General Patterns meet Particular Examples:
Knowledge meet Understanding
Patricio Lankenau
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Prescribed Title #2
“Only seeing general patterns can give us knowledge. Only seeing particular examples
can give us understanding.” To what extent do you agree with these assertions?
Word Count: 1600
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You look at the clock: 0.00; your final is tomorrow and you haven’t slept at all. You’ve
been hectically reading and re-reading your math textbook in hopes of soaking the knowledge
required to pass the final. You read the text, see the examples, read the text, and see the
examples. You begin to see the patterns, you begin to gain knowledge. Next thing you know,
the exam is on the desk in front of you and as you skim through the questions you realize you
don’t understand any of them. But how is this possible when you spent countless hours seeing
patterns and gaining knowledge?
The circumstance described above is a familiar situation to many students, and one
which hints at the idea that only seeing general patterns can give us knowledge. Furthermore,
only seeing particular examples can give us understanding. These assertions make the
assumption that there is an underlying difference between knowledge and understanding as
well as general patterns and particular examples. They raise the knowledge issue of: to what
extent is knowledge different or similar to understanding? When we consider the areas of
knowledge of mathematics, and history, it appears that knowledge and understanding
undertake different meanings in these different disciplines. They are distinct. Thus, they should
be analyzed separately.
As the assertions imply, there exists an underlying distinction between general patterns
and particular examples. Patterns are usually considered in the realm of mathematics as
repeating trends which can be identified and then predicted. When we consider general
patterns, we think of geometric patterns, such as bee hives, which seem to recur. The term
particular example implies that it must meet certain criterion. Although the criterion framework
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is not described, it most likely means that the example must be representative but unique. It
must provide representation to the underlying dynamic give the knower understanding. Some
ways in which general patterns and particular examples differ are in their contradictive
preceding adjectives: general and particular. Although literally they mean the opposite; general
refers to wide variety and particular implies a unique singular, they do overlap. Numerous
general examples can be contained or embodied in a particular. For example, one can observe
the angles between leaves in plants, which can be considered general patterns. However, one
can also reduce the general into a particular by stating that almost all leaves are offset by an
angle known as the phyllotactic angle1. Although the distinction might seem trivial, it plays an
important role in determining the difference between knowledge and understanding,
specifically in how each is obtained.
Similarly, the assertions imply that there is a difference between knowledge and understanding.
The normally accepted definition of knowledge, developed by Plato, is: justified, true belief.
This simple packing of such a large concept as knowledge has many implications, such as what
constitutes valid justification, what defines a belief, and what the concept of truth is. This
imposes the question of when knowledge becomes understanding. “I understand” is a phrase
commonly used by students, often without consideration of what it really means.
Understanding is is talked about, often lackadaisically, by many of us in our quotidian activities.
Yet, encapsulating the meaning into words is hardly done. Understanding commonly implies the
ability to apply certain knowledge. In that sense, knowledge is subordinate to understanding.
1
Prusinkiewicz, Przemyslaw, and Aristid Lindenmayer. The Algorithmic beauty of plants. New York [etc.: SpringerVerlag, 1990. Print.
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Similarly, understanding can be considered the ability to explain or pass on certain knowledge.
Although this can be refuted using counter-examples of memorization of information, it seems
to fit for certain situations. Overall, the topic of understanding is an elusive one, and not being
able to explicitly define it implies that anybody who claims they understand something can’t
possibly understand it when it’s impossible to understand what understanding even is.
However, it is important to acknowledge the nuances and implications of understanding,
especially when it is compared to knowledge.
In mathematics, knowledge can be considered the possession of information, and
understanding the ability to apply and use that information. Although this is generally agreed
upon, it is important to note that there are forms of mathematical knowledge which cannot be
applied, such as knowledge of multi-dimensional geometric figures or esoteric mathematical
concepts such as Ackerman’s function2. Part of the IB Mathematics Higher Level curriculum is
mathematical conjectures. In this unit, we learn how to recognize general patterns in order to
gain the knowledge of what function resulted in such patterns. This is concurrent with the first
assertion; the observation of general patterns was used to gain understanding. Furthermore,
after working out numerous problems and examples we could apply the knowledge gained to
more situations. This is in agreement with at least one way of defining understanding and tends
to agree with the second assertion that particular examples can lead to understanding. The
nuance lies in the presence of the word “only” in both assertions. Although the example agrees
with the assertions, it cannot be assumed that general patterns are the only way of gaining
2
A mathematically recursive function which breaks conventional arithmetic and proves that not all total
computational functions are primitive recursive
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knowledge or that particular examples are the only way of understanding. If we consider the
case of possessing the knowledge of the formula for the sum of the first N natural numbers 3
without having first observed general examples to derive it, we can see that there are other
means of gaining knowledge without observation of general patterns.
If we consider the area of knowledge of history, knowledge can be considered historical
facts, or reliable pieces of information which are commonly accepted to be true. Subsequently,
historical understanding is something which implies the ability to process historical facts and
from them, assemble coherent and meaningful interpretations of the past. This, like in
mathematics, implies that there exists a hierarchical relationship between knowledge and
understanding, one which places understanding above knowledge. Historical general patterns
can be seen through the analysis of historical documents. Good historians seek breadth of
sources in order to see the general pattern and draw knowledge. This is much like what is the
described in the first assertion. The problem arises when we look at the implications of the
second assertion in history. As previously mentioned, particular examples imply that there is a
select few (possibly one) sources or examples which are required to give us understanding.
Particularity and singularity are concepts not sought after by historians. Because of human
limitations, we are unable to objectively transcribe all of history. Assuming that what he know
or see is true, we would still not be able to transfer all of that information to future
generations. This implies that history is prey to selection bias, which is the inaccurate
representation of a whole based on a particular. A history text-book, for example, provides
information such as dates or event descriptions which might be enough to meet the test’s
3
∑
=
(
)
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requirements, but cannot be all the historical information. Although all history is prone to
selection bias, it does not mean that history is less useful. It means we have to be aware of it.
This concept of awareness ties in to the concept of understanding. If we accept the proposed
idea that understanding is on a higher level than knowledge, then we can say that in order to
reach that level is through awareness of such things as bias.
Although some of the differences explored previously might seem trivial, it is the small
nuances in the language we use that yield deeper meanings. It is important to form a mental
image, possibly a Venn diagram, to coherently recognize the areas in which general examples
and particular examples overlap as well as the areas in which they differ. Similarly, recognizing
the differences and similarities between knowledge and understanding is part of the critical
thinking, which should be sought after in order to gain a deeper understanding. It is always
important to pursue definitions to elusive ideas or words such as knowledge and
understanding, but it must be acknowledged that such packed terms cannot often be reduced
to single or even groups of words. Yet, even surface analysis provides insight that would
otherwise be overlooked.
When one looks at the ways in which the assertions: “only seeing general patterns can
give us knowledge” and “only seeing particular examples can give us understanding” hold in
two different areas of knowledge, it becomes clear that the assertions’ veracity depends on the
ways of knowing used in the specific area of knowledge. The fallacy in the assertions, and the
reason for why I do not agree with them, is the presence of the word: ‘only’. This implies that
there are no other methods of attaining knowledge outside of seeing. Additionally it implies
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that particular examples are the only gateway to understanding. Although, there are many
situations that adhere to those standards, there are many that do not. The assertions’ claims
can be validated or repudiated with anecdotal evidence, and therefore can be true within
certain boundaries. This is not true for the absolute claims which create a false binary which
must either be true or false. Providing a single example, as we have done earlier, disproves the
assertion. For this reason, it becomes clear that the assertions are true to a variable extent, and
although their absolute claims are wrong, they do provide keen insight into the ways of
knowing and understanding.