We begin this teaching document with a review of basic questions about measurement, our understandings of its meaning and applications. These questions will be revisited repeatedly, as needed as we explore, think and re-think our assumptions:

Q1.- Why was this document about measurement written?

Q2.- What is "measurement"?

Q3.- Why do we "measure" things ?

Q4.- How do we measure things?

Q5.- How are the concept and practices of measurement helpful?

Answers, for the moment:

A2.- The purpose of this “hyper” (electronic, web-based, multimedia, interactive document is to help students learn, adults and teachers review the basic reasons and utility of measurement in real-life, as well as its value in supporting the arts and sciences. This is done in what is called a Brain-Based learning approach (more natural and biology-justified, reality based detail, intuitive, instinctive forms), multi sensory (relevant, coordinated visual, tactile, auditory and possibly smell and taste) experiences, using the most powerful physical (i.e. realia) and digital electronic (i.e. virtual) technology, Internet-assisted multimedia, such as is currently available in classrooms and at home.

Even though the focus on measurement here is focused on the basic quantitative sciences (Mathematics, technology, engineering), the concepts extend the same into all areas of human activity. Learning is facilitated by carefully designed, realistic and interactive multimedia images incorporating "hyper" (touch-sensitive) images and text-links that can give access to relevant information and activities.

The activities include scenarios with coordinated visual, tactile sensing and audible effects on present-day flat-screen displays. Interaction to virtual models is provided through standard mouse-click or finger-touch action. In the future, better interaction seems possible using more convenient, “smarter”, pedagogical virtual, 3-dimensional models.

Although having reached this point in the document means that you most likely have available most, if not all the tools necessary to access the supporting experiences provided, it is still important to recognize the growing importance of the latest, most powerful tools and resources offered by today's information technology. These tools continue to grow in their abilities to use both physical and virtual information and experiences to supply meaningful real-life interactions and learning. In real life, the range of human-sensitive multimedia objects, artifacts, boards, microelectronics toys, "smart" home and work appliances and communication devices and products available keeps growing every new day.

A key piece of the interactive activities included in this document is the the multimedia Flash Player (Adobe Corp.) animation software, typically embedded in today's most common computers and web browsers. *** IMPORTANT NOTE: Although the Adobe Flash animator has been one of the most successful Internet multimedia tools used to date, its availability has been slated for discontinuation by the year 2020, and thus, a migration of the interactive activities on this document into a new platform is to be scheduled correspondingly ***.

For a preliminary sample of this, click (or touch) this link: Measuring weight (Figure 1 below). The recalled display will include the image of a scale and balance, together with interactive objects to help introduce the concept of measuring, balancing weight and solving problems, typically described in the subject of (basic) Algebra. The activity can be used to begin to sense the meaning of real-life measurement for handling physical weight. More details of the use of this display are included in Chapter II. This display is also meant to support an effective introduction of the basic concepts of balance in nature, physics, chemistry, etc. as well as the real-life connections of the “sentences” in the mathematical language of Algebra.

Fig. 1 Interactive virtual display to help teach basic concepts of measurement and balance, incorporating abstract symbols and expressions in Algebra

Virtual (computer-based) interactive displays like this introductory one, in conjunction with real-physical complements and counterparts can be used to help us better "feel" and appreciate the concepts of accuracy and prediction (figuring out before doing) value of measurement.

Because of their non-material, imaginary-core nature, virtual interactive quantitative activities can provide:

• Environments that support faster, cheaper, safer and stronger learning of basic mathematics, introducing enhanced objects and actions that can illustrate concepts and emphasize their relevance through a diversity of meaningful, coordinated, instinctive, intuitive situations. This can also be done through specially-designed strong, repeatable, simultaneous stimulation of the senses and brain,

• More and better opportunities to actively develop corresponding appreciation and subject-matter skills using instinct, common sense, disposition for reasoning, enhanced self-esteem and ability for disciplined analysis. These abilities help students develop key skills, appreciation and mastery for the development of understanding and effective identification of situations and resolution of real-life "problems", as well as connections to curricular school requirements.

A2.- What is measurement and why it is important?
The word measurement is used in two ways: As a verb and as a noun. As a verb, measurement (or measuring) are the activities of determining, with greater accuracy (accurate : with exact measure) and precision (precise :being reliable, repeatable, trustworthy), about how big or small things are. As a noun, measurement is any of the physical objects, images, sounds, and symbols used as representations to communicate reliably with others about how big or small things are, so we can understand, imagine, work with, transform and communicate what is in our environment the best possible.

Measurement, therefore, consists of the representations (including images and symbols) and the activities used to determine and communicate the magnitude, amount, or size of things and actions in the environment.

Measurement is important because things in the environment could be far or close, large or small, fast or slow, hot or cold, heavy or light, brilliant or dim, rich or poor, passionate or mellow, etc., etc. and the differences can affect us in many different, and maybe very important ways.

Measurement therefore introduce first-hand evaluation of real world quantities that are necessary for the purposes of living. Measurement supports even better living through its use in the sciences and mathematics.

Measurement can be done first-hand, or be "direct measurement", when it results from our direct observation, comparison or manipulation of things. "Indirect measurement” is done when we do not want to or cannot observe and measure things directly and instead measure other related things and actions, and then process, manipulate, calculate and evaluate that information to derive measurements of our interest.

The concept and practices of measurement are important because they provide a firm basis for a variety of essential activities in life: supporting the accuracy or exactness and precision or trust necessary to do whatever we do best: in safer, most economic, expeditious ways, i.e. not under-done or over-done, but just done right, and with the best balance of consequences (“safer, cheaper, faster”) .

To teach measurement concepts better, it is helpful to express our ideas more meaningfully and clearly within the context of the learner's real-life experiences: most effective, unencumbered (J.M.Carroll) and rewarding (Situated, real-life-connected learning - Ref: J.J.Rousseau, J.Dewey,J. Bruner, M.Montessori, J.Lave, M.J.Wenger, J.Greeno. J. J. Gibson). For this purpose, it is helpful to take advantage of a scientific approach to the environmental transformation processes called systems engineering. Systems engineering is the science of understanding, designing and constructing integrated functional things (systems) that transform the environment). The study of the mind and its processes is called Cognitive Science. Cognitive science, together with the art, science, or profession of teaching, called pedagogy and the disciplined, reliable, scientific interdisciplinary methodological practices for defining and solving problems (e.g. "How to Solve it", George Polya, 1945) altogether rely on measurement to provide the strong, reliable support needed for more rational, "smart" living.

In particular, measurement tools and practices are important to productive knowledge and skills referred these days as the "STEM" (Science, Technology, Engineering, and Mathematics) subjects. Measurement is central to the practices and disciplines referred to as "quantitative or hard sciences".

Unfortunately, and in spite of the great efforts being directed to better prepare our children in the quantitative, scientific areas, the reality is that there has been only limited demonstrable progress, as indicated by a variety of assessments, reports and analyses, including local and international tests of competency (e.g. the IES NCES NAEP and OECD PISA).

Unfortunately also, often times the adults involved in educating the children seem to be themselves casualties of troubled formative circumstances. Adults involved often struggle themselves to determine the need, timing and best ways to promote meaningful, real-life knowledge and skills in children. This is particularly concerning, given the highly global, technological, complex and competitive future world they will be living in.

As described earlier, a central objective of this document is to increase the connections between the real world and the experiences of children, as early as possible in their lives. In particular, helpful experiences would be those related to the topics in their current K-12 academic standards for the subjects of mathematics and science ("Common Core" - mathematics, and "New Generation" - science).

The concepts and practices of measurement are very important, from their simplest, possibly intuitive forms in every day routines, to the most complex ones in professional activities. The benefits of learning them properly and consistently, as soon as possible in life, therefore are quite evident.

It is also important that teachers and others who support children's education clearly understand themselves the specific nature, application and real importance of the basic concepts being taught, to better help their students make the most sense of their learning, from the beginning, and into their effective meanings. For this reason, a proper understanding of the most basic concepts (academics), supported through activation of the power of our natural body and mind systems is also helpful. These concepts are described in more detailed in what is called the "SYstems Model of Learning".

The Systems Model of Learning (Figure 2) is described next. It presents a formal view and a reminder of the basic elements and functions of the human body and mind that would support successful interaction and learning in our world. A download of a corresponding animated Power Point Show (Microsoft Corp) is also accessible through the link.

The biological basis for learning and the use of technologies.

Objective: To present a summary review of basic concepts of learning, education and educational tools (educational technology) relevant to the process of learning, and throughout the rest of the book.

A basic definition of terms is introduced first, to highlight the basic human physical biology that supports learning. Starting with the processes of collection of information from the environment through our five senses (INPUT), and into the processing, storing, interpretation, definition of patterns leading to understanding and the decision-making by nerve tissue and the brain (PROCESSING/STORAGE). The process then completes repeated system cycles through the diverse actions allowed by the muscular and skeletal means available to the body (OUTPUT), and the senses collecting again the likely changing conditions related to the developing actions and circumstances (FEEDBACK).

Although there is a myriad of sources of research information in most any topic these days, on-line tools such as Google, Yahoo, Bing, etc. are suggested as an initial step. Within the text descriptions ahead, key terms will be flagged with the symbol ?, which will activate a corresponding hyperlink to a Google web search engine (www.google.com) for the given term. The reader can then select one or more of the links suggested in the search results page.

What is learning?
For our purposes, the first term to be explored is "learning" ? . The Microsoft Encarta Dictionary® states that learning is “The acquisition of knowledge or skills”. The word "Knowledge" is defined as (http://www.merriam-webster.com/dictionary/learning) the “general awareness or possession of information, facts, ideas, truths, or principles”, whereas skills are defined more concretely as “the ability to do something well”. Why and how do people acquire knowledge and/or skills? For the general purposes of this document, we will assume that knowledge and skills are needed because they prepare people to live, and live better by dealing more effectively and efficiently with the environment, using our bodies sensorial, nerve/brain and motor systems. Knowledge and skills empower us to survive and manage our environment better, safer, faster, and more economically.

Human learning: The senses collecting information for the brain to process and use
How does learning happen?
Figure 2 displays an overview of the processes involved in learning, framed conceptually in a general systems model (INPUT, PROCESSING/STORAGE, OUTPUT) described as MMR-MSR-SSE. The diagram describes how the patterns of the world (1- Meaningful Real World) are perceived using the five natural senses of vision, hearing, smell, taste, in various representation forms, (2 - Multiple Representations) participating in simultaneous collection of information produced by stimulation of the senses (3 - Multiple Senses: INPUT; VIEWING/READING, LISTENING, FEELING) to support the productive interpretation and development (4 - Assisted through Repetition - VISUAL, AUDIBLE and TACTILE ?) of the complex patterns of the world. The brain will therefore organize, arrange, rearrange and store information (mostly in forms of constructed PATTERNS, which include LANGUAGE), that will eventually be retrieved and used to drive back corresponding responses (OUTPUT; WRITING, SPEAKING, PHYSICAL MOVEMENT) meant to TRANSFORM the real world. The contribution of the smell and taste senses is de emphasized in this overview diagram, to reflect the minimal role these senses currently have in the processes of general academic learning.

Fig. 2 A Systems Model (INPUT-PROCESSING-OUTPUT) of the complex process of learning
(click on image to download a Power Point Show of the full MMR-MSR-SSE model and references)

The full model also highlights common pedagogical practices that are helpful in the process of teaching and learning:

1.- Learning is easier and better when objects and actions are more meaningful and familiar to the learners reality and experiences

2.- More different and effective representations, diverse images and symbols used to describe objects and actions in the environment will help stimulate the learners brains and emotions more positively

3.- The more directed, stronger and coordinated stimulation of our various senses (i.e. vision, hearing, touch, taste and smell) the more impacted, deeper, effective learning

4.- Repetition of experiences allows learners to develop, refine and retain (permanent) knowledge better

5.- Allowing our brains and bodies to produce more meaningful, diverse and powerful responses to knowledge helps solidify our learning and ability to communicate and coordinate with others

The visual sense in humans –is a complex one. Below are images of the visual sense organs. These include a cross section of the human eye, starting when light captured from the environment reaches and passes through the pupil. cornea, and lens, to finally strike the light-sensitive retina in the back of each of the eye balls. The expanded view of the retina shown next displays the detail of its nerve cell arrays. These are called rods and cones. Rods and cones absorb different frequency-color radiation energy from the light received and convert that light energy into electrical signals ? , using photosensitive protein called Opsin. These signals are then transmitted to the brain as visual information through the optical nerves. Through this process, our eyes can be compared to a present-day color video camera, capturing information from the environment and communicating it electronically to a "video processing computer" in the brain.

The closer view of the retina's rods and cones tissue in Figure 3 makes more visible the distinction among them: The rods are basically fast, color-blind (i.e. "black-and-white") shape detectors of light intensity, intended to support fast object-spatial and position sensing only. The cones on the other hand are slower, but each able to selectively detect one of three different color frequencies in the received light: red, green, or blue. As the distinct intensity of signals from each of the types of cones is transmitted and processed in the brain, a unique color (out of millions of different combinations) is determined as the mix of these (primary) colors.

The optical nerves from each of the two eyes will then provide dual image perspectives as electrical information from the retina to the visual cortex zones in each eye to the back of the brain, as shown in the last image in Figure 3. The optical nerves bringing signals from the eyes to the brain are thus analogous (i.e. similar) to “data cables”, bringing visual (video) information from two color video cameras (the eyes) in a three-dimensional (3D) stereoscopic ? array able to sense depth, just like the three- dimensional video processing areas in today’s televisions and computers.

Currently, the design and construction of most electronic imaging devices that handle visual color are built with consideration to these primary colors sensed by the eyes (Red-Green-Blue, or RGB). A “subtractive” equivalent (Cyan-Yellow-Magenta, or CYM) is used for devices that mix ink to absorb, rather than produce selected color light. Further description of the RGB primary colors and mixing, as well as CYM coloring for printers is available through the "Encoding color in digital images" link included in Figure 8 ahead.

Fig. 3 The visual sense in humans – Eyes to brain

The auditory (hearing) system (Figure 4) senses sounds from the environment received through physical vibrations of molecules in the air, processing them through the outer and inner ear, drum, anvil, hammer, cochlea, etc., to create electrical signals that are transmitted to the brain through auditory nerves. Here again, in the second image, there are dual auditory paths to bring auditory information from two “microphones” (the ears) in a stereoscopic array (to sense depth), to the "stereo" audio processing areas (auditory cortex) in the brain.

Fig. 4 The auditory sense in humans

Similarly to the visual and auditory systems, the olfactory/smell and taste/flavor senses (Figure 5) also rely on various kinds of nerve cells and tissue to communicate smell and flavor sensation information to the brain. There seems to be limited spatial separation of information for olfactory information (two nostrils), and none (one palate) for taste/flavor information.

Fig. 5 The olfactory and taste sense in humans

Finally, the perception of the environment through physical feel and touch contact involves a diversity of organs and complex processes that extend through different parts of the body, notably the skin and muscles. The feel and touch systems actually consist of two different but related structures, one for input of information ("sensors"), feeling and exploring physical objects, and the other for output ("actuators"), touch and movement reactions exerting pressure and modifying the environment, with the assistance of muscles and bones. Figures 6.a and 6.b show images of feel and touch (through skeletal muscle activation) related organs in the body . These include nerve routes to and from the medulla (dorsal horn), and the brain, related to involuntary, fast-response reflective reaction ? functions, as well as slower, more rational reactions involving the brain.

Fig. 6.a Feeling and touching: a complex 2-way system

Fig. 6.b The neural paths for feeling (sensing-in) and touching action (reacting-out)

For educational purposes, the senses of feel and touch are powerful and important because they support both physical exploration of the material world, as well as active participation and interaction in the modification of the environment using muscles, hands, fingers, body movements, as well as vibration of the vocal cords to produce sounds and speech.

Making sense: The brain at work, learning and symbolizing
Information collected by the vision, auditory, smell, taste, and touch senses is brought to the brain, where highly complex, almost magical processes begin. Somehow, the brain will identify and activate its structure and chemistry to identify, classify, analyze, synthesize, connect, reconnect, organize and reorganize incoming and previously-stored information in highly complex schemes, or schemas ?. The individuals use these schemas to eventually generate conclusions, predictions ?, possibly defining meaningful reactions to deal with a situation or opportunity in the environment, either unconsciously or/and immediately (reflex ?) or consciously, through a mediated, rational analysis (intelligent ?) process.

Although much research has been dedicated to the understanding of brain structure and functions (brain research ?), much more still needs to be discovered, as it seems to be an extremely complex process with vast information collecting, organizing, storing, reorganizing, and retrieving systems, functioning and evolving constantly throughout our lifetimes.

One important concept involved in brain functioning and learning is the use of representations ? . Representations are alternative expressions of knowledge (or awareness), normally involving the use of more effective or efficient surrogate ? objects, which can include realistic or imagined objects (e.g. toys) images, sounds and motions, semi-realistic objects such as recreations, figurines, cartoons, stick-figures or icons ? and all the way to abstract and/or encrypted ? symbols ?, whose meanings are basically disconnected visually and dependent on corresponding mental translation protocols ?, often developed specifically for unique purposes. Examples of these protocols are the meanings, rules of construction and translation of common conversational human languages, mathematical systems and security protocols designed in present technologies to safeguard secure electronic communications.

The images included in Figure 6 are those of ancient representations found in archeological sites around the world. These images show counting dots and scratches, realistic images of animals, icons and symbolic narrative text. The images in Figure 7 are of symbols introduced for early electronic communications (wireless Morse code), tactile symbols for blind persons, (Braille), and hand-finger gestures for deaf or mute persons (Hand signs in Sign Language).

Fig. 6 Ancient representations with symbols, images, icons and narratives

Fig. 7 Encoding text for early electric communications, for the blind and for deaf-mute persons

More recent development in electronic computing, communications, presentation, and "robotics" (automated physical action machines) technologies have resulted in the current widespread utilization of binary coding of information. Binary coding is the simplest form of coding, using “zeros” and “ones” represented physically in electronic devices or media supporting one of two distinct electrical or magnetic states typically described as "ON" or "OFF". Because the states of the binary media can only be two (0, 1), each of these are called binary digits or bits. Strings of binary symbols are then grouped and interpreted in series or sequences though corresponding protocols. Currently, and due to the great ability and flexibility of present day electronic devices, binary coding, commonly called digital coding is used today to efficiently encode text, sound, images, and virtual-physical interactive environments. Digital coding is used today in all kinds of digital devices, “digital computers” and robots. The links in Figure 8 below can be used to call related interactive web activities for exploration of digital encoding of numbers, sound, images, and through special processing and devices, motion and physical activity simulation.

Fig. 8 Encoding numbers, text, sound, and images in binary digital code

Why are mathematics and measurement important?

What is/are mathematics?

Through time, a variety of definitions of mathematics have been proposed (e.g. mathdef1; mathdef2,mathdef3, mathdef4) Although each of these definitions can be quite valuable and motivating to learners, each include important conceptual limitations, particularly regarding the meaning and connections with related knowledge, as well as the fundamental role in basic real-life applications and skills. From a practical standpoint, "Mathematics are all the things we do to understand, predict, and transform our environment efficiently, safely, using accuracy and precision , and balancing: loose (estimation) versus tight (more formal) tools and procedures to achieve the best safety, speed and economy. In practical applications, mathematics therefore will include the creation, manipulation and application of objects, images and symbols, in both real and virtual (computer generated) forms: whatever help us understand, imagine, and transform our environment safely and efficiently, so we can live better, happier, more plentiful, longer lives.

What is measurement?
Measurement refers to all the objects and activities we make and do to help us "understand, predict, and transform our environment with optimal (best) accuracy, efficiently, promptly and reliably. We do this by providing the most balanced accuracy and precision necessary to provide the best levels of safety, speed and economy in the things we do. Measurement therefore consists of the activities and physical representations used to describe, imagine and communicate the extent, size, or magnitude and effect of things and actions in our environment. Measurement help us introduce and manage first-hand evaluations of quantities, opportunities and threats in our environment. Because of its nature, measurement is a central concepts to (quantitative) disciplines such as mathematics, engineering, technology and science ("STEM").

Measurement activities are important because they produce helpful, reliable assessments of how big or small, close or far, fast or slow, light or heavy, hot or cold, etc. things are. Without at least an intuitive assessment of these quantities, it would be impossible to exist and to be successful in our lives.

Measurement representations are important because they give us sharable, enduring, manipulative expressions of quantity. Through representations, we can record, share, and maintain measurement information through space and time. Good representations are important because they can facilitate the ways and means by which measurement information can be acquired, stored, processed, communicated, and applied.

The importance of representations was highlighted by the British philosopher Bertrand Russell, as he saw on them the key to the effective resolution of problems:

“The greatest challenge to any thinker is …. Stating (representing) the problem in a way that will allow a solution”
Bertrand Russell (1872-1970, Crainer's: The Ultimate Book of Business Quotations (1997), p. 258)

With fast evolving social, industrial, and economic progress, globalization, cooperation and competition around the world, the impact that measurement, calculation, and data analysis can have over our lives will only keep growing. All we need is to consider the many examples in which electronic, optical, and electromagnetic, technological means are used today to accurately and reliably measure, identify, and control objects and processes during the production, distribution, and consumption of the goods and services we enjoy. Consider the images below and the following question: How has measurement impacted the development of science, engineering, industrialization and socites we enjoy through the development of human civilization?

At the present time, and regarding our education, more concrete questions could be:

1. How can we provide the best materials, images, activities, and interactions with the real world to best prepare our children to build a better future?

2. How can we best use the available materials and technologies to teach, learn and implement measurement, accuracy and precision in the tools (technology, mathematics , science, engineering, industry, and commerce) that help us build safer, richer and happier lives through the 21st century and beyond?

The remaining pages of this instructional guide, together with the interactive materials presented throughout its pages, are designed and implemented with consideration to best known teaching theories and practices (Teaching and Learning with Technology in Elementary and Secondary Classrooms, F.Razo), and are intended to provide simple, yet safe, powerful and effective environments: images, sounds, simulation of movement and interaction in activities related to the concept of measurement. More than anything, it is hoped that the realistic experiences presented can help children become early and life-long witnesses and motivated users of the beauty and power of the ideas and skills that could better help them perceive, imagine, transform and enjoy their environment throughout their lives.

Why do we measure?
We measure so we can be more rational, accurate and precise, in making things in life as reliable, effective, efficient, safer, cheaper/affordable, fast and as enjoyable as possible.

When and how do we measure?
We measure when we feel it justified by our need, and by our ability to improve and balance our resources with other, possibly conflicting needs. Because, for example, making things “safer” could mean making them more expensive, slower or even impractical; or likewise making them “faster” could lead to more expensive or even impossible options; making “cheaper” things leading to less safe, slower or impractical situations …… It is then important to know more about measurement and choices, so we can better explore properly balanced alternatives for action .

To measure, objects or actions need first to be selected (identified for qualities) with a specific purpose in mind. Then, evaluated for size, or extent (quantity). Then, and once we properly understand and imagine (through properly selected knowledge, e.g. sciences, mathematics), our more desirable options, we need to properly define and transform the environment in what we want it to be (e.g. engineering), by devising and using the proper tools and procedures (technologies) .

MEASUREMENT SCALES
This section is dedicated to the understanding of measurement scales, or the "type and range of outcomes" during the process of measurement.

A measurement scale is “a graduated series or scheme of rank and order” (Merriam Webster On-Line Dictionary, 2005 ref 6), used for the purposes of identifying, comparing, ordering, and assigning things among and relative to others in the environment.

The three most common commonly cited types of measurement scales are categorical/nominal, ordinal, interval, and ratio. In practice, the use of each type of scale would ultimately define the accuracy and precision achievable for our intended purposes when dealing with items and actions in the environment.

The following are broad descriptions of the three general types of scales. After that, more detailed descriptions and multimedia interactions are included to reinforce the concepts implied in each kind of scale.

Categorical (also called nominal) scales are used to evaluate and identify items belonging or assignment to a group through the obvious presence of a qualifying characteristic, such as their color, gender, species, group affiliation, intended use, social or economical class belonging, nationality, interest preference, favorite food, etc. Categorical scales are not intended to identify the degree, amount or quantity of a characteristic in the items, objects or actions but just its presence (or absence).

The ordinal, interval, and ratio scales are used to evaluate the intensity or quantity (quantitative) of a characteristic in things with each allowing an increasing degree of accuracy (or exactness), from approximate, or looser estimate, to precisely, or tighter measurement:

• The (looser) ordinal scales provide broad, inexact, coarse assessment of apparent quantity or order. Examples are: small>medium>large, very-cold>cold>warm>hot>very-hot, liberal>moderate>conservative>ultra-conservative, etc.

• The (tighter) interval and ratio scales are more intently focused on the specific, accurate degree or quantity of a characteristic or quality possessed by items. To become more accurate, these scales rely on the use of small unit pieces or fraction-of-unit pieces as a basis for comparison in numeric sequence, to evaluate larger quantities such as length, weight, time, temperature, speed, frequency, etc.

  The main difference between interval (only) and ratio scales is the existence of a definite-mark point at which one (ratio) defines or not (interval) the absence of a real zero amount. In the interval-only scales, an arbitrary zero marking is assumed (Example: Celsius and Fahrenheit zero-temperature, start-time). In ratio scales, zero values do model zero as the complete absence of intensity in the quality being measured (Examples: Weight, distance, speed, energy). A consequence of this is that ratio scales also model a multiplicative proportionality of quantity, that is, for example, a measurement number twice as large as another does represent twice as much of the amount being measured. In an interval scale, no such proportionality is implied (for example: zero degrees Celsius does not imply the absence of heat, or 10 degrees imply twice as hot as 5 degrees).

Specialized quantitative scales also have been created for particular purposes, such as are the logarithmic scales, which are used for comparing things with large difference in size, such as the low and high sound energy levels, or the mild and strong intensities of earthquakes (e.g. Richter scale). Decibels and Richter readings would imply exponential-10 proportionality, i.e. a measurement of 2, 3 or other number will imply not twice or three times, but 100 (for 2), 1000 (for 3), or the corresponding exponent of 10 of the proportionality of the corresponding objects or events being measured. The following section includes more detailed descriptions of the basic scales.

Categorical (qualitative, nominal) scales:
Categorical scales, sometimes referred also as qualitative, or nominal, are used to identify and classify items within categories, or groups possessing (or not) some quality of interest (e.g. color, gender, nationality, political affiliation, species, etc.).

The Categorical Identification activity shown below, illustrates the process of identifying shapes within a categorical scale for basic geometric shapes. The display illustrates 13 different shapes (rectangle, ring, circle, triangle, heart, etc.) which need to be identified and matched with its corresponding category box hole. This activity illustrates how objects can be classified simply by their shape and in this case, without reference to other characteristics like say, their size, weight, color, temperature, or perhaps price or even some subjective "emotional value".

Classification is accomplished by clicking-and-dragging each of the different objects shown onto its corresponding matching category hole. A count of all correct matches is displayed and sounded out.

QUESTIONS: What makes each shape sufficiently different from others, so as to be uniquely and unmistakably identified? Could you think of a situation where there could be shape-category confusion?

The display Categorical Classification shown below uses a three color categorical scale activity to identify and classify (or group) various objects within corresponding class or category. Classification of objects in this activity is done by clicking, dragging and placing the objects shown in their corresponding color bin.

QUESTION: Does it matter if the red, blue, or gold spheres are greater, or smaller than the others, during their color classification?

Quantitative scales: Ordinal and interval
Ordinal and interval scales are used to identify, and classify objects according to the magnitude, quantity, or size of some characteristic they possess. Because these scales refer to the quantity possessed, they are called quantitative scales.

Identification of quantity
When comparing the size of items, they can be “greater-than”, “equal to”, or “smaller-than” others. The display Comparing Size illustrates this process by visually or numerically identifying the quantity of length in items, as compared with that of others.

This activity presents red and blue linking cube chains in lengths from 1 to 9. Each red and blue chain can be selected and their size compared, visually (gross) or/and numerically (fine). The length comparisons can be done by using specially labeled keys, marked with the common mathematical symbols for amount comparison (<, =, >). Correct and incorrect choices are signaled by corresponding sounds.

Ordinal (gross size) scales are useful when broad statements of comparative-order amount or content of a characteristic of items are more practical. Examples of ordinal scales are:

• Small, Medium, Large, - item size

• First/Second/Third, - group size

• Cold/Warm/Hot, - temperature

• Liberal/Moderate/Conservative, - political affiliation

• Extra-light/Light/Regular/Heavy/Extra-heavy, - weight

• Quiet/Normal/Loud, etc. - sound energy level

The Ordinal Classification display is used to order (colored) balls according to their broad size order group: Small, Medium, or Large.

During ordinal scale classification, items are identified to belong within a small number of vaguely sized groups, and therefore, once classified and becoming part of a group, the specific individual differences are blurred. Accuracy in the description of single items is therefore lost.

QUESTION: What kind of scale would we have if we merge the Medium and Large groups into one called “Bigger stuff”?

As we will see next, the blurring in identification of individual items and groups can be reduced as we begin to use scales with more, equal sized, finer, accurate scales. These are called interval scales.

Interval scales are used to identify and classify items within more accurately sized groups. Groups in interval scales are normally defined within limits of equally spaced, contiguous segments, such as (0 to 5), (5+ to 10), (10+ to 15), (15+ to 20), etc. Interval scales that include a value of zero that truly marks the point when the characteristic being measured ceases to exist (nothing there), and from there on numbers do represent relative sizes, are called ratio scales.

A practical example of the use of interval scales, i.e. those having no real zero scale point, is seen in the Celsius and Fahrenheit temperature scales: their zero points do not signify the absence of temperature, but rater arbitrary points such as the temperature at which pure water (Celsius), or salty water (Fahrenheit) begins to freeze. On the other hand, the Absolute, or Kelvin temperature scale is considered a ratio scale, since its zero is placed at the point where molecular movement in matter stops, due to the absence of energy, or heat.

Call the Interval Scale Classification - Measuring Temperature display to observe the interval points in the two most common temperature scales: Celsius, and Fahrenheit.

Click-and-drag the thermometer left and right, and observe the Celsius and Fahrenheit measurement range. Note the “freezing water” point (Celsius = 0, Fahrenheit = 32 degrees), and the “boiling water” (Celsius = 100, Fahrenheit = 212 degrees) points.

QUESTIONS: For freezing water, why does Celsius say “0”, while Fahrenheit says “32”? For boiling water, is the heat required different because Celsius says “100”, and Fahrenheit says “212” degrees? Which scale, Celsius of Fahrenheit means more heat for each additional degree?

The Ratio Scale Classification – Students Height display is used to identify and classify students into their (more accurately) defined multiplicative proportional scale groups.

QUESTIONS: What would a student height of zero mean?
How many units taller would a student measuring 60 would be, than a student measuring 30?
How many times taller would a student measuring 60 would be, than a student measuring 30?
Would the student height-scale be an interval or ratio scale?

FOR GROUP DISCUSSION: How could our lives ( safety, economy, happiness?) be if nobody, even adults knew when and how to measure and do the things we need to do?