See It, Say It, Symbolize It--Transformations of Functions

 The See It, Say It, Symbolize framework is about first making explicit the conceptual idea that you want students to see. This happens in conjunction with the relevant conceptual language with both ideas bound together by the symbolic representation of the idea. 

SEE IT 

Let's examine this idea through an example of transformations of functions. In this particular instance what we want to make explicit is the "activity" of the transformation which in this instance begins with a graphical representation of the parent function 

The prompt given to students is to "move each point in the plane 3 units to the right". For this particular  example we are going to focus attention on the transformation of five specific points. 

SAY IT 

As we ask students to describe what they are "seeing" we challenge them to use meaningful conceptual language to describe the transformation. A few examples of the conceptual language students shared is shown below. The first two example of Saying It is a restatement of the initial prompt while the third example describes the actual aspect of the functional relationship that is being transformed using terms (input and output) that speak to specific aspects of the functional relationship. 

1. Each point in the plane is shifted three units to the right.
2. The graph is formed by shifting each corresponding point on the original function relationship 3 units to the right.
3. Each input value of the original function is increased by 3 units while the output values remains the same. 

SYMBOLIZE IT

Then we challenge students to use algebraic symbols to describe the functional relationship. At the beginning we provide them with choices since the algebraic symbols are oftentimes very abstract ideas to a novice learner. Four possible options are shown below.

By doing this the cognitive load is lessened for the learner because they do not need to spend the energy on, at least initially, generating the functional relationship, but on making a connection between known input and output values from the graph of the transformation and selecting the function rule, that "fits" those inputs and outputs. For example, choosing a known input-output relationship of (3,0) there is only one function rule that "fits".

As learners gain more experience with these ideas they have the tools to  "look through" the abstract symbolic representation because they have visual images (seeing) and the necessary language (saying) to communicate the meaning. For example, consider the transformation shown below.  

The hope is that they can "see" through the graphical representation that the graph is shifted 3 units upward and "say" that the "input value remains the same, while the output value is increased by 3 units".

In summary, utilizing the See It, Say It, Symbolize It to organize instruction involves placing the algebraic symbolic representation on the back burner until students have made associations and starting with the "seeing it" utilizing the  representation that will best enable the learner to enable the main ideas of the concept to be seen explicitly.