What is a Function Approach?
In defining what a function approach is, sometimes we have to define what it
is not. Teaching function notation early is not teaching from a function
approach! Nor is it teaching from a function approach when you just move the
chapter of functions to the first or second topic of the year. To teach from a
function approach means using functions, function representations, and function
behaviors to enhance the teaching of algebraic concepts /skills such as adding
and subtracting polynomials , factoring, equation solving , systems of equations,
modeling, solving inequalities , laws of exponents, properties of inequalities ,
definitions, etc. This requires that you start with a “function implementation
module” that begins with numeric representations of functions and leads to
students learning to move freely through representations. This is followed by an
analysis of the geometric behaviors of functions integrated with studying
parameter-behavior connections. After this, one is then ready to start teaching
more traditional algebra, but not with a traditional pedagogy. This is teaching
from a function approach! A graphing calculator is required for all students at
all times – both in the implementation module and throughout the algebra course.
The Implementation Module
Start by using data pairs of real-world relationships (no traditional symbol
manipulation needed). The relationships may be of a wide variety. That is, there
is no need to segregate by function type. Rather, we integrate a variety of
relationships such as linear, quadratic, exponential, absolute value, etc.
simultaneously – just as students might encounter them in their lives. The
mathematics is simple: classify relationships by shape, and whether they seem to
be increasing and/or decreasing in nature.
This first step primes the students for later study of
functions. The data relationships and contexts help make associations to assist
recall; and students strengthen their innate abilities of pattern recognition.
Teach students to move freely from numeric to graphic
forms and make the connections between the two . This is easily accomplished by
making the data sets available to the students through calculator programs that
can be distributed to student devices via the GraphLink™ cable or through TI
Navigator™. When the programs are executed, the data is transferred to the list
editor making it available to be viewed in numeric and graphic forms (and later
in symbolic form). The question of whether students think the relationship is
increasing or decreasing can be answered by looking at the numeric
representations. Keep in mind that the data sets all come from a real-world
context, this makes answering the question rather simple. Students need to make
the association between increasing (or decreasing) numbers in the range with a
rising (or falling) graph. Names of the graphs (shapes) of data relationship may
reflect student inexperience with mathematics as they will use names like V or J
instead of absolute value and exponential.
This step primes the students for a more intensive study
of functions. The data relationships and contexts help make associations to
assist recall. Further, they get the attention of the students. The use of
technology helps with attention too. Students start making mathematical
connections between representations.
Example 1
The population of earth (shown below in billions) is increasing as confirmed by
the numeric representation (or the graph), and the shape of the graph. The
students might call the shape a J; the instructor can introduce the term
exponential when appropriate.
Teach students to move from numeric or literal forms to symbolic forms. This is
not accomplished by students learning English-mathematics conversions. Rather,
we use pattern recognition and the list editor on a graphing calculator to reach
the goal. Pattern recognition is an innate function of the human brain, while
English-mathematics conversions are language specific. Below is one simple
example. This process can be used for a variety of functions such as linear,
quadratic, and rational. In every case, we use pattern recognition and guided
discovery to create a model. We do not use English-mathematics conversions (or
regression).
Example 2
Suppose we start with 500 small candies marked with a letter on one side. We
toss them on the table. How many do we expect to have the letter facing up?
Students say about 250. You say, “How did you get that?” See the edit line in
Figure 3 for the student answer.
We eat the candies with the letter facing up, and toss the remaining on the
table. We now have about 250 candies on the table. How many do we expect to have
the letter facing up?
Students say 125. You say, “How did you get that?” Students say 250(1/2). You
say, “And where did the 250 come from?” See the edit line in Figure 2 for the
student answer.
(Note: There are now 2 factors of ½)
The candies with the letter facing up are eaten and we
toss the remaining on the table. We now have about 125 candies on the table. How
many do we expect to have the letter facing up? Students say about 63.You say,
“How did you get that?” Students say 125(1/2). You say, “But where did the 125
come from?” Students say 250(1/2)(1/2). You say, “But where did the 250 come
from?” See the edit line in Figure 5 for the student answer.
(Note: There are now 3 factors of ½)
Based on appropriate questions and discussions, at this
point most of the class will recognize the exponentially decreasing pattern and
you are ready for the introduction of symbols – see Figures 6 and 7 below.
In L1 of Figure 8, you can enter new values to see the power of abstract
mathematical symbols.
We now need to look at the graphical representation (see Figure 9) and use the
Y= editor to introduce standard x notation.
Teach the connections to real world situations to make associations that help
memory and make algebra seem “reasonable.” For example, using real-world
contexts, students find that the mathematical concepts of increasing and
decreasing are simple ideas. They need not wait for calculus to understand the
basic concepts – maybe the formal definitions, but not the concepts. Further,
students have a basic understanding of function representation. There is no need
to formalize concepts now. This is just an introduction to functions so that
algebraic concepts can be taught using functions at a later time – including a
formal definition of a function and function notation.
Students now realize that mathematical symbols like y =
−500x + 3000 and s = −16t2 + 15t + 6 have a real-world meaning. They know that
the symbols have a real-world connection, and how the symbols are connected to
the numerical and graphical representations. At this time, students have not
made sense of symbols such as 
Teach the geometric behaviors of basic functions
(increasing/decreasing, max/min, rate of change, zeros, initial condition, when
negative -rules.html">positive /negative, domain, and range). Why? If we are going to use function
concepts to teach algebra, students must know something about geometric
behaviors. For example, suppose your students know nothing about functions but
you want them to use a graphing calculator to solve the equation 2x2 + 41x =
115. What do they do? Simple, they graph the function y = 2x2 + 41x – 115 in
the 10 × 10 (or decimal ) window, find the zero of the function to be 5/2, and
call it the root of the equation and then quit. This is not enough.
How can you use rate of change and initial condition of
the linear function to teach addition of polynomials unless students know what
the rate of change and initial condition are? You can use the distributive
property to teach addition and subtraction of polynomials, but will it be with
understanding? What will be the underlying mathematical connection? What will be
the associative cues students can use to recall it next year? How have you used
the innate visualization brain processing of mathematics to your advantage? Do
you capitalize on the innate and learned number sense? So, typically we use the
distributive property and abstract symbols after we use function behaviors to
discover how we add and subtract polynomials.
How will students learn to use the behaviors of rate of
change, increasing, decreasing, maximum and minimum, etc. to find mathematical
models of data relationships without knowing anything about these behaviors?
Well, you could use the low-level cognitive skill of rote memorization. Okay,
not really. The point being is that you are teaching black box mathematics when
students are not taught function behaviors before functions are used to teach
and to do algebra. Function behaviors can be taught using contextual situations
to help students understand.
Teach parameter-behavior connections. In the natural
progression of teaching from a function approach, parameter-behavior questions
start to come to mind. I wonder why some lines are steeper than others? Why do
some parabolas open up and others down? Are the rates of change of the branches
of the graph of an absolute value function related? Why is the vertex where it
is? If your students are not asking these questions, shouldn’t you?
The answer to these questions is found by studying
parameter-behavior connections. How do you teach parameter-behavior connections?
Well, one excellent way is to use guided-discovery activities. Another is to
find embedded guided-discovery exercises in homework. Example 3 below is a
guided discovery graphing calculator activity that is typically assigned to
student groups.
