Notes (3/15/15)

  1. Yero, J. (2002). Teaching in Mind How Teacher Thinking Shapes Education. Hamilton, MT: MindFlight Publishing.
  2. Barry Fishman quote taken from “It’s Not About the Technology,” Teachers College Record, Date Published: July 06, 2006 ID Number: 12584
Chapter 1
  1. Boehmig, S. P. (2006) Pittsburgh’s South Side. Arcadia Publishing. Charleston, SC.
  2. Henry Chadwick is also credited with first devising this statistic, which caught on as a  measure of pitching effectiveness after relief pitching came into vogue in the 1900s. Prior to 1900 – and, in fact, for many years afterward – pitchers were routinely expected to pitch a complete game, and their win-loss record was considered sufficient in determining their effectiveness.
  3. Albert, J., Bennett, J. (2001) Curve Ball Baseball, Statistics, and the Role of Chance in the Game. Copernicus Books. NY. p. 2.
  4. “Programmed Learning is a learning methodology […] first proposed by the behaviorist B. F. Skinner in 1958. According to Skinner, the purpose of programmed learning is to ‘manage human learning under controlled conditions.’ Programmed learning has three elements: (1) it delivers information in small bites, (2) it is self-paced by the learner, and (3) it provides immediate feedback, both positive and negative, to the learner. It was popular in the 1960s and through the 1970s, but pedagogical interest was lost in the early 1980s as it was difficult to implement and its limitations were not well understood by practitioners. It was revived in the 1990s in the computerized Integrated Learning System (ILS) approach, primarily in the business and managerial context.” Source:
  5. TEMAC Programmed Learning Materials. (1962) Encyclopedia Britannica Press. Chicago IL.
Chapter 2
  1. I don’t remember the title of the book. It probably was his “Abstract Algebra” book that was published in 1959. Dr. Sawyer did write a book about Linear Algebra in 1972 from which I quote: “The preparation of this material was undertaken because no published books seem to meet the needs of the first-year engineers. Some books were mathematically overweight; in order to prove every statement made, long chains of propositions were included, which served only to exhaust and antagonize the students.”
Chapter 3
  1. Pelky, J. (2007) Entrepreneurial Capitalism and Innovation:  A History of Computer Communications 1968-1988. Chapter 1 Data Communications: Emergence 1956-1968 Modems and Multiplexers.
Chapter 5
  1. Although [Mary] Dolciani is not well known by the general public, she was influential in developing the basic modern method used for teaching basic algebra in the United States. Source:
  2. Madison Math was a curriculum developed by Robert Davis at Syracuse University piloted at Madison Junior High School in Syracuse where it began in 1957, to explore how learning of mathematics works, which led to his 1966 book “Discovering Mathematics: A Text for Teachers.” Don Cohen (nicknamed the Mathman) worked with Bob and was one of the Madison Project’s chief advocates and also led the Saturday Madison math sessions at PS 41 from 1966-1972. His twitter handle is: @themathman. See blog entry.
  3. Links for Pick’s Theorem: Henri Picciotto’s blog.
  5. Clipped from my blog entry (Dynamic Math Classroom #60 Nov 26, 2012.)
  6. Harold Jacobs. (1972) Mathematics: A Human Endeavor. I used edition 1. (Available for $.01)
  7. Columbia Prep School Newsletter “Authors on the CPS Faculty” by Barbara Ash, September 10, 1974. Quote from that article: “This September [1974], Mr. Charischak is beginning his 4th year as mathematics instructor for seventh and eighth graders. For Mr. Charischak the 1974 to 1975 school year also marks the first official seventh and eighth-grade mathematics program in sometime. This new program represents a synthesis of his own ideas as well as techniques borrowed from the math textbooks with which he has been especially pleased. Through his work with the Madison Project for mathematics teachers and his frequent unfortunate experiences with many texts, Mr. Charischak has developed the criteria for a successful program for seventh and eighth-grade math students. His basic ideas include the following:
  • Mathematics books for junior high school students must be composed in a style that can be appreciated by that age group.
  • The process of learning mathematics, as with other subjects should be one of discovery.
  • The concepts and rules of math must relate to reality as much as is possible. To accomplish these goals, Mr. Charischak emphasizes the use of visual and manipulative aids, along with well-written reading material which can be enjoyed by seventh and eighth graders. He plans to spend more time compiling his ideas and eventually hopes to publish his work on the development of a successful mathematics program for junior high school students.” I never did publish it.
Chapter 6
  1. Maeroff, G. (1975) Junior High is not Easy to Handle. Lakeland Ledger. Lakeland, FL.
  3. The eight students who participated in my Codes and Ciphers activities in 1977 all had above average skills in everyday math. I wish I’d had some less skilled students.  They could have revisited skills they’d had in another context (normal classroom teaching) and might have been able to overcome the difficulties they’d had with them. If I were doing the activities today I would have so many more choices in delivering them.  A Google search leaves me breathless at the possibilities.  Here’s a book I just had to order from the UK. “Secret Breakers – The Power of Three.” Not because I expect to find a lot of math here, but because it is advertised as a DaVinci Code for kids. What a rich source to embed some mathematical exploration.
  4. The open-classroom movement originated in the British public elementary schools after World War II. The movement, known then as informal education, spread slowly to the United States. In 1967 a parliamentary commission headed by Lady Bridget Plowden published a report, “Children and Their Primary Schools,” that promoted open education in all British schools. American educators who visited British schools during the late 1960s had read the Plowden report and visited classrooms where informal education dominated teaching and learning. They viewed informal education–or, as they came to call it, open classrooms or open education–as an answer to both the American education system’s critics and the problems of U.S. society. Source:
  5. Cursor Magazine CURSOR - Programs for PET Computers was the name of an early computer-based "magazine" that was distributed on cassette from 1978 and into the early 1980s.
Chapter 7
  3. The original source for this chart is long since forgotten. You can see a similar chart here: Smith, S. (2009) Craps … A Casino Game of Pure Chance.
  4. My Scratch program.
  5. Soloway, E., Lochhead, J. and Clement, J. Does Computer Programming Enhance Problem Solving Ability? Some Positive Evidence on Algebra Word Problems. In Computer Literacy, edited by Robert J. Seidel, Ronald E. Anderson, and Beverly Hunter, 171-185. New York: Academic Press, 1982.
Chapter 8
  1. Antonia ("Toni") Stone created the Playing to Win Computer Center, which was the first Community Technology Center (CTC), at Union Settlement Association's Washington Houses Community Center in East Harlem in the early 1980's. Video by S L Productions.
  2. The Fortune Society is a nonprofit social service and advocacy organization, founded in 1967, whose mission is to support successful reentry from prison and promote alternatives to incarceration, thus strengthening the fabric of our communities. Source:
  3. Toni Stone’s obituary:
  4. New York Area Educators Note: “Computers and software for classroom use can be tried out by New York area teachers and school administrators at Teachers College/Columbia University's new Microcomputer Resource Center, the first program of its kind in the Northeast. The Microcomputer Resource Center, which opened November 1, 1980 is a free service to educators confronted with the sudden popularity of computers in elementary and secondary schools. It features three microcomputers frequently purchased by schools and a cassette library of educational programs written by local teachers as well as those published by computer companies. Among the specially designed materials are a baseball game that pitches arithmetic problems, a bowling game that teaches decimals, and a dart game that is scored by a student's speed in estimating round numbers. Karen Billings, director of the Microcomputer Resource Center, explained that it was organized because computers, already in homes and business, are coming to the field of education, and teachers need a place to learn about them. “Many schools began purchasing microcomputers about two years ago,” she continued, “when technology reduced the price and size of computers to $2,000 and less for a tabletop model.” Although originally acquired for mathematics classes, the machines are now being used for all academic subjects; simulated chemistry experiments and geography drills, for example, are on cassette in the Center library. A qualified staff member, who already has taught with computers in his or her classroom, is on hand at the Microcomputer Resource Center to introduce the novice to the equipment. Teachers experienced with computers also are welcome to experiment with materials and meet with colleagues interested in improving microcomputer services in their schools. The Center contains a growing collection of books and periodicals relating to computers in education.” Compute! Magazine Issue 003.March, April, 1980.
  5. Papert , S. (1980) Mindstorms: Children, Computers, and Powerful Ideas. Basic Books. NY.
  6. Ibid. p. viii
  7. Tammet, D. (2013) Thinking In Numbers: On Life, Love, Meaning, and Math. Little Brown & C. (p. 136)
  8. Charischak, I. (1988). Creating Dynamic Stories with LogoWriter. Dynamic Classroom Press. White Plains, NY.
  9. CLIME is currently named “Council for Technology in Mathematics Education” and has been a affiliate group of NCTM since 1988. Its current publication is “CLIME Connections.”
  10. Newsletter “Scenes from a Dynamic Classroom – featuring LogoWriter and LEGO TC Logo.” (1989-1990) Current version is the blog titled “Math 2.0: Scenes from a Dynamic Math Classroom”
Chapter 9
  2. CIESE model for lesson implementation in a nutshell: Step 1: Set the stage. Engage the students. This is more than just a simple anticipatory set. It is about engagement. Step 2: Do the activity. My setting the stage was leading up to something. This is where you do the “something.” (Example will follow later in the chapter.) Step 3: Debrief. Ask the students what they learned today. It can take 5 minutes. This part is critical. Otherwise the point of the lesson (activity, etc) may be lost in the shuffle of the kids moving on to the next activity or class.
  4. Steinberg, L. (1996) Beyond the Classroom. Simon & Schuster. NY. Steinberg discusses this “lack of engagement” problem throughout the book.
  5. Gardner, H. (1991) The Unschooled Mind. BasicBooks. p. 150.
  6. Papert, S. (2000) What’s the big idea? Toward a pedagogy of idea power. IBM SYSTEMS JOURNAL. Vol 39, Nos 3 & 4.
  8. Family Fractions activity in Elizabeth
Chapter 10
  1. NCTM. (2000) Principles and Standards for School Mathematics. p. 24-25
  2. Guiding Principles for Mathematics Curriculum and Assessment.
  3. An update to the 2000 principles and standards was published in Feb, 2014 called “Principles to Actions – Ensuring Mathematical Success for All.” In this update the technology principle became Tools and Technology and included the use of manipulatives such as Geoboards and Cuienaire Rods.
  4. Source: describes a webinar titled “Where is Math 2.0?”
  5. As of August 2, 2012 Key Curriculum was acquired by McGraw Hill. See
  7. Source: “Focus in High School Mathematics Technology to Support Reasoning and Sense Making.” Chapter 5 Simulations as a Path to Making Sense of Probability, p. 69.
  8. Devlin, K. Blog entry. What is algebra?
  9. I took a BASIC program Popshot from the CURSOR magazine collection in 1980 and turned it into Place Value Popshot which helped my 6th grade students with reviewing their place values in decimal numbers in a fun way.
  10. Leron, U. (1985). Logo today: Vision and reality. The Computing Teacher, 12,26-32. Uri describes the tension caused by Papert’s initial insistence of Logo being taught free form without teachers and curriculum and those that felt that the teacher’s role in Logo environment was important as was developing curriculum materials that would support Paperts notion of powerful ideas. Papert did eventually concede that when writing “Mindstorms” he didn’t take into consideration enough the needs of teachers.
  11. In Mindstorms Papert gave his first formal definition of a microworld as a: “...subset of reality or a constructed reality whose structure matches that of a given cognitive mechanism so as to provide an environment where the latter can operate effectively. The concept leads to the project of inventing microworlds so structured as to allow a human learner to exercise particular powerful ideas or intellectual skills. ”(Mindstorms, p. 204)
  12. Steve Hargadon’s article
  13. Solomon, G. (2007) web 2.0 – new tools, new schools. ISTE. p. 2
  14. Dan Myer’s blog.
Chapter 11
  1. David Thornburg (2013) From the Campfire to the Holodeck: Creating Engaging and Powerful 21st Century Learning Environments. Jossey-Bass.
  2. Roger Schank, Teaching Minds: How Cognitive Science Can Save our Schools, Chapter 8. pgs 89-90.
  3. The idea behind Fraction Darts is an old one. Darts was created in 1973 with support from National Science Foundation. The authors were Sharon Dugdale and David Kibbey. Many versions of the program have appeared since. The version I'm currently using was written in Flash in 2005 by Jason Sayres for CIESE.  Ricky Carter and Robert Berkman collaborated with me on a Logo version of Darts that I used in my workshop for years. Jason Sayers version is easier to use and looks spiffier but it’s a closed microworld, Access to the HTML part of the code allows for some modification e.g. specific placement and size of the balloons. Altering it further would require knowledge of Flash. It would be great if someone wrote an open source version in Scratch (or equivalent) that allows for altering the code.
  4. See for a proof that the midpoint between two fractions is the average of the numerators divided by the average of the denominators.
  5. Graphing Equations and Green Globs.
  6. Gewirtz, C. (2011) Curriculum Definition Raises Red Flags. Education Week.
Chapter 12
  1. Zoombinis -
  2. Gary Stager. Learning Adventures: A new approach for transforming real and virtual classroom environments. A paper written for the ACEC 2008 conference in Perth, Australia.
  3. From the song lyrics “Gee, Officer Krupke” in West Side Story.
  4. Here’s a typical assessment of math students knowledge of math in the 4th grade. According to Conceptua Math in their youtube video ( when students hit 4th grade, many begin to lose confidence in their ability to do math. Like most vendors Conceptua math has something special that they believe flattens the fractions barrier for most students and will prep them properly to pass the tests. But how well do they do that is still an open question in my mind.
  5. “Given what we know about learning, how can new technological tools help promote great teaching and learning?" The good news and bad news about technology and learning are one and the same. Schools have not yet begun to systematically tap learning science through technology to deepen, accelerate, and nurture learning. The "bad" here is obvious. So what's the “good” news? It’s that, since we mostly haven't figured out the right way to put things together, we're in a position to make enormous progress by tapping emerging tools and technologies the right way. Hess, F. Breakthrough Leadership in the Digital Age: Using Learning Science to Reboot Schooling, p. xiii.
  6. Xerox, 1970. The Weird Number. The Fractionville activity I wrote about in Chapter 9 was inspired by this video.
  7. Read more about the Hippasus story in Brian Clegg’s “A dangerous Ratio”
  8. A professor of math education used the Weird Number video as a motivator for a lesson development assignment by his students. Here’s what they came up with.
  11. This part is based on an article I wrote for ISTE (International Society for Technology in Education) entitled “In the Spirit of Eratosthenes: Measuring the Circumference of the Earth” in 1998.
  13. According to Carl Sagan’s book “Cosmos” Eratosthenes hired a surveyor to pace the distance from Syne to Alexandria approximately 800 kms.
Chapter 13
  1. Papert, S. (1980) Mindstorms. Basic Books. p. viii
  2. Barry Fishman quote taken from “It’s Not About the Technology,” Teachers College Record, Date Published: July 06, 2006 ID Number: 12584