s? How many 4s? How many 5s? How many 6s? We counted the frequency (number of occurrences) of each outcome and made the primitive histogram shown in Table 05. https://en.wikipedia.org/wiki/Histogram.
Table 06 is a frequency distribution (https://en.wikipedia.org/wiki/Frequency_distribution)
of the outcomes of the 100 1D6 rolls up yonder. Table 06 boldly
displays the six outcomes (1, 2, 3, 4, 5, and 6), the frequency (number
of occurrences) of each outcome, and the experimentalprobability of the outcome (also known as empirical probability) shown as a fraction, a decimal, and a percent.
The least possible outcome is 1 and the greatest possible outcome is 6.
What is the mean outcome? It must be somewhere between 1 and 6,
inclusive. One way to calculate it: add the 100 outcomes in Table 04
and divide the sum by 100. Groan – that way makes us want to take a
nap. Another way: Use the data in the frequency distribution (Table
06). Multiply each outcome by its frequency and add the products. Then
divide that sum by the total number of rolls, 100 in this case. Yeah!
sum of (outcome frequency) = 1 x 14 + 2 x 20 + 3 x 21 + 4 x 13 + 5 x 15 + 6 x 17 = 346
mean outcome = sum of 100 outcomes / 100 = 3.46
Oops, you can’t roll 3.46 with a D6! That’s OK – the mean of a set of
numbers can be any real number between the least number and the
greatest number, inclusive.
The mathematical theory of probability
has its roots in attempts to analyze games of chance by Gerolamo
Cardano in the sixteenth century, and by Pierre de Fermat and Blaise
Pascal in the seventeenth century (for example the "problem of points").
We can now teach some important aspects of probability in elementary
school. Dice and dice-based games provide a hands-on, interesting, and
challenging environment for young minds.
Recently I went to a Dollar Store and purchased 10 dice for a dollar.
What a bargain. With the aid of a skillful parent, teacher, or
siblings, young children can gain insight into one of the most
important components of the discipline of mathematics! They can learn
to pose and explore fun and challenging math problems.
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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