Teaching Dimensional Analysis

If I remember my adolescent psychology correctly, Jean Piaget claimed that the average child develops abstract (“formal operational”) thinking around age 16—exactly the age most high school students take their first chemistry course.  Since 2003-04 (the year that Massachusetts’ MCAS tests became a graduation requirement for that year’s sophomores), I’ve seen a steady decline in the proportion of my students who are able to achieve the ability to think abstractly and set up and solve complex problems by the end of the year.   My experience would suggest that kids need practice to make the leap to abstract thinking.  We’re giving them less and less of that practice, and the result is that fewer and fewer of them are managing to make the leap.  I believe that this is probably the root cause behind college professors’ claims that rising scores on high stakes tests seem to have a negative correlation with actual readiness for college coursework.

On the subject of teaching dimensional analysis, this year, I tried something that seems to have helped students understand what’s going on:

I started by writing the problem on the board:

1/2 × 2/3 × 3/4 × 4/5 × 5/6 × 6/7 × … × 99/100 = ?

A couple of students came up with the answer of 1/100, and I had one of them explain how he got it.

Then, I wrote down a problem such as “find the volume in liters of 64 g of CH₄ gas at 273K and 1 atm”.  I wrote on the board:

64 g CH₄ × 1 =

Then I said that because 16 g CH₄ = 1 mol CH₄, the fraction 1 mol CH₄ / 16 g CH₄ = 1, so we could replace the 1 by the fraction.  And I did.  And then I canceled grams of CH₄, and simplified the result to 4 mol CH₄.  I made a big deal out of the fact that because we multiplied by 1, we didn’t change the actual amount of gas, and therefore 64 g of CH₄ is the same as 4 mol CH₄.

Next, I inserted another × 1 before the = sign, and repeated the process with 1 mol gas (CH₄) at STP = 22.4 L.

Then, I went back and showed them how to chain them together, emphasizing the fact that each conversion is really just multiplying by 1, showing how the unit they had dictated which conversion to use, and which part had to go where.

I think most of my students understood the discussion, and about 1/2 to 3/4 of them actually demonstrated the ability to set up problems correctly.  The smarter students were the ones who resisted the most—they were able to see the ratios in their heads, and saw writing down the chain as tedious.  When it came time to do these on a test, these smarter students were the ones who couldn’t reliably get the right answers.

Originally posted to the ap-chem discussion list.

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