The Functionator 3000: Transforming Numbers and Children

Mrs. Fisher's class was learning about arithmetic functions by pretending to operate real-world "function machines" (Reeves 2006). Functions are a unifying mathematics topic, and a great deal of emphasis is placed on understanding them in prekindergarten through grade 12 (Kilpatrick and Izsák 2008). In its Algebra Content Standard, the National Council of Teachers of Mathematics (NCTM) recommends that instructional programs enable all students to "understand patterns, relations, and functions" (NCTM 2000, p. 158). More specifically, students in grades 3-5 should be able to investigate and express functions using various representations, including words, tables, and graphs. Elementary-school-age children are typically introduced to functions as "input-output" machines drawn on paper. When using these "machines," the teacher gives students several input numbers and their corresponding output numbers, and the students are to conjecture what the machine is doing to the input numbers to produce the output numbers. Students then check their conjectures against all the number pairs they have and, if valid to that point, use one or two more input/output pairs as reinforcement of the relationship that they have discovered. Another way that elementary-school-age students may investigate functions is by assuming the role of a "robot" that asks classmates to identify the relationship between two sets of data by identifying numeric patterns as a rule (Moss et al. 2008). Both Reeves (2006) and Willoughby (1997) recommended using a physical model of a function machine with an operator who pretends to input numbers into the machine and pretends to receive output numbers, all the time calculating the output numbers mentally. When given their turn to control the machine, some students invariably get into the play-acting role required of the operator and turn the whole experience into one of mathematical learning through play. Students are encouraged to discover and then couch the input-output relationship as a function, using more and more sophisticated language as they move up the grades, eventually using such algebraic terminology as f(x)=x[superscript 2]-1 in the middle grades. The personal transformation of students from compliant observers to take-charge facilitators of their classmates' mathematical learning appears quite dramatic at times. Such was the case with Sydney, a quiet child in Fisher's third-grade accelerated mathematics class at Maximo Elementary School, a Title I school in St. Petersburg, Florida. Sydney often went about her work unobtrusively, content to let others take the lead in whatever mathematics topic the class was discussing. However, a change came over her at one point in the school year. This is the story of that change. The authors tell her success story because the more opportunities elementary school children have to blossom, the more chances that teachers have of helping them achieve to their fullest capacity. A bibliography is included.