One thing I’ve noticed in my introductory chemistry classes is that students who have trouble with the class almost invariably have real troubles with math. It’s not always that they are completely unable to do math – hand some of them an equation to rearrange in terms of x and they can do it with no trouble. The problem is that these students don’t connect all that math stuff they learned in math class to anything else. In short, they don’t know that numbers mean something.
Case in point: One of the first things we go over in introductory chemistry is the concept of significant figures – the idea that when you write down a measured number, you should write it in such a way that the person reading the measurement knows how good of a measuring device was used to get the number. When you calculate with these measured numbers, your calculator often starts adding on extra digits, making the numbers look more impressive than they actually are – meaning that you need to round the numbers after doing the caclulation to reflect that.
(Example: Punching in 10.0 / 3.00 on a calculator gives you 3.333333333333333 … But the measurements really aren’t that good, and should be rounded to the same number of figures that the original measurements were known to. So the answer is rounded to 3.33 .)
Rounding should be no trouble for a college student, right? Wrong. Students can round to the nearest whole number with no problem. They can also usually round off to the nearest tenth or hundredth without much difficulty, though they are prone to merely chopping off the number rather than actually rounding – so 4.59 becomes 4.5 rather than 4.6 . But having them round to a place bigger than the ones place and all hell breaks loose.
Let’s say you estimate the cost of a project for your home, adding the estimated costs of all the parts of the project together. You come up with a total of $2576.08 . Now since this is all estimation, you decide to record the estimate to the nearest hundred dollars. You round off at the hundreds place and write down an estimated total of $2600. You don’t make those zeros in the ones and tens place go away and write the estimated cost as $26, because that would be silly – right? You’d never be able to walk into Lowe’s and buy $2600 worth of materials for $26. Yet this is precisely the mistake that altogether too many students make. They have an idea in their heads that rounding is the chopping off of numbers, and by golly they will do it – no matter whether the answer they get makes any sense or not. They don’t know that the numbers that they calculate have a meaning, and they don’t check their answers to see if they make sense.
Now these students don’t actually misround the money example I gave above (or at least I hope they don’t), but they do make exactly the same mistake with things like masses, volumes … well, anything else except money.
A typical example goes something like this. A student calculates the mass of a chemical that should be produced in a reaction. Their calculator gives them a bunch of extra digits, and they need to round the answer. Let’s say the student gets a mass of 106.75730235 grams, but the student needs to round the answer to two significant figures (that’d be the tens place in this number). The student writes down a mass of 11 grams as their answer instead of a mass of 110 grams. Yet I bet that not a one of the students that do this would accept $11 from me as a full payment if I owed them $110!
I wish I knew what to do about this. I show them examples – including the money one above. I show them the different masses in the laboratory. Some of them do figure it out, but there are a few who just can’t seem to learn that rounding isn’t just taking an axe to a number. What’s more disturbing to me (since many of these students want to go on to become nurses) is that some of them also never realize the mistake they’re making since they never check their calculations.
My wife calls it "learning in silos" – where students don’t transfer what they learn in a class to, well, anything else. They understand that dollars are represented by numbers and that the numbers mean something, but they don’t transfer that to grams. Or liters. Do we not tie numbers early enough in the schools to real things? Or do we turn math into a video game by making students use TI-83 calculators all the way from high school?
I’m in denial that college students would actually round 106.75730235 to 11 instead of 110. That seems like such a fundamental skill of life and to blow it often enough for you to take notice is crazy.
It’s by no means a new kind of mistake for the intro students to make. Over the past five years, I’ve had at least one student have that kind of number trouble each semester.
I’ve seen it on the increase over the past three semesters or so, though. I’m not sure if it’s an artifact of something new going on in the high schools (an emphasis more on doing math with only numbers rather than working with measurements?) or whether it’s a coincidence.
[…] (This is what comes of treating real-world problems as math exercises without stopping to think that these numbers mean something.) […]