Dear Matt,

My curriculum calls for teaching beginning lessons on functions in Algebra 1. My students get confused with all the terms and notation. Do you know of any ways of introducing this topic that will help them understand?

Response:

:

Students learn about functions the same way they learn about other mathematical topics – beginning with experiences and situations, having opportunities to talk about the experiences and situations, first with informal language and gradually with more formal talk, and only then using more abstract representations and notation. Starting with the abstract symbols will only serve to keep them confused.

At the Algebra 1 level, emphasis should be on developing the concept of function. Since function concepts are sometimes abstract in the way they are presented, making it difficult for many students to grasp the idea, it is important to use models that make the ideas accessible. Mark Sand, of Northwest Missouri State University, uses the model of a mail carrier to make functions concrete for his students. The letters form the domain; the mailboxes the range. Any letter without a clear, unambiguous address would not fit the “mail carrier” function. Letters are matched to boxes in a specific way based on their addresses. It becomes clear in discussion of this model that all letters in the function can be delivered to exactly one address. In addition, many letters may be delivered to some mailboxes, while other mailboxes may have no letters. The important characteristic that makes the mail carrier a function, however, is that each letter CAN be delivered, and will go to ONLY ONE address. Additional ideas such as one-to-one function, in which each mailbox receives one letter, constant function, where all the letters go to one mailbox, or onto function, where there are no empty mailboxes, can easily be grasped in this model. Composition of functions can be tied to forwarded mail; inverse functions could be a “Return to Sender” activity. Another notion that flows from this model is that every function does not have to be about numbers, nor does every function need an algebraic “rule”, an idea students may bring with them from earlier grades. Though there are many other models that can be effective, this one seems to hold much potential. (For a short description of Sand’s ideas, see “A Function Is a Mail Carrier” in Algebraic Thinking: Grades K-12, NCTM, 1999.

Some Algebra 1 students may be ready for the more formal function notation; others will not grasp this until later courses. Finally, if students can understand that a situation that can be represented by a function is one where there is predictability they will have a good beginning understanding of this key concept.

Matt

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