Transformations of mathematical and stimulus functions

Abstract

Following a pretest, 8 participants who were unfamiliar with algebraic and trigonometric functions received a brief presentation on the rectangular coordinate system. Next, they participated in a computer-interactive matching-to-sample procedure that trained formula-to-formula and formula-to-graph relations. Then, they were exposed to 40 novel formula-to-graph tests and 10 novel graph-to-formula tests. Seven of the 8 participants showed substantial improvement in identifying formula-to-graph relations; however, in the test of novel graph-to-formula relations, participants tended to select equations in their factored form. Next, we manipulated contextual cues in the form of rules regarding mathematical preferences. First, we informed participants that standard forms of equations were preferred over factored forms. In a subsequent test of 10 additional novel graph-to-formula relations, participants shifted their selections to favor equations in their standard form. This preference reversed during 10 more tests when financial reward was made contingent on correct identification of formulas in factored form. Formula preferences and transformation of novel mathematical and stimulus functions are discussed.

Figure 1. Stage 1 included giving rules and rules with exemplars by way of a PowerPoint® presentation.

Stage 2 involved computer-interactive conditional discrimination MTS training.

Figure 2. A square root function where A2 is in standard form, B2 is in factored form, and C2 is the graphical representation of this function.

Figure 3. One of the 40 tests of novel formula-to-graph relations.

The solid lines represent the basic cube function (y  =  x³), and the dashed lines indicate the possible transformation when the formula becomes more complex. A participant who identified a novel variation of the formula that included a negative constant 4 within the argument and a negative constant 4 following the argument would select C as the correct comparison item.

Figure 4. One of the 30 tests of novel graph-to-formula relations.

The solid line represents the basic exponential function (y  =  x³), and the dashed line indicates the transformation when the formula becomes more complex. A participant who identified that this function included a negative coefficient of x, a positive constant 6 within the argument, and a positive constant 4 following the argument would select F as the correct comparison item in its standard form or D as the correct comparison item in its factored form.

Figure 5. The first 10 formulas as displayed in standard (left) and factored forms (right).

Two additional similar sets of 10 formulas in standard and factored forms were used during the experiment.

Figure 6. The 40 probe formulas employed in the test of novel formula-to-graph relations.

The subscripts of y identify the sequence in which these formulas were presented by the computer program.

Figure 7. The top block shows the correct and incorrect responses on the pretest.

Problem numbers are listed along the x axis for each of the 8 participants. Accurate responses contain the digit 0; errors are shaded blocks containing 1. The four blocks beneath show the same participants' error patterns following training and classification by error patterns.

Figure 8. Participants tended to select formulas in their factored form during the first 10 novel graph-to-formula relations.

All 8 participants tended to shift their selection preferences in favor of the standard forms of the formulas following the manipulation of rules suggesting that mathematicians favored formulas expressed in their standard form. This trend shifted when the experiment indicated that reinforcement would be doubled for identifying formulas in their factored form.

Figure 9. The function y  =  sin(x) has a reciprocal of cosecant function y  =  csc(x).

Taking the reciprocal of the cosecant function, y  =  1/csc(x) is obtained, and in this example, the reciprocal of a reciprocal is combinatorially entailed and y  =  sin(x) is the same as y  =  1/csc(x).