Saturday, July 16, 2016

On Using Scientific Notation

This week I have been reading with great interest a discussion in a chemistry teachers' Google Group. It started with a post about whether or not teachers find it acceptable when students give answers as 3.6e-4 instead of 3.6 x 10-4. There were a couple of back and forth ideas about whether or not this is acceptable practice on its face value:

Then the conversation shifted away from that and to a more important idea: does writing an answer as 3.6e-4 tell a teacher something about a student's understanding of the math and/or the chemistry?

This is where I was really drawn in. In my experience, students arrive to chemistry in tenth or eleventh grade with little understanding of scientific notation despite its inclusion in the 8th grade standards of the Common Core (emphasis mine):

Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology.

When we begin the mole, we have to use scientific notation because the numbers of atoms in a small sample of matter will be so large. Until then, has there really been a point in their mathematical education where they really needed scientific notation? I argue that even though it is probably taught in middle school math, it isn't really needed and, after that initial treatment, it also isn't revisited or practiced. Am I wrong? Why teach it in 8th grade if students don't need it?

What I observe when I teach this is that a lot of students want to use, and persist against my advice, to use the buttons for "x 10^" instead of EE, but shouldn't we expect that? They see 3.6 x 10-4 and that looks like the operations they have been putting into a calculator for many years, so they do what they know. Maybe what is missing is a conversation about when that will work (when they multiply) and when it will not (when they divide without parentheses) because that gets at mathematical understanding too. Eventually, most students become adept at punching the designated buttons into their calculator, but in March and April when we have been using it for months, I still have students who ask what they did wrong to get 0.00036 when I give an answer as 3.6 x 10-4, so I know the understanding isn't there.

Chemistry creates the headache that requires scientific notation to be the aspirin. Too often, though, perhaps my focus has been on getting us over the math hurdles so that we can be successful on the chemical ones. This same Google Group debates the merits of significant figures about once a year. Every year by May I make the same threat: Next year I am not teaching the sig fig rules. Instead I will require students to write down every digit they see in their calculator. After a few days of that torture, when they start begging to have some rounding rules and then I unpack the sig fig rules. Would it work? Is there something there that could help with scientific notation?

In short, how can I simultaneously help students be successful in chemistry and better understand math? Do you have ideas? I am all ears. Please comment!

Related aside: There are two iOS apps that I like for helping students see the magnitude of the powers of ten: TickBait's Universe and Universal Zoom. While they won't necessarily help students understand what a number in scientific notation represents, I love the conceptual way they represent it.

Related aside #2: Funny that it's called scientific notation, right? Not mathematical . . .


  1. I had that issue. First I went to our math people, they swore up and down that they taught what the e meant. Then I tell my students when they start having to do mole conversions, this is when it show ups; "stands for exponent," on what it means, give examples and then hit them with the killer.

    "if you put the "e" in the answer, I do not care how beautiful the rest of your work is, I will give you a zero because you obviously do not know the answer and are just copying. There will be no partial credit. Be smarter than the calculator."

    1. I agree that the students have been taught scientific notation. And that if they are penalized for writing the wrong thing, they will [probably] start to write the right thing. My worry, though, is that writing the wrong thing indicates a deficit in their understanding. A penalty doesn't address that. What does?

    2. Instruction addresses the deficit. Most do know the idea. Most students after a few practice problems and board work do get the idea. The penalty only happens if a student has not been paying attention.

  2. The idea that students have been taught scientific notation is an interesting one. In the mathematics books I teach from, there is no real context for the lesson. By "real context" I mean to say that typically the book has students convert the average distance from the sun for several planets from kilometers to millimeters or something that shows how to use scientific notation rules but lacks any real physics or chemistry muscle. There might be a good case to be made for having students give answers in a 3-column table: calculator result, scientific notation, and full numeric notation. As an example: 3.6 e -14: 3.6x10^-14; 0.000000000000036.

    And to the point of worrying that the kids are not using the calculator in the most efficient manner by their refusal to embrace the EE key, let them do it the way that makes sense to them. I cringe when my daughter cuts meat with a knife and fork because she does not do it the way I do it--the right way. She is inefficient and makes chunks that are terrible looking. The food slides all over the plate and I have considered eating at a different time so I don't have to witness the carnage.

    1. I too am all for kids arriving at answers in different and creative ways. And in doing problems in a way that makes sense to them. The trouble in this particular case is that when a student types in 3.6 x 10^-4 ÷ 1.2 x 10^-6, the calculator will not calculate what they mean because they don't use the EE button. Instead of the correct answer, they will be off a factor of 1,000,000 becaus ethe calculator will have multiplies a quotient of the first 3 things by 10^-6.

      My struggle here isn't really about how to get them to use their calculators or what they should record on the paper. I think I do a pretty good job at both of those things. My worry is what do we do to help students understand what scientific notation represents. To provide, as you put it above, a real context so they can understand and appropriately judge their answers.