To B or not to B

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 $f(x) = \sin(ax)$ in their calculus class is the same as the function $A(t) = \sin(\omega t)$ 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 $V = Bh$, or no letters at all. Indeed, formulas should always have words associated with them. Naked formulas like $V = bh$ mean nothing by themselves without surrounding words, such as in the sentence

The volume $V$ in cubic inches is given by $V = bh$, where $b$ is the area of the base in square inches and $h$ is the height in inches.

Changing the two occurrences of $b$ to $B$ changes the meaning of this sentence not one whit.

[Thanks to Jason Zimba for suggesting the title of this post.]

About Bill McCallum

I was born in Australia and came to the United States to pursue a Ph. D. in mathematics at Harvard University, met my wife, and never went back. I am a professor at the University of Arizona, working in number theory and mathematics education.
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9 Responses to To B or not to B

  1. jzimba says:

    Admittedly, there are conventions in mathematics, and conventions are helpful…just as long as one doesn’t forget that they are conventions.

    Your point that “Naked formulas…mean nothing by themselves without surrounding words” applies strongly to science, and to the learning of it. Each symbol in a formula like “F = ma” has a detailed meaning that must be understood in order to apply the formula correctly. For example, “F” in the formula refers not simply to “force,” but specifically to the net, external force on a system. (Often when I’ve taught the Second Law, I’ve taken the trouble to carry around a lot of subscripts, as in “F_{net, ext} = m_{tot}a_{cm}.”)

  2. nancymclaughlin says:

    We humbly request some advice for grade 8 content and sequence. We know that CCSS writers do not dictate sequence, but would like to call upon the expertise who designed the Progressions and extracted content accordingly. PARCC designations of major and supporting are considered when we think that bivariate data goes logically with functions. Is this on the right track? Also the major clusters for Expressions and Equations namely connections between proportional relationships and linear equations would go well with solving linear equations. We would put the bivariate and functions before the proportions and linear equations. Would you? Thank you in advance for your consideration for advise on grade 8.

  3. Ken Mullen says:

    I agree completely. The pure mathematical meaning is independent of the letters used, and students should be exposed to different ways of doing things so that they can build a bigger picture. They have to see it enough that they can use Repeated Reasoning.

    But, there are also other kinds of meaning that we give to mathematical expressions. You might call them conventions. I think conventions can make mathematics easier to understand. If you consistently use B to represent the area of the base, then you can take shortcuts when you are communicating. It can make communication more efficient and also more effective. Of course, the person you’re communicating with needs to know the conventions, and you need to know that they know the conventions. Still, conventions should be broken sometimes. We learn something by breaking a new trail. And we need to make sure students know that they can break new trails, too.

    • Bill McCallum says:

      Thanks Ken, these are useful comments. And I’m certainly not objecting to conventions here, including the one about $b$ versus $B$. But its important to distinguish between mathematical laws and conventions. The distributive property is a law; order of operations is a convention (one, I hasten to add, that I have no intention of abandoning!).

  4. Ken Mullen says:

    Units are interesting here, too. V = lwh most often gives cubic units. My wife studied hydrology, where they sometimes use the V = bh approach to get volume of water in acre-feet.

  5. Bill McCallum says:

    I agree [with nancymclaughlin] that you could interweave the standards on Functions with the standards on Statistics and Probability (which in Grade 8 consists of just one cluster, “Investigate patterns of association in bivariate data”). But I don’t think I would put that before cluster B in Expressions and Equations, “Understand the connections between proportional relationships, lines, and linear equations.” That’s the cluster where students understand the meaning of slope, why a straight line has constant slope and why the graph of a linear equation in two variables is a straight line. This seems to me to underpin all the work with linear functions.

  6. jzimba says:

    Math works even if there is no gravitational field. Maybe the convention ought to have been “A = Fd,” where F is the area of any face and d is the third dimension of the prism.

    • calvinballing says:

      … and A stands for Amount of Volume, presumably. I agree that b vs. B is a convention, and that students should understand what counts as a convention and what doesn’t. However, is there a good reason to flout convention on this particular issue? As has been pointed out, teachers are not told what symbols to use based on this formula, so why not tell teachers to use V=Bh in the classroom but write the standard as jzimba suggests:

      “The volume A in cubic inches is given by A=Fd, where F is the area of the base in square inches and d is the height in inches.”?

  7. Bill McCallum says:

    I’m of two minds on the issue of flouting convention. On the one hand, there was no intention to do so, and curriculum writers are free to use the convention; the standard in no way dictates how the formula is written in a textbook. On the other had, the depth of feeling about this makes me think there is an important point to be made here. Some people do really seem to believe that it is a mathematical error to write $V =bh$, and that is in itself a deeply erroneous view of mathematics.

    But, perhaps we’ve exhausted this discussion. Go forth and $B$ whatever you want to $b$!