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"Oh my God, I can see it." The young English professor realized, knew in her heart, that she was seeing numbers in a way she never knew existed, an instinctual way. Using her innate skill, shared with other mammals and birds, she was subitizing ordered three-dimensional shapes that represented large numbers. She was not seeing quantity for she now understood that quantity often only applies to numbers represented in base ten, the standard, the decimal number system. No, she was seeing numbers , really seeing their shapes, organized simplistically in various base systems to help trigger her latent visual talent. She was subQuanning.
To subQuan is to suddenly identify the numbers of each place shape in descending size. If the container size was 'based' on ten then she could say that she can recognize very large quantities within seconds. She was having a Rain Man experience when she saw 7,832 colored plastic boxes and stated such in less than five seconds. Later, she saw 23546 in container shapes based on seven, or simply, base seven (SEE PICTURE). SubQuans are verbalized in the same fashion as one would say a phone number - one digit at a time. That is Digit-speak. The subQuan, 23546b7, is spoken as two-three-five-four-six base seven. It is NOT spoken as twenty-three thousand, five hundred and forty six because it is a subQuan, not a number. Twenty, thousand, hundreds, and forty are unique words only valid in base ten. In fact, the decimal equivalent, the number, of the subQuan 23546 base 7 is six thousand one hundred ten.
The Theory of subQuan
The foundation for the theory of subQuan starts with subitizing - the reality that birds and mammals instantly recognize small numbers of things precisely. Two is not one, nor is it three. Newborns, as young as a few days old, have shown this instinctual ability. Dr. Stanislas Dehaene wrote a book titled, The Number Sense, that defines his proof that subitizing is innate. Later research projects have validated Dehaene's thesis of inherent number sense.
While this foundation has taken many, including Dehaene, down a path of research into the idea of approximate number sense (ANS), Cooper Patterson has instead focused on that small precise number sense (PNS). Organizing objects or 'things' into two columns seems to easily extend the ability to precisely subitize from three to more than nine. The 2008 Adobe Flash program at ItOnlyTakes1.org still allows visitors to validate their abilities, even though the restrictions to advance have been disabled (SEE EXAMPLE IN PICTURE). But the program doesn't stop there. It allows the visitor to subitize increasingly larger groups of items, eventually reaching four digit numbers. Hundreds of volunteers, from the young age of four to the upper seventies, have all shown the ability to subQuan quite quickly, many around 0.5 seconds per place shape, or 2 seconds to identify the four digit numbers.
The distinct organization of objects and the repetitive use of subitizing each place can create confusion with the true meaning of subitize so a new word was coined: subQuan.
In 2009, Cooper turned to virtual reality to continue examining the phenomena of subQuan, and discovered other extremely relevant, yet equally as easy to see, patterns.
The first pattern quickly emerged when a number of unit cubes were aligned and stacked into similar-sized groupings. The resulting three-dimensional arrangements exhibited similar shapes. Of course, the familiar large cube and square were instantly recognized by everyone who has ever seen a cube or a square. However, the one-dimensional alignment of unit cubes (Ones) defied labeling, so another word was derived quickly by participants and coined: Seg. The four primary shapes are pictured on the left. The number of Ones per Seg is determined by a base number. The number of Segs in a Square are also determined by the same base number. In fact, each larger shape is composed of the same base number of smaller shapes. For example, a Seg in base seven is composed of seven Ones. This Seg would be grouped together with six other identical Segs to form a Square of seven Segs, or simply, a seven Square. Seven Squares would be stacked to create a seven Cube composed of 7 x 7 x 7 unit cubes. Forming Shapes is the same for any base and provides the rational for creating sheets to aid in identifying subQuans and the hands-on formation of 3D shapes. These sheets are appropriately named Base Number Sheets. (SEE PICTURE)
Creating ever expanding groups, larger than Cubes, reveals a repeating pattern of Cube-Seg-Square shapes, albeit based on larger and larger Cubes. This second easily recognized pattern leads to the concept of Place Shapes, instead of the more abstract and base dependent concept of Place Value.
The third, and most important, instantly recognizable pattern to date can occur when viewing different quantities in different bases simultaneously. The pattern is actually a pattern of patterns, or a metapattern. This metapattern reveals itself when the subQuans for different bases are identical! For example, in each picture below there is the pattern: three Cubes, four Squares, one Seg, and five Ones. And thus the correspondingly derived subQuans: 3415b6, 3415b7, 3415b8, 3415b9, 3415bA. Each of the quantities in the sequence, 803, 1237, 1805, 2525, 3415, form the exact same number of each shape when displayed in base six, base seven, base eight, base nine, and base ten sheets respectively yielding the easily observable subQuan metapattern, 3415 base x.
This sequence of pictures for the numbers above depicts an unbelievable connection to modern Algebra and higher level mathematics when viewed symbolically along with the shapes expressed exponentially. A snapshot of the equation board, shown below, for these pictures shows the relevance of subQuanning: polynomial equations can be seen!
Therefore, subQuanning enables almost everyone to quickly see the relationship between a sequence of numbers and a general expression for base x. Hopefully, the understanding that any quantity NOT in the sequence can be calculated is equally clear. The importance of laying a foundation of numbers based on our instinctual ability to see subQuans will hopefully provide the motivation necessary for learning higher level mathematics rather than the blind faith memorization inherent in our current decimal- and symbol-centric mathematical instruction!
There are two powerful concepts covered in more detail in the book, Seeing Numbers that is freely available in its pre-published form on this site.
- The idea that numbers do not capture all the various understandable forms that Place Shapes can take. These other forms contain extremely valuable information that also leads to algebraic equations, but benefits from understanding differences or change. (See chapter on change)
- The concept of negative numbers is introduced in a manner already familiar to almost any child: something missing. (See chapter on rectangles)
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