Levels of Abstractions

Within the Infinite Series Lesson, there are three levels of abstraction. The first, and the least abstract, is when students cut a sheet of paper to represent the fractions within the infinite series in order to begin conceptualizing the sum of this infinite series of fractions. The second level of abstraction occurs when students create a two dimensional diagram representing the sum to a greater degree (or using a greater number of terms). Lastly, students use symbols to think about and represent the infinite series, which would be the last level and the most abstract.


Specialized Symbolism and Technical Terminology

Towards the end of the lesson, students are exposed to mathematical terms and symbols; abstract representations of the series simulation or model they interacted with earlier in the lesson. During this phase, students are engaged in the third stage of Bruner's Stages of Representation, the symbolic stage. This gives students the opportunity to learn the following academic language specific to math:

Name
Symbol
Meaning
How it is read
Summation*

The terms (or numbers) of a series following this symbol are added together
"the sum of"
Infinity

The series, function, or concept that is being address continues forever and has no end
infinity
Limit
lim
The point which a function or series approaches as the input approaches a given value (i.e. In this lesson, it would be the limit of the series as it approaches infinity)
"the limit of"
Exponent
x2
Raising a number to a power or multiplying it with itself the number of times indicated by the number in superscript. For example, x2 = x * x and x3 = x * x * x
"x squared" or "x to the power of"
nth term
n
A specific term, or number in a series, where n is replaced with numerical value of the position where that term is found in a series. For example, if n = infinity, then we are talking about the infinite term of the series, which is the case with the infinite series used in this lesson.
"the nth term"
Series
1+2+3...
A set of numbers, which can be obtained by following a specific pattern. For example, the infinite series 1/2 + 1/4 + 1/8 + 1/16... is the same as the sum of (1/2)n
"the series"
* The summation symbol would be presented with the concept of the nth term; however, that cannot be properly displayed on the webpage.


Abstract Ideas

In the Infinite Series Lesson, students work with the abstract ideas of limits, infinity, and adding an infinite number of terms in a series. Without a simulation or model that can help students visualize these ideas, students may find these abstract ideas extremely challenging, especially the idea that the sum of an infinite series can have a finite value. Students may erroneously assume that any series or function that goes on infinitely has no value or has a value of infinity. Without giving students the opportunity to explore and create their own understanding of this concept, it can be difficult to dispel this misconception. Thus, the use of models in this lesson can help students grapple with the abstract idea of a limit and how an infinite series can have a finite value.