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

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 automaticity 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 preArabic 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 nongenerative 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 nongenerative 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. Twentyseven 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 incrementationbyonesystem (ione) reveals a repetition of the numbers 0123456789 in an infinite loop. This incrementation loop could start at any number in hte loop. For example, starting with 6 would mean 6789012345677 ... 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 incremementation 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 nonbase incrementor track.
INCREMENTOR1
You notice the wonderful thing about the 1x table is that every number is part of the incrementor home track and there are no nonhome tracks. All ten of the baseone incrementors are used. No wonder you can start from any number and fluently do the 1x table incrementation. You will notice that incrementation1 and incrementation9 are opposites of each other.

INCREMENTOR2
The even numbers are the incrementor home track (green in the table), and the odd numbers (orange) are the only nonhome or alternative track. The even incrementors (5 of them) are used for the home track and the odd incrementors are used for the alternative / nonhome track. Because there is only one nonhome track and it only has 5 incrementors, it is relatively easy to master both the home and nonhome track incrementation. You will notice that incrementation2 and incrementation9 are opposites of each other.

INCREMENTOR3
The home track is in dark green. All 10 the incrementors are use for the home track and for each of the nonhome tracks. There are two sub homebased tracks (light green on the table). To explain what is meant here, look at the dark green column, and only focus on the base1 numbers (not the base10's)  you have 0, 3, 6, 9, 2 (the base1 number at 12), and the next number would be 5 (the base1 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 hometrack incrementation cycle is used, just starting at a different location. Each of the three columns uses the hometrack cycle 036925814703 etc. with a different starting point. So, on whichever number you start, the base1 numbers form an infinite loop. You will notice, incrementation3 and incrementation7 are opposites of each other.

INCREMENTOR4
This comment goes for Base4, Base6, and Base8. The hometrack and the sub based hometrack(s) use the even numbers loop 04826048 etc. and the alternative 5 numbers loop uses the odd numbers 15937159 etc. You will notice, incrementation4 and incrementation6 are opposites of each other.

INCREMENTOR5
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 16 and 27 and 38 and 49. With only two numbers in a loop, it makes it easy to increment from any random number. For example 53, you default to the 38 alternative loop, e.g. 5358, 63, 68, 73, 78, 83, 88, etc.
0 5 

1  6

2  7 
3  8 
4  9

INCREMENTOR6
See the comment for Incrementor 4. Here the 5 hometrack even incrementors use the even numbers loop 06284062 etc. and alternative loop uses the 5 odd numbers 1739517 etc.

INCREMENTOR7
The home track is in dark green. All 10 the incrementors are use for the home track and for each of the the nonhome tracks. There are two sub homebased tracks (light green on the table). To explain what is meant here, look at the dark green column, and only focus on the base1 numbers (not the base10's)  you have 0, 7, 4, 1, 8 (the base1 number at 28), and the next number would be 5 (the base1 number at 35), etc. Each of the sub hometrack cycles use the hometrack incrementation cycle, just starting at a different location. So, on whichever number you start, the base1 numbers form an infinite loop. You may already know that incrementor7 and incrementor3 loops are opposites of each other.

INCREMENTOR8
See the comment for Incrementor 4. Here the 5 hometrack even incrementors use the even loop 08642086 etc. and the alternative loop uses the 5 odd numbers 1975319 etc.

INCREMENTOR9
There are eight subhome 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 subhome tracks uses the same loop, just starting at a different location. You may already have noticed that incrementor9 and incrementor1 loops are opposites of each other.

INCREMENTOR10
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 DEMO 

The same goes for 3, 4, 5, ..., 9 
iONE As we know, once a child is familiar with incrementor1 (also called ione), 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:
iTEN The incrementationbytensystem (iten) 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 iten (or izero) system the number never varies, e.g. 212 ,22, 32 ,42 ,52 ,62 , etc. This system is also highly generative from the outset and easily internalized.
iTWO Another system that is quick to acquire is the itwo system. This system has two incrementation wheels, even and odd: 02468, and 13579. We are all familiar with these systems, and it is noticeable that most can count from any random number onwards using the ione, itwo or iten systems.
iFIVE With only two numbers in each loop, memorization is easy.
iFOUR/iSIX Since these two systems are similar to the iTWO 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.
iTHREE/iSEVEN These two systems use all ten numbers (09), so the list is long, and the pattern in new for the subhome 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 (0123456789) 
i nine (0987654321) 
itwo (02468) and
(13579) 
ieight (08642) and
(19753) 
ithree (0369258147) 
iseven (0741852963) 
ifour (04826) and
(15937) 
isix (06284) and
(17395) 
i five (05) (16) (27) (38) (49) 
i ten (00) (11) (22) (33) (44) (55) (66) (77) (88) (99) 
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 ithree system wheel would look like this: 0369258147. 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 0369258147 ithree 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 ione system, rather than
additiononthefly. 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 inum 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 inums 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: 
ione  inine  ithree 
itwo  ifour  iseven 
iten  isix  
ifive  ieight 
Benefits
With the proposed incrementationbyn system (inum), the
underlying pattern is internalized, just like the ione 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 (ione through iten), and then deal with
addition, subtraction, multiplication and division, or is the current
system of teaching the ione 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 automaticity 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 inum system improve the automaticity 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 twoyear 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.
The experiimental method can be seen here: FLASHMATH
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
incrementationbyonesystem). Then all subjects will take a test at
the completion of Kindergarten to assess their ability to negotiate the
incrementationbyonesystem (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 (210). This test will
determine the automaticity 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 allEnd 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 110 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
120.
(3) Gradually expanding to add larger numbers (4>x<11) to a
random number between 120.
(4) Then expand adding any number (1>x<11) to any number below
100, and
(5) An algorithm for multidigit 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
automaticity, e.g. 3 692581470 3 692581470 3 692581470... Software will
be developed to allow each student to develop a thorough
selfverification 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 oneonone 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 fiveitem 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 automaticity of the Inum 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. Fulltime
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
preservice 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: Attritioncommon 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 midstream.
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
automaticity of each incrementation sequence. A student in the
experimental group would have to actively coach a nonexperimental
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 agreedupon 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 nonexperimental
situations
This is an exciting part of the generalizability issue. It is exactly
the implementation of this treatment in nonexperimental 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 (110) 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 automaticity 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 automaticity of addition and multiplication of the students. It is automaticity 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 nonreflective 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 inum counting system in curricula world wide.
Copyright
© 2000 – 2008, Jacques du Plessis  (Last edit Nov 4, 2008)
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.