The Iconic Arithmetic Calculator set to decimal digit mode shows computation using our conventional base-10 numbers with one single modification. Depth-value replaces place-value. The Image shows three iconic decimal numbers that have been added by putting them into the same container. One of these numbers is negative. Positive digits are represented as digits, while negative digits have a negative sign attached in front. The sum now need to be standardized for ease of reading. Depth-value eliminates the sequential restrictions of place-value notation. This small change allows iconic decimal numbers to retain the physical grounding provided by the Additive Principle. Digit number facts must still be memorized for sums up to ten; Ungrouping serves to decompose sums greater than ten. Other than grouping size, the rules of iconic arithmetic are the same for all bases.
The physical parallelism of depth-value notation makes the algorithms of arithmetic much simpler. With iconic containers, as many numbers as desired can be added at the same time. Positive and negative numbers can be added together at the same time. The additions at each level of nested containment can occur in parallel; there is no need to add sequentially from right to left. “Carrying” and “borrowing” both become grounded in the physical transfer of units into containers.
The addition shown in the image,
The decimal unit iconic calculator: 4936+(–275)+615=5276
The Nature of Digits
Digits encode the magnitude of unit ensembles. As labels for (or pointers to) ensembles, digits are linguistic conveniences rather than mathematical objects. The digit
Addition makes sense for unit ensembles. The addition of digits, such as in the addition tables memorized by young children, consists of a collection of 100 linguistic conventions that permit reference to the entirely obvious and intuitive results of putting units together. Of these conventions, 45 refer to actual ways that units can be combined, while 55 are artifacts of the conventional symbolic language. Specifically, neither addition to zero nor the commutativity of the order of adding things together exist in our physical world, both are baggage imposed by the attempt to define addition with abstract symbols rather than with physical actions.
There are two major benefits provided by the digit shorthand notation. We gain words and symbols to describe visual, physical ensembles,and we gain algorithms to help us to maintain a counting of ensembles when they are put together. The costs of digits include disconnection from the physical ground of addition, the overhead of memorization of 100 linguistic sentences that describe the addition of two digits, and onerous algorithms that have embedded within them the presumption that people are computers. Digits provide expressibility at the cost of cognitive effort. They also have the unintended consequence, perpetrated by our school systems but thoroughly supported by our culture, of removing sensibility from arithmetic. Perhaps the most fascinating and bizarre aspect of arithmetic using digits is that it has contributed to our culture completely forgetting the meaning of addition. Instead we have fallen into a form of idolatry, worshipping the memorization of arbitrary symbols rather than respecting natural understanding and concrete processes.
Iconic Addition with Digits
In physically grounded arithmetic, positive and negative numbers are added by putting them into the same container. Three rules (Group, Merge, Cancel) convert the result of addition into a standard form most convenient for communication. These rules are illustrated below by animations.
Grouping by Tens
Grouping puts ten units together in one container, exchanging multiplicity of units for multiplication by bounding. Since digits already represent unit ensembles, there are many ways to combine digits to add to ten. Digit combinations must be memorized. For a base-10 number system, when digits add to less than ten, the memorized result of their addition is sufficient. When digits add exactly ten, the memorized result is combined with a convention to generate a new form that is ten. Using place-value, the new form is a one in the column to the left of the original digits. Using depth-value, the new form is a container with one unit that stands in place of the entire group of ten. When digits add to more than ten, memorization, a new form and a residual are needed. The residual is a digit that represents what remains after a group of exactly ten has been removed.
The strategy of iconic arithmetic is to reduce the cognitive load of memorization of digit combinations and to minimize the complexity of the algorithms that assist counting. The tactic is to decompose digits so that groups of ten are formed. The decomposition can be visualized as returning digits to their unit ensemble meaning and collecting ten units, or it can be symbolized as the decomposition of available digits until they match one of the five digit facts for addition to exactly ten. Decomposition is not a new type of transformation, it is Ungrouping in order to access the units that are symbolically grouped together in the form of each digit. Of course, more than two digits can be added at the same time. The extreme of this occurs with unit ensembles, for which ten individual units must be placed together to form one base-10 group. Unit ensembles, however, show their magnitude directly so they impose little cognitive overhead in seeing “how many”. Digits hide magnitude, and thus require the cognitive effort of memorization.
Ignoring commutativity and addition to zero, there are 95 combinations of two or more digits that add to less than ten. The usual strategy to reduce this complexity is to add digits two at a time. Of the two digit sums, 20 add to less than ten, 5 add to exactly ten, and 20 add to more than ten. Using digits, the 20 pairs that add to less than ten must be memorized. Fortunately, they all involve smaller numbers in combination. An alternative is to count the sum rather than to memorize the sum, and children and adults often “count up” (on their fingers for example) in preference to recalling, essentially reducing digits to their unit ensembles.
A major objective for the design of humane arithmetic is to minimize cognitive load, which effectively means to minimize memorization. The iconic strategy is guided by the behavior of unit ensembles. The 20 more difficult digit facts that add to more than ten (such as
The net result is that common arithmetic requires memorization of 45 two-digit facts, while the iconic approach requires 25, all of which are the simpler facts that add to ten or less. The first animation below shows adding to less than ten. The second animation shows adding exactly to ten, and the third animation shows adding two digits that sum to more than ten. Each of these animations is an example of the base-10 digit grouping rule, which is stated below.
Digits grouping to less than ten: 3+4=7
Digits grouping to exactly ten: 7+3=10
Digits grouping to more than ten: 8+5=13
Here are two more examples of digit Ungrouping to prepare for Grouping by tens. The display objective is to construct decompositions that leave the group of ten about to be formed in close spatial proximity. The first example groups four digits together to make ten. It is not necessary to add digits two at a time when several smaller digits can conveniently be collected to form a group of ten. The two-at-a-time restriction for addition is symbolic rather than iconic thinking.
Grouping with more than two digits: 4+2+3+2=11
Next, several digits are added, in the process forming many groups of ten. Since the sum is
Multiple digit grouping: 8+9+8+3+6+7+7=48
Merging Boundaries
Merging combines boundaries that are in the same container. Any number of boundaries can merge at the same time. The Merge rule is independent of how unit ensembles are represented and it is possible to have many Merge actions occurring at the same level of nesting. This first animation shows a simple example of merging.
Merge boundaries by adding 26 to 50
This next example of merging shows many groups forming at the same nesting level, followed by many concurrent merges.
Lots of merging: 98+98+98+98+98+98+98=686
Canceling Digits
The Cancel rule annihilates positive and negative units. Since digits are unit ensembles, positive and negative digits of the same magnitude can also Cancel. The first example shows Canceling single digits.
Canceling digits: 6+(–6)=0
The second example of the Cancel rule for digits shows canceling occurring at several different levels of nesting.
Lots of Canceling digits: 69465–45064–24001=400
Exploring Decimal Digit Addition
The primary features of iconic addition are parallelism and the alternation of Group and Merge that achieves standardization. Parallelism is much more important than the efficiency it offers to do many things at the same time. Actions that can occur in parallel are independent. It is not necessary to do all available actions in one step, doing parallel actions at the same time is an optional efficiency. The great benefit of parallelism for students is that because of independence, it does not matter when or in what order each of the composite steps are performed. This means that effort can be packaged into convenient chunks, and that an entire addition can be completed by repeating the same action many times. Compare this to the sequential nature of symbolic addition. The addition of five three digit numbers requires adding five digits sequentially while remembering the running total and then carrying any digit overflow to the next column to the left, which makes the next addition consist of six digits. The process is repeated for the second and third columns while the impact of errors, slips, and forgetfulness accumulates. With iconic addition, each small step can be isolated, performed and then independently completed. The next iconic step starts again from the beginning, placing no burden on memory. Adding place-value numbers requires continuous and concerted concentration. Adding depth-value numbers requires many small steps, each disconnected from the others.
We’ll begin with a simple example,
A simple parallel addition: 24+35+13=72
The three numbers above are added by placing them into the same container, then they are standardized for ease of reading. From a conventional perspective, standardization appears to be doing what is called addition. Iconically, addition has already taken place. In conventional notation, the operations of arithmetic are confounded with the effort of making numbers readable. By keeping these different concepts separate, operations can be connected to intuition, while the management of readability can be specifically associated with the essentially arbitrary selection of representations. Students of symbolic arithmetic get confused because what they are taught to call “addition” has nothing to do with what adding actually means. We do not need to learn how to add. We need to learn how to manage the tools and representations of mathematical communication, tools which have a great diversity of efficiency and power. For example, we can choose to say “five”, or we can choose to say “the set of all sets that have precisely five members”. Both convey the same idea, but the latter should be used only when communicating with an audience that is sufficiently sophisticated to understand why the extra detail. For some occasions, holding up a hand with fingers spread is the appropriate way to say “five”. None of these forms of communication are about adding.
In the above video, boundaries are first merged. The innermost digits add to less than ten, so they are grouped. At the same time, the outermost digits add to more than ten. One digit (the
In the next example every level exhibits activity. Three of the four levels form a new group of ten, and they all Merge concurrently.
8256+3375=11631
Adding negative numbers works exactly the same as adding positive numbers. When all numbers have the same polarity, that polarity is global to the addition problem, so that none of the mechanisms change. When a positive number is subtracted from a negative number, the “minus sign” is attached to the number being subtracted. In all cases, subtraction is the addition of negative numbers. Below, as an illustration, the above addition is conducted using negative numbers.
–8256+(–3375)=–11631
The remaining three videos show visually interesting combinations of the Group/Merge alternating pattern of standardization. The first video shows a lot of parallel group and merge activity.
7887+9898+8778+8989=35552
This next example is quite similar, except that another round of grouping occurs since merging digits into each level creates new groups that are greater than ten.
9+99+999+9999+99999=111105
Finally, a well-known addition pattern.
123456789+987654321=1111111110
Decimal Digit Subtraction
Subtraction, in the form of adding negative numbers, introduces two additional dynamics: Canceling and Mixed Polarity numbers. Canceling occurs early, leaving each nested level with digits that share the same polarity. Since digits are shorthand for unit ensembles, it is often necessary to Ungroup digits to Cancel units. When two digits of opposite polarity occupy the same container, the digit with the greatest absolute magnitude is Ungrouped to match exactly the digit to be canceled. The following video illustrates simple digit Ungrouping.
Ungrouping digits for Canceling: 2–7=–5
When many digits of opposite polarity are added together, several canceling pairs can occur. Rather than add up all the positive digits and then add up all the negative digits, and then do one large subtraction, the Cancel technique opportunistically identifies small collections of digits that Cancel each other. There are four Cancel matches in the next example. The display aligns them so that they can be easily identified, but a student could just cross off the Canceling pairs to leave the minimum of addition work to do.
Canceling by pairing: 7+6–7–6–8–9+8+8+2=–11
Sometimes canceling pairs may not be apparent, so that several digits need to be Ungrouped. Here three decompositions lead to four groupings that Cancel.>
Canceling by digit Ungrouping: 9–1+9–7–4–6–6+9=3
Sometimes even the appropriate set of decompositions may not be apparent. In this example, the internal algorithm that the Calculator uses to find canceling digits does not succeed in canceling all negative digits. Two pairs are identified and Canceled, but another round of canceling must then take place to complete the job. A student too could identify cancellations in smaller steps, perhaps leaving leftovers to contribute to later Cancel actions (just as the Calculator algorithm has). The goal is not maximal efficiency (which is desirable for computers), rather the goal is to take small comfortable bites of the problem that once completed do not need to be remembered or reversed (which is desirable for people).
Canceling in two passes: –8+6+4+6–5=3
When one number is subtracted from another, digit Ungrouping can occur at every level of nesting. The initial Cancel action substantively reduces the complexity of the subtraction being performed.
Digit Ungrouping: 4837458–1622313=3215145
Mixed Polarity Numbers
Depth-value numbers support different polarities at different levels of nesting. Canceling digits at each level of nesting independently can leave a positive digit at some levels and a negative digit at other levels. The second dynamic introduced adding negative numbers to positive numbers is the standardization of any intermediate mixed polarity levels that may arise. The goal is a final result of single polarity. Elimination of mixed polarity levels is achieved by migrating deeper units outward to Cancel shallower units of opposite polarity. Here all action is driven by the polarity of the innermost digit, which will dominate the sign of the final result. Units of the dominant polarity migrate outward by a three step sequence: Ungroup followed by Unmerge followed by another Ungroup. The first Ungroup decomposition separates one unit from the more deeply nested digit. This single unit is Unmerged to the next outer level, the iconic analog of “borrowing”. Finally the Unmerged container is Ungrouped into constituent digits sufficient to Cancel the outer digit of opposite polarity. The next video shows an elementary example of the Ungroup, Unmerge, Ungroup sequence. Since we don’t want to confuse symbolic with iconic concepts, let’s call this sequence Unmixing.
Unmixing: 20–3=17
Unmixing is not restricted by any particular type of polarity. In this example, negative units propagate outward to Cancel positive units. This will occur when the net result of an addition is negative. The example also shows concurrent Unmixing at two different levels of nesting.
Unmixing several levels concurrently: 507–8040=–7523
This example shows concurrent Unmixing on three different levels, combined with digit Canceling at each level.
Unmixing and Canceling: 846132–54705=791427
When one or more empty levels separate the unit to be Ungrouped from the digit to be cancelled, the Unmixing action must propagate across the empty level. This is achieved by repeating the Unmixing step until the dominant unit reaches the digit to be canceled.
Unmixing across an empty level: 2005–60=1945
Here is an example that includes three concurrent Unmixing actions. The innermost Unmix is immediately adjacent to the cancelled unit. The outermost Unmix must travel across two empty levels.
Concurrent Unmixing across several empty levels: 815369024–52392025=762976999
Now an example in which every level but the innermost requires Unmixing. In this extreme circumstance, each level must be processed sequentially.
Extensive Unmixing: 483745–397988=85757
We have been looking at the dynamics of iconic addition when digits of different polarities are added together. The examples isolate two particular standardization sequences. Digit Ungrouping is motivated by Canceling within each nested level. Unmixing facilitates Canceling across different levels. In most cases, both types of standardization sequences occur at the same time, with Grouping and Merging of digits of like polarity also occurring concurrently. We close with three examples that illustrate these standardization processes operating in coordination with one another. The first example shows the Grouping process resolving a mixed polarity number.
Grouping to resolve mixed polarity: 22–16–9–9=–12
The next example shows Grouping operations occurring concurrently for each polarity, prior to Unmixing.
Grouping of different polarities: 182–19+361–18–18=488
The final example includes all of the processes used in standardization acting in concert.
All of the standardization processes: 6282–509+5171–918–828=9198
Iconic arithmetic with conventional digits is a hybrid system, combining the familiarity of our conventional base-10 digit notation with a depth-value organization of magnitude. The design imperative is to maintain the Additive Principle. The three standardization rules of Group, Merge, and Cancel dominate transformations. Negative numbers add the necessity to Ungroup and Unmerge, in order to make Canceling available in all cases. Digits add a special requirement of needing to be Ungrouped so that their constituent units can be accessed. Since digits are a shorthand notation for a unit ensembles, no additional transformation processes are needed. The overhead of using digits then comes down to memorization of 25 digit combinations, each of which can be accessed by reverting to the unit ensembles that the digit combinations identify. A primary benefit not mentioned here is that this system also maintains the Multiplicative Principle, so that every operation of arithmetic remains simple.