Big+b,+little+b-+volume+discussion+worth+reading

[] =To B or not to B= Posted on [|January 28, 2013] by [|Bill McCallum] Once every few months or so I receive a message about the following standard: > 6.G.2. Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = lwh and V = bh to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems. See if you can guess what people think the problem is before reading on. The most recent message said quite sternly that V = bh was NOT correct, and that it MUST BE V = Bh. This point of view is one of the starkest examples I know of the obstacles we must overcome in restoring the culture of mathematics in schools. The notion that certain letters MUST always stand for the same thing across different formulas is itself a mathematical error, a profound misunderstanding of how symbols are used. It’s like thinking the word blue must always be written in blue. And the misconception is not harmless. Students who come to college with it do not fare well amid the profusion of symbols in their science classes, unable to see that the function f ( x )= <span class="mi" style="font-family: MathJax_Main-Web; font-size: 17px;">sin <span class="mo" style="font-family: MathJax_Main-Web; font-size: 17px;">( <span class="mi" style="font-family: MathJax_Math-italic-Web; font-size: 17px;">ax <span class="mo" style="font-family: MathJax_Main-Web; font-size: 17px;">) in their calculus class is the same as the function <span class="mi" style="font-family: MathJax_Math-italic-Web; font-size: 17px;">A <span class="mo" style="font-family: MathJax_Main-Web; font-size: 17px;">( <span class="mi" style="font-family: MathJax_Math-italic-Web; font-size: 17px;">t <span class="mo" style="font-family: MathJax_Main-Web; font-size: 17px;">)= <span class="mi" style="font-family: MathJax_Main-Web; font-size: 17px;">sin <span class="mo" style="font-family: MathJax_Main-Web; font-size: 17px;">( <span class="mi" style="font-family: MathJax_Math-italic-Web; font-size: 17px;">ωt <span class="mo" style="font-family: MathJax_Main-Web; font-size: 17px;">) in their physics class. Part of the power of algebra is that you can choose any letter you like to represent a quantity, as long as you specify what the letter stands for, and you can re-use that letter with a different meaning in a different problem. That said, the standard does not dictate the use of any specific letters; indeed, the core meaning of the standard is not about formulas at all, but rather about finding the volume of a rectangular prism by multiplying its length, width, and height, or by multiplying the area of its base by its height. So teachers and curriculum materials can use whatever letters they like, including <span class="mi" style="font-family: MathJax_Math-italic-Web; font-size: 17px;">V <span class="mo" style="font-family: MathJax_Main-Web; font-size: 17px;">= <span class="mi" style="font-family: MathJax_Math-italic-Web; font-size: 17px;">Bh, or no letters at all. Indeed, formulas should always have words associated with them. Naked formulas like <span class="mi" style="font-family: MathJax_Math-italic-Web; font-size: 17px;">V <span class="mo" style="font-family: MathJax_Main-Web; font-size: 17px;">= <span class="mi" style="font-family: MathJax_Math-italic-Web; font-size: 17px;">bh mean nothing by themselves without surrounding words, such as in the sentence > The volume <span class="mi" style="font-family: MathJax_Math-italic-Web; font-size: 18px;">V in cubic inches is given by <span class="mi" style="font-family: MathJax_Math-italic-Web; font-size: 18px;">V <span class="mo" style="font-family: MathJax_Main-Web; font-size: 18px;">= <span class="mi" style="font-family: MathJax_Math-italic-Web; font-size: 18px;">bh, where <span class="mi" style="font-family: MathJax_Math-italic-Web; font-size: 18px;">b is the area of the base in square inches and <span class="mi" style="font-family: MathJax_Math-italic-Web; font-size: 18px;">h is the height in inches. Changing the two occurrences of <span class="mi" style="font-family: MathJax_Math-italic-Web; font-size: 17px;">b to <span class="mi" style="font-family: MathJax_Math-italic-Web; font-size: 17px;">B changes the meaning of this sentence not one whit.