Information Age Education
   Issue Number 225
January 15, 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) 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.

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.

Read about dice on the Internet: Wikipedia Dice https://en.wikipedia.org/wiki/Dice 

Most of the dice we use are regular polyhedra known as Platonic solids.

Table 01 Platonic solidWikipedia https://en.wikipedia.org/wiki/Platonic_solid

Table 1

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.

DDA 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.
  • D6D6s 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 gambling games such as Craps https://en.wikipedia.org/wiki/Craps.
  • 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.
D&D’s dice notation was adopted by creators of other role-playing games such as RuneQuest, Tunnels & Trolls, many more. Go fish:
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.

Table 2

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 hexahedron or 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.
Congruence (Geometry) https://en.wikipedia.org/wiki/Congruence_(geometry)
  • The faces are numbered 1 to 6 with numerals or with pips (dots). 
Numeral DiceWhile 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!

Digit Die

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]
Die Rolling Note

(End of chapter 1)

Chapter 2: 1D6 Probabilities and Statistics (excerpt)

Rolling 1D6 is a probability event https://en.wikipedia.org/wiki/Event_%28probability_theory%29

A probability event has one or more possible outcomes  http://en.wikipedia.org/wiki/Outcome_(probability).

Probability event: Roll 1D6

Roll 1D6 Probability

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.

Table 3

Casino DiceWe 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. 

•    Amazon https://www.amazon.com/gp/gw/ajax/s.html

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.

Table 4

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 5

Dice Data Note

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 experimental probability of the outcome (also known as empirical probability) shown as a fraction, a decimal, and a percent.

•    Empirical probability https://en.wikipedia.org/wiki/Empirical_probability

Table 6

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.
Ahoy Teacher

(end of section copied from chapter 2)

Final Remarks by David Moursund


Quoting from the Wikipedia https://en.wikipedia.org/wiki/Probability_theory:

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.

References and Resources

Albrecht, R. (2017a). Robert Albrecht. IAE-pedia. Retrieved 11/12/2017 from http://i-a-e.org/free-books-by-bob-albrecht.html.

Albrecht, R. (2017b). Free eBooks by Bob Albrecht. IAE-pedia. Retrieved 12/28/2017 from http://i-a-e.org/downloads/free-ebooks-by-bob-albrecht.html.

Albrecht, R. (10/30/2017). Dice probabilities & statistics 01. Eugene, OR: Information Age Education. PDF File: http://i-a-e.org/downloads/free-ebooks-by-bob-albrecht/298-dice-probabilities-statistics-01-1/file.html. Microsoft Word File: http://i-a-e.org/downloads/free-ebooks-by-bob-albrecht/297-dice-probabilities-statistics-01/file.html.

Moursund, D. (2017). Improving math education. IAE-pedia. Retrieved 12/31/2017 from http://iae-pedia.org/Improving_Math_Education.

Moursund, D. (2016). Learning Problem-solving Strategies Through the use of Games: A Guide for Teachers and Parents. Eugene, OR: Information Age Education. Microsoft Word 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. PDF 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. (11/27/2011). Using Math Games and Word Problems to Increase the Math Maturity of K-8 Students. Eugene: OR: Information Age Education. Microsoft Word File: http://i-a-e.org/downloads/free-ebooks-by-bob-albrecht/218-using-math-games-and-word-problems-to-increase-the-math-maturity-of-k-8-students-2.html.  PDF File: http://i-a-e.org/downloads/free-ebooks-by-bob-albrecht/219-using-math-games-and-word-problems-to-increase-the-math-maturity-of-k-8-students-3.html.

Free Educational Resources from IAE

Moursund, D. (2017). Free educational videos. IAE-pedia. Retrieved 7/14/2017 from http://iae-pedia.org/Free_Educational_Videos.

Moursund, D, (2017). Free open source software packages. IAE-pedia. Retrieved 7/14/2017 from http://iae-pedia.org/Free_Open_Source_Software_Packages.

Moursund, D. (2017). Open source and open content educational materials. IAE-pedia. Retrieved 7/14/2017 from http://iae-pedia.org/Free_Open_Source_and_Open_Content_Educational_Materials.

Moursund, D. (2017). TED talks. IAE-pedia. Retrieved 7/14/2017 from http://iae-pedia.org/TED_Talks.

IAE publishes and makes available four free online resources:
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. 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.

Email: moursund@uoregon.edu.

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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.