Incrementology – Expanding the Counting Concept for Fundamental Math
A Research Design by Jacques du Plessis

 


"Thus, the task is not so much to see
what no one yet has seen,
but to think
what nobody yet has thought
about that which everybody sees."

Arthur Schopenhauer (1788-1860)

  

Introduction

It is not easy to truly present research that would revise the fundamentals in the learning of numbers or letters. Many centuries and generations of scholars have reflected on these basic issues, so the chances seem to be infinitesimally small that the fundamental pedagogical conceptions of letters or numbers could be altered. The fundamentals of letters and numbers are indeed the essence of what will become reading, writing, and math. Revisiting the premise of how we learn these fundamentals is always of relevance. To justify interest in this research, I compare fundamental learning to growing a tree. When a tree is young, small adjustments can easily affect the fundamental growth and final structure of the tree, but once the tree is mature, similar small adjustments will have a much reduced effect. If we were to make some changes to a very young sapling, we approach the spirit of fundamental change. A Redwood or Baobab could effectively be redirected to the world of Bonsai – speaking of the profound impact of acquisition of fundamentals.

In his book Introduction to Mathematics (1911) , Alfred North Whitehead presents us with an excellent reason why we should consider revisiting the premises of fundamentals. He says,

 

"By relieving the brain of all unnecessary work,
a good notation sets it free to concentrate on more advanced problems,
and in effect increases the mental power of the race.

Before the introduction of the Arabic notation, multiplication was difficult,
and the division even of integers called into play the highest mathematical faculties . . .

Civilization advances by extending the number of important operations which we can perform without thinking about them."


  

I suggest rereading the quote above. It expresses the vision and mission driving this research. The objective is to determine if indeed an area of useful automatic skill development at the fundamental level has been overlooked. The following example illustrated the key role of one system above another. Let's pretend to go back to the days of togas and sandals and the pre-Arabic numeral system with the following math problem: Please give me the total for CDXXXVII + XXVIII + CCXLVII = ? Try to solve this without resorting Arabic numerals. As you can see, Roman numerals make the task considerably more brain intensive. This research will uncover areas in math where we have not applied rote learning to free our brain of unnecessary work in order to solve bigger problems.

The Alphabet and Numbers — A Reflection About Fundamental Development

Although this research is about numbers, it is useful to compare and contrast the learning of numbers and letters. I compare counting, incrementation by one, with the memorization of the alphabet. The alphabet is a memorized list. The alphabet is internalized through rote learning. There is no logic, rhyme, or mental construct that will explain the sequencing of the letters so as to make it possible to infer the next letter. Sound types are not clustered. The frequency of usage cannot be abstracted from the order. In short then – the order of the alphabet is a historical phenomenon and for the intents of the learner that is just is the way it is.

How then is the abstract list internalized? Firstly we accept the abstract nature of the order and we chunk the sequence as is. This is done through rote learning, sometimes aided by songs and mnemonics.

Initially, the letter A is the access node to the list. Ask a young child what comes after N in the alphabet, and he or she will likely start at the only available access node, A , and process the whole list to discover what comes after N . It takes a good while to develop more access nodes on the A – Z continuum. To many adults not every letter in the alphabet ever becomes an access node. In other words, if I were to ask someone to tell me what comes after < random letter >, it would not be possible to recall the next letter intuitively every time. Many people would at times be forced to backtrack one, two, or more letters to generate the appropriate response.

With a non-generative list, the first node is the primary access node (and often the only access node). Such a list is called an Alphabetic List in this paper. Similar to letters, numbers are also memorized as a rote list initially. This is evident as toddlers try to count. They often jumble the order and omit some numbers, proving the non-generative nature of the list to them. Sometime after having memorized enough numbers to expose the generative traits, the generative quality of the list becomes clear. Every number inside the memorized sequence becomes an access node. A list with such a generative ability endows every node with the quality of an access node. A child, having memorized numbers without understanding the generative attributes clearly will have much difficulty to intuitively treating any number as an access node. The child who understands and can use the generative ability of numbers, is able to treat any number as an access node. This generative type of list is called a Numeric List.

The Numeric List - Generative Possibilities

For young children, the early stages of recalling numbers is not a generative process, and it is recalled from memory, like with the alphabet. A child starts out with a memorized sequence. Then with some children, once they can count to 20 and above comfortably, a shift occurs – the generative aspect of counting becomes apparent, i.e. the numeric list is not processed like an alphabetic list anymore. Now with 20, 30, 40, 50, 60, 70, 80, and 90 memorized, the child can count (increment by one) from 1 to 100. The algorithm is understood and the list concept fades and it becomes a generative experience with the algorithm driving an automatic fluency in generating the next incrementation. With this generative ability in place, I could ask the same child, "What comes after 27?" and the answer would be given with no difficulty. Twenty-seven will become as good an access node as one or any other number. It might sound odd to describe counting in terms of an incrementation by one or as a one times table, but seeing counting in this light is fundamental to a whole new system that this research introduces.

What follows the achievement of counting (incrementation by one) in current math education for young children? After counting, basic addition follows, and then comes subtraction, then multiplication, and then division. By this time the math environment is becoming complex and the fundamentals are supposed to be in place.

This research introduces and suggests a new set of activities to slip in between the mastery of counting and the introduction of addition, subtraction, etc. For now I call this Incrementology. Other than using the term counting, the term incrementation or enumeration are also used.

We have already established that incrementation by one becomes intuitive. Although it is technically correct to say that for each utterance in the counting process, the previous number is incremented by one, this is not what is happening in the mind. A subconscious generative process is in place. An analysis of the incrementation-by-one-system (i-one) reveals a repetition of the numbers 0-1-2-3-4-5-6-7-8-9 in an infinite loop. This incrementation loop could start at any number in the loop. For example, starting with 6 would mean 6-7-8-9-0-1-2-3-4-5-6-7-7 ... as the infinite loop, and the incrementation if you start at 16, or 36, or 14,336 since the incrementation wheel would be used in the same way, i.e. the next number would conclude with a seven, followed by an eight, etc. So, if I started at 36, it would follow 37, 38, 39, 40, 41, etc. We know this as counting; a fundamental tool that this paper proposes to be under utilized. The graph below visually illustrates the generative process of counting (incrementing by one).
Incrementing by 1 (Counting) has 1 home loop and no alternative loops
The Infinite Loop: 0 - 1 - 2 - 3 - 4 - 5 - 6 - 7 - 8 - 9 - and then it starts over ...
(Since there is only one loop -- the home loop) Light Blue numbers indicate the loop
00 → 01 → 02 → 03 → 04 → 05 → 06 → 07 → 08 → 09 →
10 → 11 → 12 → 13 → 14 → 15 → 16 → 17 → 18 → 19 →
20 → 21 → 22 → 23 → 24 → 25 → 26 → 27 → 28 → 29 →
30 → 31 → 32 → 33 → 34 → 35 → 36 → 37 → 38 → etc.

The next two paired graphs below illustrate incrementation by one in its default position (zero at the top) and the wheel then turns perpetually one digit to the right (clockwise). If the wheel (the second wheel below) is turned to a random digit, (e.g. six in this case), can the user take off from that random spot without a hitch? The six could be 6, or 16 or 36, or 56, or 126, or any other number ending in a six. With incrementation by one, that is a given.

The graphs below (starting with one, and ending with 10) indicate the following. The darker green column indicates the incrementor home track and the  light green column(s) are also based on the home track cycle. The orange column(s) indicate the alternate non-base incrementor track.


INCREMENTOR-1

You notice the wonderful thing about the 1x table is that every number is part of the incrementor home track and there are no non-home tracks. All ten of the base-one incrementors are used. No wonder you can start from any number and fluently do the 1x table incrementation. You will notice that incrementation-1 and incrementation-9 are opposites of each other.
  1 x INCREMENTATION DEMO

 

INCREMENTOR-2

The even numbers are the incrementor home track (green in the table), and the odd numbers (orange) are the only non-home or alternative track. The even incrementors (5 of them) are used for the home track and the odd incrementors are used for the alternative / non-home track. Because there is only one non-home track and it only has 5 incrementors, it is relatively easy to master both the home and non-home track incrementation. You will notice that incrementation-2 and incrementation-9 are opposites of each other.
2 x INCREMENTATION DEMOS
(EVEN & ODD NUMBERS)
EVEN ODD

 

INCREMENTOR-3

The home track is in dark green. All 10 the incrementors are use for the home track and for each of the non-home tracks. There are two sub home-based tracks (light green on the table). To explain what is meant here, look at the dark green column, and only focus on the base-1 numbers (not the base-10's) -- you have 0, 3, 6, 9, 2 (the base-1 number at 12), and the next number would be 5 (the base-1 number at 15), etc. Now look at the middle column, and you will notice the 2 at the top, and then a five below it, then 8, .... This means, the home-track incrementation cycle is used, just starting at a different location. Each of the three columns uses the home-track cycle 0-3-6-9-2-5-8-1-4-7-0-3- etc. with a different starting point. So, on whichever number you start, the base-1 numbers form an infinite loop. You will notice, incrementation-3 and incrementation-7 are opposites of each other.
  3 x INCREMENTATION DEMO

 

INCREMENTOR-4

This comment goes for Base-4, Base-6, and Base-8. The home-track and the sub based home-track(s) use the even numbers loop 0-4-8-2-6-0-4-8- etc. and the alternative 5 numbers loop uses the odd numbers 1-5-9-3-7-1-5-9- etc. You will notice, incrementation-4 and incrementation-6 are opposites of each other.
4 x INCREMENTATION DEMOS
(EVEN & ODD NUMBERS)
EVEN ODD

 

 

 

INCREMENTOR-5

The home track has only two numbers, zero and five. There are four alternative tracks, each with only two numbers. The alternative track pairs are 1-6 and 2-7 and 3-8 and 4-9. With only two numbers in a loop, it makes it easy to increment from any random number. For example 53, you default to the 3-8 alternative loop, e.g. 53-58, 63, 68, 73, 78, 83, 88, etc.
5 x INCREMENTATION DEMOS
(FIVE NUMBER PAIRS)
0-5 1-6
 
2-7 3-8

4-9  
 

 

 

 

INCREMENTOR-6

See the comment for Incrementor 4. Here the 5 home-track even incrementors use the even numbers loop 0-6-2-8-4-0-6-2- etc. and alternative loop uses the 5 odd numbers 1-7-3-9-5-1-7- etc.
6 x INCREMENTATION DEMOS
(EVEN & ODD NUMBERS)
EVEN ODD

 

INCREMENTOR-7

The home track is in dark green. All 10 the incrementors are use for the home track and for each of the the non-home tracks. There are two sub home-based tracks (light green on the table). To explain what is meant here, look at the dark green column, and only focus on the base-1 numbers (not the base-10's) -- you have 0, 7, 4, 1, 8 (the base-1 number at 28), and the next number would be 5 (the base-1 number at 35), etc. Each of the sub home-track cycles use the home-track incrementation cycle, just starting at a different location. So, on whichever number you start, the base-1 numbers form an infinite loop. You may already know that incrementor-7 and incrementor-3 loops are opposites of each other.
  7 x INCREMENTATION DEMO

 

INCREMENTOR-8

See the comment for Incrementor 4. Here the 5 home-track even incrementors use the even loop 0-8-6-4-2-0-8-6- etc. and the alternative loop uses the 5 odd numbers 1-9-7-5-3-1-9- etc.
8 x INCREMENTATION DEMOS
(EVEN & ODD NUMBERS)
EVEN ODD

 

INCREMENTOR-9

There are eight sub-home tracks (light green on the table). The home track is in dark green. All 10 the incrementors are used for the home track and each of the sub-home tracks uses the same loop, just starting at a different location. You may already have noticed that incrementor-9 and incrementor-1 loops are opposites of each other.
  9 x INCREMENTATION DEMO

 

INCREMENTOR-10

There is one home loop, with just one number in it, and there are nine alternative loops, also each having just one number in it. This makes it very easy to increment from any random number. As you already know, if you were to start with a number like 53, you just stick with the 3 since there is 0 incrementation, e.g, 63, 73, 83, etc.
10 x INCREMENTATION DEMOS
(TEN PAIRS OF SINGLE NUMBERS)
0-0 1-1
 
2-2 etc.
The same goes for 3, 4, 5, ..., 9

 



Which incrementation loops are easier or more difficult to memorize?

i-ONE    As we know, once a child is familiar with incrementor-1 (also called i-one), any number becomes an access number in the system. That is so obvious, that is sounds odd to describe counting in this way. Yet, inside this automation logic lies the magic. An inspection of the other incrementation systems reveal the following:

i-TEN    The incrementation-by-ten-system (i-ten) is simplistic: 0 to 0, and 1 to 1, and 2 to 2, and 3 to 3, and ... , 8 to 8, and 9 to 9. In the i-ten (or i-zero) system the number never varies, e.g. 2-12 ,22, 32 ,42 ,52 ,62 , etc. This system is also highly generative from the outset and easily internalized.

i-TWO    Another system that is quick to acquire is the i-two system. This system has two incrementation wheels, even and odd: 0-2-4-6-8, and 1-3-5-7-9. We are all familiar with these systems, and it is noticeable that most can count from any random number onwards using the i-one, i-two or i-ten systems.

i-FIVE    With only two numbers in each loop, memorization is easy.

i-FOUR/i-SIX    Since these two systems are similar to the i-TWO system, the rearranged order takes a little getting used to, but with only five numbers in each loop (odd and even), the mastery is not too difficult.

i-THREE/i-SEVEN    These two systems use all ten numbers (0-9), so the list is long, and the pattern in new for the sub-home tracks have to be memorized, even though it is the same cycle as the home cycle, but because the departure point is at different points, it takes getting used to. These two systems are the most difficult to master.

Each system's loops (home and alternative) and it's mirrored system

i -one (0-1-2-3-4-5-6-7-8-9)
1 home loop, 0 alternative loops, 1 master run, 0 sub runs

•

i -nine (0-9-8-7-6-5-4-3-2-1)
1 home loop, 0 alternative loops, 1 master run, 8 sub runs

i-two (0-2-4-6-8) and (1-3-5-7-9)
1 home loop, 1 alternative loop, 1 master run, 1 sub run

 •

i-eight (0-8-6-4-2) and (1-9-7-5-3)
1 home loop, 1 alternative loop, 1 master run, 7 sub runs

i-three (0-3-6-9-2-5-8-1-4-7)
1 home loop, 0 alternative loops, 1 master run, 2 sub runs

•

i-seven (0-7-4-1-8-5-2-9-6-3)
1 home loop, 0 alternative loops, 1 master run, 6 sub runs

i-four (0-4-8-2-6) and (1-5-9-3-7)
1 home loop, 1 alternative loop, 1 master run, 3 sub runs

•

i-six (0-6-2-8-4) and (1-7-3-9-5)
1 home loop, 1 alternative loop, 1 master run, 5 sub runs

i -five (0-5) (1-6) (2-7) (3-8) (4-9)
1 home loop, 4 alternative loops, 1 master run, 5 sub runs

•

i -ten (0-0) (1-1) (2-2) (3-3) (4-4) (5-5) (6-6) (7-7) (8-8) (9-9)
1 home loop, 9 alternative loops, 1 master run, 9 sub runs


Just like in the table above, these wheels illustrate the mirrored systems as paired opposites

The following explanation can be extended to all the systems. The i-three system wheel would look like this: 0-3-6-9-2-5-8-1-4-7. Once you have memorized these numbers, the next step would be to plug them into real incrementation. Details about how to do so will be discussed below under the topic The Experimental Acquisition Strategy. If I were to give you a random number, say 71, and you were to increment by three, the '1' on the 0-3-6-9-2-5-8-1-4-7  i-three wheel would be your access point and you would loop from that point on, e.g. 71>74>77>80>83>etc. Once you can increment subconsciously, the access point may as well have been 81, 61, or 11.    

Examples:
81:       81 > 84 > 87 > 90 > 93 > etc.
71:       71 > 74 > 77 > 80 > 83 > etc.
11:       11 > 14 > 17 > 20 > 23 >etc.
     

Once the learner has mastered an incrementation wheel, every node becomes an access node -- from a random number the learner would be able to count with the specific incrementation from that point on. The key concept is that it would be an exercise in counting, just like with the i-one system, rather than addition-on-the-fly. Like with regular counting, students would also have to keep track of the changes of the tens digit.
After having mastered multiplication tables with traditional teaching methods, students become empowered to incrementally add a chosen number (from 1 through 10) to another number as an access node (departure point) only if that random number divides exactly into the system number, e.g. if you give me 21, I can count in threes or sevens, but if you give me a prime number like 29, it is markedly more difficult to add threes or sevens to that number.

All the links in the table below provide access to each of the i-num systems. -- >
These links are provided for two reasons: (i) as a resource to prove to yourself that not every system is part of your counting system, and (ii) to see for yourself how much slower and less efficient you are in those i-nums where you can not count.

To be able to use these pages to train yourself to count in these systems, the first phase should be relatively easy, the second phase more challenging, and the third phase should be the most difficult.

First Phase: Second Phase: Third Phase:
i-one i-nine i-three
i-two i-four i-seven
i-ten i-six  
i-five i-eight  

Benefits

With the proposed incrementation-by-n system (i-num), the underlying pattern is internalized, just like the i-one system (counting) is naturally internalized by little children. The effect of this instructional course is intended to enhance students' numbers sense significantly.

Once a specific number system has been internalized in an incremental direction, the addition by that system has already been mastered and the multiplication table will not be a new concept since it's structure is already internalized. Multiplication is in essence a subset of this proposed system.

Subtraction on this number system will be as easy as it is to learn how to count decrementally. Dividing by the system number will be simplified since it is in essence structured on this proposed system and on multiplication.

Quo Vadis?

The question is: should instruction initially focus on all the incrementation systems (i-one through i-ten), and then deal with addition, subtraction, multiplication and division, or is the current system of teaching the i-one system, followed by addition, subtraction, multiplication and division still superior? This research will help address that question.

I -- RESEARCH QUESTION

Does the mastery of the 1 through 10 incrementation systems enhance the learner's automatic skill development of addition and multiplication ability, using any whole number as an access node?

The same question rephrased would be: Compared to traditional math education, does the i-num system improve the the automatic skill development with which students do addition, subtraction, multiplication, and division?

If I ask you to impose a given incrementation to a random number, will you be able to do so faster if you have mastered the incrementation system for the number you are adding compared to just working hard at becoming proficient at adding in general?

II - SUBJECTS, SELECTION, AND ASSIGNMENT

Population

- TO what population will the research be generalized to?     
The population to whom this study will be directed will be beginner students of arithmetic. This population would typically consist of first, second, and third grade pupils in primary school.

- How will a sample from that population be selected?

- How will the subjects be assigned to conditions?

     At the end of the Kindergarten year, all parents will be informed of the new methodology and their permission will be asked should their child be chosen to be in the experimental group. Children of parents not giving their blessing will unavoidably end up in the control group receiving standard instruction. Two teachers of each of the first two grades (Grade 1, Grade 2) at two primary schools will be selected to participate. That amounts to a total of eight teachers. Intact classes will be used for the experiment. Based on the anticipated loss of numbers per class, the experimental groups for first grade will be about 10 to 15% larger than the control groups since no more students will be added to the control groups for two years. All new students in either the first or second grades will either move into a class not participating in the study, or should that not be feasible, they will be placed in the control group which would offer instruction quite similar to what they have had.

- What procedures and/or documentation will assure that subjects in each condition are comparable before application of the Independent Variable?

A class using the experimental method will stay on that method for both grades. At each school one teacher will be randomly assigned to the experimental method and the other to the traditional method. These teachers will then alternate methods with the new incoming class. For each teacher the experiment will be conducted over a two-year period, with the same teacher having an experimental group one year and a traditional group the next year. Teachers of the control group will document the arrival of new students so that their results can be compared with the rest of the control group for possible variation in performance.

III - MEASUREMENT

- Describe each measure and when it will be given.
- Describe how each measure relates to the research question.

Sameness in Treatment
Initially both groups will receive the same treatment in kindergarten in their quest to be able to count (learning the incrementation-by-one-system). Then all subjects will take a test at the completion of Kindergarten to assess their ability to negotiate the incrementation-by-one-system (counting). This will be the pretest. The post test will be given at the end of second grade and the test will include incrementation by all the other numbers (2-10). This test will determine the automatic skill development of responses in doing addition from any random number. Three performance tests will be given to each group: the first half way during first grade, the second at the end of second grade and the third half way through the second grade (see list below), and only the incrementation systems already mastered by the experimental group will be included in this test.

Assessment Schedule:
Kindergarten: Same treatment for all-End of year assessment will be the pretest.
Grade 1: Mid year and end of year performance assessment
Grade 2: Mid year performance assessment and end of year post test.

Progression of Mastering Addition
In the traditional method, once pupils can count they are taught to add and subtract.
All four teachers will come together to evaluate their methods and then to decide on a consistent method for teaching both the traditional and the experimental methods.
The final objective will be to add any number from 1-10 to any random number.

The traditional group will follow the following curriculum:
(1) Learn how to count.
(2) Learn to add small numbers (x>5) to a random number between 1-20.
(3) Gradually expanding to add larger numbers (4>x<11) to a random number between 1-20.
(4) Then expand adding any number (1>x<11) to any number below 100, and
(5) An algorithm for multi-digit problems.

The Experimental Acquisition Strategy
On a macro level, the experimental group will master each incrementation system and then practice addition of that incrementation system on any random number below 100.

At the micro level, the following five steps will be followed:
(1) The incrementation wheel will be automated as a memory experience. The numbers will be taught as a song. Other means might also be employed to memorize the numerical string to a high degree of automatic skill development, e.g. 3 692581470 3 692581470 3 692581470... Software will be developed to allow each student to develop a thorough self-verification of this objective.
(2) The next objective would be to break up the string so that every number would become an access node. With the string above ( 3 692581470 3 692581470 3 692581470...), the student might be comfortable to recite the string, starting with 3, but what if the student is asked to start with 8, or 4? The objective of this step would be addressing that problem, so that any number would be a departure point.
(3) Then the rote learning will be applied to a generative environment. Initially with visual aides, the student will count in the specified incrementation system. This is the final objective -- that the student can count in the specified incrementation system, starting at any number. This objective might be broken down into subsets. An example of that would be to add just the incrementation number to a random number. Another objective would be to give the student any number between 0 and 40, and using the memorized incrementation wheel, the student will increment from that point on by the incrementation number. Initially the incrementation wheel may be used as a memory aide for this step as well. Finally, the students should be able to rapidly increment with the specified incrementor from any random number to any other number.

Subject Assessment
Two types of tests will be given.

Type A: Add to a random number a randomly displayed incrementor. Only the incrementors that had been taught up to that point will be included.

Type B: Do a running incrementation to 60, starting from a randomized departure number and incrementing by the incrementation number.
Both types of assessment will be conducted in a similar fashion. The subject and the evaluator will have a one-on-one session and the session will be video taped. The evaluator will be another teacher, preferably unfamiliar with the experimental method and not familiar with any of the subjects. Experimental and control subjects will be tested in a random fashion.

All the subjects will get a five-item trial run. The test will be conducted as follows: A random number (the departure number) will be displayed on a white card. The evaluator will operate the display of the departure number. A stack of colored cards with the incrementors face down will be placed in front of the student. The teacher displays a new departure number and the subject turns over the incrementor card and both the task for the type A or type B evaluation will be done with this modus operandi, i.e. the subject will either add the incrementor to the departure number, or do a running incrementation to 60. Once the five trial run items are completed, the evaluator will explain the evaluation process to the subject. The evaluator will instruct the student do go through all the incrementor cards as fast as possible. Afterwards, the video will be watched and each session will be evaluated for speed and accuracy. The second session will be conducted in the same way and type B instructions will be done. Subject performance will not be tallied during the evaluation to reduce the stress factor.

IV - INDEPENDENT VARIABLE

- The conclusion will be about which specific thing (cause)?
The independent variable would be "curriculum". It would have two levels (1) incrementation, and (2) traditional curriculum. It is projected that the automatic skill development of the I-num system of the experimental group will make a positive difference in how rapid new learners of addition (or multiplication) will be able to demonstrate their ability.

Features Common to All Treatments
1. The same time (one hour) of group instruction per day
2. The same instruction of counting
3. The same evaluation method
4. The same exposure to all teachers (all will be teaching both methods)

Features Unique to Group 1 (The Experimental Group)
1. Presentation of the incrementation systems. First half on teaching an incrementation system and the other half of the time to learn addition and subtraction.

Features Unique to Group 2 (The Control Group)
1. Absence of teaching the incrementation systems. Full-time on teaching addition and subtraction.
- Specific similarities between conditions. What factors have been controlled and how have they been controlled?

Both groups will receive the same amount of instruction (time). Both groups will be evaluated in the same way. Teachers will receive pre-service training to ensure their teaching strategies are alike inside both the control and experimental groups, so that two teachers teaching the same treatment will be as similar as possible in how they conduct their instruction of the particular treatment. The teachers will be selected because they are willing to engage in the experimental treatment and because their students on average do acceptably well in math. If any teacher has consistently much lower results on average, such teachers will not be considered to participate.

- How will it be assured and documented that the features of each condition were implemented properly?
Prior to the start of the research all teachers involved will agree to the curriculum for both groups. Each activity will be planned by all teachers and committed to a specific day of instruction. The teachers will meet once a week to compare how their classes are progressing and to adjust the schedule accordingly. At the end of each instructional session, each teacher will document her instruction and that will be reviewed by the research coordinator to ascertain compliancy with the program. The research coordinator will also attend the weekly teachers meeting to ensure that they are all on the same page. She will visit the classes periodically to ensure that she is apprised of the situation.

V - SUMMARY OF POTENTIAL THREATS

Describe each threat of internal validity and address the following:
(a) the threat in general,
(b) why is it/not a threat to your conclusions,
(c) if the threat is plausible, why other measures were not taken to rule out this threat.

Threat 1: Rivalry or demoralization of the control or experimental group
Since parents will be consulted about the detailed content of the curriculum, their written permission will be needed to allow their children to participate in the study as potentially part of the experimental group. Since all the students in the same class will receive the same treatment the experimental group will consist of only students whose parents agreed to the experiment.

Since all the student in the same class will be receiving the same treatment, there should not be a problem with rivalry in the class. Teachers will be instructed not to make it known that there are two different treatments going on at the same time to avoid raising expectations in one or the other group. Thus, if the students are treated as normal and simply just taught in their respective methodologies, there should not be a serious problem with rivalry or demoralization due to social factors.

Threat 2: Attrition-common factors
Since this is a regular school activity as part of the curriculum, attrition should be similar in all groups. This should not pose a serious threat to the study.

What if the incrementation method appears too weird for some parents and they change their kids to a different class? A pilot study will be conducted with children of this age group, most likely in a homeschooling setting. If these children will have to prove that there is merit to this method by performing as well or better than children in the current school curriculum. This evidence will be presented to the parents to convince them that the method is sound. Part of the parental agreement will be that they will not demand a change mid-stream.

What if it is appealing and some parents want to switch their kids into an experimental class? The results will not be published until the study is completed, thus will this threat be eliminated.

Threat 3: Lack of treatment fidelity/diffusion (leakage of instruction)
The prescribed curriculum that all the teachers designed and agreed to use collectively will include that each teacher will follow and each teacher will document their daily instruction. The research controller will compare the teachers' progression to verify that they are all progressing satisfactorily and that they are in compliance with the agreed curriculum. This will help to minimize the chance of treatment diffusion. There is a slim chance that students could be talking to each other about the differences in how they are studying math, but that is not viewed as a serious threat to the respective treatments since the intensive rote learning makes the difference in the automatic skill development of each incrementation sequence. A student in the experimental group would have to actively coach a non-experimental student to create the leakage effect. Teachers will specifically commit not to discuss their teaching with the teachers of the other treatment group.

Threat 4: Differences in how the groups are treated
Since the same teacher will be teaching both treatments in turn, this threat should be minimized as well. The teachers of the same treatment of the same grade will meet one a week to reestablish the sameness of their teaching methodologies Other than these teachers, they will commit not to discuss their math instruction with any other teacher. Hopefully this will minimize differences in the curriculum that would imply differences in instruction.

Threat 5: Selection process resulting in unequal groups
The schools will be carefully selected. This might favor schools in specific parts of the countryside where there is least amount of attrition. The division of class sizes is usually fairly equal. Since attrition is not seen as a serious threat in this case, the groups should stay roughly the same. New students entering the school and class after kindergarten will automatically be enrolled in the control class, and their progression will not be included in the research study. This might cause the control group classes to be somewhat bigger than the experimental groups, since both groups will experience some attrition, but only the control group will receive new pupils, making it somewhat more difficult to manage. This factor might be controlled by trying to add newcomers to classes other than the experimental and control classes if possible.

Threat 6: Random variability in samples
Although there is a lack in random assignment of students to either the control or experimental groups, this is not seen as a serious threat, because there is no compelling evidence to suggest that the actual assignment of pupils to one class or another would have any bearing on their math abilities since it will not be decided which method the class will follow till after the class selections had already been made. Teachers will be teaching both control and experimental treatments in consecutive academic years. Their initial assignment to either group will be random.

Threat 7: Maturation across the course of the study
Maturation will affect both groups, since they will have about the same amount of time on task. The difference between the two groups will not affect the maturation process and the experimental design should thus be able to discriminate if that treatment made the crucial difference.

Threat 8: Testing caused changes in the participant
This is not a threat since the instruction and the integrated measurement will be the same for all subjects. The effects of testing are not expected to be significant at all, and all subjects will be assessed in the same way.

Threat 9: Instrumentation changes
This is not a threat since the instruction and the integrated measurement will be the same for all subjects. All teachers will be bound to comply with the agreed-upon method of instruction and assessment.

Threat 10: Regression toward the mean
Since kindergartner classes should have about the same spread of abilities and previous exposure to math, it is unlikely that there will be regression toward the mean. The pretest will occur only after the pupils of both treatment groups have received the same treatment (counting). This pretest should confirm the spread of abilities throughout both treatment groups.

Generalizeability of the Results

The focus population at which this study will be directed will be beginner students of arithmetic. This population would typically consist of kindergartners through roughly second grade students.

A: To groups beyond the actual study
If this study was to be conducted in Northern Utah , the results would likely be similar if the instructional methodologies are very similar to the schools in which the experiment was conducted. That might widen the population beyond those who use the exact same text, but open it up to schools abroad that follow a similar curriculum. The similarity in instruction would include both the actual materials that the students will be using, as well as the methods of instruction employed by the teachers. In such cases it would be relatively safe to predict that as far as this aspect of generalizability is concerned, the treatment should be expandable to such schools throughout the nation and to many other countries. Should there be schools where math is approached in a profoundly different way (conceptually: counting, then addition, then subtraction, then multiplication, then division), it would be inappropriate to assume generalizability to these groups, and further research would be required to support that assumption.

B: To settings beyond the study
The local schools do not have a very unusual setting in comparison to most schools throughout the nation. Thus we are looking at an environment with similar teacher methods and instructional materials, but providing a different setting. The usage of this method is implemented in a very natural school setting and it might very well be that it is transparent to the learner that this is an experiment. This enhances the generalizability of the study. Beyond school settings the setting could include homeschooling and less conventional instructional settings, e.g. in hospitals, and one room schools. I see no threat to the implementation of this treatment and the ability of this treatment to succeed very well in these settings, provided that the instructors are adequately familiar with the training method and that the method of the experimental treatment group reflects no unusual or confined way in which the methodology is applied.

C: To implementations of the treatment in non-experimental situations
This is an exciting part of the generalizability issue. It is exactly the implementation of this treatment in non-experimental settings that is projected to provide the real benefit. If the results of the research support the projections, those who have been trained with this methodology will be able to do addition and multiplication with an automated fluency as if it were a counting exercise.
    Since the study was conducted in a very natural setting, there are no obvious artifacts present in the experimental treatment that can be judged to impact the study. The only possible distractor could be the occasional presence of a research evaluator, but this is deemed to be a minor distraction.

D: To variations on the specific treatment of this study
A variation already mentioned in the introduction of this study where an incrementation is not taught and then addition is practiced for that incrementation system, but all the incrementation systems (1-10) are taught and then the addition of each incrementation system is practiced. Another approach might be to teach the complementary systems. Complementary systems are reversed to each other, e.g. 1 and 9, 2 and 8, 3 and 7, 4 and 6, with 5, 1, and 10 being unique. If the complementary systems are chunked together, addition and subtraction could then be instructed and practiced together. The strength of this method is in the understanding of the concept and the underlying principles. If these principles and conceptual framework is not violated, teachers teaching with some form of variation should not pose a problem to the validity of the results.

E: To other measures and other outcome variables
The assessment is designed to prove student automatic skill development in addition and multiplication, and it is not partial to any particular method. Should other measures provide excellent results, the assessment of this treatment should be equally valid to compare the experimental treatment with such other measures.

The measure that is being tested is the automatic skill development of addition and multiplication of the students. It is the automatic skill development that is at the heart of this measure and it can be expanded to measures of subtraction and division.

F: Implications of this research
The research premise is that an expanded emphasis on the rote memorization of the cycles of the ten single number loops will empower the learner to develop an non-reflective automated ability to support addition, multiplication, and also subtraction and division. This memorization process will replace the present cognitive development of the ability to rationally calculate multiplication, addition, subtraction or division. It is postulated that the automated ability will enhance speed and accuracy when demonstrating the skill.

If the superior development of fundamental math skills can be empirically proven, it would compel math educators internationally to take note of the results and to incorporate the theory behind the i-num counting system in curricula world wide.

Copyright © 2000 – 2017, Jacques du Plessis -- (Last edit February 4, 2017)
   Modular 10 arithmetic (increment size 3) basic number theory

The term “enumerology” was very new to me. When I read this article, my primary goal was to capture the meaning of the term, enumerology. According to the authors, enumerology is a research methodology to capture social processes by which numbers are generated and the effect of these processes on behaviors and thoughts. While statistics is aimed at precise counting to aggregate data with estimation, enumerology is directed at “understanding how counting actually occurs, how organizations function, and the place of counting in everyday life” (Garfinkel, 1967; Bogdan & Ksander, 1980, p. 302). In my understanding, enumeration refers to describing how the quantitative data are produced and what they mean. The definition of enumerology sounds like a great alternative approach in debates between qualitative and quantitative methodology in social science. Especially, the assumptions regarding to counting are interesting. I often notice that numbers of survey findings released through the mass media are differently understood by different levels of audiences. Yet, I feel that enumerology considers about qualitative data behind numeric features, not about quantitative data themselves. Ultimately, the enumerological approach is intended to show complicate interactions and relationships around a specific social context. In statistics, counting stands for objective representation of phenomena. Quantitative approach advocates might attempt to criticize that enumerology ignores objective and standardized criteria in counting.