Counting Drumbeats: Should our number syllables represent quantity?

Excerpt from Doug Clements "Subitizing: What is it? Why teach it?'

"Three pictures hang in front of a six-month-old child. The first shows two dots, the others show one dot and three dots. The infant hears three drumbeats. Her eyes move to the picture with three dots."

IMAGINE: This ability of recognizing syllables, movement, and number is born within us. We have the capacity to recognize a four digit number instantly if it consisted of four syllables. Our language does not facilitate our natural instinct. 4,367 is a ten-syllable number. No wonder our brains get confused! With subQuan and the slight change of two single digit numbers -ro for zero and -ven for seven, we are able to make these larger numerosities accessible to younger children: four three six -ven. Visualizing numbers in their shapes allows yet deeper understanding (as seen below).
 -ro, one, two, three, four, five, six, -ven, eight, nine, repeat....
We can now discuss numbers in various containers or bases. We don't always package things in tens. The picture below was a fun adventure in Walmart at all the different container sizes we use.
Chinese language has made it easier on their children with counting. Instead of introducing a whole new set of words after ten, they say ten one, ten two, and so on. We could do the same. Remember the push to change 9-1-1 emergency number to the single digit saying of it rather than nine-eleven? That's because in an emergency, when the brain is stressed, it can't find the eleven key on the phone. The only numbers on our digital devices are 0-9, so logically saying one-ro for ten, one-one for eleven, and one-two for twelve would make it much easier for our children to understand number. We could then catch up sooner with the Chinese as seen in this graph below.
Interesting thoughts based on well-documented research. The investigation continues...

Retrieved from the ning....

Cooper's Blog Posts:

Using games to create engagement in education, business, and government
posted 8/5/2011

3D Game Lab continues to bring incredible value to the Dream Realizations effort to help people understand numbers.  In Tom Chatfields 7 Ways to Reward the Brain, he leads the viewer through seven key game concepts that when combined in addressing a given issue or problem create engagement!

DR is focused on creating engaging activities around the fundamentals of number sense to ensure that as many people as possible will be motivated to discover for themselves that they can understand and change their lives by increasing their knowledge of numbers. Numbers, in themselves, are boring or scary to most people because they simply haven't had the right foundation for understanding them. It is very difficult to engage people who have already tried to understand numbers via mathematics and ended up thinking that they were not 'bright' enough to get it or that there is any value in their personal lives to tackle this subject. 3D Game Lab is providing a game foundation exhibiting many, if not all, of Chatfields 7 Ways to Reward the Brain discussed in the video. DR's entire staff is currently participating in a three week intensive study of 3d Game Lab.

Please stay in touch to see the exciting results. We hope you will take time to not only view the video but also to visit 3D Game Lab's website.

3D GameLab and teaching subQuan
posted 8/3/2011

According to many experts, Jim Gee included, the United States education system has the potential to undergo a dramatic paradigm shift due to the demand of innovation in face of world-wide competition. I have seen the potential for games in education since the mid 1960's and now the day of game-based education is dawning.

As Jim points out in his video, kids want to participate, they want to be engaged. Games are engaging and they require their own levels of knowledge acquisition. Unlike school, games do NOT separate learning from assessment. Games are always assessing. Furthermore, reference material (on-line help, user guides, developer notes) all make sense once one has stated to play the game. The game allows you to play first, then access relevant data once you know that you don't know something. Jim calls this 'language on demand'. I agree with Jim that vocabulary reading and writing develop as students interact with one another and with research material. The language is built as needed. Repeated internet searches on a new topic, say 'subQuan', reinforces not only the spelling of subQuan but also the fundamental concepts since they appear on a search's findings. Also, the language is reinforced by top gamers as they have a high set of standards on who is considered 'top' and using the right vocabulary is an excellent differentiator of expertise. I actually used this understanding of terms to rapidly separate applicants in a Silicon Valley billion dollar firm. There were certain vocabulary terms that were just fundamental to certain levels of expertise.


At Dream Realizations and ItOnlyTakes1, education and games are inseparable. SubQuan is a fundamental concept that the eye 'sees'. It doesn't require learning of a process, once you see, you see! Putting this new concept into a game environment is fundamental to our goals. We are all engaged in Boise State University's beta course on 3D Game Lab this August, 2011. We look forward to teaching and learning with our students. We will be initially using Second Life to accomplish this and, hopefully, 3D Game Lab as our engine.


Watch Jim Gee:

subQuan question worth investigating
posted 6/20/2011

Is there a proof that there is only one complement that is contained for any shape? I.e. if you have the number 7, the segs complement is 10 -3 (1, -3), the squares complement is 100 -90 -3 (1,-9,-3), the cubes complement is 1000 -900 -90 -3 (1, -9, -9, -3), ...  The negative complement or the second segs, squares, cubes, etc complement requires free mode (when the value in any shape(base place) exceeds the base).  [NOTE: Please comment if this is partially confusing so that I know what to clarify.]

Subitize is not required for subQuan!!!
posted 4/10/2011

Wow! I never expected to discover that the ability to subitize was NOT foundational to the ability to subQuan. I always focused on Stanislaus Dehaene's neuropsychological research on number sense as the explanation for the efficacy and quickness of subQuan. However, both my co-researchers became aware that counting, albeit much slower, was just as effective as subitizing in determining the subQuan of a number. In retrospect, the invention of place-value was not beholden to the methodology of determining the quantity within each place-value. SubQuan continues to reveal very fast and powerful visualization of mathematical structures resulting from its definitional representation of numbers in every base system but it no longer depends on the ability to subitize. This does not mean that Dream Realizations will abandon its efforts to promote subitizing. We still believe assessment of pre-college mathematical abilities should be based on velocity, perhaps even acceleration, rather than distance. [To clarify this last statement, assessment is dominantly based on arriving at a correct answer (distance) rather than the time it takes to derive the answer (velocity).]

Changing the website: what was I thinking? Does this affect our other technology decisions?
posted 4/10/2011

Over the last Christmas season we discussed an idea of mine to make the website visually driven rather than text driven so we made some changes. Over the past three months we did not get the results we were hoping for so we are in the process of moving back to the Ning-based website. I am hoping that this adventure into visualization does not deter those who are following us early on. Upon reflecting, I believe traditional web-surfing for 'just the facts' lends itself to text-based web design such as Ning. However, I still firmly believe that when building foundations for knowledge acquisition and social learning relationships that visually driven immersive technologies, such as Second Life, will become more and more dominant and any visualization facilitating this movement will be very successful. Therefore, expect our actual coursework to appear first in our laboratory in Second Life, then in an OpenSim environment for easy transfer of scripts, textures, objects to give access for young dreamers. We will utilize those immersive environments to create machinima (Videos made within an immersive environment) to provide exposure for non-immersive collaborators. Hopefully, we will then translate our findings into the emerging touch technologies of iOS. [Just a reminder, please help however you can. We eagerly accept any and all input and ideas, especially from our young dreamers. No matter how young, get involved. It's your future.]

Four Steps to Polynomial Derivation (steps 3 and 4) posted 12/3/2010

3) Differences: Discover the relationship that differences have with metapatterns.

When we examine the quantities creating one digit metapatterns and then two digit metapatterns, we can discover a simple way to discover the metapatterns from the data by using differences.

In the data associated with these metapatterns, we would have discovered by looking at different metapatterns that the first difference determines the value of the 'seg' and that when that value is removed from the data, the resulting number is the value of the 'remainder'. Having access to these sheets enables very quick discovery of this fact.

4) Polynomial Derivation: Develop an expression for the pattern from differences.

When this examination of metapatterns continues into squares and cubes, the pattern of differences reveals itself in all its glory. A spreadsheet is easily created to automate the differences, determine whether we have a seg, square, cube, seg of cubes, et al., and to determine the coefficient which turns out to be the value within the metapattern. Removing the effect from the data and repeating continues to give us significantly smaller shapes until we get to the remainder. Expanding this metapattern to base x then reveals the equation for the metapattern.

The metapattern 1234 (pronounced one-two-three-four) has the data that generated it placed into the spreadsheet and the equation +1x^3 +2x^2 +3x^1 + 4x^0 is automatically generated. Having established the simple rules of deriving a polynomial from metapatterns, new data that is not conducive to Base Number Sheets, such as positive and negative non-integer coefficients are still derivable by the rules. View-only spreadsheet at .


The ability to visualize numbers, SubQuanning, observe the metapatterns, see the importance of differences, and then derive the polynomials lay the foundation for a rich, powerful, and easily understood foundation for science, technology, engineering, and mathematics (STEM)!!

Four Steps to Polynomial Derivation (steps 1 and 2)
posted 12/3/2010

1) SubQuanning & 3-D SubQuanning: Learn how to name quantities in containers.

The foundation of numbers must start with recognition. Traditionally this is taught at the same time as the alphabet. (See 'Teaching Counting') Regardless of where an individual is in their understanding of numbers, the recognition of numbers by SubQuanning is absent due to SubQuanning's recent discovery. In 2008, was developed as a prototype to enable number recognition by Subquanning. A visit to this site reveals five discoveries, named Discovery 0 through Discovery 4. For each Discovery there are three buttons: discover, hone, and master.

The discover button reveals what is to be covered. The hone button generates a more varied sample for discovery, and the mastered button is not currently enabled. Briefly, Discovery 0 establishes a base line for an individuals recognition and key-press response times as the numeric symbol is displayed along with a SubQuanned quantity. Discovery 1 allows individuals to discover that they can recognize quantities instantly (This is very close to the concept of subitizing except for the organized layout.). Discovery 2 displays two base A (ten) digits to scale. Discovery 3 displays two digits in a random base, and Discovery 4 displays three normalized base A integers.

Either using physical or immersive manipulatives enables the discovery of a 3-D naming convention: segs, squares, cubes, seg of cubes, square of cubes, cube of cubes, seg of cube of cube, ...

The naming discovery starts with recognizing the cube and square shapes then reducing the dimensionality to one, usually garnering the name 'line' for the first container. However, lines are infinite which leads to the term 'line segment' and when reduced to the monosyllable 'seg'. It is not a big step to recognizing a similar big seg to the left of the cube which is appropriately called a seg of cubes. The smallest unit, which does not fill a container is accurately called 'remainder'.

2) Metapatterns: Observe that different quantities in different sized containers can have the same name.

Perhaps the easiest of all steps is recognizing the same SubQuan in different bases. Although the quantities are different, the same expression of SubQuan can be identical. The SubQuanned metapattern, 56, pronounced five-six, is readily apparent in the following three bases.

Avatar Physical Interaction
posted 6/2/2010
I continue to think about how t make avatar interactions more real and personal. I definitely believe that creating a range-of-motion animation where we move the left arm around in a hemisphere and the left arm around in a hemisphere we can identify the ideal animation to engage so that avatar to avatar interactions align. For example a tall and short avatar will interact with different animations than two tall or two short avatars.


Rebecca's Blog Posts:

DR @OSU ELC on AdobeConnect
posted 5/29/2012

The theory of subQuan - for Emergent Learning Commons

Cooper and I presented on Adobe Connect today.  If you would like to peruse the slides, click the link below.  I will attach the recorded link once it's been uploaded.  Great fun today!!

The theory of subQuan

Subitizing: What is it? Why teach it? (1999)
posted 5/28/2012

Excerpts from Douglas H. Clements

Subitizing is "instantly seeing how many." From a Latin word meaning suddenly, subitizing is the direct
perceptual apprehension of the numerosity of a group. In the first half of the century, researchers believed that
counting did not imply a true understanding of number but that subitizing did (e.g., Douglass [1925]). Many saw
the role of subitizing as a developmental prerequisite to counting. Freeman (1912) suggested that whereas
measurement focused on the whole and counting focused on the unit, only subitizing focused on the whole and the
unit; therefore, subitizing underlay number ideas. Carper (1942) agreed that subitizing was a more accurate than
counting and more effective in abstract situations.

In the second half of the century, educators developed several models of subitizing and counting. They
based some models on the same notion that subitizing was a more "basic" skill than counting (Klahr and Wallace
1976; Schaeffer, Eggleston, and Scott 1974). One reason was that children can subitize directly through
interactions with the environment, without social interactions. Supporting this position, Fitzhugh (1978) found
that some children could subitize sets of one or two but were not able to count them. None of these very young
children, however, was able to count any sets that he or she could not subitize. She concluded that subitizing is a
necessary precursor to counting. Certainly, research with infants suggests that young children possess and
spontaneously use subitizing to represent the number contained in small sets and that subitizing emerges before
counting (Klein and Starkey 1988).

But how is it that people see an eight-dot domino and 'just know" the total number? They are using the
second type of subitizing. Conceptual subitizing plays an advanced-organizing role. People who "just know" the
domino's number recognize the number pattern as a composite of parts and as a whole. They see each side of the
domino as composed of four individual dots and as "one four." They see the domino as composed of two groups
of four and as "one eight." These people are capable of viewing number and number patterns as units of units
(Steffe and Cobb 1988).

Children use counting and patterning abilities to develop conceptual subitizing. This more advanced ability to
group and quantify sets quickly in turn supports their development of number sense and arithmetic abilities.

The spatial arrangement of sets influences how difficult they are to subitize. Children usually find
rectangular arrangements easiest, followed by linear, circular, and scrambled arrangements (Beck-with and Restle
1966; Wang, Resnick, and Boozer 1971). This progression holds true for students from the primary grades to

Certain arrangements are easier for specific numbers. Arrangements yielding a better "fit" for a given
number are easier (Brownell 1928). Children make fewer errors for ten dots than for eight when dots are in the
"domino five" arrangement but make fewer errors for eight dots when using the "domino four" arrangement.
For young children, however, neither of these arrangements is easier for any number of dots. Indeed, twoto
four-year-olds show no differences among any arrangements of four or fewer items (Potter and Levy 1968).
For larger numbers, linear arrangements are easier than rectangular arrangements. It seems, then, that most
preschool children cannot subitize conceptually. Instead, they count one by one. By school age, they can learn
to subitize conceptually, although first graders' limits for subitizing scrambled arrangements is about four or five
items (Dawson 1953).

If the arrangement does not lend itself to grouping, people of any age have more difficulty with larger sets
(Brownell 1928). They also take more time with larger sets (Beckwith and Restle 1966).

Use conceptual subitizing to develop ideas about addition and subtraction. It provides an early basis for
addition, as students "see the addends and the sum as in 'two olives and two olives make four olives' " (Fuson
1992, 248).

Children can use familiar spatial patterns to develop conceptual subitizing of arithmetic. For example,
students can use tens frames to visualize addition combinations (fig. 4). Such pattern recognition can assist
students with mental handicaps and learning disabilities as they learn to recognize the five- and ten-frame
configuration for each number. "These arrangements ... help a student first to recognize the number and use the
model in calculating sums. It is this image of the number that stays with the student and becomes significant"
(Flexer 1989). Visual-kinesthetic finger patterns can similarly help, especially with the important number
combinations that sum to 10.

"Subitizing is a fundamental skill in the development of students' understanding of number" (Baroody
1987, 115). Students can use pattern recognition to discover essential properties of number, such as conservation
and compensation. They can develop such capabilities as unitizing, counting on, and composing and decomposing
numbers, as well as their understanding of arithmetic and place value-all valuable components of number sense.

Baratta-Lotton, Mary. Mathematics Their Way Menlo Park, Calif.: Addison-Wesley Publishing Co., 1976.
Baroody, Arthur J. "Counting Ability of Moderately and Mildly Handicapped Children." Education and Training of
the Mentally Retarded 21 (December 1986): 289-300.
______ Children's Mathematical Thinking. New York: Teachers College Press, 1987.
Beckwith, Mary, and Frank Restle. "Process of Enumeration." Journal of Educational Research 73 (1966): 437-43.
Brownell, William A. The Development of Children's Number Ideas in the Primary Grades. Supplemental Educational
Monographs, no.35. Chicago: Department of Education, University of Chicago, 1928.
Carper, Doris V. "Seeing Numbers as Groups in Primary-Grade Arithmetic." Elementary School Journal 43(1942):
Chi, Michelene T. H., and David Klahr. "Span and Rate of Apprehension in Children and Adults." Journal of
Experimental Child Psychology 19(1975): 43439.
Clements, Douglas H. "(Mis?)Constructing Constructivism." Teaching Children Mathematics 4 (December 1997):
Clements, Douglas H., and Leroy G. Callahan. "Cards: A Good Deal to Offer." Arithmetic Teacher 34(1986): 14-17
Davis, Robert B., and R. Perusse. "Numerical Coinpetence in Animals: Definitional Issues, Current Evidence, and a
New Research Agenda." Behavioral and Brain Sciences I (1988): 561-79
Dawsan, Dan T. "Number Grouping as a Function of Complexity." Elementary School Journal 54 (1953): 35-42.
Douglass, H. R. "The Development of Number Concept in Children of Preschool and Kindergarten Ages." Journal of
Experimental Psychology 8 (1925): 443-70.
Fitzhugh, Judith I. "The Role of Subitizing and Counting in the Development of the Young Children's Conception of
Small Numbers." Ph.D diss., 1978. Abstract in Dissertation Abstracts International 40(1978): 4521 B4522B.
University microfilms no.8006252.
Flexer, Roberta J. "Conceptualizing Addition." Teaching Exceptional Children, 21(1989): 21-25.
Freeman, Frank N. "Grouped Objects as a Concrete Basis for the Number Idea." Elementary School Teacher 8 (1912):
Fuson, Karen C. "Research on Whole Number Addition and Subtraction." In Handbook of Research on Mathematics
Teaching and Learning, edited by Douglas A. Grauws, 243-75. New York: Macmillan Publishing Co., 1992.
Gelman, Rochel, and C. R. Gallistel. The Child's Understanding of Number Cambridge: Harvard University Press,
Ginsburg, Herbert. Children's Arithmetic. Austin, Tex.: Pro-ed, 1977.
Klahr, David, and J. G. Wallace. Cognitive Development. An Information Processing View Hi I lsdale, N. J.: Lawrence
Enbaum Associates, 1976.
Klein, Alice, and Prentice Starkey. "Universals in the Development of Farly Arithmetic Cognition." In Children's
Mathematics, edited by Geoffrey B. Saxe and Maryl Gearhart, 27-54. San Francisco: Jossey-Bass, 1988.
Mandler, G., and B. J. Shebo. "Subitizing: An Analysis of Its Component Processes." Journal of Experimental
Psychology: General 111(1982): 1-22.
Markovits, Zvia, and Rina Hershkowitz. "Relative and Absolute Thinking in Visual Estimation Processes."
Educational Studies in Mathematics 32 (1997): 29-47.
National Research Council. Everybody Counts. Washington, D. C.: National Academy Press, 1989.
Potter, Mary, and Ellen Levy. "Spatial Enumeration without Counting." Child Development 39 (1968): 265 72.
Schaeffer, Benson, Valeria H. Eggleston, and Judy L. Scott. "Number Development in Young Children." Cognitive
Psychology 6(1974): 357-79.
Silverman, Irwin W., and Arthur P Rose. "Subitizing and Counting Skills in 3-Year-Olds." Developmental Psychology
16 (I 980): 539-40.
Salter, Aletha L. J. "Teaching Counting to Nursery School Children." Ph.D. diss., 1976. Abstract in Dissertation
Abstract International 36 (1976): 5844B. University microfilms no.
Steffe, Leslie P., and Paul Cobb. Construction of Arithmetical Meanings and Strategies. New York: Springer-Verlag,
Steffe, Leslie P., Patrick W. Thompson, and John Richards. "Children's Counting in Arithmetical Problem Solving."
In Addition and Subtraction: A Cognitive Perspective, edited by Thomas P Carpenter, James M. Moser, and
Thomas A. Ramberg. Hillsdale, N. J.: Lawrence Earlbaum Associates, 1982.
von Glaserfeld, Ernst. "Subitizing: The Role of Figural Patterns in the Development of Numerical Concepts." Archives
de Psychologie 50(1982): 191-218.
Wang, Margaret, Lauren Resnick, and Robert F. Boozer. "The Sequence of Development of Some Early Mathematics
Behaviors." Child Development 42 (1971): 1767-78.

The Number Line is a Cultural Construct
posted 4/28/2012

Michel Paul, from the Math 2.0 discussion group, posted a great article link about research documenting that the number line is not an innate construct, but rather a cultural one.  The following is the initial post:

OK, this has me curious. Animals other than humans can subitize. Chimps can subitize better than humans. There does seem to be a kind of fundamental number sense that is hardwired into some animals that include us. 
However, this article points out the very interesting fact that the number line we consider so fundamental is a cultural construct.
"After confirming the Yupno participants' understanding of numbers with piles of oranges, the researchers gave the number-line task to ..."

OK, so they verified an 'understanding of numbers' before testing the number line concept. 
Wouldn't subitizing have a lot to do with how we can, for example, correspond our fingers with groups of objects? So maybe that's why chimps can subitize better than us? They can manipulate their toes as though they were fingers?
-- Michel

And to that is my response:

I noticed that it is the ordinal process that seems to be in question, not the natives ability to understand quantity.  This leads me to believe that teaching our kids first how to count instead of random (or in our case organized- subQuan) subitizing leaves them at a disadvantage with their innate ability to understand number before symbols are introduced.  Is our culture programming over our natural capabilities?  Have we simply jumped too quickly to the abstract in our educational system and not solidified the correspondence of our fingers (and toes - heehee) with groups of objects.

    "These findings suggest that how we think about abstract concepts is even more flexible than previously thought and is profoundly affected by language, culture and environment," said Nunez.

As a culture, we have done the same thing with locking ourselves to one base system and linking our number vocabulary to this system.  At Dream Realization, we are able to focus on single digit vocabulary (not tens, hundreds, thousands...) just as you would input into any interface: remote, ATM, phone (six five four).  This way we can talk about containers of different sizes (bases) and show how numbers make beautiful patterns.  Some would call this an abstract concept, but in actuality it makes more sense than keeping these blinders focused solely on base ten.

    "Our familiar notions on 'fundamental' concepts such as time and number are so deeply ingrained that they feel natural to us, as though they couldn't be any other way," added former graduate student Cooperrider. "When confronted with radically different ways of construing experience, we can no longer take for granted our own. Ultimately, no way is more or less 'natural' than the Yupno way."

Subitizing is natural.  Dehaene, in his book "The Number Sense", discusses research projects involving crows, horses, rats, chimps, pigeons, cats, dolphins, parrots, and Macaques.  He also noted some of those animals having additive abilities (even exposing the hoaxes).  He calls their ability of animals to understand number "fuzzy math", they resort to approximate counting without words or symbols. 

SubQuan is relying on that natural ability of the eyes.  Notice below how grouping the numbers by digits, each "shape value" can be subitized and a large number not only quickly recognized, but also if the pattern is repeated in multiple bases (from data) then there results a metapattern: the foundation of algebra.  In this picture a quartic equation is shown in shapes and equations, with the help of a base-translator controller.  Thanks to the designs of D. Cooper Patterson, this is possible in a virtual world.

I was also quite fascinated by the representation of time (past, present, & future) of the indigenous people.  Definitely a cultural thing.  Read the article to find out more!!  :-)

First sQi3d Sushi Bar Group Presentation
posted 4/10/2012

Dream Realizations just finished a guest presentation with Rawlslyn Francis from Florida State College at Jacksonville via Second Life live at the 23rd International Conference on College Teaching and Learning (

We talked about our tools and content, but most importantly, we highlighted our sQi3d Sushi Bar and its capabilities.  With a maximum capacity for 42 avatars, we can give synchronous lessons and group presentations. If you would like to schedule a presentation to your institution, now would be a great time!

Come to Boise State EdTech Island to visit our group learning station located at:

In other news, we have new videos up on YouTube.  Check out the DR playlist:

Thoughtful Ponderings from a Cognitive Engineer
posted 3/23/2012

Cooper's post to the Math 2.0 group:

I am not a math professional. I am an engineer by schooling, twice in fact. However, I have a problem right away with the referenced article by Peter Gray.

"The first step in coming to grips with math is to knock it off its pedestal. The real-life problems that are important to us are problems like these: Whom should I marry? Should I marry? Should gays be allowed to marry? What career should I go into and how should I prepare for it? If I invent gizmo X, will people buy it? Should corporations have the same constitutional rights as individuals? What's the best way to unplug the toilet? Math plays little if any role in solving such problems, nor do such problems have clear-cut right or wrong answers, demonstrable by some formula."

He advocates knocking math off its pedestal based on his alleged belief that math doesn't address a list of real problems he states. Hmmm, let me apply my engineering eye to these and tell you what I see.

"Whom should I marry? and Should I marry?"
It seems prudent to me, growing up in a family of seven kids and seeing many marriages not only within but without, that by pure observation of those that are successful versus not, certain 'math' concepts come to mind.
1) Compatibility: HOW MANY activities do they agree on (from vacation to cleaning to raising children, etc.)
2) Handling finances: nothing is more MATHEMATICAL than handling money and, I will leave this exercise to the more robust researches in the group, that money problems are one of the top causes of divorce.
"Should gays be allowed to marry?"
Hmmm, sounds like a religious question. Psalm 1 comes to mind. But back to math and more specifically the numbers involved: how many gays want to get married? What is the correlation between married gays and the rest of us? How does this affect our economy? Every single attribute in this debate can be assigned a numeric consequence except the religious view. And for those, I recommend you start with Psalm 1 before you go into the law making business. Just a thought, not a request for a religious discussion in Math 2.0. 
"What career should I go into and how should I prepare for it?"
If your still reading, bless you. This is the ultimate statement of ignorance about math and life. Your career should: 
1) make you happy (you better have a list of importance on what makes you tick. You should CORRELATE an estimation of your desires with those that your career choice offer. Just taking an exam for determining your fit for a career involves substantial mathematics on analyzing your psychological makeup. 
2) provide for you. Again, back to finances. Will you come out positive in life. Nothing could be more 'math' than this!!!
"If I invent gizmo X, will people buy it?"
OMG. I have started six companies and worked for thirteen others. We were always analyzing what it took to get people to 'buy' an idea or product and it was ALL MATH: % of people you could reach, % of people that would be interested once reached, % of people that would watch a presentation, and so on. OMG, how could anyone answer this without math!!!!
"Should corporations have the same constitutional rights as individuals?"
This is getting old but I will persevere. WHAT DO THE NUMBERS SAY? How many conflicts will they have? How will resources change do to this? What is the MAGNITUDE of this resource shift? How will this affect the economy? In otherwords, what is the financial bottom line on every strata of society? Where does the money flow from this decision, because if money doesn't change its flow, then no one will be pushing for it. Hmm, that just might be my opinion, but possible a wise one coming from decades of observing mathematically based solutions versus emotionally based solutions.
"What's the best way to unplug the toilet?"
Obviously asked with the purest ignorance of engineering, and thus math.
Two choices: physical or chemical. 
Two concerns: financial and capability to perform the act.
I hope the real math, the engineering of the plunger that eventually leads to common acceptance then to common sense. Or maybe its the size of the toilet outlet, 2 1/2" output toilets clog much, much easier than 3". But what sane person would want to know that math when purchasing a new toilet because they got tired of unplugging their old toilet!!!

"Math plays little if any role in solving such problems, nor do such problems have clear-cut right or wrong answers, demonstrable by some formula."
Well, I guess someone that make statements like these did not get a GOOD math education. Otherwise, they would see math applications in every single thing they say or do. Maybe just not the math they were taught.

Math 2.0 people. You have your work cut out for you. Math is not on a pedestal. Is is under your feet. IT IS THE FOUNDATION of a full and productive life, even if you just want to produce smiles and happiness.

And a selfless plug. Dream Realizations, ItOnlyTakes1, and Cognitive Engineering Laboratories (CERLabs) are dedicated to bringing recent and on-going research, on how easily the brain 'sees' very large numbers, number shapes and algebraic expressions when presented properly, to the masses. Please check out and join us at Dream (don't miss the 's' on the end.)


I believe my intent has been misunderstood. I do not believe that everything SHOULD be viewed through a mathematical lens. I intimated that everything COULD be viewed mathematically and took objection to the article that gave several examples where the author claimed that math was irrelevant and therefore shouldn't be worshiped. I agree it shouldn't be worshiped. Math is not on a pedestal or should ever be, but it is incredibly foundational. Not only to us but apparently to birds and other mammals as they all seem to have developed the ability to recognize and instinctually operate on small numbers. That is the area of my research and my life's recent work. That is what I believe should become the core of mathematics education: recognize numbers, especially their place shapes, observe how numbers can be used,to predict the future, and observe how meta patterns form the basis for polynomials and calculus. What students do after that is up,to them, but they will know that numbers can help them predict the future.  (I'm hoping I rattled the hornet's nest again.). Sweet dreams.

DR @VWBPE: The Poster Site, part 3 and more... 
posted 3/17/2012

In addition we are also running a continuous competition that will go on after the conference, so take a look at our sign to the right of the bar.  We would appreciate having you participate.


Special thanks go out to the institutions that have helped house our learning stations: University of Washington Avalumni, Boise State EdTech Island, Oregon State University Pixel Mtn., University of Oregon CLIVE, and ARVEL @CAVE Island.  If you or your institution would like to host one of our learning stations, please contact

We also offer individual and group tours and lessons of our material.

DR @VWBPE: The Poster Site, part 2 The sQi3d Controller  posted 3/17/2012

The sQi3d Sushi Bar may seem like just a quaint place to sit, but those placemats are not your ordinary placemats.  They serve up numbers like you’ve never seen before. 


The sQi3d (subQuan interactive 3d controller) talks with our 3d placemats.  Visitors may play with different modes: subQuan, quantity, base, shape, color.  The device will speak to a specific placemat (6-A) or multiple mats if base ‘x’ is chosen.


A look at our menu to the left of the bar will highlight the sQi3d’s capability.  We welcome you to play and discover patterns.


The 3d base sheets start on the far left at base six and runs the length of the bamboo bar to base ten (A) on the far right.  You can imagine the prim usage, so we had to disable 2d mode in addition to not displaying bases two through five.


Here we have a quartic equation: 5x4 +6x3 -4x2 -2x +8, but don’t let me kill your synapses. Come discover for yourself.  You can “see” it!

DR @VWBPE: The Poster Site, part 1
posted 3/17/2012

Dream Realizations, in conjunction with ItOnlyTakes1 and CERLabs, participated in VWBPE12 in many ways.  With four presentations and a very creative Poster site, we provided the audience with the ability to “see” numbers, how their “new view” applies to Algebra and beyond, and shared the interface on which Cooper Macbeth has been so diligently working.


Notice the sQi3d Sushi Bar (pronounced ‘squid’) as our backdrop coinciding with this year’s Japanese theme: Be Epic!  Our subtle and yet powerful programming lay hidden for only the intrepid and curious visitor.  Let me give you a tour of our relaxing and “eye-opening” Poster Site.


The first view of our platform is a space for rejuvination and collaboration with our tai chi poseballs and soothing spa-like atmosphere.

In one corner, you will find a small bamboo table with a blue orchid bonsai centered between two incredibly effective artifacts.  To the left is what we call The Hands.  The Hands is a binary counting tool that teaches color, shape, container (base), and number.  This device was offered to all our session attendees on Saturday’s 3-hour Workshop.  To the right is our standard Base Number Sheet (BNS).  This sheet happens to be in base 6 and is controlled by a HUD or a sQi3d Controller.  Above the table, on our only media prim, is our Prezi introducing subQuan and our need for it in math education, which can be found at


Moving along the back wall, you will find information about our 501(c)(3) Dream Realizations, contact points to follow our developments and get involved with us (, and two screens showing our activities around SL.


Don’t forget to spend some luxurious time in the spa!  But let’s turn our attention now to the heart of our undertaking: The sQi3d Sushi Bar.

What is Number Sense? Is it the same as number concept?
Posted 2/14/2012

Thanks to Mitzi on the LinkedIn group Math, Math Education, Math Culture for posting this question and a special thank you to Colin McAllister for letting me know about it!  Here are a few of the responses for you to peruse:

Colin McAllisterMitzi, It seems to me that "number concept" is an abstract or intellectual understanding, and that "number sense" is a skill that one applies to perceiving and manipulating numbers, i.e. counting. The simple answer is that they are not the same. They differ in the same way as the dictionary definitions of "concept" and "sense" differ. A computer programmer might use number concept when designing a program algorithm, and number sense when checking the output of a program for obvious errors.

This reference may be of relevant: subQuan (sub'-kwän) is the ability to perceive at a glance a quantity much larger than seven by organizing the items into rows, columns, and containers. (From the Latin: subitas quantitas)
There are links on that wiki page to subQuan research and to a Ning network

Henry SchafferMy favorite example of number sense is "The Back of the Envelope".

@Colin - subQuan seems to emphasize the visual perception which I think is part of number sense.

Lynne IpinaPerhaps you would be interested in the "strands" of numerical fluency from the report "Adding It Up: Helping Children to Learn Mathematics. See pages 115-117:

Rebecca Reiniger Mitzi, Sounds like you have hit the same problem that I have in investigating the various definitions of "number sense". I have seen it also as number instinct and quantity sense. I would love some of your research finds where they discuss the variances.

We here at Dream Realizations are tapping into the natural ability of animals and humans to recognize small quantities by subitizing. Stanislas Dehaene collected research findings that show birds, mammals, and babies can distinguish from 3-5 objects and with training can reach up to 15. We use this natural sense, or instinct, and organize the objects, so that your eye can see larger numbers. Numbers have shape. Not only in base ten, but in ANY base. This opens up a world of possibilities as Algebra becomes visible. I put together a Prezi to explain this a bit further:

If you can call color a concept, then yes, number is a concept. But if color is a sense, then so is number.

@Colin - Thank you for the accolades! We are very concerned about the focus on counting because it obscures the instinctual ability to subitize, which is proving to have a profound impact on understanding Algebra and higher mathematical concepts. We had the darndest time getting 5th graders to STOP counting and let their eyes do the work. :-)

@Henry - You are absolutely right! The visual perception is emphasized in subQuan, but it is also auditory: three syllables = three digits = three terms of an algebraic expression. Very cool concept: to exploit our natural number sense!

Check out the group page if you're interested in more or adding your own comment:

Or feel free to comment here!

CERLabs Learning Environment Prototype
Posted 1/18/2012

Now that we are speaking at CO12, Connecting Online Conference, Cooper has been working tirelessly on the design and implementation of a learning environment in Second Life.  We currently have eight Universities willing to house these rotundas where we will also advertise their programs.  We hope to greatly increase the virtual world traffic for the benefit of all entities.  Our first prototype can be found at the University of Washington: Contact us for a tour as it is still under construction.

This first lesson, Place Shape vs. Place Value, will be the main topic of our presentation at CO12.  You can see an explanation of this concept during the first part of our Prezi.  We continually brainstorm additional lessons that take advantage of the 3D realism in Second Life.

A fascinating thing about 3D lessons is the active involvement of participants in their own learning.  These environments are conducive to: 1) teachers wishing to understand the concept of subQuan in order to apply it in their classrooms, 2) college students desiring to have a deeper understanding of why algebra works, or 3) anyone interested in learning how to use a virtual environment to enhance the education process.  Avatars participating in the lesson receive a HUD, which allows them to interact with the self-paced lesson.  Try your newfound skills and discover the shape of numbers!

If you can join us at CO12, please do: WiZiQ on February 3rd @ 6pm PST.  Follow this link to sign up.

If you would like to learn more or get involved in lesson design, please do not hesitate to notify Cooper, Anna-Marie, or me.

subQuan Intro via Prezi
Posted 12/14/2011

I've been working on a Prezi introduction for DR and subQuan to give viewers and idea of our direction. It is still in rough form as I need new pictures in accordance with the new BNS updates that Cooper is so diligently working on. It's quite a challenge to get everything up and running after back surgery and a root canal. Ugh! Our prayers are with you, Cooper!!

Anyway, take a peek and give me some feedback about cohesiveness, understandability, and flow. Also, let me know if it induces motion sickness. I will continue to make changes and upgrades and post my revisions here to keep you up to date!

Thank you in advance for your comments!

  Here's the link!

3D GameLab
- The Future of Gaming
Posted 8/6/2011
          Learn by doing!  What a concept!  I think kids are interested in producing, not just consuming.  I have heard some individuals say that games are only fun for a while then people/children will get bored and want to do some higher-level learning.  I think that it all works together: another tool.  How boring to do one activity every time, with games the versatility is so broad.

            I have often thought that mathematics is one of these heavy textbook driven courses: giving more information than is used at the time of student need.  That is why I am feeling called into the mission of changing the paradigm of math education.  I believe that by changing the focus of math in the elementary grades, we could, conceivably, have math through Algebra taught before children get to middle school.

            Imagine not being required to take 4 years of high school math.  Not having math as a stand-alone class except as an elective.  Incorporating math into all other disciplines.  Imagine students with the ability to grasp estimates and have a great number sense.  This is exactly what Dream Realizations is working toward and I believe that gaming is a very large part of it all.

I really like the idea of games that are not just for playing, but also for creating and producing: interaction and emotion.  They leave you vulnerable to brainwashing, as David Perry explained here:

So, we need to be careful with content, but to be able to have a deep, emotional pull that substantiates situational, experiential learning is a powerful tool for education.  This leads the educator to be careful, thorough, and thoughtful in design of games and tools used in the classroom.

I’m not sure if I want to be involved in something virtual that can be classified as “better than life”.  I would rather be involved in something that enhances life and makes some of it easier and more rewarding in the internal aspect.

Games and their rewards are motivating for humans and being able to measure all those points of data allows for specified reward system to build and engage learners.

Tom Chatfield – “7 Ways to Reward the Brain”

The power of virtuality

1.  Experience bars measuring progress

2.  Multiple long- and short-term aims (calibrated slices)

3.  Rewards for effort (credit for trying, too)

4.  Rapid, frequent, clear feedback (learn the lesson and move on)

5.  An element of uncertainty (this lights the brain up – transforms the levels of engagement - Dopamine)

6.  Windows of enhanced engagement (memory & confidence)

7.  Other people!

Business – recycling and education (real time energy meters)

Education – grand continuity (small tasks and calculated randomness with rewards)

Government – financial rewards to remove obesity and more

ENGAGEMENT – individual & collective

Hello all!  Posted 12/3/2010
Welcome to Dream Realizations! We are looking forward to some interesting conversations regarding Math Visualization! I will be posting some research abstracts in the future to talk about and disseminate as to how they relate to subQuanning and I would love your opinions.

Have a blessed day!

Forum retrievals from the ning....

posted May 1, 2013, 11:10 AM by Rebecca Lynne Patterson

Can some students not learn Algebra? posted 11/26/2011 by Rebecca

Dan Meyer posted an interesting and highly commented on blog post  this week:

"Last April, fourteen of Palo Alto High School's twenty math teachers petitioned their school board [pdf] against raising graduation requirements to include Algebra II:

We live in an affluent community. Most of our students are fortunate to come from families where education matters and parents have the means and will to support and guide their children in tandem with us, their teachers. Not all of them. [..] We are concerned about the others who, for reasons that are often objective (poor math background, lack of support at home, low retention rate, lack of maturity, etc) can't pass our Algebra II regular lane course. Many of these are [Voluntary Transfer Program] students or under-represented minorities."


Many of the comments to this blog post talk about the usefulness of Algebra and specifically Algebra II.  As it is taught currently, I would tend to agree that's it's abstractness is not necessarily used on a daily basis.  I do, though, believe that if algebra were more easily understood, it could be applied in more situations in order to increase successful outcomes.

This is where DR comes in!  We are needed now more than ever.

Brain Calisthenics for Abstract Ideas
posted 6/10/2011 by Rebecca

I read a fantastic NYTimes article this week:


It talks about how our intuition is underused and that the human eye can recognize patterns quickly in our subconscious even before we understand it ourselves: perceptual training.  Quoted studies include:

1) Matching equations to graphs (quickly)

2) Liking a better deck of cards when gambling

3) Distinguishing painting styles [I like this one the best!]* 

4) Fraction building (quickly - repetition)

5) Slime-mold for global warming experiment


Patterns, patterns, patterns....they are what life is all about.  It is so nice to find research that is, yet again, substantiating the direction of Dream Realizations. 

*3) Distinguishing painting styles [I like this one the best!]  Seeing multiple painting styles instead of just concentrating on one artist alone allows the eye to pick up the pattern naturally.  I relate this to base systems: if we teach all base systems, then the eye can pick up the patterns for itself and focus is off the single painter (base 10).  The world has opened up with possibilities!!


Definitely worth the read!

DR is developing a public grant to bring subQuan to the iPad
posted 4/9/2011 by Cooper

I believe the touch interface will enable the creation of an intuitive and very real graphical user interface (GUI). This GUI allows us to build seamless transitions in the user input screen based on the level of knowledge and special needs of the user. This is ideal for young students learning numerical values, words, and symbols. I also believe this might provide a solution to a problem dyslexics have with numbers.

Replies to This Discussion

Reply by Rebecca Reiniger on April 12, 2011 at 9:49pm

Just got intel on this website today: .  This may be a way to get our programming publicly funded.  We will need to dive deeper into the ramifications and upfront costs of such an endeavor. 

Math Visualization
posted 12/4/2010 by Rebecca

What does "math visualization" conjure up in your mind?  Is it only physical manipulatives?  Or is it drawing out a word problem into pictures?  We regularly step back from math and try to visualize its usage, but have we stepped back far enough to see the patterns?  Just a few random thoughts to start off a Saturday morning.  Happy weekending, all!

Replies to This Discussion

Reply by Alexsis Kamala on December 4, 2010 at 2:51pm

Seeing the patterns of numbers takes looking at the shapes numbers make, not at the symbols such as 1, 2, 3 etc. These symbols don't have the beautiful patterns that subquanning makes. When the numbers are all laid out in subquan sheets, the patterns become very apparent. It makes the other things we do in math seem so easy.

Reply by Maria Droujkova on December 11, 2010 at 7:13am

For me, visualization means RE-presenting. That is, the action of constructing, making, building, growing a visual representation of mathematical ideas based on some other representation: words, formulas, tables and so on.

One of my favorite math visualization activities is storytelling - where visualization happens in people's minds. Some math storytelling is assisted by pictures, for example, the Sona (Africa) sand drawings. We did a math club activity about them.

Reply by Maria Droujkova on January 8, 2011 at 6:17pm


I usually invite kids to talk and draw their visual and sound synesthesias. Some have very particular colors for their numbers, and also piano notes.

Smell and taste are harder to represent. You can use consistently smelling substances such as coffee and chocolate, I guess. We will try some of this when my math club resumes. This will be good "apple math" activity - that's what we call snack time math.

Reply by Maria Droujkova on January 9, 2011 at 2:26am

Cute name :-)

This discussion reminds me of the phonics article Alexander recently linked at Math Future:

He advocates for made-up systems for learning English reading. My daughter, as his son, is English-Russian bilingual, and I can only agree! And we may need a similar made-up system for learning math reading. I know many people who used such private languages with their kids. Imagine groups of parents using the same language, much like "Baby Sign" movement. Won't that be powerful?

Reply by Rebecca Reiniger on January 12, 2011 at 8:01pm

That was a fantastic video, JayCee.  I watched the second (47 min) as the first didn't allow me to view for some reason.  The human brain is really amazing.  I don't think we've even scraped the surface of our own capabilities and with neurological research coming to the fore, I don't think it will be too long before we make some huge discoveries thanks to PET and fMRI technology.

Thanks so much for posting this one!

Jonathan Crabtree said:

I taught mathematics using creative visualisation more than 20 years ago. Right brain learning is an excellent way to pre-process information before the left brain processes the symbols.

However I am now wanting to explore the development of multi-sensory learning of mathematics.

So has anybody else created a 10 value musical scale or associated quantities with tastes yet? I'm also keen to know if anybody else uses a 10 point color scale based on our visible spectrum.

Here is a short video of Daniel Tammit with David Letterman.

In his book 'Born on a blue day' Daniel talks about synaesthesia. If there is a way we can help children retain this ability, we will go a long way to helping them enjoy numbers.

Thank you for watching!


Reply by Cooper Macbeth on January 15, 2011 at 5:37am

Thanks for sharing this. I have seen other presentations on BBC and also read his book. It was disturbing to see how much we had in common as he has been diagnosed with autism and that is not what one seeks. Mais, c'est la vie. I do not know how much of subQuanning is related to his ability but I do know the wonders that are coming out of it. We have 40+ year-olds who hated math, didn't really understand Algebra and within 30 minutes they are looking at subQuan metapatterns and telling us the equations, like 3x^3+2x^2+6x+5 in less than 30 seconds. To be honest, we have no idea how far subQuanning is going to take us but it definitely get the response, "Oh my God", a lot. Sorry for my delay in replying, since we are still building the website to replace this Ning as fast as we can, I haven't checked it much. Please forgive me....Cooper

Brains vs. Computers
posted 12/4/2012 by Rebecca

The brain can be compared to computer hardware. Graphics Processing Units (GPUs) have become more widely useful in computer and video processing. “Using GPUs … gives the computer a processing capacity that competes with supercomputers over twice its size” ( , paragraph 2). The vector (3D) graphics implementation in computer hardware has increased capacity and facilitated a greater speed in computing. "GPUs are redefining high performance computing," said Jen-Hsun Huang, president and CEO of NVIDIA. "With the Tianhe-1A, GPUs now power two of the top three fastest computers in the world today. These GPU supercomputers are essential tools for scientists looking to turbocharge their rate of discovery" ( , paragraph 7). Designers are finding in the computer industry that graphics give computers a stronger foundation for processing.

            Dehaene (Jossey-Bass, 2008) describes a major upheaval in the mental arithmetic system during the preschool years when “progressing in math means storing a wealth of numerical knowledge in memory” (p. 281). This storage of numbers in the brain is different than the storage of vocabulary. Verbal memory can be powerful with its cohesiveness and connectedness. Humans possess associative brains; we can open a memory from a vague recollection, something computers have yet to master, which makes learning vocabulary easier than learning the multiplication table. Each calculation within the multiplication table hangs independent of the other multiples when memorizing and that’s where the brain’s associative nature fumbles and numbers jumble inside our head, resulting in mistaken memories, which correlates to wrong answers. So, why are we still relying on this outdated practice of memorization?

Jossey-Bass, I. (2008). The Jossey-Bass reader on the brain and learning. San Francisco: Jossey-Bass.

Virtual Worlds and Mathematics
posted 12/4/2010 by Rebecca

How can we incorporate what has been done in the past with virtual worlds, how the brain learns, and using this 3-D environment to tap into the instincts with which individuals were born and see mathematics as we have never seen it before: patterns unrecognizable without the advancement of technology?

Welcome to our Blog

posted May 7, 2012, 3:51 PM by Cooper Macbeth   [ updated May 16, 2013, 10:24 AM by Rebecca Lynne Patterson ]

Here are the old blog posts from our original Ning.

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