It’s a Binary World!
Full, Active Content Version is available at www.animath.net/frbinary.html
An activity-driven version is available at: www.animath.net/frbinary-sequence.html
Partial intro lesson: http://www.animath.net/temp-lessonplan-Binary-1.html

Felipe H Razo Ph.D.

California State University East Bay
Hayward, Ca., 94544, USA

ABSTRACT

Counting and numbers, as a way to deal with the right amounts of things and their changes around us, in the best ways possible, seem to have been with us forever. For this reason, through time, we have learned to deal with the environment by imagining quantities of objects, phenomena and their changes as consisting of groups of smaller, equal-sized, similar “unit” parts, joined, grouped and ungrouped together conveniently and consistently to act together. We have learned to use helpful objects and symbols to aid us in imagining and communicating amounts and relationships. Over time, we have learned to use convenient material objects, tokens, artifacts, symbols, images, numerals and numbers organized in effective forms/order arrangements, understandings, practices and in languages, including mathematics and supported by various tools of technology.

Historically, having ten fingers in our hands, readily available to help describe, communicate and manipulate quantities and numbers seems to have motivated our preference for counting subtance as items in small and large buildup-order of groups of increasing sizes by ten. This natural circumstane has resulted in the widely used decimal, or “Base 10” counting system we know today. Recently, however, rapidly evolving electronics and industrial technologies have resulted in a shift to almost all quantity information, operations and communications not being performed in Base 10 nor directly by humans, but instead by devices and machines using a numeric system that is more efficient for the purposes of automation. This numeric system groups items and multiples in groups of two units rather than ten, and is conceptually the simplest possible. This is called the "binary" or Base 2 system.

Today, quantity and other information is overwhemingly created and managed in the Base-2 binary numeric system. Translation into and out of the decimal system is mostly done, as necessary for the benefit of traditional, historic human communications. Indeed: it is a binary world out there.

And although today there are some concerns about if and how new and more powerful concepts of counting and computing (e.g. “quantum computing”) being seen to transform the world of binary technologies, the question is: How proactive and consistent do we need to be in educating ourselves and our children to teach us and incoming generations learn about and master bases other than ten? In particular the binary Base 2 applications, as effective and efficiently as possible, from the earliest ages. This article proposes this to be not only desirable and feasible, but helpful for the development of motivation and confidence in quantitative thinking and acting within future (and fast evolving) “digital societies.”

1.- INTRODUCTION

Through history, our survival has depended on our abilities to perceive, imagine, and transform the environment [1]. These abilities have been related to the intelligent use of images, symbols and numbers, to make things more efficient, safer, cheaper, faster, better. The better the representations and manipulations of objects and actions, the better we can understand, and manage them.

We use representations to help us understand, imagine and transform the environment by counting, measuring size, form and qualities of objects, groups, and actions. We measure by considering quantities as collections or "counts" of simple, separate (discrete) identifiable, identical size, consecutive “basic unit”amounts of "like substance" that are joined together to make up some unified amount or “total” quantity [3]. Through careful selection, representation and manipulation of these representations, counting and measurement can be made as accurate (free of error) and precise (reliably repeatable outcomes) as possible, so we can do the things we do better. Through convenient images, symbols, arrangements and manipulations, we can record, communicate and process information in more effective (purposeful) and efficient (cheaper, faster, and safer) ways [5].

Through history, different civilizations have designed a variety of number systems for counting, to keep track of the size or effect of things, by aggregating (putting together), disaggregating (separating) or modifying quantities of substance, using consistently various symbols and procedures to represent and imagine what is possible to stay safe and satisfy our needs and abilities. As mentioned previouly, the most successful number systems in our past seem to have taken advantage of us having five, ten fingers (digits) in our hands, readily available to represent, communicate and manipulate simple quantities.

The Ancient Egyptians (about 2,500 BC) used strokes and icons of special objects in units and groups that are multiples of ten (https://discoveringegypt.com/egyptian-hieroglyphic-writing/egyptian-mathematics-numbers-hieroglyphs/). During the old Babylonian empire (about 1,800 BC), scribes carved scratches on clay tablets to record numbers representing terms of formal commercial transactions. The old babylonian culture seemed to have focused particularly in groupings of 10, 12 and 60 units (https://blogs.scientificamerican.com/roots-of-unity/ancient-babylonian-number-system-had-no-zero/). Research in history indicates that our current scales for measuring angle degrees (30, 90, 180, 360 degrees) and time (12, 24 hours, 30 days) were influenced by the Babylonian counting systems and evidence exists that the calculation of diagonals of rectangular triangles in trigonometry was performed by the Babylonians using their 60's counting system some four thousand years ago (two millennia before the greek philosopher Pythagoras declared his triangle-sides theorem).

The Chinese (about 500 BC) used a variety of formal and informal sets of characters to represent numbers for different occasions, mostly using groupings of 10 (http://www.mandarintools.com/numbers.html). Indeed the common abacus design seems to reflect much of the structure of counting and number manipulations in ancient China.

The Greek (https://en.wikipedia.org/wiki/Greek_numerals) and the Roman (https://www.cuemath.com/numbers/roman-numerals/) number systems used elaborate combinations of symbols and letters to describe their numbers, while the Mayans (about 1,000 AD) used dots and lines, mostly to relate to moon cycles, seasons, and passage of time in general.

The Mayan counting system emphasized using the numbers 1, 5 and 20 as a basis for counting the number of days in their month. The Mayan system also refered to other time events and preceded most other systems in recognizing and explicitly including the number zero (conch,) to describe the absence of quantity (https://www.google.com/search?q=mayan+number+system&tbm=isch&chips=q:mayan+number+system,g_1:history:gmsOcjJITWI%3D,online_chips:mayan+mathematics:Lf0XMV-1uNI%3D&hl=en&sa=X&ved=2ahUKEwiZjMO3_Kr4AhUIATQIHYo_DrkQ4lYoBnoECAEQLQ&biw=1845&bih=972).

The ancient Roman system relied on letter symbols to represent groups in numbers approaching or following around 5 (V=5) or 10, such as X=10, L= 50, C=100, D=500, M=1000. The non-trivial, “additive” or “subtractive” form of placing adjacent symbols in groups of the Roman number system presented obvious shortcomings, particularly as compared with the contemporary, simpler, more effective Hindi-Arabic, base 10 or "decimal system" we are most familiar with and described next.

Eventually, and due to advantages of having ten fingers (digits) in our hands, as well as the simplicity of uniform, consistent groupings (i.e. bundling) of units and symbol usage (nine groups and a zero-null symbol) through size, number systems converged into the current decimal system. This system, which has also improved through time is still prevalent today, although mostly for common human events is called the Hindu-Arabic, or “Base 10 system”.

NOTE: A "base" in a number system is the maximum number of unit items that can be counted up in each ordered digit position, before being considered full, bundled and counted as a unit-bundle part in the next, higher order-position digit. Such positions are known as “place values”. This process to be illustrated more clearly, visually in multimedia animations inside following sections of this document.

In the Base 10 or decimal counting system , there are only ten basic symbols (1, 2, 3, 4, 5, 6, 7, 8, 9, and zero), which are used repeatedly in a series of ordered positions to signify the number of units (or "bundles") of increasingly higher value, each position (place value) to the left being ten times greater than the previous one (….10000, 1000, 100, 10, 1.0, 0.1, 0.01, ….., etc.). The quantities indicated by the stated digit in each position has therefore a value ten times greater (or smaller) than the ones following next to the right or left (i.e. ….., x1000, x100, x10, x1, ÷10, ÷100, ÷1000, …. etc).

Recently, however, accelerated developments in science and technology, particularly in electromagnetism, electronics and materials have led a shift to a number system based in only two symbols, typically represented by a zero, and a one (0, 1). Conveniently, these two symbols correspond to each of two distinct, reliably defined and manipulated physical states of electronic, magnetic and/or optical materials and devices. The system is called binary, or Base 2 number system, and it is conceptually the most efficient way of using a distinctive physical characteristic of objects to represent information [7]. For numbers beyond 1, the binary system uses combinations of its two basic symbols, arranged in sequential space-ordered positions, representing multiples of its Base 2. Similar to the decimal system, the positions represent "bundles" that increase in value from the right to the left by multiples (or fractions) of 2 (i.e. …, 32, 16, 8, 4, 2, 1, ½, ¼, ⅛, 1/16, 1/32, …..etc.).

Because of the massive gains in efficiency and speed in producing, manipulating, and communicating number and other information, binary representations are used today for practically all automated operations, with translation into the decimal system done only when human communication and participation is absolutely necessary.

Still, and in spite of its importance, relative simplicity, and potential learning benefits, today the binary system seems to be a topic absent from most general curricula of K-12 educational institutions. The question is: Why? and then, what are the consequences of this absence?

2. TEACHING DIFFERENT NUMBER BASES

Up to now, teaching to count and manipulate quantities in our elementary and high-school (K-12) classrooms has been focused exclusively on the Base 10 system. While this would have been appropriate in the past, when there was no justified alternative, how appropriate is this narrow focus for today and tomorrow's changing times and needs?

Understandably, up to now, a focus on the Base 10 system has helped educate citizens to operate competently in a decimal world. But with dramatic shifts in technology, industry, socioeconomic and even personal, home activity towards heavy reliance on automated measurement, communications, calculation and control of processes, the abilities to design, produce, and apply corresponding products and new tools can very well determine the future personal, industrial and commercial survival, success and dominance of individuals, organizations and nations.

As key quantitative abilities begin to develop (or not) in our minds and bodies during the earlier years in our lives [8], perhaps it would be helpful to reconsider our basic elementary education, boldly but carefully. Like most successful, large-scale projects, the objectives, ways and means for developing stronger foundations of basic modern number and computation thinking in our children should be opportunely provided.

What would it take, and what could it mean to add the topic of bases other than 10 in the early, elementary and high school (preK-12) curricula?

First, to promote general numeracy, proven successful approaches, such as starting with smaller numbers and repeated, multisensory counting practice should always be considered. Then, providing more opportunities to leverage student motivation for learning in context [9, 10], participative skill development [11], and community building [12] should be considered, such as proper use of available technologies to increase student involvement and understanding through multiple and more accessible media [13], visual sharing, group chanting, participation and interaction experiences. More issues related to the use of current technologies for learning and teaching can be found at: Teaching and Learning with Technology in Elementary and Secondary Classrooms (DOCUMENT, FRazo, 2010)

Following ahead are examples of teaching activities for the concepts of place value, number bases, and the Base 2 number system. Although these activities were designed and field tested with careful consideration to the various factors affecting learning, also included in the TPACK Technical, Pedagogical, and Content Knowledge frame [17], up to now there have been no opportunities for large-scale testing of these activities in order to demonstrate sufficient statistical significance of their effects. Still, the evidence gathered so far through repeated experiences [14, 15, 16] point to consistent increases in student appreciation of the process of counting under different base systems, as well as of their understanding of the invariant concepts for number systems and operations under different bases.

3. REPRESENTATION

As an example of different representations of a number in alternative groupings or bases, the number 13 is shown below, side-to-side, in corresponding place-value representations in Bases 10, 5, and 2. The same 13 items are therefore shown grouped in corresponding bundles of multiples of ten, five, and two items.

"13" in base 10 = "23" in base 5 = "1101" in base 2

Fig 1 Different Bases – Grouping 13 Items by 10’s, 5’s, and 2’s

The images and links in Figure 2 below display interactive web pages on the Internet featuring 13 items to be counted in virtual animations of counting objects (cubes or pebbles), as they are gradually counted and grouped consistently in increasingly larger size bundles. These activities can be used in standard Internet-enabled computers, interactive touch-screen panels and larger "smart boards" so as to view, access and interact with (virtual) objects and scenarios, while counting, observing, modifying and learning the ensuing patterns repeatedly, inexhaustibly, efficiently and safely.

Fig 2. Representing the number 13 with counters in groupings of 10, 5 and 2

The following link is to an image of a physical operating electronic counter, designed and built by a 10th grade student, using a common binary counter microchip and common, standard components: (binary counting electronic panel). The panel was designed and built for students to experience and internalize the counting process in binary and decinal forms, using physical electronics, industrial components.

4. FRACTIONS

The topic of fractions, typically a challenging one when introduced in the middle elementary and higher grades, could also be supported by relating them to their representation in bases other than 10. In Figure 3, the decimal number 23.675 is used to present the place value concept for fractional amounts (right of the units point), both in Bases 10 and 2. In Figure 4, an ancient clay tablet can be used to illustrate how an old civilization used highly accurate fractional representations of whole and fractional quantities in their then-prevalent Base 60 numeric system. The 2 decimal places (1/10, 1/100) in the decimal counter can represent only one-hundredth-of-a-unit (0.01) block increments, whereas in the binary counter with 3 fractional places (½, ¼, ⅛) can only represent quantities in one-eight of-a-unit block increments (0.125). The binary representation that uses only 3 fractional digits is known as binary point eight, and has been used extensively in the approximated, high-speed measurement of remote (telemetry) high voltage electrical infrastructures.

Fractions in Base 10 www.animath.net/cnt10wfracx.html	Fractions in Base 2 www.animath.net/cnt2wfracx.html For practice: http://www.animath.net/shownums1.html

Fig 3 Representation of the number 23.675 with fractional parts in Base 10 and in Base 2

To reduce the limitations fraction representations due to arbitrary place-value partitioning (e.g. “thirds”-0.333…, , pi-3.1415, all “irrational numbers”, ...) or to the limited number of digits provided by available computing devices (e.g. device available “decimal counters”, “binary word” bits, ...), more elaborate forms have been developed, such as doubling the number of digits, approximating, or multiplying numbers by a scaling factor (“logarithmic” scaling in decimal devices, "double word" or "floating point" in binary computers). These and other important topics could be made appropriate and feasible, in some simplified, illustrative form, at least for initial understanding and motivational discussion, beginning in the primary and secondary grades.

The use of place value concepts to represent and manipulate fractional quantities would show much significantly, appropriate, and ancient than suspected, as students observe and explore samples like the interactive web page of Figure 4 below, where a clay tablet, made some 4,000 years ago in current-day Irak, during the old-Babylonian Empire. The clay tablet displays symbols and calculations describing the size and relationship between sides in right triangles involving fractional parts. The numbers, are stated in a then-prevailing Base 60 number system, and correspond to a calculation commonly assumed to have been first stated as the "Pythagorean Theorem", some 2,000 years later. The buttons to the right of the page allow the students to see how the Babylonians represented the square root of the numbers 2, and 1,800 represented in the Base 60 system.

Fig 4 Fractions from our ancestors, some 3,700 years ago

Next, the activity in Figure 5 below, titled "Inter Bases Converter" lets students enter a number in Base 10, to be seen expanded, approximated as possible in available digits for two other equivalent base representations, for comparison including up to five fractional digits.

Fig 5 Equivalent Representation with fractional parts in Different Number Bases

Fraction representations are also quite useful in the binary, base-2 system. The activity in Figure 6 presents an interactive process called binary successive approximation method, which is widely implemented with “digital” circuitry to describe electrical quantities representing measurements in the environment. Measurement quantities represented in binary forms are often referred to as digital measurements. Digital measurements of from the environment (size, weight, temperature, pressure, etc.) are created by first producing corresponding electric voltages in physical devices called sensors or transducers (converters to electrical voltage) and then evaluated in special devices called digital meters.

Digital meters typically compare received voltages to some internally generated reference value, typically by a process of either accumulation (ramping) of known-size unit pulses, or by comparing with an accumulation of all, successively smaller binary fraction reference voltages within the instrument capacity range (successive approximation) while not exceeding the received voltage level. The successive approximation method, is illustrated in the interactive activity of Figure 6 below. A measuring comparison sequence begins from the left by successively comparing the received external voltage with succesive binary fractions of the internal voltage range, adding each value to the internal reference-total only while the received voltage is not surpassed, until all the binary digit fractions (½, ¼ ,1/8, 1/16, 1/32, etc.) are exhausted.

In the interactive activity of Figure 6, the electrical voltage of a measurement (sensor) source is stated through a slide-knob (left) in the first page and then, in the second page, a successive approximation process is allowed. The process shows how selecting successively smaller binary segments of voltage results in a fast measurement value. The process stops when a match condition is detected, or when all the binary fraction digits are exhausted.

Fig 6 Binary successive approximation method for electrical measurement

5. ADDITION, SUBTRACTION, AND MULTIPLICATION

Figure 7 below illustrates the process of addition and subtraction of two numbers in Base 2, step by step, using both symbols and visual virtual objects. The animated activity displays meaningful objects (pebbles) depicting digit-by-digit ordered place value joining and regrouping in traditional right-to-left order into larger bundles (carrying), similarly as it is commonly done in the traditional Base 10 system. Today, addition, subtraction and all simple and complex quantity operations are performed with numbers, and overwhelmingly with tools consisting of electromagnetic and electronic arrays and procedures (algorithms) supported by basic electronic components and assemblies.

Fig 7 Symbolic and physical addition and subtraction in Base 2

Starting in our elementary education, adding (joining) of quantities has been represented by combining (adding) corresponding digits of numbers one by one at their “place values,”creating and combining increasing size-pieces into increasingly larger “bundles” from the right-most to the left-most positions in sequence. For subtraction (taking away) procedures, calculating tools have been using an effective method called “complementation,” to represent removing substance or “undoing” quantities by adding a corresponding complementary (negative) numbers. Complementation consists in modifiying the number to be subtracted, by replacing each of its digits by their corresponding complement: The complement of a digit being the number that when added will bring the position value to its maximum.

Therefore, in base 10 the possible digit values are from 0 to 9. So the maximum number in each location (place value) is 9. The complement of a digit 0 is 9, for 1 is 8, for 2 is 7, for 3 is 6, for 4 is 5 and so on: for numbers 6 would be 3, 7 would be 2, 8 would be 1 and 9 would be 0. Replacing each and all digits in a number by their complement will yield the complement of the full number. In the binary, base 2 system, the complements for the two digits, 0 and 1 are just their opposite: 1 and 0.

In today’s “digital binary arithmetic” electronic technologies, negative numbers are created simply reversing (“flipping”) each and all the ones in it to zeros and viceversa. Flipping is llustrated in Figure 7 by using the “+/-“ button for the subtrahend. The button therefore creates the negative representation needed to subtract a number by addition using standard addition tools (“adders”.) In electronic adder circuits, flipping is done by toggling the “off” or “on” voltage for the bits held in their basic electronic (“flip-flop”) semiconductor component setup.

Due to compliance with representing positive and negative numbers, when a result is positive, binary adders ignore the propagating carry rippling over to the left bits (into a “bit bucket”,) an additional unit (+1) is added to the first bit-digit position. This is also illustrated in Figure 7.

In addition to the extraordinary speed and economy allowed by increasingly effective electronic technologies, complementing numbers make it unnecessary to provide separate, dedicated additional "subtraction” and other related circuits and algorithms (i.e. division, roots, etc.) in electronic digital binary computing.

Figure 8 below presents a step-by-step execution of the standard right-to-left symbolic process of multiplication (repeated addition) in Base 10, which is traditionally used to teach multiplication to elementary grade children. A replication of the same process with symbols (0,1) in Base 2, an then a visual animation of binary multiplication with small numbers (<31, <15).

The activity illustrates the same process using symbols and virtual pebbles and the concept of register (temporary holding number) as well as shifting, or sliding digits through place values, during the right-to-left steps of the process. Due to the extended number of digits required in binary representations, and the reduced visual screen space required by the related animation, the pebble activity uses only eight digits (bits), called “a byte”, allowing the multiplication of only two small numbers (less than 15, and 31), accumulating the products at the top (accumulator register).

Fig 8 Multiplication with pebbles in Bases 10 and 2

6. THE PAST, THE PRESENT, AND THE FUTURE

Through time, tools have been developed to make things easier, faster, cheaper, and safer. This has also been the case for the important life-needs of counting, measuring, and more and more, for controlling and transforming the environment. Evolving from sticks and stones and life in the caves, today we find everywhere massive amounts of information and tools that can make the lives of each one of us as rich as that of royalty in previous times. In figure 9 below, four older, widely used mechanical devices, including our fingers are shown, which represented and calculated numbers in base-10 .

Fig 9 Primitive mechanical devices representing numbers and automating calculations in base-10

More recently, the dramatic transition from mechanical, Base 10 number representation and computation, to a Base 2 electronic and electromagnetic world has propelled, and continues to revolutionize society. Figure 10 includes images of a Base 10 /decimal mechanical accounting machine (left), which was popular in administrative offices around the middle of the twentieth century (National Cash Register Model 32, circa 1950), the panel of an early vintage electronic Base 2 /binary digital computer (center, Digital Equipment Corporation PDP-8, circa 1965), representing the face of the sweeping electronic technologies that came to replace mechanical computing, with devices such as large scale integrated (LSI ) binary circuit semi-conductor microchips (right,) billions of times more powerful, and thousands of times smaller than their predecessors, in effect revolutionizing all facets of our industry, commerce, leisure and education.

Fig 10 Images of mechanical and electronic tools introduced during the second-half of the 20th Century to automate basic arithmetic operations

At about the middle of the twentieth Century, the NCR Model 32 "accounting machine" carried accumulated additions for numbers in some 20 totals in its mechanical gears, performing parallel additions in about one second. Today and into the twenty-first Century, binary digital processing microchips can hold billions of numbers, and perform additions within billionths of a second, while their vast storage, presentation and telecommunications capabilities are still increasing. We definitely have come a long way, and in the future we will most likely be in situations, places and with needs that we can hardly imagine, but there is no question that we need to be ready and prepared for it as much as we can be. Even today, increasingly more and more different types of devices, some extremely small, affordable and efficient are already being used to assist all kinds of old and new industrial, commercial, leisure, and common household operations. All these use almost exclusively the binary system for representation and operation of information in their applications, the result is that almost 100 percent of the computational activity is being performed automatically by electronic means and it is in the binary, Base-2 numeric system. Unfortunately, on the other hand, even the latest K-12 mathematics curriculum ("Common Core", 2012 in California ), currently in the process of implementation, does not even mention the Binary, Base-2 number system.

7. BASES OTHER THAN TEN IN THE K-12 CLASSROOM

During previous years, various activities, such as those presented in this document have been tried and tested, though informally, to investigate their potential to teach elementary children understand and appreciate number systems other than Base 10 [16]. The results have been positive. Not surprisingly, success in the use of virtual interactive multimedia activities seem to also be the result of proper attention to pedagogically proven approaches to teaching traditional standard Base 10 concepts.

Our experience is that introducing the Base 2, usually together and in parallel understandings with those traditionally focused Base 10 concepts usually increases interest and eagerness in the students to learn the basic concepts of place value and operations with the assistance of new technologies, even among lower elementary grade children. The fact that the binary-Base-2 numeric system provides the actual conceptual foundation to the design and implementation of today's digital computers and communication usually adds a healthy touch of reality and excitement.

8. CONCLUSIONS

As it is clear that the concepts of counting, measurement, place value, number operations and numeracy in general will continue to hold great importance in supporting, teaching and learning quantitative skills, it is critical that we reinforce our education on these topics, to make them more relevant and motivating.

As technologies evolve, it has also become apparent that practically all number operations in the real world are becoming automated, and as a result of technological developments, numbers today and in the future are becoming less focused on the traditional decimal, Base 10 system, but instead in the binary, Base 2 system, and in electronic or electromagnetic forms that are most appropriate for data coding, computing, storage, and telecommunication devices.

In preparing our children for a life of design, production and implementation of competitive automation solutions, it will seem therefore helpful that we increase our attention to exposing and promoting a variety of other useful, non-decimal number systems, particularly in the binary system. NOTE: more recently (21st century) much talk has emerged about something described as “quantum computing,” and poised as a technological development to potentially surpass the power of binary computing, as we know it.

Because the basic concepts involved in the binary system are shared, consistent, and simpler than those in the decimal system, it would also seem appropriate to consider reviewing the curricula, so the binary system can begin to complement, support and strengthen our teachings on basic numeracy, from the earlier grades.

As the power of digital technologies grow and become more accessible, interactive multimedia is also becoming more available as teaching tools, therefore there seems to be an urgent need to support and enrich teaching, successful traditional and new pedagogical practices. This will assist in preparing the children more effectively to be successful in the binary (or whatever) world they will be living in, and whichever technologies become most useful in their future.

BIBLIOGRAPHY

[1] Croll, E. Parkin, D., Bush Base Forrest Farm: Culture, Environment, and Development, Rutledge, New York, 1992.

[2] Smith, D. Jekuthiel, D. & Ginsburg, D. Numbers and Numerals. W. D. Reeves, 1937.

[3] Schifter; D., A Constructivist Perspective on Teaching and Learning Mathematics, Phi Delta Kappan, Vol. 77, 1996.

[4] Stevens, S., On the Theory of Scales of Measurement, Science Vol. 103, No. 2684, 1946

[5] Dantzig, T. Number: The Language of Science. Macmillan Company, 1954.

[6] Ortenzi, E. Numbers in Ancient Times. J. Weston Walsh, 1964.

[7] Shannon, C.E., A Mathematical Theory of Communication. Bell System Technical Journal, July, 1948.

[8] Elkind, D., Formal education and early childhood education: An essential difference, Phi Delta Kappan Journal. Phi Delta Kappa International, 1986.

[9] Lave J., Cognition and practice. Cambridge University Press. Cambridge, Ma. 1993.

[10] Greeno J., Situations, mental models, and generative knowledge. In Klahr & Kotovsky, Complex Information Processing, Hillsdale, NJ. 1989.

[11] Reitman, W. R., Cognition and thought, an information processing approach. Wiley and Sons, NY, 1965.

[12] Vygotski L., Mind in Society: The Development of higher psychological processes. M Cole, et. Al (Eds), Cambridge, Ma, Harvard University Press, 1978.

[13] CISCO & The Metiri Group, Multimodal Learning Through Media: What the research says. Global Lead, Education. CISCO Systems, Inc. 2008.

[14] Razo, F., Teaching fractions in remedial mathematics using virtual manipulatives. Conference of the International Journal of Arts and Sciences Proceedings, Volume 1 (11), Las Vegas, Ne, 2008.

[15] Razo, F., Preparing teachers for the realities of technology in elementary classrooms. California Council on Teacher Education - Conference, San Diego, Ca, 2009.

[16] Razo, F., Using the Brigance II test to assess the effect of virtual manipulatives in teaching basic numeracy. Unpublished research report, Hayward, Ca, 2009.

[17] TPACK, http://www.tpack.org/ Open Internet information site, 2009.