# An analysis of the introduction of the graphing calculator which has changed the structure of teachi But the metacognitive engagement is particularly important as mathematics becomes more abstract, because students will have few clues even when a solution has gone terribly awry if they are not actively engaged in sense making.

Because mathematical relationships are generalized in algebra, students must operate at a higher level of abstraction than is typical of the mathematics they have generally encountered previously. Are such skills sufficient for a correct solution? Few students view algebra as a powerful toolkit that allows them to solve complex problems much more easily.

University of Iowa A correlational analysis of the NAEP grade 8 math test data found a positive relationship to achievement, across student subgroups, for: Calculator access. What graph-reading and table-building skills are required? Overall, the effects of calculator use were found to be small, but detectable.

How might students approach and respond to this problem?

## Teaching mathematics for understanding

While we can list each set of values, it is very efficient to say that for all values in gallons which we call x by convention , the total cost which we call y by convention , is equal to 2x. The second principle of How People Learn argues that students need a strong conceptual understanding of function as well as procedural fluency. Algebraic tools allow us to express these functional relationships very efficiently; find the value of one thing such as the gas price when we know the value of the other the number of gallons ; and display a relationship visually in a way that allows us to quickly grasp the direction, magnitude, and rate of change in one variable over a range of values of the other. Functions are even found in such familiar settings as baseball statistics and metric conversions. Within mathematics education, function has come to have a broader interpretation that refers not only to the formal definition, but also to the multiple ways in which functions can be written and described. First, and most superficially, the student likely carelessly mislabeled the coordinate for the y-intercept 0, 3 rather than 0, —3. Conceptually, students need to understand that these are different ways of describing the same relationship. This result is not surprising given that algebra is often taught in ways that violate all three principles of learning set forth in How People Learn and highlighted in this volume. But the metacognitive engagement is particularly important as mathematics becomes more abstract, because students will have few clues even when a solution has gone terribly awry if they are not actively engaged in sense making. As functional relationships become more complex, as in the growth of a population or the accumulation of interest over time, solutions are not so easily calculated because the base changes each period. Box suggests that a problem described in its everyday manifestation can be solved by many more students than the same problem presented only as a mathematical equation. Conclusion: African-Americans, Latinos and low SES students appear to benefit as well as other students from calculator access and emphasis on reasoning. Functions are important as well to interpretations of local and world demographics and population growth, which are critical for economic planning and development.
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Teaching and Learning with Graphing Calculators