The foundation of “The Max Area Pattern” and Its
Training Significance
Yi Xu
Dept. of Physical EducationNanjing Normal College, Sichuan Neijiang
Abstract- by reference and simplifying the volume expressions of Barrel Theory, this paper puts forward the Double
Bottleneck Concept, the Weights of plate as well as the K-value concept on the flex coefficient of the items reflecting
the comparative needing of the factors. It also advances the automatic arrangement on the position of each item and the
Max Area Pattern representing the athletic diathesis and ability. It also analysis training pattern in different
time-limitation by the Max Area Pattern based on the concept of Balanced-Development.
Keywords- the Max Area Pattern; the Double Bottleneck Concept; the flex coefficient; the Balanced-Development;
After "barrel theory”, rooted in Japan's Toyota on
corporate management experience, was introduced in the
sports community; the "bucket model" was used to
express the image of the sport quality and the
relationship between exercise capacity. Although
Professor Feng Yan [1], Donghua Qu, Youkuan Shi [2],
Kai Li [3] have put forward different views, they simply
point out the relationship of the sport quality and sport
performance from the way of concept, and failure to
provide an operational methods to quantify and evaluate
their relationship.Given this situation, this paper
attempts to establish a model of "maximum acreage" to
quantify and analysis relationship between the sport
quality and sport performance, and then discuss training
mode choice under the different aging.
2. The foundation of "maximum
area" model
A.Introduction of "bucket model" simplified
Simplifying the "bucket model" volume expression
to "area model" expression can more easily quantify the
relationship between the performance determinants and
overall athletic performance. Expanding the barrel and
array the wood boards constituting a barrel in turn and
then making a straight line XY to mark the water level
height, the acreage of quadrilateral XOZY (seeing
Figure 1) is the bucket volume, and that is the
collectivity sport level. For the mode of "barrel
reasonable place" (oblique), the area expression can also
be simplified as "area model": arrange the wood boards
from high to low and mark a water height through
making a horizontal straight line XY, to round the
boards with O as the original point to make the line XY
rise until getting a maximum area of quadrilateral
XOZY. Here the area of quadrilateral XOZY (seeing
Figure 2) is the bucket volume of reasonable placing
barrel, that is, the overall athletic performance.
B.The factor analysis of "maximum area" model
Some factors that influence its overall athletic
performance is an effect of corresponding to a single,
and having no impact on athletic performance with other
interrelated factors. Such as the reaction of starting
sprint and the height of high jump athletes, their impact
on athletic performance has nothing to do with other
factors. This type of factor is called constant factor, the
corresponding, a class of factors of mutual influence and
mutual restriction that have an impact on athletic
performance are known as variable factors. "The biggest
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ht © 2012 SciRes.159
area" model is discussing the relationship between the
variable factors and athletic performance.
C. The introduction of weight values
A variety of variable factors have different impacts
on athletic performance, and some impacts are great and
some impacts are minimal. Taking the sprint for an
example, strength of speed and strength absolute has a
great impact, and technology is on the lesser of its
impact, but relatively tactical awareness has much
smaller impact; the mastery of knowledge like sports
bio-mechanics and aerodynamics theory has smaller
impact; while as to the pingpong, relative to the strength
of speed and strength, the technical and tactical
awareness have a much greater impact. It is clear that a
same kind of factor in different sports weights
differently; and in the same sport, different factors
weights differently.
"Barrel theory" incorrectly set an equal of
determinants impacting on the overall athletic
performance (seeing Figure 1); the "maximum area"
model has designed to introduce a weight value to
accurately reflect the weights of determinants impacting
on the overall athletic performance, by way of the
corresponding weights size of each element as plate
width (seeing Figure 3).
There are already many reports in the literature [4]
[5]of corresponding studies on a single degree of
correlation between a variety of factors and overall
athletic performance, and its findings can be used as
reference data of the weight value of each factor plates
when establishing the mode of "maximum area".
D. Multi-criteria of evaluating plate height
Different projects require different standards on the
quality development of various levels. For example, a
people with a normal development level of physical and
intellectual, weighting the needing-standard of chess
athletes, certainty has a higher development degree of
physical plates in the mode of “maximum area” than the
development degree of intellectual quality plates; while
weighting the one in the standard of sprinter, he
certainly has a lack of physical plate development in
“maximum area” mode and has a higher development
degree in the intellectual quality. It can be seen that a
factor in an absolute level of development has different
relative level of development in different sports, that is,
the height of reflected plates are different. Clearly, each
value of plates tested from the normal standard of
requirement must be amended through some kind of
mathematical treatment and then can be used as the
height of mode plate.
The multiple relationships between the normal
standard of requirements and the standard of items on
each factor is called the scalable coefficient K.
K = the average of mass demand ÷the average of
relative project demand, Namely: plate height = scalable
coefficient K ×measured standards of the mass
For example, the speed plate is respectively 0.5 and
1 by the way of the normal requirements of standards for
an ordinary people and relative demand of standards for
sprint, and the scalable coefficient of speed factor in
sprint K = 0.5 ÷ 1 = 0.5. The factor plate K of different
project can be found in mathematical statistical methods.
The plate height corrected by the K value is truly
reflecting the relative level of development of various
factor plates in the project.
At present the theories of training are still using the
general-purpose calculations of normal requirements
standards when assessing the development level of each
quality plates, and fail to take into account of the
particular nature of different projects for the needs of
various quality development levels, as a result, some
level value of quality plates, that the relative
development is well above the needed in the project, has
been valued too low, and wrongly making a long board
as a short board.
E.Method of founding the "maximum area" model
The development level of each variable factor is the
plate height of a single element, and the weight of that
impacting on the overall athletic performance is the plate
width of a single factor. Arrange the plates from high to
low and make a straight line XY within the plate,
maximizing the area of quadrilateral XOZY under this
line, and the maximum area is the overall level of the
sport (seeing Figure 3). It can be found in Figure 3 that,
in the particular oblique state, if putting one plate or few
plates after the point U1, even if it is not the lowest one
piece of wood, the capacity may be increased. For
example, in Figure 3 after raising the height of plate g to
a height more than plate d, the capacity increases (seeing
Figure 4).
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It can be seen from Figure 3, the straight line XY,
affecting the size of maximum area, is decided by the
two points U1 and U2, and the two points is called a
bottleneck point. There is only one form of distribute the
location of the point bottleneck: one each on either side
of the midline. The change of any point of a bottleneck
may lead to the size changes of maximum area.
Of course, those factors, of which the development
degree has overtopped the height of left bottleneck, due
to the restricting of other factors, will have no affect
athletic performance no matter how much further
increase. For example, grasping the Einstein's theory of
relativity is probably no helpful for a swimmer to
improve their sports performance.
3. Rethinking the concept of
"maximum area" model to "balanced
A. Some errors in previous related theoretical studies
On the one hand, m in any previous studies and
literatures, it is known that the balance is relative and
temporary, and non-equilibrium is absolute in the whole
training process of competitive ability factors; at the
same time, it also proposed that in the exercise training,
ways of using "balance compensation" [6 ]should be
paid attention to and to enhance and improve the
vulnerable factors, which to balance the development of
competitive ability factors, so as to achieve the effect of
improving competitive ability.
Previous studies misunderstand the "balanced
development theory” to the "equivalent development"
pursuing the same development level of factors, which
ignore the absolute nature of non-equilibrium of the
factors development, as well as reality of limited
development space for some vulnerable factors. Facing
of the situation that "vulnerable factors "can not change
and improve in a short term, and the mean of " balance
compensation " is invalid, scholars bring forward a
mode of " non- balance compensation "in the process of
researching the " theory of balanced development ", and
that is a supplementary method when the " theory of
balanced development "can not get the desired effect [6]
[7] Clearly, the conclusions of these studies have
disguisedly recognized the limitations of "balanced
development" theory in practical applications of exercise
traini ng.
In fact, the nature of "balance compensation" and
"non- balance compensation" is not to compensate the
lack of use of certain factors, but to enhance the overall
activity level and select different directions in exercise
training. The two is just the application model showing
different" relatively balanced development “under the
guidance of "relatively balanced development”, and
there is no essential difference.
B. The concept of "relatively balanced development"
revealed by the model of “maximum area"
In the model of "maximum area", making t O as the
origin point, OY as the horizontal axis, OX as the axis to
establish a Cartesian coordinate system, and finding the
midpoint B of segment XY (
), the point B is
called the equilibrium point. It can be seen from Figure
3 that, the length of segment OZ is a constant value OZ,
the area of quadrilateral XOZY =
, and the
abscissa of equilibrium point B is a constant value
Obviously, the area of quadrilateral XOZY depends on
the value of the vertical coordinates of equilibrium point
B. All means and methods used in the training, the
ultimate goal of that is to increase the area of
quadrilateral XOZY by changing the state of line XY
and heightening the position of the vertical axis of
equilibrium point B.
In precondition of non-changing position of the
equilibrium point B, the slope change of the straight line
XY will not change the area of quadrilateral XOZY, but
rather reflect the gap of the relative level development
among various factors plates, and the different training
ideas and development mode used to achieve the same
movement level.
In the model of "maximum area", impacted by the U
bottleneck points of 1 and U2, the the excess
development of non-bottleneck-point plates only shows
that under the premise of the "maximum area" as well as
the certain level value of other plates, the plate only
need to develop to an enough level, and the higher part
of straight line XY is wasted. It is also a premise that a
certain dynamic stabilization with the certain level value
of each plates to evaluate the development level of each
factor whether reasonable or not. In a system of
relatively dynamic stability, the development levels of
various factors are different, but forming a composition
relationship of more reasonable between each other and
high utilization of factors, and this training model is
called "relatively balanced" development.
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From the model of "maximum area", it can be seen
that to there is not only one way to increase the
"maximum area" by heightening the plate with
bottleneck point, yet no necessary to raise the lowest
plate. To develop the plate of non-bottleneck and change
the position of sorting plates can also increase the
"maximum area." The change (change of line XY) of
"the maximum area" and the level value of other plates,
and the plate position can be automatically change the
sort, and then the plate relatively developed surplus
before can also become lack. For example, the more
development level of plate g 3 from Figure3 of low sport
level becomes lack in Figure 4 of the high sport level.
The actual development level of the plates of
bottleneck points U1 and U2, which show a relative lack
of development, is not lack of U1 and U2 section, the
actual level of development is not dropped behind.
Bottleneck point just reminds us that in the current
relatively stable system, if the bottleneck point plate
may be raised, benefits can be immediately got. But it
does not mean the best option is to enhance the
bottleneck point. In fact, at a training phase of high level,
the development of certain factors (especially physical
factor) has tended to a limit, even if devoting an amount
quantity to its training, the level of the factor is difficult
to improve and the increase range of athletic
performance is minimal, obviously which is not worth
the candle. At this time even if the factor is the one
shorter and wide plate, it can not get a lot of training.
Here, it is naturally become a more reasonable choice to
explore the potential of other plate factors (these factors
can be theoretically increased unlimitedly), even if it is
one longer and narrower plate, the overall athletic
performance can be improved by making it longer.
Similarly, the development level of the bottleneck
points U1 and to the right plate are both absolutely in a
lack state, but in the current state of the stable
development, based on the function of "maximum area"
model automatic sorting, it is expressed relatively
surplus. If the rule of automatic compositor is not
followed, and putting these plates forward, a lack state
of development of restraining “the greatest area"
increase will also be shown.
Clearly, the bottleneck points U1 and the right
section both have a two-side nature of development level
of absolute lack and absolute spare, as well as a two-side
nature of relative shortage and relative spare. To
determine a development level of one factor is to put it
in a dynamic stabilization system within a certain value
of development level of the other factors, and to get a
conclusion of relative shortage, appropriate and spare,
of which the absolute gap is nothing to do with the
development level of this plate relatively in this system.
C. The utilization of the plate is the standard to evaluate
the equilibrium level
"The maximum area" model can clearly reflect the
status and role of various factors placed when
determining the overall athletic performance by dint of
the weights and the area ratio of plates in the
quadrilateral XOZY. As "the greatest area" model can
automatically sort, the level that plates are higher than
the position of XY, has intuitively and accurately show
the relative-balance development level of various plates.
Seeing from the Figure 3, the development of blocks a, b,
c is relatively advanced, while the plate of d, e, k, is
relatively lagging behind.
The overall balance of plate development can be
better reflected by using the overall utilization ratio of
the "maximum area" model. For example, the maximum
area of the development model of Figure 3 and Figure2
are the same, but the overall utilization rate in Figure
1 is 76.4% and the overall utilization rate in Figure 5 is
79.2%. So, the relatively balanced development of
various factors in Figure 5 better than that in Figure1.
The utilization of other plates is improved by the
way of "Balance compensation" theory using an
approach of "same development", the results of that is
unsatisfactory in the training practice of high level
athletes. It is not the best way of improving utilization to
develop disadvantaged sections. The utilization of other
plate can be better improved by changing the bottlenecks
points to tend a balanced development.
When setting the training programs, the training
efficiency of training imputing and producing has to be
fully calculated, and to get the maximum training returns
with the minimum investment in the principle of
effectiveness and economy. The theory of
non-balanced-compensation emphasizes the role of
compensation for advantaged factor to disadvantaged
factor, but in practice, by nature it has achieved the
purpose of increasing the maximum area through
development of excellent plates, changes of the straight
line XY, and increasing the utilized area of the excellent
plate, ignoring the training effectiveness.
4. The significance of sports training
of "maximum area" model
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A. accurately reflecting the role of timing
1) The upper limit of the training inputs
Many physical factors, such as strength, endurance,
etc., because of physical constraints, need to be restored
and recovered excess, so the. Skills factors, constrained
by physical ability and time, the amount of training
imputed once or within a certain period of time also has
a certain limit. Intelligence factors, especially the game
experience gained, due to the constraints of events
arranged, the training in a certain period of time is much
limited. Even in the training arrangement is very
scientific and rational, but after all, there is a certain
limit for people’s time and energy. Therefore, the upper
limit of the training inputs must exist as it is restricted
by the timing of completing the training objectives.
The different training objectives will also constraint
the amount of training inputs. For example, as a
professional athlete, he had to take the limit of impacting
sport as a target, and the training inputs must be fully
excavated, so as to tend closely to a endurance limit of
personal physical and psychological; while the training
mode of making sport as a hobby can only improve their
athletic ability at the premise of non-affecting work and
study, and the training inputs will certainly be much
2) The impact of time on the plate height
There is a purpose in exercise training. No matter the
short-term goal and the long-term goal, as long as the
target exists, there is bounded a time stamp. Therefore,
any setting of the training program must have a limit of
specific length of time for completion. There are mainly
two aspects of the time factor impacting on the training
effect. First, in a certain length of time, there is a upper
amount limit of investment of the overall training of
various factors. Second, many physiological factors,
over time, the level of which will naturally reduce, and
the investment in the training of these factors can only
play a role slowing the reducing speed level.
3) The role of the effect of aging on evaluating the
training effect
"Barrel theory" fails to consider the timeliness of
exercise training; it ignores the time length needed to
improve some low-level factors and features of
development of certain factors inevitable drooping after
the development and maintenance of a certain length of
time. "The maximum area" model can not only reflect
the result of increased levels of factors, but also may
reflect the results of factors reducing level. For example,
over time, as the improvement of the technical level and
abundant experienced competition for a table tennis
player, the height of corresponding plate will increase,
while the decline level in physical fitness, the height of
corresponding plate will reduce. "The biggest area"
model can bring the positive effects which produced
from upgrading skills and enriching extensive
competition experience and the negative impact resulted
in the decline level of fitness, and then accurately reflect
the final result.
B The effect of "double choke points" in the selection
process of exercise training
There must be a change of the plate height in the
maximum area model or sorting turns when raising the
height of one or more plate after a certain point to the
above height, and which will increase the horizontal or
vertical axis value and lead to a maximum area increase.
(seeing Figure 3, Figure 4). Similarly, the plate height in
the maximum area mode or the sorting turns will change
if one of more factor levels in front of the bottleneck
point is lower than the one. it will definitely lead to the
reduction of maximum area as well as the value of
bottleneck point of horizontal or vertical coordinate.
In the new model, the factors that constituted a
bottleneck point may be the original factors and maybe
the other factors of which the level of development is
lower or higher than the original factors. For example,
raising the height of plate g in Figure 3 above the height
of plate g, the sort of plate changes and the bottleneck
point U1 also changes from the plate d to plate g. (seeing
Figure 4).
C.The determination of training inputs the direction and
movement training
Enhancing the plate g in Figure 3, of which affect the
weight value greater and plate l , of which affect the
weight value smaller respectively to a same height, the
maximum area will increase 13.85% (Figure 4) when
enhancing the plate g, while to enhance the plate 1 can
only increase the maximum area of 3.08% (Figure 6). In
the initial training phase, a variety of factors are easier to
be improved, obviously, the factor g, which to invest the
same amount of training into effect on athletic
performance factor weights greater, can achieve a much
better actual result compared with the factor 1, of which
the weight value of the direction impacting athletic
performance smaller. (Especially in order to obtain
significant results in a short term).
At a high level training phase, certain factors
(especially physical factor) has developed to a limit,
even if a large amount of training is invested in, the
factor levels are difficult to improve, and the increase of
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athletic performance is minimal, which is obviously not
worth the candle. Then even if the factor plate is shorter
and wider, no more training will be put in, but to explore
the potential of other factors (these factors can be
unlimitedly enhanced theoretically). At this time,
even if it is a longer and narrower one, it can also
become longer to improve the overall athletic
performance to
extent. This can be verified from the psychological,
technical movements, tactical awareness of the world's
best players and that occupies a large proportion
training.Take Figure 7 as an example, it reflects the
development of all aspects of a high quality high jump
athlete, the development of the leg explosive g,
technology development f have been closer to the limit,
and there is a certain improvement room for
psychological e.At this point, to enhance the
psychological e, the game experience, d, and the other
plate h and i impacting small on weight but can be
enhanced to the height above point U1, also have an
effective for improving the of athletic performance
(seeing Figure 8). In this upgrade process, even if the
level of legs explosive g decreases in certain, the
improvement and maintain of the athletic performance it
will not be affected (seeing Figure 8).
Of course, as to those factors, the development of
which is higher than the height of the left bottleneck,
due to the restrictions by other factors, no matter how
much further increased, will not affect athletic
performance. For example, grasping the Einstein's
theory of relativity is probably no helpful for a swimmer
to improve their sports performance.
D. Choice of "relatively balanced development" concept
and the training mode
1) different combination of training under the modes of
"relatively balanced development"
Seeing the Figure 3 and Figure 5, the height of four
plates corresponding d, e, f, g in the two models are
different, but the "maximum area" are equal. Therefore,
each factor of "the largest area" model can have a
relatively different balanced combination mode and
getting the same value of "maximum area", that is the
athletic performance is equal. In other words, all roads
lead to Rome, different development paths and models
can achieve the same training objectives. It full obeys
the principle of giving priority to efficiency and purpose
when choosing these different development model.
E. The timing effect and training mode under the
concept of "relatively balanced development"
1) Selection of Short-term training mode
In the initial training phase, a variety of factors are
easier to be improved, obviously, comparing with the
investments to factors weighting small impact on
athletic performance, to invest the same amount of
training into factor, of which weights greater on will
obtain actual much better results (especially in the short
term in order to obtain significant results.) At this time,
the training mode shown is naturally inclined to the
approach of "balance compensation".
2) Selection of long-term training mode
When making the long-term training programs, the
balanced development of high level sport often require
non-balanced development in low-level campaign, that
the advance development is basic. This development
mode, even though in the short term can improve
athletic performance to maximization, but will prevent
the development of movement to a certain extent level,
and a plate can not be enhanced with an appropriate
speed. The formed bottleneck point will restrict the
utilization of other plates in a long term, which will
cause a serious imbalance in the actual development,
and has seriously hampered the situation of improving
sport level, and so as to achieve the maximization of
final training effect.
At the high level training phase, certain factors
(especially physical factor) has tended to a limited
development, and the potential factors is focused to
improve, and then the training model clearly
demonstrated is the "non-value-compensation" approach,
namely: developing the skills of players, making the
formatting of individual technical and tactical style
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5. Steps of evaluating training
program with application of
"maximum area" model
Follow the steps should be followed when
quantitatively evaluating the training program. First, the
length time of training must be determined. Secondly,
according to a given time, and fully taking into account
the training purpose, the restore speed of body, training
time, space, climate, and many other factors, to establish
different training distribution inputs program, and
correctly assess each program element of the plate
probably obtaining the maximum amount training of
investment. Third, to determine the changes that maybe
produced under the program of different distribution
training inputs, including the rise and fall changes
caused of time factors. Fourth, to establish the
"maximum area" model with training effect of different
training inputs distribution program. Fifth, to compare
and analys the "maximum area" model got in step four,
and to choose the optimized solution.
6. Conclusions
For the studies of degree of correlation between
non-physiological factors and the athletic performance,
there is no specific result to refer to; as to the calculation
of specific values of scalable coefficient K, it has not yet
been carried out; what factors should be included of the
composition of the "maximum area" model has yet to be
explored. Therefore, although the "maximum size"
model can explain some phenomenon in the exercise
training in a reasonable way, and provide a reference
and theoretical basis for the training mode selection, but
there is still some distance away from practical
application, which is still in the conceptual stage.
This article discusses the "maximum size" model
established by the structure of "double choke point". The
possibility of establishing "Maximum size" model" with
“three bottleneck point", "four choke point" and
"multi-choke point", and what the requirements of the
abscissa range for "choke point" are stille to be further
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