After all, if a machine is doing the calculating for you, why bother to learn it?Īfter more than three decades of research, the findings are clear. Still, some teachers wondered whether this shift in instruction in AP calculus and across the K-12 curriculum would have a negative effect on students’ instrumental understanding. Frederick Peck, CC BY Does the tech help or hurt? But zooming out (at right), the graphs of the two functions are basically identical. In the local area (-6 < x < 6) shown at left, the graphs of y = x² + 5 and y = x² are clearly different. Through exploration such as this, students gained relational understanding of infinite limits. By zooming out, students can see that globally, the graphs are basically identical. By zooming in, students can see that in any local area, the graphs are clearly different. Rather, the focus of instruction shifted so that students could learn mathematics through the graphing calculator.įor example, by using the graphing and zoom features of the graphing calculator, students could compare and contrast the local and global behavior of functions such as y = x² and y = x² + 2. This was not just so students would learn how to use the calculator. The AP program required teachers to use graphing calculators in their courses. After 1995, there was a marked shift away from this instrumental understanding and toward questions that probed for relational understanding.Īs exams evolved, so too did teaching philosophies. Prior to 1995, AP calculus exam questions probed almost solely for the students’ ability to use rules to find derivatives and integrals of functions. The impact is quite clear in the Advanced Placement (AP) calculus program, which started to require graphing calculators in their courses and on their exams in 1995. While the calculator takes care of the “how,” students can focus on “why.” People with a relational understanding don’t just know how to invert and multiply, they know why such a procedure results in the quotient of two fractions.Īdvocates for graphing calculators in school saw promise in the tool’s ability to help students develop relational understanding. In contrast, “relational understanding” is a kind of connected, conceptual understanding. ![]() In theory, then, you should be able to reverse the conversion and derive a Fahrenheit temperature from ans.The first scientific graphing calculator. Another way to write the equation for temperature conversion in KAlgerbra is to use the times function: times(5/9, 70-32).Īs you complete math problems, established variables are listed in the right column of the calculator, including the ans value, which is updated with the answer to the completed equation. ![]() There are also special functions for common math operations in KAlgebra, and when you type any letter into KAlgebra, a tooltip provides potential auto-completion for available functions. In KAlgebra, as in most programming languages, division is represented by a forward slash and multiplication by an asterisk, so the equation to, for example, convert 70° Fahrenheit is (5/9)*(70-32). In the temperature conversion example, the symbols are pretty common, so you probably already know that ÷ represents division and × represents multiplication. You may or may not know the meaning of every special math symbol, but as long as you know that a special symbol has a specific meaning, then you can look it up. This is generally how equations and mathematical functions are expressed: they use special symbols like ÷ and × as well as variables like n, and then they identify what variable represents what kind of value. ![]() For instance, to convert Fahrenheit degrees to Celsius, the equation is: (5÷9) × (n-32), where n is Fahrenheit. ![]() When representing an equation in KAlgebra, you must do some minor translation of math symbols as they're often written by hand to how they're represented on a computer. But before rushing into 3D space, start with some basic syntax. Unlike any of the graphing calculators I've ever used, it's also a 3D plotter. KAlgebra is, like many of the famous graphing calculators used in schools, both a scientific calculator and a 2D plotter.
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