SubQuan+and+friends

=SubQuan and Friends: Subitizing with Unitizing= toc Editing note: this wiki keeps versions well. Edit anything and everything! To contact this essay's authors, find their names in the History tab at the top of this page and use the wiki message system. If you want to leave a comment or a question for in-line discussion that isn't a part of the article, color and sign it - MariaD

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Introduction
Subitizing, the ability to instantly recognize small quantities, has been known from ancient times. Even illiterates could play cards or dice with counters places in special groups.

In the early twentieth century, Maria Montessori was the first European mathematics educator to make a system using subitizing beyond counting accessible to young children. Such systems existed before. Indian Vedic mathematics, Oriental, Egyptian, Persian abacus-based mathematics, Mayan nepohualtzintzin system and other counting systems which are hundreds or thousands of years old incorporate subitizing.

In most settings, children are taught to count even small quantities of objects, even if they can subitize. In this video by Brendan Murphy, you can observe typical behavior: kids forgetting to count when they get into the heat of doing math. Watch the little hand on the right showing three, and the "teacher kid" on the left showing four. media type="youtube" key="zChoUnHtSWw?fs=1" height="385" width="640"

In the last few decades, several individuals and groups of people developed systems that connect subitizing with unitizing. In these systems, subitized quantities are grouped into "meta-units" that are still instantly recognizable. This approach supports strikingly beautiful and mathematically sophisticated activities. In this article, we review several of these systems. In particular, we focus on uses by different types of students, the potential of the system to support advanced mathematical topics, accessibility to young children, and the potential for mathematical and artistic creativity.

Overview of systems
In this section, we briefly describe each system and its main features. In the following sections, we analyze and compare systems along several dimension. We start with systems that apply to several arithmetic operations. Then we review two projects that focus on one topic, but are relevant because of their qualities. We end with systems that are used for algebra and other mathematical areas "beyond arithmetic."

Arithmetic system: Maya numerals (circa 400BC)


The Mayan system used a combination of subitizing to four, a symbol meaning "five" (a line), and positions, including a placeholder that played the same role as 0 does in Arabic numerals. The Mayan system's base was 20. It also had separate hieroglyphic symbols sometimes used for powers of 20. Addition and subtraction in this system was rather easy and visual, but the system did not straightforwardly support multiplication or division algorithms.

Arithmetic system: Chinese abacus (circa 1000)


The abacus uses strings of beads up to five for subitized quantities, single beads in the top row to stand for units of five, and the positional system of rods to stand for powers of ten. Because of this mix of visual and symbolic representations, it takes conceptual understanding and training of memory before abacus is understood. Abacus is used for arithmetic computations. 

Arithmetic system: Russian abacus (1700s)
This abacus uses strings of 10 beads, visually separated into subitizable fives by the black beads in the middle. Because the connection between the Arabic number system and this abacus is more straightforward and obvious, this version is used as a manipulative in Western schools. The modern 10-bead versions often separate fives by color. This abacus is sometimes further simplified to match elementary school programs by only having two rods.

Arithmetic system: Montessori beads (1900)


Montessori bead strings come in groups from one to ten and are color-coded, because most people can't subitize to ten. This approach requires memorization of color-number combinations.

Arithmetic system: Cuisenaire (1952) and Dienes (1960s)


This system, developed by Cuisenaire, is similar to Montessori, but uses colored rods, where the concept of number is based on length, rather than quantity. Here is Gattegno using it to teach young kids algebra: media type="youtube" key="ae0McT5WYa8?fs=1" height="385" width="640"

Dienes blocks, available in bases other than ten as well, use length and quantity to represent number.

Arithmetic system: Fractal abacus (1993)


In 1993, [|David Gibson published an article] describing a visual system based on fractals. Unfortunately, we are unable to locate any further information about David or his system. It uses color-coding (instead of position, as in abacus) to represent place values. This system can be modified for different bases.

Arithmetic system: Intrinsic units (2000)


This system combines familiar objects that represent quantities, and the idea of a fractal. It was used by Maria Droujkova and others in math clubs and online projects. Artists frequently make fractals of this sort. Combining fractal systems (exponents), such as the owl shown above, and simple repetition (multiplication), as the dogs shown above, is delicate and tricky for young children.

Arithmetic system: Metamorphic numerals (2011)
These artisan numbers, handcrafted by Mark A. Jaroszevicz (MAJ Design), represent the positional system in 3D. Mark writes:

The series **"[|Metamorphic Numerals]"** is best described as a multi-sensory approach to understanding math. Counting with our fingers is an ancient method that all people around the globe can relate to and communicate through. "Metamorphic Numerals" revisits the past by using measured clay spheres to form three-dimensional "pinched numeral symbols" that are calibrated to the gram. The sculptures translate Hindu Arabic numerals.

The basic shapes are created by the use of your fingers, and the number is differentiated by its mass in relation to the number of fingers used.

Numbers 1-9:

Equation 999x3:

Factorization diagram: You can count on monsters (2003)


This is less of an attempt to make a system for many topics in mathematics, and more of an artistic project, representing prime factorization as colorful monsters. It is created by Richard Evan Schwartz. Quantities are symbolized by meaningful (creature-like) shapes as well as colors. Combination of shapes means multiplication - a development coming up in several systems of the last two decades.



[|Poster information], [|book information], [|explanation of the system].

Factorization diagram: Primitives (2008)


This is a system for prime factorization, developed by Alec McEachran and described in an article, a Flash interactive and a poster, all linked from [|his site]. Alec and Richard Evan did not know of one another's work until an exchange of emails with one of this article's authors, despite visual similarities. The system focuses on the topic of prime factorization. As the monsters project, this is not a completely subitizeable system - you can see the prime 17 in the picture above.

Factorization diagram: Brent Yorgey's and friends' factor trees (2012)
Beautifully minimalist, these diagrams inspire math artists and software authors alike. You can find many listed at Brent's blog, [|Math Less Traveled].

[|An animated color version is here.]

Algebra systems: Algebra tiles and Mortensen's "More than Math" (circa 1980s)


Algebra tiles build on Cuisenaire ideas: in addition to color-coding for measure, they introduce the second dimension (using squares) to represent the second power. Note that Montessori used similar ideas, representing 100 as a square and 1000 as a cube. These systems can be used to represent linear and quadratic equations. References: [|Geoff White's page]; [|NLVM algebra tiles].

__Algebra system: SubQuan__
This picture of Cooper Patterson (Cooper Macbeth) and Rebecca Reiniger (Ute Frenburg) was taken in Cooper's Dream Realizations laboratory in Second Life (Nov 2010). The Subquan metapattern, displayed in bases 7-A, is 1234.

SubQuan, from Latin meaning 'sudden quantity', is organized subitizing. When objects are placed into containers, subitizing by digits is possible, which allows the human eye to see very large quantities very quickly and to also see their "shape": seg, square, cubes, seg of cubes, square of cubes, etc. In addition, the container size can change, permitting individuals to see quantities in various bases. This leads to recognizing identical patterns between bases, otherwise known as metapatterns, which reveal polynomic expressions. So far we have found that children as young as 10 years old can "see" polynomials. The visual nature of subQuanning transforms mathematics, removes the stumbling block students have been hidden behind, and leaves many rote memorization techniques in math education in question. Incorporating two more steps entitled "differences" and "polynomial derivation" and any polynomial equation can be found from a set of data points. Dream Realizations, building remediation on subQuanning, is still in its infancy and is interested in all assistance. Please contact: Dream Realizations Come dream with us!

__**Videos**__
[|Cooper introduces itonlytakes1.org] Cooper gives a Virtual tour of his Lab

__**Kiosk in Second Life**__
SubQuan Presentation board in Second Life

__How SubQuan works with different ages of students__
__//**Early Education:**//__ Stop relying on counting! Not only can the human eye see between 4-7 objects without counting, the eye can also see the absence of those same amounts. Training our instinctual number sense through subQuanning could alleviate reliance on rule-based processes in mathematics.

__//**Elementary Education:**//__ Why do we have to memorize the multiplication table with its 100 facts? That's 45 facts, not including the repeats, 0's, or 1's. Can you imagine if we asked kids to memorize the first 45 squares and then everything else could be derived from there? Why do we teach addition and subtraction, it's the same thing? Why don't we call it combining and use color combinations for clarification?

__//**Secondary Education:**//__ Solving equations using FOIL: why are we spending so much time on teaching so many various methods to do the same thing and most of this time is focused on quadratics? With subQuanning and polynomial derivation, these students could be computing nth degree polynomials, which makes the concentration on FOIL seem ludicrous.

__//**Post-secondary Education:**//__ Two words : math remediation! For years students have been taught the same methods for the same concepts and it still hasn't sunk in. These are called the lost kids, the math-phobes, the self-doubters. But it is not they that are blind, they can see, if given a chance. Given a chance to approach math from a completely different direction authorizes students to take possession of their learning. The shields on their eyes are removed and the fog of math dissipates.

__//**Engineering:**//__ Think of this - the ability to visualize a 27th degree polynomial.

Early childhood vectors: A is for Alligator
One can view quantities as consisting of a part that shows the numerocity, and a part that shows quality or scale. Clyde Greeno calls the numerocity part "the numerator" and the quality/scale part "the denomination." For example, a place setting may contain 1 knife, 2 spoons and 3 forks and can be considered a three-dimensional vector. Settings for five people will need 5k+10s+15f, and this "vector algebra" is quite accessible to young kids.

"Zoo-gebra" game is played with a box full of toy animals: H is for Hippo, G is for Giraffe, Z is for Zebra, C is for Camel; B is for Bear ... a {G,C,B,Z,H}-basis vector-space. It requires digit and English-letters alphabet, possibly posted on the wall, and for the warm-up, all sing the alphabet song. PLAY One player at a time: close eyes; grab a handful of animals; put those on the table. NATURAL RESPONSE: typically, children sort toys into kind-classes before ever being prompted to do so. CHALLENGE: write down whatever combination you have ... and do/say it right, to win that many raisins. [Too young? use alpha-numeric tiles. Old enough? use whiteboards.] RESULTS: children's "written" linear combinations, like 4C+3Z+1B+0Z+0H. Return toys to "cage" --> next player, etc. OPTIONS: if 2-digit vocabularies are owned (0-99), then vector addition and subtraction happen without being "taught" ... likewise for scalar multiplication (as repeated addition) and for scalar (remainder) division by digits (divisor = the number of players). The child internally achieves a personal theory of linear algebra ... all from learning to play kindergarten zoo-gebra. "Hide-and-seek" equation game is also great with animal counters. Some animals go for a walk (say, two elephants and three antelopes) and then kids all close their eyes, except for the kid who sets up the problem. When they open the eyes, only 1 elephant and 2 antelopes are out, the rest hiding. How many are hiding? This is a well-loved game that can be played with toys or even with kids themselves hiding in a playhouse. It can incorporate written, drawing or oral responses, depending on the age, or kids showing the answer using extra counters or prepared animal portrait cards.

Fractions and SubQuan connection
An obvious extension is into the world of fractions. The five major models of fractions are: quotients, parts&wholes in sets, multiple units of measure, stretch/shrink operators, and intensive quantities. The vector approach focuses on fractions as units of measure, with the numerator showing the number of units. For example, 3/4 means "three 1/4ths." Cooper Patterson: "I like to use a relaxed Latin-to-English translation of numerator and denominator with high school students of all abilities as it appears to facilitate their 'overcoming' brute memorization to afford some understanding of fractions (as attested by their explanations of 'why' 1/5 + 2/3 = 13/15). The relaxed translation for numerator is 'the numberer': the one who quantifies named-things. The denominator is 'the namer': the one who names things." Another way for a word play: //num**B**erator// and //denomination teller//. Word-oriented kids will find these especially illuminating.

In subQuanning, the denominator is the base system (the containers) and the num(b)erator is the subQuan with respect to that base system. Consider, for example, this picture in base seven:

In vector notation, it is: 2C + 6B + 5A Where A, B, and C are powers of seven. Here 2, 6 and 5 are "num(b)erators." Visually, it is often easier to focus on "pattern errors" (missing pieces) to instantly recognize, or SubQuan, the quantity. This process does not lend itself well to formulas, but here is a possibility: 2C + (BASE - 1)B + (BASE - 2)A

Fractional bases - Egyptian fractions and beyond
Continuing the same base beyond zeroth power, we obtain fractions. One of the best resources for visualizing it for base 10 is [|Universcale].

Here is the visualization for[| Base 2 by xkcd]; unlike Universcale, it's a static picture, so zoom is impossible and the author used logarithmic scale. Kids as young as four or five have no problems understanding log scale visualizations of this sort (MariaD experiments). The same author also has a [|log scale depth visualization].

Egyptians used base two fractions thousands of years ago, and some curricula still include "Egyptian fraction" representation of numbers. Of course, binary numbers in computer science are base two.

With toddlers and young kids, base two (positive powers) can be explored using parents, grandparents, great-grandparents and so on. This is a screenshot from a Natural Math interactive family tree. Negative powers of two can be explored with folding or cutting a piece of paper, or with "pizza math."

"Vector Algebraic Theory of Arithmetic"
Some of these ideas can be found in Clyde Greeno's article "Vector Algebraic Theory of Arithmetic": media type="custom" key="7951494"

Young children and multiplicative reasoning
When multiplication is based on addition, which in turns is based on counting, this "tower of concepts" becomes unstable and young children can't climb it. In terms of metaphor studies (Lakoff and Nunez), the metaphors are linking rather than grounding. Most Western curricula reflect this phenomenon, placing multiplication after addition in the learning sequence, and introducing it to children around the age of seven or eight. Exponentiation and proportions are introduced even later, around adolescence.

Illustration from 2005 PME presentation by Maria Droujkova

However, possibilities for the growth of the number sense are now better understood. In the 1970s researchers started to look at how young children can instantly perceive quantities (subitize) and estimate composite units of length. Piaget experimented with children before the age of two, but they weren't followed up, until neuroscientists in the 1990s developed methodologies for working with newborns. There are more studies in 2000s 

The studies show that it is indeed difficult for children to build the number system based on the counting scheme (Steffe). However, one can base number systems directly on multiplicative reasoning, such as in Confrey's work on the splitting conjecture, or on subitizing, such as SubQuan. This is a shorter and straighter path than going through counting, and it can be done with babies and toddlers.

Confrey's work (and other experiments based on splitting) relied on the belief that babies and toddlers have some sort of "multiplicative sense" in them. The studies of the last few years look at early multiplicative and proportional abilities in more detail. For example, it makes a perfect sense that humans (probably animals too) have some sort of mechanism for estimating proportion just for orienting themselves in space by using their perspective-based eyesight.

Here is an example of such a study (thanks Dor): //"Developmental Science Volume 14, Issue 1 Page 1 - 161 The development of numerical estimation: evidence against a representational shift (pages 125–135) Hilary C. Barth and Annie M. Paladino// //How do our mental representations of number change over development? The dominant view holds that children (and adults) possess multiple representations of number, and that age and experience lead to a shift from greater reliance upon logarithmically organized number representations to greater reliance upon more accurate, linear representations. Here we present a new theoretically motivated and empirically supported account of the development of numerical estimation, based on the idea that number-line estimation tasks entail judgments of proportion. We extend existing models of perceptual proportion judgment to the case of abstract numerical magnitude. Two experiments provide support for these models; three likely sources of developmental change in children’s estimation performance are identified and discussed. This work demonstrates that proportion-judgment models provide a unified account of estimation patterns that have previously been explained in terms of a developmental shift from logarithmic to linear representations of number."//

Computer generated displays are required to map the three dimensions of life onto a 2-D display. Recent advances in this 3-D modeling and using polarized lenses or other left-eye right-eye splitting techniques have enabled very accurate reproduction of 3-D simulations onto 2-D screens. The recent movie, Avatar, is an excellent example of the state of the art in 3-D modeling. The reason this discussion is appropriate is that cognitive engineers are using our knowledge of computer technology to help us understand the mind. We are mapping areas of the mind that are responsible for specific 'computer-like processes' (See Dehaene's __The Number Sense__ for mapping regions related to mathematics.). Initial work done in 1981 on real-time computer graphic 3-D displays by Daniel Cooper Patterson revealed the distortions caused in the human visual system when basing the 3-D to 2-D mapping on a logarithmic algorithm. Indeed, the visual system is 'put at ease' when the mapping is done on an inverse distance algorithm. These two algorithms actually match up exactly at the integer exponents but not in-between which may be the cause for the eyes confusion. Research needs to be done to determine if it is the inherent design of the senses and their connections within our minds that is the foundation of our number sense and not a developmental stage nor an instructional consequence.

Magnitude collapse
[|From a Math Future discussion] about I[|hor Charischak's blog post].

When dealing withlarge numbers, people switch to logarithmic scale (unless they train themselves not to) and, for example, 10^9 is perceived as "the next number after" 10^8 or even 10^6. Millions, billions, it's all up there somewhere! Just looking at that counter whirl in hundreds, I was shocked at the result of my computation of when the counter will hit 10b.

Psychologists claim that natural human intuition only works well to compare quantities if the ratio stays within hundreds: http://www.psychologytoday. com/blog/alternative-truths/ 201003/understanding- earthquake-magnitude-88 I severely missed that billions/hundreds ratio of speed change in the counter.

The next question is, how can young kids learn to deal with this limitation? This involves meta-cognition based on the number sense: I knew not to trust my judgment about the App Store counter, because I recognized the number ratio is beyond my intuition range. I know for a fact I can subitize but can't subitize **. But very young kids give you very assured (and wrong) answers about quantities way beyond their subitizing range! Unless you take steps to help them learn not to do it.

This "magnitude collapse" can lead to very problematic behavior. I think this example comes from "How to lie with statistics" but I am not sure: company managers would spend half an hour arguing about spending $500 vs $800 on a shed, but sign deals in the millions range easily because they can't parse such large numbers.

Significance of computer tools
Only now, in the last few years in fact, people are starting to develop computer tools, that can make inherently multiplicative work available to kids. There is no physical tool that can model exponentiating (fractal, subquan, or other) for arbitrary bases. Montessori beads and Dienes blocks come closest, but they don't model the ACTION of splitting or grouping (moving from one power to the next/previous) in quite the same way counters model, well, counting. We are very hopeful for Kinect and touch-screen tools, especially multitouch, to open multiplicative/fractal/subitizing world to babies and toddlers.

Authors
Add your name if you edit this in any way!
 * Maria Droujkova
 * Rebecca Reiniger
 * Cooper Patterson
 * Milo Gardner
 * Clyde Greeno